University of Wollongong
Research Online
University of Wollongong Thesis Collection
University of Wollongong Thesis Collections
1998
Pneumatic conveying performance characteristics
of bulk solids
O. C. Kennedy
University of Wollongong
Recommended Citation
Kennedy, O. C., Pneumatic conveying performance characteristics of bulk solids, Doctor of Philosophy thesis, Department of
Mechanical Engineering, University of Wollongong, 1998. http://ro.uow.edu.au/theses/1591
Research Online is the open access institutional repository for the
University of Wollongong. For further information contact the UOW
Library: research-pubs@uow.edu.au
PNEUMATIC CONVEYING PERFORMANCE
CHARACTERISTICS OF BULK SOLIDS
A thesis submitted in fulfilment of the
requirements for the award of the degree
DOCTOR OF PHILOSOPHY
from
UNIVERSITY OF WOLLONGONG
by
0. C. KENNEDY, BE
Department of Mechanical Engineering
1998
This is to certify that this work has not been submitted for a degree to any other
university or institution.
Oliver C. Kennedy
1
SUMMARY
This thesis is concerned with the analysis of the performance of bulk solid pneumatic transport
systems. A bulk solid parameter, with the potential to characterise materials on the basis of
suitability for this m o d e of transport, is proposed. Procedures for producing conveying
characteristics are developed and linked to conveying models and design requirements. A n e w
model based on the power requirements of the pneumatic conveying process is presented.
The deaeration behaviour of bulk solids is analysed in some detail. Theoretical aspects,
appropriate test rig arrangements and experimental arrangements are discussed. A characterising
parameter, the normalised time constant, is proposed. Extensive experimental data are used to
clarify the significance of this performance indicator and to assess its capacity to predict the
pneumatic conveying potential of a given bulk solid.
The process of generating conveying characteristics for the bulk solid/pipeline configuration
from experimental data is discussed. The computerisation of this process and the application of
numerical techniques are developed and demonstrated. The advantages and possibilities are
explored, and limitations noted.
The implementation of a solids friction factor (Barth type) model within the conveying
characteristics procedures noted above, is developed and demonstrated. The application of scale
up procedures within this model environment is discussed and tested against experimental data.
Determination of conveying characteristics for stepped pipelines is demonstrated, as is a method
for the optimised design of stepped pipeline configurations.
A "power" based pneumatic conveying model is proposed. This model considers the power
available from the isothermal expansion of the gas d o w n the pipeline, and the power consumed
by the various components of the system. Features of this model include a solids friction factor
correlation, the treatment of bends, a n e w analysis of the air-only component and a method for
the identification of invalid conveying regions. The generation of conveying characteristics
curves for single and stepped pipeline systems within this model is demonstrated. The
performance of the model is tested against a selection of experimental data. Trends inherent in
the model, and its scaling behaviour, are examined. Potential areas for further development of
the model are discussed.
ii
PAPERS
The following papers were written or contributed to by the author during the course of this work
Wypych, P.W., Kennedy, O.C., Stebbins, K. and Arnold, P.C., Pneumatic conveying of coal.
Int. Coal Engng. Conf. Sydney, N.S.W., 19-21 June, 1990. Organised by the IEAust. Preprints
of Papers, pp. 73-78. Published in Mech. Engng. Trans, of the IEAust.. Vol. M E 1 5 , No. 4, D e c ,
1990, pp. 255-261.
Wypych, P.W., Kennedy, O.C. and Arnold, P.C., The future potential of pneumatically
conveying coal through pipelines. Bulk Solids Handling. Vol. 10, No. 4, Oct., 1990, pp. 421427. "
Kennedy, O.C, Wypych, P.W., and Arnold, P.C., The effect of blow tank air injection on
pneumatic conveying performance. Pneumatech 3. Int. Conf. on Pneumatic Conveying
Technology. Jersey, Channel Islands, U.K. 24-26 Mar., 1987. Organised by the Powder
Advisory Centre, London, U.K. Proc. of the Tech. Program, pp. 267-279.
Kennedy, O.C, Whitaker, E.A., Wypych, P.W., and Arnold, P.C, Computer-aided
determination of pneumatic conveying flow characteristics, scale-up data and reliable conveying
limits. 3rd Int. Conf. on Bulk Materials Storage. Handling and Transportation. Newcastle,
N.S.W., 27-29 June, 1989. Organised by the IEAust. Preprints of Papers, pp. 150-153.
Wypych, P.W., Kennedy, O.C. and Arnold, P.C, Pneumatic conveying of pulverised & crushed
R.O.M. coal. Pneumatech 4. Int. Conf. on Pneumatic Conveying Technology. Glasgow,
Scotland, 26-28 June, 1990. Organised by the Powder Advisory Centre, London, U.K. Proc. of
the Tech. Program, pp. 159-173.
Kennedy, O.C, Wypych, P.W., and Arnold, P.C, Pneumatic conveying technology improving
Australian bulk handling operations, Paper for presentation at Bulk 2000. London, U.K. 29-31
Oct., 1991. Organised by the LMechE, London, U.K.
Arnold, P.C, Wypych, P.W., Kennedy, O.C. and Whitaker, E.A., Long-distance pneumatic
conveying of fly ash and pulverised coal. N E R D D P End of Contract Report. Project No. 793,
July 1988. Dept. of Mech Engng., Uni. of Wollongong, 99 pp.
Arnold, P.C, Wypych, P.W., Kennedy, O.C and Whitaker, E.A., Dense-phase pneumatic
conveying of pulverised coal and fly ash. A E R B Final Report Aug., 1988. Dept. of Mech
Engng., Uni. of Wollongong, 99 pp.
Conference Papers Accepted/Pending
Kennedy, O.C The deaeration behaviour of bulk solids - the time constant as a characterising
parameter. 6th Int. Conf. on Bulk Materials Storage. Handling and Transportation. Wollongong,
N.S.W., 28-29 September, 1998.
Kennedy, O.C A power based model of the pneumatic conveying process. 6th Tnt. Conf. on
Bulk Materials Storage. Handling and Transportation. Wollongong, N.S.W., 28-29 September,
1998.
Ill
Acknowledgments
The author wishes to thank his supervisors, Associate Professor P. W. Wypych and Professor P.
C Arnold, for their support, encouragement and advice over this course oftinswork. Their
assistance has been invaluable, not only directly to the author, but also indirectly in that without
their energetic establishment of the facilities in the Bulk Materials Handling Laboratory m u c h of
this work would not have been possible.
Many members of the professional and laboratory staff associated with the BMH Laboratory
have also provided m u c h appreciated assistance over the years this work was in progress. In
particular the author acknowledges the very competent and willing technical support provided by
laboratory staff M r D. M . Cook, M r I. Frew, M r s W . Halford and M r I. M c C o l m . The
invaluable support of M r A. Kent and M r K. Maywald in the development and maintenance of
data acquisition systems, vital to the experimental work, is gratefully acknowledged.
The encouragement and insightful discussions offered many of the professional staff and fello
postgraduate students involved in this area has enriched the author's experience of this project.
M r K. Stebbins, M s B. Whitakker and Dr R. Pan in particular acknowledged.
Finally, the author gratefully acknowledges his parents, for undertaking the arduous task of
editing this manuscript.
IV
TABLE OF CONTENTS
SUMMARY
PAPERS
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
NOMENCLATURE
CHAPTER 1
INTRODUCTION
CHAPTER 2
TEST ARRANGEMENTS
CONFIGURATIONS AND DATA
2.1
2.2
2.3
CHAPTER 3
3.1
3.2
3.3
3.4
3.4.1
3.4.2
3.5
3.6
3.7
3.8
3.9
3.10
CHAPTER 4
4.1
4.2
4.3
4.4
4.4
4.5
CHAPTER 5
Page
i
ii
iii
iv
vii
Test Facilities
Test Materials and Pipeline Configurations
Bulk Solid Deaeration Characteristics
9
12
14
DEAERATION CHARACTERISTICS AND
POWDER CLASSIFICATION
16
Deaeration Behaviour
A Simplified Theoretical Analysis
Deaeration Time Constant
Test Standards - Apparatus Design and Test Procedures
Fluidising Column Diameter and Mass of Solids
Plenum Chamber Volume and Venting Procedure
Effect of Plenum Chamber Volume
Normalisation of Deaeration Time Constant
Time Constants for a Range of Materials
Comparison of Theoretical and Experimental Time Constant
Extrapolation of time constant values from a data single point
Conclusion
17
20
22
22
23
23
25
27
32
37
40
41
CONVEYING CHARACTERISTICS
GENERATION TECHNIQUES
42
Manual Generation - Interpolation Processes
Three-Dimensional Surface Fitting
3-D Surface Modelling - Stepped Pipelines
3-D Surface Modelling - Scaling Capabilities
3-D Polynomial approach
3-D Surface Modelling - Conclusions
43
44
48
48
48
49
SOLIDS FRICTION FACTOR MODEL
50
5.1 Pneumatic Conveying Model
5.2
Application of the Pneumatic Conveying Model
5.3
Scaling Methodology
5.4
Stepped Pipelines - Optimised Design
5.5
Conclusions
50
51
54
55
57
CHAPTER 6
6.1
6.2
6.3
6.4
6.4.1
6.4.2
6.4.2.1
6.4.2.2
6.4.2.3
6.4.3
6.4.4
6.4.5
6.5
6.5.1
6.5.2
6.5.3
6.6
6.6.1
6.6.2
6.6.3
6.6.4
6.6.5
6.6.5.1
6.6.5.2
6.6.5.3
6.6.5.4
6.6.6
6.6.6.1
6.6.6.2
6.6.7
6.7
6.8
6.9
6.9.1
6.9.2
6.9.2.1
6.9.2.2
6.9.2.3
6.9.2.4
6.9.2.5
6.9.3
6.9.4
6.10
POWER BASED PNEUMATIC CONVEYING MODEL
Pressure Drop Based Models - a review
Power Based Model - Introduction
Power Source - Isothermal Expansion
Air Component
Superficial Gas Velocity
Interstitial Air Flow - Equivalent Diameters
Area Analyses
Equivalent Diameters
Air Velocity - Interstitial and "Free" Air
Pressure Drop and Power Consumption (¥##)
Air Component - Conclusions
Air Component - Example
Solids Component
Solids Velocity
Mass of solids in pipeline
Power Requirements - Solids Transportation
Bends
Bends - Power Consumption
Bends - Power Consumption - "fixed velocity"
Bends - Power Consumption - "average velocity"
Bends - Power Consumption - "integration approach"
Bends - Power Consumption - "segment approach"
Bends- comparison of approaches - Mass in bend
Bends - comparison of approaches - solids exit velocity
Bends- comparison of approaches - power consumption
Bends- comparison of approaches - pressure drop
Bends - implementation
Bends - implementation - bend velocity factor
Bends - implementation - bend location
Bends - some conclusions
Vertical Sections
Change in Kinetic Energy
Solids Friction Factor
Model P C I
Buckingham % Theorem Approach
Buckingham % Theorem - Model P C 2
Buckingham 7C Theorem - Model P C 3
Buckingham % Theorem - Model P C 4
Buckingham % Theorem - Model P C 5
Buckingham 7i Theorem - Model P C 6
Solids Friction Factor - S o m e general conclusions
Solids Friction Factor / VSR optimisation
Power Based Model - Conclusions
60
60
65
66
67
68
72
72
73
74
76
82
83
85
85
97
105
107
108
109
109
111
112
113
116
119
122
125
126
130
133
134
135
135
136
143
144
147
149
151
153
154
156
157
CHAPTER 7 POWER BASED PNEUMATIC CONVEYING MODEL 158
- IMPLEMENTATION & EVALUATION
7.1 Conveying Characteristics - General Shape Obtained 159
7.2
Pipeline Geometry - Trends in Performance
7.3
Variation with Pipe Length
7.4
Variation with Pipe Diameter
7.5
Variation with Number of Bends
7.6
Review of Model Performance -Preferred Versions
165
167
174
180
186
vi
CONVEYING LIMITS POWER CONSUMPTION DISTRIBUTION
192
Conveying Limits
Power Consumption Components
Conclusion
192
197
206
SCALING - LONG DISTANCE CONVEYING
- STEPPED PIPELINES
208
Scaling Characteristics
Scaling Characteristics - Diameter Variation
Summary of Scaling Trends with respect to Diameter
Scaling Characteristics - Length Variation
Summary of Scaling Trends with respect to Length
Some Scaling Examples
Long Distance Conveying - Stepped Pipelines
Conclusions
208
213
223
224
234
235
241
247
CHAPTER 10 POWER BASED M O D E L - REVIEW AND FURTHER W O R K
248
CHAPTER 11 REFERENCES
254
CHAPTER 8
8.1
8.2
8.3
CHAPTER 9
9.1
9.2
9.2.1
9.3
9.3.1
9.4
9.5
9.6
APPENDICES
APPENDDC A
APPENDIX B
APPENDIX C
Experiment Data, Materials, Pipeline Configurations
Trends in Model Performance
Sample Conveying Characteristics
259
284
309
Nomenclature
A
Pipe cross-section area (m 2 )
A'
Coefficient used in equations 6.5.15 and 6.5.16
Af
Cross-section area ofpipe occupied by fluid (usually air) (m 2 )
Ag
"Free" area available for gas flow above the bed of bulk solid (m 2 )
As
Cross-section area ofpipe occupied by solids (m 2 )
Asm
Cross-section area ofpipe occupied by solids, at mean pressure conditions (m 2 )
B'
Coefficient used in equations 6.5.15 and 6.5.16
Bm
Mass of bulk solid in bend (kg)
BVF
Bend velocity factor, equations 6.6.30 and 6.6.31
C'
Coefficient used in equations 6.5.15 and 6.5.16
d
Particle diameter
D
Pipe internal diameter
d50
Median particle diameter
De
Equivalent diameter of a circular section giving the same area as Ag- (m 2 )
Dh
Equivalent hydraulic diameter, corresponding to the Ag - Eq. 6.4.8 (m 2 )
Dh'
Modified hydraulic diameter (m) - Eq. 6.4.15
dSv
Equivalent particle diameter - based on a sphere having the same surface area to
ratio as the particle (also known as the Sauter mean diameter)
/
Friction factor (Fanning Equation)
fn
Friction factor (Darcy-Weisbach Equation)
Vf
Fr
Froude Number = ——=
v.
Frm
Froude Number at mean conditions =
JgD
V
Frs
Particle Froude Number =
g
gc
H
h
h0
K
L
m *
ntf
Ms
ms
pe
pt
pm
Po
PoB
Pof
Pos
R
RB
Acceleration due to gravity, * 9.81 ms' 2
Gravitational constant
Holdup or Slip ratio
Spring back height after impact (ref Szikszay impact factor - Sz, and h0)
Fall height prior to impact (ref Szikszay impact factor - Sz, and h)
BankoffK Factor
Pipeline length (m)
solids to air mass flow ratio
Air mass flow rate (kg s"1)
Mass of solids in pipeline (kg)
Solids mass flow rate (kg s"1)
gas pressure, at pipeline exit (or bend exit) (Pa)
gas pressure, at pipeline inlet (or bend inlet) (Pa)
gas pressure, mean pressure (or bend inlet) (Pa)
Power ( W or k W )
Power consumed by bulk solid in traversing a bend ( W )
Power requirement for transporting air ( W or k W )
Power requirement for transporting solids ( W or k W )
Universal gas constant = 287.1 (m s"2 K"1)
Radius of bend (m)
VfpfD
Reynolds Number, based on pipe diameter = — — - —
Re
Is*
viii
S
Sz
T
vx
ve
Vf
Vfa
Vfaf
Vfat
Vfin
Vj
Vs
Vse
Vsi
Vsm
VSR
VSR'
Slip velocity (m s'1)
Szikszay's impact factor = h/h0
Temperature ( K )
Free settling velocity of particle ( m s"1)
Specific gas volume, at pipeline exit ( m 3 kg')
Superficial air velocity ( m s")
Average "actual" air velocity, allowing for presence of solids ( m s"1)
"Free" air velocity in the area above the bulk solid ( m s'1)
Interstitial air velocity ( m s"1)
Superficial air velocity at m e a n conditions ( m s'1)
Specific gas volume, at pipeline inlet (m 3 kg"1)
Average solids or particle velocity ( m s"1)
Average solids velocity at exit conditions ( m s"1)
Average solids velocity at inlet conditions ( m s"1)
Average solids or particle velocity at m e a n conditions ( m s"1)
Ratio of average solids velocity to average "actual" air velocity
Slope of the solids to air velocity ratio, Eq. 6.5.12
GREEK SYMBOLS
a,P,y
5H
ApB
/dpy
Aps
Ap,
e
</>B
Af
XKK
Zs
hot
It,
Hn
p,
Pfm
PKFO
pip
ps
Terms in the solids friction factor correlation (powers of dimensionless parameters)
Change in height along the pipeline - positive for exit above inlet ( m )
Pressure drop resulting from bend (including accel. zone) (Pa or kPa)
Pipeline air pressure drop due to air-only flow (kPa)
Pipeline air pressure drop solids component (kPa)
Total pipeline air pressure drop (kPa)
Pipe roughness ( m )
Angle around pipe bend (radians)
Air-only friction factor
Coefficient of additional pressure drop due to particle-to-particle collisions Morikawa & Tanaka equation
Friction factor associated with solids
Friction factor for the combined air and solids mixture
Dynamic viscosity of air (*10(0000976r-47624) (N s m 2 )
Mechanical friction factor
Air density (kg m 3 )
Air density at m e a n conditions (kg m 3 )
Dispersed particle or solids density (kg m 3 ) - Morikawa & Tanaka equation
Bulk density, in loose poured condition (kg m 3 )
Particle or solids density (kg m 3 )
/ Introduction
1
Chapter 1
INTRODUCTION
The transport of bulk solids through a pipeline, driven by an expanding gas medium, results in a
very complex two phase interaction between the gas, the particles of the bulk solid and the pipe
wall. This complexity means that the development of reliable analytical tools to assist in the
design and optimised operation of these systems is a formidable challenge. The impetus to
expand and refine these tools remains, since in m a n y applications the unique features of this
m o d e of bulk solids handling lead to it being the preferred transport system.
The complexity of this process stems from the number of parameters that can influence the
performance of the system.
A multitude of particle properties, such as size, shape, density and electrostatic potential,
significantly affect the interactions between the gas and solids components of the flow.
Additional properties derived from the behaviour of the particles in the presence of the gas add
to the parameters to be considered w h e n developing models for the conveying process; these
properties include the fluidised bulk density, drag coefficient and pick-up, saltation and free
settling velocities.
The analysis of the gas component, on the other hand, would seem to present fewer difficulties.
It would be reasonable to consider the behaviour of this phase to be more predictable than the
solids, since the constituent particles of this phase (molecules) are of a vastly smaller scale than
the solid particles. Hence the possibility of treating the gas as a homogeneous mass is feasible.
However there are numerous situations within the conveying process that warn against too
simplistic a treatment of the gas phase. The turbulent behaviour of the gas at the margins of the
bulk solid/gas and pipe wall/gas interfaces, the variation in gas velocities along the pipeline, the
changing and non-circular flow channel available to the gas, the variations in flow above and
within a moving bed of particles, pressurefluctuationsat pipe transition points and around bends
are some of the examples of this complexity. Local pressure and velocityfluctuationscan have
a significant influence on the re-entrainment of bulk solid particles.
The moisture content of the gas may also be a significant factor in the performance of the
conveying system. Excess moisture could be considered a problem because of its potential to be
absorbed by the bulk solid. In m a n y cases this can have major consequences for the flow
properties, such as cohesiveness and adhesiveness, of the bulk solid. Compression of
atmospheric air, with its subsequent cooling to ambient conditions will generally result in a
relative humidity in the region of 1 0 0 % , with free moisture condensing in the air supply system.
Consequently, moisture traps are routinely used and frequently the air is dried further (via
refrigerant or desiccant driers) to provide the desired quality of air to the system. Dry air, on the
other hand, exacerbates any tendency the bulk solid m a y have to the build up of electrostatic
charges, affecting the conveying performance of the bulk solid, and increasing the explosion
hazard potential of the system.
2
1 Introduction
Particles also interact with other particles and the wall, via collisions and inter-particle forces.
Additional particle properties, such as coefficient of restitution, hardness, friability and V a n D e r
Waals forces, and bulk properties, such as internal and wall friction characteristics, degree of
agglomeration and cohesiveness, have a significant input to the overall performance of the
system. Particle to particle collisions can provide the impetus for re-entrainment of stalled
particles.
The degree of homogeneity within the bulk solid is also important in the behaviour of the
conveying system. A wide particle size range means that s o m e fractions of the bulk solid will
experience conveying conditions exceeding required rninimum pick-up velocities.
The
remainingfractionswould be less consistently entrained in the gas flow, relying on m o r e random
events such as collisions, local flow restrictions resulting in higher local gas velocities and
duning.
Considering the complexity and interaction of the factors outlined above, it is not surprising that
the analysis of pneumatic conveying behaviour and the development of accurate theoretical
models is an extremely difficult task. S o m e of the relevant material properties, for example
particle shape, m a y be definable in a descriptive manner but pose difficulties in the
determination of suitable numerical quantities that allow their inclusion in a theoretical model.
Estabhshing the complexity of this process does not, however, eliminate the need to provide
reliable predictions of pneumatic conveying systems. O n the contrary, the complexity of these
systems reinforces the importance of developing accurate analytical tools to assist in their design
and operation.
The most reliable means of estimating the pneumatic conveying behaviour of a given bulk solid
in a particular pipeline configuration, is to carry out test work on the bulk solid in a pipeline as
similar as possible to the required pipeline. Providing a comprehensive set of tests has been
conducted, a good indication of the performance of the bulk solid under a range of conveying
conditions will be available. The performance of the proposed system is then estimated directly
based on the results obtained if the pipeline used closely matches the intended configuration, or
more typically, based on the data collected but translated via the application of a scaling theory
to suit the intended pipeline.
In the analysis of conveying test results, three stages may be identified. Firstly, the transient
period associated with the initial introduction of the bulk solid into the pipeline, and secondly, a
steady state region where the performance of the system settles d o w n to a consistent and reliable
state; in this region the major operating parameters (air mass flow rate, total pressure drop,
solids mass flow rate) should have reached steady values whereby the rate at which the solids are
being fed into the system matches the rate at which the bulk solids are discharged from the
pipeline. Thirdly,_there is a second transient region for batch type systems as the conveying
cycle is concluded and the majority of the bulk solid is purged from the pipeline. Examples of
this basic data are given in Figures 1.1, 1.2 and 1.3.
1 Introduction
3
0.16
0.14
I
0.12
--
0.1
--
S
£
©
0.08
0.06
--
33 0.04
--
0.02
--
-
50
100
150
200
250
300
250
300
Time (s)
Figure 1.1 Typical conveying test data - air mass flow rate
300
50
100
150
200
Time (s)
solids in blow tank
Figure 1.2
solids returning to hopper
Typical conveying test data - solids mass flow rate
/ Introduction
180 -r—
9
120 -- / Vx.
2
Q
2
V)
2
a.
u
•a
o
H
20 •'• *
0 -| 1 1 1 1 1
0
50
100
150
200
250
300
Time (s)
Figure 1.3
Typical conveying test data - total air pressure drop
A preliminary observation of Figures 1.1 to 1.3 reveals one of the challenges associated with the
interpretation of test data, namely the identification of representative steady state conditions.
This process requires a degree of judgement. Fluctuations of the key parameters (air and solids
mass flow rates, pressure drop) need to be considered. Steady state conditions should reflect
reliable and repeatable operating states for the pneumatic conveying system.
The pipeline should be "full", meaning in this instance that the mass of bulk solid in the pipeline
should have stabilised to a reasonable extent. This certainly applies to the total mass in the
pipeline, but should ideally also apply to individual sections of the pipeline, though this is less
easy to measure. A s a consequence, data from the initial andfinaltransient periods of the cycle
will generally be disregarded.
Trends in the air pressure drop also need to be assessed and a view formed as to whether the
conveying regime is sustainable, or whether a continuation would in fact lead to a blockage. In
this context it should also be recognised that it is not unusual for there to be a pressure rise
towards the end of the cycle, resulting from the gradual clearing of solids from the line and the
consequent increase in air velocity as the pipeline and pressure vessel vents. These higher
velocities can rapidly re-entrain material that had settled out in the pipe, leading to the observed
rise in pressure.
Similarly, it is not unusual for blow tank fed systems discharging into an initially empty line, for
the initial rate of solids flow into the pipeline to be higher than the average rate that will be
achieved. This is a result of the large initial pressure differential between the pressure vessel and
the pipeline, which in turn facilitates a surge of bulk solid into the pipe. The subsequentrisein
pressure acts to restrain this surge, leading to the solids flow rate converging to the steady state
value appropriate to the selected operating conditions. A corresponding peak in the pressure
drop signal at the beginning of the conveying cycle is also likely to be observed, with the
pressure reducing as the flow stabilises.
/ Introduction
5
Given that this degree of caution and discernment needs to be exercised, even in the analysis of
accurately obtained test data, it is useful to collect data across a range of pneumatic conveying
conditions rather than focussing on a previously determined desirable operating point. In this
w a y a broad understanding of the pneumatic conveying performance of the bulk solid m a y be
obtained - and the likelihood that an anomaly in the behaviour of the material or the analysis of
the data will skew the results is minimised. Suggested procedures for conducting tests and
analysing test data leading to the determination of conveying characteristics or the production of
a state diagram, are discussed in some detail by W y p y c h and Arnold (1985).
The focus in the preceding discussion is on the steady state performance of the bulk solid in the
pipeline. In m a n y cases the behaviour of the conveying system in the transient regions is critical.
A sudden discharge of material into the pipeline at the commencement of a conveying cycle, if
not appropriately controlled, can readily result in blockages in the pipeline. Similarly, at the
completion of the conveying cycle, there is likely to be an increase in air velocity as the pressure
vessel and pipeline vent. A rapidrisein air velocity in this fashion can again result in blockages
as material that had previously settled out from the conveying system, is rapidly re-entrained.
Alternatively it m a y lead to unacceptable levels of particle degradation or pipeline wear. Despite
the potential significance of transient behaviour, it is preferable that the system design and
operation be governed by the steady state operation levels that can be achieved, rather than
restricted by transient flow events. Problems that m a y occur in the transient regions can be
addressed by the appropriate monitoring and control at start up. Therefore it is reasonable that
the emphasis, in recording and analysing conveying data, be on the steady state rather than the
transient behaviour.
The operation of a pneumatic conveying system is best summarised by the conversion of the set
of steady state operating conditions into a family of curves showing the relationships between
the three major parameters of interest, ie the air mass flow rate, the total system pressure drop
and the solids mass flow rate. The general form of this diagram is shown in Figure 1.4, for a
material that is capable of conveying in the dense phase m o d e and has a smooth transition
between the lean and dense phase regions.
350
j
300
--
250
--
'A?
1
a.
2
Q2 200 -3
SB
ON
150
••
•-
3 100 -•
\
50 -•
0 A
••'!
1
1
1
1
1
1
1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Air Mass Flow Rate (kg/s)
Solids Mass Flow Rate (kg/s)
Air only 1 2 3 — —4
Figure 1.4 Typical form of the conveying characteristics diagram for a pneumatic
conveying system
6
I Introduction
It should be noted that the smooth transition between lean phase and dense phase regions, ie the
smooth trough or valley shown in Figure 1.4, cannot be automatically assumed. This transition
is one of the key regions for potential instability in pneumatic conveying systems, and is
concerned particularly with horizontal conveying and where saltation m a y occur fairly abruptly
as the superficial air velocity decreases. The flow regime in this case moves from a fully
suspended state to a settled bed with dilute phase conveying occurring above this bed. In this
case a step increase in pressure drop m a y be observed, as noted by Zenz and Othmer (1960) and
Klintworth and Marcus (1985). This is consequently a region of some uncertainty in terms of
flow regimes and signalled byfluctuatingpressure drops, pipeline vibration and the occurrence
of pipeline blockages.
The precision of such a step change is likely to be blurred by the non-homogeneity of the bulk
solids, resulting in a range of saltation velocities and a more gradual switch between suspension
and non-suspension flow. For materials that are shown to be successfully conveyed in the dense
phase region, this step m a y not be observed, eg the materials presented by Marcus et al (1990),
pp 199-201 and thefinecoal discussed by Geldart and Ling (1990).
Various formats to express the conveying characteristics behaviour of a bulk solid have been
proposed in the literature. The "state diagrams" presented by Rizk (1982) use average air
velocity (X axis), pressure drop per meter (Y axis), with constant values of solid mass flow rates
plotted as curves or contours on these axes. Meyers and Marcus (1985) also used this set of
parameters and went on to construct a "dimensionless state diagram" based on log(Fr) ( X axis),
log(m*) (Y axis); plotting curves of constant solid mass flow rate on these axes gave straight
lines.
An overview of the above approach is given in Marcus et al (1990), with an extension to
considering the replacement of the air velocity parameter (generally taken as the average air
velocity) with a dynamic pressure term: p/v2/2. This modification is considered to have a
normalising effect on the data, resulting in a more consistent presentation of the results.
Certainly it addresses the major difficulty associated with the use of air velocity as a key
parameter. The expansion of the air d o w n the pipeline means that a range of air velocities is
acting in the system at any one time, rendering the selection of a relevant "average" velocity a
difficult matter. The inclusion of the air density term in die dynamic pressure parameter to some
extent balances the increase in velocity as the air expands.
The presentation of pneumatic conveying data in a generalised and dimensionless form has been
pursued further by a number of authors, including W y p y c h and Arnold (1987), Myler et al
(1986) and Molerus and Wirth (1991). W y p y c h and Arnold in their paper describe a method to
produce a form of generalised pneumatic conveying characteristics and demonstrate the results
using pneumatic conveying data for a fly ash/cement mix. The pressure drop due to the solids
component is plotted on the Y axis; this must be estimated from the total pressure drop and
assumptions m a d e regarding the calculation of the air-only pressure drop. Air mass flow rate
divided by pipe diameter squared (m/ D'2) is plotted along the X axis. Curves related to the
solids mass flow rate are plotted against these axes. In this case however a modified term
composed of the following elements is proposed:
msV
D2*105
where L' is pipe length, modified to account for the bends, D is the pipe internal diameter and
the exponent 2.8 is a value W y p y c h and Arnold suggest should be used w h e n considering scaleup across pipe diameters. Quite good similarity is shown for the two pipe configurations
discussed, indicating the potential usefulness of tins approach. It would be interesting to see
whether this consistency is maintained across a broader range of bulk solids and pipe
configurations.
1 Introduction
7
A possible drawback with the use of dimensionless state diagrams, is that the link to the original
test rig m a y be weakened. Given the complex interactions involved in pneumatic conveying,
and the limited extent to which w e can reliably extrapolate from this data to predict the
performance of other systems, the accuracy of the predictions should be expected to diminish
significantly as the parameters of the proposed installation depart from those of the test rig.
Consequently it is important that basic information (length, pipe diameter, vertical lift, number
of bends etc) applying to the original test rig be known, so that in the determination of the
performance of some other proposed system, an understanding of the degree of extrapolation,
and hence some indication of the likely accuracy, is retained..
Another interesting approach, proposed by Matsen (1982), results in phase diagrams strongly
reminiscent of the phase diagrams found in thermodynamics, showing the state of substances.
A simpler approach used by Mills and Reed (1980), involves the use of the air mass flow rate
along the X axis (or alternately volumetric flow rate at a nominated pressure and temperature),
the solid mass flow rate along the Y axis, with contours or curves showing constant total air
pressure drop being plotted against these axes. A particular feature of this arrangement is that
the solids to air mass flow ratio (m*), or phase density, is readily determined with constant
values of this parameter appearing as straight lines on the diagram.
An alternative approach, and the format that will be used in this thesis, is to replace the air
velocity parameter on the X axis with the air mass flow rate, with total pressure drop being
shown on the Y axis, ie as shown in Figure 1.4. Curves or contours showing the relationship
between these two parameters for constant values of solid mass flow rate m a y then be drawn on
these axes. Under steady state conditions, as discussed by Mills et al (1988), the air mass flow
rate is constant along the pipeline and m a y be more clearly linked to the conveying conditions
being recorded during a given experimental test than terms which vary, such as the air velocity.
Given the air pressure at any point, it is then also a simple matter to determine other relevant
data such as the superficial air velocity, the value of dimensionless parameters such as the
Reynolds, Froude and M a c h numbers.
To a large extent the suitability of a particular format is influenced by the system being
examined and the degree of control or the more readily controllable parameters, available to the
operator. It is usually possible to set the air mass flow rate, for example via the selection of
choked flow sonic nozzles. If, in addition, good control over the solids mass flow rate is also
available, perhaps by the incorporation of a rotary valve feeder, the graphical format used by
Mills and Reed is indicated. If, on the other hand, a pressure vessel is used to feed the bulk solid
into the pipeline, the ability to "dial in" a given solid mass flow rate is reduced. For a given
system, the achieved solids flow rate will be largely influenced by the system pressurisation
prior to discharge and the proportional splitting of the total air supply between the conveying air
going directly into the pipeline, and the air passing through the pressure vessel. In this case the
format used in Figure 1.4 is likely to be more relevant. Consequently, since pressure vessels
typically feature in dense phase conveying regimes, it could be argued that this format is likely
to be m o r e suitable where dense phase pneumatic conveying is to be studied. The advent of high
pressure rotary valves and hybrid systems such as combined blow tank/screw feeders, however,
means that this conclusion is not absolute,
In general it could be suggested that the presentation of the pneumatic conveying characteristics
or the state diagram, should be in a format that assists a plant operator to set up operating
conditions to achieve a desired result. The format is also of considerable interest to
experimenters, researchers and designers and they will no doubt develop a preference for
particular arrangements. However they will also have the resources to translate readily between
the formats, and hence the specific presentation is likely to be less critical.
8
/ Introduction
This family of curves (providing they reliably represent the steady state operation of the system)
is perhaps the most useful tool a designer or operator can have in the design of successful
conveying systems.
The steady state operating levels achieved by a particular system are a function of both the
material properties of the bulk solid being conveyed {including individual particle and bulk
properties) and the geometry (pipe length, diameter, number and type of bends, vertical sections)
of the conveying system itself. Ideally these steady state operating conditions would be
predictable from a theoretical analysis of basic particle and bulk properties of the material,
consideration of the behaviour of air as it expands d o w n a pipeline, with these analyses being
m a d e within the restrictions of a given pipeline configuration. A s indicated previously,
successful development of this theory has been hampered by the complexity of the interactions
involved and the wide variety of bulk solid properties that are encountered. Whilst worthwhile
progress has been made, eg extensive work has been carried out in thefieldof pickup velocity,
saltation velocity and m i n i m u m transport velocity (a useful collection of correlations and
equations is given by Plasynski et al., 1991), the present reality is that the most reliable means of
establishing the likely performance of a bulk solid in a pneumatic conveying system is to collect
test data for the material in a similarly sized test pipe loop. A model, generally incorporating an
empirical correlation component (frequently associated with the solids friction factor), m a y then
be applied to predict the performance of the proposed system.
Having obtained a set of conveying characteristics for a particular bulk solid/pipeline
combination, a plant operator is then in a position to set key parameters (air flow rates, system
pressure, solid flow rate) at levels that produce an optimised and desired outcome. T h e key
consideration in most instances will simply be to achieve reliable transport and eliminate
blockages. In addition, the pneumatic transportation system will need to achieve a throughput
that matches the requirement of the overall process - preferably at the same time minimising
wear on system components and degradation of the bulk solid. Meeting the latter two objectives
will also effectively rninimise the overall energy consumption of the pneumatic conveying
system.
This thesis concentrates on several aspects of the pneumatic transport of bulk solids, including:
1. The development of a deaeration characteristic for bulk solids and some assessment of its
relevance in rating bulk solids for handling in pneumatic systems.
2. The basic process of developing conveying characteristics curves from experimental data.
Empirical three dimensional surface fitting techniques and solids friction factor model
approaches are investigated. Once a consistent and objective process had been developed
and to some extent "automated" via a suite of computer programs, a number of additional
possibilities emerge and are pursued, including:
3. The incorporation of scale-up procedures into the process, resulting in the capability to
produce conveying characteristics curves for the n e w configuration.
4. Extending the process to stepped pipelines, which are of major importance in the
consideration of long distance pneumatic conveying installations.
5. Introducing some design optimisation, for example the determination of the optimal lengths
of each diameter section of a stepped pipeline system.
6. The overall pneumatic conveying process is considered from a power consumption
perspective, and a model developed based on this principle. T h e possibilities and insights
offered by this approach, including scale-up predictions, the identification of potential
blockage zones, the development of conveying characteristics curves and the relationships
between the major parameters (mf, ApT, ms) for variations in pipeline length, pipe diameter
and number of bends, are explored.
2 Experimental Arrangements - Configurations and Data
9
Chapter 2
TEST ARRANGEMENTS
CONFIGURATIONS & DATA
The data analysed in this thesis were obtained in a set of test rigs established at the Bulk
Materials Handling Laboratory, Department of Mechanical Engineering, University of
Wollongong. Extensive pneumatic conveying test facilities have been developed in this
Laboratory over recent years as part of the active research and consultancy activities. It is not a
primary purpose of this thesis to develop test facilities, instrumentation and testing procedures;
this important foundation work has previously been established. A n outline of the arrangements
is given here; further details of these procedures and facilities m a y be found in W y p y c h and
Arnold (1985), Wypych, Armitage and Arnold (1987) and W y p y c h (1989).
2.1 Test Facilities
Air Supply for the test facility is provided by one, or a combination of three rotary screw type
compressors. T w o air receivers, at a m a x i m u m pressure of 700kPag, assist in mamtaining a
stable air supply to the system during high demand intervals. Water traps eliminate condensate
downstream from the compressors. In addition, the air is refrigerator-dried andfilteredupstream
of the receivers, ensuring that a clean dry air supply is available to the blow tanks and conveying
pipeline. The supply air is returned to ambient temperature conditions prior to introduction to
the pneumatic conveying system. The quality of the air is important, given the variety of
materials that m a y be handled in the test rigs and the need to ensure that there is reasonable
scope to convey moisture sensitive bulk solids.
Air mass flow rate is measured by either orifice plate or annubar. A set of orifice plates using D
and D/2 pressure differential tappings, designed in accordance with BS1042, is available,
allowing the measured air flow range to be matched to the expected operating conditions.
Similarly, the annubar installation consists of a manifold bank which provides convenient
selection of the desired measurement range. Figure 2.1.1 shows a schematic arrangement of this
air supply system.
Positive pressure bottom discharge blow tanks are used to discharge the bulk solid into the
pipeline. For the longer distance pipeline configurations, a tandem blow tank arrangement,
consisting of two l m 3 pressure vessels, is used to ensure that sufficient material is available to
"fill" the pipeline and maximise the possibility that a steady-state conveying regime is
established in each test run. The blow tanks arefittedwith aerated fluidising cones in the lower
sections to promote steady discharge throughout the full batch cycle. The incorporation of this
feature in the pressure vessels had been found (Kennedy et al, 1987) to be particularly effective
at mamtaining constant flow rates of the bulk solid towards the end of the cycle, counteracting
the tendency for the flow rate to tail off, with the consequent reduction in the duration of the
steady-state conveying periods.
2
10
Experimental Arrangements - Configurations and Data
Control of the discharge rate of the bulk solid into the pipeline is achieved indirectly b y
adjusting the distribution of the air supply between direct injection into the pressure vessel (top
air andfluidisingcone air) and conveying air fed directly into the pipeline.
To test rigs, via orifice
plate or annubar manifold
Figure 2.1.1
Schematic A r r a n g e m e n t of Air Supply
Since the purpose of the test facility is to enable numerous test runs to be conveniently
undertaken, the typical arrangement of the pipe loops and receiving silo is such that the receiving
hopper is mounted directly above the pressure vessel/s. With this configuration, the refilling of
the blow tanks is simplified. T h e potential to automate the system particularly with the tandem
blow installation to provide continuous conveying, is incorporated although not fully
commissioned at the time. T h e difficult flow property characteristics of the fine powders used in
this work means that it is not possible to ensure filling the pressure vessels within the time
interval between discharge cycles. The requirement that the test facility needs to beflexibleand
available to convey a wide range of products under a variant of conveying conditions, also
means that there is little purpose in fine tuning the system for one product and one operating
point.
Two basic pipe loop configurations were used in collecting the data analysed in this thesis. The
longer distance systems utilised a ground level circuit of approximately 1 3 0 m per loop. At the
discharge end of this pipeline, a vertical lift of approximately 7 m raised the bulk solid to the
2
Experimental Arrangements - Configurations and Data
U
height of the receiving hopper above the pressure vessel(s). For m a x i m u m length tests, nine
circuits of this pipe route were available, comprising up to three different pipe diameters in a
stepped pipe configuration. Earlier tests had four pipe diameter sections and some test data from
this arrangement are included. However, it was subsequently concluded that the overall
arrangement would be improved by reducing to three steps and adjusting the lengths of each
stage. Interconnecting pieces allow considerable freedom in assembling shorter pipeline
configurations. The circuit type layout used in this test rig to obtain the large overall length
meant that a relatively large number of bends was used. The bend radius in this section was lm.
The circuit is laid out in a rectangular partem, the long side being approximately 6 0 m and the
short side approximately 5 m . Figure 2.1.2 gives a general arrangement of the testrigset up.
The second test rig arrangement provided shorter overall lengths and smaller diameter pipes.
The lift in this case, approximately 4 m , was located towards the beginning of the pipeline. The
bend radius was also smaller, at 0.254m.
Reverse jet filter bag installations fitted in or above the receiving hoppers carried out the
solids/air separation function.
The pressure vessels and the receiving hoppers were independently mounted on load cells.
During each conveying test, the change in mass indicated by these transducers w a s used to
determine the solids mass flow rate. Where a complete batch cycle was being analysed, the total
mass of solids currently in the pipeline was deduced. Stability in this parameter was an
additional indicator that the system was operating under steady-state conveying conditions.
Filter
Receiving Hopper
Blow Tank
Fluidising C o n e
Pressure Tapping Point
P
_I
Figure 2.1.2
Schematic Arrangement of Pneumatic Conveying Test Rigs
Air pressure along the pipeline was measured via transducers mounted in sockets in the pipe
wall, perpendicular to the flow. A permeable m e m b r a n e fitted flush with the inner pipe wall
separated the bulk solid in the flow from the pressure transducer. Details of this mounting
configuration are given in W y p y c h and Arnold (1985). This arrangement was found to be quite
robust and effective, however attention must be paid to the condition of the permeable
membrane. Since the actual pressure being monitored is the air pressure in the cavity enclosed
12
2
Experimental Arrangements - Configurations and Data
by the membrane and pressure transducer, deterioration or gradual blockage of the permeable
membrane will have the effect of preventing pressurefluctuationsin the conveying pipe being
transmitted through to the transducer. Variation in the condition of the permeable membrane
adds to the potential error in the data being recorded during a test. Regular calibration of the
pressure transducers is advisable and possible if the conveying pipeline isfittedwith a solenoid
operated shut-off valve at the end of the pipeline. The performance of all the transducers can
then be readily checked by simply pressurising the pipeline to a known level and comparing the
outputs. If the pipeline is then rapidly vented, the rate of pressure decay recorded for each of the
pressure transducers gives a good indication of the state of the permeable membranes. Decay
rates significantly slower than the norm for that test indicate potentially faulty permeable
membranes which can then be identified for replacement. These procedures were adopted to
minimise experimental variation arising from this source.
2.2 Test Materials and Pipeline Configurations
A large amount of experimental data was analysed during the course of this thesis as various
aspects were developed, such as the computerisation of conveying characteristics generation,
conveying model implementation, analysis and design of stepped pipelines,. For the majority of
the work, data from eight materials transported through a number of pipe configurations were
used. Details of these materials are listed in Tables 2.2.1 and 2.2.2. The Tables include
abbreviated designations for each of these materials and testrigconfigurations and are used
subsequently in the text to simplify the identification of the materials/pipeline configuration
details and minimise repetitive detail. Where data from different bulk solids and pipeline
configurations, other than those listed in the following Tables, are being analysed, the relevant
details are given at that point.
Table 2.2.1:
Test Materials
Designation
Material
A
kgm" 3
Pip
Associated
kgm"
Test Rigs (see
Table 2.2.2)
2000
800
TR1,TR2
2080
800
TR1, TR2
PF3
PF4
Pulverised Coal, sieved to
7 0 % < 75 urn
Pulverised Coal, sieved to
8 4 % < 75 u m
Pulverised Coal
Pulverised Brown Coal
2170
1500
750
520
PF5
Pulverised Briquettes
1571
482
PF6
Pulverised Brown Coal
1495
520
FA1
Fly Ash
2230
874
FA2
Fly Ash
2197
634
TR1, TR2
TR16, TR17,
TR18, TR19
TR20, TR21,
TR22, TR23
TR24, TR25,
TR26, TR27,
TR28
TR3 toTRll
inclusive
TR12, TR13,
TR14, TR15
PF1
PF2
It is noted that the stereopycnometer method was used to obtain the particle density data listed in
Table 2.2.1. The magnitude of the values reported for materials PF1, PF2 and PF3 suggests that
these particles contained significant porosity. It is anticipated that a specific gravity method
would give lower values.
2 Experimental Arrangements - Configurations and Data
Table 2.2.2:
Test Rigs - Pipeline Configurations
Designation
TR1
TR2
TR3
TR4
TR5
TR6
TR7
TR8
TR9
TRIO
TR11
TR12
TR13
TR14
TR15
13
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
stepped pipeline
TR16
stepped pipeline
TR17
TR18
TR19
TR20
a section of TR16
a section of TR16
a section of TR 16
stepped pipeline
TR21
TR22
TR23
TR24
a section of TR20
a section of TR20
a section of TR20
stepped pipeline
TR25
TR26
TR27
TR28
a section of TR24
a section of TR24
a section of TR24
a section of TR24
L
(m)
D
65
48
25.4
52.5
168.5
110.5
43.7
46.65
20.15
48.85
28.7
95.5
139.2
172.1
382.2
553.5
562.4
382.4
558
391.2
233.66
105
105
105
105
105
105
105
105
105
69
81
69
558
391.2
233.66
563.4
381.2
252.7
563.4
381.2
252.7
146.1
390.3
261.1
150
146.1
390.3
261.1
150
(mm)
(m)
3.9
3.6
7
0
0
0
0
7
7
7
7
Number
of Bends
6.3
9
4
5
2
0
2
0
2
2
4
4
5
12
17
69
81
69
81
105
0
6.75
0
0
0
17
12
16
12
8
69
81
105
0
0
0
16
12
8
69
81
105
0
0
7
17
12
8
69
81
105
0
0
7
17
12
8
60
69
81
105
0
0
0
7
4
12
8
5
60
69
81
105
0
0
0
7
4
12
8
5
6.65
6.75
Bend
Radius (m)
0.254
0.254
-
It m a y be noted that in a number of cases in Table 2.2.2, distinct pipeline configurations are
formed as subsets of the complete pipeline. This can be an efficient means of maximising the
data recorded during a test run. Providing the pressure drop across each pipe section is recorded,
the conveying conditions resulting from one set of solids and air mass flow rate values across a
number of different pipe length and diameter sections can be monitored. In the case of stepped
pipelines, if the data are intended to be applied to a specific conveying model, this procedure is
likely to be essential so that model parameters dependent on pipe details can be determined.
For each bulk solid/pipeline configuration, the intention of the tests is to obtain conveying dat
from steady-state operating conditions over a sufficient range of conveying regimes to enable a
3 0009 03192487 6
2 Experimental Arrangements - Configurations and Data
14
conveying characteristics plot to be generated. The resulting set of experiment data includes
details of the pipe configuration in use (length, diameter, number of bends etc), and the steadystate conveying parameters (air and solids mass flow rate, and system pressure drop) that form
the foundation of subsequent calculations. The information required in this data set varies to
some extent depending on the analysis carried out. If the purpose is tofita three dimensional
surface to the experimental data, then the air and solids mass flow rates and the total system
pressure drop is sufficient. Where the intention is to use the data to determine parameters for a
conveying model, additional information relating to bulk solid properties and the pipeline
arrangement m a y be required. The following Table 2.2.3 shows an example of the data set used
in a power based conveying model, described later in this thesis. Appendix A lists the full set of
experimental data associated with the bulk solid/pipeline configurations listed in Tables 2.2.1
and 2.2.2.
Table 2.2.3:
Material:
Sample Experiment Data Set
Test Rig:
PF3
TR1
Particle Density: 2170kg m 3
Loose Poured Density: 750kg m 5
Ambient Pressure: 10 lkPa
Ambient Temperature:
No.
points
21
data Pipe ID
(mm)
Pipe
(m)
25.4
65
Air Mass Flow Rate
(kg s"1)
.0100
.0088
.0071
.0061
.0039
.0021
.0095
.0086
.0072
.0059
.0037
.0023
.0090
.0083
.0067
.0053
.0037
.0100
.0079
.0067
.0059
Pressure Drop
(kPa)
82
82
80
79
81
81
116
116
115
115
116
115
156
156
155
151
155
190
186
183
184
Length
15 ° C
Lift
(m)
No. of Bends
3.9
9
Solids Mass
Rate (kgs"1)
.238
.265
.301
.309
.333
.276
.418
.443
.489
.505
.537
.395
.668
.688
.724
.747
.741
.820
.875
.902
.896
Flow
Bend Radius
(m)
.254
Exit Pressure
(kPa,)
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
2.3
Bulk Solid Deaeration Characteristics
It has not been a primary focus of this work to relate particular bulk solid characteristics to
pneumatic conveying performance and to build conveying models on that basis. However, in
the course of the research some attention was paid to thisfield;in particular, the significance of
2
Experimental Arrangements - Configurations and Data
15
the deaeration behaviour of bulk solids has been examined as a possible aid to the assessment of
the suitability of these materials for this m o d e of transport. Clearly, good air retention capacity
is of assistance in pneumatic conveying systems, especially long distance applications. Perhaps
the most direct measure of this characteristic is the rate at which a fluidised sample of the
material deaerates.
Since there is little material in the literature regarding this characteristic, there was a need to
develop test procedures and analysis techniques that m a y have a general application. S o m e
interesting results were obtained and a measure of this characteristic is proposed, namely the
"time constant" based on an exponential pressure drop over time across a bed of deaerating
material. A s a result of the active industrial consultancy work conducted by the Bulk Materials
Handling Laboratory, it was possible to collect data on this characteristic for a wide range of
materials. T h e measure has proved to be a useful comparative indicator for bulk solids and is
c o m m o n l y included in the Laboratory's standard series of material property tests.
Since this aspect developed into a substantial study in its own right, the detailed discussion of t
test apparatus, procedures, analysis and results is presented in Chapter 3.
3. Deaeration Characteristics and Powder Classification
16
Chapter 3
D E A E R A T I O N CHARACTERISTICS A N D
P O W D E R CLASSIFICATION
Geldart (1973) has previously identified thefluidisationbehaviour of bulk solids as a significant
bulk property, and proposed the division of bulk materials into four classes (Groups A, B, C and
D ) based on particle size and particle to air (orfluidisingmedium) density difference. These
groups are conveniently shown graphically in Figure 3.1.
10000
o
!
N^
1000
B
\
D
*
Q
A
I
c
s
*
**•
100
10
100
1000
10000
Particle Size (|am)
Figure 3.1
Geldart Fluidisation Classification Diagram
The bulk solids within each group cover a range of particle sizes and densities, and share the
same generalfluidisationbehaviour. For example, Group A powders, which m a y range in size
from 2 0 p m to 1500um and a particle density varying from 200 to 7000kg m"3, are easily
aerated, exhibit a limited tendency to form bubbles and a considerable degree of expansion
duringfluidisation.The bed collapses slowly when thefluidisationair supply is shut off.
The potential of classifying bulk solids via their fluidisation behaviour has been of interest to a
number of workers and whilst some discrepancies have been noted, particularly for powders
falling in the Geldart Group C Classification as indicated by Zenz (1984), it is fair to say that
Geldart's Classification has been accepted as a basic starting point. Molerus (1993), for
example, has taken Geldart's Classification and investigated dimensionless ratios built up from
key forces associated with the behaviour of powders in the fluidisation process, in an effort to
clarify the location of boundaries separating onefluidisingregime from another. This would
3. Deaeration Characteristics and Powder Classification
17
appear to be a worthwhile approach in that as one mechanism loses its dominant influence, eg
cohesive forces for Group C powders, and an alternative mechanism takes over, eg fluid drag
forces, the appropriate dimensionless ratio should clearly signal this change. Nevertheless, the
application of this procedure is certainly less straightforward than Geldart's system and involves
the determination/estimation of constants specific to each material. The result is a band, rather
than a precise boundary, which is probably a good outcome and more likely to accord with
reality, though in their generation these bands tend to be referred back to the original Geldart
Classification for s o m e degree of comparison/verification.
Dixon (1979), when exarnining the tendency of bulk solids to form slugs during vertical lift
pneumatic conveying, developed a similar set of four groups of powders that broadly correspond
to the Geldart Classification and are plotted on the same set of axes. Pipe diameter is used in the
development of these classifications, in addition to fluid density and particle properties such as
density, terminal settling velocity, voidage and m e a n size.
Since both fluidisation and slugging are concerned with bulk solid behaviour in conjunction with
afluidisingmedium, typically air, it is not surprising that attempts have been m a d e to use the
Geldart and Dixon Classifications to characterise the suitability of bulk solids to handling in
pneumatic conveying systems. It has been suggested that Group A powders, for example, are
good candidates for pneumatic conveying; Group B powders are likely to be difficult,
generating high pressures and severe pipeline vibrations in dense phase conveying modes.
Group D solids are potentially good performers though requiring higher gas velocities. The
cohesive properties of Group C powders pose particular problems for pneumatic conveying and
m a y require specialised equipment, eg flexible walled pipelines, for successful handling; see
Marcus et al {1990) pp 78-80 and Dixon (1979).
Similarly, the deaeration behaviour of the bulk solid may also be used to form the basis of a
classification system. Indeed, deaeration is conceivably more relevant than fluidisation
behaviour, particularly w h e n considering the suitability of bulk solids to handling by pneumatic
transportation. It could be anticipated that the longer it takes a bulk solid to deaerate, the longer
it will retainfluid-likeproperties and the easier it will be to m o v e in a pneumatic conveying
pipeline.
3.1 Deaeration Behaviour
The deaeration of afluidisedbulk solid is basically the venting of air from the interstitial spaces
between the particles, and m a y include escape of bubbles of air and/or the collapse of channels
of air depending on the m o d e offluidisationof the material. Visually, this is accompanied by
the collapse of the height of the bulk solid and m a y include the spouting of fine particles from
vents that m a y continue on for some period after the apparent final collapse of the bed.
When a differential pressure meter is set up to record the air pressure in the bed, either across a
fixed height within the column offluidisedmaterial or across the whole column of material, the
deaeration of the bulk solid is observed as a decay of differential pressure over time. T h e initial
pressure, during fullfluidisation,should result in a pressure which, if taken across the area of the
column offluidisedmaterial, is sufficient to balance the gravitational force acting on the bulk
solid.
As die bulk solid collapses, the interstitial gaps diminish and the particles settle and rest upon
one another. A s indicated by the pressure readings, the air continues to escape at a diniinishing
rate after the apparent conclusion of the collapse of the bed. The bulk solid is effectively acting
as afilter,resisting the escape of the gas during this process and, with only the relatively small
pressure differential expelling the air, the pressure decay m a y require an extended period of
time. Particle size, packing arrangement, density and shape would be expected to have
significant effects on the rate of release of this air.
18
3. Deaeration Characteristics and Powder Classification
Deaeration has not received the same degree of attention as fluidisation from researchers, no
doubt becausefluidisationis integrally involved in m a n y important industrial processes and
hence an understanding and optimisation of the process confers direct benefit. S o m e early work
in this area was in fact carried out in the context of fluidised bed catalytic processes, in
recognition of the fact that "a tendency toward deaeration is inherent in any fluidised system",
Diekman and Forsythe (1953). In this instance, the relationship between the viscosity of the
mixture (gas/solid) and the degree of deaeration is of interest. Deaeration is measured in terms
of the collapse in bed height versus time of afluidisedbulk solid, and a plot showing deaeration
time against voidfractionis produced. This is shown to be a linear relationship for the product
tested (cracking catalyst). Mainwaring and Reed (1987) discussed previous work by Sutton and
Richmond (1973) where the determination of a constant deaeration rate factor is proposed, again
based on the reduction in bed height method. In m a n y instances this leads to the same
conclusion reached by Diekman and Forsythe, ie that the deaeration process is substantially
linear.
Measuring the collapse in bed height is not the most convenient experimental method, in that the
determination of the position of the top surface m a y be difficult (due to dust loads obscuring the
clear observation of the surface, and an uneven surface topology) and often the collapse itself
m a y be very rapid. It is likely that a high speed visual record of the process will need to be
m a d e for subsequent analysis. Consequently, Mainwaring and Reed concluded that exarnining
the differential pressure across the bed , as a means of measuring and recording the deaeration
process, warranted investigation.
The rate of pressure decay over time is readily recorded on a chart recorder. Mainwaring and
Reed decided tofitan hyperbolic function of the following type to the resulting curve:
Ap
-f-xt = A.
h
Eq 3.1.1
where the constant At is the relevant parameter identifying the deaeration behaviour for the
material being tested.
More recently, Fargette et al (1997) also proposed using the decay in pressure differential, rather
than collapse in bed height, to indicate deaeration behaviour. In this paper the time taken for the
pressure differential across the bed to reach zero was suggested as a characterising parameter, to
allow comparison of different powders. There would not appear to be procedures adopted to
allow for different bed heights or plenum chamber volumes.
An alternative approach, in terms of selecting a function to describe the deaeration process, is
suggested by the observation that a substantial portion of the decay processfrequentlyfollows an
exponential decay pattern, as indicated in Figure 3.1.1. This exponential decay rate has been
found to provide a consistently good fit to the deaeration behaviour for a wide range of
materials; it is very simple tofitto the data and is considered to be preferable to the hyperbolic
function noted above. In some instances the initial period of deaeration does not follow this
trend, and m a y have a convex upward initial deaeration behaviour similar to that exhibited by
the Pulverised Coal sample shown in Figure 3.1.2. This is likely to be the result of a nonhomogeneous initial state, with air pockets trapped in thefluidisedbulk solid escaping from the
column of material and in effect prolonging thefluidisingphase beyond the shut off of the air
supply.
3. Deaeration Characteristics and Powder Classification
19
2.5
-A
2 --
I
V
S3
w
1.5 4-
s
?s
I
1 --
fc
0.5
1
1
0.5
1—
1.5
2
3.5
2.5
4.5
Time (s)
Figure 3.1.1
Exponential pressure decay during deaeration
Calciner Ash (time constant = 0.65 s)
8.000
7.000 --
•
6.000 --
Exp. Data
¥
5.000 : :
5 ;
1
T| 4.000
2
U
|
&
* * \
• \
•\
•\
•V
3.000
a, 2.000 -1.000 --
0.000
1
1
i-
H
1
I I
10
20
30
40
50
60
' I * -1
70
80
90
Time (s)
Figure 3.1.2
Deaeration Behaviour - Pulverised Coal showing initial "convex up'
deaeration rate (time constant = 22.8 s)
Another exception to the smooth exponential decay of pressure differential across the bulk solid
occurs when the collapse of the bed includes one or more abrupt slumps. The wall of the
fluidising column will always provide some support to the bulk solid as it deaerates. In some
instances this friction/normal force interaction results in a series of step like collapses of the
3. Deaeration Characteristics and Powder Classification
20
material. This creates a pressure surge in the column, as indicated in Figure 3.1.3, with the
exponential decay generally resuming after each "disturbance". A more detailed digitisation of
the experimental data used to develop the curve in Fig 3.1.3 would show that the smooth
portions of the decay are also m a d e up of a series of saw-tooth steps, an indication of the degree
of frictional interaction between this material (pulverised brown coal) and thefluidisingcolumn
wall. Other effects, notably electrostatic charge build up, m a y also be contributing to this
behaviour. This figure also demonstrates the initial "convex u p " rate of pressure decay noted
previously.
5.000
4.500 --
1.000 -0.500
0.000
20
40
60
80
100
120
140
160
180
200
Time (s)
Figure 3.1.3
Deaeration Behaviour - Pulverised Briquettes (Brown Coal)
showing irregular peaks in deaeration rate
Despite these anomalies, the overall exponential decay pattern provides a very useful indicator
of this aspect of the material's behaviour. In situations of interest in pneumatic conveying, (for
example the rate of deaeration of a bulk solid where it m a y have been deposited from the
conveying m e d i u m after a bend), the general trend in deaeration behaviour, (indicated b y the
exponential decay pattern) is of more relevance than the exceptions to this trend that m a y be
observed in, and be specific to, the deaeration of the material in afluidisingcolumn. Hence it is
valid to extract characteristics for the material based on the exponential decay portion of the
material's deaeration behaviour and if necessary, "smoothing" out any steps resulting from an
intermittent collapse of thefluidisingcolumn. This is not to say that further examination of
these other effects m a y not provide useful insights and valuable material characterising
properties in their o w n right; however it is proposed that a focus on the overall exponential
decay pattern is a worthwhile approach.
3.2
A Simplified Theoretical Analysis
During the deaeration of a column offluidisedbulk solid, three sources m a y be identified for the
air escaping from or through the material:1. F r o m the plenum chamber beneath the bulk solid, since this is at a higher than ambient
pressure. In this analysis, this volume is designated as volume Vj; it is testrigdependent.
2. F r o m the collapse of the expanded interstitial spaces between the particles resulting from the
fluidisation of the bulk solid, ie the air occupying the difference in volume between the
3. Deaeration Characteristics and Powder Classification
21
fluidised and deaerated states of the material. In this analysis, this volume is designated as
volume V2; it is material dependent.
3. From the interstitial spaces between the particles once the bulk solid has collapsed from its
expandedfluidisedstate, as the pressure drops from thefluidisedvalue at any particular
point in the material, to the ambient pressure. In this analysis, this volume is designated as
volume Vi, it is material dependent (affected by both the amount of material and the
difference between the particle and bulk densities).
It is the intention of this Section to develop a relationship between time and the pressure dr
across the bulk solid, during the deaeration process, preferably providing some support to the
exponential decay that is generally observed experimentally, particularly after the initial phase.
The second source of air identified above, resulting from the drop in volume occupied by the
bulk solid during deaeration, will not be considered at this point. This air flow is substantial
during the early phase of deaeration and may well contribute significantly to the initial convex
upwards pressure decay observed for some materials, since the higher air flow will maintain the
pressure differential across the bulk solid. It is likely to contribute significantly to the rate of
deaeration if collapse in bed height is used as the analysis method. It will havetittleto no
impact, however, once the collapse in bed height is largely completed, which is the case for the
majority of the time when pressure decay across the bed is being recorded. For this reason, as
well as the simptification it brings to the analysis, this source is not considered here.
If the pressure at the base of the bulk solid, and hence in the plenum chamber, is Pi, the mass
air in the plenum chamber, ie in the volume Vi is:
p v
i\
Mx - —~
(making use of the ideal gas equation)
Eq 3.2.1
Similarly, the mass of air in the interstitial spaces (using an average of the pressures Pi a
is:
u
(Px + Pjv*
M =
Eq32 2
>
2RT
Therefore the total mass (M/j)of air associated with the volumes Vj and V3 is:
+
=Z
W7^
Eq3.2.3
RT+ 2RT A T^'
If a control volume surrounding the plenum chamber/bulk solid system is defined, the rate at
which the mass of air within this volume falls is:
-jf- = -PatmAvA Eq3.2.4
where "A" is the area of the top (free) surface of the deaerating material, and vA is the veloc
the air as it leaves this surface. Making the assumption that this velocity is directly proportional
to the pressure drop (Ap), ie vA - k Ap, where a value for k may be obtained from the slope of
the initial portion of afluidisationplot for the material, and differentiating Eq 3.2.3 with respect
totime,it m a y be shown that:
Ap 2V1+V3 4
Integrating both sides gives:
\n{Ap) = —~f!Lj-t +const Eq 3.2.6
resulting in the following expression:
2Vy+V3
Ap=K.e
Eq 3.2.7
22
3. Deaeration Characteristics and Powder Classification
const,
(an exponential pressure decay, N B K is a constant, AT = e
)
The reciprocal of the coefficient of the independent variable (time) in the exponential term may
be used to define the rate of decay, and characterise the deaeration behaviour of the material.
For a time dependent exponential decay, this constant is normally k n o w n as the time constant.
The above analysis suggests that the time constant m a y be determined from:
time constant = — nqj.z.s
A time constant in seconds will be obtained if the following units are used for each of the terms:
term consistent units
3
volumes Vi and Vi
areav4
pressure patm
variable k
time constant
m
m
Pa
m ( N s ) " orms" Pa"
s
3.3 Deaeration Time Constant
In defining any material property in a form useful for subsequent classification of materials, it is
important that some numerical measure be found for this property. This not only provides a
convenient means for separating materials into different groups, but makes the property
accessible for use in mathematical models. A n exponential decay suggests the use of the
familiar time constant parameter, which increases the attractiveness of this approach to
deaeration analysis.
The time constant is the major parameter in an exponential fit. It is directly linked to the rate of
decay, and in effect is the time it takes for the differential pressure across the bed to decay from
any given initial pressure to a value approximately one third (actually 1/e) this value.
If the differential pressure meter tappings are located in such a way that they are always within
the column of deaerating material as it collapses, then this pressure differential also represents a
pressure gradient through the bulk solid, since it applies across a fixed bed height. If, on the
other hand, the pressure tappings are arranged so that the low pressure tapping is always above
the material, then the pressure differential m a y be evaluated in terms of a pressure drop per unit
mass of material. The experimental work conducted here has been consistently based on the
former approach, ie with the pressure tappings wholly within the bulk solid.
3.4 Test Standards - Apparatus Design and Test Procedures
A s this work was being planned, and the options for assessing the deaeration performance of a
bulk solid were being examined, it became apparent that it was necessary to establish some
standardised procedures. Initially this issue arose when preliminary tests aimed at repeating
some test work reported by Mainwaring and Reed (1987) were being conducted. It appeared
likely that basic features of the test apparatus, such asfluidisingcolumn diameter and plenum
chamber volume, would have an impact on the results obtained. S o m e of this information was
readily available in the published data. S o m e key parameters were not, in particular details
relating to the plenum chamber and venting procedures, raising the question of the sensitivity of
the results to these physical parameters and test practices.
A number of potentially significant parameters were identified.
• fluidising column diameter
• plenum chamber volume
• mass of solids (or height of bulk solid in thefluidisingcolumn)
• venting procedure
3. Deaeration Characteristics and Powder Classification
23
3.4.1 Fluidising C o l u m n Diameter and M a s s of Solids
Since the air must escape from the top surface of thefluidisedmaterial (and the base of the
material as well if the bottom of the column is vented), for a given mass of material, it would
seem likely that the diameter of thefluidisingcolumn would affect the result. A possible
relationship that m a y be of relevance is the ratio of thefluidisedvolume of the bulk solid to the
available surface area for the air to escape.
nr2h
Volume
—
~—
=
5- = h
(or 0.5/2 for top and bottom surface venting)
KY
Free Surface
This ratio eliminates the column diameter and suggests instead that the height of the fluidised
material (and consequently the mass of the bulk solid in the test apparatus) is important. A
fluidising column of « 1 0 0 m m diameter was selected. This diameter column was the standard
size used in the Materials Handling Laboratory at the University of Wollongong for fluidisation
tests, and appears to be a c o m m o n size used by workers in this field.
3.4.2 Plenum Chamber Volume and Venting Procedure
Determining a suitable plenum chamber volume is less obvious. T o nunimise the effect of
pressurised air from the plenum chamber venting through the material and extending the
apparent deaeration time constant, the smallest possible plenum chamber is desirable.
The impact of the air in the plenum chamber could also be reduced by venting the plenum
chamber simultaneously with the shut off of thefluidisingair supply. Whilst this would
minimise the effect of the plenum chamber volume, it alters the physical process significantly by
allowing the escape of the air in two directions. In determining the suitability of either approach
(ie venting of the air only from the top of thefluidisingcolumn, or allowing venting from both
the top and bottom of the column), it is worthwhile giving some consideration to the actual
physical processes to which the results of this test m a y be applied. In terms of pneumatic
conveying suitability, the situations of interest are likely to include the rate of deaeration of
fluidised material that has settled out from the conveying stream at a pipeline change, eg bend or
pipe diameter increase, or alternatively in a batch conveying system, and the ability of material
retained in the pipeline to remain in afluidisedcondition between runs. For bulk storage
situations, the relevant situation would be the rate at which material loaded into a bin releases
interstitial air and packs moretightlyinto the bin.
In all these situations, the air mixed in with the bulk solid is basically required to vent th
the free top surface of the material. This gives sufficient reason to adopt a standardised
deaeration procedure that does not employ venting of the plenum chamber during the actual
deaeration event.
One difficulty associated with providing a low plenum chamber volume is ensuring an even air
distribution to thefluidisingcolumn. A test rig intended to address these concerns was
developed, based on a small plenum chamber located within a larger air supply volume that
could be isolated via a cylindrically shaped sliding valve. A ring of "port holes" was provided to
promote an even air supply to the permeable membrane, as shown in Figure 3.4.1, and the
design was effectively free of any pressure pulse in the small plenum chamber as the sliding
valve was closed.
24
3. Deaeration Characteristics and Powder Classification
Permeable Membrane
Minimised Plenum Chamber
Slide Valve
Large Plenum Chamber
Figure 3.4.1
Air Inlet
Zl —
Pneumatic Cylinder
Fluid isarion/Deae ration C o l u m n , minimised P l e n u m C h a m b e r V o l u m e
(slide valve version)
The complexity of the design, uncertainty with regard to effectiveness of the sealing arrangement
between the cylindrical sliding valve and the minimised plenum chamber volume, and concerns
about the impact of vibrations resulting from the closing of the shding valve on the deaerating
column of bulk solid, led to the development of a far simpler but effective design. In this case
the large air supply volume is eliminated completely and the minimal plenum chamber is built
up from three pipeflangessandwiching a pair of permeable membranes, Figure 3.4.2. The two
membranes ensure evenfluidisation,assisted by a baffle at the air inlet to prevent direct impact
of the air jet on the bottom membrane. The simple design ensures effective sealing of the
bottom end of the fluidising column when the air supply is shut off; actuation of the supply
valve achieves both outcomes simultaneously.
Permeable Membranes
Baffle
Minimised Plenum Chamber Volume
Variable Volume Plenum Chamber
Figure 3.4.2
Supply Air
FIuidisation/Deaeration C o l u m n , minimised P l e n u m C h a m b e r V o l u m e
(simplified version)
3. Deaeration Characteristics and Powder Classification
25
3.5
Effect of Plenum C h a m b e r V o l u m e
This design provided an effective n u n i m u m plenum chamber volume, however the significance
of the plenum chamber volume on the overall results remained to be established. T o this end a
variable volume chamber was connected to the base of thefluidisingcolumn, as indicated in
Figure 3.5.1.
Variable Volume Plenum
Supply Air
Figure 3.5.1
Fluid isation/Deaeration Column, Variable V o l u m e Plenum C h a m b e r
Arrangement (simplified version)
The plenum chamber volume could now be readily altered by adjusting the amount of water in
the secondary or variable volume chamber. The niinimum plenum chamber volume that could
be achieved (occurring when the variable volume plenum chamber was isolated from the test rig)
was set by thefreevolume of the dual permeable membrane base,« 167ml.
The deaeration behaviour for a range of materials was then observed and the results analysed to
produce a time constant for each material. These tests were repeated for plenum chamber
volumes ranging from 167ml to 5,655ml. The results, as indicated in Figure 3.5.2, show a
substantially linear relationship for each material between the plenum chamber volume and the
time constant associated with the rate of deaeration of the material. It is noted that in each case
thefluidisingcolumn ( 1 0 0 m m diameter nom.) was loaded with a sample of the material,
sufficient to create a deaerated height of « 4 0 0 m m . The amount of material in the fluidising
column was not altered during the subsequent tests while the plenum chamber volume was being
varied. The deaerated height and the mass of solids being tested was therefore held constant.
Recalling E q 3.2.8 developed in Section 3.2,
2VX+V,
time constant
IPatmAk
the term Vi represents the plenum chamber volume, and V3 the interstitial volume between the
particles in the collapsed bed condition. For a fixed mass of material being tested, V3 is a
constant. A linear relationship between the plenum chamber volume and the time constant
would therefore be expected, as indicated by the experimental results shown in Figure 3.5.2.
3. Deaeration Characteristics and Powder Classification
26
1000
2000
3000
6000
5000
4000
Plenum Chamber Volume (cc)
Bulk Solid
•PVC Powder
+
Monarch
sand
- } K — R a w Zircon
Figure 3.5.2
-Calcined
Alumina
- ^ — Pulverised
Coal
-Alumina
Time Constant variation with Plenum C h a m b e r V o l u m e
The linearity displayed in Fig 3.5.2 also suggests that it is possible to extrapolate the time
constant value backwards to obtain an estimate of the value of this parameter at a zero plenum
chamber state. Carrying out a linear extrapolation of this sort on the data for the six materials
shown in Fig 3.5.2 gives the time constants listed in Table 3.5.1.
Table 3.5.1 Zero Plenum Chamber Volume Time Constants
kgm" 3
d50
urn
Time Constant (seconds)
(extrapolated for zero
plenum chamber volume)
Monarch Sand
2679
214.6
0.39
Time Constant
(seconds)
(minimum plenum
chamber volume)
0.39
P V C Powder
1522
152.1
0.77
0.95
R a w Zircon
4628
117.9
1.12
1.15
Calcined Alumina
4105
107.2
4.29
4.34
Pulverised Coal
1834
73.9
9.62
9.9
Alumina
4076
93.1
9.25
10.5
Material
Ps
A standardised test procedure therefore, could be to conduct a series of deaeration tests over a
minimum of three different plenum chamber volumes, determine the time constants of the
exponential decay and then, using a linear extrapolation, estimate the time constant for the zero
plenum chamber condition.
It m a y also be observed that the time constant obtained for the smallest plenum chamber volume
condition is in each case very close to the extrapolated zero plenum chamber volume condition.
3. Deaeration Characteristics and Powder Classification
27
In practical terms it is certainly more convenient to carry out several tests at this stage and, from
the average of the results, estimate the deaeration time constant for the material, than it would be
to carry out the multiple tests and extrapolation procedure outlined above. The results of this
work indicate that useful deaeration data m a y be obtained from a single series of tests, provided
a testrigeffectively incorporating a niinimum plenum chamber volume is developed, and that a
series of variable plenum chamber volume tests verify that an extrapolation of thetimeconstants
obtained show a close correlation between the zero and minimum plenum chamber volume
conditions.
3.6 Normalisation of Deaeration Time Constant
These results, however, do not reveal any information on the significance of the height of the
material in thefluidisingcolumn on the deaeration behaviour observed. Additional test work
evaluating this aspect was carried out using alumina powder. In this casefiveseries of tests were
conducted. In each test series, the mass of solids in thefluidisingcolumn, and consequently the
deaerated height of the material, was held constant, and deaeration time constants for the range
of previously used plenum chamber volumes determined.
The results of these tests are presented in Table 3.6.1, and Figure 3.6.1. The linear relati
between the deaeration time constant and the plenum chamber volume, is again observed, over
the range offluidisingmasses tested. It is also apparent that there is a consistent increase in the
time constant with increasing mass in thefluidisingcolumn.
Table 3.6.1
Deaeration Test Results for Alumina, for various powder masses and
plenum chamber volumes.
Plenum Chamber
Volume
(cc)
167
656
1072
1489
1905
2322
2739
3155
3572
3989
4405
4822
5238
5655
Deaerated Height of
Alumina Powder (m)
>
Time Constant (s)
Mass of Alumina in Fluidisina Column
3 kg
1kg
2kg
1.8
3.49
5.12
6.34
2.15
4.3
7.38
4.98
2.45
5.66
8.41
2.74
6.35
3.04
9.45
7.03
3.34
10.48
7.72
11.52
3.64
12.56
8.4
3.94
13.59
9.09
4.24
9.77
14.63
4.54
10.45
15.66
4.84
11.14
16.7
5.14
11.82
17.74
5.44
18.77
12.51
5.74
0.125
0.240
0.354
4kg
8.69
10.14
11.38
12.62
13.85
15.09
16.33
17.57
18.8
20.04
21.78
22.52
23.75
24.99
5 kg
11.11
12.89
14.41
15.92
17.44
18.96
20.48
21.2
23.52
25.03
26.55
28.07
29.59
31.11
0.469
0.584
3. Deaeration Characteristics and Powder Classification
28
35
0 -|
1
1
1
1
1
1
0
1000
2000
3000
4000
5000
6000
Plenum Chamber Volume (cc)
Figure 3.6.1 Variation of Deaeration Time Constant with Plenum Chamber Volume,
for various Masses of Bulk Solid in Fluidising C o l u m n (Alumina Powder)
Fig 3.6.1 indicates firstly that the mass of solids has a major influence on the time constant
obtained, and secondly that the relationship is substantially linear. The linearity of this
relationship is supported by E q 3.2.8 developed in Section 3.2:
2VX+V3
tune constant =
—
2palmAk
Recalling that the term Vi represents the plenum chamber volume, and Vj the interstitial volume
between the particles in the collapsed bed condition, for a fixed plenum chamber volume, Vj is a
constant, therefore a linear relationship is suggested between Vj (the mass of bulk solids being
tested), and the time constant.
The linear nature of this relationship suggests that a normalised time constant, obtained by
dividing the time constant for any given test by the mass of the bulk solid used in that test, m a y
provide a consistent means of reporting this parameter. In effect, this results in an estimated
time constant for 1kg of the bulk solid. Processing the alumina powder data in this fashion
results in the curves shown in Figure 3.6.2 for a selected set of plenum chamber volumes, across
the range of masses of powder tested.
It may be more relevant, however, to consider normalising the deaeration time constant data on
the basis of the deaerated height of the bulk solid in thefluidisingcolumn. Variations in particle
density between materials and particle packing arrangements would result in time constants per
kg of material referring to substantially different geometric situations being compared ie volume
to free surface area ratio. O n the other hand, a time constant normalised on the basis of
deaerated height would require that similar situations are being compared. It would seem likely,
therefore, that a time constant normalised on the basis of the deaerated height of the material
would prove more useful since one of the major purposes of obtaining a deaeration parameter is
to facilitate comparisons between materials. Figure 3.6.2 shows data from the same series of
tests as Figure 3.6.3, normalised on the basis of the deaerated height of the material in the
column.
3. Deaeration Characteristics and Powder Classification
0.5
1.5
2
2.5
3
3.5
29
4.5
Mass of Bulk Solid (kg)
Plenum Chamber Volume (cc)
•0
—•—565
4822 — B — 5 6 5 5
Figure 3.6.2
-ic—1489
X
2322
X
3155
-3989
Variation of Deaeration Time Constant with Mass of Bulk Solid in
Fluidising C o l u m n (Alumina Powder)
Time Constant Normalised on the basis of Mass.
60
50 -•
a o
e g 40
m
Vi
2 ew
41
Q
3 30
•a S
.
1u
"3 £
g b 20
5 H
z
10
1.5
0.5
2
2.5
3
3.5
4.5
Mass of Bulk Solid (kg)
Plenum Chamber Volume (cc)
-656
Figure 3.6.3
-A—1«9 —X—2322
X • 3155
-3989
4822 — -
Variation of Deaeration Time Constant with Mass of Bulk Solid in
Fluidising C o l u m n (Alumina Powder);
Time Constant Normalised on the basis of Deaerated Height.
-5655
30
3. Deaeration Characteristics and Powder Classification
A comparison of time constants, normalised by both of these methods, is given in Table 3.6.2
In consideration of these two factors - plenum chamber volume and mass of solids (or the
deaerated height of the material in the column) - it is clear that if a consistent and useful
parameter for the deaeration behaviour of a bulk solid is to be obtained, the influence of these
aspects must be taken into account. The experimental data suggests two trends in deaeration
behaviour that m a y be utilised:
1. The deaeration time constant varies linearly with the volume of the plenum chamber.
2. The deaeration time constant varies linearly with the mass of bulk solid (or height of
deaerated powder) in thefluidisationcolumn.
In recognising these characteristics, it is recommended firstly that a linear extrapolation be
carried out to estimate the deaeration time constant at zero plenum chamber volume, and
secondly that the resulting time constant be normalised by dividing by the height of the
deaerated powder in the column.
Applying this procedure to the alumina powder results in the data presented in Table 3.6.2
Table 3.6.2
Extrapolated Zero Plenum C h a m b e r V o l u m e and Normalised Deaeration
Time Constants for Alumina Powder.
(zero
Mass of Alumina Time Constant
plenum chamber volume)
Powder
(kg)
(s)
1
2
3
4
5
1.7 (1.8)*
3.2 (3.5)
4.7 (5.1)
8.2 (8.7)
10.5 (11.1)
Normalised Deaeration Time Constant
mass basis (s/kg)
1.7 (1.8)*
1.6 (1.7)
1.6 (1.7)
2.1 (2.2
2.1 (2.2)
height basis (s/m)
13.4 (14.4)*
13.4 (14.6)
13.3 (14.5)
17.5 (18.5)
18.0 (19.0)
* minimum plenum chamber volume deaeration time constants shown in (brackets), for
comparison with the extrapolated zero plenum chamber volume values.
The above results show a consistent normalised deaeration time constant value, varying a
m a x i m u m 0.3 seconds (or 1 7 % ) from the average value of 1.8 seconds (mass basis) or 2.9
seconds (19%) from the average of 15.1 seconds (height basis normalisation). This is a
significant result when the variation in mass of solids (a factor of 5) and the variation of plenum
chamber volume (a factor of 34) is taken into account. For the testrigused, the time constant
obtained for the "minimum plenum chamber volume" is seen to be close to the extrapolated zero
plenum chamber volume condition for both the experimental data and normalised forms.
Figure 3.6.4 shows the variation of the normalised deaeration time constant for the complete set
of Alumina Powder data (Table 3.6.2 represents the extreme left of this set of data). If the
process of obtaining a normalised time constant is effective, this data would be expected to
collapse to a single line. The reasonablytightbounds found for the data in Fig 3.6.4 support the
use of this procedure.
3. Deaeration Characteristics and Powder Classification
1000
2000
3000
4000
5000
31
6000
Plenum Chamber Volume (cc)
Mass of Bulk Solid (kg)
Figure 3.6.4
Variation of Normalised Deaeration T i m e Constant (height basis) with
Plenum C h a m b e r Volume.
Returning to the original set of deaeration data for a range of materials (sand, alumina, P V C ,
pulverised coal etc), normalisation of this data results in the plot shown in Figure 3.6.5.
It may be noted in Fig 3.6.5, in comparison with the previous presentation of this data in a "raw
or pre-normalised form in Fig 3.5.2, that there has been some adjustment in the relationships
between the various materials. In particular, pulverised coal is n o w seen to have a longer
deaeration time constant than alumina, which in general would be the expected result for the two
materials. Obtaining an accurate relative rating between materials is important in terms of
characterising materials, hence the appropriate normalisation of the deaeration time constant has
a real significance.
The relative rating for these six materials is examined in further detail in Table 3.6.3. Here ar
listed the deaeration time constants (extrapolated, from a linearfitto the data, to the zero plenum
chamber volume condition) for both the raw and normalised data,. A rating has been established
for each material (for both the raw and normalised data), based on the smallest time constant in
each case (the monarch sand material provided these values for both the raw and normalised sets
of data). This serves to highlight further the relative difference in deaeration behaviour between
the materials, in particular the alumina and pulverised coal as indicated by the normalised time
constants.
3. Deaeration Characteristics and Powder Classification
0
1000
2000
3000
4000
5000
6000
Plenum Chamber Volume (cc)
Bulk Solid
•^—Monarch
— • — P V C Powder—£—Calcined
Sand
Alumina
$ K — R a w Zircon —#—Alumina
Figure 3.6.5
—X"—Pulverised
Coal
Normalised (height basis) Deaeration Time Constants for a range of
materials
Table 3.6.3 Rating the Deaeration Behaviour for various Materials, based on the Zero
Plenum C h a m b e r Volume Deaeration Time Constant (Raw and Normalised)
Material
Monarch Sand
P V C Powder
R a w Zircon
Calcined Alumina
Alumina
Pulverised Coal
R a w Data
Time Constant
(s)
.39
.77
1.12
4.29
9.25
9.62
Rating
1
2
2.9
11
23.7
24.7
Normalised Data
Rating
Time Constant
(s/m)
.78
1.53
3.2
8.58
18.49
27.5
1
2
4.1
11
23.7
35.3
3.7
Time Constants for a Range of Materials
Table 3.7.1 lists deaeration time constants obtained for a wide range of materials, normalised on
the basis of deaerated height. It should be noted that these data were in many cases collected
while the materials were being processed for other purposes in the University of Wollongong
Bulk Solids Handling Laboratories. In a number of instances the tests were conducted prior to
the point at which the significance of the deaerated height of the material, as a normalising
parameter, was recognised. In these instances, a deaerated height has been estimated from other
test parameters. For almost all these materials, the deaeration time constant reported is for the
rninimum plenum chamber volume configuration, rather than an extrapolated zero plenum
chamber volume. For two materials, the aluminium powders, testriglimitations did not allow
the minimum plenum chamber volume configuration to be used. In these cases an estimated
time constant was determined as outlined in Figure 3.7.1.
3. Deaeration Characteristics and Powder Classification
Table 3.7.1
Material
33
Normalised Deaeration T i m e Constants for a R a n g e of Materials
Dixon Slugging
Normalised
Geldart
Classification
Deaeration Time Fluidisation
Constant (s/m)
(D= 100mm,
Classification
P=2Bar)
monarch sand
duminium powder (cohesive)
P V C powder
calcium carbonate
calciner ash
sugar (extra fine)
P V C powder (corvic)
raw zircon
pulverised spray dried kaolin
electro-static precipitator dust (1)
baghouse dust
tertiary alumina
calcined alumina
milled extrafinesugar
highfluoridetertiary alumina
gypsum
superphosphatefines(coarse comp.)
aluminium powder
superphosphate fines
fly ash (coarse)
alumina
pulverised coal (Cooranbong)
ground flux
pulverised coal (K coal)
fly ash
pulverised coal (Munmorah - B mill)
Alumina (A Lobb tests - Lobb, 1995)
pulverised coal (Stewarton)
pulverised coal (Hunter)
pulverised briquettes (brown coal)
pulverised coal (Hunter 6 0 % < 75 u m )
mill dust
fly ash (Gladstone, < 10 p m )
milled bath
pulverised coal (Curragh)
pulverised coal ( 7 0 % < 75 urn)
pulverised coal ( 8 0 % < 75 u m )
pulverised coal (Curragh 84)
pulverised coal (Munmorah 3 C mill)
electro-static precipitator dust (2)
0.8
1.4
1.5
1.8
1.9
2.1
2.7
3.2
3.6
4.2
7.4
7.8
7.9
8.8
10.1
10.2
10.2
11.2
12.1
14.5
18.5
19.6
20.7
21.8
24
27.4
36.7
40.7
42.7
45.6
56.6
65.5
66.2
68.3
95.2
114
122.1
156.2
157.7
209.4
B
C
B
C
on A/B boundary
on A/B boundary
C
C
C
C
B close to A/B
on A/B boundary
B close to A/B
on A/B boundary
B
C
C
C
B
C
C
C
on A/B boundary A
on A/B boundary A
A close to A/C
on A/B boundary
A
A
C
A
A
A close to A/B
A close to A/C
A
A
A
C
A
A
LA
C
A
C
C close to A/C
A towards A/C
A towards A/C
A
C
A
A
A - towards A/C A
C
C
C
C
C - towards A/C
C - towards A/C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
N B : A s noted previously, the laser diffraction technique has been used in determining the
median particle size data, which was subsequently used in obtaining the Geldart Classifications
for the materials listed in Table 3.7.1. It is possible for this method to indicate a smaller particle
size than m a y be obtained by sieving, particularly for fine materials with a tendency to form
agglomerates. W h e r e this occurs the material would m o v e to the left on the Geldart Chart, for
example possibly moving from a Group A to a Group C classification.
34
3. Deaeration Characteristics and Powder Classification
Several points can be m a d e regarding these data. Firstly the materials that would generally be
expected tofluidisewell, retain their fluidised state and perform well in dense phase pneumatic
conveying systems eg pulverised coals and fine fly ashes which have significantly higher
deaeration time constants. Similarly materials that would be expected to deaerate rapidly, such
as sand and P V C powder, did so and had short time constants.
Secondly, some fine materials with strong cohesive properties (indicated by major channelling or
the formation of spherical agglomerates duringfluidisationeg calciner ash) that might otherwise
be expected to deaerate slowly (on the basis of the small size of the constituent particles), had
relatively short time constants. This difference was marked, and m a y prove to be of
considerable value in borderline cases, eg assessing which of the Geldart and Dixon Group C
materials are in fact good pneumatic conveying candidates, and which m a y prove difficult.
S o m e interesting contrasts are exposed. For example, the two electro-static precipitator dusts
included in the sample set, exhibited widely differing deaeration behaviour which, as indicated
in Table 3.7.2, is certainly not suggested by the corresponding basic particle size and density
data.
Table 3.7.2
Parameter
Comparison of electro-static precipitator dust data
electro-static precipitator dust electro-static precipitator dust
(2)
(D
209.1
4.2
normalised time const, (s/m)
dsv particle size ( m m )
9
7.2
3
3290
2817
particle density (kg m )
Thirdly, the time constant parameter showed a wide range in values (in excess of two orders of
magnitude) for the materials tested. The extent of this variation is significant in terms of
distinguishing or classifying materials.
Table 3.7.1 also shows classifications of these materials based on the Geldart's Fluidisation
Classification and the Dixon Slugging Diagram. For this purpose, particle density was obtained
by an air comparison pycnometer method, using a Quantachrome Stereopycnometer, M o d e l
SPY-3. Particle size was obtained by a laser light scattering method .on a Malvern series 2600
Particle Size Analyser, Model 2602C. This device reports a number of characteristic particle
diameter values, notably the median particle diameter d 50 and, of interest in terms of these
classifications, the Sauter m e a n diameter (or D[3,2]). This diameter is derived from the m e d i u m
diameter and the specific surface area measured for the particles, and is equivalent to the surface
area to volume diameter (dsv) used by Geldart (Lapple, 1968; Malvern Instruments Instruction
Manual, 1991).
In considering the significance of the resulting classifications, it should not be forgotten that all
these materials, to a greater or lesser extent, consist of a range of particle sizes and the use of a
representative particle diameter is a necessary compromise. Other factors in the measurement
process add further uncertainty to the determination of a characterising diameter. For example,
the sizing method is volume rather than mass based and consequently does not take account of
any variation in particle density of the constituents of the bulk solid. Similarly, the
stereopycnometer method used in determining the particle density should effectively determine
the true density of the solid, taking into account any pores in the particles. The question could
well be asked as to whether this or the apparent density of the particle is of relevance during the
fluidisation and deaeration processes.
It is interesting to observe the regular division in material classifications (Geldart Classification)
shown in Table 3.7.1. A s m a y be expected, the majority of the bulk solids having short
deaeration time constants were classified as Group B, though s o m e were Group C. N o
3. Deaeration Characteristics and Powder Classification
35
materials tested were of the Group D type; these would be expected to have very short
deaeration time constants, however they would be unsuited to testing influidisationfacilities
being used. Group A materials consistently had significantly longer time constants and are
grouped in the 10 to 40 second deaeration time constant bracket. The Group C materials,
however, did not fall neatly into a bracket of deaeration time constants and instead were located
at either the low end or the high end of the spectrum.
The Dixon Slugging Classification for these materials, also shown in Table 3.7.1, in most
instances matches with the Geldart Classification. For the purposes of deterrrnning these
classifications, a pipe diameter of 1 0 0 m m , an air pressure of 200kPaabS and a voidage of 0.45
was assumed.
Of particular interest are the materials that received a Group C classification but also
demonstrated long time constants. Quite a number of the bulk solids in the long time constant
region were in the Group C classification. A number of these materials are of a type that would
generally be expected to convey well in dense phase pneumatic conveying regime (pulverised
coals and fly ash), suggesting that the other Group C materials that performed similarly under
the deaeration test would have similar pneumatic conveying potential. The ability of the
deaeration test to distinguish markedly between the cohesive and the goodfluidisingGroup C
materials, demonstrates the insight that this procedure is able to provide.
The preponderance of Group C materials in the high deaeration time constant zone in Table
3.7.1 suggests that a large proportion of Group C materials have long deaeration time constants.
This outcome is more likely to be a result of the particular set of materials tested than a general
conclusion regarding the behaviour of the fine particulate bulk solids found in the Group C
classification. M a n y of the materials listed were also in the process of being assessed for their
pneumatic conveying potential, and hence were more likely to be in the good fluidising/air
retention category.
Figure 3.7.1 shows the data from Table 3.7.1 in a graphical format. The chart indicates that
Group A and B materials fall within reasonablytightregions along the deaeration time constant
axis, with all Group B materials having time constants below * lOseconds (including materials
on the A / B boundary) and Group A materials generally within the 10 to 40 seconds range. O n
the other hand, Group C materials range from the low end of the time constants expected from
Group B materials to somewhat beyond the highest of the Group A materials (a m a x i m u m time
constant above 200 seconds). This reinforces the points m a d e previously and adds weight to the
proposal that the deaeration time constant m a y be a more valid and reliable means of assessing
thefluidisationand pneumatic conveying potential of a bulk solid than the simple Geldart
Classification. The retiability of the deaeration time constant as a predictor for these purposes
remains to be established.
3. Deaeration Characteristics and Powder Classification
36
5-i
-. »ft x *—
B
©
...
— •
U
• ••
en
M
• ' •
o
CC
a
^ •-
r— t—
0-
0.1
1
10
100
1000
Normalised Deaeration Time Constant (s/m)
Bulk Solid Classification
• B BA/B A A
Figure 3.7.1
XC
Normalised Deaeration Time Constants for a range Geldart Classifications
Based on experience with the pneumatic handling of a range of the materials listed in Table
3.7.1, the normalised time constant m a y be used to m a k e the following preliminary
classifications.
Table 3.7.3
Preliminary Classification based on the Deaeration T i m e Constant
Potential Pneumatic Conveying Handling Characteristics
Normalised T i m e
Constant (s/m)
Composed of either coarse heavy materials orfinecohesive materials. The
<5
coarse materials m a y be suitable for lean phase conveying, slug flow, or
low velocity plug flow for the more rounded, non-interlocking particles
(eg wheat grain). The fine cohesive materials are likely to be difficult to
handle by pneumatic conveying, resistingfluidisationand entrainment in
the air stream.
Potentially difficult to convey, particularly in dense phase and at low
5-15
velocities, likely to cause severe pipe vibrations and blockages under these
conditions. M a y perform well in bypass systems and be suitable for
handling by air slides.
Transition range, includes some materials that m a y be expected to perform
15-25
satisfactorily in pneumatic conveying systems and others that are likely to
require bypass systems.
>25
These materials have good air retention characteristics and are likely to be
the most suited to dense phase pneumatic conveying, particularly as the
normalised time constant exceeds 40 s/m.
The option of successfully handling a bulk solid in an air slide is interesting in that it combines
the requirement that the product be readilyfluidisable,with the need for reasonably quick
deaeration so that some control of the flow can be obtained by shutting off the air supply.
3. Deaeration Characteristics and Powder Classification
37
Aliunina is very successfully handled in air slides, suggesting that materials with similar
deaeration time constants m a y be similarly suited to this m o d e of transport.
3.8 Comparison of Theoretical and Experimental Time Constant
In Section 3.2 the following theoretical expression was developed for the time constant:
2V1+V3
tune constant
E q 3.2.8
lPatmAk
It is the purpose of this Section to apply this expression to estimate the value of time constant
using experimental data to determine values for V\, V$ and k. The value obtained m a y then be
compared to the experimentally determined time constant.
Two materials have been used in this comparison, alumina and PVC powder. An air decrease
fluidisation test was conducted with these materials to obtain the required base data. Table 3.8.1
lists relevant results from these tests. Figures 3.8.1 and 3.8.2 show the corresponding
fluidisation plots.
Table 3.8.1 Results of Fluidisation Tests - Time Constant Estimation Procedure
Parameter
Alumina
P V C powder
3.951
2.363
mass (kg)
3
4075
1522
ps (kg m" )
0.472
0.55
deaerated height of material (m)
3
1.666 E-4
1.666 E-4
Vi, plenum chamber volume (m )
2.963 E-3
3.0304 E-3
V3, interstitial volume (m )
2
8.3323 E-3
8.3323 E-3
A, area(m )
100900
101000
Patm, atmospheric pressure (Pa)
s
a,
¥
••6
«
u
O
V
u
3
W>
t»
e
3
4
5
6
0-
Superficial Air Velocity (cm/s)
Figure 3.8.1
Fluidisation Plot - P V C Powder
3. Deaeration Characteristics and Powder Classification
38
1
3
2
4
Superficial Air Velocity (cm/s)
Figure 3.8.2
Fluidisation Plot - Alumina Powder
As noted in Section 3.2, the relationship between the velocity of the air escaping from the bulk
solid and the pressure drop, is assumed to be of the form, v A = k Dp. A value for k m a y be
obtained from the low flow end of thefluidisationplot, since this analysis is primarily concerned
with the deaeration behaviour of the bulk solid once thefluidisedbed has substantially
collapsed. Figures 3.8.3 and 3.8.4 show these data and straight linefits,assuming a zero
intercept, for the P V C and Alumina powder respectively. It should be noted that the axes are
reversed from the format in Figures 3.8.1 and 3.8.2 (velocity is along the vertical axis), and that
total pressure drop, rather than pressure gradient, is shown along the horizontal axis.
0.0035
0.0030 -^ 0.0025
'5
| 0.0020 +
< 0.0015 -'3
1 0.0010
a
3
™
0.0005 0.0000
100
200
300
Pressure Drop (Pa)
Figure 3.8.3 Velocity vs Ap - PVC Powder
400
500
600
700
3. Deaeration Characteristics and Powder Classification
/—s
0.0030 -
39
.-""•
^
s 0.0025 -
^ ^ m
'3
©
0.0020 •
<
0.0015 -
n^.
"a
•E 0.0010 -
^*-*^
m^>
a
e3n 0.0005 0.0000 -
500
1
1000
1
1500
1
2000
2500
Pressure Drop (Pa)
Figure 3.8.4
Velocity vs A p - Alumina Powder
These data allow the following values for k to be determined
P V C powder
k = 4.9668 E-6 m 3 N'1 s"1
Aliunina powder
k - 1.2932 E^m^'s- 1
Time constant values m a y n o w be determined for these two materials using E q 3.2.8.
Deaeration tests were also conducted for these materials and experimental values for the time
constant obtained for comparison with these values.
Table 3.8.2
Time Constant Comparison
Parameter
P V C powder
time constant (Eq 3.2.8)
4.0 s
time constant (experimental)
1.5 s
Alumina
15.2 s
18.5 s
This shows reasonable agreement between the calculated and experimentally obtained values for
the time constant. A more accurate analysis, including some allowance for the collapsing
interstitial volumes and/or a more refined relationship between the escaping air velocity (VA) and
the pressure drop, would be expected to provide even closer results.
It is interesting to note that if the value for the slope k is adjusted to match the calcul
constant value with the experimentally obtained value for the P V C powder (resulting in k =
1.3323 E-5 n^lSrV1), this corresponds closely to the slope from the origin to the commencement
of the constant pressure drop region of thefluidisationplot. This is demonstrated in Figure
3.8.5. In thisfigure,the vertical axis shows the total pressure drop across the fluidised bed,
rather than the pressure gradient as in Figures 3.8.1 and 3.8.2, to match the requirements of the
equation vA = k Ap.
3. Deaeration Characteristics and Powder Classification
40
3000 -,
A2500 - ""
/
*
•
,|J
^^^ *
•
/ fb
^
Su
a
©
u
a
/
2000 -
1500 -
y
*
^P
f
m
§
J
a
»
&
*
A
4>
U
8
J^
/A
/^P
/#
V
Jv
BT
t
1000 t
0
500 ^F *•
0 -
1
1
1
0.01
0.02
0.03
\
0.04
1
0.05
1
0.06
1
0.07
1
0.08
0.09
Superficial Air Velocity (m/s)
-fluidisation
Figure 3.8.5
k = 1.3323E-5
k = 4.9668E-6
Velocity vs A p - P V C P o w d e r , comparing " A " values
A similar result is obtained for the Alumina powder, however the steeper more linear initial
section of thefluidisationplot for that material m e a n s that the effect is less apparent.
It may be that the approach suggested above, ie taking the value of k from the slope from the
origin to the c o m m e n c e m e n t of the steady-state pressure drop, provides a m o r e representative
indication of the relationship between the air velocity and the pressure drop, and hence a m o r e
accurate value for the calculated time constant. Alternatively this m a y b e a result peculiar to
these two sets of data. Further investigation over a wider range of materials would b e required
to clarify this.
3.9 Extrapolation of time constant values from a data single point
A final point to b e m a d e here is that the consistent trends observed in the deaeration behaviour
of powders suggest that it should be possible to estimate with reasonable accuracy the zero
plenum chamber volume time constant for a specific material, given the deaeration time constant
obtained from an existing test rig with a k n o w n , and perhaps substantial plenum chamber
volume. Figure 3.9.1, taken from the linearised data for the six materials presented previously
(Fig 3.6.5) demonstrates this.
41
3. Deaeration Characteristics and Powder Classification
80.00
0.00
1000.00
2000.00
3000.00
4000.00
5000.00
6000.00
Plenum Chamber Volume (cc)
Bulk Solid
+
Monarch
Sand
-5K—Raw Zircon
-PVC Powder —£—Calcined
Alumina
A
Aluminium
-Alumina
(1)
Figure 3.9.1
-X—PulvCoal
O Aluminium
(2)
Normalised Deaeration T i m e Constant Data - Extrapolation of Zero
P l e n u m C h a m b e r V o l u m e T i m e Constant from a Single Data Point
The materials Aluminium (1) and Aluminium (2) have known deaeration time constants at a
plenum chamber volume of 3255cc. Since there is a consistent converging trend observed for
the six materials where deaeration data versus plenum chamber volume is k n o w n (and no crossovers are apparent in this trend for these materials), it would seem likely that a reasonable
estimate for the zero plenum chamber volume deaeration time constant could be found for the
aluminium powders by following the trends in the surrounding materials back to the y axis.
3.10 Conclusion
The deaeration behaviour of powders has been analysed and a method proposed for
characterising this behaviour, based on an exponential decay time-constant model.
Experimental procedures to obtain these data have been developed and tested for a range of
materials. Trends in these data have been examined and a method has been investigated for
normalising the data to allow consistent comparison between materials. A simplified theoretical
analysis has been carried out, with the results generally supporting the trends observed
experimentally.
Deaeration time constants for a large number of materials have been obtained, and converted to
a normalised time constant format to assist in the comparison and classification of these
materials. A preliminary classification of the suitability of these materials for handling by
pneumatic systems (air slides and pneumatic conveying systems) has been proposed, based on
the deaeration time constants found and the typical performance characteristics exhibited by
these materials. For example materials that perform well in air slide systems were found to have
normalised deaeration time constants in the region of 5 to 15 s/m. Materials that perform
particularly well in pneumatic conveying systems, with the capacity for conveying in dense
phase, were found to have normalised deaeration time constants in the region of 40 s/m and
above. The proposed classification system is presented in Table 3.7.3.
4
42
Conveying Characteristics - Generation Techniques
Chapter 4
CONVEYING CHARACTERISTICS
GENERATION TECHNIQUES
Conveying characteristics plots such as that shown in Figure 4.1 are a very useful w a y of
presenting pneumatic conveying performance data for a bulk solid/pipeline combination. T h e
inter-relationships between the three major conveying parameters (solids and air mass flow rates
and the system pressure drop) are clearly delineated, and the possible operating points to meet a
desired performance objective (say a particular solids mass flow rate) are apparent. Other
significant performance aspects, such as gas and solids velocities and overall power
consumption, can also be estimated for desired operating points. With the conveying
performance data presented in this format a system designer is in a position to match optimally
the hardware selected with the material characteristics and the transport requirements.
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FUR HRSS FLON RRTE IK9/1]
CONTOURS: 11J Solrds Mass Flowrate Ikg/sJ
Figure 4.1 Conveying Characteristics Plot: Bulk Solid: F A 2 , Pipeline Arrangement: T R 1 2
The basis for generating conveying characteristics plots is a set of steady state conveying data for
the particular bulk solid/pipeline combination concerned. These data are usually the result of a
planned series of experiments, with subsequent numerical analysis, such as that set out by
W y p y c h and Arnold (1985). The data m a y also be the result of scaling estimations and not
directly obtained from experiment. The purpose of this Chapter is not to review the procedures
for obtaining this basic data, but rather to discuss its conversion into the conveying
characteristics format.
4
Conveying Characteristics - Generation Techniques
43
4.1
M a n u a l Generation - Interpolation Processes
The manual generation of conveying characteristics plots is basically an exercise in educated
interpolation. T h e raw experimental data are plotted in the desired format, as for example
shown in Figure 4.1.1. Contours, representing in this case lines of constant solids mass flow
rate, are drawn through the data. Whilst this process is straight forward, the experimental
variation inherent in the pneumatic conveying data, together with the curvature in the contours
resulting from the lean to dense phase transition, m e a n that there is a requirement for the analyst
to exercise a considerable degree of judgement in assembling thefinalset of curves.
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Figure 4.1.1
Steady State Conveying Data: Bulk Solid: F A 1 , Pipeline Arrangement: T R 3
The process of obtaining and subsequently analysing the pneumatic conveying data to obtain the
raw steady state data, provides ample scope for experimental variation. Each of the parameters
being measured has an error band associated with it. T h e difficulties of dealing with a two phase
process and the consequent need to use afilterm e d i u m to isolate the pressure transducers from
the solids component, interfere with the direct measurement of this parameter. A s the filter
m e d i u m becomes less permeable during the course of the experiment, there is a variation in the
extent of this interference. Typically, where the solids mass flow rate is being recorded at both
the discharging pressure vessel and the receiving hopper, a difference will be found between
these two values, raising the issue of which should be chosen as representative of the test. There
will also be an experimental error associated with the air mass flow rate measurement. In
addition, having obtained the data for one batch run, the task of identifying a satisfactory steady
state zone remains. The assessment of test data and selecting an appropriate section for this
purpose provide a further opportunity for experimental variability to be included in the data set.
Consequently, when manually locating the lines of constant solids mass flow rate, the raw data
provides a guide to, but not necessarily the absolute location of, thefinalcontours. The process
of interpolation should result in contours that are consistent with the raw data and with each
other, and follow, unless in clear conflict with the data, the expected trends for this physical
process.
Despite these qualifications, the process is capable of producing good results, particularly as t
analyst gains experience in the field. However it is a laborious task and remains open to a
degree of subjective influence. Consequently at an early stage in this project, options for
computerising this task were explored.
44
4 Conveying Characteristics - Generation Techniques
Initially a simple process of linear interpolation was employed. The conveying characteristics
plot m a y be considered as a three dimensional surface, with the solids mass flow rate values
indicating the height of the surface at the respective X (air mass flow rate) and Y (total pressure
drop) coordinates. A grid was established across the conveying characteristics plot area, and at
each node on this grid the three nearest experimental data points were located. These three
points were used to determine the equation of the c o m m o n intersecting plane. This equation
then allowed a value of the solids mass flow rate at the nodal point to be estimated (or
interpolated). The repetitive nature of these calculations was well suited to computerisation and
values for the solids mass flow rate across the nodal network could be rapidly established.
The end result was an extremely erratic set of data, and totally irrelevant contours, highhghting
the fact that the required interpolation process is far more subtle than arigidadherence to the
raw experimental data. A more flexible analytical tool was required, one that developed curves
based on, and fitting as closely as possible to, the complete data set but able to absorb
experimental variation without excessively distorting thefinalcontours.
4.2 Three-Dimensional Surface Fitting
A three-dimensional form of the least squares error minimisation analysis offered a possible
mathematical technique that could overcome the disadvantages of a simple interpolation scheme.
With this approach, a function isfittedto the whole data set, with the parameters of the function
chosen so as to minimise the total error between the experimental data values and the
corresponding values predicted for those data points. In determining the total error, the
summation is based on the square of each error term rather than the raw error. This ensures that
each error contibutes a positive quantity, and prevents positive and negative errors from
cancelling out, leading to a misleading solution.
With this approach, the resulting function is not considered to have any fundamental physical
link to the pneumatic conveying process; it is simply an empiricalfitto the data that produces a
shape that bestfitsthe data set. A s such, it only has relevance to the extent that it accurately
represents the trends in the data, and should not be considered reliable in regions outside the area
covered by the data, ie the function should be useful for interpolation purposes but not for
extrapolation. Nevertheless, the method offers the potential for rapidly generating conveying
characteristics plots for a set of experimental data that should be internally coherent (the spacing
of the contours and their relationship to each other should be consistent) and largely removed
from subjective influences.
Several options for the three dimensional function were investigated, including a power series
and converting the data from Cartesian to polar and spherical coordinates. The most successful
option, however, proved to be a simple three dimensional polynomial function of the form:
Z=(aX+bY+c)"
E q 4.2.1
where the power "«" was an integer value between 2 and 4. The parameters X and Y refer to the
air mass flow rate and total pressure drop respectively in standard Cartesian format with Z
referring to the solids mass flow rate. The format of E q 4.2.1 is in fact a shorthand
representation of the actual formula used, which is the expansion of the polynomial product. For
example, in the case of "n" = 2, the actual equation used was:
Z = aX + bX+ cY2 + dY + eXY +f
E q 4.2.2
Therefore in this case the least squares error minimisation needs to determine values for the six
constant coefficients that result in the bestfitbetween the function and the experimental data. In
general, there are 3n constant coefficients to be determined, found as a result of minimising
Equation 4.2.3.
LiError)2 = ^(aXK^)+bYK^)+c)" -ZKexp)] Eq4.2.3
where there are "A" experimental data points.
4
Conveying Characteristics - Generation Techniques
45
Taking partial differentials of E q 4.2.3 with respect to each of the 3n constant coefficients in
turn, and setting these equations to zero, allows a matrix of 3n equations to be assembled, which
on solution give the required function. This process and the presentation options that flow from
it are discussed in some detail in the paper: "Computer-Aided Determination of Pneumatic
Conveying Flow Characteristics, Scale-Up Data and Reliable Conveying Limits", Kennedy et. al
(1989).
Applying this technique to sets of experimental data demonstrated that it had potential to
generate plausible sets of curves, as indicated in Figure 4.2.1.
r i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i*
.00
0-01
0-02
0.03
0.04
0.05
0.06
0.07
AIR MASS FLOW RATE
IKO/SJ
CONTOURS: Solid* Mass Flo«rote Ikg/sJ
Figure 4.2.1
Conveying Characteristics Curves, 3-D Surface Fitting (n = 3)
Bulk Solid: P F 1 , Pipeline Arrangement: T R 1
It is also apparent from Figure 4.2.1 that as the contours move towards the limit of the area
covered by the experimental data, there is the potential that they will diverge in spurious
directions unrelated to the physical interactions actually taking place. This illustrates the point
m a d e previously that this method is unsuited to extrapolation.
Figure 4.2.1 was by no means an extreme case, on the contrary it represents a well behaved
instance from the point of view of generating plausible conveying characteristics curves. Figure
4.2.2 shows the results of this surfacefittingtechnique applied to a second data set, again with a
third order polynomial selected as the describing function. The result in this case is a somewhat
more irregular set of curves and no indication of the concave upwards transition from lean to
dense phase conveying behaviour.
Several facilities were built into the program to address this situation, with the aim of prepar
more acceptable set of conveying characteristics curves. Firstly the power of the polynomial
function could be adjusted. Generally a third order polynomial proved to be the most suitable,
however if it is desired to restrict the variation generated by the function, a second order
polynomial m a y be used. Figure 4.2.3 shows the outcome of that choice on the same set of data.
46
4
Conveying Characteristics - Generation Techniques
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AIR MASS FLOW R A T E
lKg/sJ
CONTOURS! Solids Mass Flouroie
Ikg/sJ
Figure 4.2.2 Conveying Characteristics Curves, 3-D Surface Fitting (n = 3)
Bulk Solid: Fly Ash, Pipeline Arrangement: D = 5 9 m m , L = 1 6 2 m
400.
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CONTOURS: Solids Mass Flowrate
Figure 4.2.3
Ikg/sJ
Conveying Characteristics Curves, 3-D Surface Fitting (n = 2)
Bulk Solid: Fly Ash, Pipeline Arrangement: D = 5 9 m m , L = 1 6 2 m
This has the effect of imposing a more uniform trend across the area, though it would appear to
be at the expense of ruling out a reasonable capacity to follow the lean to dense phase transition.
Figure 4.2.4 shows the effect of extending the order of the polynomial in the other direction.
Clearly this provides too m u c h scope for the polynomial to follow the variability in the
experimental data rather than identifying a relevant overall trend, with a counter-productive final
result.
4
Conveying Characteristics - Generation Techniques
Al
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CONTOURS: Solids Mass Flourote
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Figure 4.2.4 Conveying Characteristics Curves, 3-D Surface Fitting (n = 4)
Bulk Solid: Fly Ash, Pipeline Arrangement: D = 5 9 m m , L = 1 6 2 m
The other approach explored is to allow the analyst to add additional data points to the data
to encourage the least squares error minimisation process to produce a more acceptable outcome.
This technique, whilst not ideal, is to some extent analogous to the judgements m a d e when
producing a set of conveying characteristics curves manually where the intention is to produce
curves of the generally expected shape within the constraints of the experimental data. In Figure
4.2.5 it can be seen that a series of data points have been added to the data set along the margins
of the plot to impose a preferred shape on the surface.
400
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Figure 4.2.5
Conveying Characteristics Curves, 3-D Surface Fitting (n - 3),
modified data set
Bulk Solid: Fly Ash, Pipeline Arrangement: D = 5 9 m m , L = 1 6 2 m
4
48
Conveying Characteristics - Generation Techniques
S o m e success has been achieved with the modifications to the data set applied in the latter case,
with improved curves being generated and with these curves holding a more stable shape right
across the plot area. However it must be acknowledged at the same time that the objectivity of
the conveying characteristics generation process has been compromised by this addition to the
analysis armoury, which is contrary to one of the key goals in developing this technique.
4.3 3-D Surface Modelling - Stepped Pipelines
O n e virtue of the 3-D surfacefittingtechnique is that it can be used just as readily with data from
a long distance stepped pipeline as with data from a single diameter pipe configuration. Since
the function is a purely empiricalfitto the experimental data set, there is no need to take into
account the different contributions of each pipe section. A difficulty w h e n applying the
technique to longer distance conveying systems however is that the data are invariably restricted
to the lean phase region, hence there is little likelihood that useful indication of the transition to
dense phase conveying will be obtained. Figure 4.3.1 indicates the type of results that m a y be
expected. In this case the results are quite acceptable in the region where experimental data have
been collected, however the shape of the curves in the lower air mass flow region has no
relevance. There is also little in the data to suggest where the transition from lean to dense phase
conveying behaviour should commence, hence any modification of the data set to force this
outcome is likely to be quite arbitrary.
500
400
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C O N T O U R S : S o l i d s Mass F l o « r a t e
Figure 4.3.1
tkg/sJ
Conveying Characteristics Curves, 3-D Surface Fitting (n = 3)
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0 (stepped pipeline, 1 2 0 0 m )
4.4 3-D Surface Modelling - Scaling Capabilities
Frequently it is desirable to use experimental data obtained on a test rig to predict the
performance of different pipeline configurations. It would be useful if this 3-D surface
modelling approach could be adapted for that purpose. Since the function obtained by this
method is purely empirical and has no concrete physical relationship with the conveying
mechanisms, it cannot be scaled directly to produce a n e w set of contours.
It is possible however, to apply scaling procedures, such as the scaling equations suggested by
Mills et al (1982) or W y p y c h and Arnold (1987), to the original set of experimental data, and
then to use this scaled set of data to determine a n e w polynomial function. Inclusion of this
capability within a computerised conveying characteristics generation package is attractive as it
eliminates the bulk of tedious and repetitive calculation involved in carrying out these
4
Conveying Characteristics - Generation Techniques
49
computations manually, and consequently allows the designer to examine a wider range of
alternative pipeline configurations.
Whilst this facility was incorporated into this set of programs, it will not be discussed further
here. Other limitations of this method of generating conveying characteristics suggest that a
different approach m a y be warranted, resulting in substantial modifications to the analytical
procedures being introduced.
4.5 3-D Surface Modelling - Conclusions
The predominant conclusion drawn from this work is that computerised generation of conveying
characteristics plots is possible. It has the advantage of being a substantially objective process,
and eliminates the tedious manual calculations that would otherwise be required. This in rum
allows the analyst to explore m o r e aspects of the conveying process, and pipeline configuration
alternatives, than m a y otherwise be feasible.
On the other hand, some significant shortcomings are identified, predominantly the tendency to
produce erratic curves around the margins of the conveying characteristics plots, and the routine
requirement to add d u m m y points to the data set to force a satisfactory overall shape. Though
various alternative calculation options were tested to try and improve the stability and reliability
of the results, ultimately these shortcomings caused the abandonment of this technique as the
primary means of conveying characteristics generation. The positive features of this approach
were retained however and incorporated in the solids friction factor version of the conveying
characteristics generation model described in the following Chapter.
50
5
Solids Friction Factor Model
Chapter 5
SOLIDS FRICTION FACTOR MODEL
The development of computerised techniques to generate conveying characteristics plots
outlined in the previous Chapter demonstrate that harnessing this resource could assist
considerably in the analysis and presentation of pneumatic conveying data. The collection,
analysis and presentation of pneumatic conveying experiment data are a data and calculation
intensive process. Tools which have the capacity to stteamline this process have the potential to
assist in the identification of significant patterns in behaviour that m a y otherwise be buried under
the mass of data. The development of a consistent objective treatment of the data is also of
value in this respect.
The purely empirical approach followed in the previous Chapter shows significant limitations
requiring, attimes,substantial intervention from the analyst to drive the computation towards a
more acceptable outcome. It seems likely that the lack of a physical model is the major reason
for this occasional irregularity in the conveying characteristics contours, flowing from the
variation in the experimental data.
In an effort to improve the consistency of the analysis, a solids friction factor conveying model
has been included in the algorithm. In a sense this model acts as afilterbetween the raw
experiment data and the 3-D surfacefittingprocess. The experimental data are applied to the
solidsfrictionfactor model, and the characterising parameters of that model are determined,
relevant to the particular bulk sohd/pipeline configuration under consideration. The model is
then used to generate conveying data to which a 3-D surface can befitted.This ensures that a
more consistent set of data points would be presented to the surfacefittingprocess, resulting in
more consistent conveying characteristics curves.
5.1 Pneumatic Conveying Model
The pneumatic conveying model selected for this purpose is based on an approach put forward
by Barth (1958) of the following form
2
/
v pfVfAL
Apt=Apf+Aps=(Af+m*As)
^
Eq5.1.1
The basis of this and other versions of pressure based pneumatic conveying models is discussed
in some detail in Chapter 6 as a background to the development of the power based pneumatic
conveying model, and hence is not duplicated here. It is clear however that the model is strongly
based on single phase fluid flow theory, with the inclusion of the solids concentration and solids
friction factor terms to account for the presence of the bulk solid in the flow.
The air component of Equation 5.1.1
Ap = A
f \ fl 2D Eq 5.1.2
is determined independently of the presence of the solids. Consequently J^is the superficial air
velocity, calculated on the basis that the full pipe cross-section is available for flow. Standard
5
Solids Friction Factor Model
51
expressions for the friction factor Af m a y be used. In this case a version of the Swamee-Jain
equation (Swamee et al, 1976)
1.325
Af —
Eq 5.1.3
' 5.1A >
with the pipe roughness term removed and the Reynolds Number evaluated at mean pipeline
pressure conditions is employed.
The only unknown term in Eq 5.1.1 is therefore the solids friction factor Xs. The steady-s
conveying data are then applied to this equation and a series of values deduced for this solids
friction factor. Using these values, a correlation analysis is carried out to determine a suitable
expression for Xs based on a set of relevant dimensionless conveying parameters. The
expression is of the form:
X. = A x m *a xFr
r
'V
Eq 5.1.4
V A
where Frm and pfm are the Froude Number and the air density respectively, evaluated at mean
pipeline pressure conditions. In practice, since the particle density is constant for a given bulk
solid, Eq 5.1.4 is simplified to:
Xs=Axm*a
xFrf xPfinr
Eq 5.1.5
with ps being absorbed into the constant coefficient (A). The dimensionless parameters used in
Eq 5.1.4 have been used by other researchers in a similar format, for example Weber (1982),
Szikszay (1988) and Wypych (1989). Pipe geometry parameters such as (L/D) are frequently
included. These m a y have relevance if it is intended to develop a general equation for the solids
friction factor, however in this application where the data are typically sourced from a single
pipe geometry, such parameters would effectively be constant throughout the correlation
analysis. If they are to be included, any power term determined for this parameter would have a
spurious value, and be highly likely to distort the solids friction factor values if the equation is
applied to a different pipe configuration.
5.2 Application of the Pneumatic Conveying Model
The procedure adopted in applying this model to conveying data and the generation of
conveying characteristics is to process the experimental data and obtain the terms in Equation
5.1.5. Once this expression is available it is possible, via an iterative process, to calculate
conveying parameters for a range of conditions across the proposed conveying characteristics
range. Programs have been written to implement this procedure, with the user being able to
specify the range of air and solids mass flow rates for which the model would determine the
conveying data points.
As an example of this process, consider the results when applied to data obtained from PF5
(pulverised briquettes) on testrigTR20. The solidsfrictionfactor correlation gives the
following expression:
~ «.«/»
^.-.2328
„ -1.3674
-1574
T-.
- ~ -•
Xs=0.593xm*
xFrm
x pfm
Eq5.2.l
Figure 5.2.1 shows the range of Xs values produced by E q 5.2.1, corresponding to the
experimental data points from PF5/TR20, as a function of the superficial air velocity. It m a y be
noted that these data have been obtained on a stepped pipeline system (refer to the pipeline
arrangement details given for T R 2 0 in Appendix A). In such cases the solids friction factor
correlation requires that pressure drop data for each pipe diameter section be provided
separately. If this is not done, it is not possible to determine the values of the dimensionless
parameters, eg Frm (since a pipe diameter value is required), and hence proceed with the
analysis.
5
52
Solids Friction Factor Model
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0.00000
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20.00
30.00
40.00
50.00
70.00
60.00
80.00
Superficial Air Velocity (m/s)
Figure 5.2.1
Solids Friction Factor Values
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0
Having determined these model parameters, the model can now be applied to generate a fresh set
of data points across the conveying characteristics range. The intention with this process is to
smooth out any experimental variation in the raw data set, and also to obtain a broader spread of
data points to be applied to the 3-D surfacefittingprogram. Following through this analysis
with the data originating from PF5/TR20, the model generated data points obtained are shown in
Figure 5.2.2 .
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i ii
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M o d e l Generated Data Points
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0
Figure 5.2.2 demonstrates that this procedure has the potential to generate a well distributed set
of data, well suited to the generation of conveying characteristics using the 3-D surface fitting
technique described in Chapter 4. Figure 5.2.3 confirms this expectation, and there would
appear to be a good correspondence between the original data points, also shown on Figure
5.2.3, and the contours of constant solids mass flow rate produced by this analysis. A statistical
analysis of these data indicates that in this case a high level of accuracy has been achieved, with
the correlation coefficient between the experiment and model calculated solids mass flow rate
values being 0.99, and the average error being -0.5% .
J
600.
53
Solids Friction Factor Model
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r.r..r .T, —
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CONTOURS! UJ Solrds Mass Flovrate Ikg/sJ
Figure 5.2.3
Conveying Characteristics Curves, 3-D Surface Fitting (n = 3)
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0
O n e of the benefits of having transformed the raw experimental data into the form illustrated by
Figure 5.2.3 is that the generation of additional conveying parameters is quite straight forward.
Providing these parameters can be calculated from the basic set (air mass flow rate, pressure
drop, solids mass flow rate) it is a simple matter to extend the computer algorithm to provide the
additional output. Figure 5.2.4 illustrates this facility by the overlay of contours of the Froude
N u m b e r on the conveying characteristics plot for PF5/TR20. Other parameters, such as the
Reynolds Number, superficial air velocity or specific power consumption could be similarly
generated.
600.
i
i
i
i
1 " "J
V
500.
_
N(J
•
400.
'• "
'1 "
Pip* D*1o1 1.-
**y
*/
/
-
^
-
^
^
— •
Die.
Cnn)
Cnl
SB
563
<o'f
•^iSa
fir!
"
51
3B1
105
253
^ ^
- r»*.
^&£"
y
'ibr*
1197
^4?*^ •
•V
7^-0*
Ltngth
^ • * f r
-
DL
a
a
Si^i•J0 j
300-
en
•^iiU_L^jS—
<n 208.
LU
CC
0-
•
1
IML-Jfc^V
100. -
0
6.0
-L4 i /
1,,,,
0.1
,,, , 1 1 .
0.2
0.3
0.4
RIR MASS FLOW RRTE IKg-'s)
CONTOURS: U J Solrds Moss Flovrate 1kg/sJ
12J Froude Number at pfpelfne tnlet
Figure 5.2.4
Additional Conveying Parameter Contours - Froude N u m b e r
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0
5
54
Solids Friction Factor Model
5.3
Scaling Methodology
The use of this model carries with it some implicit scaling characteristics. It is apparent from E q
5.1.1 that altering the length (AL) in the equation will tend to have a linear impact on the total
pressure drop.
,
, pfVfAL
J ;
Apt=Apf+Aps=(A{+m*Xs)
E q 5.1.1
2 D
The outcome with respect to changes in pipe diameter is less clear since the superficial air
velocity also includes the pipe diameter. Variations to the air and solids friction factors in
response to changes in pipe geometry will also impact on the scaling performance of the model,
preventing a simple expression of these relationships.
Nevertheless, once an expression has been obtained for the solids friction factor, Eq 5.1.1 can be
applied to any desired pipe geometry (though the stability and usefulness of the resulting
function cannot be guaranteed). The capacity to predict accurately the performance of the bulk
solid in different pipeline configurations, is the basic goal of scaling calculations. Consequently
the model that has been developed and discussed in this chapter can be applied for this purpose.
Figure 5.3.1 illustrates the result of this process. The base data in this case are takenfromthe
6 9 m m diameter pipe section, and the solidsfrictionfactor expression derived from these data is
subsequently used to predict the performance of the full 1.2km pipe length. The result in this
case is again very good, with the correlation coefficient between the solids mass flow rate values
obtained experimentally for the full pipeline, shown in Figure 5.3.1, and the values predicted by
the model being 0.99, and the average error between these two sets of values being 1.75%.
600F1pc
0*1B1U
Die- Length
[nl
563
3B3
253
1197
100.
-
i i i i I i i i • I i i • i i .., • I
3
0-1
0.2
... I ....
0.3
i ... ,
0-4
RIR MASS FLOW RRTE IK9/si
CONTOURS: tlJ Solrds Mass Flovrate lko,/sJ
Figure 5.3.1
Scaled Conveying Characteristics:
Bulk Solid: PF5, Pipeline Arrangement: T R 2 0 (scaled from T R 2 1 )
If the results shown in Figure 5.3.1 were typical of the scale-up performance of this model, it
could be concluded that a more than adequate model has been produced and that for all practical
purposes no further refinement is required. Unfortunately that did not prove to be the case, and
more commonly the scaling performance fell well short of that standard. .Figure 5.3.2 is more
typical of the results obtained. The contour lines shown in Figure 5.3.2 are the result of the
scaling analysis; actual conveying data points for that pipeline are marked directly on the plot.
In this case it is apparent that there is a considerable disparity between the experimental and
calculated values of solids mass flow rate. The statistical analysis indicates a cross correlation of
0.82, and an average error of 6 2 % .
5
Solids Friction Factor Model
55
150. ' / 1 i i I i i i i I i i i i I i i i i | i i i i | i i i i | i i i i | i i i i
P1p? Delel I ',
01i-
Ltngth
(nnJ
Citl
52
48
2.91
a
cc
a
cc
sn
Toi.t 4B
3.67
100. -
3.7E
3.62
ugyfrq—^^_
li
L31
zi.E9 4.flfl
<ti 3.
50.
LU
-mag—tvti——
cc
0,56
B.5B
8-59
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I
tf.00 0.01
0.02
0.03
0.04
0.05
0.06
0.07 0.09
RIR HR55 FLDN RRTE lff9/|]
CONTOURS: 11) Soltds Mass Flovrate Ikg/sJ
Figure 5.3.2
Scaled Conveying Characteristics:
Bulk Solid: P F 1 , Pipeline Arrangement: T R 2 (scaled from T R 1 )
The reasons for this inconsistency in scaling performance are likely to be complex. It would
seem likely that the scaling characteristics embedded in E q 5.1.1 are too simplified; there is no
direct consideration of bends or vertical pipe sections, for example. It m a y also be the case that
the concept of extending a single phase flow equation into the more complex two phase situation
is not appropriate. It would seem that for some situations, errors in the scaling characteristics of
E q 5.1.1 m a y cancel out and a good result be produced, whereas in others they do not.
5.4 Stepped Pipelines - Optimised Design
Previous figures showing conveying characteristics plots for PF5/TR20, eg Figure 5.3.1,
demonstrate that this model is quite capable of carrying out pneumatic conveying calculations
for stepped pipeline configurations. If the experimental data are derived from a stepped pipeline
installation, the solidsfrictionfactor correlation requires that pressure drop data for each pipe
diameter section be provided separately. If this is not done, it is not possible to determine the
values of the dimensionless parameters, eg Frm, and hence proceed with the analysis.
Once an expression for the solidsfrictionfactor is available, the calculation for the pressure drop
for a stepped pipeline is only marginally more difficult than equivalent calculations for a single
diameter pipe section. For the stepped pipeline case, the calculations c o m m e n c e with the exit
pipeline section, since the k n o w n pressure is the exit pressure, usually taken as the ambient
pressure. For a given air and solids mass flow rate the pressure drop for that section is
determined using E q 5.1.1. This process is complicated by the need to include an iterative
facility since m a n y of the parameters required (such as Frm, V, pfa, etc) are also a function of the
pressure drop. Nevertheless, once the required accuracy has been achieved and a pressure drop
determined for the exit pipe section, the inlet pressure for this section forms the k n o w n exit
pressure for the immediate upstream section. The iterative process is then repeated until the
pressure drop across the full pipeline has been determined.
Once a reliable algorithm has been developed for a single diameter pipe section, a relatively
simple extension of this algorithm allows multiple pipe sections to be handled. The preceding
figures giving conveying characteristics data for the PF5/TR20 combination, demonstrate the
viability of this mechanism.
5
56
Solids Friction Factor Model
A n e w element introduced by multiple pipe section configurations is the need to select the
optimum length for each pipe section. Before any progress can be m a d e with this concept, the
criteria to be used in defining an optimum arrangement need to be established.
In estabhshing a basis for these criteria, a rational point to commence would be to consider the
exit pipe section. The critical question to be resolved is, what is the optimum length of this
section? The primary effect of increasing this length, for a given air and solids mass flow rate, is
an increase in the total pressure drop through that section. Consequently the air velocity at the
inlet section of the pipe will fall, and at some point result in excessive deposition of the bulk
solid and potential blockages. Therefore one criterion could be the superficial air velocity.
Once this has fallen to some pre-determined value, the pipe diameter could be reduced, so that
the velocity is restored to a safe value. A similar analysis would then be applied to the upstream
section and in this w a y thefinalpipeline configuration be determined.
In considering this issue, Wypych and Arnold (1989) suggested using the mhumum Froude
Number as the criterion for stepping the pipe diameter. For fine powders, such as pulverised
coal andflyash, a value of 6 was suggested as appropriate.
For the purposes of implementing and evaluating this concept, the proposal of Wypych and
Arnold was adopted. A s an example, an optimised three segment pipeline equivalent to T R 2 0
for bulk solid PF5 has been determined. The same pipe diameters and total length as test rig
T R 2 0 were specified, and the design optimisation based around a bulk solids throughput of
2kg/s. Figure 5.4.1 shows the conveying characteristics generated for this optimised pipeline
configuration. Since a minimum Froude N u m b e r of 6 is used as the pipe stepping criterion, an
additional contour showing this parameter is also drawn on the plot.
P1pt Delel I '.•
01a.
"0-0
0.1
0.E
0.3
Lrnjtli
Inn!
(nl
BB
375
81
521
105
30]
0.4
RIR CIRSS FLOW RRTE IKVsl
CONTOURSs UJ Solrds Mass Flovrate Ikg/s)
12J Froude Number at prpeltne fnlet
Figure 5.4.1
Scaled Conveying Characteristics:
Bulk Solid: P F 5 , Pipeline Arrangement: Optimised Equivalent of T R 2 0
The results for this optimised pipeline are quite similar to the conveying characteristics
determined for the original pipeline. A closer inspection (compare with Figure 5.4.1 with 5.4.2)
indicates that the lines of constant solids mass flow rate are approximately 30kPa lower across
the range for the optimised pipeline. The adjustments to the pipeline that achieve this outcome
are indicated in Table 5.4.1.
5
Solids Friction Factor Model
57
Table 5.4.1 Comparison of Pipe Configuration, T R 2 0 and Optimised Lengths
Pipe Diameter
(mm)
TR20
69
81
105
Pipe Length
(m)
Optimised Pipeline
563
381
253
375
521
301
600.
Pipe D H e 1 l »
500- V
01o-
Ltngth
Cnn)
(nl
SB
c
w
400
0O
a
61
3B]
105
2 S3
1197
300
LU
cc
=J
in
UJ
az
0-
200
100.
RIR HRS5 FLOW RRTE I K B ' ^
C O N T O U R S : UJ S o l r d s Mass Flovrate Iko/sJ
(2J Froude Number at pfpelfne fnlei
Figure 5.4.2
Scaled Conveying Characteristics:
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0
In this case the modifications to the pipeline would result in a marginal increase in capital cost,
since the suggestion is to increase the lengths of the larger diameter pipe sections. However the
potential for running the system at a reduced power requirement, and the reduced wear as a
result of the rninimised conveying velocities, should more than offset that cost. A more serious
problem m a y have been unearthed if it were determined that the lengths of larger pipe diameters
were excessive, as this would indicate that the diameter stepping criterion has been exceeded. In
this case, the system m a y be prone to blockages.
Providing the underlying model and predictions are sound, the capability to optimise pipe
lengths in this manner is a very valuable design aid.
5.5 Discussion and Conclusions
The preceding Sections in this Chapter indicate that the combination of a basic pneumatic
conveying model, with computerised analysis, conveying characteristics generation and design
facilities, can provide a powerful set of tools for the design and operation of pneumatic transport
systems. The stability of the conveying characteristics generation process is improved by the
introduction of the Barth model, and it is possible to reproduce the lean phase to dense phase
transition. The requirement to insert d u m m y data points into the analysis is eliminated,
improving the overall objectivity and consistency of the analysis. This approach is also capable
of carrying out scaling calculations and analysing stepped pipelines. The ability to optimise the
design of stepped pipeline systems is a further powerful extension.
5
58
Solids Friction Factor Model
Against these advantages a number of limitations must be acknowledged. Chief among these is
that the model is not stable in all cases, in some instances demonstrating a very limited capacity.
Figure 5.5.1 illustrates a form of conveying characteristics observed with a number of data sets
where the solids mass flow rate contours show a tendency to form closed loops. It is not clear
whether this is a limitation of the model or the surfacefittingtechnique, however the possibility
of obtaining this outcome diniinishes the level of confidence that can be held in the overall
technique.
Pip* DHoll t
e
0-
-
Die-
L>nstk
Cnn)
Inl
5Z
48
To*.i
4B
a.
a
cc
a
UJ
cc
=3
tn
in
LU
cc
CL
II
0.02
0.03
0.04
0-05
0-06
0-07
RIR MR55 FLOW RRTE \K%f %\
CONTOURS: L1J
Figure 5.5.1
S o l r d s Mass F l o w r a t e
tkg/sJ
Conveying Characteristics Curves
Bulk Solid: P F 3 , Pipeline Arrangement: T R 2
Similarly, whilst in some instances the model reproduced the lean phase to dense phase
transition characteristics expected, in others it failed to do so. Figure 5.5.2 shows a set of
conveying characteristics curves for another long distance stepped pipeline where this transition
is not evident. The results still appear to be quite satisfactory in the region where the
experimental data are available, which is perhaps the major requirement. Even in the cases
where this transition is reflected, eg PF5/TR20, the strength of this trend diminishes in the lower
solids mass flow rate curves in a manner that would not be expected. It would appear from these
sorts of results that the Barth model does not have a fundamental characteristic that ensures that
the dense phase processes are reproduced. This is perhaps not surprising since this model is an
extension of single phase flow theory, which is more closely reflected in lean phase conditions.
5
Solids Friction Factor Model
59
500.
g f 1p* 0>1«1 1»
Die.
[nl
79 5B
59
146
390
31
SB)
105
150
400. Q-
CL
a
cc
a
UJ
ir
=3
L»n$th
Inn]
300
Tot.i B4E
200
<n
LU
100. -
cc
0-
"0-0
0.1
0.2
RIR HR55 FLOW RRTE IKg/s)
CONTOURS: UJ Solrds Mass Flovrate Ikg/sJ
Figure 5.5.2
Conveying Characteristics Curves
Bulk Solid: P F 6 , Pipeline Arrangement: T R 2 4
The scaling performance of the model is another area where the overall performance of this
model raises some concern. Whilst accurate in some cases, it can be quite inaccurate in others as
indicated in Figure 5.3.2. Inconsistency of this sort reduces the rehability of the model in a
crucial fashion, and again suggests that there m a y be underlying inadequacies in the manner in
which the physical processes are being represented.
An additional positive aspect of this analysis is that the combination of computerised analysis
techniques with models of the pneumatic conveying process facilitates examination of those
models across a range of conveying conditions and with a large number of data sets. A s a
consequence the strengths and weaknesses of the models can be examined in a depth that would
not be feasible if the analysis were to be carried out manually. O n e instance of this enhanced
analysis capacity is the ability to calculate and overlay additional parameters on the conveying
characteristics plots, demonstrated in this chapter by the use of the Froude Number. Inclusion of
additional parameters of this sort is a relatively trivial exercise once the basic network of data is
available in digital form.
One parameter that also received this attention was the power consumption during the pneumatic
conveying process. O n e aim in configuring a pneumatic conveying system is to nmiimise the
power consumption of the process, for obvious direct cost reasons, but also for indirect reasons
such as reduced wear and particle breakdown. Therefore it appeared that contours of this
parameter, or perhaps the specific power consumption ( k W kg'1 m' 1 ) m a y provide a valuable
insight into the conveying system. During the development of this idea, the proposition was
conceived of basing an entire pneumatic conveying model around the power consumption
requirements of the physical processes involved. The development of this power based
pneumatic conveying model will be the focus of the remaining chapters of this thesis.
60
6
Power Based Pneumatic Conveying Model
Chapter 6
POWER BASED PNEUMATIC CONVEYING MODEL
O f the methods of bulk solid transportation available to industry, pneumatic conveying is
perhaps the least transparent. Exactly what is happening inside the blow tank, rotary valve,
pipeline, bend, receiving hopper, filter upstand and so on is, to a great extent, hidden. T h e
extent of this uncertainty leads to a corresponding concern regarding the feasibility of pneumatic
conveying as a transportation option, at the investment/design phase, and uncertainty w h e n
seeking to commission or optimise an existing system.
Consequently there has always been a need to develop a set of guidelines or procedures to ensure
the reliable design and operation of these systems. Experience has been, and remains, a key
ingredient, ie previous knowledge of the performance of similar materials in similar
circumstances. General guidelines relating material properties, safe conveying velocities, system
pressure drops etc, such as that outlined by Lee (1984), have been established and n o doubt
remain a powerful tool in the armoury of consultants in this field. These guidelines, of
necessity, err on the conservative side and have limited value w h e n there is a strong need to
optimise the performance of the system.
More elaborate step-wise procedures, eg those set out by Latincsics (1982), have been developed
to refine this design procedure. These are a very considerable simplification of the physical
processes and, in the absence of some corresponding experimental data, do not carry sufficient
authority, nor convey adequate insight, to allow optimal design to be carried out.
A reliable analytical model is therefore an important element in the successful application of
pneumatic conveying in industry. A model is required that allows systems to be designed and
operated close to their optimum potential, clarifying any actual competitive advantage these
systems m a y have over alternative solutions to a bulk solid handling process. Such a model
should not only reliably calculate relationships between key system parameters (pressure drop,
conveying rate, air usage) but also shed light on the trends in performance of the system w h e n
one variable or another is altered. Considerable efforts have been m a d e in this area, though it
must be acknowledged that at this stage an adequate, purely theoretical, model covering the
range of conveying regimes has not yet been established.
The primary purpose of this Chapter is to review the major model type that has occupied
researchers in this area, and then to describe an alternative "power based" model that has been
developed during the course of the thesis.
6.1 Pressure Drop Based Models - a review
The development of reliable models to predict the behaviour of pneumatic conveying systems
has been of major interest since the onset of research work in thisfield,and has grown in
6 Power Based Pneumatic Conveying Model
61
significance and relevance in proportion to the increasing importance of this mode of transport
to industry.
These models typically take the form of a relationship between the total air pressure dr
the pipeline, or possibly the pressure drop per unit length, and a series of parameters that are
considered to contribute significantly to this pressure drop. A n early example of this type of
relationship is that proposed by Hinkle (1953), as reported by Zenz and CMhmer (1960), which
identifiesfluidacceleration, particle acceleration,fluid-to-pipefriction loss and a relationship for
the particle-to-pipe friction loss, as significant terms for horizontal conveying. Expressions
relating to pressure drop attributed to bends and vertical conveying may also be included,
extending to consideration of pressure losses associated with varying and filtration/separators
such as that outlined Marcus et al (1990).
All these approaches borrow extensively from single phase fluid dynamics in establishing
basic framework of the models. In particular the Darcy-Weisbach Equation (Olson, 1973):
~J~ = ~~f)—^— Darcy-Weisbach Equation
or the similar Fanning Equation (Govier and Aziz, 1972):
dPf
2fpV2
—r— = — — —
Fanning Equation
ax
D
figure prominently in establishing the key frictional pressure drop term for both the gas and solid
components.
A major challenge in seeking to apply either of these Equations is to account appropriat
the presence of solids in the two phase mixture. The friction factor is almost inevitably the term
that will be involved in this treatment, and an approach put forward by Barth (1958) has been
broadly utilised, for example Weber (1981) and Wypych (1989), resulting in:
Ap=Apf+Ap=[Xf+m*Xs) f f
This approach clearly proposes calculating the pressure drop contributions from the air
solids components separately, with both terms following the Darcy-Weisbach formulation.
In considering the air component, Weber (1981) recommended the use of a simple empirical
formula proposed by Blasius to obtain a value for the air-only friction factor:
k\ =0.316x5/— Blasius Equation
}
VRe
Wypych and Pan (1991) examined a number of expressions for the air-only friction factor (Af) in
the light of experimental results, recommending the use of either the Blasius Equation or
relationships proposed by Swamee et al (1976) for smooth pipes, with the latter equation
preferred for rough or new pipe conditions, with the important proviso that mean conditions
(based on average air density) be used.
Af = F—~, 1.325
'
r=rr
Swamee-Jain Equation
i.7£> ' R e 0 9 .
The determination of the air-only pressure drop contribution would appear to be a relatively
straightforward task, in comparison to the corresponding calculation for the solids component. It
is worthwhile reflecting however, that in two phase flow more complex interactions are taking
place, influencing for example, the cross-sectional area available for the airflow,the apparent
pipe "roughness" and relative velocity between the gas and the surrounding solid and pipe wall.
Consequently, if the air-only pressure drop is calculated as outlined by the above equations,
6
62
Power Based Pneumatic Conveying Model
these additional effects must be absorbed elsewhere, typically in the solids frictional loss
component.
The solids friction factor (Xs), in the model proposed by Barth, is combined with the solid to air
mass flow ratio parameter (m*). A theoretically based expression for its value, developed from
basic properties of the bulk solid, has not been successfully produced. The approach that has
typically been adopted is to extract values for Xs by the analysis of experimental data, and then
to develop an empirical power function based on a set of dimensionless parameters, via a
correlation analysis. Numerous instances of this procedure are reported in the literature. For
example, in his review of hydraulic and pneumatic conveying, Weber (1981) refers to the
following correlation produced by Stegmaier (1978) for a group offinepowders:
X =2.1x/w*-°-3xFr-2xFra5x
d_
Stegmaier's solids friction
\D)
factor coefficient equation
This correlation was used by Chambers and Marcus (1986) to review experimental data from a
range of sources, resulting in a predicted pressure drop within a factor of two of the actual value.
Wypych and Arnold (1987), in analysing a group of power station fly ash samples, suggested
that the correlation of pipe-friction coefficient due to solids could be substantially improved by
including the mean air density and effective pipe length terms, as outlined below:
K
9 x l 0 6 x F r - ° - 6 x F r : U 8 2 2 - Pfin
~08911
• ,. •
• ,^-°-85
These additions enabled a considerablytightercorrelation for the data presented to be achieved,
however the dimensionless status of the relationship is lost with the inclusion of the air density
(Pfin^
in its place,
term. This could no doubt be addressed by considering a term such as v p; J
however the impact of this on the overall correlation would need to be evaluated. With the
inclusion of the pipeline length, W y p y c h and Arnold's correlation indicates that pipe diameter
has a minor impact on the solids friction factor coefficient.
A variation on this approach proposed by Szikszay (1988) and Weber (1991), considers a
friction factor for the mixture as a whole rather than separating the gas and solid components. A
similar basic equation (Darcy-Weisbach type) is used, with Szikszay including the solids to air
mass flow ratio term:
± = \ xm*x P/2DJ
XV
fin
'ot
Szikszay's Equation
dl
K* = Fr^xm*b>
KD)
and Weber leaving it out:
Pf-xVfXL
Ap = Alotx
2D
Xot = const.xFra x Fr* x m *d x\ —
\DJ
Weber's (1991) Equation
^
KP/J
The inclusion of the solids to air ratio term (m*) in the mixture friction factor, however, means
that in terms of this component the correlations are equivalent. It is noted that Weber's
correlation also makes use of two additional terms.
6
Power Based Pneumatic Conveying Model
63
Weber makes several points with regard to this approach. Firstly, by combining the friction
factor terms, an inconsistency that can arise in the correlation can be avoided. Since, in the
method described previously, the solids friction factor is the "residue" that is found after the
remaining terms are calculated, it is possible (particularly at lower solids loadings) for negative
values for Xs to apparently apply. Clearly this is a physical impossibility, and is related to an
inappropriate weighting being given to other components, for example the air only pressure
drop. Secondly, Weber reports an improved correlation being obtained for the combined
friction factor case with a mean linear error being reducedfrom« 6 4 % to « 3 7 % for a set of fine
materials (data from Stegmaier, 1978). This is further explored with a set of simulated data
where the performance of the original model and the combinedfrictionfactor model are
compared. This mixture friction factor approach would appear to demonstrate a marked
reduction in systematic error, which tends to confirm that the original model treats some of the
pressure loss components inappropriately - with these errors showing up more strongly at the
"extremes" of the conveying regime.
Ideally, in any correlation for the friction factor (mixture or solids and gas
resulting equation would exhibit substantial stability in terms of the powers obtained in the
correlation when data for different materials are analysed. If this was achieved, a general model
could be applied with a reasonable degree of confidence to predict the performance of a given
system, without resort to conveying test work on the bulk solid concerned. Unfortunately this is
not the case and, as noted by Weber (1982), to obtain improved accuracy it is necessary to
obtain experimental data and to carry out the correlation on the specific material. Szikszay
(1988), for example, gives the results of a correlation analysis on a number of coarse bulk solids,
as outlined in Table 6.1.
Table 6.1: Comparison of "mixture" friction factor correlation results (Szikszay, 1988)
M
°del:
dl " *** X m *X
2D
'
***
-Frfinxm*bx
-T
KDJ
Material
a
b
c
Sand (d = 0.5mm, SS pipe)
-0.58
-0.95
0.32
Sand (d = 0.5mm, P V C - pipe)
-0.5
-1.05
0.42
Styropor (d = 0.5mm, SS pipe)
-0.42
-0.77
0.48
Styropor (d = 2mm, SS pipe)
-0.64
-0.095
0.39
Polystyrol (d = 3mm, SS pipe)
-0.054
-0.088
0.62
Rubber (d = 1.9mm, SS pipe)
-0.36
-0.55
0.75
Lewatit (d = 0.76mm, SS pipe)
-0.2
-0.92
0.75
|
It is apparent that a wide range of powers may be obtained. In some cases the m * term has
effectively no input on the behaviour of the model (where b approaches -1); for other materials
it retains a significant input. Curiously, Szikszay does not use a constant coefficient term in the
correlation, as is the usual practice. This function is in fact served by the (AID) dimensionless
parameter which would remain constant for a given pipeline/bulk solid combination. Care needs
6 Power Based Pneumatic Conveying Model
64
to be exercised when carrying out correlations that the dimensionless parameters have a
meaningful variation across the experimental data. If the term remains effectively constant
throughout the test work, a valid or useful correlation reflecting the impact of this parameter
should not be expected.
The variability of the correlation power terms across materials indicates that either the basic
model inadequately reflects the operation of the system, or that dimensionless parameters
selected do not include some key features of the bulk solid's behaviour, or, quite likely, a
combination of these possibilities. Szikszay introduced several additional parameters in an
effort to improve the usefulness of these correlations, specifically an "impact factor" (Sz),
analogous to the coefficient of restitution, a mechanical friction factor (mR), and the particle to
fluid density ratio.
rj\° ( „ Y
P
i
\e
- - x (Sz x p.R J .... Szikszay' s "extended" Equation
ypf)
This approach certainly seems reasonable, in that some account is taken of material-specific
properties. Inclusion of an impact factor would be supported by the work reported by Molerus
(1993, ppl28), where pneumatic conveying experiments under weightless conditions ( N A S A
space laboratory experiments) indicated that the pressure drop for moderate solids loading cases
"originates exclusively in particle-pipe wall impacts". Nevertheless, at this point it must still be
recognised that a completely satisfactory set of dimensionless parameters has yet to be found. It
should be noted that when applied to test data for a single material, the [Sz x p:R) term is again
a constant and a friction factor correlation cannot be expected to give a relevant result for this
parameter.
Szikszay is by no means alone in seeking to include a consideration of impact phenomena in the
pneumatic conveying model. A n alternative approach was put forward by Morikawa and
Tanaka (1988) for example, that essentially set aside separate pressure drop contributions that
may be attributed to particle-to-particle and particle-to-wall collisions respectively. Separate
pressure drop coefficients were assigned and the pressure drop contribution for each component
was assumed to follow the same basic Darcy-Weisbach formulation generally adopted by the
friction factor models. For example, for the particle-to-particle collisions, the pressure drop
contribution is given as:
dx p^ x v2
=
x
"PKK *KK "77x
o
Morikawa and Tanaka Equation
There may be some merit in this approach, however distmgmshing between these specific
components experimentally and hence obtaining some verification would be very difficult, and it
is not clear why the Darcy-Weisbach formulation is appropriate for these interactions.
Kt
=Fra
fin
xm
J £
There are two remaining points that could be made in general regarding the application of the
Darcy-Weisbach pressure drop style of model. Firstly it would appear to be more suited to
dilute phase conveying regimes; this is the predominant focus of authors reporting in this field,
and no doubt reflects thefluiddynamics foundation of the method. Use of this style of model to
generate pneumatic conveying characteristics plots, as reported in Chapter 5 of this thesis, is
more successful in the dilute phase region, and there does not appear to be a consistent inbuilt
tendency in the model to reproduce the increase in pressure drop at lower air mass flow rates as
the conveying system moves into the dense phase region.
Secondly, and again in relation to pneumatic conveying characteristics, when considering the
effectiveness of various models, it is worthwhile to tike account of not only the mean error but
also to consider the error in relation to the particular "location" within the conveying regime. In
other words, to consider where across the conveying regime the model is most effective, and
where it is falling down. Certainly the overall error, as represented by say the mean deviation or
6
Power Based Pneumatic Conveying Model
65
the standard deviation of the error, is a useful comparative statistic. However, examination of
the performance of the model across the possible conveying regime adds valuable insight into its
strengths and weaknesses.
This has by no means exhausted the field of models proposed over the past thirty five years or so
to handle two phase flow. Numerous variations and alternative formulations have been put
forward, and comparisons of these correlations presented in the literature, for example
Muralidhara et al (1979). Nevertheless the major approach to generating useful analytical
models has been reviewed. A c o m m o n theme has remained that for accuracy sufficient to
deliver an optimum system design, material specific test data and correlation analysis are
required, and it can perhaps be concluded that despite the extensive work and progress that has
been reported, the authors themselves would not be satisfied that a generally applicable,
comprehensive and accurate model has been produced.
6.2 Power Based Model - Introduction
Optimising the rate of energy consumption has long been recognised as one of the key
underlying themes in the successful design and operation of pneumatic conveying systems.
Zenz and Othmer (1960, pp 319) state that it is desirable from a power consumption point of
view to operate a pneumatic conveying system at as low a velocity as possible. It has also been
stated, Kraus (1980), that the design of a pneumatic conveying transport system is "actually an
energy balance that accounts for all the conceivable forms of energy that m a y exist in a proposed
system".
The development of a mechanical energy balance for a two-phase system is presented in some
detail by Govier and Aziz (1972, pp 300-305), leading to an expression relating pressure drop
across a straight section of pipe to changes in potential and kinetic energy of the phases plus an
undefined term accounting forfrictionallosses and other irreversibihties. Power balance issues
and some basic equations are also presented by Marcus et al (1990, pp 209-214).
Other authors, for example Beck (1986) and Marcus, Reed and Chambers (1986) have featured
energy use rninimisation as an essential element in developing an optimal pneumatic transport
system. O n e means of assessing the best operating conditions to use for a given installation is to
determine the specific energy requirements (from experimental data) and to overlay curves for
constant values of this parameter on the conveying characteristics diagram, as demonstrated by
Mill and M a s o n (1984).
This interest is not surprising since power consumption is a major ongoing running cost for
pneumatic conveying systems. It is an area where pneumatic conveying is generally
acknowledged to be at a disadvantage when compared to alternative systems, such as belt
conveyors and bucket elevators. In addition, any excess power that is being consumed will show
up directly in additional maintenance costs due to pipeline wear and avoidable damage to the
transported bulk solid.
Despite this interest and the recognised importance of energy consumption in this field, it wou
appear that the analysis has not been developed to the point where it could be used as a basis for
an analytical model for pneumatic conveying. Indeed even where energy losses have been a
particular consideration Borzone et al (1990), the analysis of transport energy losses has
proceeded from the Fanning Friction pressure drop model.
The exploration of the potential of such an approach has formed a significant component of this
thesis, and a description of the basis of the model is the primary focus of this Chapter.
The underlying principle is very simple; a straightforward application of the conservation of
energy principle. Firstly a set of experimental data is collected. The power available for the
66
6
Power Based Pneumatic Conveying Model
transport of the bulk solid/air mixture is identified and balanced against the rate of change in the
kinetic and potential energy of the gas and solid components, the power loss associated with the
frictional losses in the air component, and the power losses associated with the solid component
(frictional, impact, abrasion, breakage). The basic relationship m a y be expressed as follows:
Power = mf[AKEf + APEf) + ms{AKEs + APES) + (air frictional power loss)
+ (solids frictional power loss)
E q 6.2.1
Values for the change in kinetic and potential energy are determined from experimental data.
The power loss associated withfrictionaldrag in the air component is based on the standard
Darcy-Weisbach equation, though some attempts have been m a d e to compensate for the
presence of solids. The solids frictional power loss component is not based in the DarcyWeisbach formulation as proposed in the model originating with Barth, but rather on a simple
solidsfrictionapproach. It is based on slidingfriction,and includes specific consideration of
bends. The value of the solids friction factor is an "unknown" in the analysis. A n expression for
this factor is determined via a correlation analysis of experimental data.
Details on the assumptions and procedures used for each of these components, and the
implications of these approaches, are provided in the following Sections.
6.3 Power Source - Isothermal Expansion
The primary source of power for the transport of the bulk solid in pneumatic conveying systems
is the compressor or blower, and the selection of an appropriate sized unit is an important step in
the design process. In estimating a power requirement for the prime mover of the system,
allowance needs to be m a d e for efficiency losses in the electrical and mechanical drive
components, and in the air supply network. It is not, however, the starting point of this analysis.
As far as the transport of the bulk solid is concerned, the power available to carry out this task is
limited to that stored in the compressed air (positive pressure systems being considered here,
though the principle applies equally to vacuum systems) and delivered to the inlet end of the
pipeline. This energy is released by the expansion of the gas d o w n the pipeline. T o determine a
value for this available energy, it is necessary to select the most applicable expansion process, ie
isentropic, adiabatic, polytropic, isothermal, frictional adiabatic ....
An isentropic expansion is unlikely in view of the extensive irreversibilities encountered in the
pneumatic conveying process, particle-to-air friction, particle-to-wallfriction,particle-to-particle
friction, breakage, erosion, noise and heat generation etc, and the ample opportunity for heat
transfer during the substantial period it takes for air to travel through a typical system. A n
isentropic expansion would result in a temperature drop in the order of 50°C, for a moderate
pressure drop system of around lOOkPa, which does not accord with practical experience.
For pneumatic conveying systems handling bulk solids at ambient temperatures, there is no
significant apparent temperature differential d o w n the pipeline. This is no doubt a result of the
combination of two factors, heat generated internally by the irreversibilities in the transportation
process (friction, impacts etc) and heat flow between the environment and the pipeline system.
If it is desired to exclude external heat transfer, but to include irreversibilities, ie frictional
adiabatic flow, Stebbins (1989) has shown that the temperature drop to be expected would be
quite small - in the order of 1°C (assuming Xf « 0.017). This magnitude of temperature drop is
plausible.
The expression for frictional adiabatic flow is reasonably complex however, and given the
unrestricted heat transfer conditions generally applying between the pipeline and the surrounding
6
Power Based Pneumatic Conveying Model
67
environment, its use in this situation is not warranted. The more relevant process to consider is
simple isothermal expansion. If, during the flow through the pipeline, insufficient heat is
generated internally to maintain a steady temperature, then heat transferfromthe surroundings
will take place as a result of the temperature differential, effectively adding to the energy
available to transport the material. On the other hand, if excess heat is generated via the
frictional mechanisms, this will be lost to the surroundings. In either case the energy/power
available to carry out the conveying task can be closely approximated by the isothermal
expansion of the supply gas.
This power may be simply determined from:
Power = mf\e p-dv power, expansion of a compressible substance
which, following evaluation for an ideal gas undergoing an isothermal process, b
P
Power = mfRT'In
power, isothermal expansion... Eq 6.3.1
\PeJ
This simple expression defines the available power that will be used as the starting point of this
analysis.
V
For T = 15°C, this evaluates to: Power = 84.16m, In kW, and for a system operating at a
1
\PeJ
pressure drop of say 200kPa, and exit pressure at lOlkPa, this approach would imply that
4.7kW (per kg s"1 of supply air) is available to ensure the transport of both air and solids.
Note: This is approximately half the power quoted at various times by researche
Thames Polytechnic, London - for example Mills and Mason (1984), when estimating
compressor power requirements. In this instance an approximate allowance is being made for
inefficiencies in the compression stage. Such an allowance is not applicable when considering
the actual power available at the pipeline. Inefficiencies will in fact diminish the power
available, and will be accounted for in thefrictionfactor calculations that follow.
6.4
Air Component
As has been outlined in the introduction to this Chapter, the usual approach that has been taken
in accounting for the air flow requirement in pneumatic conveying, has been to employ an
equation of the Darcy-Weisbach or Fanning type, and effectively ignore the presence of the
solids in that process.
Intuitively this would appear to be neglecting the dominant characteristic of p
conveying, that is the presence of the solids component. It would be expected that this twophase mixture would modify the wall friction characteristics and restrict (particularly at high
solids loadings) the cross-sectional area available for the airflow.A significant proportion of
the gas solid interaction is now likely to be along a moving rather than stationary boundary. Air
flow is now likely to be a combination of flow above a moving bed of material and some
interstitialflowbetween the solid particles. The increasing velocities down the pipeline as the
gas expands would also result in a change in available cross-sectional area (increasing as the
pressures fall) so that even a constant diameter pipe system takes on some of the characteristics
of a tapered (expanding in the direction offlow)pipe. The variation in conveying conditions at
specific locations and over time, even when an overall steady state conveying regime may be
judged to have been established, means that it is unlikely that the transition between the
variations outlined above will be smooth and predictable.
6 Power Based Pneumatic Conveying Model
68
The extensive list of imponderables given above is perhaps one of the soundest reasons for
staying with a simple single phase analysis of the gas flow component, and leaving the more
complex interactions to be absorbed into the solids friction factor correlation. It is possible
however, that such a simplification also limits the potential of the subsequently developed
models to reproduce effectively the behaviours demonstrated by actual conveying systems.
This Section does not propose to unravel completely the mysteries of air flow through a two
phase rnixture. However some extensions of the air-onlyflowanalysis are presented; these seek
to account for important variables such as the reduced area available for flow, consequent
changes in velocity, altered boundaries for the air only flow (wetted perimeter concepts), the
influence of interstitial flow and power sharing options.
6.4.1 Superficial Gas Velocity
The standard approach, as indicated above, is to determine the influence of the air alone
component based on a superficial velocity that ignores the presence of the solids. This may be
readily determined from:
Vf =~J- Eq 6.4.1
f
PfA
If it is desired to take account of the presence of solids, and to determine an "actual" superficial
air velocity, it is necessary to take account of the area taken up by the solids component:
mf
Vfa=
/ / . x
Eq 6.4.2
Eq 6.4.2 recognises the two-phase nature of the situation which, even in terms of a rationa
structure for this section, introduces some difficulties. Determination of a value for the crosssectional area occupied by the solids component (As) requires an estimate of the solids velocity,
which may involve establishing a relationship between the air and solids velocities. Therefore it
can be concluded that the "actual" superficial air velocity and the solids velocity are mutually
dependent and that the development and evaluation of Eq 6.4.2 would be more appropriately
carried out together with the solids velocity. This organisational arrangement has been adopted,
consequently the further development of this analysis is presented in Section 6.5 which deals
more particularly with solidflow,and is not repeated in detail here.
In summary, however, if the simphfying assumption presented in Section 6.5 is adopted, ie that a
ratio between the "actual" superficial air velocity and the solids velocity can be found (Vs= VSR
. Vfa), then Vfa can be determined from the following equation:
(
Vfa
~
m,
A \VSR.ps
mf
•+•
p f)
Eq 6.4.3
The superficial air velocity determined from Eq 6.4.3 diverges markedly from the value obtained
from Eq 6.4.2 as the solids loading increases, as indicated in Figure 6.4.1.
6 Power Based Pneumatic Conveying Model
69
20.00
100
150
200
250
Solids to Air Mass Flow Ratio (m*)
-Vfa
Figure 6.4.1
Comparison between Vf and Vfa for a range of solids loadings for:
m f = 0.07kg s"\ ps = 2300kg m"3, Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m
If the air-only component of pressure drop is determined using the Darcy-Weisbach equation
outlined in Section 6.1:
, PfVf^
Ap.
Eq 6.4.4
' " f
2D
using the Swamee-Jain equation for Af (and taking s to be zero), and determining the air
velocities and density at mean pressure conditions, then it is apparent that the use of Vfa in place
of Vf\viH have a major effect. This is illustrated in Figure 6.4.2.
80.00
•3 10.00 +
0.00
20
•+•
•+•
+
40
60
80
100
120
Solids to Air Mass Flow Ratio (in*)
•Vfa
Figure 6.4.2 Air-Only Pressure Drop, based on Vf and Vfa for a range of solids loadings
for: m f = 0.07kg s"1, ps = 2200kg m 3, Apt = 150kPa„, VSR = 0.5, D = 5 0 m m , L = 400m
Since this is a power based method, it is necessary to convert the pressure drop values
determined above into equivalent power consumption values. Following the approach
6 Power Based Pneumatic Conveying Model
70
previously outlined in Section 6.3, these calculations are based on an isothermal expansion of the
air, ie the use of E q 6.3.1:
V
power, isothermal expansion... E q 6.3.1
Power = mfRT\n
\P,J
Since this is a two-phase flow situation, and the pressure drop associated with the air-only flow
is only a part of the total pressure drop, a decision needs to be taken with regard to the values to
use for the pressures pt and pe. The difference between these two terms (ie pt - pe) is the pressure
drop found by the application of E q 6.4.4. There are three basic options available:
i) let /?e be the pipeline inlet pressure, therefore pe=pt- Apf
ii) let pe be the pipeline exit pressure, therefore pi=pe + Ap,
iii) base the calculation around the mean pressure conditions, therefore # = Pfm + Ap,/2 and
Pe =Pfin - Apf 12
The first method will indicate lower power consumption figures, the second higher values, and
with the third approach resulting in values between the other two. This variation in calculated
values is illustrated in Figure 6.4.3, using the same range of conveying data previously displayed
in Figure 6.4.2. The most consistent approach would appear to be method (iii), ie to base
calculations on the mean pipeline pressure, in which case E q 6.3.1 becomes:
pt +pe + Apf
Pof = mfRT\A —
-^
E q 6.4.5
1
XPi+Pe-APf)
where pt and/»e are the overall pipeline inlet and exit pressures respectively.
If methods (i) or (ii) are adopted it is perhaps also more consistent to use inlet or exit air
conditions respectively in determining the pressure drop, E q 6.4.4, before proceeding to
calculate the corresponding power consumption values. That is the approach that has been
followed in obtaining the results shown in Figure 6.4.3.
4000.50
©
a
9
S
v>
Ua
o
m.
3500.50 -3000.50
2500.50
2000.50
w
1500.50
"3
Ou
• mm
1000.50
500.50
<
0.50
40
60
80
100
120
Solids to Air Mass Flow Ratio (m*)
•P inlet
• P mean
•Pexit
Figure 6.4.3 Air Only Power Consumption, based on Vfa and Inlet, Mean and Exit
Pipeline Pressures, for a range of solids loadings for:
m , = 0.07kg s'1, p s = 2200kg m 3, Apt = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
On examining the spacing between the three curves in Figure 6.4.3 it is also apparent that the
impact of the air-only component is not linear d o w n the pipeline, with the contribution to power
consumption increasing towards the exit end. This is related to the velocity squared relationship
6
71
Power Based Pneumatic Conveying Model
in the pressure drop equation, though this is complicated b y the higher solids velocities towards
the exit end o f the pipe which in turn allows a greater area for the air flow, consequently
reducing the rate of increase for Vfa. These effects are illustrated in Figure 6.4.4, where the
pipeline has been broken u p into a series of segments for the purposes of calculating the air flow
p o w e r consumption. A n overall linear pressure drop d o w n the pipeline has been assumed, and
in each segment the pressure drop and p o w e r consumption calculations have been based o n the
m e a n condition for that segment. Corresponding inlet m e a n and exit p o w e r consumption results
for the complete pipe, are included for comparison.
3500.50
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
400.00
Distance Along Pipe
•Cumulative Power
Consumption
Inlet
Mean
Exit
Figure 6.4.4 Air Only Power Consumption, based on Vfa, cumulative power consumption
determined by analysing the pipe as a series of segments (and determining the power
consumption in each segment at its mean condition). Limits based on and Inlet, Mean and
Exit Pipeline Pressure assumptions for the full pipe length, are also shown for comparison.
mf= 0.07kgs\m s = 5.0kgs1,ps = 2 2 0 0 k g m 3,Apt = 300kPa g , VSR = 0.5,
D = 5 0 m m , L = 400m
Figure 6.4.4 indicates that a more detailed analysis, such as the segment approach, is likely to
result in a higher p o w e r consumption being determined for the air-only component than that
found b y assuming m e a n pressure conditions for the calculations (as s h o w n b y the end point of
the cumulative p o w e r consumption curve). T h e m e a n pressure method, however, is s h o w n to b e
substantially closer to the segment m e t h o d than either the inlet or exit pressure approaches.
Consequently, in view o f its simplicity compared to incorporating a "segmented" method, the
m e a n pressure m e t h o d been preferred at this stage.
Returning to the difference in power consumption values obtained by calculating the air-only
pressure drop based o n either Vf or Vfa, and accepting the m e a n pressure m e t h o d (iii) described
above, Figure 6.4.5 illustrates the significant difference that emerges at higher solids loadings.
This rise in p o w e r consumption (and pressure drop) is in accordance with what would intuitively
b e expected, and m a y contribute to the concave upwards shape of conveying characteristics
curves in the l o w air flow, high solids loading region.
6
72
Power Based Pneumatic Conveying Model
1800.00 •,
a
o
'mm
1600.00 •
1400.00 -
-mm
a,
E
s
a
o
1200.00 -
tt
400.00 -
e
1000.00 •
800.00 •
600.00 -
200.00 0.00 -
PH
a
0
1
1
1
50
100
150
_
1
200
250
Solids to Air Mass Flow Ratio (m*)
i_
53
Figure 6.4.5
Air Only Power Consumption, based on Vf and Vfa for a range of solids
loadings for: mf = 0.07kg s \ ps = 2200kgm 3, Apt = 150kPa g , VSR = 0.5,
D = 5 0 m m , L = 400m
6.4.2 Interstitial Air Flow - Equivalent Diameters
The analysis described in Section 6.4.1 results in an "actual" superficial air velocity, taking
account of the presence of the solids component. The approach suggested has a number of
favourable aspects, including the certainty that the upper limit for the solids velocity (discussed
in detail in Section 6.5) is equal to the "actual" superficial air velocity (Vfa), or alternatively that
the maximum possible value for VSR is 1. This is not necessarily the case if the standard
superficial air velocity (Vf) is used.
An anomaly with this approach however is that in determining the pressure drop, and the
subsequent power consumption, the full pipe diameter has been used (in the Darcy-Weisbach
Equation, and the Reynolds Number and friction factor evaluations). It would appear possible,
therefore, that this analysis can be further refined. In particular there are two areas that m a y be
addressed:
1. the determination of cross-sectional areas available for the solids and air flow components,
and calculating equivalent pipe diameters for the calculation of pressure drop
2. the possibility of sphtting the air flow into a component generally moving above the bulk
solids (at a higher velocity) and a component moving in the interstitial spaces between the
solid particles.
6.4.2.1 Area Analyses
To address the first point requires an analysis of the interactions between the air and solids
component. This has been carried out in some detail in Section 6.5 (Solids Component), with
the following result for the cross-sectional area occupied by the solids:
A
s =
m*x Axpf
m 'xpf + psxVSR
Eq 6.5.20
(see Section 6.5 for development)
This area is based on the particle density of the bulk solid, consequently a bed of material (at say
a loose poured or fluidised bulk density) would occupy a somewhat larger area, further reducing
the area above the bulk solid available for "unrestricted" air flow. The determination of crosssectional areas corresponding to various bulk densities (tapped, loose poured,fluidised),as an
extension of Eq 6.5.20, is straight forward, for example:
6 Power Based Pneumatic Conveying Model
73
Ps
Aslp =
As for the loose-poured bulk density case, and
Pslp
ps
A^ =
As for thefluidisedbulk density case.
Psf
It follows therefore that there is a residual area that may be considered to be "above" the flowi
bed of bulk solid that allows a relatively unrestricted path for a portion of the air flow. If, for
example, it is assumed that the material being conveyed is in afluidisedstate, the "free" area
above the bulk solid is simply:
Aff = A-Atf Eq 6.4.6
Clearly an equivalent area can be readily obtained, based on the loose poured bulk density of the
bulk solid, and the treatment similarly extended to other bulk density formats (tapped,
compressed etc). It is noted, however, that the following discussion and the results presented are
based on thefluidiseddensity assumption.
6.4.2.2 Equivalent Diameters
Having identified the "free" area available for the flow of the gas (Ag), it is possible to consider
alternative options in the calculation of the air only pressure drop. A simple approach is to find
the equivalent diameter that will yield the same cross-sectional area as Ag, ie:
A,= \ K )
E q 6.4.7
A second approach is to recognise that the free cross-sectional area above the bulk solid is
unlikely to approximate a circular shape. In fact it m a y be represented in an idealised form by
the upper sector of a circle, as shown in Figure 6.4.6.
Figure 6.4.6
Pipe Cross-section
A common approach in the analysis of non-circular flow sections is to determine an equivalent
hydraulic diameter to take account of the increased proportion of fluid/wall interactions
occurring. This diameter m a y be determined from Equation 6.4.8.
4A
D.=
— —
perimeter
Eq 6.4.8
To evaluate Eq 6.4.8 it is necessary to determine a value for the perimeter of the area Ag. A
consideration of the geometry shown in Figure 6.4.6 results in the following relationship
between A, / ^ a n d 6.
6
74
Power Based Pneumatic Conveying Model
2trxA.
2<9-sin(20)-
Eq 6.4.9
•= 0
A n iterative process is required to solve for 0in Eq 6.4.9. Having found 0, the perimeter of the
top section may be determined from:
Eq 6.4.10
perimeter = D x [n - 0 + sin(#))
As a general point it may be noted that De and Dh are both smaller than D, with De being larger
than Dh as indicated in Figure 6.4.7. The difference between De and Dh increases as the
fluidised bulk solid occupies a larger proportion of the pipe cross-section. The smaller the
equivalent diameter, the higher the pressure drop and power consumptionfigureswill be.
0.0002
0.0004
0.0006
0.0008
0.001
Cross-section occupied by Fluidised Bulk Solid, Asf
Figure 6.4.7
0.0012
0.0014
A
(m 2)
Equivalent Area and Hydraulic Diameters (De and Dh) versus ASf
D = 5 0 m m (A = 0.00196m2)
6.4.2.3 Air Velocity - Interstitial and "Free" Air
The previous analysis has identified the simple superficial air velocity, and an actual superficial
air velocity taking into account the presence of the solids component. Having determined the
areas associated with the bulk solid and the "free" air above the material bed, it is n o w possible
to extend the analysis to consider a separate velocity for each of these regions.
It would be reasonable to assume that the air flowing in the region above the bulk solid will h
a higher velocity than the air moving through the interstitial spaces. The actual value of this
interstitial velocity would be difficult to determine experimentally; however, once it is known it
is a relatively simple matter then to calculate the air mass flow rate in this region, to subtract that
value from the overall air mass flow rate andfinallyto calculate a velocity for the air flowing
above the bulk solid.
To allow this analysis to progress, two relationships were developed to estimate the interstitial
air velocity.
1. The interstitial air velocity was assumed to lie midway between the solids velocity and the
actual superficial air velocity, ie.
K+Vfi,
Vfat -
9
VfaH + VSR)
7.
Eq6.4.11
6
Power Based Pneumatic Conveying Model
75
2. The interstitial air velocity was assumed to range linearly from Vs at very low solids
concentrations, to Vfa where the fluidised bulk solid occupies the full pipe cross-section, ie
Vfai =K+ ~f(vfa - V.) Eq 6.4.12
Option 1, Eq 6.4.11, is reasonable in the sense that the lower limit for the interstitial ai
would be the solids velocity and the upper limit would be Vfa, Clearly the interstitial air velocity
(Vfm) cannot be below Vs, if it was it would be expected that the solids velocity would fall. If Vfai
is equal to Vfa, then the air moving through the bulk solid and the air passing above it would have
the same velocity, which is also unlikely. Therefore E q 6.4.11 represents a reasonable and
simple assumption. A difficulty can arise, however, when thefluidisedbulk solid begins to
occupy the majority of the cross-sectional area. Under these circumstances the shortfall in air
mass flow not passing through the bulk solid accumulates and must flow in the diminishing area
above the material bed. Consequently high air velocities (and corresponding pressure drops and
power consumption values) are calculated.
The second option, Eq 6.4.12, was developed to address that problem. With this relationship,
the area occupied by thefluidisedbulk solid approaches the full cross-section of the pipe, the
interstitial air velocity approaches Vfa, and hence there is no increasing discrepancy or
discontinuity to contend with. The higher pressure gradient occurring under these conditions
also supports the contention of an increasing interstitial air velocity. At the other extreme where
there is a very low solids loading, it has been assumed that the velocity in the immediate vicinity
of the solids particles is moving at the same velocity as those particles. A linear relationship has
been assumed between these limits, basically for simplicity. The lack of any evidence to support
the use of a more complex relationship at this stage is also relevant.
As indicated earlier, the velocities within and above the bulk solid are linked by the requi
that the total air mass flow rate is the sum of these two components. Equation 6.4.13 quantifies
this relationship.
Pfinx[(Asr-As)xV,ai+Ag*V,ar} = m, Eq 6.4.13
where V,a,is the velocity of the "free" airflowingabove the bulk solid.
Figure 6.4.8 shows the relationships between these air velocities for Options 1 and 2, over a
range of solids loadings.
6
O.oo -I
0
1
20
Power Based Pneumatic Conveying Model
1
40
1
60
1
80
'
100
'
120
Solids Loading (m*)
VfaiOptl —•—VfaiOpt2 —±—Vfaf Opt 1 —)£— VfafQpt2
Figure 6.4.8
Air Velocity - interstitial (Vfai) a n d "free" (Vfaf), Options 1 a n d 2, for a
range of solids loadings, based o n m e a n conditions:
mf= 0.07kg s 1 ,p s = 2200kg in 3 , Apt = 1 5 0 k P a g , VSR = 0.5,D = 5 0 m , L - 4 0 0 m
It may be noted from Figure 6.4.8 that the interstitial air velocity determined from Option 2 is
lower than that obtained from Option 1 at low solids loadings. W h e r e thefluidisedbulk solid
occupies half the pipe cross-section, the velocities are equal (in this instance at m * « 80), and
above this Option 2 results in higher velocities. The increasing free air velocity resulting from
Option 1 at higher solids loadings is also demonstrated in this figure.
6.4.3 Pressure Drop and Power Consumption (Vfaf)
The previous analysis in Section 6.4.1 examined the pressure drop and power consumption
relationships based on the superficial and "actual superficial" air velocities. T h e subsequent
analysis allows this to be extended to include equivalent diameters for the air flow channel, and
the interstitial and free air components. This Section will continue the approach of basing the
calculations around the m e a n pressure conditions.
A second basic element in tins phase of the analysis is the assumption that only the power
consumption associated with the conveying of the free air component needs to be considered.
The is because in the case of the air flow through the spaces between the particles, the frictional
drag and corresponding power "loss" is predominantly transferred directly to the bulk solid.
Consequently in terms of transporting the bulk solid, this power is not lost to the conveying
mechanism, but usefully engaged. Therefore it would be an exercise in double accounting to
subtract this component from the power available to m o v e the bulk solid. This approach is
based on the further assumption that the air-to-particle surface interactions significantly outweigh
the air-to-pipe wall interactions in the ^ r e g i o n identified in Figure 6.4.6.
From the foregoing approaches, a number of different combinations of assumptions and options
can be assembled, generating a number of different air flow models. T o obtain an insight into
the impact of these air flow models on the overall power based conveying model, it is relevant to
prepare curves showing the trends inherent in the various air flow models over a range of
conveying conditions. Table 6.4.1 is intended to clarify the definition of the various models,
and assist in the interpretation of the following Figures.
6
77
Power Based Pneumatic Conveying Model
Table 6.4.1 - Air Flow Model Definitions
Model No.
Diameter
Velocity
Interstitial Air Velocity
Model 1
D
Vf
Not relevant
Model 2
D
Vfa
Not relevant
Model 3
De
Vfaf
Eq6.4.11
Model 4
De
V,af
Eq 6.4.12
Model 5
Dh
v*t
Eq6.4.11
Model 6
Dh
Vfaf
Eq 6.4.12
Figure 6.4.9 shows the pressure drops calculated for Models 1-6 across a range of solids
loadings.
20
40
60
80
100
120
Solids Loading (m*)
•model 1
-model 2 — £ — m o d e l 3
-model 4 — $ — m o d e l 5
-model 6
Figure 6.4.9
Air Only Pressure Loss, for a range of solids loadings, based on mean
conditions, Models 1-6:
mf= 0.07kg s1, ps = 2200kg to3. Ap, = 150kPa g , VSR = 0.5, D = 50mm, L = 400m
It is evident from Figure 6.4.9 that the models making use of equivalent diameters (M
6) result in significantly higher pressure drops than the simpler models based on the full pipe
diameter and the superficial (and actual superficial) air velocity. Figure 6.4.10 converts the data
shown in Figure 6.4.9 into the equivalent power consumption form.
78
6 Power Based Pneumatic Conveying Model
3000.00
500.00 --
0.00 -I 1 1 1 1 1
0
20
40
60
80
100
120
Solids Loading (m*)
+ model 1 —•—model 2 —£—model 3 —\*~- model 4 —JK—model 5 —•#—model 6
Figure 6.4.10 Air Only Power Consumption, for a range of solids loadings, based on
m e a n conditions, Models 1-6:
m,= 0.07kg s'\ ps = 2200kg m -3 , Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
The power consumption data for models 1 to 6 follow a similar trend to that shown in the
pressure drop comparison, though the magnitude of the difference between models 1 and 2, and
the equivalent diameter models (3-6) is somewhat less at the higher solids loading end. This is
largely due to the fact that models 3 to 6 consider only that component of the total air flow that
passes above the bulk solid in determining the power consumption. At higher solids loadings
this component is reduced. The second method for calculating the interstitial air velocity (Eq
6.4.12 - used in models 4 and 6) is also seen to reduce the power consumption (and pressure
drop) at higher solids loadings. This difference becomes more pronounced as the bulk solid, in a
fluidised bulk density condition, occupies a greater proportion of the pipe cross-section, as
shown in Figure 6.4.11 for models 5 and 6. Consequently this trend has the potential to distort
the overall model performance in that region, and when converted to pressure drop format can
readily exceed the total air pressure drop.
6
Power Based Pneumatic Conveying Model
79
9000.00
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
Asf/Area(%)
-model 5
-model 6
Figure 6.4.11 Air Only Power Consumption versus Area occupied by the Fluidised Bulk
Solid, based on m e a n conditions, Models 5 and 6:
mf= 0.07kg s"1, pa = 2200kg m'3, Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
There are a number of other variations that may also be explored. For example, for the same
reasons that the air drag and power loss associated with the interstitial gas flow has been ignored,
it m a y be argued that in the case of the models using the hydraulic diameter concept, only the
proportion of the power loss that m a y be attributed to the air/wall drag should be considered.
A n estimate of this can be achieved by multiplying the power values obtained in models 5 and 6
( arc length
to obtain values for model 7 and 8 respectively, for example:
by the ratio of
\ perimeter J
Dx(ir-O)
Power^n = Z ) x ( ; r - 0 + sin(0)) *PowermodBli
Eq6.4.14
A n alternative approach intended to provide a similar outcome, ie to consider air/pipe wall rather
that air/bulk solid interactions, m a y be achieved by using only the pipe wall to air arc length,
rather than the entire perimeter, in deterrnining a modified hydraulic diameter.
D.
AAff
Dx(n-O)
E q 6.4.15
A n additional two models based on this approach, and the two alternatives for interstitial air
velocity, (models 9 and 10) have also been investigated.
A final possibility that has been explored is the use of the loose poured bulk density in plac
the particle density in these calculations. This is in effect treating the particles and the air
surrounding them when in their typical packed state as a homogeneous mixture. A n y interstitial
air flow then only occurs in the expanded gaps between the particles when the bulk solid is
fluidised. The layer of air around each particle is assumed to be moving at the same velocity as
the particle. O n e reason for including this variation is that in evaluating the performance of the
overall power based model, the use of this density instead of the particle density in many
instances appeared worthwhile. This approach could be applied to any of the previous ten
6
80
Power Based Pneumatic Conveying Model
models that have been proposed, doubling the total number of models, however it has been
restricted to the criteria used for models 5 and 6.
Consideration of these variations has resulted in the development of an additional six models,
their definitions being set out in Table 6.4.2 .
Table 6.4.2 - Air Flow Model Definitions
Density
Power
factor
arc length
perimeter
Model No.
Diameter
Velocity
Model 7
Dh
Vfaf
Interstitial
Velocity
Eq6.4.11
Model 8
Dh
Vfaf
Eq6.4.12
arc length
perimeter
Ps
Model 9
Dh
Vfa,
Eq6.4.11
not relevant
Ps
Model 10
D'H
Vfaf
Eq6.4.12
not relevant
Ps
Model 11
Dh
vM
Eq6.4.11
arc length
perimeter
Pslp
Model 12
Dh
Vfaf
Eq6.4.12
arc length
perimeter
Pslp
Air
Ps
Figure 6.4.12 illustrates the result of applying the reduced power assumption set out in E q
6.4.14. In this case the power loss for air flow as calculated by models 1, 2 and 5 - 8 are shown.
Models 1 and 2 are included for reference purposes. Models 5 and 6 show the results
determined by the hydraulic radius method (with the two interstitial air velocity options).
Models 7 and 8 correspond to models 5 and 6 respectively, with the power output modified as
suggested by E q 6.4.14.
3000.00
40
60
80
100
120
Solids Loading (m*)
-model 1
•model 2
•models — X — m o d e l 6 — ^ — m o d e l 7
-model 8
Figure 6.4.12 Air Only P o w e r Consumption, for a range of solids loadings, based on
m e a n conditions, Models 1,2 and 5 - 8 :
m , = 0.07kg s \ ps = 2200kg m'3, Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
6
81
Power Based Pneumatic Conveying Model
A s would be expected, models 7 and 8 generally follow the trends shown by models 5 and 6,
being reduced by the arc length/perimeter factor. O n e interesting outcome, however, is the
small initial drop in power consumption at low solids loadings, where the results for models 7
and 8 even fall below the predictions of the basic Vf model (model 1). This is a direct result of
the power reducing factor introduced for these models, and raises the question of the variation of
this parameter with solids loading. Figure 6.4.13 displays this variation for m* values ranging
from 0 to 200, which corresponds in this instance to the pipe varying from an "empty" to nearly
"full" state ("empty" and "full" in this case refer to the proportion of the cross-sectional area
occupied by the fluidised bulk solid).
20
40
60
80
100
120
140
160
180
200
Solids Loading (m*)
•Asf/Area
•Arc Length/Perimeter
Figure 6.4.13 Area and Length Ratios versus Solids Loading:
mf= 0.07kg s"1, ps = 2200kg m"3, Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
It is apparent from Figure 6.4.13 that the initial fall in the arc length/perimeter factor is quite
sharp. Hence there is an initial fall in power consumption shown for models 7 and 8, despite the
overall trend for increasing power consumption as m* increases. A s the pipe cross-section
approaches the full state, the arc length/perimeter factor approaches a limit of 0.5.
Figure 6.4.14 indicates the behaviour of models 7 to 12, with models 1 and 2 again included for
comparative purposes.
6
82
Power Based Pneumatic Conveying Model
0.00
20
40
60
80
100
-X—
model 8 — g g — m o d e l 9
120
Solids Loading (m*)
-model 1
•model 10
-model 2
model 11
-model 7
-model 12
Figure 6.4.14 Air Only Power Consumption, for a range of solids loadings, based on
m e a n conditions, Models 1,2 and 7 - 1 2 :
mf= 0.07kg s"\ ps = 2200kg m"3, Ap, = 150kPa g , VSR = 0.5, D = 5 0 m m , L = 4 0 0 m
Models 7 to 12 exhibit similar performance characteristics to each other. This indicates that the
modified hydraulic diameter approach (models 9 and 10) provides a similar outcome to the
power reduction factor method used for models 7 and 8. All these models show the initial fall in
power loss at low m* values. The models using the simpler interstitial air velocity assumption
(Eq 6.4.11 - models 7, 9 and 11) result in higher and increasing power requirements at the high
solids loading end, this being more pronounced for models 7 and 9. If the solids loading is
increased further, these models follow the same trends shown in Figure 6.4.11 for models 5 and
6. It is interesting to note that the use of the loose poured bulk density instead of the particle
density (models 11 and 12) did not have a major impact on this power consumption component.
6.4.4 Air Component - Conclusions
In considering the contribution of the air component to the power requirements in pneumatic
conveying, it is apparent that there is a wide variety of options available. Twelve models have
been developed, and their behaviour explored, in this Section. There is ample scope for further
models to be proposed, eg consideration of the variation in conditions along the pipeline
(extending the work illustrated in Figure 6.4.4), making allowance for the moving air/bulk solid
boundary, and the development of more comprehensive formulations for interstitial air velocity.
Even within the options and assumptions proposed in this Section, the set of possible models
could be significantly increased by considering all the possible combinations.
At this point there is no experimental evidence available to support directly one of the models
over the others, since test data report the combined effect of all the components (air, solids
friction, bends and vertical sections). However there is, in the general shape of conveying
characteristics curves obtained for materials that are able to be conveyed easily across the lean to
dense phase transition region, reason to prefer the Vfa model (model 2) over the simple Vf model.
It would also seem reasonable to base calculations on mean rather than inlet or exit conditions, if
a single point is to be used.
6
Power Based Pneumatic Conveying Model
83
In going beyond these assumptions it is necessary to rely to a large extent on judgement as to
what is likely to represent best the complex, turbulent and chaotic interactions that are actually
taking place. O n e w a y of assessing this is to examine the performance of the overall pneumatic
conveying model with k n o w n conveying data, w h e n different air-only models are included. The
development of the equivalent diameter and hydraulic diameter approaches occurred in this
context and appear to have some merit w h e n compared to using the full pipe diameter for these
calculations. The hydraulic diameter concept is attractive in that it at least recognises (if oversimplifying or idealising) the basic geometry of the situation.
Separation of the interstitial and "free" air components is also a simplification of the actual
processes taking place. Nevertheless, ignoring this aspect is likely to be a less representative
simplification, hence its inclusion is recommended. T w o alternatives for dealing with the
interstitial air velocity have been presented. The simplest model, taking an average of the solids
and actual superficial air velocity, appears to give reasonable values until the area occupied by
the fluidised bulk solid exceeds approximately 6 0 % of the pipe cross-section (refer Figure
6.4.11). Beyond this point this approach is likely to introduce instability into the model. The
second option for dealing with the interstitial velocity (Eq. 6.4.12) effectively addresses this
instability problem and hence is preferred. Experimental work in this area, whilst no doubt
presenting challenging problems in terms of mstmgiiishing the various components, would be
worthwhile.
Finally, the proposition that only that proportion of the power consumption that relates to the
"free air" component moving above the bulk solid should be considered, is a logical extension of
the power based pneumatic conveying model (and perhaps shows one of the strengths of this
approach), in that these elements can be separately identified and accounted for. The
introduction of the power reducing factor (Eq 6.4.14) to the hydraulic diameter based models is
a further logical extension of this approach, and evaluation of its effect on model performance
appears warranted.
All of the models proposed are based on major simplifications of the physical interactions takin
place. This is inevitable and necessary, given the complexities of these interactions, if a
workable model is to be produced. A n y inadequacies in the model selected will naturally
influence the overall performance of the pneumatic conveying model. These shortcomings will
to some extent be absorbed and compensated for by other elements of the overall model, notably
in the solids friction factor correlation. Therefore the extent to which the air-only component
assumptions are adversely affecting overall model performance, m a y not be apparent.
Consequently the analysis of trends in performance for the various models is a valuable aid in
selecting the most promising approach.
In conclusion, therefore, having considered a range of options, developed a number of models
and explored to some extent their likely impact on overall pneumatic conveying model
performance, it is suggested that model 8 provides a reasonable and consistent means of
incorporating the air-only power consumption component. Variation from and adjustment of
this aspect of the power based model should nevertheless be considered in the light of the overall
performance of the model.
6.4.5 Air Component - Example
Despite the simplifications of the actual physical processes involved, the recommended model
remains somewhat complex and the results from a number of the equations developed in this
section need to be combined in order to estimate the magnitude of the air-only power
consumption component.
To assist in clarifying this approach, the relationships involved in the application of Model 8
collected here, together with a typical sample calculation.
6 Power Based Pneumatic Conveying Model
84
For the purposes of this example, the power consumption associated with the air component for
the following conveying regime will be determined:
pipe diameter 100mm
pipeline length
bulk solid particle density
bulk solidfluidiseddensity
air massflowrate
solids to air massflowratio
ambient pressure
total pressure drop
temperature
VSR
250m -3
2575kg m
750kg m
0.15kg s'
110
101 kPaabs
350kPag
20°C
0.65
P 101 + 0.5x350 3
m
1. The air density at mean pressure is: pfin = — = 0 2 8 7 1 x 2 9 3 =
2. The pipe cross-sectional area is: A = 0.007854m2
3. The area taken up by the solids component may be obtained from Eq 6.5.20, ie
m*xAx pf
A
s = —i
~^E = 0.001393m2
m*xpf + psx VSR
Ps .
2
4. The area occupied by thefluidisedbulk solid (from A^ =
As ) = 0.00478 lm
Psf
5. The area "above" thefluidisedbulk solid = A-A4= 0.003073m2
6. The actual superficial air velocity (from Eq 6.4.2) V,a = 7.08m s"
7. The solids velocity (noting that VSR = 0.65) Vs = 4.6m s"1
8. The interstitial air velocity (from Eq 6.4.12 for model 8)
Vfai=Vs+-f(Vfa-Vs) -6.11ms"1
9. The velocity of the air passing "above" the fluidised bulk solid (Eq 6.4.13, rearranged)
m4
LP
r-Ur-^)*^
fin
VM =
.
= 8.148m s"1
A
ff
10. Considering the geometry of the partially occupied pipe section (Figure 6.4.6), and iterating
to solve for 0 (Eq 6.4.9), gives 6 = 99.9°
11. To estimate the friction loss associated with the air passing through this section, the
perimeter of the area above the bulk solid needs to be found and an hydraulic diameter
determined. The perimeter is made up of the arc section:
arc length = D x (71 - 0) = 139.8mm, and
the straight section = D x sin (0) = 98.51mm, therefore the perimeter is 238.31mm.
AA
12. The hydraulic diameter (Eq 6.4.8), Dh
— = 51.58mm
perimeter
13. The pressure drop associated with this flow can now be estimated from Eq 6.4.4
PfVJAL
Apf = Af — — — , with lf= 0.01899 (from the Swamee-Jain relation), and using Vfaf and
Dh for ^/and D respectively, giving Dp,= 10.02kPa.
6 Power Based Pneumatic Conveying Model
85
14. Power consumption corresponding to this pressure drop may then be calculated (Eq 6.4.5).
f Pt + Pe+ Apf 1
P o f = mfRT\n\
•
, where the air mass flowrate is that
1
A
'
\Pi+Pe- PfJ '
associated with the flow above thefluidisedbulk solid, ismf= pfxAgx Vfaf= 0.0821kg s"1.
Therefore Pof= 250.9W.
15. Finally, applying the assumption in Model 8 that only portion of Po/may be attributed to
„ , „
arc length
pipe wall frictton mteractions, ie Pof = Pofx
:
= 147.2W .
1
]
perimeter
This power consumption component may then be included in the overall analysis of the
pneumatic conveying system.
6.5 Solids Component
The bulk solid undergoes numerous complex and energy absorbing processes as it is transported
down the pipeline. Air drag, collisions (particle-to-particle and particle-to-wall), breakage,
tumbling or rotational motion, duning, suspended high speed flow and so on may all be
experienced. It does not seem likely that in the reasonably near future it will be possible to
replicate accurately and account for all these processes in a theoretical model. Neither is it
apparent that it is possible to separate out the contributions of these components in experimental
data.
Consequently the usual practice is to lump all of these effects together and seek to acco
them via an expression that includes a friction factor, values for which are deduced from
experimental data. If a set of assumptions is put in place with regard to the air flow component
(as outlined in Section 6.4 for example), it is possible to obtain an appropriate value for this
component (and hence extract the corresponding friction factor) for a single experimental point.
The challenge lies in determining an expression that gives a suitable value for this component
across the range of conveying conditions. Since numerous effects are involved, and different
behaviours are likely to dominate in different conveying regime zones, the satisfactory
determination of this expression is a very difficult matter.
The difficulty in finding an expression that performs well across the range of conveying
conditions may lie in either the basic form or type of expression that has been adopted for this
component, or in the parameters used in the correlation analysis to determine an expression for
the friction factor, or a combination of both. The approach used in this model differsfromthe
usual formulation used in the pressure drop type models described in Section 6.1. In that case
the contribution (in terms of a pressure drop) is assumed to follow a similar partem to single
phasefluidflow,and a slightly modified Darcy-Weisbach equation is used. In this power based
model, the power requirement associated with the transport of the bulk solids is primarily
assumed to be a function of the mass of solids in the pipeline, gravity, a friction factor and the
average velocity of the solids. Accordingly:
Pos = MsxgXAsxVs
Eq 6.5.1
There are significant issues related to the evaluation of each of these parameters. In pa
the determination of the mass of solids in the pipeline, and the solids velocity.
6.5.1 Solids Velocity
The interactions inherent in pneumatic conveying mean that virtually all of the parameters are
interconnected, and solids velocity, air velocity and mass of solids in the pipeline, are a case in
point.
The solids velocity is naturally related to the velocity of the driving medium (air). The
force between the air and the particles resulting from the velocity differential between these two
phases is the primary mechanism for moving the bulk solid through the pipeline. Momentum
6 Power Based Pneumatic Conveying Model
86
transfer between colh'ding particles m a y play a significant role on a local scale (in particular
contributing to the re-entrainment of particles from a stationary or semi-stationary bed) however,
except for the case of conveying in a downward direction where gravitational forces contribute
strongly, ultimately all the energy required for transport must be transferred from the conveying
an. Consequently the slip velocity, or relative velocity between the two phases, is of some
importance and isfrequentlydiscussed in the literature. It m a y be defined simply as:
S = Vfa-Vs
E q 6.5.2 (ref Govier and Aziz, 1972, pp308-312)
In this context the velocity considered for each of the phases is generally defined as the average
velocity of that component, taking into account the cross-sectional area available to it. This is in
contrast to the usual approach taken in pneumatic conveying when considering the airflow,of
determining a superficial air velocity based on the full cross-sectional area of the pipe.
Alternative approaches (Govier, 1972) used to characterise this parameter include the holdup or
slip ratio, which is the ratio of the velocities of the two phases:
V
1
Slip ratio - E q 6.5.3
H = fa
Vs VSR
and the Bankoff K Factor which, in terms of the parameters relevant to this thesis, m a y be
defined as:
A.(\
^
Bankoff K Factor - E q 6.5.4
M u c h of the discussion on slip velocity in the literature centres around vertical conveying, which
presents simpler mechanisms for analysis, allowing a balance to be made between aerodynamic
drag forces and gravity, with reduced need to take account of wallfrictionand impact forces, for
example Singh (1982). In this instance particle terminal velocity has understandably a particular
relevance. It is not clear, however, that particle terminal velocity would retain the same
significance for horizontal conveying since the mechanisms and forces resisting the aerodynamic
drag are quite different.
Provided that the readily measurable parameters, air mass flow rate and pipeline pressure, are
known, the superficial air velocity m a y be determined from:
mf
Vff =— A
E q 6.5.5
Pf
This Equation ignores the presence of solids. If it is desired tofindan "actual" average velocity
for the air component (Vfa), the following approach m a y be used:
m,'r
m,
+
—
f
rE q 6.5.6
A=Af + As =
-^-+-jf
PfVfa \PsK
(
mr + •m f
Eq 6.5.7
V/a
~ A VSR. ps
pf
or
H.m.
Vfa
~ A \ Ps
•+
mf
Eq 6.5.8
pf)
Comparison of the two air velocities (Vfand Vfa) over a range of solids loadings shows a very
significant difference, particularly at higher loadings, as indicated in Figure 6.5.1 .
6
Power Based Pneumatic Conveying Model
87
25.00
20.00
1
3 15.00
o
"3
>
mm
10.00
5.00
20
40
60
80
100
120
140
160
Solids to Air Mass Flow Ratio (m*)
•Vfa
•Vf
Figure 6.5.1
Comparison between V, and V,„ for:
m/= 0.1kg s'\ ps = 1650kg m'3, Ap, = 250kPag, VSR = 0.5 (H = 2), D = 50mm
45
40 -35 -^30 -•
1
W
o
"S3 20 ->
u
< 15 -10 -5 -0 --
+
0.1
0.2
0.3
0.4
0.5
0.6
Solids to Air Velocity Ratio (VSR)
Vfa
Vf
Figure 6.5.2
Comparison between Vf and V,a for:
1
mf= 0.1kg s , ps = 1650kg m3, Ap, = 250kPag, m* = 50, D = 50mm
0.7
0.8
6
88
Power Based Pneumatic Conveying Model
Similarly, the average "actual" velocity of the air component can be seen to rise steeply as the
slip ratio increases (or V S R decreases), as indicated in Figure 6.5.2 . These Figures illustrate the
magnitude of the effect that the introduction of the solids component has on the air velocity, and
confirm the necessity to take this into account w h e n seeking a relationship between the air and
solids velocities.
To incorporate this into a model requires that some relationship be adopted between the solids
and air velocities. Marcus et al (1990, pp98-103) discuss an empirical equation based on work
by Hinkle (1953) relating average particle velocity to air, pipe and particle parameters as set out
in Equation 6.5.9.
V. Vfx (l-0.6Sxd"* x p°
5
x pf2 x D-0.54
E q 6.5.9
E q 6.5.9 is interesting in that it suggests, for a given pipeline/bulk solid combination, that the
solids velocity is unaffected by the solids loading. This m a y be a result of the correlation being
primarily developed from a consideration of lean phase data. Nevertheless, b y taking the result
given from E q 6.5.9, a value for Vs can be obtained and substituted into Equation 6.5.10 (which
is a rearrangement of E q 6.5.7) to determine Vfa.
VfxpsxAxVs
fa
E q 6.5.10
(psxAxVs-ms)
This draws the solids mass flow rate into the relationship, and also allows values for the slip
velocity ratio to be determined. Figures 6.5.3 to 6.5.5 illustrate the relationships implicit in Eqs
6.5.9 and 6.5.10. (It should be noted that thesefigureswere developed with a view to exannriing
the variation of the relevant parameters as the bulk solid was conveyed d o w n a nominated
pipeline - hence the 0 to 500 kPag pressure range.)
100
200
300
400
500
Pipeline AirPressure (kPag)
Figure6.5.3
ValuesoiV s ,V f and Vfa for:
D = 1 0 0 m m , djo = 1 0 0 p m , m f = 0.3kg s 1 ,p s = 1650kg m " 3 , m * = 100 and
assuming a pressure range of 0 - 500kPa„
600
6
100
Power Based Pneumatic Conveying Model
200
300
400
89
600
500
Pipeline Air Pressure (kPag)
Values of m*
- * — 90
Figure 6.5.4
•150
Variation of VSR with m * for:
D = 1 0 0 m m , d so = 100pm, mf= 0.3kg s"\ ps = 1650kg m~3,
and assuming a pressure range of 0 - 500kPa g
It can be seen that Vs is very close to the superficial air velocity, and that average "actual
velocity (ie flow through the interstitial spaces) is somewhat higher. The difference between Vf
and Vfa reduces at lower values of m * , but increases for bulk solids of larger particle size.
Figure 6.5.4 indicates that the solid to air velocity ratio (VSR) is expected to be relatively
constant across a range of pressures, for low to moderate solids loadings (say up to m * = 50),
but to drop significantly at higher m * values. This is at odds with a statement by Marcus et al
(1990, pp 103) that at "high loadings
the solids and gas act more fluid-like to reduce the slip
velocity", and m a y again be a result of the data set used to obtain the correlation set out in E q
6.5.9, and hence m a y deviate from behaviour that could be exhibited byfinerfluidisedpowders
conveyed in a dense phase regime.
6
90
10
Power Based Pneumatic Conveying Model
15
20
25
30
35
Vfa (m/s)
Figure 6.5.5
Relationship between Average "actual" air velocity and solids velocity for:
D = 1 0 0 m m , d 50 = 100pm, mf= 0.3kg s"1, ps = 1650kg m"3, m* = 50 and
assuming a pressure range of 0 - 500kPa g
Figure 6.5.5 indicates a linear relationship between Vfa and Vs, for the conditions shown. This
linearity is retained at high solids loadings (m* values) and for larger sizes of particles, however
at the lower velocity end the solids velocity m a y drop sharply. For example, if the basic
arrangement illustrated in Figure 6.5.5 is retained but d 50 is increased to 3 m m and m* to 100, the
relationship shown in Figure 6.5.6 is obtained.
6
10
15
Power Based Pneumatic Conveying Model
20
25
30
91
35
40
Vfa (m/s)
Figure 6.5.6
Relationship between Average "actual" air velocity and solids velocity for:
D = 1 0 0 m m , dso = 3000pm, W / = 0.3 kg s"\ ps 1650kg m , m * = 100 and
assuming a pressure range of 0 - 500kPa g
A n intuitive understanding of the physical interactions involved would support a drop off in
solids velocity at some air velocity, particularly marked for larger particles as they settle out and
form a stationary or semi-stationary bed on the bottom of the pipe. Whilst the behaviour of this
function provides such a drop off at the lower end, it would be interesting to determine whether
the relationship continues to hold for higher solids loadings, as indicated in Figure 6.5.7. In one
sense the increase in actual air velocity shown in Figure 6.5.7 is expected at lower solids
velocities, since the reduced interstitial space being available for the passage of the air will
require a corresponding increase in velocity. The extent to which this should be accompanied by
some recovery or stabilisation in solids velocity (as is demonstrated in Figure 6.5.7) would be of
some interest. However it is unreasonable to expect the correlation of E q 6.5.9 to reflect
accurately the behaviour of the bulk solid in this region. Under these conditions the material is
more likely to be transported in a form of plug flow and a relationship for slug velocity, such as
that proposed by Muschelknautz and Krambrock (1969), M i and W y p y c h (1994) or Pan and
W y p y c h (1995), is more appropriate.
6
92
Power Based Pneumatic Conveying Model
2 -|
|
1
1
1
1
1
1
1
1
1
20
22
24
26
28
30
32
34
36
38
40
Vfa (m/s)
Figure 6.5.7 Relationship between Average "actual" air velocity and solids velocity for:
D = 1 0 0 m m , d 50 = 3000pm, mf= 0.3kg s"\ p. = 1650kg m"3, m* = 150 and
assuming a pressure range of 0 - 500kPa g
Experimental results reported by Beck (1986) for three materials (sand, -5mm limestone and
3 m m , approximately cubic, P V C chips) support a linear relationship between the air and solids
velocity. For the P V C chips, data for bends, horizontal and vertical conveying sections are
presented, with the following Equation 6.5.11 being suggested as providing a representative
relationship for that particular bulk sohd/pipeline configuration.
V. = 0.47 x Vfa - 4.5
E q 6.5.11
Applying Eqs 6.5.9 and 6.5.10 to Beck's pipeline configuration and carrying out a linear
regression on the resultant plot gives:
V. = 0.46 xF/a+2.3 Eq 6.5.12
There is surprisingly close agreement, with regard to the slope portion, between these two
Equations - though there is a marked difference in the intercept (with the Hinkle type expression,
E q 6.5.9, effectively suggesting a velocity 7 m s"1 higher across the range). Clearly there is a
difficulty with E q 6.5.12 at low air velocities, with the indication that Vs will actually exceed Vfa,
as Vfa falls below 5 m s"1.
In terms of obtaining a simple expression for solids velocity to apply in this model, this linear
relationship between the actual or interstitial air velocity and solids velocity, is attractive. It
would be even more convenient, as will be seen in a later discussion of the iterative procedures
used in the determination of an equation for the solids friction factor, if this expression could be
further simplified to a directly proportional relation, ie the elimination of the constant intercept
term, as shown in Equation 6.5.14, as compared to Equation 6.5.13.
6 Power Based Pneumatic Conveying Model
Vs = VSR'xVfa + const.
Eq 6.5.13
V. = VSR x V
E q 6.5.14
fa
93
Amongst the limited range of experimental results reported in the literature on this topic, the
study by Midttveit (1988) into the accuracy of solids velocity measurement in pneumatic
conveying using cross correlation of capacitance signals, gives some insight. Results for a small
number of velocity measurements for an alumina powder being conveyed in horizontal and
vertical pipe sections, were reported. Figure 6.5.8 presents these results in terms of m* and VSR.
In this case it would appear that the assumption of a fixed value for VSR would not result in
major errors. It m a y be noted however, that the data suggests a trend for the VSR term to
increase with increasing solids loading.
1.00 -r0.90 --
-Horizontal Section
-Vertical Section
0.80 -0.70 -0.60 •8 0.50 -•
0.40 -0.30 -0.20 -0.10 -•
0.00 --
10
Figure 6.5.8
20
30
40
m*
50
60
70
80
Relationship between VSR and m* Midttveit (1988) Results for Alumina
Powder
If Eq 6.5.14 is assumed to apply, then Eq 6.5.7 may be used to determine Vfa,
f
mf
m
V
'--A [VSR. p.
\
pf)
E q 6.5.7
and Vs found from E q 6.5.14. If an equation of the form used by Beck (Eq 6.5.13) is substituted
into E q 6.5.6, the quadratic shown in Equation 6.5.15 is obtained.
Vfa2(A.rs. r,VSR') + V^Ar* r, VSR') - mf. rs.const = 0 .... Eq6.5.15
or
A'.Vfa2(L)+B'.Vfa + C'=0
E q 6.5.16
where: A' = (A.r, r,VSR'); B' = (A.rr r,VSR'); C' = - mf. rs.const
Values for Vfa are found via the standard quadratic roots Equation 6.5.17, with the positive root
providing the meaningful physical solution.
Vf.
fa
~
-B'±yl&2-AxA'xC
2A
Eq 6.5.17
Values for Vs can then be obtained by the application of E q 6.5.13
6
94
Power Based Pneumatic Conveying Model
T o assess the feasibility of using a simple proportional relationship between Vs and Vfa (Eq
6.5.14), in contrast to the straight line relationship (Eq 6.5.13), Figures 6.5.9 to 6.5.10 have been
produced comparing air and solids velocities obtained by the two methods.
35.00
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Air Mass Flow Rate (kg/s)
-Vfa (Beck)
Figure 6.5.9
-Vfa (prop.)
A
Vs (Beck) —X—Vs(prop.) — X — Vf (superf.;
Relationship between air velocities and solids velocities, based on the linear
"Beck" equation and the simple proportional relationship. Pressure and
m * held constant, air mass flow rate varied.
for: VSR' = 0.47,const = -4.5, VSR = 0.3,D = 5 0 m m , p s = 1 6 5 0 k g m 3,
m* = 50 and pipeline pressure = 250kPa g
6
Power Based Pneumatic Conveying Model
95
50.00
45.00 -40.00 -35.00
1
30.00 -25.00 --
u
o 20.00
I 15.00 10.00
5.00 0.00
50
100
150
200
+
+
250
300
350
400
450
Pipeline Pressure (kPa)
-Vfa (Beck)
-Vfa (prop.]
-£—Vs(Beck) — X — Vs(prop.) — a ( — V f (superf.)
Figure 6.5.10 Relationship between air velocities a n d solids velocities, based o n the linear
" B e c k " equation a n d the simple proportional relationship. Air m a s s flow
rate a n d m * held constant, pipeline pressure varied.
for: VSR' = 0.47, const = -4.5, VSR = 0.3, D = 5 0 m m , ps = 1650kg m' 3 ,
m* - 50 and W/= 0.1kg s"1
As noted in the captions for Figures 6.5.9 and 6.5.10, the values for the slope and constant in Eq
6.5.13 are those suggested b y Beck for the 3 m m P V C chip materials. It can be seen that it is
possible to produce quite similar relationships using the simple proportional equation (Eq
6.5.14), in this case with VSR = 0.3 being used.
In the case of the material analysed by Beck, it would seem reasonable that the value of the
intercept (-4.5) is related to the particle size and particle terminal velocity. A n extrapolation of
VSR
the equation basically suggests that at any air velocity below
,
, or 9.6m s in Beck's
-const.
example, the solid particles would have zero velocity. Whilst the equation m a y not be expected
to accurately m o d e l behaviour at that limit, nevertheless the underlying trend does reflect that
relationship. Consequently it is reasonable to expect that a finer powder would have a smaller
intercept value which results in closer agreement between relationships produced b y E q s 6.5.13
and 6.5.14. Figures 6.5.11 and 6.5.12 demonstrate the potential for a close correlation between
results predicted b y these t w o relationships.
6
96
Power Based Pneumatic Conveying Model
35.00
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Air Mass Flow Rate (kg/s)
-Vfa (Beck)
•Vfa (prop.)
•Vs (Beck) — X — V s (prop.) — X — V f (superf.)
Figure 6.5.11 Relationship between air velocities and solids velocities, based on the linear
"Beck" equation and the simple proportional relationship. Pressure and
m * held constant, air mass flow rate varied.
for: VSR' = 0.5, const = -1.5, VSR = 0.44, D = 5 0 m m , p. = 1650kg m"3,
m* = 50 and pipeline pressure = 250kPa g
6 Power Based Pneumatic Conveying Model
97
50.00
45.00
40.00
35.00 4-§ 30.00 -h 25.00
'3
£! 20.00 -r
15.00 -10.00 +
5.00
0.00
+
50
100
150
200
250
300
350
400
450
Pipeline Pressure (kPa)
-Vfa (Beck)
-Vfa (prop.) — £ — V s ( B e c k )
—X—Vs(prop.)
—X—Vf(superf.;
Figure 6.5.12 Relationship between air velocities and solids velocities, based on the linear
"Beck" equation and the simple proportional relationship. Air mass flow
rate and m * held constant, pipeline pressure varied.
for: VSR' = 0.5, const = -1,5, VSR = 0.44,D = 5 0 m m , ps - 1650kg m"3,
1
m * == 50 and m / = 0.1kg s"
In consideration of the similarity of these relationships, as shown in Figures 6.5.9 to 6.5.12, it is
concluded for the purposes of this model development it is satisfactory to assume the simpler
proportional relationship between the average "actual" air velocity and the average solids
defined in E q 6.5.14. This simplifies calculations related to the determination of the mass of
solids in the pipeline, and the iterative correlation analysis of experimental data which allow a
"bestfit"value for VSR for a given material to be found. It is acknowledged however, that in
further developing and refining this model this is a key area that should have additional attention.
6.5.2 Mass of solids in pipeline
A s indicated in E q 6.5.1, the power associated with the transport of the solids component in this
model is considered to be directly proportional to both the solids velocity and the mass of solids
in the pipeline. In deterrnining the mass of solids in the pipeline, the simphfying assumption
developed in the previous Section, ie that the solids velocity at any point in the pipeline is
directly proportional to the average "actual" air velocity at that point, (Vs= VSR . Vfa), will be
applied.
The incremental mass of solids occupying an incremental length (dx) of the pipeline may be
determined from:
dMs - Asx psxdx
Eq6.5.18
where:
S = T/OD T/ E(l 6-519
A
PsxVSRxVfa
6 Power Based Pneumatic Conveying Model
98
which may also be expressed as:
m*xAx pf
A, m*xpf + psx VSR
Eq 6.5.20
Therefore:
m. •dx
VSRxVfa
and noting from equation 6.5.7 that:
(
>
1 m
mf
+ •
VSR.ps
pf)
Eq 6.5.21
dM =
Eq 6.5.7
f
and replacing p>f with ——, dMs may be written as:
RI m x A
dx
Eq 6.5.22
dMs =
RxT ms
VSRxmfx
+ ~
P
Ps
To determine the mass of solid in a pipeline of length "L", it is necessary to integrate Eq 6.5.22
over the range of x from 0 to "L". Providing that the pneumatic conveying system is operating
under steady-state conditions, and is basically an isothermal expansion process, the only
parameter in Eq 6.5.22 varying with x is the pressure. If it is further assumed that the pipeline
pressure varies linearly with x, ie
P = Pi+j(pe-Pi) Eq 6.5.23
Eq 6.5.22 becomes:
msx A
dM s =
dx
Eq 6.5.24
VSRxmfxRxT
m
__
/
,
s
PS
Pi+j{Pe-Pi)
therefore, by collecting constant terms with respect to this integration:
t
A"
M, = J
^n
dx
°- + E"
M.
_1f
C"
0 B"+C E"+D" E" x
or
M
f
C"
,
Fu+G"x
f D"x
F'+Gux
t
D"x
dx
B"+C" E"+D" E" x
6
Power Based Pneumatic Conveying Model
99
where:
A"=m„ xA
B"=VSRxmfxRx T
C"= Pt
DM- [Pe-p)
L
E"=«^_
F"=VSRxmfxRxT+
msxPi
G"= msxp.
L*PS
Evaluation of these integrals gives:
A'xL (A"xC" A"xFu} (F"+G"xL^
t«v> _..„
17ll2 In
M,= E"
„,. + KETxE"
„,. _..- D"xE
_,.
or
r
LA+
K= Ps
E q 6.5.25
F"
[PM] Pi [vSRpsmfRT + msPl) Id 1 i
\Pe-Pi)
mr
m
s(Pe~p)
(vSRpsmfRT+msPi)_
Eq 6.5.26
Eq 6.5.26 provides an estimate for the mass of solids in a pipeline of constant diameter for a
given conveying arrangement (pipeline length, air and solids mass flow rates, pressure drop,
particle density etc). A simpler approximation for this quantity m a y be obtained by deterrnining
values for the air and solids velocity at the average pressure in the pipe section, and calculating a
value for the total mass based on the assumption that these values apply throughout the pipeline,
ie:
Ms = LxAsmxps
E q 6.5.27
where
Axm*xpf.
A^^—r
— ^
E q 6.5.28
4
~ m*xPfin+VSRxps
(NB: this result m a y be obtained by the substitution of the expression for Vfa given in E q 6.5.7
into E q 6.5.19, and A ^ is the cross-sectional area at the mean pressure conditions).
Consequently, an approximation for the mass of solids in the pipeline may be determined from:
Lx p x Axm*xpAu
m*xPfin+VSRxps
There is some attraction in using Eq 6.5.29 in place of Eq 6.5.26 in a model where numerous
iterative computations are involved. T o assess the differences in the results produced by these
two Equations, the following Figures (6.5.13 to 6.5.19) have been produced.
100
6
500
Power Based Pneumatic Conveying Model
1000
1500
2000
2500
Pipeline Length (m)
-Integration Method
•Averaging Method
Figure 6.5.13 Mass of solids in pipeline versus pipe length - comparison between the
results obtained by the integration and averaging methods.
for: D = 1 0 0 m m , ps = 1650kg m"3, Ap, = 250kPa, mf= 0.1kg s"1,
ms = 5.0kg s'1 (m* = 50) and VSR = 0.5
Figure 6.5.13 shows a linear relationship between mass of solids in the pipeline, and the pipeline
length for both methods of calculation. A comparison between the two methods shows that the
percentage difference remains constant across the range in pipe lengths, with the averaging
method giving a value 1.26% higher for the example presented. A n increase in pressure drop or
solids loading results in an increase in this difference (at Ap, - 350kPa, percentage difference =
1.92%; at m* = lOOkPa, percentage difference = 1.97%).
6
Power Based Pneumatic Conveying Model
101
1200
50
100
150
200
250
300
Solids to Air Mass Flow Ratio (m*)
-Integration Method
•Averaging Method
Figure 6.5.14 M a s s of solids in pipeline versus solids to air mass flow ratio - comparison
between the results obtained by the integration and averaging methods.
for: D = 1 0 0 m m , L = 200 m , ps = 1650kg m"3, Ap, ~ 250kPa,
m / = 0.05kg s 1 , VSR = 0.7
Figures 6.5.14 and 6.5.15 examine the difference in mass of solids calculated by these two
methods as a function of the solids to air mass flow ratio. Constant pipeline pressure drop
(250kPa), air mass flow rate (0.05kg s"1), pipeline length (200m) and solids to air velocity ratio
(VSR = 0.7) have been assumed for the purposes of developing these twofigures.The impact of
variations in these parameters is studied in subsequent figures.
As can be seen in Figure 6.5.14, the value determined by the averaging method shows a gradual
increase over the integration method with increasing m*. This difference plateaus at around
2.7%, as m* exceeds 400 (see Figure 6.5.15), though values in this range and beyond have tittle
practical relevance.
6
102
Power Based Pneumatic Conveying Model
3.00
100
200
300
400
500
600
Solids to Air Mass Flow Ratio (m*)
Figure 6.5.15 Difference between the mass of solids in the pipeline versus Solids to air
mass flow ratio - comparison between the results obtained by the
integration and averaging methods.
for: D = 1 0 0 m m , L = 200m, ps = 1650kg m"3, Ap, = 250kPa, mf= 0.05kg s"1, VSR = 0.7
In a similar manner, Figures 6.5.16 and 6.5.17 explore the difference in mass of solids calculated
by the two methods as a function of varying pressure drop d o w n the pipeline. Again the
averaging method consistently predicts a higher mass of solids in the pipeline, at around 500kPa
this has reached approximately 3.7%.
6
103
Power Based Pneumatic Conveying Model
1800
100
200
300
400
600
500
Pipeline pressure Drop (kPa)
-Integration Method
-Averaging Method
Figure 6.5.16 M a s s of solids in pipeline versus pipeline pressure drop - comparison
between the results obtained by the integration and averaging methods.
for: D = 1 0 0 m m , L = 500m, ps = 1650kg m*3, mf= 0.1kg s \ ms = 5.0kg s"1, VSR = 0.7
9.00 •d
8.00 -
(%)
ation
3
7.00 -
& <a
a
&a -3
6.00 5.00 -
5
DjD
I#
4.00 -
2 Si
S -«
3.00 -
#
O
2.00 -
U u.
1.00 0.00 - *
1
1
1
1
1
1
1
500
1000
1500
2000
2500
3000
3500
4000
Pipeline Pressure Drop 0*Pa)
Figure 6.5.17
Difference between the mass of solids in the pipeline versus pipeline
pressure drop - comparison between the results obtained by the integration
and averaging methods.
for: D = 1 0 0 m m , L = 500m, ps = 1650kg m"3, mf= 0.1kg s"1, ms = 5.0kg s"1, VSR = 0.7
104
6
Power Based Pneumatic Conveying Model
A s the pressure drop is increased further, the percentage difference again reaches a plateau
region. T o demonstrate this plateau, the pressure drop considered in Figure 6.5.17 is extended
well beyond regions of practical interest, and it can be seen that the percentage difference
approaches 1 0 % at the upper range. For practical regions of conveying however, this difference
is limited to around 5%.
Figures 6.5.18 and 6.5.19 show the variation in mass of solids in the pipeline as a function of the
solids to air velocity ratio (VSR). The range of interest, naturally, is from 0 to 1, and as can be
seen in Figure 6.5.18 the two approaches produce similar resultsrightacross this range. Figure
6.5.19 in fact reveals that there is a peak in the percentage difference between these two methods
occurring at VSR * 0.15, where the difference is 2.75%. This subsequently reduces to around
1.26% at VSR =1.
0.2
0.4
0.6
0.8
Solids to Air Velocity Ratio (VSR)
•Integration Method
-Averaging Method
Figure 6.5.18 M a s s of solids in pipeline versus solids to air velocity ratio - comparison
between the results obtained by the integration and averaging methods.
for: D = 1 0 0 m m , L = 500m, ps = 1650kg m"3, mf= 0.1kg s"\ ms = 5.0kg s"1, Ap, = 250kPa.
6 Power Based Pneumatic Conveying Model
105
3.00
0.2
0.4
0.6
0.8
Solids to Air Velocity Ratio (VSR)
Figure 6.5.19 Difference between the mass of solids in the pipeline versus Solids to air
mass flow ratio - comparison between the results obtained by the
integration and averaging methods.
for: D = 1 0 0 m m , L = 500m, p. = 1650kg m" 3 , m f = 0.1kg s"\ m s = 5.0kg s"\ Ap, = 250kPa.
Overall it can be concluded that the difference between the two methods is relatively minor,
particularly in the regions of interest for practical pneumatic conveying, and consequently it is
justifiable, at least in the initial development and evaluation phases of this model, to proceed on
the basis of using the mass of solids as estimated by the averaging method.
6.5.3 Power Requirements - Solids Transportation
The foregoing analysis prepares the w a y for the determination of power requirements for the
transport of the bulk solid. A s indicated in E q 6.2.1 there are three power consumption
components associated with the solids:
1. the power required for the change in kinetic energy of the solids
mr
E q 6.5.30
(Vsl-v2)
power^
2. where the solids velocity at the inlet and exit ends of the section being considered are
determined by the methods outlined previously in this Section
3. the power required for the change in potential energy of the solids
powerPE = m s xgx(change in height)
E q 6.5.31
4. the power lost in the sliding friction interactions between the bulk solid and the pipe wall
The following discussion, and the Equations presented (Eqs 6.5.32, 6.5.33 and 6.5.34) relate to
the third of these components, and more particularly friction encountered in the straight line
parts of the pipeline. The important contribution or impact of bends in the pneumatic conveying
line is considered separately in the next Section 6.6.
The development of an expression for the power requirement for the transportation of the solids
component m a y follow a similar path to that outlined for the mass of solids in the pipeline, ie
setting out the power requirement for an incremental mass:
dPos = XsxgxVsxdMs
E q 6.5.32
then integrating this expression along the pipeline. The solids friction factor (Xs) is usually
determined from experimental data, with a correlation analysis resulting in a combination of
dimensionless parameters that best indicate this aspect of the pneumatic conveying behaviour.
The dimensionless terms used typically include parameters such as air velocity (in say Froude or
6 Power Based Pneumatic Conveying Model
106
Reynolds number terms), and would in fact vary down the pipeline. It would be ideal therefore,
to consider Xs and Vs as variables within the integration, and evaluate the expression accordingly.
For example, if the solids friction factor which is assumed to be a function of the Froude number
and m* are making use of Eq 6.5.22, the power requirement could be determined from the
evaluation of:
V
Pot = const}
fa
Jgd.
(m*)1
msxAx Vfa
dx
RxT ms
mf x
P— + —
Ps;
Eq 6.5.33
where the powers "a" and "6" are constants determined from a correlation analysis. A particular
difficulty with this approach is that the constants "a" and "6" are unknown, hence a general
solution would need to be obtained that subsequently allows a correlation analysis to determine
the best values for "a" and "6".
The standard approach (and the one used in this model) however, is to determine values for the
dimensionless parameters used in the solidsfrictionfactor correlation at some specific condition
(typically mean pipeline conditions). This simplifies the analysis andtiesin with the approach
developed in the previous Section to estimate the mass of solids in the pipeline from mean
conditions. A reasonable extension of this would include the use of the mean solids velocity in
place of Vs in Eq 6.5.32, resulting in:
Pos=AsxgxVsmxMs
Eq 6.5.34
A minor adjustment made to this Equation in the power based model is to reduce the total mass
of solids by the amount estimated to be in the bends (as discussed in the following Section), with
this component being treated separately. The magnitude of this reduction is likely to be
relatively small, and thus is not expected to have a significant effect on the performance of the
model. Nevertheless the adjustment is included, primarily on the basis of consistency.
6
Power Based Pneumatic Conveying
Model
107
6.6
Bends
O n e of the well recognised features of pneumatic conveying is the wear experienced by various
components of the system. Coupled with this is the corresponding breakage or attrition of the
bulk solid constituents during transport. It is also well recognised, as stated by Burnett (1992) in
his review of wear in pneumatic conveying pipelines, that pipeline bends are the critical
component in this respect and experience the highest level of erosive wear. Consequently
monitoring the condition of the pipeline in the vicinity of bends (often there is a secondary wear
zone downstream of the bend resulting from the rebound of the particle stream), maintenance
and replacement of bends are a significant activity associated with the operation of a pneumatic
transport system. Bend configurations used range from sweeping long radius bends to short
radius bends, butt welded bends, mitred bends and various types of blinded tee bends where the
bulk solid being transported forms a buffer to reduce the erosion rate, in m u c h the same manner
as a rock box in chutes. The reduced wear rates offered by blinded tee type bends, however, are
likely to be accompanied by a greater propensity for blockages - hence bend configuration needs
to be matched to the characteristics of the bulk solid being conveyed. The literature also
contains discussion on the merits of long versus short radius bends, with conflicting
recommendations being made, reflecting the variations in bulk solid characteristics. Marcus et
al (1990, pp 116-120) report the work of several investigators indicating that lower pressure
drops were obtained for short radius bend, however this trend was reversed for larger particles.
It can reasonably be concluded, that bends have a significant influence on the overall
performance of pneumatic conveying systems and that specific treatment of these components is
warranted in pneumatic transport models.
Obviously the primary function of the bend is to change the direction of the particle stream, a
the consequent concentration of particle impacts in this region results in the high level of
pipeline erosion and particle breakage referred to above. This is also a zone of relatively high
energy consumption and a number of authors have commented on this aspect, including the
predominantly qualitative considerations by Brauer (1979) and the proposal of dimensionless
erosion energy parameters by Soo (1977). In addition to particle properties (hardness and shape
in particular), pipe wear (and hence energy consumption) has been found to be highly dependent
on particle velocity. Models based on kinetic energy as the primary driving mechanism for the
erosion process (ref. Burnett, 1992) suggest that wear is proportional to the particle velocity
squared. Experimental data suggests that wear is proportional to higher powers of particle
velocity, with Burnett in his review noting work by Mills and M a s o n (1979) on a pilot plant
incorporating mild steel bends and studying penetrative erosion indicating powers 3.5 to 4.5.
This increased rate of energy consumption is also accompanied by a localised rapid increase in
the rate of pressure drop, as for example indicated in results presented by Reed and Bradley
(1991). Consequently the treatment of bends in pneumatic conveying models generally focuses
on the inclusion of terms to provide specifically for the pressure drop associated with the bend,
as it is apparent that it would be inappropriate to assume that these effects can be absorbed in the
correlations based on the straight pipe equations. O n e approach that has been adopted,
borrowing from single phase flow procedures, is to nominate a representative equivalent length
of straight pipe to use in place of the bend in the model. Govier and Aziz (1972) for example,
present a table of equivalent length in pipe diameters for various pipefittings(90° long radius
bend rate 20 diameters, for example) in single phase flow, but note in relation to two phase flow
that precise procedures for deterrnining appropriate equivalent lengths are not available.
Experimental work at Thames Polytechnic reported by Reed at al (1986) produced a bulk solid
specific correlation giving equivalent lengths for bends over a range of conveying line inlet
velocities. This would certainly be a more reliable system design aid, though requiring detailed
experimental test work for each material. A n alternative procedure presented by Reed and
Bradley (1991) suggests an equation to be used to estimate the pressure loss that m a y be
attributed to bends. The equation is of the form:
6
108
Ap = OSKp^
Power Based Pneumatic Conveying
Vi
where p^p
Model
is a "suspension" density for the bulk solid, and
experimental data are used to determine a relationship for the coefficient "K" for each bulk
solid/bend geometry combination required.
It is also noted in the literature that the pressure drop that may be attributed to bends is not
limited to the physical location of the bend and that in fact a substantial proportion of the
pressure drop will actually be experienced in a downstream re-acceleration zone as the air stream
re-establishes the steady state velocity of the particles. The length of this zone is open to some
conjecture, with Argarwal et al (1985) commenting on the work of several researchers reporting
values ranging from 8 to 188 pipe diameters.
In setting out the approach taken in this model, it should be first stated that acceleration zones
are not considered as a separate component. That is not to say that work involved in changing
the velocity of the bulk solid is ignored. A s far as this model in general is concerned, the overall
velocity increase of the conveyed materials is taken into account by the change in kinetic energy
in the system. In the analysis of bends, the energy lost by the bulk solid in traversing the bend
(through frictional interactions with the pipe wall) is estimated. This is directly related to a
velocity loss, which is subsequently m a d e up in the downstream re-acceleration zone. In this
way the pressure loss associated with the energy consumption at the bend m a y be considered to
extend beyond the immediate bend location to the downstream re-acceleration zone. In terms of
a power based analysis, however, it is not necessary to propose a second energy consumption
component for the re-acceleration zone, as the energy requirement for this has already been
accounted for in thefrictionallosses around the bend and the overall kinetic energy change in the
system. In effect, this analysis includes the re-acceleration zone, in that it assumes that the solids
velocity has returned to its "steady state" value after the bend.
The magnitude of the pressure drop can be readily estimated once a power loss value has been
determined, as will be demonstrated in following Sections.
An underlying aim in this model is to develop terms for the various energy consuming
components in pneumatic conveying, leaving a m i n i m u m number of parameters as unknowns.
This approach has resulted in the solids to air velocity ratio (VSR), and the solids friction factor
(Xs) being the unknown parameters. B y iterating through a range of VSR values, and at each
value carrying out a correlation analysis to determine an expression for Xs> it is possible to select
an optimum VSR value and Xs combination for a given material. In reaching this position, a
number of simplifying assumptions must be made. O f relevance to the treatment of bends is the
assumption that the basic relationship between the normal forces (involved in the bulk solid and
pipe wall collisions) and the consequent frictional interactions and energy losses (erosion,
particle breakage) are the same for bends as for straight sections of pipe. Therefore the same
friction parameter (ie Xs as determined by the friction factor correlation) is applied to bends as is
assumed for the straight sections.
6.6.1 Bends - Power Consumption
The basic approach used here is to assume that thefrictionalenergy losses associated with the
transport of the bulk solid around the bend are directly related to the magnitude of the normal
force acting between the bulk solid and the pipe wall. In linear horizontal conveying this force
originates from gravitational attraction. In transportation around bends, the normal force is
predominantly a result of the centripetal acceleration generated as the material changes direction.
For example a bulk solid moving at a velocity 1 0 m s"Taround a 0.5m radius bend experiences an
acceleration of 2 0 0 m s~2, compared to the 9.81m s'2 resulting from gravitation attraction. The
combined acceleration (assuming the bend lies in the horizontal plane) is 200.2m s"2. Clearly it
is likely that the forces resulting from centripetal acceleration will dominate, and need to form
the basis of the analysis. The following analysis takes this approach, ie determining power
6
Power Based Pneumatic Conveying Model
109
losses based on the magnitude of the centripetal forces. In application to the model, the
additional input from gravitational force is included by vectorial addition.
6.6.2 Bends - Power Consumption - "fixed velocity"
Based on the centripetal force model outlined above, the power consumed in overcoming
frictional resistance as the bulk solid is conveyed around a bend may be determined from:
V2
P o B = B m x X s x - f - x Vs
Eq 6.6.1
Where Vs may be determined from:
»i
m
VSR xmf
*- + •
Pf
)
\Ps
at any required position in the pipeline.
Eq 6.6.2
The mass in the bend (Bm) can be found by determining thetimeit takes for a bulk solid particle
to traverse the bend (tB), and multiplying this value by the solids massflowrate.
RBx<j>B
h ~ y
Eq 6.6.3
s
where RB is the bend radius and ^ is the angle (in radians) around the bend (usually nf2).
Rn x (hn
Bm=msx-^Eq 6.6.4
s
Which allows the expression for the power loss that may be attributed to the bend to be written
as:
PoB = Xs x V2 x m s x 0 B
Eq 6.6.5
The initial attempt to incorporate this component into the power based pneumatic conveyin
model followed the methodology outlined in Eqs 6.6.1 to 6.6.5. Consequently a value for the
solids velocity is estimated at the bend, with this velocity subsequently being used tofindvalues
for the centripetal acceleration, mass in the bend and the power consumed. The solids velocity is
essentially treated as afixedvalue in these calculations. This ignores the effect that the frictional
interactions would have, resulting in a fall in solids velocity as the bend is traversed. However it
is considered that assuming afixedvelocity for the purposes of this portion of the analysis (for
example in deterrnining the centripetal acceleration) would not significantly impair the
performance of the model, given the relatively small proportion of mass involved in the bends.
In applying the model to experimental data, and in particular when applying the correlations
obtained to predict the performance of alternative pipeline configurations containing different
numbers of bends, it appears that the "fixed velocity" approach m a y be overestimating the
power consumption component that should be attributed to the bends. Consequently the options
for, and impacts of, including the fall in solids velocity around the bend, are investigated.
6.6.3 Bends - Power Consumption - "average velocity"
One possibility is to accept that there will be a drop in velocity, in other words that there is an
inlet solids velocity (Vsi) and an exit solids velocity (Vse). A n improved estimation of the impact
of the bend is likely to be obtained if the average of these two velocities is used to determine the
various parameters.
The same basic procedure outlined in Eqs 6.6.1 to 6.6.5 is followed. The solids velocity
inlet is obtained by Eq 6.6.2 (given values of pipeline pressure, air and solids mass flow rate and
VSR). Thetimefor a bulk solid particle to traverse the bend is now:
2 x RB x $B
ts=
v +V
Eq 6.6.6
' si ~ v se
and the mass of bulk solids in the bend is:
6
110
Bm=mx
m
s
Power Based Pneumatic Conveying Model
2xRBx0B
— - — — —
v +V
'si
E q 6.6.7
' se
and the expression for the power loss becomes:
xmsx</>B
P0B = A. X
E q 6.6.8
T o evaluate these Equations it is necessary to determine a value for the solids velocity as it exits
at the bend. The power required to traverse the bend (as set out in E q 6.6.8) can only c o m e from
two sources: (1) a fall in the kinetic energy of the bulk solid and (2) drag from the conveying anas it passes through the bulk solid. Certainly there will be some loss of kinetic energy from the
bulk solid during transit through the bend. Similarly there will be some air drag on the bulk
solid in the bend itself, though it is likely that a significant portion of the transfer of energy from
the air to the bulk solid to re-establish a steady state velocity, will occur in the re-acceleration
zone downstream of the bend. In terms of analysing solids velocity within the bend, the
approach that has been adopted here is to assume that all the energy required is taken from the
bulk solid kinetic energy. B y equating the fall in kinetic energy with the power consumed
around the bend it is possible to obtain an expression for the exit velocity, and hence evaluate
Eqs 6.6.6 to 6.6.8:
m.fc-^2) = ^ x ' a
' ' no
xmsx</>£
E q 6.6.9
resulting in:
( 2-
\
Xsx
E q 6.6.10
Vse = Vsi X
\2 + Xs x 0B J
Substituting the relationship from equation 6.6.10 into 6.6.8 results in:
(
PoB=AAsxmsx0BxV2x
1
Eq6.6.11
K2 + XS<PB J
Similarly E q 6.6.10 can be substituted into E q 6.6,7 to give:
w 5 xi? B x^ B x(2 + ^ x ^ )
Eq6.6.12
2xF,
T o determine the pressure drop associated with thisfrictionalpower loss (including the pressure
loss experienced in the re-acceleration zone), the power consumption can be equated to the
power provided by the isothermal expansion of the air, ie:
P<>B =KA % * xm,x^= mfRTln —
Eq6.6.13
\PeJ
which m a y be rearranged to give:
-PoR
mj-RT
&PB=Pi-Pe=PiX
1-2
E q 6.6.14
It is interesting to note that the above expressions indicate that the power loss and pressure drop
associated with bends are independent of bend radius (though not bend angle). Hence no insight
is gained on the relative performance of long and short radius bends from this approach,
suggesting this aspect m a y be more material dependent (coefficient of restitution, for example).
6
Power Based Pneumatic Conveying Model
111
6.6.4 Bends - Power Consumption - "integration approach"
The "average velocity" method discussed in Section 6.6.3 is relatively straightforward to
implement in the model, however since it approximates the various parameters involved, an
integral Equation is developed to assist in establishing the extent of these approximations.
Considering an incremental section of the bend:
V + 8V
s
s
V
Figure 6.6.1
B
Incremental section of pipeline bend
The mean velocity through this section is Vs + dVs 12 (NB SVS is a negative quantity in this case),
and thetimefor a particle to pass through this element is:
Rp x Sd>R
$tB=7
' !:
Eq 6.6.15
V.+ &v.
V
and the incremental mass occupying this element is:
RB x 8<f>B
SBm=m
1
dVs
Eq6.6.16
V.+
2 ;
Therefore the increment of power consumption required in this element is
V
r
8 PoB =msxXsx
svY
\F „ + -2^ )x8(/>
B
Eq 6.6.17
Equating this power loss with the rate of transfer of kinetic energy:
SKE s2m-(Vs+SVs)2]
SPoB=^f^[V
Eq6.6.18
St
which may be expanded to:
\2
m.*\x{v. + ^
><S^B=msx{vs+^y(-Vs)
Eq6.6.19
and simplified to:
-V.
ASS0B =
v
SV
Eq 6.6.20
s
2
As ^approaches df, and SVS approaches dVs,
-8V
SK
V„ +
-dV.
V,
Eq 6.6.21
6
112
Therefore
Power Based Pneumatic Conveying Model
\d(/)B =
~dK
V.
which upon integration yields:
Eq 6.6.22
Equation 6.6.22 can be readily solved for Vse, giving:
V„
V.
Eq 6.6.23
oK4*
A similar interpolation process to determine the power consumption due to frictional interactions
may be developed. However, remembering the assumption made in obtaining an expression for
the exit solids velocity (ie equating the fall in kinetic energy of the solids with the power lost due
tofriction),the power loss term m a y be written as:
m
,n
m,xV2
s r ,
i „, j. Eq6.6.24 \v2 -V \ = —
—
PoB=~T l si
B
se\
2
rt
.2K4B
Recalling the expression previously developed for the mass in the bend,
8 mB=ms
RBxS0B
Eq 6.6.16
vJJL
~dVs
w e can obtain:
k*V.
V^ -m. xR„xdV,
mxR
(1
5 _ '"s " "B R
and the relationship:
Bm=l
d<j)B =
A.XV;
V
se
OT
m, xRn
Kfa -l
1
Eq 6.6.25
v.
si-
Eq 6.6.26
The pressure drop calculation corresponding to this integral analysis is the same as that
previously outlined in Eq 6.6.14, ie,
f ~PQB V
,mfxRxT\
= Pi x i-e
&PB
with the value for PoB in this instance being obtained
from Eq 6.6.24
6.6.5 Bends - Power Consumption - "segment approach"
As a comparison between the average velocity approach and the integration analysis developed
in the previous section, a segment approach to bend analysis is developed. The Equations
established for the average velocity method are applied across each of the segments in turn.
Consequently the key Equations become:
2 - Xsx A$B
V„ =V„,si j x
Eq 6.6.27
\2 + A xA(f> )
%
B
where the subscript"/' indicates the inlet and exit solids velocities for they 'th segment
( 1 V"PO
t
Z>A, xmsx
j=U
Eq 6.6.28
2+
XMB)
6
Power Based Pneumatic Conveying
113
Model
and
B
m= T,msxRBx
<j>Bx(2 + A,xA<f>B)
E q 6.6.29
2xV^
j=U
The pressure drop can again be estimated from E q 6.6.14, using the power loss value determined
from E q 6.6.28. The summations required for these Equations m a y be readily carried out on a
spreadsheet.
The analyses outlined in Sections 6.6.1 to 6.6.4 are now compared graphically to evaluate the
suitability of the various methods. Results for the fixed velocity, average velocity, integration
analysis and segmented methods are given, showing the variation in results obtained across a
range of m, m*, pipeline pressures, Xs and VSR values. Ninety degree bends have been assumed
in this analysis, and were divided into 1° sections for the purposes of the segment method (this
high resolution should result in close agreement with the integration method). The normal forces
acting in the bends are based solely on the centrifugal acceleration in this analysis, as set out in
the previous Section.
6.6.5.1 Bends - comparison of approaches - mass in bend
Results have been obtained for a number of parameters related to the performance of pneumatic
conveying systems in bends, eg mass of bulk solid in the bend, solids exit velocity, power
consumption and overall pressure drop. These have been collected into separate Sections in
order to facilitate the comparison. Figures 6.6.2 to 6.6.6 show the variation of the mass of bulk
solid within the pipeline for variations in solids friction factor, air mass flow rate, pressure at
bend inlet, solids loading and solids to air velocity ratio.
1.70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Solids Friction Factor
-fixed vel.
Figure 6.6.2
-average
- £ — integration
-segments
M a s s in bend vs Xs - comparison between the results obtained by the fixed
velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s~\ ms - 5.0kg s"\
Press. = 200kPag, VSR = 0.6, RB= l m
The mass of bulk solids in the bend is basically related to the velocity of the solids compone
that region. The slower the bulk solid is moving, the higher the mass of bulk solids will be for a
given solids mass fiowrate. Consequently, as seen in Figure 6.6.2, an increasing solids friction
factor, which results in a reduced exit velocity, is associated with an increasing mass
accumulating in the bend. The only exception is the result obtained by the "fixed velocity"
6
114
Power Based Pneumatic Conveying Model
method which, not surprisingly, shows a constant mass of bulk solid in the bend regardless of
the friction factor value. Also of note is the increasing divergence of the "average velocity"
method from the remaining two methods at the higherfrictionfactor end of this Figure.
The remaining four Figures in this Section show a similar trend in the relationship between the
mass of bulk solid in the bend versus air mass flow rate, pressure, solids loading (m*) and solids
to air velocity ratio for the four analysis methods examined. In each case the "fixed velocity"
method produces significantly lower values for this parameter across the range. The trends
indicated can again generally be understood in terms of the effect on the solids velocity.
1.900
s 1.700 -•
a 1.500
u
B
•o
1.300
"o
1.100
s
mm
0.900
o
I0>
3 0.700
0.500
0.045
-i-
•+-
+
+
+
•+-
+•
-+•
0.055
0.065
0.075
0.085
0.095
0.105
0.115
0.125
Air Mass Flow Rate (kg/s)
•fixed vel.
Figure 6.6.3
•average
-integration - — X — s e g m e n t s
Mass in bend vs mf- comparison between the results obtained by the fixed
velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, ms = 5.0kg s"\ Press. = 200kPag,
VSR = 0.6, RB = lm, Xs = 0.3
6
Power Based Pneumatic Conveying Model
115
1.500 -r-
a
1.300 ••
"8 1.100 -S
a
43
0.900 -•
S
-9— 0.700 ••
©
«
0.500 ••
0.300
125
•+-
-+-
-+-
-+-
-+-
-+-
•4-
-f-
-+-
•+-
4-
•+-
•+-
150
175
200
225
250
275
300
325
350
375
400
425
450
475
Pipeline Pressure at bend inlet (kPa abs)
-fixed vel.
Figure 6.6.4
-average
-integration — ^ — s e g m e n t s
Mass in bend vs Pressure at Bend - comparison between the results
obtained by thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s"1, ms = 5.0kg s"\ VSR = 0.6,
RB= lm, ^, = 0.3
3.000
50
100
150
200
250
Solids to air mass flow ratio (m*)
-fixed vel.
Figure 6.6.5
-average
-±—integration — - ^ — s e g m e n t s
Mass in bend vs m* - comparison between the results obtained by the fixed
velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, m / = 0.1kg s"1, VSR = 0.6,
Press. = 200kPa g /? fl = lm, X* = 0.3
6
116
Power Based Pneumatic Conveying
Model
3.500
St 3.000
'mmT3
B
ii
Jm
2.500
a
T3
•mm
.tfi
.. 2.000
"o
3 1.500
mm
mm
O
<*> 1.000
ml
mm
0.500
-fixed vel.
Figure 6.6.6
-average
-£—'integration — X — s e g m e n t s
M a s s in bend vs VSR - comparison between the results obtained by the
fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s"1, ms - 5kg s"\
Press. = 200kPa g /f*= lm, Xs = 0.3
Figure 6.6.5 demonstrates a different relationship in that the increase in mass in the bend is
related to the higher solids loading, rather than to a reduced solids velocity through the bend. In
fact increasing m* (while holding other parameters constant) results in a higher velocity as
indicated in Figure 6.5.1, but not to a sufficient extent to counterbalance the increase in solids
loading.
One conclusion that may be drawn from these Figures is that the "fixed velocity" method of
estimating this quantity is substantially lower in each case, whereas the other three methods are
in quite close agreement. The only exception to this similarity is seen in Figure 6.6.2, where the
mass determined by the average velocity method diverges significantly below the results
determined by the other methods at higher solids friction factors. The particularly close
agreement between the integration and segment methods is also noted.
6.6.5.2 Bends - comparison of approaches - solids exit velocity
This Section presents an additional five figures, following the format used in the previous
Section, though in this case examining the predicted solids exit velocity.
Exit velocity is strongly dependent on the solids friction factor as seen in Figure 6.6.7, with the
average, integration and segment methods being in close agreement. The results obtained from
the fixed velocity method are again distinct from the others, which is not surprising given the
basic assumption of that method. The curves shown for the fixed velocity method represent the
solids velocity at the bend inlet for the other methods.
6
Power Based Pneumatic Conveying
117
Model
10.500
9.500
s
8.500
7.500
5
6.500
o
5.500
£ 4.500
•a
T3
3.500
2.500
1.500
0.500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 0.45
0.5
0.55
0.6 0.65
0.7 0.75
0.8
0.85
Solids Friction Factor
-fixed vel.
Figure 6.6.7
-average
T|J—integration — X — s e g m e n t s
Exit Velocity vs Xs - comparison between the results obtained b y the fixed
velocity, average, integration a n d segment methods.
for: D = 5 0 m m , ps = 1650kg m" 3 , ntf- 0.1kg s"1, ms = 5.0kg s"1,
Press. = 200kPag, VSR = 0.6, RB= l m
12.500
10.500
1&
8.500
w
e
>
6.500
•ml
•a
UJ 4.500
t»
TS
mm
Xfl 2.500
0.500
0.04
0.06
0.08
0.1
0.12
Air Mass Flow Rate (kg/s)
-fixed vel.
Figure 6.6.8
-average
-±—integration
-segments
Exit Velocity vs ntf - comparison between the results obtained b y the fixed
velocity, average, integration a n d segment methods.
for: D = 5 0 m m , ps = 1650kg m ' \ ms = 5.0kg s"1, Press. = 200kPag,
VSR = 0.6, RB= l m , X, = 0.3
6
118
Power Based Pneumatic Conveying Model
20.500
18.500
3" 16.500 ••
s
r; 14.500 ••
'S 12.500 -t
©
£ 10.500 •
'^ 8.500 ••
w
»
o
6.500 •+•
4.500
2.500 ••
0.500
150
200
•+•
•+•
-f-
250
300
350
400
450
Pipeline Pressure at bend inlet (kPa abs)
-fixed vel.
Figure 6.6.9
-integration — X — s e g m e n t s
-average
Exit Velocity vs Pressure at Bend - comparison between the results
obtained by thefixedvelocity, average, integration and segment methods.
for:D = 5 0 m m , p s = 1650kgm" 3 ,m f = 0.1kg s"\ ms = 5.0kgs\ VSR = 0.6,
RB= lm, Xs = 0.3
16.500
14.500 ••
f
12.500 ••
10.500 -•
O
8.500 •m>
6.500 ••
mm
<fl
TS
•©
Xfl
4.500 ••
2.500 ••
0.500
100
125
150
175
200
225
Solids Loading (m*)
-fixed vel.
- average
— £ — integration
-segments
Figure 6.6.10 Exit Velocity vs m * - comparison between the results obtained by the fixed
velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s"\ VSR = 0.6,
Press. = 200kPag, RB= lm, X, = 0.3
6
Power Based Pneumatic Conveying
Model
119
14.500
12.500 -•
-55
s 10.500 -.Ml
mi
o
>
•mm
SB
"©
1/5
8.500 ••
6.500 -4.500 ••
2.500 ••
0.500
0.05
0.1
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Solids to Air Velocity Ratio (VSR)
-fixed vel.
Figure 6.6.11
-average
-integration —^-—segments
Exit Velocity vs VSR - comparison between the results obtained by the
fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m" 3 , ntf= 0.1kg s"1, ms = 5kg s~\
Press. = 200kPa g ) J R j B = l m , Xs = 0.3
Of particular note in reviewing this set of Figures is the remarkably similar results obtained for
the average, integration and segment methods. Since solids velocity is a major factor in the
performance of bends, this suggests that there is likely to be scope for simplifying calculations
w h e n taking into account the effect of bends (for example by utilising the Equations developed
for the average velocity method).
6.6.5.3 Bends - comparison of approaches - power consumption
This Section presents an additional five Figures, following the format used in the previous
Section, though in this case examining the power consumption indicated by each of the analysis
methods. Power consumption is the key parameter for this power based pneumatic conveying
model, consequently the relative performance of these four methods is of s o m e relevance.
6
120
Power Based Pneumatic Conveying Model
700.500
600.500 ••
500.500 •o
'£400.500 -•
E
gs 300.500 -•
5
200.500 ••
I
*
100.500 +
0.500
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
Solids Friction Factor
-fixed vel.
- integration
-average
-segments
Figure 6.6.12 Power Consumption vs X.s•- comparison between the results obtained by
the fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, m,= 0.1kg s"1, ms — 5.0kg s"1,
Press. = 200kPag, VSR = 0.6, RB= l m
400.500
350.500 -•
300.500 - ••g 250.500
E
9 200.500 +
B
rj 150.500 ••
£ 100.500 -•
©
OH
50.500 +
0.500
0.04
0.06
0.08
0.1
0.12
Air M a s s F l o w R a t e (kg/s)
-fixed vel.
-average
-j_—
integration
-segments |
Figure 6.6.13 Power Consumption vs ntf - comparison between the results obtained by
the fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, ms = 5.0kg s~\ Press. = 200kPag,
VSR = 0.6, RB = lm, X. = 0.3
6
121
Power Based Pneumatic Conveying Model
900.500
800.500
fe 7°0-500 ••
o
600.500 -•
g 500.500
s
a 400.500
o
^ 300.500 4
a
| 200.500 -•
04
100.500 +
0.500
250
300
350
400
450
Pipeline Pressure at bend inlet (kPa abs)
-fixed vel.
-integration — X — s e g m e n t s
-average
Figure 6.6.14 Power Consumption vs Pressure - comparison between the results
obtained by thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s"1, ms = 5.0kg s"1, VSR = 0.6,
RB= lm, Xs = 0.3
3000.500
C* 2500.500 -•
.2 2000.500 ••
E
1500.500 -•
a
o
U 1000.500 -•
t
I
500.500 -•
0.500
100
125
150
Solids Loading (m*)
-fixed vel.
-average
-£—integration — 5 ^ ~ ~ s e 9 m e n t e
Figure 6.6.15 Power Consumption vs m* - comparison between the results obtained by
thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, m,~ 0.1kg s"1, VSR = 0.6, Press. = 200kPa g
RB= lm, X, = 0.3
6
122
Power Based Pneumatic Conveying
Model
450.500
400.500 -/**v
tB
350.500 -•
©
300.500 -•
-*—
a 250.500 ••
s
a 200.500 •S
©
Ua 150.500 -•
100.500 -•
04
50.500 ••
0.500 -I W T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Solids to Air Velocity Ratio (VSR)
— + — f i x e d vel.
—•—average
-—A—integration — X — s e g m e n t s
Figure 6.6.16 P o w e r Consumption vs VSR - comparison between the results obtained by
the fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, ntf= 0.1kg s"1, ms — 5kg s"1,
Press. = 200kPag,#/;= l m , Xs = 0.3
The general conclusion that can be drawn from this set of Figures is that, as foreshadowed in the
previous comments on solids exit velocity, the average, integration and segment methods show
remarkably similar results across the range for all the variable parameters considered. This
indicates that it should be satisfactory to select the simplest expression w h e n incorporating a
bend power consumption component into the model. Thefixedvelocity method on the other
hand produces significantly different results, reinforcing the observations m a d e earlier regarding
the inadequacy in model performance with regard to bends w h e n this approach is used.
With regard to more specific comments on power consumption variations, one aspect that may
be noted is that higher velocities lead to higher power consumption rates as shown at the lower
pressure end of Figure 6.6.14 and at the higher air mass flow rate and VSR ends of Figures
6.6.13 and 6.6.16 respectively. M a n y factors interact in pneumatic conveying however, and
some aspects of this are demonstrated in Figures 6.6.12 and 6.6.15. Increasing the solids friction
factor (Figure 6.6.12) results in higher drag forces and consequently a higher power
consumption. However it also results in reduced velocity which, whilst increasing the mass
actually in the bend at any point in time, leads to a levelling off of this parameter. Similarly in
Figure 6.6.15 power consumption is seen to increase with solids loading. This is understandable
both in terms of the increased mass of solids traversing the bend, and the increased velocities
that are actually implied, given the other parameters that are being held constant. In m a n y
installations, higher solids loadings are achieved by a reduction in air mass flow rate which is
accompanied by a reduction in velocities and a consequent reduction in power consumption. A
more complete review of the significance of bends in lean phase as against dense phase
conveying regimes requires examination of experimental data and conveying characteristics.
6.6.5.4 Bends - comparison of approaches - pressure drop
ThefiveFigures in this Section show the variation in pressure drop based on the power available
from isothermal expansion of the gas (Eq 6.6.14), and the power consumption values identified
in the previous Section. Consequently the comments m a d e in Section 6.6.5.3 could also be
applied here (and hence will not be repeated). O f interest, however, is the magnitude of the
pressure drop values obtained which are seen to be reasonable. It should be noted that these
6
123
Power Based Pneumatic Conveying Model
pressure drops include the total power consumption associated with the bend, and hence also
incorporate the downstream re-acceleration zone.
25.50
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
Solids Friction Factor
-fixed vel.
-average
-_\— integration
segments
Figure 6.6.17 Pressure Drop vs Xs - comparison between the results obtained by the fixed
velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m'3, m,= 0.1kg s"\ nts = 5.0kg s"1,
Press. = 200kPag, VSR = 0.6, RB= l m
10.50
9.50 8.50 ••
a
04
7.50 •
6.50
a
5.50 -
TS
a
4.50 --
2 3.50 Q
2.50 --
I
1.50 ••
0.50
0.04
0.06
•+•
-+-
0.08
0.1
0.12
Air Mass Flow Rate (kg/s)
-fixed vel.
-average
- integration
-segments
Figure 6.6.18 Pressure Drop vs m,- comparison between the results obtained by the
fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, ms = 5.0kg s"1, Press. = 200kPag,
VSR = 0.6, RB=lm,Xs
= 0.3
6
124
Power Based Pneumatic Conveying Model
14.50
% 12.50 ••
TS
5 10.50 - EQ
ss
Z
°i
a
TS
a. 6.50 --
2
?
4.50
s-
a
«
0.50
2.50 -
150
200
-+-
+
-+•
250
300
350
400
450
Pipeline Pressure at bend inlet (kPa abs)
-fixed vel.
-average
-jfc—integration
-segments
Figure 6.6.19 Pressure Drop vs Pressure at Bend- comparison between the results
obtained by thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m 3, m,= 0.1kg s"1, ms = 5.0kg s"\ VSR = 0.6,
RB=lm,Xs = 0.3
90.50
4, 50.50 a
"2 40.50
©
Q
30.50 +
§ 10.50 OH
50
100
125
150
Solids Loading ( m * )
-fixed vel.
•average
- A — integration
-segments
Figure 6.6.20 Pressure Drop at Bend vs m* - comparison between the results obtained by
thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, m,= 0.1kg s"1, VSR = 0.6,
Press. = 200kPag,/?i?= 1 m , Xs = 0.3
6
Power Based Pneumatic Conveying Model
125
16.50
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Solids to Air Velocity Ratio (VSR)
-fixed vel.
-average
-^—integration — ^ - — s e g m e n t s
Figure 6.6.21 Pressure Drop at Bend vs VSR - comparison between the results obtained
by thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"3, W / = 0.1kg s"1, ms = 5kg s",
Press. = 200kPagtRB- lm, X, = 0.3
6.6.6 Bends - implementation
O f the components involved in this power based pneumatic conveying model, viz change in
kinetic energy, change in potential energy, air flowfrictionpower loss, solidsfrictionpower loss
along straight pipe and solidsfrictionalpower loss around bends, only the latter two include the
solidsfrictionfactor parameter. This is an unknown term, and a specific correlation, based on
the analysis of experimental data, is required to determine an expression allowing its calculation
for each bulk solid. Consequently there is a need to extract values for the solids friction factor
from experimental data.
This may be achieved by:
1. Deterrnining the total power available (from the total pressure drop, and the power that can
be generated by the isothermal expansion of the air flow)
2. Subtracting the power "losses" associated with the change in kinetic energy of the air and
solids components, the change in potential energy and the air flow friction losses
3. Equating the result to the solids friction power loss terms (straight pipe and bend
components)
Ideally this results in an expression of the following general form:
Xs x [solids velocity x normal force in straight pipe + solids velocity x normal force in straight
pipe] = Isothermal Power - [(ms+ m/)x(AKE + A P E ) + air friction loss]
or
Isothermal P o w e r - (m3 + m f j x (AKE + A P E ) + Air Friction Loss
A =
[solids velocity x normal force in straight pipe + solids velocity x normal force in bends]
Consequently it is desirable to extract the solids friction factor term from the straight pipe and
bend solids power loss components. For the straight line component this is easily arranged as
the basic Equation has X. as a distinct coefficient, ie
Pos = AsxgxVsmxMs
Eq6.5.32
where Ms is independent of Xs, and m a y be obtained from E q 6.5.29. The situation regarding the
treatment of bends is less straight-forward however, as indicated in the following Equations:
i)
Power loss determined by the average bend velocity approach
6
126
Power Based Pneumatic Conveying Model
1
V
Eq 6.6.11
Power loss determined by the integration approach
ii)
m.xVl
PoB =
1
1
Eq 6.6.24
2X 4B
e
s
Clearly it is not possible to extract Xs as a distinct coefficient from either of these Equations, and
in fact Xs will remain as an implicit term in thefinalexpression.
6.6.6.1 Bends - implementation - bend velocity factor
To allow the analysis to proceed, it is necessary to adopt a simplifying approach of some kind.
In this case a simplification based on Eq 6.6.8 (the average bend velocity method) has been
selected.
fv.+v Y
Sl
se
PoB=Xsx\
j xmsx<f>B
Eq6.6.8
where the average velocity is replaced by the product of a "bend velocity factor" (B VF) w
solids inlet velocity:
PoB =Xsx(BVFxVsif
xmsx(f>B
Eq6.6.30
where:
l+V
V t
BVF = 0.5x
Eq 6.6.31
To apply Eq 6.6.30 it is necessary to have some understanding of both the appropriate range of
values of BVF, and its variation over different conveying conditions. Broadly, BVF can vary
between 0.5 (implying a zero solids exit velocity) and 1 (exit velocity equals the inlet solids
velocity). The followingfiveFigures 6.6.22 to 6.6.26, show the variation in more detail.
1.000 -•
0.500
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
Solids Friction Factor
-fixed vel.
-average
-integration — X — s e g m e n t s
Figure 6.6.22 Bend Velocity Factor vs Xs- comparison between the results obtained by
thefixedvelocity, average, integration and segment methods.
for: D = 50mm, ps = 1650kg m"3, mf= 0.1kg s"1, ms = 5.0kg s"1,
Press. = 200kPag, VSR = 0.6, RB= l m
6
Power Based Pneumatic Conveying Model
0.500 -I
0.04
1
0.06
1
0.08
111
1
0.1
1—
0.12
Air Mass Flow Rate (kg/s)
+
fixed vel.
B
average
£\ integration —X—segments
Figure 6.6.23 Bend Velocity Factor vs ntf - comparison between the results obtained by
the fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m ^ , ms = 5.0kg s"\ Press. = 200kPag,
VSR = 0.6, RB= lm, Xs = 0.3
0.500 -I
150
1
200
1
250
1
300
1
350
1
400
1
450
Pipeline Pressure at bend inlet (kPa abs)
+
fixed vel.
—•—average
—£—integration — X — s e g m e n t s
Figure 6.6.24 Bend Velocity Factor vs Pressure at Bend - comparison between the results
obtained by thefixedvelocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m"\ mf= 0.1kg s"1, ms = 5.0kg s"\ VSR = 0.6,
RB=lm,Xs
= 0.3
6
128
Power Based Pneumatic Conveying
Model
1.000 -•
&
0.900 ••
S
0.800 -•
u
tsmi
o
I 0.700 ••
TS
a 0.600 -•
0.500
25
50
75
100
125
150
175
200
225
Solids Loading (m*)
-fixed vel.
-average
-integration
-segments
Figure 6.6.25 B e n d Velocity Factor vs m * - comparison between the results obtained by
the fixed velocity, average, integration a n d segment methods.
for: D = 5 0 m m , ps = 1650kg m" 3 , mf= 0.1kg s"\ VSR = 0.6, Press. = 2 0 0 k P a g
RB- l m , A s = 0.3
1.000 -•
S
0.900 --
fa
S 0.800 -•
u
o
0.700 ••
5a
T3
a 0.600 --
ta
0.500
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Solids to Air Velocity Ratio (VSR)
-fixed vel.
-average
-integration
-segments |
Figure 6.6.26 B e n d Velocity Factor vs VSR - comparison between the results obtained b y
the fixed velocity, average, integration and segment methods.
for: D = 5 0 m m , ps = 1650kg m" 3 , mf= 0.1kg s"\ ms = 5kg s \
Press. = 2 0 0 k P a & RB = l m , Xs = 0.3
In reviewing these Figures, it is apparent that the bend velocity factor, as herein defined, is
independent of variations in all the major conveying parameters that have been examined here,
other than the solidsfrictionfactor as indicated in Figure 6.6.22 . Consequently the value of
BVF could be read directly from the curves presented in Figure 6.6.22, for the relevant analysis
method. O n c e again the average velocity, integration and segment methods produced very
similar results, though at higher Xs values the average velocity method begins to diverge from the
other two. Since a value of 0.3 has been assumed for Xs for Figures 6.6.23 to 6.6.26, the value
of BVF has been found to be slightly above 0.8 .
6
129
Power Based Pneumatic Conveying Model
This raises the question of what range of values of Xs is likely to be encountered with this
method. Figure 6.6.27 gives one typical example. In this case the solids friction factor is plotted
against solids loading to illustrate the variation. This Figure shows values of X. varying from 0.1
to 0.9. Values above 1 are also found for Xs, particularly under lean phase conveying conditions
where the energy efficiency of pneumatic conveying is less favourable. This trend is also
indicated in Figure 6.6.27, at the lower m* region.
1
0.9
•
0.8 4
m
©
tS 0.7 +
fa 0.6 -•
a
©
fa 0.4 •01
•••
3 03 f
7
•
0.2 +
0.1
0
• • •••
-+-
—J—
-H—
—I—
-+-
50
100
150
200
250
300
Solids to Air Mass Flow Ratio (m*)
Figure 6.6.27 Variation of Solids Friction Factor with Solids to Air Mass Flow Ratio;
Bulk Solid: Pulverised Coal, 2 Pipeline Configurations: (1) 52.5mm dia,
4 8 m long and (2) 25.4mm dia, 6 5 m long.
The introduction of the BVF parameter and Eq 6.6.31 does not completely resolve the di
of having Xs as an implicit term in the overall power balance relationship. Assumption of a
particular value for BVF, say between 0.7 and 0.8, is an improvement over the use of the fixed
velocity approach (which has been clearly indicated in the previous Sections to overestimate
significantly the bend impact). A fixed value of BVF however, would fail to follow the variation
in bend influence over the range of conveying conditions, and hence would add to the degree of
uncertainty in the resulting model. Nevertheless it is suggested that the above method, ie
selecting a fixed value for BVF, is the best initial approach. The possibility of subsequently
refining the model (and Xs correlation) remains bytakinginto account the variation of BVF with
X. in repeated iterations.
In this respect use can be made of Eq 6.6.31, together with the expression obtained for
exit velocity from the integration method (Eq 6.6.23), yielding:
(
-X <b.
BVF = 0.5x i+e
V
Eq 6.6.32
Therefore in carrying out the initial analysis of experimental data to determine a correlation for
Xs, a preliminary value may be selected for Xs, say 0.4. This expression may then be used to
determine a more appropriate value for BVF, via Eq 6.6.32, and the correlation analysis
repeated. A number of iterations of this procedure will result in a convergence of the parameters
in the Xs correlation, and an improved estimate of the power consumption attributable to the
solids transit through the bends.
130
6
Power Based Pneumatic Conveying
Model
6.6.6.2
Bends - implementation - bend location
A n additional issue that is considered with regard to the implementation of the bend component
in the power based pneumatic conveying model, is the extent to which the actual location of the
bends needs to be included in the analysis.
It would simplify the application of the model if it could be assumed that all bends may be
considered as contributing to power consumption on the basis of m e a n pipeline conditions. In
this case, the effect of a bend at that location could be determined and an appropriate multiple of
the result obtained to account for the actual number of bends. This approach is also attractive in
that the solidsfrictionfactor correlation is obtained for m e a n pipeline conditions but it is less
clear h o w appropriate it is for use elsewhere along the pipeline.
Section 6.6.5.3, and in particular Figure 6.6.14 provide some insight into this issue. If, for a
given steady state conveying operating condition, the assumption is m a d e that the pressure drop
d o w n the pipeline is linear, then the variation of the power consumption that m a y be attributed to
a bend with pipeline pressure is of importance. If this variation is linear then it is satisfactory to
consider all the bends to be operating at m e a n conditions. If not, then s o m e degree of error will
be introduced by such an assumption.
For the case presented in Figure 6.6.14, an approximately linear relationship exists at higher
pressures (say above 250kPag). Overall, it is not a linear relationship, with the non-linearity
becoming more pronounced at lower pressures (and higher velocities). The following two
Figures 6.6.28 and 6.6.29 examine this influence in further detail, based on experimental data
for a particular conveying installation. The installation had in fact four 90° bends. For the
purposes of exploring this bend location approach, and the trends that are contained within it, the
effect of bend numbers ranging from 1 to 20 (hopefully an unrealistically high number for a
practical installation of this length) has been calculated. T w o approaches for bend location have
been analysed, (i) assuming all the bends are at the "mean" pipeline condition and (ii) assuming
that the bends are equi-spaced d o w n the pipeline.
These Figures indicate that carrying out the bend analysis on the basis of all the bends being at
the m e a n condition underestimates the impact of the bends. Whilst the difference in results is
minor below around five or six bends, the divergence increases above that point, with this being
particularly evident in the pressure drop version of this relationship, Figure 6.6.29. The
difference between the two approaches is also increased at higher total pipeline pressure drops.
Both these factors indicate that lumping all the bends together at the m e a n conditions is likely to
be less satisfactory w h e n analysing long conveying systems, as these are likely to have both
higher pressure drops and larger numbers of bends. Both options have been explored in the
model implementation; generally the equi-spaced assumption is preferred.
6
131
Power Based Pneumatic Conveying Model
1200.5
a
1000.5 -
a
•o
a
S «
s
15
% a
a a
© co
U
800.5 600.5
400.5
200.5 -
a
I
8
9
10 11
12
13 14 15
16 17
18 19 20
fa
Number of Bends
•equi-spaced
-mean
Figure 6.6.28 Power Consumption vs N u m b e r of Bends - comparison between the results
obtained by assuming (a) that all the bends are at the "mean pipeline
condition, and (b) that the bends are equi-spaced d o w n the pipeline.
Solids friction factor correlation based on pulverised coal conveying data,
pipeline con fig.: D = 52.5mm, L = 48m, RB= 0.254m, w / = 0.02 kg s"\
ms = 4.15kg s"\ Press. Drop = 200kPa g , VSR= 0.7
120.5
0i
T3
100.5 -
a
a
CO
o
•M
80.5 -
a
T3 ss
60.5 - -
n
40.5 -
fa
20.5
Q
8
9
10
11
12
13
14 15
16
17
18
19 20
Number of Bends
-mean
-equi-spaced
Figure 6.6.29 Pressure Drop vs N u m b e r of Bends - comparison between the results
obtained by assuming (a) that all the bends are at the "mean pipeline
condition, and (b) that the bends are equi-spaced d o w n the pipeline.
Solids friction factor correlation based on pulverised coal conveying data,
pipeline config.: D = 52.5mm, L = 48m, RB= 0.254m, ntf= 0.02 kg s"\
ms = 4.15kg s"\ Press. Drop = 200kPa g , VSR= 0.7
The divergence at higher bend numbers indicated in Figures 6.6.28 and 6.6.29, results from
significantly higher power consumption, and corresponding higher pressure drops, associated
with bends located towards the exit (and high velocity) end of the pipeline. Figures 6.6.30 and
6.6.31 illustrate this by presenting the power loss and pressure drop associated with a single
bend as a function of its location from the inlet end of the pipeline. The "integration method"
6
132
Power Based Pneumatic Conveying Model
has been used in performing these calculations, with a pipe length of 4 0 0 m and a total pressure
drop of 350kPabeing assumed.
1200.00
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
400.00
Bend Location (metres from inlet)
Figure 6.6.30 Power Consumption per Bend vs Distance from Pipe Inlet
for: D = 5 0 m m , ps = 1650kg m"3, ntf= 0.1kg s"1, ms = 5kg s"1, L = 4 0 0 m
Total Press. Drop = 350kPa, RB= lm, Xs = 0.3
14.00
wl 12.00
%
"W
T3 10.00
a
a
mla
ii
8.00
a.
a 6.00
u
o
aa
u
4.00
01
a
w
u
fa
2.00
0.00
0.00
50.00
100.00
150.00
200.00
250.00
300.00
350.00
400.00
Bend Location (metres from pipe inlet)
Figure 6.6.31 Pressure Drop per Bend vs Distance from Pipe Inlet
for: D = 5 0 m m , ps = 1650kg m"3, mf= 0.1kg s"\ ms = 5kg s"1, L = 4 0 0 m
Total Press. Drop = 350kPa, RB = lm, X5 = 0.3
An obvious alternative to the two approaches demonstrated in Figures 6.6.28 and 6.6.29 is to
evaluate the impact of bends on the pneumatic conveying system performance based on the
actual location of the bends along the pipeline. The development of a model intended to
represent a real physical system always involves a series of compromises that seek to simplify
the complex interactions that actually take place (and hence make the calculation procedures
manageable) with minimal adverse impact on the performance of the model as a whole. The
initial approach of calculating bend effects based on mean conditions is a case in point. The
analysis suggests it is a reasonable approximation, however if the bend numbers are large and/or
there is a preponderance of bends at the exit end of the pipeline, this method is likely to
6
Power Based Pneumatic Conveying
Model
133
underestimate the impact of the bends on overall system performance. Determination of bend
effects based on actual location would be the ideal approach, where these are known, though at
the cost of some increase in complexity of calculation, and requiring additional detail for data
input.
The proximity of bends to each other is a further issue that would influence the magnitude of the
bend effect. If the downstream bend lies within the acceleration zone of the previous bend (ie
before the solids velocity has recovered) the centripetal forces and corresponding drag forces
would be reduced. This should result in a reduced power loss for that bend, though it would also
be accompanied by an increase in mass of solids in the bend, and a greater risk of blockage.
Incorporation of this aspect would require the establishment of the length of the acceleration
zone of each bend, and determination/estimation of the rate of acceleration of the solids.
The models developed in this thesis, however, have not sought to include either of these options.
Incorporation of these aspects m a y well prove to be a worthwhile future refinement, however at
this stage of model development and evaluation, this degree of simplification in the treatment of
bends w a s considered appropriate.
6.6.7 Bends - some conclusions
This Section has seen a quite extensive discussion of the influence of bends in pneumatic
conveying, and the development of some interesting and useful relationships. These
relationships allow the modelling of bends to be extended beyond an "equivalent length" or
similar simplification, and to take account to some extent of the prevailing operating conditions.
One outcome of the analysis presented is the suggestion that the major effects of the bends
(power requirement, pressure drop, solids exit velocity) are independent of bend radius. This is
no doubt a simplification of the situation, as a fairly orderly progression of the bulk solid around
the outer surface of the bend has been assumed. Impact behaviours,ricochets,coefficient of
restitution characteristics etc have not been directly considered. If these features were included in
the bend analysis, it is quite possible that bend radius would emerge as an active parameter.
To some extent the solids friction factor correlation takes these additional energy loss
interactions into account, however not in a manner that allows any suggestions to be m a d e as to
the optimum bend geometry for a particular bulk solid. O n the other hand, the analysis does
suggest that where possible it is preferable to locate bends earlier in the pipeline route rather than
later.
The analysis has highlighted the need to take into account the drop in solids velocity resulting
from the increased solidsfrictionexperienced in the bend. A number of possible methods of
determining this have been outlined and corresponding relationships developed. The magnitude
of the reduction in solids velocity around the bend is of interest w h e n considering the potential
conveying difficulties that m a y be experienced at bends. This can be particularly evident in high
density and slug flow regimes where a solids velocity reduction has the potential to increase the
solids bulk density to the extent that compaction and arched stressfieldsm a y develop across the
pipe, with consequent blockage opportunities. Lean phase conveying regimes avoid this
problem, but encounter the more prevalent high solids velocity situation, with the accompanying
high power consumption, pipeline erosion and bulk solid attrition.
In Section 6.6.1 the contribution from gravity to the frictional forces around the bend is briefl
discussed. The subsequent analysis is based solely on the centrifugal force component, though it
was noted that gravity effects m a y be included by vectorial addition to the acceleration term.
For bends lying in a horizontal plane, this results in the following bend power loss relationship, a
modification of E q 6.6.30.
6
134
Power Based Pneumatic Conveying Model
10.5
(BVFxVj
PoB = Xsx
•+gd
Rl
Eq 6.6.33
xm„xR„x
The power loss for bends oriented in a vertical plane (solids friction losses only) may be
estimated from the following Equations:
where the exit velocity is upwards:
(BVFxVstf
PoB
=msxRBx<f>BxlXsx
RB
|
+
g
Eq 6.6.34
V2 +
ii. where the exit velocity is down:
PoB=msxRBx<{>Bx<
/Lx
(BVFxvJ
g
RB ~>I2
_g_
Eq 6.6.35
V2
In the development and evaluation of this model all bends have been assumed to lie in the
horizontal plane, and Eq 6.6.33 has been used. This simplifying assumption was in recognition
of the fact that the majority of the bends in the pipeline configurations used were in the
horizontal plane and also, as indicated in Section 6.6.1, that the solids velocity component
generally dominates the gravity term in the expression. This permitted some lessening of the
complexity required in the computer models of the process.
Though this treatment has been extensive, it is warranted because bends are a very signific
element within a pneumatic conveying system. They provide the basis of the versatility in
routing that is attractive in this mode of transportation. They also provide a significant
proportion of the problems that need to be resolved in the successful operation of the system.
As is outlined in the following two Chapters, which present an evaluation of the power based
pneumatic conveying model, bends also consume a major proportion of the total power
requirement under a wide range of conveying conditions.
6.7 Vertical Sections
The treatment of vertical lift or fall in this model is quite straight forward, and has been briefly
outlined previously in Section 6.5. The change in potential energy of the bulk solid between the
beginning and the end of the pipeline can be readily determined from the difference in elevation
of these two points in the conveying system. A fall in height naturally results in a power gain,
rather than a loss, for the system.
Both the transporting medium and the bulk solid should be considered in this context, thoug
a typical application it would be expected that the solids component will dominate. The
relationship considering both these phases is:
PopE = (ms + mf) xgx (change in height)
Eq 6.7.1
The frictional interactions between the bulk solid and the pipe wall are fundamentally diff
for vertical and horizontal sections. Equation 6.7.1 does not make any allowance for these
losses, and the validity of assuming that frictional losses in vertical sections may be treated in the
same way as horizontal sections is debatable. Nevertheless that is the assumption that has been
made at this stage of the model development; in other words the length of vertical pipe is
included without differentiation from horizontal pipe in total pipeline length, for the purposes of
evaluating frictional losses. It is noted that this is an area where further refinement may be
warranted.
6
Power Based Pneumatic Conveying Model
135
6.8
Change in Kinetic Energy
The treatment of kinetic energy changes in the system is also quite straight forward. Again, both
the bulk solid and the transporting medium should be considered, with the bulk solid component
expected to dominate. Use m a y be made of the solids to air velocity ratio (VSR), and Eq 6.8.1
developed - where Vfai and Vfae represent the actual superficial ar velocities at the inlet and exit of
the pipe respectively.
r
m, x VSR2 mf
PO
E q 6.8.1
KE = -±—
+ -r- *(*£-*£)
V
Solids Friction Factor
6.9
The determination of a relationship for the solids friction factor is a key element of this and
other models. It is one of the main areas where the specific properties and handling
characteristics of the particular bulk solid m a y be taken into account. The only other opportunity
for differentiating one bulk solid from another in the model lies in the densities (particle and
bulk) of the material.
As has been noted previously, values for the solids friction factor may be determined from the
analysis of test data. For the power based model, this m a y be expressed as:
Isothermal P o w e r - [ms +mf j x ( A K E + APE) + Air Friction Loss
X =
[solids velocity x normal force in straight pipe + solids velocity x normal force in bends]
Eq 6.9.1
where estimates for the terms on therighthand side of the equation m a y be obtained by the
methods outlined in this Chapter. It should be noted that the solids friction factor (Xs) has been
extracted as a c o m m o n factor from the terms in the right hand side denominator in the
development of E q 6.9.1. This formulation indicates that the solids friction factor is specifically
related to the slidingfrictionpower consumption components. This would include behaviours
not identified in the other terms such as particle breakage and pipeline wear.
The solids friction factor is not a constant, and will be found to vary over a range of conveyi
conditions - as for example indicated in Figure 6.6.27, and further demonstrated in the following
series of Figures in this Section. Therefore it is desirable and necessary to develop an expression
that follows this variation in values to a reasonable degree of accuracy. To this end a power
relationship of relevant dimensionless parameters is usually sought, via a correlation analysis
(say least squares error rninimisation). The relationships proposed by Stegmeir, W y p y c h and
Arnold, Szikszay and Weber noted previously in this Chapter are typical in this respect.
In the development of this model, the same basic procedure has been followed. There are
several options available in selecting the set of dimensionless parameters to be used. One
approach is to assemble a smorgasbord of potential dimensionless parameters, and allow the user
to explore the suitability of various combinations of these in obtaining a correlation for Xs. That
is the initial arrangement adopted and will be referred to as conveying model PCI.
136
6
Power Based Pneumatic Conveying Model
6.9.1
Model P C I
The parameters selected in this model include a number of terms that other authors have
previously identified as being useful in solids friction factor correlations, such as m*, Fr and
Many of the traditional dimensionless parameters widely used in the analysis of fluids
may be considered as ratios of relevant forces, and that approach has been used to extend the
selection in use here. The list of parameters available in this model is set out in Table 6.9.1 .
Table 6.9.1 - Dimensionless Parameters, Model P C I
Comments
a c o m m o n l y used
dimensionless parameter
ratio of inertia force to gravity
force
No.
1
Parameter
solids to air mass flow ratio
2
Froude N u m b e r (at m e a n
pressure conditions)
3
pressure area to friction
surface area
4
Reynolds N u m b e r
5
M a c h N u m b e r (at m e a n
pressure conditions)
gas compressibility under
isothermal conditions, ratio of
inertia force to
compressibility force
6
pressure coefficient (Cp)
ratio of pressure force to
inertia force
7
density ratio (air density at
m e a n pressure conditions)
8
9
10
pressure to viscous force
ratio (mean pressure
conditions)
upstream to downstream air
density ratio
Definition
m*
6 refers to the wetted
perimeter of the pipe section
(Figure 6.4.6), a unit pipe
length is assumed - (related to
ratio of pressure force to
friction forces)
diameter basis, ratio of inertia
force to viscous force
indicative of extent of
expansion of air (acceleration
rate)
pressure force to gravity
force ratio
Ten dimensionless parameters are made available in this model. Ideally only a small number of
parameters should be used in a correlation. If this results in a goodfitto the experimental data,
then the confidence with which the resulting equation can be applied to n e w data points is
enhanced. Increasing the number of parameters in use in any correlation should give a closer fit
to the experimental data, indeed if the number of independent parameters used equals the
number of data points, a perfectfitshould be possible. Conversely, in this case the likelihood
that this correlation wouldfitany subsequent data point is correspondingly diminished.
To illustrate the relationships between the solids friction factor and the various dimensionless
parameters set out in Table 6.9.1, the results obtained for a set of test data on pulverised coal are
6
137
Power Based Pneumatic Conveying Model
presented in Figures 6.9.1 to 6.9.10. These data were collected on a pipeline configuration
composed of the components set out in Table 6.9.2. A total of 30 conveying points are
included, covering a wide range of solids loadings.
Table 6.9.2
(ill
(mm)
Length
(m)
(m)
No.
bends
52.5
48
3.5
4
Pipe diam.
250 •
of Bend radius
(m)
0.254
•
*
. 200 •
£ 150 •
a
s
fa
« 100
50 -i
1
0.5
0
1
1
Z. 1
1.5
•
-+-*
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.1: Parameter 1 versus solids friction factor, Model P C I
o*
la
fa
' 15 •
•
•
•
Ol
O
Param
|j 10 •
• •
••
•
•• • •
•
•
•
• •
•
£ 25 •
"O
a
§ 20 -
0
1
1
1—
—1
0.5
1
1.5
2
Solids Friction Factor (from experimental data)
Figure 6.9.2: Parameter 2 versus solids friction factor, Model PCI
2.5
6
138
Power Based Pneumatic Conveying Model
1
1.5
2
2.5
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.3: Parameter 3 versus solids friction factor, M o d e l P C I
o
Z01
©
e
100000
90000 -
•
80000 70000 60000 -
•
• •
50000 *
40000 • •
«
|
u
30000 20000 -
«8
fa
10000 ••
0 0
• • • •
1
0.5
1
1.5
Solids Friction Factor (from experimental data)
Figure 6.9.4: Parameter 4 versus solids friction factor, M o d e l P C I
0.07
o
J=
w
«
0.06 ••
0.05 • -
§ 0.04 - •
Ifi
m
ii
0.03
*ii
»
sas 0.02 -
u
« 0.01 -
fa
0
0.5
1
1.5
Solids Friction Factor (from experimental data)
Figure 6.9.5: Parameter 5 versus solids friction factor, M o d e l P C I
2
2.5
6
0
139
Power Based Pneumatic Conveying Model
0.5
1
1.5
2
2.5
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.6: Parameter 6 versus solids friction factor, Model P C I
0.0007
0.0006 o
0i
2 .2
0.0005 • -
^*
••*• ••
•
•Si 0.0004 •
0.0003 ••
£ 5
4> T3
0.0002 - -
E
2
0.0001 -
mm
— i —
1
0.5
1.5
Solids Friction Factor (from experimental data)
Figure 6.9.7: Parameter 7 versus solids friction factor, Model P C I
0
0.5
1
1.5
2
Solids Friction Factor (from experimental data)
Figure 6.9.8: Parameter 8 versus solids friction factor, Model P C I
140
6
Power Based Pneumatic Conveying
0.5
1
Model
1-5
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.9: Parameter 9 versus solids friction factor, M o d e l P C I
Solids Friction Factor (from experimental data)
Figure 6.9.10: Parameter 10 versus solids friction factor, Model PCI
It is apparent from these Figures that the strength of the relationship between these ten
dimensionless parameters and the experimentally derived solids friction factor are quite variable,
for this particular set of experimental data. Parameters 4, 6, 7 and 9 appear to have a relatively
poor correlation, the Reynolds N u m b e r being one of these terms. Flow at high Reynolds
Numbers over rough surfaces is held to have little correlation to the Reynolds N u m b e r (Olson,
1973 pp 451); this would be supported by the results shown in Figure 6.9.4. S o m e a m o n g the
others demonstrate promising relationships, for example parameters 1 (m*), 3 (pressure area to
friction area) and 10 (pressure force to gravity force ratio).
One difficulty with this approach is that there is a large number of combinations that could be
explored. Ten options, for example, are available for a single parameter correlation, forty five
for a two parameter correlation, two hundred and forty combinations for a three parameter
correlation, and so on. It is not a particularly encouraging prospect. Experience m a y well
suggest that a number of these terms are more reliable indicators of the variation of Xs (for
example m*), and a reasonably constant set of parameters m a y be settled on. Figure 6.9.11
shows the result of the correlation using thefirstthree dimensionless parameters.
6
m\.0
Power Based Pneumatic Conveying
141
Model
•
2-
Solids Friction
Factor - calculated
from correlation
/+
1.5 -
y}+
1-
0.5 -
0 - *-—
1
0.5
1
1
h1.5
1
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.11: Predicted versus experimental solids friction factor (correlation based on
dimensionless parameters 1,2 and 3, M o d e l P C I )
The Equation obtained in this case is
-1.0589 _ -0.0076 „ 0.007628
4 = 147.24 XTU^TI-
K-i
E q 6.9.2
where Jt\, n 2 and n 3 are the dimensionless parameters 1, 2 and 3 respectively. Relevant
information regarding thefitof E q 6.9.2 to the experimental data includes:
correlation coefficient
0.995
m e a n error
0.08%
standard deviation in error
3.91%
VSR
0.9
The value of the solids to air velocity ratio (VSR) noted above is selected so as to maximise the
correlation coefficient for the solids friction factor equation. Selecting a VSR value for a
particular bulk solid/experimental data set is part of the model optimisation process, and is
described in Section 6.9.4.
In considering Eq 6,9.2 it may be noted that the parameters K2 and /r3 apparently have little
input (a power of zero effectively removes the term from the Equation). Consequently there is
some attraction in simplifying this Equation by considering only the m* dimensionless
parameter, which clearly from Figure 6.9.1 has a good correlation with this set of data. In this
case the Equation becomes:
Xs = 6.96 x x? -5714
E q 6.9.3
and the relevant information regarding thefitof E q 6.9.3 to the experimental data includes:
correlation coefficient
0.994
m e a n error
0.11%
standard deviation in error
4.68%
VSR
0.95
This is a good fit to the data, however the magnitude of the VSR value suggested is a concern.
A s indicated previously, there is further discussion on the selection of the VSR value in a later
Section (6.9.4), however at this point it m a y be noted that the VSR value listed is selected on the
basis that it gives the best correlation coefficient for the As equation.
142
6
Power Based Pneumatic Conveying
Model
A second point that m a y be m a d e with regard to using m* alone, in m a n y instances the range of
m* values obtained, particularly in long distance applications, is quite limited. This m a y impact
adversely on the reliability of using this term alone. Figure 6.9.12 compares the solids friction
factor values obtained from E q 6.9.3, with the experimental data obtained.
Solids Friction
Factor - calculated
from correlation
0.5
1
1.5
2
Solids Friction Factor (from experimental data)
Figure 6.9.12: Predicted versus experimental solids friction factor (correlation based on
dimensionless parameter 1, M o d e l P C I )
An alternative approach that may be taken is to review the series of graphs relating A* and the ten
dimensionless terms, and selecting those that have the more obvious strong connections. In this
instance, parameters 1, 3 and 8 would merit consideration as a useful combination. A
correlation using these terms gives the following result:
1.0812 _ 0.7298 _ 0.0609
^=56.6lxff-Lmuff.
71,
E q 6.9.4
8
where n\, n^ and n% are the dimensionless parameters 1, 2 and 3 respectively, and the relevant
information regarding thefitof E q 6.9.3 to the experimental data includes:
correlation coefficient
0.996
mean error
0.07%
standard deviation in error
3.73%
VSR
0.9
This demonstrates a very good relationship, in fact the best of the three relationships examined
so far. O n the other hand it could also be noted that all of the proposed combinations
demonstrate a more than adequate correlation, and there is little between them. The low power
for the third term again suggests it m a y be contributing little to the overall correlation. For
comparison the graphical representation of this correlation is shown in Figure 6.9.13.
6
143
Power Based Pneumatic Conveying Model
2.5 --
Solids Friction
Factor - calculated 1.5
from correlation
0.5
0
0.5
1
1.5
2
2.5
3
Solids Friction Factor (from experimental data)
Figure 6.9.13: Predicted versus experimental solids friction factor (correlati
dimensionless parameters 1,3 and 8, Model PCI)
A further point may be made when considering the possible dimensionless parame
apply. Some parameters that may well be of interest (such as the particle diameter to pipe
diameter ratio used by Stegmeir, Wypych and Arnold, Weber and others) can be misleading if
used in a correlation analysis of test datafroma single pipeline system. If there is no variation
in the parameter across the range of conveying conditions, then it effectively acts as a constant in
the correlation. No relevant relationship can consequently be expected, and any power obtained
for this parameter from the correlation is unlikely to do anything other than give erroneous
values for the solidsfrictionfactor at other pipeline configurations. Preferably, the terms being
used should vary in value across the range of conveying conditions represented in the test data.
The subsequent correlation should then have the best opportunity to reflect the relationship that
the parameter has to the solids friction factor.
6.9.2
Buckingham 7t Theorem Approach
An alternative method for developing a solids friction factor correlation is to employ the
Buckinham n theorem, as for example described by Olson (1973, pp 264 - 268) to identify the
relevant dimensionless parameters.
In this approach, to obtain the most significant dimensionless groupings, the
the system that appear to be most closely related to the mechanism being studied are identified.
The dependent variable is then assumed to be a function of these parameters. For example it
may be considered that the pressure drop per unit length may be a function of the following
terms:
&Pt
IT,
~
„
mass >
— .AvnD,p.,K.pt.?f.-^i)
Eq6.9.5
where mass in this and the following four correlation models refers to the mas
in the pipeline; volume refers to the pipe volume.
In this instance there are eight terms (including the dependent variable) to be considered, with
the dimensionless parameters to be constructed from combinations of them. Using the
6
144
Power Based Pneumatic Conveying
Model
Buckingham 71 theorem, a dimensionless analysis will result in at least one set of (n - m )
independent dimensionless terms that can be assembled - where "n" is the number of terms (8)
and " m " is the number of fundamental dimensions (3). Consequently in this instance there
should be five dimensionless groupings. There are a number of guidelines suggested for setting
up the basic groupings. These include the selection of three (= m ) repeating variables, ideally
composed of a geometry term, afluidterm and a flow characteristic term, that are included in
each of the dimensionless groups. For the function set out in E q 6.9.5, D, mf and Vf m a y be
nominated as repeating variables. A dimensionless analysis results in the groupings listed in
Table 6.9.3. This set of parameters has been used to form a second version of the pneumatic
conveying model - designated here as P C 2 .
Table 6.9.3 - Dimensionless Parameters, Model PC2
TCi
It 2
Apt
pfxVfxL
DxVfx ps
pf
7C3
DxK
7t4
pf xVf
DxVfXpf
71 5
pf
DxVfx mass DxVf x Asx ps
pf x volume
pf x A
6.9.2.1
Buckingham 71 Theorem - M o d e l P C 2
A series of models have been developed using this basic approach. Model P C 2 is thefirstof
these, with the solids friction factor correlation being formed from a power function of the five
parameters set out in Table 6.9.3. O n e simphfying aspect of this method is that the correlation is
based on all the identified dimensionless groups; it is not necessary to examine all the possible
combinations. Applying model P C 2 to the same set of data previously used for model P C I ,
results in the following relationship:
.
— 1 O O A s / — 0.6480
1.692 _-0.7837
-1.2213,^ -0.6829
T,
, n <•
4 = l.U/o x/Tj
n2
;r3
nA
7t5
E q 6.9.6
The relevant information regarding thefitof E q 6.9.6 to the experimental data includes:
0.996
correlation coefficient
mean error
0.08%
standard deviation in error
3.93%
VSR
0.95
Figures 6.9.14 to 6.9.18 show the variation of each of these five parameters with the
experimentally derived friction factor.
6
145
Power Based Pneumatic Conveying Model
JUUUU '
•
* •
25000 T
•
20000
4
tU
|
15000 •
V,
2
£
10000 •
5000 0 -
1
0.2
0
1
0.4
•
*••
• •
•
••
H
1
1
0.6
0.8
1
-1
1.2
1
1.4
1
1.6
,
•.
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.14: Parameter 1 versus solids friction factor, Model P C 2
160000000
140000000 ••
120000000 100000000 ••
s
80000000 -
2
60000000
fa
40000000 20000000 ••
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Solids Friction Factor (from experimental data)
Figure 6.9.15: Parameter 2 versus solids friction factor, Model P C 2
180000000 •
* —
160000000 140000000 •
•
120000000 •-
a
100000000
m
E
80000000 •
es
60000000 •
• •
*>••
m.
«
fa
40000000 •
•
20000000
0-
0
1
0.2
1
0.4
•
H
1
1
H
1
1
H
0.6
0.8
1
1.2
1.4
1.6
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.16: Parameter 3 versus solids friction factor, Model P C 2
6
146
Power Based Pneumatic Conveying
Model
100000 •-
•
90000 ••
•
80000 70000 •
u
«
•
• •
• • •
•
60000 50000
e
40000 -
2
30000
fa
20000 •
10000 • •
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.17: Parameter 4 versus solids friction factor, M o d e l P C 2
6000000
••
5000000
in 4000000 ••
%•'
m
m>
|
3000000 -
« 2000000 fa
1000000
0
— I —
0.2
0.4
•
•
•
• •
*
•
• •
•
•
0.6
0.8
1
•
•
•
1.2
•
1.4
1.6
1.8
2
Solids Friction Factor (from experimental data)
Figure 6.9.18: Parameter 5 versus solids friction factor, M o d e l P C 2
It is interesting to note that whilst none of these figures indicates a particularly close relationsh
between the dimensionless parameters and the solids friction factor, the overall correlation
remains good. The following Figure 6.9.19 shows thefitof this Equation to the experimental
data.
6
Power Based Pneumatic Conveying
147
Model
Solids Friction
Factor - calculated
from correlation
1
0.5
2
1.5
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.19: Predicted versus experimental solids friction factor (Model P C 2 )
6.9.2.2
B u c k i n g h a m 71 T h e o r e m - M o d e l P C 3
Since the purpose of this analysis is to develop an expression for X\. a more focussed approach
m a y be to include the solids friction factor with a set of select parameters on the basis that they
m a y influence its value. A possible set of parameters is set out in Equation 6.9.7 .
/
mass ")
^ ^ / . A A . / V . ^
*«•«
A Buckinham p analysis of these parameters results in a total of three dimensionless terms, Xg
and two independent parameters. Table 6.9.4 gives the definitions of these two terms.
Table 6.9.4 - Dimensionless Parameters, Model PC3
7li
Ps
Pf
7t2
mass
As x ps
volumex pf
Ax pf
The relationship developed by the application of Model P C 3 , using the same set of experimental
data is:
1
1 < £ 0 w — 0.1008 _ -0.5929
T-. S r\ O
Xs= 3.569 X K X
7t2
E q 6.9.8
Relevant information regarding thefitof E q 6.9.8 to the experimental data includes:
0.993
correlation coefficient
m e a n error
0.15%
standard deviation in error
5.44%
VSR
0.95
Figures 6.9.20 and 6.9.21 show the variation of each of these parameters with the experimentally
derived friction factor.
6
148
0
0.2
Power Based Pneumatic Conveying
0.4
0.6
0.8
1
1.2
Model
1.4
1.6
Solids Friction Factor (from experimental data)
Figure 6.9.20: Parameter 1 versus solids friction factor, Model PC3
mM. -
•
200 •
•
••
«3 150 •
•
ii
s
2 100 •
fa
50 0-
\
,
L
1
—«
1-
1
1
H
1—
•
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1
H_ -•
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.21: Parameter 2 versus solids friction factor, Model PC3
Parameter 1 in this model is the particle to fluid density ratio which, as has been found
previously, appears to have little relevant variation across the range of solids friction factor
values. This is not surprising since the particle density exceeds the fluid density by a factor
exceeding 10 3 , and this ratio does not vary greatly across the range of pressures. The second
parameter, however, appears to have a closer relationship.
Figure 6.9.22 compares the experimentally derived and calculated solids friction factors.
6
149
Power Based Pneumatic Conveying Model
m.O -
•
./
2 -
Solids Friction
Factor - calculated
from correlation
1.5 ^S^
S+
1-
0.5 -
1
1
0.5
1
0 - <LC-
1—
1.5
1
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.22: Predicted versus experimental solids friction factor (Model P C 3 )
6.9.2.3
Buckingham 71 Theorem - Model P C 4
A further three models have been developed in the same manner, where A, forms one of the
dimensionless terms at the outset. For model PC4, the following parameters have been selected:
(
mass
E q 6.9.9
Xs=f\Vf,D,L,ps,-
LJ
A Buckinham 71 analysis of these parameters results in a total of three dimensionless terms, Xs
and two independent parameters. Table 6.9.5 gives the definitions of these two terms.
Table 6.9.5 - Dimensionless Parameters, Model PC4
Ttl
712
L
D
mass
As
2
LxD xps ~ D2
The relationship developed by the application of Model PC4, using the same set of experimental
data, is:
Xs = 1.0 x ^ r 0 3 6 0 2 ^ " 0 5 6 6 4
E q 6.9.10
Relevant information regarding thefitof E q 6.9.10 to the experimental data includes:
correlation coefficient
0.989
mean error
0.18%
standard deviation in error
6.08%
VSR
0.95
Figures 6.9.23 and 6.9.24 show the variation of each of these parameters with the experimentally
derived friction factor.
6
150
uuu -
Power Based Pneumatic Conveying
•
4*
• 4MB* » * * • • •«•
900 -
Model
800 700 •
m.
ii
mm
ii
s
600
500
400
2
300 •
«
fa
200 100 •
0•
1
1—
1
1
1
_i
0.2
0.4
0.6
0.8
1
1.2
1—
1
1
1.4
1.6
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.23: Parameter 1 versus solids friction factor, Model PC4
U.I* •
•
0.12 •
&
a
t
0.08 •
ft 0.06 •
M
M
fa 0.04
0.02 •
0 -
0
0.2
V.
0.4
0.6
0.8
1
,V
•,
- i — * -
1.2
1.6
1.8
1.4
2
Solids Friction Factor (from experimental data)
1
1
1
1
1
Figure 6.9.24: Parameter 2 versus solids friction factor, Model PC4
Parameter 1 in this model is the pipe length to diameter ratio, which is constant for any data set
that comes from a single pipeline configuration. Consequently it does not contribute any useful
data to this correlation, and naturally exhibits no variation across the range of solids friction
factor values. The power term shown against this number could well be quite misleading, and if
the model is used to predict the performance of a different diameter system it m a y give quite
erroneous results. Checking the model with data for two and four pipeline configuration systems
(ie with data sets that have two and four values of LID respectively) gives powers for the 7t\
parameter of 0.3958 and 0.2825; significantly different considering it is the opposite sign from
the corresponding term in E q 6.9.10.
This does not mean that the model has no potential - basically it requires data that includes a
range of variation in this parameter, before reliable results can be expected from the correlation
analysis.
The second parameter, however, appears to have a close relationship. The overall correlation in
this instance is reliant completely on this second parameter.
Figure 6.9.25 compares the experimentally derived and calculated solidsfrictionfactors.
6
151
Power Based Pneumatic Conveying Model
2.3 -
• y/
2 -
Solids Friction
Factor - calculated
from correlation
4ys{
* • s+
1.5 -
1 -
0.5 -
0 - mtmW
1
1
-\
1
0.5
1
1.5
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.25: Predicted versus experimental solids friction factor (Model P C 4 )
6.9.2.4
Buckingham 7C Theorem - M o d e l P C 5
For model P C 5 , the following parameters have been selected:
K =A
mass
fr, , Ff •> ^
volumesollds
, volume
Eq6.9.11
pipe J
The volumesoMs term in this case have been taken to m e a n the volume occupied by the solids in
the packed bulk density condition.
A Buckinham 71 analysis of these parameters results in a total of three dimensionless terms, Xs
and two independent parameters. Table 6.9.6 gives the definitions of these two terms.
Table 6.9.6 - Dimensionless Parameters, Model PC5
Tli
7t2
volumesoluh
Asp
volumepipe
A
mass
As x ps
1} x pf
pfx L2
The relationship developed by the application of Model P C 5, using the same set of experimental
data, is:
\ = 0.00434 x V 1 0 0 4 ^ 4 9 2 6
E q 6.9.12
Relevant information regarding thefitof E q 6.9.12 to the experimental data includes:
correlation coefficient
0.993
mean error
0.13%
standard deviation in error
5.44%
VSR
0.95
Figures 6.9.26 and 6.9.27 show the variation of each of these parameters with the experimentally
derived friction factor.
152
6
0
0.2
Power Based Pneumatic Conveying
0.4
0.6
0.8
1
1.2
Model
1.4
1.6
Solids Friction Factor (from experimental data)
Figure 6.9.26: Parameter 1 versus solids friction factor, Model PC5
0.00025
0.0002 +
V
$ 0.00015 4
s
•\
2 0.0001 +
cs
fa
0.00005 0
•*
•••
—»—
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.27: Parameter 2 versus solids friction factor, Model PC5
Both parameters appear to have a close relationship to the solids friction factor. Figure 6.9.28
compares the experimentally derived and calculated solids friction factors.
Solids Friction
Factor - calculated
from correlation
0
0.5
1
1,5
2
Solids Friction Factor (from experimental data)
Figure 6.9.28: Predicted versus experimental solids friction factor (Model PC5)
2.5
6
153
Power Based Pneumatic Conveying Model
6.9.2.5
Buckingham 7t Theorem - Model P C 6
For model PC6, the following parameters have been selected:
\=A
Vf,D,pf,-
mass
volumesolids
L
'volume pipe j
Eq6.9.13
The volumesoios term in this case has been taken to mean the volume occupied by the solids in
the packed bulk density condition.
A Buckinham % analysis of these parameters results in a total of three dimensionless terms, X*
and two independent parameters. Table 6.9.7 gives the definitions of these two terms.
Table 6.9.7 - Dimensionless Parameters, Model PC6
7Il
volumesolids
Asp
volumepipe
A
mass
712
LxD2
As x ps
pfXD2
x pf
The relationship developed by the application of Model PC6, using the same set of experimental
data, is:
Xs = 2.9359 x ^-° 0 5 3 8 ^ 2 -° 5 3 9 4
E q 6.9.14
Relevant information regarding thefitof E q 6.9.14 to the experimental data includes:
correlation coefficient
0.993
mean error
0.13%
standard deviation in error
5.21%
VSR
0.95
Figures 6.9.29 and 6.9.30 show the variation of each of these parameters with the experimentally
derived friction factor.
u.o 0.5 •
•
t 04 '
|0.3-
a °-2i
0.1 -
00
1
0.2
V
1
0.4
1
0.6
1
0.8
1
1
,
• , •
1
H
1.2
1.4
1.6
1.8
Solids Friction Factor (from experimental data)
Figure 6.9.29: Parameter 1 versus solids friction factor, Model PC6
*-
6
154
200
180
160
N 140
u 120
| 100
2
Power Based Pneumatic Conveying
Model
•
80
«
fa
60
40
20
1 1 1 10 1 1 1 1 *A
0.2
0
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Solids Friction Factor (from experimental data)
Figure 6.9.30: Parameter 2 versus solids friction factor, Model PC6
Both parameters appear to have a close relationship to the solids friction factor. Figure 6.9.31
compares the experimentally derived and calculated solids friction factors.
2.5
2
Solids Friction
Factor - calculated
from correlation
0.5
0
0
0.5
1
1.5
2
2.5
Solids Friction Factor (from experimental data)
Figure 6.9.31: Predicted versus experimental solids friction factor (Model PC6)
6.9.3 Solids Friction Factor - Some general conclusions
In reviewing the role of the solids friction factor in this pneumatic conveying model, it m a y be
seen that it is specifically linked to the two power loss components that are evaluated on the
basis of sliding solids friction (friction occurring in the bend and straight pipe sections). These
components are based on a simple coulombfrictiontype mechanism . This is a simplification of
the interactions occurring during pneumatic conveying, and does not include all the energy
absorbing interactions taking place. Particle to wall and particle to particle collisions resulting in
breakage and wear are examples of significant areas of energy loss. Whilst these phenomena are
very complex and chaotic in nature, it is reasonable to propose that the s u m of their effects is to
increase the "frictional" losses through the system. The identification of these losses with a
solids friction factor represents one means of incorporating them into the model.
Since the magnitude of this component (solids friction losses) of power consumption is
determined by subtracting the non-solids friction terms from the total input power, obtaining a
6
155
Power Based Pneumatic Conveying Model
correlation for X.fromexperimental data will result in a relationship that to some degree at least
takes the less well defined behaviours (pipe wear, particle impacts and breakage) into account.
The variation of Xs over a range of conveying conditions supports this assertion, in that
velocities (where impact mechanisms including particle breakage and pipeline wear would be
expected to increase markedly) the solids friction factor increases significantly, as illustrated in
Figure 6.9.32. At higher velocities, the friction factor typically exceeds unity, which would
require either a strongly adhesive interaction between the bulk solid and the pipe wall (a
behaviour that is not indicated by standard wall friction tests for the material), or the presence of
the other energy absorbing mechanisms noted above.
6
8
10
12
14
16
Actual Superficial Air Velocity (m/s)
18
20
Figure 6.9.32: Variation of Solids Friction Factor with Air Velocity (Pulverised Coal,
Model P C 6 )
There is some spread in values of Xs shown in Figure 6.9.32, reflecting the range in conve
conditions (two pipeline configurations are included in the data). There is, however, a four to
five fold increase in the solids friction factor value across this range of velocities. A friction
factor based purely on a slidingfrictionfactor mechanism would not be expected to increase
with velocity.
The solids friction factor is perhaps the major area that allows the model to differentiat
the behaviours of different bulk solids. Extracting a relationship for this parameter from a set of
experimental data will reflect the impact of its frictional properties - coefficient of restitution,
friability, arching tendencies and internalfrictionetc - on the pneumatic conveying performance
of the material, and the extent to which these properties are brought into play in the particular
pipeline geometry in use. Consequently the expression obtained is of considerable importance
in the ultimate application of the model. This relationship is influenced by the appropriateness
of the set of dimensionless parameters selected for use in the correlation. A number of
groupings have been suggested in the six models discussed here. These by no means exhaust the
possibilities, though it is suggested that they provide a useful beginning.
Another, and perhaps more important, influence on the relationship obtained is found in th
underlying basis of the model. Mechanisms adopted for the treatment of the air flow
component, solids flow around bends, assumptions with regard to the interstitial velocity of the
fluid, and the relationship between the particle andfluidvelocity, have a fundamental impact on
the performance of the overall model. N o amount of refinement in the selection of
156
6
Power Based Pneumatic Conveying
Model
dimensionless parameters to be used in the solids friction factor correlation will be able to
reverse serious flaws in these assumptions. Nevertheless, where the model is inadequate, the
solidsfrictionfactor correlation will inevitably (and automatically) endeavour to compensate for
it.
A significant effort has been made in the development of this model to explore the ramifications
of the assumptions made, and to expose the trends inherent in the various options. Consequently
there is some confidence in the overall structure of the model as it has been developed. O n e
outcome that tends to support this view is the general "reasonableness" of the solids friction
factor values obtained by this power based model. Leaving aside the specific pneumatic
conveying aspects of the problem, and consideringfrictionfactors that would be anticipated for
the interaction between a bulk solid and a steel plate, values in the order of 0.2 to say 0.7 m a y be
expected. General results produced by this model follow the data presented in Figure 6.9.32
(say 0.2 to 2.5 as the range of friction factor values). The lower range of values (which
correspond to lower velocity and higher solids loading conditions) is of the order that would be
expected from a slidingfrictionprocess. The higher values can readily be explained by the
introduction of additional impact, attrition and wear mechanisms, and occur at the higher
velocity regimes.
Considered from this standpoint, values generally reported in the literature for the "solids
friction factor" seem extremely low. For example, Szikszay (1988) indicating values from *
0.005 - 0.03 for the total friction factor, or W e b e r (1982) reporting * 0.0001 - 0.0035 for the
solidsfrictionfactor for a range of materials, are typical. Clearly these values are a result of the
models employed, and no doubt are justified in that context. However it could well be argued
that the magnitude of the values obtained indicate that either the parameter being reported is not
closely related to the solids friction factor, or that the overall model does not closely reflect the
physical processes taking place.
The prospect of producing a general solids friction factor correlation is of interest. In this thesis,
application of the model has been based on developing a specific correlation for each bulk solid,
which m a y then be used to determine the conveying characteristics for a nominated pipe
geometry. It is feasible to input experimental data from a range of materials and to obtain an
overall solidsfrictionfactor correlation. T o do so, however, would require an extensive review
of the dimensionless parameters used in the correlation. At present, key bulk solid
characteristics, such as static and kinematic friction coefficients, coefficient of restitution, a
friability index, hardness, size and particle shape, are ignored. This is quite suitable w h e n
obtaining a correlation for a specific bulk solid since they remain constant throughout the range
of experimental data and hence cannot usefully add to the correlation. Their overall influence,
however, is reflected in the solids friction factor values obtained and hence is an intrinsic part of
thefinalcorrelation. If a range of bulk solids is to be included in the analysis, these parameters
are no longer constant and it would be important that these terms be included in the
dimensionless parameters.
6.9.4 Solids Friction Factor/VSR optimisation
A final area to discuss with regard to the selection of the solidsfrictionfactor correlation is the
corresponding selection of the VSR value. In effect, there are two unknowns to be determined in
the analysis of a set of experimental data - the relationship between the solids and fluid velocity
(VSR), and the solids friction factor correlation. T o facilitate the analysis, these parameters m a y
be considered to be typical of the bulk solid being analysed, and hence be determined from an
analysis of a set of experimental data.
This is a simphfying assumption, and in particular with regard to the imposition of a fixed value
for VSR for a bulk solid, m a y be too sweeping an assertion. S o m e discussion on the
implications of this assumption is set out in Section 6.5.1, and the justification for taking this
6
Power Based Pneumatic Conveying
Model
157
approach established. Nevertheless, since it is necessary to apply this relationship across the
range of conveying conditions, it would appear likely that this is an area that could benefit from
further refinement.
Having made this simplifying assumption, however, it is still necessary to select the most
appropriate value for VSR. The value of VSR has a major effect on the behaviour of the model.
A low value, for example, gives a lower solids velocity. A s a consequence the total mass of
solids in the pipeline increases, the cross-sectional area occupied by the bulk solid increases, the
centripetal acceleration in the bends decreases and so on.
The procedure adopted in the application of this model has been to commence from a minimum
initial value of VSR (0.05), to evaluate the various terms (such as As and As[ etc) on the basis of
this value at each data point, and finally to carry out the solidsfrictionfactor correlation. The
effectiveness of this VSR value is then assessed by comparing the experimental and computed
values of Xs, with the correlation coefficient between these sets of values being noted. The value
of VSR is then increased by 0.05, and the analysis repeated. This procedure is repeated until the
limit of VSR = 1 is reached (values above this indicate that the solids are exceeding the actual
superficial air velocity and hence are not tenable). The VSR value corresponding to the
m a x i m u m correlation coefficient is then accepted as being the most appropriate to use for this
material.
Some interesting points become apparent in this analysis. At low VSR values it is not unusual
for the analysis of experimental data points at higher solids loadings to show that the area
occupied by the bulk solid exceeds the pipe cross-section. This indicates that the VSR value
being considered is not physically possible. Consequently, where this condition is detected for
any of the data points, the analysis for that VSR value is terminated, the VSR value is
incremented and the calculations recommenced
For the same reasons, low VSR values can result in very restricted space for the air flow
component, leading to a high pressure drop. If this exceeds the total pressure drop for the
system at the data point being analysed, there is little purpose in proceeding with the analysis at
that VSR value. Hence the calculations are terminated, and the VSR term is again incremented.
As a consequence, the optimal selection of the VSR value and the solids friction factor
correlation are carried out together, and are to a significant degree mutually dependent. At this
point no attempt has been m a d e to refine the value of VSR beyond the 0.05 step. A n iterative
search procedure could be implemented for this purpose, however it would not appear to be
warranted. Frequently there is not a great deal of difference between the correlation coefficients
found for adjacent VSR values, hence there would be limited gain in exaniining intermediate
points at the cost of additional computing time and inconvenience.
6.10 Power Based Model - Conclusions
The preceding Sections in this Chapter have set out in some detail the framework upon which
this model is based. It is basically a very simple model, though in implementation there is ample
opportunity to enter into complex inter-relationships. S o m e effort has been m a d e to explore
various analysis and modelling options, and the trends that follow upon the adoption of
particular assumptions and flow mechanisms.
The usefulness and potential of this approach is best assessed by the application of the model to
data collected from pneumatic conveying systems. This assessment is the primary focus of the
following chapter.
158
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Chapter 7
POWER BASED PNEUMATIC CONVEYING MODEL
IMPLEMENTATION & EVALUATION
The model described in the previous Chapter m a y be used to generate conveying characteristics
plots, via the iterative application of a series of calculations. The overall process m a y be
separated into two distinct phases.
Firstly, the identification of the appropriate solids friction factor relationship is a fundamental
component of the computations. Where this is available from a previous analysis, the relevant
parameters m a y be entered directly. Alternatively the program m a y include a correlation
analysis phase to allow the solidsfrictionfactor relationship to be determined from a set of
experimental data, with the resulting relationship being used in the following computations.
The second phase is devoted to the establishment of the conveying characteristics curves for the
bulk solid. These curves relate to a particular pipeline configuration, hence it is necessary to
specify relevant aspects (length, pipe diameter, number of bends, bend radii, vertical piping,
steps in pipe diameter etc) of the system of interest. It is also necessary to specify the range over
which the conveying characteristics are to be determined. In this case (since the X and Y axes
have been chosen to show the air mass flow rate and total air pressure drop respectively) this
means indicating the m a x i m u m values required for the air mass flow rate and total system
pressure drop.
The program may then establish a grid network across the XY region that has been identified for
this set of conveying characteristics curves and seek to determine, at each node on this grid, the
set of air mass flow rate, total pressure drop and solids mass flow rate terms that satisfy the
requirements of the model. This is a reasonably intensive iterative process. In fact the iterations
need to be carried out in two levels. Initially, for a given air mass flow rate (or grid line parallel
to the Y axis), a solids mass flow rate is assumed and the model iterates through pressure drop
values until the matching pressure drop is found. The "matching pressure drop" is the pressure
drop that the model predicts for the particular combination of solids and air mass flow rates.
This pressure drop is then compared to the required value for that node on the grid. If necessary,
the value of the solids mass flow rate is subsequently adjusted with the intention of bringing the
pressure drop closer to the required value. In other words, there is an overall iteration in the
solids mass flowrate parameter and, within each one of these values, iterations in the pressure
drop are performed so that the model is satisfied. It can be reasonably concluded that the
method is not suited to manual calculation, but is reliant on the application of computer
resources.
A useful way to assess the potential of this model is to observe the shape of the conveying
characteristics generated, noting the extent to which they are able to reproduce or match
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
159
experimental data. A particular challenge for any model in this respect is to follow the
transformation in performance as the conveying regime moves from the lean phase region into
the dense phase region. The reversal in the relationship between the total pressure drop and the
air mass flow rate across this transition region (ie for a given solids mass flow rate, the pressure
drop falls with decreasing air mass flow rate in the lean phase region, but increases in the dense
phase region) is the significant feature intinsregard. The accuracy of the model, as indicated by
the difference between the calculated and experimental data, is also of major importance.
7.1 Conveying Characteristics - General Shape Obtained
A s noted in the introductory Sections of this Thesis, for powdery bulk solids of the type well
suited to pneumatic conveying, and for conveying characteristics set out in the air mass flow
rate/total pressure drop format, the lines of constant solids mass flow rate are generally expected
to show a concave upwards shape. In simplistic terms, this behaviour can be inferred from a
tendency of the system to follow air-only flow behaviour at low solids loadings (ie an increasing
pressure drop at higher air flows), whereas the "choking" effect of the solids component at high
solids loadings, drives up the pressure at the low air mass flow rate end of the curve.
Reproducing this behaviour from a model based on the Darcy-Weisbach pressure drop approach
(as outlined in Section 6.1) is difficult since the underlying tendency of that equation is to
produce substantially higher pressure drops at higher velocities, resulting from the velocity
squared term. The task of the solids friction factor correlation in that case is to produce not only
an appropriate result under lean phase conditions, but to counteract that trend at higher solids
loadings and lower air velocities. O n the other hand, the power based approach that has been
outlined in Chapter 6, to an extent separates each of these influences and allows them to
contribute independently to the total pressure drop. A s can be seen from the following Figures
in this Section, one outcome is that the expected shape of conveying characteristics is
reproduced with ease.
A number of different approaches were developed in Chapter 6 for various aspects of this
overall power based conveying model. S o m e of the components, such as solids friction in
straight pipe sections and changes in kinetic and potential energy, have a constant treatment
throughout the range of models. For other aspects, such as the calculation of the air-only
component, the treatment of bends and the selection of dimensionless parameters to use in the
solids friction factor correlation models, a range of alternative analysis procedures are
developed. It is possible to develop programs employing various combinations of these
assumptions, and subsequently to explore the relative performance of the resulting models. T o a
significant extent this has been done, however because of the number of potential combinations
it is not practical nor particularly beneficial to reproduce the range of output here. Instead, a
preferred set of assumptions has been selected, based on both the discussions and conclusions
reached in Chapter 6 and from an evaluation of the output that has been obtained, to illustrate
and explore the performance of the overall model. In taking this course it is recognised that
further investigation is likely to identify different combinations and refinements that enhance the
performance of the model. The current set of assumptions will, however, allow the potential of
the model and its underlying trends to be explored.
The preferred set of assumptions include the following:
1. Air-only power requirement based on Model 8, as defined in Table 6.4.2
2. Interstitial air velocity calculated from Equation 6.4.11
3. Air power consumption modified as per Equation 6.4.14
4. Bend power consumption calculated from Equation 6.6.33
5. Bend velocity factor (BVF) calculated from Equation 6.6.32
6. Bends assumed to be equi-spaced along the pipeline
7. The six sets of dimensionless parameters for the solids friction factor correlation developed
in Sections 6.9.1 and 6.9.2 (PCI - P C 6 )
160
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
A s a consequence of the six sets of dimensionless terms noted above, six models are examined
based on options 1 to 6 and one of the sets of dimensionless terms P C I to P C 6 . The first of
these models (based on PCI), allows up to five dimensionless terms to be selected from a range
often possible parameters. The analysis in Section 6.9.1 suggests that a simple expression based
on one term (m *) m a y provide a good relationship with the solids friction factor. That approach
has been adopted for the first model in this Chapter.
The resulting models are referred to as PCI to P6, and it is noted that these differ only in the
dimensionless terms used in the solids friction factor correlation. Figures 7.1 to 7.6 show
conveying characteristics developed from these six models, based on experimental data obtained
for a pulverised coal (PF1). The coal had a particle density of 2000kg m"3, and a loose poured
bulk density of 800kg m"3. The bulk solid had a 7 0 % < 7 5 m m size range. T h e pipeline used
(TR2) had the following configuration: D = 52.5mm, L = 4 8 m , Lift = 3.5m, 4 Bends, Bend
Radius = 0.254m.
The evaluation of the fit of each model to the experimental data has involved the consideration
of two aspects. Firstly, an expression is obtained for the solids friction factor via the procedures
outlined in Chapter 6 and a least squares error rninimisation analysis. The solids friction factor
values determined by the application of this Equation are then compared with the values deduced
from an analysis of the experimental data. The accuracy offitis represented by the correlation
coefficient between these two sets of data, and also by both the m e a n "error" (or difference
between these values) and the standard deviation in the error values. T o allow a standardised
comparison between different sets of data, the error in each case has been determined as a
percentage of the corresponding experimental data point. Consequently the m e a n error and
standard deviation in the error are expressed as percentages. Thesefiguresgive an indication of
the success of the solids friction factor correlation part of the process. A part of this process is
an iteration through the possible VSR values . Generally the VSR value resulting in the best cross
correlation is selected, though this default option m a y be overridden.
Secondly, the fit of the overall model to the original conveying data (the set of air mass flow
rate, pressure drop and solids mass flow rate data points) is assessed in a similar manner. A
visual indication of thisfitis given by the location of the experimentally obtained solids mass
flow rate data points on the conveying characteristics curves obtained by the application of the
model. These data have been arranged so that the decimal point indicates the location of each
datum point. A more objective indication of the fit is determined by comparing the
experimental and predicted values of the solids mass flow rate values at each of the experimental
datum point (ie at the coordinates of the air mass flow rate and pressure drop defined at each
datum point). The predicted solids mass flow rate is deterrnined by interpolation from the
surrounding grid point values established by the model. The correlation coefficient, m e a n
"error" (%) and standard deviation in error (%) are again used to represent the closeness of the
relationships. A second and similar comparison is also made, based on the pressure drop
predictions. In this case the pressure drop deterrnined by the model at each experimental datum
point (air mass flow rate and solids mass flow rate) is estimated by interpolation, and compared
with the corresponding experimentally determined pressure drop. The same set of statistical
parameters is again determined.
These data together with the VSR value that gave the best correlation for the solids friction factor
expression, and the parameters deterrnined for that equation, are given with each of the
conveying characteristics plots.
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
200.
P1P« Detail*
Din- Length
C it»3 liul
mi 48
a
'
CL.
48
Te(.
Q_
o
cc
a
LU
cc
:D
en
•n
UJ
0=
Q_
Figure 7.1.1
RIR MRSS FLOW RRTE thg/ti
Conveying characteristics based on Model PCI - for pulverised coal,
•0.57669
VSR = 0.9
^ = 7.00849 x n[
(NB: n\ = m*)
Fit parameters
correlation coefficient
m e a n "error" (%)
standard deviation in error (%)
200.
X, correlation
0.994
ms
0.1
5.0
0.5
7.4
Apt
0.987
12.5
0.995
i i i i t i i i i i i i i i i i r' i i i i i i i i I I
\
i
8.2
i i
i
i
Pip«
D«talU
Din. Length
Llklll llnl
52 49
T»1.i
o
az
49
100. -
a
LU
cc
UJ
• i i i i i i i i i i i i i i f •i i i i i • i i » i i I i i i i
-00
Figure 7.1.2
0-01
0-02
0-03 0-
0.05
0-06 0- 07
RIR MR5S FLOW RRTE [K9/9I
Conveying characteristics based on Model P C 2 - for pulverised coal,
0.70743 _ 1.92746 _ -0.85704 _ -1.46436 _ -0.70658
VSR = 0.9S X, = 1 . 0 2 3 1 3 x < / u / 4 X
tf
nA
n*
Fit parameters
correlation coefficient
mean "error" (%)
standard deviation in error (%)
Xs correlation
0.995
ms
Apt
0.995
0.1
4.1
0.2
6.8
0.978
13.1
9.8
162
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
200.
P1P4 O^tBlIf
Dig* Lrng+h
dill) hill
52 46
Toi.i
46
0_
O
or
100.
-
a
LU
DC
ZD
m
UJ
az
OL.
0-01
0-02
0-03
0-04
0-05
0-06
0-07
RIR MR5S FLOW RRTE tKg/si
Figure 7.1.3
Conveying characteristics based o n M o d e l P C 3
for pulverised coal, VSR = 0.95 \ = 1.25823 x x°-25m7T?5m2
Fit parameters
correlation coefficient
mean "error" (%)
standard deviation in error (%)
ks correlation
0.992
0.17
5.84
ms
4Pr
0.993
0.976
12.85
9.47
0.4
8.1
200.
Pt pa Detalli
Ola. Length
dm) hiil
52 46
T<H.
o
cc
100.
Q
LU
az
zn
CO
UJ
0=
D_
0-01
0-02
0.03
0.04
0-05
0-06
0-07
RIR MR55 FLOW RATE iKg/si
Figure 7.1.4 Conveying characteristics based on Model PC4
for pulverised coal, VSR = 0.95 Xs = 1.0 x /r/3444;^58011
Fit parameters
correlation coefficient
mean "error" (%)
standard deviation in error (%)
X, correlation
0.99
0.2
6.13
m.
0.992
0.01
8.28
M
0.975
12.96
10.47
48
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
200.
Plp« D o t a l h
Dia- Length
dm) 1 nt t
52 48
0_
T»t.i 48
O
CC
a
LU
az
cO
iO
100. -
RIR MR5S FLOW RRTE iKg/si
Figure 7.1.5 Conveying characteristics based on Model PC5 for pulverised coal, VSR = 0.95 Xs = 0.01251 x x[°25mx^Mm
Fit parameters
correlation coefficient
mean "error" (%)
standard deviation in error (%)
X. correlation
0.992
0.17
5.84
ms
Apt
0.993
0.976
12.85
9.47
0.4
8.1
200. i i i i i i i i
F ' 1 M Dotal li
Dig. Length
diul hiil
52 48
0-
a
CC
•
LU
cc
T«H . I
100.
=3
en
UJ
az
Q_
0-5.B
I I I I I I 1 I I I I I
I I I I I I I
0.02 0-03 0-04 0-05 0-06 0- 07
RIR MRSS FLOW RRTE iKt)/?]
Figure 7.1.6 Conveying characteristics based on Model PC6
for pulverised coal, VSR = 0.95 A, = 1.8908 x x? m2n? m6
i i i i i i i i i J_L.
-00
0-01
Fit parameters
correlation coefficient
mean "error" (%)
standard deviation in error (%)
Xs correlation
0.992
0.16
5.66
ms
Ap,
0.993
0.37
0.979
12.82
9.35
8.0
48
164
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
It is apparent from Figures 7.1.1 to 7.1.6 that all six models produce conveying characteristics of
a similar form. The concave upwards shape is demonstrated in each case, and the statistical data
indicate a good correlation between the results predicted, and the experimental data on which it
is based. The results are not identical however, for example the contours of constant solids mass
flow rate in Models P C 2 and P C 4 are noticeably lower - particularly at the higher solids flow
rates - than the other models. It m a y be concluded, therefore, that the choice of dimensionless
parameters to be used in the solids friction factor correlation (the only distmgmshing element
between these six models) m a y have a significant impact on the overall performance.
Similar results are obtained from the analysis of other sets of experimental data. Appendix C
presents data and conveying characteristics for some thirteen such sets. Clearly a multitude of
plots would be generated if each of the models is used to generate conveying characteristics for
each of these sets of data. In order to limit the amount of output to a reasonable quantity,
models P C I and P C 6 only have been used to generate the plots in Appendix C.
Generally it may be concluded that each of the models is successful in generating a set of
conveying characteristics curves that accurately represent the performance of the bulk
sohd/pipeline configuration on which each is based. A smooth transfer between the lean phase
and dense phase conveying regimes is obtained, and the correlation statistics indicate that there is
a close relationship between the experimental data and the performance predicted by the models.
There is, however, little in the data presented to prefer one of the models over the rest. One of
the main purposes in developing a pneumatic conveying model is to predict the performance of
pneumatic conveying systems in pipeline configurations other than those used to obtain the test
data; in other words, use test data to design other conveying systems reliably. Perhaps the most
important feature to be taken into account in selecting a preferred model is the rehability of the
results obtained when the model is applied to alternative pipeline configurations. O n e means of
assessing this capability in a model is to obtain conveying data for a particular bulk solid on a
number of pipeline configurations, and then to use the correlations obtained for one pipeline to
predict the performance of the others. A n assessment of the fit m a y again be conducted, and the
performance of the model evaluated. Scale-up analysis of this sort is included in Chapter 9.
Whilst useful and necessary in providing a final check on the suitability of the model, the
overview gained by this approach does not indicate which elements of the model are
contributing positively or negatively to thefit.It is quite possible, given the complexity of the
interactions occurring in pneumatic conveying, for a model to treat a number of the physical
interactions inappropriately with these inaccuracies to an extent of cancelling out each other.
Consequentiy it m a y appear that the model performs well for a particular configuration while
containing significant errors that will show up when applied to alternative pipeline systems.
An alternative approach is to take a conveying point from an established set of data, and vary
one of the conveying elements (system operating parameters such as m, ms, or pipeline
configuration features such length, diameter or number of bends) in a controlled manner, and
observing the subsequent impact this has on performance. The underlying trends in each of the
models can then be observed. This is likely to provide a more reliable basis for selecting a
particular model. The effect of variations in system operating parameters such as mf and ms is
reflected in the conveying characteristics curves presented previously. It is apparent from these
that all of the models performed well in this respect when developing conveying characteristics
for the pipeline system corresponding to that used to obtain the experimental data.
Consequently the following Section focuses on the trends inherent in the models when pipeline
configuration elements are varied. The trends are shown as variations of system pressure drop
for constant mf and ms as the selected pipeline parameter is changed. It would be similarly
possible to select an alternative set of constant conveying parameters, say, pressure drop and air
mass flow rate constant, and observe the variation in solids mass flow rate.
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
165
7.2
Pipeline Geometry - Trends in Performance
Once a solids friction factor correlation has been obtained for a particular pipeline configuration
for models P C I to P C 6 , it can be readily applied to alternative pipeline arrangements (different
lengths, diameters, number of bends, proportion of vertical lift etc). This possibihty can be
applied to reveal the underlying trends in the model if, say, the pipe length is varied while the
other parameters (diameter, number of bends, air and solids mass flow rates) are held constant.
The followingfigurespresent the results of this analysis for the bulk sohd/pipeline configuration
used to generate the conveying characteristics curves in Figures 7.1.1 to 7.1.6.
One aspect of this approach is that the analysis focuses on a particular location within the
conveying characteristics range. The initial set offiguresis based on a "mid-range" location (mf
= O.OSkgs"1, ms - 3kgs_1); subsequent analyses examine trends in regions that are more clearly in
dense phase and lean phase regions.
800
50
100
150
200
Pipe Length (m)
Figure 7.2.1
Pressure Variation with Pipe Length - underlying model trends
Bulk Solid: T R 1 , Pipeline Arrangement: T R 2 , W / = 0.03kg/s, ms = 3kg/s
In Figure 7.2.1, pipe length is the varying parameter. General expectations would be firstly that
the pressure drop would increase with pipe length, and secondly that this increase would tend to
be approximately linear and proportional to pipe length. It m a y be noted that the six trend
curves converge at around L = 50m. The tendency of the models to agree on pressure drop at
that length (the approximate length of T R 2 ) is consistent with the goodfitto the data achieved
by each model.
Away from this region, there is some variation in pressure drop predicted. Model PC5 in
particular indicates a rapid rise (exponential increase) in pressure drop with length. This extent
of pressure increase is not supported by test rig experience, suggesting that Model P C 5 would
not form a reliable basis for predicting the performance of other pipeline systems.
Models PCI, PC3 and PC6 show similar trends in this case, and indicate a reasonable rate of
pressure increase. Models P C 2 and P C 4 also indicate a linear rate of pressure drop increase with
increasing pipe length, however it would appear to be at an insufficient rate.
166
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
400
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure 7.2.2
Pressure Variation with Pipe Diameter - underlying model trends
Bulk Solid: T R 1 , Pipeline Arrangement: T R 2 , mf= 0.03kg/s, ms = 3kg/s
Figure 7.2.2 shows a similar trend for all models, with the pressure drop decreasing with
increasing pipe diameter as would be expected. In each case there appears to be a levelling out
at larger diameters, in particular for PC4. The convergence at a diameter of 52.5mm (TR2
diameter) is again noticeable.
160.00
140.00 « 120.00
§3
~ 100.00 +
p.
©
80.00
a
3
60.00 4
01
01
£ 40.00
20.00 4
0.00
6
8
10
12
14
16
Number of Bends
Figure 7.2.3
Pressure Variation with N u m b e r of Bends - underlying model trends
Bulk Solid: T R 1 , Pipeline Arrangement: T R 2 , m,= 0.03kg/s, ms = 3kg/s
The variation in predicted pressure drop with number of bends, as indicated in Figure 7.2.3, is
similar and shows a substantially linear increase in pressure drop as bend numbers increase for
an oi the models. This is m agreement with the corresponding suggestions in Chapter 6, eg
figure 6.6.29. The convergence of pressure drop indications at around 4 bends (TR2
configuration) is again evident.
From this brief review of trends it is apparent that model performance may be substantially
attected by the choice of dimensionless parameters used in the solidsfrictionfactor correlation.
this is particularly the case, perhaps surprisingly, in the variation of pressure drop with pipe
length, as highlighted by model PC5. Recalling the establishment of PC5, pipe length was one
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
167
of the parameters specifically included in the development of the dimensionless parameters.
Pipe length in model P C 5 was replaced by pipe diameter in model P C 6 . Consequently the
rehability of model P C 5 is called into question. Similarly, though in a less dramatic fashion,
models P C 2 and P C 4 m a y not perform adequately with respect to variation in pipe length.
These Figures have provided a useful initial indication of the performance of the models,
warranting an extension of the analysis to different sets of bulk sohd/pipeline configurations. It
is again noted that only one specific operating point (mf and ms value) was examined in these
Figures. In addition to exploring the performance of the models for different bulk solids and
pipelines, in each case a "base point" has been selected in the dense and the lean phase
conveying regions (as indicated by the location of the conveying point with respect to the
pressure m i n i m u m zone on the conveying characteristics curves). Model performance can then
be examined w h e n either lean phase or dense phase mechanisms are dominating.
In a final adjustment to the analysis procedure, it has been assumed that there are no bends in th
pipeline, where performance trends with respect to variation in pipe length and diameter are
being examined. This is intended to eliminate the possibility that variation in the effect of bends
between the models m a y obscure the other relationships being specifically examined in those
cases.
For the purposes of this phase of the analysis, eight sets of experimental data, obtained from
different pipeline and/or bulk solid combinations, have been used to develop the model solids
friction factor parameters. In each case trends in performance for both lean and dense phase
conveying regimes have been explored for the variation of three pipeline parameters (length,
diameter and number of bends). Accordingly forty eight Figures are required to convey the
complete set of results. These Figures are not all reproduced here, but are presented in full in
Appendix B.
7.3 Variation with Pipe Length
A s noted previously, it would be reasonable to expect system pressure drop to increase as the
pipeline length increases (other variables being held constant). In terms of the application of this
modelling approach, a first order consequence of the increased length would be an increased
total mass of solids in the pipeline, resulting in additional solids friction power losses. This
would tend to give some degree of linearity to the relationship between the total system pressure
drop and pipe length. The increased pressure resulting from the above consideration however
also affects other elements of the model, for example adjusting the m e a n pipeline conditions
which flow on to aspects such as solids velocity, cross-sectional area taken up by the bulk solid
and the air-only pressure drop calculations. Depending on the solids friction factor correlation
being applied, it would also affect the solids friction factor value (though not in the case of the
simplified P C I model in use here where Xs is assumed to be a function of m* only).
These interactions introduce a significant degree of complexity into the behaviour of the model.
However again it would be reasonable to expect that the outcome (in terms of causing the
relationship between system pressure drop and pipe length to vary from a linear function) would
be for the rate of pressure drop to increase with length - in other words for the curves to show a
concave upwards shape. It is difficult to conceive of circumstances that would allow a levelling
off, or decrease, of system pressure drop with increasing pipe length.
A final general point that may be made is that as pipe length tends towards zero, the system
pressure drop should also tend towards zero.
The figures in Appendix B, Section 1 examine the performance of the models in some detail in
both the lean phase and dense phase regions. It is noted that an additional version of model
P C I , designated P C I A, has been introduced. The distmgwshing feature of this model is that the
168
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
solids friction factor correlation has been extended to include the mean Froude Number,
ie Xs = const. m* Frm . This addition to the suite of models being examined is in response to
the consistent performance demonstrated by the simple P C I model and is intended to explore
whether an extension of that correlation would prove beneficial. The selection of the Froude
Number as the extra parameter is suggested by itsfrequentadoption by other researchers, eg
Rizk (1982), Weber (1982 and 91), Morikawa (1982), Szikszay (1988) and W y p y c h (1987), in
the context of solidsfrictioncalculations, though it is noted that the development of the solids
friction factor component in those cases is somewhat different from the power based models
being examined here.
The Figures in Appendix B Sections 1.1 and 1.2 (dense and lean phase conveying conditions
respectively) generally show trends supporting the expectations outlined previously. The
following two figures are typical of the results obtained. ( N B zero bends in the pipeline have
been assumed in the application of the models).
o -I
i
1
1
0
50
100
150
200
Pipe Length (m)
Figure 7.3.1 Pressure Variation with Pipe Length - Dense Phase Conditions
Base data: Bulk Solid: PF1, Test Rig: T R 2 (L = 4 8 m )
Conveying Conditions: mf= 0.012kg/s, ms = 3kg/s (dense phase for base data)
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
169
300
50
100
150
200
Pipe Length (m)
Figure 7.3.2
Pressure Variation with Pipe Length - Lean Phase Conditions
Base data: Bulk Solid: P F 3 , Test Rig: T R 2 (L = 4 8 m )
Conveying Conditions: ntf- 0.05kg/s, ms = 2kg/s (lean phase for base data)
Whilst Figures 7.3.1 and 7.3.2 basically reinforce the conclusions drawn previously, models
PC2, P C 4 and P C 5 give some cause for concern. Model P C 5 shows a rapid rate of pressure rise
that seems out of scale with the other models. The curves for P C 2 and P C 4 show a convex
upwards shape, an unlikely result for the pressure drop versus pipe length relationship. A n
examination of all thefiguresin Section 1 Appendix B shows that all of the models apart from
P C I show this trend to some extent in one or more of the sixteen configurations presented. A
distinguishing feature of model P C I is that the solids friction factor is constant (for fixed m/and
ms, since it is a function of m* only) throughout the range of lengths, therefore it illustrates the
inherent trend of this power based approach to modelling pneumatic conveying with respect to
pipe length for a fixed solids friction factor.
Figures 7.3.1 and 7.3.2 represent the more "well behaved" results, in other cases models PC2
and P C 5 demonstrate a contrary trend, as indicated in Figures 7.3.3 and 7.3.4, effectively ruling
them out as reliable models.
170
Power Based Pneumatic Conveying Model
50
100
150
200
250
- Implementation & Evaluation
300
Pipe Length (m)
Pressure Variation with Pipe Length - Dense Phase Conditions
Base data: Bulk Solid: F A 1 , Test Rig: T R 3 (L = 168.5m)
Conveying Conditions: mf= .07kg/s, nts = 15kg/s (dense phase for base data)
Figure 7.3.3
It is clear from the above Figure that model PC2 gives an incorrect relationship between system
pressure drop and pipe length, indicating that an erroneous solidsfrictionfactor correlation has
been formed for this set of data. The convergence of all the models at around the base length for
this set of data (L = 168.5m) demonstrates the ability of each of the models (mcluding P C 2 ) to
satisfy the experimental data, and hence to produce conveying characteristics curves that are
accurate for that configuration. However in this instance P C 2 treats changes in pipe length
incorrectly, and consequently system pressure drop calculated based on this correlation could not
be relied upon.
Model PC5 again shows a rapid rate of pressure drop with respect to pipe length in comparison
to the other models, and reaches a conveying limit before 200m. This is generally the result of a
"pipe fulP' condition being reached. The curve produced by model P C 4 shows the nonpreferred convex upwards shape.
Frequently, as in this case, models P C I , P C 3 , P C 6 and P C 1 A produce similar results to one
another and appear to be indicating a reasonable relationship.
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
171
400
350
300 •
mi 250 •-
a
£ 200 g 150 -•
&
* 100
XX
X X x *-*-*-><
50
0
50
100
+
4-
+
150
200
250
300
350
400
Pipe Length (m)
Figure 7.3.4
Pressure Variation with Pipe Length - Lean Phase Conditions
Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (L = 172.1m)
Conveying Conditions: m / = 0.25kg/s, ms = 4kg/s (lean phase for base data)
In Figure 7.3.4, model PC5 indicates an incorrect pressure drop/pipe length relationship,
particularly at smaller lengths. Model P C 2 failed to converge in the solids friction factor
correlation analysis for this set of data, hence no curve is presented for that model in this case.
With the exception of P C I A, the models converge at around the base data pipe length (L =
172.1m).
Apart from concluding that it is a result of the solids friction factor correlation obtained, i
immediately apparent w h y the curve for model P C I A should be offset from the remaining
curves (indicating approximately 60kPa higher pressure drop at the base data pipe length).
Exanjining the corresponding Figure for pressure drop versus pipe diameter (Figure B 2.2.8) it
m a y be noted that at the base data pipe diameter of 6 9 m m , model P C I A predicts a pressure drop
approximately 60kPa above the other models. This corresponds to the offset noted in Figure
7.3.4, suggesting that a diameter effect (introduced with Frm) is the basis for the difference. It
m a y also be noted that the same offset is not found in the corresponding Figure for that material
showing pressure drop versus pipe length under dense phase conditions (Figure B 2.1.8).
It is interesting to note that both models PC3 and PC6 show a slight convex upwards shape;
P C I is the only model to retain a concave upwards shape.
Tables 7.3.1 and 7.3.2 are presented as a means of comparing the results obtained for the eight
sets of experimental data for dense and lean phase conveying regimes respectively. T o allow
this comparison, a simple power function has beenfittedto the pressure drop versus pipe length
relationship, of the form Ap = K L. These tables give the values of the power "n" obtained, as
well as the correlation coefficient (denoted "cc") as an indication of the closeness of the fit
between the Ap versus L relationships, resulting from the application of the models, and the
above power function. Average values of the power "n" obtained in this fashion for each of the
sets of data, are also given.
Clearly, for a linear relationship between these two parameters, the power "n" would have a
value of one. A concave upwards shape would be indicated by a value of "n" greater than one
and conversely a value of "n" less than one would result from convex upwards shape.
172
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Table 7.3.1:
Comparison of the relationship between system pressure drop and pipe
length predicted by models P C I to P C 6 , dense phase conveying regime Power Function assumed: Ap = K L
Data
PF1 - TR1
PF1-TR2
PF2 - TR1
PF2 - TR2
PF3 - TR1
PF3-TR2
FA1 - TR3
FA2 - TR12
average:
Table 7.3.2:
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
PCI
PC2
PC3
PC4
PC5
PC6
1.078
.992
1.052
.995
1.382
.992
1.136
.993
1.176
.994
1.098
.992
.
.
1.109
0.755
.999
.628
.999
.783
1.653
.953
1.526
.992
2.389
.981
1.619
.942
1.688
.991
2.114
.944
1.363
.962
-1.1
1.151
.987
1.056
.995
1.493
1.041
.986
.717
PC1A
1.011
.995
1.062
.994
1.327
.994
1.052
.996
1.071
.997
1.072
.993
.926
.993
.973
1
1
1
1.407
1.136
1.062
1
.99
1.335
.994
1.157
.281
1
_
_
.441
1
-.399
.998
1.202
.971
-.344
1
-
.99
1.035
.995
1.408
.988
1.231
.984
1.119
.996
1.207
.979
1.013
.988
.707
-
1
0.236
1.104
1
.565
.999
.73
.999
.499
.999
.448
.999
.807
.999
0.652
.98
1.21
.984
1.144
.995
1.277
.97
average
1.13
0.95
1.46
1.04
0.93
1.21
0.78
0.57
1.01
Comparison of the relationship between system pressure drop and pipe
length predicted by models P C I to P C 6 , lean phase conveying regime n
Power Function assumed: Ap = K L
Data
PF1-TR1
PF1-TR2
PF2 - TR1
PF2-TR2
PF3 - TR1
PF3-TR2
FA1 - TR3
FA2 - TR12
average:
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
PCI
PC2
PC3
PC4
PC5
PC6
1.15
.994
1.068
.997
1.178
.997
1.085
.997
1.175
.996
1.049
.998
1.005
.996
-
.773
2.064
1
.96
.628
1.196
.991
1.013
.998
1.477
.979
1.159
.995
1.076
.998
1.307
.984
.96
-
1.153
.994
1.001
.999
1.197
.997
1.339
.984
1.052
.999
1.259
.989
.986
.997
.654
1
1
1.08
0.661
.999
1.084
.438
1
-
.523
1
-.247
.914
1.139
.993
-.113
.742
0.348
1
1.808
.992
.774
2.3
1
.983
2.89
.945
1.802
.994
2.83
.996
1.949
.977
-.36
.815
1.91
.589
1
.772
1
.571
1
.472
1
.708
.996
.658
PCI A
1.128
.997
1.078
.997
1.178
.998
1.073
.998
1.148
.998
1.043
.999
1.02
.998
.919
1
1
1.111
1.073
1
average
1.24
1.00
1.35
1.24
0.97
1.31
0.90
0.59
1.08
It is apparent from the correlation coefficient values obtained that the simple power function
assumed does generally fit the data well. It is also apparent that the value of "n" is, in general
and on average, slightly above one. For a number of the data sets, some of the models (notably
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
173
P C 2 , P C 4 and P C 5 ) depart from this trend, reflecting the occasional erratic trends highlighted
earlier. This variation in performance is perhaps better shown graphically in Figure 7.3.5 .
Experimental Data Sets
Figure 7.3.5
M o d e l performance with respect to changing pipe length - variation in the
value of " n " (from Ap = K L ) for the seven models and eight sets of
conveying data, (dense and lean phase conveying results combined)
Figure 7.3.5 combines the results listed in Tables 7.3.1 and 7.3.2, and the "n" value for the le
phase conveying regime for each data set is plotted immediately after the corresponding dense
phase value. The wide variation in values resulting from the application of models P C 2 and P C 5
is clearly shown. The instances where negative values for "n" are recorded (indicating a clearly
erroneous relationship between pipe length and pressure drop) are also apparent. In a number of
cases model P C 2 failed to produce a successful correlation, resulting in missing data points in
the figure. Model P C 4 shows a more stable "n" value throughout tins set of data, though the
value is consistently well below one (« 0.65). The similarity of the results obtained for model
P C I , P C 3 , P C 6 and P C I A, which cluster around the value of one, is also apparent in this figure.
Another interesting aspect of the performance of the six models that may be extracted from the
data presented in Tables 7.3.1 and 7.3.2, is the variation in this relationship (between system
pressure drop and pipe length) between the lean and dense phase conveying conditions. T o
illustrate this aspect, the ratio between the values of "n" obtained for dense and lean phase on the
same sets of data, has been calculated and the results plotted in Figure 7.3.6. If this ratio is close
to one, it m a y be concluded that the impact of pipe length on model behaviour is substantially
the same for both conveying regimes.
174
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Experimental Data Sets
Figure 7.3.6 Model performance with respect to changing pipe length - variation in the
value of "n" (from Ap = K l " ) obtained for dense and lean phase conveying regimes.
Figure 7.3.6 indicates that in general, the models show the same trends in pressure drop versus
pipe length for lean and dense phase conveying conditions. Models P C 2 and P C 5 are again
distinguished by the extent to which they depart from this result.
In reviewing this aspect of model performance, it may be concluded that models PC2 and PC5,
in particular, are unlikely to provide a sound basis for modelling pneumatic conveying systems;
on the other hand models P C I , PC3, P C 6 and P C I A appear promising.
7.4 Variation with Pipe Diameter
It would be reasonable to expect the system pressure drop to decrease as the pipeline diameter
increases (other variables being held constant). In terms of the application of this power based
modelling approach, the dominant result of the increased diameter would be an increasing crosssectional area available for the flow of air and solids in the pipeline, resulting in reduced
velocities and lower friction power losses.
It may be noted however that a reduced solids velocity would result in a proportional increase in
the total mass of solids in the pipeline. Consequently, providing the solids friction factor does
not change, the frictional power loss associated with the solids component (apart from the
velocity squared component in the bends) would tend to remain constant. The air-only
component would fall significantly in response to the increase in available area, as a result of its
dependence on a velocity squared term. A reasonable outcome from the application of this
model would therefore be a reduction in system pressure drop as the pipe diameter is increased,
exponentially approaching a constant value basically set by the solids friction power loss as the
contribution of the air-only component diminishes. Variation in the value of the solids friction
factor across the range of configurations being considered m a y be expected to cause some
divergence from the trends described above.
If the model incorporated a more elaborate treatment of the solids velocity than currently
employed, in particular identifying the circumstances where bulk solids begin to fall out of the
conveying stream and blockages begin to form, it is possible that system pressure drop would
even rise with increased diameter. Given the simphfying assumptions used at this stage in the
application of the model however, it is difficult to conceive of circumstances that would lead to
a significant rise of system pressure drop with increasing pipe diameter.
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
175
The figures in Appendix B, Section 2 examine the performance of the models in some detail in
both the lean phase and dense phase regions for eight sets of experimental data. These figures
generally show trends supporting the expectations outlined above. The following two figures are
typical of the results obtained. ( N B For the purposes of focussing on pipe diameter effects and
excluding pipe length and bend variations, the pipeline configurations analysed in the application
of these models have no bends).
140
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure 7.4.1
Pressure Variation with Pipe Diameter - Dense Phase Conditions
Base data: Bulk Solid: P F 1 , Test Rig: T R 2 (D= 52.5m)
Conveying Conditions: ntf- 0.012kg/s, ms = 3 kg/s (dense phase for base data)
600
0.065
0.085
0.105
0.125
0.145
0.165
0.185
0.205
Pipe Diameter (m)
Figure 7.4.2
Pressure Variation with Pipe Diameter - Lean Phase Conditions
Base data: Bulk Solid: F A 1 , Test Rig: T R 3 (D = 1 0 5 m m )
Conveying Conditions: mf= .25kg/s, m, = 15kg/s (lean phase for base data)
Figures 7.4.1 and 7.4.2 again show a convergence of system pressure drop at around the base
data pipeline diameter in each case. The tendency for the pressure drop to level off at larger
diameters is evident, with the exception of P C 4 which shows an increasing pressure drop trend
176
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
as the diameter increases beyond the base data diameter. This feature of the performance of P C 4
is typical of the results shown in Appendix B.
Models PCI, PC3, PC6 and PC1A are in close agreement in both Figures, which is again
representative of the full set of results shown in Appendix B Section 2, though in some instances
P C 1 A diverges from this group. The results for P C 2 and P C 5 are less consistent, as indicated in
the following three figures.
0.03
0.04
0.05
0.06
0.07
Pipe Diameter (m)
Pressure Variation with Pipe Diameter - Dense Phase Conditions
Base data: Bulk Solid: PF3, Test Rig: T R 2 (D = 52.5mm)
Conveying Conditions: ntf = 0.01 kg/s, ms = 2 kg/s (dense phase for base data)
Figure 7.4.3
300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure 7.4.4
Pressure Variation with Pipe Diameter - Dense Phase Conditions
Base data: Bulk Solid: PF3, Test Rig: T R 1 (D = 25.4mm)
Conveying Conditions: mf= 0.003kg/s, ms = 0.5kg/s (dense phase for base data)
Figures 7.4.3 and 7.4.4 show two contrasting cases of the performance of model PC2. In Figure
7.4.3 the system pressure drop deterrnined by model P C 2 shows a significantly increasing trend
for increasing diameter; on the other hand in Figure 7.4.4 model P C 2 decreases with increasing
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
111
diameter asymptotically to a relatively low level, which appears unrealistic, bearing in mind that
this should represent the solids power loss component for the system.
50 --
" -4-.-4
^,
0 -I 1 1 1 1 1
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Pipe Diameter (m)
Figure 7.4.5 Pressure Variation with Pipe Diameter - Dense Phase Conditions
Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (D = 6 9 m m )
Conveying Conditions: ntf= 0.05kg/s, ms = 4kg/s (dense phase for base data)
Figure 7.4.5 shows an extreme case for increasing pressure drop with diameter (PC5),
contrasting with the results from the same model for the previous two figures. Figure 7.4.5 also
provides an example of the occasional divergence of model P C I A from the other results. Model
P C 2 failed to produce results for this set of data.
In a similar manner to Tables 7.3.1 and 7.3.2, Tables 7.4.1 and 7.4.2 have been prepared to
allow a comparison of the results obtained for the eight sets of experimental data for dense and
lean phase conveying regimes respectively. The functionfittedin this case is of the form:
Ap - A + B e" 0 (A, B and n are constants), recognising the general trend of the curves to
asymptotically approach a constant value. The "fit" in this case is approximate in that the above
function isfirstdifferentiated with respect to "D", to eliminate "A", prior to taking natural
logarithms of both sides and hence evaluating "n". In this process a numerical approximation of
the differentiation of Ap is carried out, with the accuracy of this calculation being reliant on the
size of the step change in "D". It is emphasised that that function is used purely to provide a
means of numerically comparing the behaviour of the models, and is not suggested as a means
of modelling the physical processes.
Tables 7.4.1 and 7.4.2 give the values of "n" obtained, as well as the correlation coefficient
(denoted "cc") as an indication of the closeness of thefitbetween the Ap versus D relationships
resulting from the application of the models, and the above exponential function. Average
values of the power "n" obtained in this fashion for each of the sets of data are also given.
For the function to have the expected exponential decay relationship, the exponent "n" would be
a negative number, as is the case for the majority of the data listed in the following Tables. The
reciprocal of the exponent "n" in this equation is analogous to the time constant in an
exponentially decaying function. Therefore the larger the numerical value of "n" the more
gradual the rate of decay of the function with respect to pipe diameter. A positive exponent
would indicate the unlikely trend of increasing pressure drop with increasing pipe diameter,
recorded by models P C 4 , P C 2 and P C 5 on occasion. Whilst, in m a n y of the instances listed
178
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
below, the correlation coefficient suggests that the exponential decay function does represent the
model' trends accurately; it is less successful than the power function applied to the pressure
drop/pipe length relationships analysed in the previous section.
Table 7.4.1:
Data
PF1-TR1
PF1-TR2
PF2 - TR1
PF2-TR2
PF3 - TR1
PF3-TR2
FA1-TR3
FA2 - TR12
average:
Table 7.4.2:
Data
PF1-TR1
PF1-TR2
PF2 - TR1
PF2-TR2
PF3 - TR1
PF3-TR2
FA1 - TR3
FA2 - TR12
average:
Comparison of the relationship between system pressure drop and pipe
diameter predicted by models P C I to P C 6 , dense phase conveying regime.
Function assumed: Ap = A + B e
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
PCI
PC2
PC3
PC4
PC5
PC6
-172.3
.985
-86.7
.978
-196.8
.993
-95.4
-
-173.6
.985
-103.1
.993
-196.2
.993
-100.2
-9.72
.986
2.65
.491
-9.71
.973
-259.2
.989
-8.99
.975
9.29
.567
9.79
.753
12.31
.571
-31.7
-72.4
.968
-45.2
.961
-70.4
.971
-86.5
.976
-58.1
.947
-74.3
-172.6
.971
-96.1
.991
-176.4
.984
-89.2
.982
-191.9
-
-42.4
.979
-
-46.2
.99 _j .981
-191.9 -122.3
.994
.99
-101.9 4.86
.871
.99
-25.7
-48.4
.989
.974
-60.2
.984
-119.2 -46.3
.99
-194.6
.992
-95.9
.981
-48.3
.987
-61.5
.984
-121.7
.98
-28.8
.974
13.74
.891
-52.7
.99
-106.8
.995
-44.4
.987
-61.8
.985
-117.4
PCI A
-45.7
.969
-157.3
.999
-52.9
.914
-43.1
.959
-53.2
.946
-46.5
.951
-24.3
.953
-29.9
.992
-56.6
average
-107.72
-75.45
-117.07
-102.83
-117.28
-58.75
-30.02
-31.22
-80.04
Comparison of the relationship between system pressure drop and pipe
diameter predicted by models P C I to P C 6 , lean phase conveying regime.
Function assumed: Ap = A + BtnD
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
cc
n
PCI
PC2
PC3
PC4
PC5
PC6
-161.3
.975
-96.9
.982
-173.4
.987
-100.9
.987
-163.3
.981
-91.1
.979
-49.7
.982
-59.7
.989
-112.0
-
-164.7
.984
-102.5
.986
-171.3
.987
-102.5
.986
-179.5
.989
-104.1
.988
-24.3
.953
-64.2
.984
-114.1
-347.5
.993
-164.3
.998
-296.9
.998
-174.0
.997
-323.1
-85.3
.948
-70.2
.955
-86.2
.951
-93.5
-163.8
.984
-96.53
.982
-166.8
.989
-110.1
.987
-163.0
-
-71.61
.982
-
-65.3
.981
-185.6
.991
-176.2
.99
-48.5
.995
-
-109.4
1
-180.1
.997
-87.4
.998
-83.7
.998
-207.1
.97
-73.8
.927
-78.8
.978
-38.9
.959
18.06
.864
-63.58
.98
-92.5
.978
-47.8
.981
-64.1
.983
-113.1
PCI A
-66.6
average
-164.87
.93
-124.8
.994
-110.9
-103.83
-167.58
.95
-61.9
.956
-67.1
.928
-77.5
.966
-34.24
.956
-40.2
.981
-72.91
-101.17
-165.06
-114.33
-47.26
-48.97
-114.13 I
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
179
It is apparent from Tables 7.4.1 and 7.4.2 that there is more variability across the models with
regard to the relationship between system pressure drop and the pipe diameter than for the
corresponding trends with respect to pipe length. Models P C I , P C 3 and P C 6 again show
consistent and relatively stable relationships that match anticipated trends. These characteristics
are shown graphically in the following Figure 7.4.6. The results for the dense and lean phase
data have been combined in thisfigure,with the lean phase results immediately following the
dense phase results for the corresponding data set. Hence, particularly for the more stable
models, the data appears to be grouped in pairs.
Experimental Data Sets
Figure 7.4.6
M o d e l performance with respect to changing pipe diameter - variation in
the value of " n " (from Ap = A + B e"0) for the seven models and eight sets of
conveying data, (dense and lean phase conveying results combined)
Using this method of comparison, model PC4 shows the most variability with respect to pipe
diameter. A s noted previously, it can be seen that models P C I , P C 3 and P C 6 follow similar
trends. There is a surprising degree of difference between model P C I and P C I A. In a number
of cases model P C 2 failed to produce a successful correlation, resulting in missing data points in
the figure. Model P C 5 is perhaps the most stable throughout this set of data, though the final
two data points indicate an increasing pressure drop trend with increasing pipe diameter.
The variation in this relationship between the lean and dense phase conveying conditions is also
of interest. The ratio between the values of "n" obtained for dense and lean phase on the same
sets of data has been calculated, and the results plotted in Figure 7.4.7. If this ratio is close to
one, it m a y be concluded that the impact of pipe diameter on model behaviour is substantially
the same for both conveying regimes.
180
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
•0.50
Experimental Data Sets
Figure 7.4.7 Model performance with respect to changing pipe diameter - variation in
the value of M n " (from Ap = A + B e"0) obtained for dense and lean phase
conveying regimes.
Figure 7.4.7 indicates that models P C I and P C 6 result in very similar trends in performance with
respect to pipe diameter for lean and dense phase conditions ("n" « 1). With the exception of
the second last data set, model P C 3 shows similar results. Model P C 4 shows a large difference
between the "n" values obtained for the lean and dense phase conditions. A review of Table
7.4.1 and the relevantfiguresin Appendix B shows that in m a n y instances this model displays a
trend for increasing pressure drop with increasing diameter. Model P C 5 is relatively consistent
in the ratio of "n" values, with Figure 7.4.7 indicating that the "n" value obtained for dense
phase conveying conditions is approximately 8 0 % of the value obtained for lean phase
conditions. This in turn suggests that the rate of pressure drop with increasing diameter is
slightly higher under dense phase conveying conditions.
In an overall review of this aspect of model performance, it may be concluded that the trends in
model behaviour are less stable with respect to pipe diameter than they are with respect to
length. Models PCI, P C 3 and P C 6 again appear to be the more reliable, with models P C 2 and
P C 4 being more variable and attimesproducing doubtful trends.
7.5 Variation with Number of Bends
The relationship between system pressure drop and the number of bends in the pipeline
configuration is perhaps the most straightforward of the parameter variations examined in these
three Sections. Clearly each additional bend incorporated in a pipeline would be expected to
increase the pressure drop across the system (all other parameters being held constant). From the
perspective of this power based model, the frictional power losses associated with changing the
direction of the bulk solid directly add to the total power requirement of the system and hence
result m a n increased pressure drop. This approach considers the centripetal forces acting as the
bulk solid traverses the bend, hence the power loss is proportional to the square of the velocity
of the solids component. It m a y be expected therefore that under higher velocity lean phase
conditions the influence of the bends would be particularly significant.
hi the application of these models, the solids friction factor is not directly altered by changing
the number of bends in the pipeline configuration. For the seven models analysed in the
preceding two Sections, none of the dimensionless parameters used in the solidsfrictionfactor
correlations employs terms directly dependent on the number of bends in the pipeline.
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
181
Therefore in changing the number of bends there is no immediate adjustment to the solids
friction factor, leading directly to a modification of the pressure drop deterrnined by the model.
However bends do have a major impact on the performance of a pneumatic conveying system,
and form a significant component of the overall system power consumption in this power based
modelling approach. The effect of bends is therefore felt in two ways. Firstly in the initial
correlation analysis carried out to obtain an expression of the solidsfrictionfactor, the more
bends there are in the pipeline the smaller the solidsfrictionfactor needs to account for the
power losses attributed to the solids component. Therefore the number of bends significantly
effects the values in the solids friction factor equation. Secondly, in the modelling process any
terms affecting the velocity of the solids component (notably system pressure along the pipeline)
have a magnified impact on the power consumption at the bends (through the velocity squared
characteristic). Therefore the presence of bends has a strong impact in the iterative process of
deterrnining the total pressure drop.
In summary, it would be expected that an increased number of bends would lead to a higher
system pressure drop. If, in the model considered, mean pipeline conditions could be applied to
the analysis of all bends, this increase in pressure drop would tend to be linear. The simphfying
assumption used with these models treats the bends as being equi-spaced along the pipeline.
Under this assumption the velocity squared term means that bends towards the exit end more
than compensate for the reduced power consumption of corresponding bends at the inlet end.
The pressure drop versus number of bends curve would therefore tend to have a concave
upwards slope. Variations from these characteristics would be a result of solidsfrictionfactor
expression.
The figures in Appendix B, Section 3 examine the performance of the models in some detail in
both the lean phase and dense phase regions for eight sets of experimental data. These figures
generally show trends supporting the expectations outlined above, indeed in this case there are
no instances of contradictory trends (pressure drop falling with increasing number of bends) as
there were for the two previous analyses. The following two figures are typical of the results
obtained. N B . For the purposes of focussing on bend effects and excluding pipe length and
bend variations, the pipeline diameter and length have been held constant at the base data
pipeline configuration in each case.
115
j
110 -•
105
•
,m.«
«
\^ 100 -•
• ± — PC3
X— PC4
a
95 -i-
X— PCS
90 -•
-#—PC6
PC1A
Qo
9
m
901
t
u
Pk
85 ••
80 -•
75 1
70 4
0
2
4
6
8
10
12
14
16
Number of Bends
Figure 7.5.1 Pressure Variation with Number of Bends - Dense Phase Conditions
Base data: Bulk Solid: P F 1 , Test Rig: T R 2 (D= 52.5m)
Conveying Conditions: m,= 0.012kg/s, ms = 3kg/s (dense phase for base data)
182
Power Based Pneumatic Conveying Model - Implementation & Evaluation
6
8
10
12
Number of Bends
Pressure Variation with N u m b e r of Bends - Lean Phase Conditions
Base data: Bulk Solid: PF1, Test Rig: T R 2 (4 Bends, RB = .254m)
Conveying Conditions: ntf- 0.05kg/s, ms - 3kg/s (lean phase for base data)
Figure 7.5.2
Figures 7.5.1 and 7.5.2 show the generally linear relationship typical between system pressure
drop and number of bends. They also indicate the markedly higher influence of bends under
lean phase conveying regimes. The complete list of Figures in Appendix B generally shows
similar trends for the seven versions of the power based model, thoughfrequentlywith a greater
divergence in the curves than shown in the above two Figures. Model P C I A in particular in
some instances shows a significantly flatter slope, indicating that the additional dimensionless
parameter (Frm) can have a major impact on the overall model performance. The following
three Figures illustrate some of this variety.
6
8
10
12
14
16
Number of Bends
Pressure Variation with N u m b e r of Bends - Dense Phase Conditions
Base data: Bulk Solid: PF1, Test Rig: T R 1 (9 Bends, RB = .254m)
Conveying Conditions: mf= 0.003kg/s, ms = 0.5kg/s (dense phase for base data)
Figure 7.5.3
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
6
8
10
12
14
183
16
Number of Bends
Pressure Variation with N u m b e r of Bends - Dense Phase Conditions
Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (5 Bends, RB = l m )
Conveying Conditions: mf= 0.05 kg/s, ms - 4 kg/s (dense phase for base data)
Figure 7.5.4
500
6
8
10
12
14
16
Number of Bends
Pressure Variation with N u m b e r of Bends - Lean Phase Conditions
Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (5 Bends, RB = l m )
Conveying Conditions: W / = 0.25kg/s, ms = 4kg/s (lean phase for base data)
Figure 7.5.5
Tables 7.5.1 and 7.5.2 have been prepared to allow a comparison of the results obtained for
eight sets of experimental data for dense and lean phase conveying regimes respectively. Given
the dominant linear trend in these graphs, a simple linear function of the form:
Ap = b + m (number of bends), has been selected for this purpose (b and m are constants).
Tables 7.5.1 and 7.5.2 give the values of "m" obtained, as well as the correlation coefficie
(denoted "cc") indicating the suitability of above Equation to represent this trend. Average
values of the power " m " obtained in this fashion for each of the sets of data, are also given.
184
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Table 7.5.1:
Data
PF1 - TR1
PF1-TR2
PF2 - TR1
PF2-TR2
PF3 - TR1
PF3-TR2
FA1 - TR3
FA2 - TR12
average:
Comparison of the relationship between system pressure drop and number
of bends predicted by models P C I to P C 6 , dense phase conveying regime.
Function assumed: Ap - b + m (number of bends)
PCI
m 1.212
cc 1
m 2.317
cc 1
m 1.223
cc 1
m 2.564
cc 1
m 1.451
cc 1
m .86
cc 1
m 1.926
cc 1
m 2.251
cc 1
m 1.725
PC2
PC3
PC4
PC5
PC6
-
1.272
1.36
1.273
1.351
PCI A
.562
-
1
1
1
1
1.992
2.249
2.247
2.294
2.33
2.025
1
1
1.277
1.191
.954
1
1
1
3.209
.999
1.516
3.361
.998
1.569
1.939
.813
1
1
1
.774
.988
1.012
.847
1
1
1
1.821
2.004
2.084
1.425
1
1
1
1.87
1.69
.998
1
2.175
1.45
3.446
.998
1.512
2.424
1.556
1
1.209
.987
2.998
2.001
1
1
-
2.0
2.21
1
1
1.255
average
1.17
1.15
2.73
1
1.719
2.147
2.449
2.498
1
1.8
1
1.6
1
1.9
.155
.999
1.92
1.1
1.41
0.95
2.04
Table 7.5.2:
Comparison of the relationship between system pressure drop and number
of bends predicted by models P C I to P C 6 , lean phase conveying regime.
Function assumed: Ap = b + m (number of bends)
Data
PF1-TR1
PCI
m 6.075
cc 1
m 13.74
cc .999
m 6.473
cc 1
m 12.34
cc .999
m 6.274
cc 1
m 8.191
cc 1
m 15.23
cc .998
m 23.59
cc .994
m 11.49
PF1 - TR2
PF2 - TR1
PF2-TR2
PF3 - TR1
PF3-TR2
FA1 - TR3
FA2-TR12
average:
PC2
PC3
PC4
PC5
PC6
-
6.039
6.305
6.039
6.28
PCI A
3.806
-
1
1
1
1
1
10.01
.997
12.35
10.91
13.35
12.61
1
1
1
1
-
6.542
4.781
6.597
1
1
1
9.072
.999
6.367
.997
12.0
.996
16.41
19.74
.963
6.285
10.48
17.39
6.872
.999
19.63
13.72
.999
5.708
9.723
1
.98
.96
1
6.36
6.286
6.451
4.233
1
1
1
1
1
10.06
.998
14.51
.998
17.71
.999
11.7
6.904
10.06
.998
14.51
.998
17.71
.999
11.5
10.39
.998
15.03
.998
18.2
.999
11.93
8.19
1
-
10.8
1
11.29
1
19.7
.999
9.6
average
5.76
12.38
6.16
1
14.05
6.04
9.40
1
11.84
.999
6.204
17.19
1
7.9
10.64
14.12
Tables 7.5.1 and 7.5.2 indicate that linear functions generally provide a very goodfitto the Ap
versus number of bends data. The slope ("m") is always positive, confirming the expected
relationship of increasing pressure drop with increased numbers of bends. Models P C 4 and
r-ciA tend to have a smaller slope than the other models. It is also clear that the slope is
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
185
considerably higher for the lean phase conveying case than for the comparable dense phase
regime (average value 10.64 compared to 1.69).
Figure 7.5.6 shows the variation in the slope of these curves in a graphical format. In this case
since there is a consistent and substantial difference in slope between the dense phase and lean
phase conveying regimes, the first half of the Figure plots the slopes obtained from the dense
phase data, followed by the corresponding lean phase data. This Figure suggests that there is a
reasonable degree of consistency amongst all the models with regard to the treatment of bends,
though there appears to be an increased amount of variability in the lean phase data. The
occasional departure of model P C 1 A from the results of the other models is also evident.
o
<a
s
I
Experimental Data Sets
Figure 7.5.6 Model performance with respect to changing number of bend in the pipeline variation in the value of " m " (from Ap = b + m (number of bends)) for the seven models
and eight sets of conveying data, (dense and lean phase conveying results combined)
Figure 7.5.6 also highlights the extent of the increase in slope from the dense to the lean phase
regimes. The consistency of this increase is explored in Figure 7.5.7, where the ratio of the " m "
values (lean/dense in this case) is plotted. It is apparent from this Figure that the obvious peaks
in the lean phase end of the Figure 7.5.6 line up with the smaller peaks in the dense phase end,
and the ratio of their values, is quite consistent. The magnitude of this ratio is significant,
indicating an increase in slope in the order of six to eight times for the lean phase conveying
regimes. It is noted that the actual value of this ratio is likely to be dependent on the specific
locations within the dense and lean phase regions selected for these calculations, and it could not
be presumed, for example, that ratio of 6:1 could be consistently applied for a particular bulk
solid.
186
7
Power Based Pneumatic Conveying Model
- Implementation & Evaluation
Experimental Data Sets
Figure 7.5.7 Model performance with respect to changing number of bend in the pipeline variation in the value of " m " (from Ap = b + m (number of bends)) for the seven models
and eight sets of conveying data, (dense and lean phase conveying results combined)
7.6 Review of Model Performance - Preferred Versions
The preceding three Sections 7.3, 7.4 and 7.5 demonstrate that the dimensionless parameters
selected for use in the solids friction factor correlation have a major impact on the overall
performance of this model. This is the case despite the fact that the same basic assumptions
regarding the physical processes taking place during conveying have been applied in each of the
seven model versions analysed.
These Sections also highlight the possibility that a particular model may treat one aspect of
pneumatic conveying performance in an appropriate manner (say pipe length), whilst predicting
doubtful trends for another parameter (say pipe diameter). It is also quite possible for a number
of dubious trends to cancel out for a specific pipeline configuration, masking the likelihood that
that model will generate quite erroneous conveying characteristics for alternative pipeline
arrangements. Figures 7.3.3 and 7.3.4 illustrate instances of that situation. These shortcomings
in the modelling process are generally not immediately apparent w h e n the model is used to
generate conveying characteristics from a set of experimental data. Each of the seven models
analysed normally works well in that respect. However w h e n the models, mcorporating a solids
friction factor correlation derived for a particular pipeline configuration, are applied to a
different pipeline arrangement, the previous work has shown that in a number of cases some
dubious trends m a y be introduced.
One reason for the incorporation of erroneous trends lies in the lack of, or inadequate, variation
in value of the dimensionless parameters used in the solids friction factor correlation. Model
PC4, for example, uses I —J as one of its two dimensionless parameters. Where the solids
friction factor correlation is carried out using data from a single pipeline configuration (the
typical situation) this term would remain a constant throughout the analysis. In these situations
this term should properly be included in the constant coefficient and not treated as an
independent variable. Where the term is included in the solids friction factor equation it cannot
contribute in any reliable fashion to the evaluation of the solids friction factor for any pipeline
configuration other than the one applying to the experimental data set. Even where particular
effort has been m a d e to collect data from a range ofpipe configurations (say three or four), this
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
187
parameter still takes on only a very limited set of values and cannot be expected to contribute
usefully in the correlation analysis.
Model PC2 illustrates a second area of difficulty that can arise when carrying a solids friction
factor correlation. This model uses the largest number of dimensionless parameters (five,
compared to two for the others or only one for model P C I ) in the correlation analysis. A s the
number of dimensionless parameters is increased, the potential to fit accurately a specific set of
data improves, however the possibility of incorporating false trends and hence incorrectly
predicting the performance of alternative configurations also increases. This model is the least
reliable (indicated by the failure to produce trend results in a number of cases) of the seven
versions analysed in Sections 7.3 to 7.5 .
Combining the results from Sections 7.3 to 7.5 it is possible to select model versions which are
likely to perform more reliably. Models P C 2 and P C 5 demonstrate incorrect trends at time with
respect to length variation, with model P C 4 also producing some doubtful results. Models P C 2 ,
P C 4 and to a lesser extent P C 5 generate some dubious trends with respect to diameter variation.
Most models appear to perform reasonably with respect to variation in the number of bends,
though in a number of cases the increase in pressure drop predicted by model P C I A is very low.
Model versions PCI, PC3 and PC6 emerge as the more promising options. Simplicity and
stability recommend the use of model PCI. The increase in complexity offered by models P C 3
and P C 6 result in some divergence from model P C I , particularly towards the more extreme
ranges of the parameters being analysed (D, L , number of bends). It is possible that this will
allow the model to reproduce more accurately the behaviour of the physical system. This m a y
also be accompanied, however, by a decrease in robustness of the model.
Having carried out the solids friction factor correlations for the seven models for a number of
sets of experimental data, it is also possible to examine the variations in value of the resulting
parameters in the friction factor expression, ie const., a,fi,y, in:
\ = const.XTT".71%.71%.nl...
E q 7.6.1
Ideally, the treatment of the physical processes in the model, and selection of the dimensionless
parameters (nv it 2--.etc) would be such that the values of the constants in E q 7.6.1 (const
,a,p,y,
) would be the same for a given bulk solid regardless of the pipeline configuration
used to obtain the experimental data for the correlation analysis. In practice, the simphfications
and assumptions accepted in modelling the pneumatic conveying process together with possible
inadequacies in the dimensionless terms chosen, result in variations in these Equation
parameters. The following tables give an indication of the extent of this variation, using the
results of the correlation analysis applied to the sets of data examined in Sections 7.3 to 7.5.
Conveying data from three pulverised coals and two fly ashes has been used in obtaining the
results presented in these tables. It m a y be noted that several additional sets of data have been
included for bulk solids PF1, P F 2 and PF3 where the solidsfrictionfactor correlation has also
been carried out on the combined sets of conveying data where the same material has been
transported through two test rigs.
188
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Table 7.6.1:
Values of constant coefficient (const), from solids friction factor Equation:
X, = const.xn? . nf. TC\ .n^...
for models P C I to PC6, P C I A
Data
PFl-TRl
PFl - TR2
above 2 combined
PF2 - TRl
PF2-TR2
above 2 combined
PF3 - TRl
PF3-TR2
above 2 combined
FAl - TR3
FA2 - TR12
average:
Table 7.6.2:
14.221
5.096
10.557
PC2
PC3
1.023
1.258
18.266
283.2
33011
21.175
.009434
11092
1200.6
1.478
0.004014
.013149
.99559
1066.6
5.17
1.895E-5
0.002475
0.63
4245.52
0.73
.009349
1.0049
14.778
8.16
10.10
13.548
6.176
7.384
5.82
9.35
PC4
0.02696
0.9975
0.9449
PC5
PC6
0.0125
1.891
.00037
0.0448
1.391E-7
25.951
163.95
PC1A
8.374
6.94
2.748
10822
4.58
0.000382
0.03205
161.6
3586.4
11.305
1.875
0.03205
0.03205
646.2
2.837
6.728
0.03205
9.161E-5
488.46
2.982
5.749
4.361
208221
18929.20
2.856E-4
1445.91
0.463
5.28
7.773
Values of a, from solids friction factor equation:
Xs = const.xn". K% . nl .nl...
for models P C I to PC6, P C 1 A
Data
PFl - TRl
PF1-TR2
above 2 combined
PF2 - TRl
PF2-TR2
above 2 combined
PF3 - TRl
PF3-TR2
above 2 combined
FAl - TR3
FA2 - TR12
average:
Table 7.6.3:
PCI
7.008
PCI
-0.57669
-0.63125
-0.58938
-0.66623
-0.62621
-0.68362
-0.69127
-0.65873
-0.67083
-0.76296
-0.56451
-0.65
PC2
PC3
PC4
PC5
PC6
0.70743
0.250429
0.026685
-0.49241
-0.3444
-0.25028
-0.02663
-0.20802
PC1A
-0.59758
0.055676
0.524621
-0.56962
-0.53757
1.051399
0.321591
-0.58342
-0.60247
0.926989
-0.52749
-0.6604
0.54241
0.653959
0.594834
0.204723
1.308147
0.536066
1.203577
0.72
-1.00372
0.007788
-0.85531
-0.62402
0.313256
-0.64421
0.085644
1.658071
-0.12
-0.22261
0.273772
-0.38399
-0.23691
0.356211
-0.43338
-0.2736
0.361867
-0.47314
-0.25299
-0.15
-0.45326
1.005596
-0.00816
-0.43627
0.623903
-0.31302
-0.52743
-0.08537
-1.65857
-0.19
0.687313
-0.24292
0.687755
-0.0497
-1.60251
0.20
-0.60059
-0.60123
-0.71136
-0.36279
-0.58
Values of p, from solids friction factor equation:
Xs = const.x n°. n$. nl .nsA...
for models P C 2 to PC6, P C I A
Data
PFl-TRl
PF1-TR2
above 2 combined
PF2 - TRl
PF2-TR2
above 2 combined
PF3 - TRl
PF3-TR2
above 2 combined
FA1-TR3
FA2-TR12
average:
PC2
1.927458
1.106235
1.448875
0.893522
-0.12881
2.213692
1.03092
1.61507
-0.74
PC3
-0.59912
-0.70623
-0.69545
-0.81348
-0.69412
-0.81965
-0.75448
-0.66806
-0.78281
-0.82004
-0.3327
-0.70
PC4
-0.58011
-0.56652
-0.56818
-0.65145
-0.60544
-0.58751
-0.70854
-0.59986
-0.68235
-0.71126
-0.50399
-0.62
PC5
-0.34885
-0.67961
-0.1242
-1.81913
-0.68589
-0.16255
-1.37833
-0.35507
-0.16503
-0.73472
1.32595
-0.47
PC6
-0.3916
-0.76218
-1.21823
-1.8624
-1.05944
-1.73760
-1.43971
-0.42194
-1.46596
-0.76552
1.266304
-0.90
PCI A
-0.59758
-0.56962
-0.53757
-0.58342
-0.60247
-0.52749
-0.6604
-0.6006
-0.60123
-0.71136
-0.36279
-0.58
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
189
Table 7.6.4: Values of /, 8, e, from solids friction factor equation:
X\ = const.xn". n$.nl. n5A .n\ for model P C 2 (not used in other versions)
Data
PFl - TRl
PF1-TR2
above 2 combined
PF2 - TRl
PF2 - TR2
above 2 combined
PF3 - TRl
PF3-TR2
above 2 combined
FAl - TR3
average:
Y
8
€
-0.857036
failed to solve
-0.359059
-0.746854
failed to solve
-0.412727
0.044076
-1.422947
-0.336948
-0.957286
-0.62
-1.464363
failed to solve
-0.240509
-0.691034
failed to solve
0.105297
1.101959
-1.585198
-0.129957
-0.561003
-0.43
-0.705681
failed to solve
-0.793037
-0.768811
failed to solve
0.833507
-0.84889
-0.689874
-0.851339
-1.074368
-0.53
A s can be seen in the above Tables 7.6.1 to 7.6.4, there is a considerable variation in the values
of these terms. This is reasonable given the different sets of dimensionless parameters used in
the correlations, and the different bulk solids involved. For the stability of the models, it would
be hoped that for a particular version, the values would cluster around a standard set of values.
In particular it m a y be expected that the power terms (a, ft, y,
) would have a consistent sign,
indicating that thefrictionfactor would trend in the same "sense" in response to an increase in
the corresponding dimensionless parameter for all bulk solids. T o simplify the examination of
this aspect, four Figures have been prepared based in the above Tables. Since actual values of
the terms (const. ,a,fi,y,
) across the seven models vary too widely to be usefully plotted on
the same graph, the values of each term for each model (for example the a terms for P C I ) have
been "normalised" by dividing by the average value obtained for that term. Where the model
produces terms that have a reasonable degree of consistency, the resulting normalised values
should cluster around unity.
12 • 1
\
1
1
Experimental Data Sets
Figure 7.6.1 "Normalised" values of (const.), from X\ - const.x n". n$ nl .nl...
190
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
Experimental Data Sets
Figure 7.6.2
"Normalised" values of a, from Xs = const.xn". n£ • nl. nl
Experimental Data Sets
Figure 7.6.3
"Normalised" values offi,from Xs = const.xn".n£. nl. n.
7
Power Based Pneumatic Conveying Model - Implementation & Evaluation
191
Experimental Data Sets
Figure 7.6.4
"Normalised" values of y, 8 and s, from Xs =
for model P C 2
const.xn".n$.nl.nl.ne5
Figures 7.6.1 to 7.6.4 certainly highlight the variability present in the terms obtained after th
correlation analysis. Model P C I again demonstrates a notable degree of stability, warranting a
continued exploration of its capacity. Thefluctuationin sign of the power terms for the
remaining models is noted, even for the versions (PC3 and P C 6 ) preferred after the trend
analysis. In considering this variation it should also be noted that the procedure applied in the
correlation analysis includes trialing a range of VSR (solids to air velocity ratios) values and
selecting that value that gives the best correlation coefficient for the set of experimental data
being analysed. This is intended to provide an objective means of selecting a VSR value
appropriate for each material. Changes in the VSR value significantly affect m a n y aspects of the
model, and would account in part for the variation in the above figures.
Having concluded this Section on comparing the trends inherent in various versions of the power
base model, it is necessary to select one or more versions to carry forward for the purposes of
examining other aspects of the models performance. T o this end models P C I and P C 6 have
been selected. It is not suggested that these models represent the ultimate options; it is hoped
that further work will reveal refinements to the model that will increase its capacity to predict the
complex processes occurring in pneumatic conveying.
192
8
Conveying Limits - Power Consumption Distribution
Chapter 8
CONVEYING LIMITS
POWER CONSUMPTION DISTRIBUTION
In the course of analysing a pneumatic conveying system via the power based model that has
been described, a number of interesting features of the conveying process become accessible.
This Chapter considers two of these aspects. The first is one means of identifying regions in the
conveying characteristics map where reliable conveying becomes unlikely. The second feature is
the ability to extract information on the power requirements of various components of the
transport system (air-only, bends, solids friction etc) under different conveying regimes, and to
determine the specific power consumed in moving the bulk solid. A s a result of the discussion
and conclusions reached in the previous Chapter, where results are presented, these will be based
on analysis using models PCI or PC6.
8.1 Conveying Limits
Conveying characteristics curves, such as those shown in Figure 8.1.1, provide very useful
information for optimising the operating parameters for a conveying system to achieve a
required bulk solid throughput.
5 kg/s
4 kg/s
3 kg/s
2 kg/s
1 kg/s
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.1.1 Conveying Characteristics Plot for: Bulk Solid: PFl, Pipeline Arrangement:
T R 2 , Model P C I
8
Conveying Limits - Power Consumption Distribution
193
There are a number of boundaries implicit in these diagrams delmeating the viable regions for
reliable conveying; these m a y not be immediately apparent.
A lower limit is defined by a line corresponding to the pressure drop versus air mass flow rate
relationship w h e n air-only is flowing through the pipeline. Feeding solids into the air stream
would inevitably increase the total system pressure drop, hence preventing any operating points
below this line. In this model, the pressure drop defined by the Darcy-Weisbach, E q 6.4.4,
would form this lower boundary, with possible minor contributions from the kinetic and
potential energy changes attributed to the air. It would be possible to establish a more accurate
boundary in this region if desired by, for example, taking into account the effect of bends as
proposed by W y p y c h and Pan (1991). This boundary assumes considerable importance for
vacuum conveying systems, since the available envelope for reliable conveying can be
significantly narrowed as this pressure differential approaches the vacuum capacity of the prime
mover. However for the positive pressure conveying systems that are the main focus of this
Thesis, this lower boundary is not normally relevant.
There will be an upper boundary defined by the performance/operating characteristics of the air
supply system that will be of relevance for a positive pressure system. However this boundary
also is not the primary focus of this discussion.
The significant boundary in this instance is that found at the left hand end of the lines of constan
solids mass flow rate on the conveying characteristics plot. It is recognised that at some point
along this line, conveying will become "rough", withfluctuatingpressures, possible shaking or
shuddering of the pipeline, occasional blockages and so on. ha other words, there will be a
boundary defming the limit of reliable conveying. The location of this boundary is most
accurately deterrnined experimentally, by carrying out numerous tests that approach these
conveying limits over a range of different operating conditions. This can be a reasonably
arduous experimental task and is specific to the pipeline on which the tests are conducted. If
those data are then to be used tojjrjsdict^heJiehaviour of different pipeline configurations, it is
difficult to translate this boundary with certamryX^ccurate knowledge of the location of this
boundary becomes important if it is necessary to operate a pneumatic transport system at a
m a x i m u m efficiency/least wear point.
The process of applying the assumptions in this power based model to a set of experimental
data, and developing the corresponding conveying characteristics, requires recognition and
handling of a simple physical constraint that forms an appropriate "upper limit" boundary along
the upper left hand comer of a conveying characteristics plot. This constraint is primarily faced
in the analysis of the air-only flow component. The methodology of sphtting the air flow into an
interstitial component (flowing between the fluidised bulk solid particles) and a "top air"
component flowing above the bulk solid, requires identification of these areas prior to their
subsequent use. A s the air flow is reduced for a given solids mass flow rate (ie proceeding to the
left along a contour of constant solids flow) it is clear that the area available for the "top air"
flow reduces. A rational upper limit would be found where this area has reduced, or is close to
zero, ie where thefluidisedbulk solid completely fills the pipe cross-section. The location of
this point is dependent on the air and bulk solid mass flow rates, the pressure in the pipe at the
point being considered and perhaps critically, the average solids velocity at that point. This last
aspect is critical in the sense that it is not apparent from simply considering a conveying
characteristics plot. Additional data and/or assumptions regarding the behaviour of the bulk
solid are required. The VSR factor linking the average solids velocity to the air velocity used in
this power based model allows the required additional information to be calculated. This is an
essential element of other parts of the model (solids friction, bend losses, changes in kinetic
energy etc), however it also allows this approach to determining an upper limit boundary to be
identified.
194
8
Conveying Limits - Power Consumption Distribution
Figure 8.1.2 shows air-only and upper limit boundaries added to the conveying characteristics
plot presented in Figure 8.1.1. In this case the upper limit boundary has been set at the point
where thefluidisedbulk solid occupies 9 5 % of the pipe cross-section.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.1.2
Conveying Characteristics Plot showing Air-Only and Upper Limit
Boundaries: Bulk Solid: PFl, Pipeline Arrangement: T R 2 , Model P C I
As can be seen on Figure 8.1.2, the upper limit boundary deterrnined by this method ties in a
region where unstable conveying m a y well be expected to occur. The same approach clearlv
may also be applied by other versions of the model. Figure 8.1.3 shows conveying
charactenstics and air-only and upper limit boundaries for the same material and pipeline
combination, using model PC6.
Pyemic
200.00 ••
-upper limit
-5 kg/s
-4 kg/s
-3.5 kg/s
3 kg/s
-2.5 kg/s
2 kg/s
1.5 kg/s
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.1.3
Cojmytog Charaeterfetie, F U showing Air-Only and Upper Limit
Bonndane,: B„]k Solid: PFl, Pipeline Arrangement: T R 2 , Model P C 6
8
195
Conveying Limits - Power Consumption Distribution
Both models produce very similar conveying characteristics curves, and the locations of the
upper limit boundaries also appear to match. Figure 8.1.4 specifically compares these two
boundaries from the above conveying characteristics, confirining the similarity.
210 --
-upper limit PC1
upper limit PC6
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Air Mass Flow Rate (kg/s)
Figure 8.1.4
Comparison of Upper Limit Boundaries: Bulk Solid: PFl,
Pipeline Arrangement: T R 2 , Models P C I and P C 6
Figures 8.1.5, 8.1.6. and 8.1.7 show a corresponding set of results for a different pulverised
flowing through a smaller diameter pipeline configuration. The location of the experimental
data points used to determine the conveying characteristics is also shown on the first two of these
Figures. A similar outcome is observed in this case, with the upper limit boundaries following a
very similar path, though the curve found using the P C 6 model is approximately lOkPa higher
than the P C I equivalent.
250
-4>—-upper limit
1 kg/s
0.8 kg/s
0.6 kg/s
0.5 kg/s
0.4 kg/s
0.3 kg/s
0.2 kg/s
0.1 kg/s
- # — air only
A
data points
200 --
a 150 --
2
100 -•
£
50 ••
0.002
0.004
0.006
0.008
0.01
Air Mass Flow Rate (kg/s)
Figure 8.1.5
Conveying Characteristics Plot showing Air-Only and Upper Limit
Boundaries: Bulk Solid: P F 3 , Pipeline Arrangement: T R l , Model P C I
196
8
Conveying Limits - Power Consumption Distribution
+
upper limit
1kg/s
0.8 kg/s
0.6 kg/s
0.5 kg/s
0.4 kg/s
— 0 ^ kg/s
-0.2 kg/s
0.1 kg/s
-$)— air only
^
data points
0.002
0.004
0.006
0.008
0.01
Air Mass Flow Rate (kg/s)
Figure 8.1.6
Conveying Characteristics Plot showing Air-Only a n d U p p e r Limit
Boundaries: B u l k Solid: P F 3 , Pipeline A r r a n g e m e n t : T R l , M o d e l P C 6
-upper limit PC1
-upper limit PC6
0.002
0.004
0.006
0.008
0.01
Air Mass Flow Rate (kg/s)
Figure 8.1.7
C o m p a r i s o n of U p p e r Limit Boundaries: B u l k Solid: P F 3 ,
Pipeline Arrangement: T R l , Models P C I and P C 6
It may be concluded from these results that a useful indication of the location of one boundary
defining an upper limit for reliable conveying, can be deduced from the parameters determined
during the application of this power based model. The simple linkage of this boundary to basic
physical properties of the bulk solid, and flow behaviour within the system, strengthens the
significance and reliability of the approach. In addition, application of the model to a different
pipeline arrangement does not require any deliberate translation of this boundary to the n e w
system. Instead, the boundary can be directly deterrnined for the n e w system, using the same
procedures on the n e w set of pipeline configuration parameters (length, diameter and number of
bends).
°
8
Conveying Limits - Power Consumption Distribution
197
It should be noted, however, that whilst it is certain that restrictions of this nature will form one
of the constraints limiting the reliable conveying range of pneumatic conveying systems, the
approach developed here incorporates a considerable degree of simplification. A s noted
previously, the identification of this boundary follows from consideration of the area available
for "top air" flow. These air flow calculations have been m a d e at m e a n pipeline pressure
conditions. The pressure used in the calculations has a major impact on the result, in that it
determines the air density and velocity and, through the VSR factor, fixes the solids velocity.
Therefore these calculations are indicative of mean pipeline conditions and the "upper limit"
found relates to a full pipe condition being approached at that point along the pipeline.
Applying the same calculations at other locations along the pipeline will result in different
alignments of this boundary. For example, at higher pressure regions (towards the inlet end)
these calculations would lead to a reduced average bulk solid velocity, resulting in an increased
proportion of the pipe cross-sectional area being required for the solids component. The end
result would be a lowering of this "upper limit" boundary. Other local effects, in particular
bends where the velocity of the solids component is likely to be markedly reduced, m a y result in
potential blockage zones through the same mechanism and m a y require further relocation of this
conveying limit boundary.
Despite the simplifications inherent in the approach undertaken, it is suggested that it is
reasonable to base these calculations on the mean pipeline condition since the development of
the model parameters (in particular the solids friction factor correlation and the selection of a
VSR value) is also based on that condition. Therefore the approach remains consistent.
As has been noted previously regarding other aspects of this model, improved understanding of
the physical relationships within the two phase flow regime (the variation of solids velocity with
respect to air velocity being a prime area) would allow a considerable further refinement in the
location of this boundary. Nevertheless this approach does offer a useful insight into real
constraints on the conveying process, and provides a rational basis for defining a reliable
conveying boundary. It m a y also be of value in identifying and assessing potential trouble spots
along a pipeline. For example, where the location of a number of bends in close proximity
causes the solids velocity to fall to a level that m a y result in a pipe-full condition being reached.
In the case of stepped pipelines, this approach m a y be adapted to indicate the suitable locations
for changes in pipe diameter.
8.2 Power Consumption Components
The reliable conveying boundary limit discussed in Section 8.1 is one instance of valuable
information underlying, but not immediately obvious from, conveying characteristics plots. The
power based pneumatic conveying model allows a second set of useful information to be
extracted in the process of generating conveying characteristics data. The application of this
model relies on the estimation of the power requirements of each of the elements of the
conveying system, with these being combined to determine the pressure drop. Therefore it is
relatively straightforward to separate and record the magnitudes of these components and assess
their impact on the overall operation of the system. This is not generally the case for other
models (for example those due to Bath, Szikszay, Weber) where m a n y of these components are
combined under the umbrella of the solids or total friction factor.
The model being discussed is based on balancing the power available to carry out the
transportation task (in the form of the isothermal expansion of the driving medium), with the
power consumed by a number of physical processes identified within the pneumatic conveying
system. These processes include the power requirement for the air-only component, though it is
noted that some effort has been m a d e to restrict the power loss in this case to that portion that
m a y be attributed to frictional drag between the air and the pipe wall. Thefrictionaldrag
between the air and the bulk solid is not considered a loss as it contributes positively to the task
of moving the material. The changes in potential and kinetic energy for the gas and solids
198
8
Conveying Limits - Power Consumption Distribution
components between the entry and exit of the pipe arrangement are also considered. These have
been lumped together in the following Figures; generally the solids component dominates in this
pairing. The other major area of power consumption is related tofrictionbetween the pipe wall
and the bulk solid. The contributions from the straight sections of pipeline and the bends are
considered and plotted separately so that their significance over a range of flow settings may be
appreciated. These components are brought into balance with the input power by the solids
friction factor correlation analysis together with the selection of an optimum VSR factor.
The approach adopted in presenting the variation of these power consumption components has
been tofixone of the three major operating parameters (say ms), cause another (say mf) to vary
through a range of values and use the model to calculate the value of the remaining parameter (in
this case Apt), and at the same time recording the magnitude of the power consumption
components listed above. The selection of the fixed, variable and calculated operating
parameters nominated above is perhaps the most relevant in that it follows the behaviour and
power consumption requirements of the system along a contour of constant solids mass flow rate
through the range of dense to lean phase conveying regimes. The results for two other
arrangements however (fixed pressure/variable air mass flow rate and fixed air mass flow
rate/variable solids mass flow rate) are also given.
Figures 8.2.2 to 8.2.7 show the results of this analysis for a pulverised coal (PF3) conve
through test rig TR2. The conveying characteristics for this particular bulk solid/pipeline
arrangement are given in Figure 8.2.1. The power component data plotted in Figures 8.2.2 to
8.2.7 have been deterrnined whilst traversing on selected paths through this set of conveying
characteristics. It will be noted that these power components have been presented in terms of
their percentage of the total power consumed rather than as an actual energy value. This is to
prevent the high power requirements at the lean phase end of the spectrum causing the behaviour
at the dense phase end to be suppressed by the graph scale.
A curve showing the specific transport energy (J kg'1 m"1) has been included on the power
consumption components plots to indicate the variation of this quantity in conjunction with the
trends in percentage consumption of power of the various components. With the exception of
Figure 8.2.3 where the specific transport energy curve has again been plotted for reasons of
scaling, separate plots of the actual transport energy requirements across the range of conveying
conditions, are provided to clarify the actual magnitudes of energy required.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.2.1
Conveying Characteristics: Bulk Solid: PF3, Pipeline Arrangement:
TR2, Model P C I
8
199
Conveying Limits - Power Consumption Distribution
Figure 8.2.2 is perhaps the most informative of this set of graphs in that it shows the distribution
of power consumption requirements along a contour of constant solids mass flow rate (in this
instance the 2kg/s contour). Consequently the relative significance of each of the terms can be
observed as the conveying process moves from a dense phase to a lean phase state. Using the
pressure m i n i m u m point on the 2kg/s contour from Figure 8.2.1 to indicate this point, this dense
to lean phase transition occurs at around m / = 0.018kg/s.
KE
PE
Solids Friction
Bend Friction
Air Component
Spec. Energy
a
o
£
B
ii
E
ii
*3
&
m
ii
\m
©
a.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.2.2
Power Consumption Requirements
Bulk Solid: P F 3 ; Pipeline Arrangement: T R 2 ; Model: P C I ;
Constant solids mass flow rate: ms = 2kg/s; [note: the specific energy curve
denotes the energy required to transport 1kg of material l m (units: J kg"1 m'1)]
A number of interesting features are apparent in Figure 8.2.2. At the lower air mass flow rate
region (including the dense phase region), the dominant power consumption component is the
solids friction in the straight sections of the pipe, absorbing up to 7 0 % of the total power. It
would appear likely, therefore, that it is predominantly the behaviour of this term that causes the
contours of constant solids mass flow rate to curve upwards at the low air mass flow rate end of
the conveying characteristics plot, Figure 8.2.1. The other major component that emerges to
dominate in the higher air mass flow rate regions (lean phase, high velocities) is the bend friction
power loss. The magnitude of this term is perhaps surprising, particularly as there are only four
bends in this pipeline. It is notable that bend friction requires an increasing proportion of the
total power consumed at the same time that this total power consumption is increasing. It m a y
be further noted that this increase in total power consumption, as indicated by the specific
transport energy curve, is increasing at an exponentiallyrisingrate, emphasising the influence of
bends in the pneumatic conveying process.
Considering the remaining curves shown in Figure 8.2.2, at the low air mass flow rate end the
potential energy component absorbs a significant proportion of the total power requirement.
However for the constant solids mass flow rate case applying in this Figure, the power
requirement associated with changes in potential energy is substantially constant. Therefore the
prominence of this term at the low air flow end region is a consequence of the reduced total
power requirement in this region rather than any increase in power consumption resulting from
potential energy changes. The kinetic energy term shows a steady increase across the Figure as
the air mass flow raterises,in this case reaching a m a x i m u m 1 4 % of the total power. This is a
200
8 Conveying Limits - Power Consumption Distribution
natural consequence of the increased air mass flow rate and higher pressure drops leading to
bigger velocity differentials between the pipeline inlet and exit. The air-only component of
power consumption indicated by this Figure is perhaps surprisingly small across the range of
conveying conditions,risingto around 5 % of the total power consumption at the high end of the
air mass flow rate. It should be recalled, however, that this component is based only on the air
to pipe wallfrictionof the "top air" portion of the total air flow.
Finally, the specific transport energy curve also gives some relevant insights into the conveying
process. This curve indicates the energy required to transport one kilogram of the bulk solid one
metre along the pipeline, and shows afifteenfold increase in specific power consumption over
the range presented in the Figure. For this case (constant pipe length, constant solids mass flow
rate), the total energy requirement would be a direct multiple of this curve. This is clearly a very
large variation and, since it would be directly related to operating and maintenance costs of the
system, indicates that the selection of the operation conditions for a given pneumatic conveying
system is likely to have a major impact on its overall viability. It is also clear that the pressure
minimum point on the solids mass flow rate curve is not related to a similar m i n i m u m on the
specific transport energy curve. This aspect is shown more clearly in Figure 8.2.3, which
focuses on the solids mass flow rate line and the specific transport energy curve.
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
-DpfkPa)
-Specific Energy (J/kg m )
Figure 8.2.3 Power Consumption Requirements: Bulk Solid: P F 3 ,
Pipeline Arrangement: T R 2 ; Model: P C I ; constant solids mass flow rate: m s = 2kg/s
Figure 8.2.3 shows the specific transport energy continuing to fall as the air mass flow rate is
reduced below the pressure minimum point on the 2kg/s solids mass flow rate contour.
Therefore, m terms of minimising operating costs, pipeline wear and product attrition, there
would appear to be no particular merit in seeking to operate around the pressure minimum
point. This Figure suggests that the most advantageous operating point would be as far into the
dense phase region as possible, consistent with reliable conveying. The pressure minimum point
becomes of significance in selecting operating conditions if the requirement is to maximise the
solids throughput for a given total systems pressure drop. Figure 8.2.5 illustrates this point for a
total pressure drop of 60kPa.
The Mowmg four figures provide supplementary information to that demonstrated in Figures
ATA I Z
^ not b e ^s^ssed in the same detail. They illustrate the trends
predicted by the model when the alternative major conveying parameters are held constant
respectively. The results displayed in Figure 8.2.4 are based on a constant system pressure drop
8
Conveying Limits - Power Consumption Distribution
201
and, in a similar manner to Figure 8.2.2, show the variation in power consumption patterns
across both the dense and lean phase regions. Consequently the curves follow similar trends to
those in Figure 8.2.2. The dominance of the solids friction components (straight pipe and bends)
is again evident, the specific transport energy curve shows an even greater rate of increase with
increasing air mass flow rate. Falling solids mass flow rate values at the high air mass flow rate
end of the Figure (indicated in Figure 8.2.5) contribute to the steeplyrisingspecific transport
energy requirement at that end.
KE
PE
Solids Friction
Bend Friction
X—Air Component
+ — S p e c . Energy
88
e
S
ii
E
£
*3
a
m
4)
BH
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
Figure 8.2.4
Power Consumption Requirements
Bulk Solid: P F 3 ; Pipeline Arrangement: T R 2 ; M o d e l P C I ;
Constant pressure drop: Apt = 60kPa; [note: the specific energy curve
denotes the energy required to transport 1kg of material l m (units: J kg'1 m 1 ) ]
In contrast, the total power requirement increases in a linear fashion with air mass flow rate, a
shown in Figure 8.2.5. This is expected and is a direct result of the isothermal expansion power
equation used as the basis for this model. It should be noted that this value should not be used to
select the prime mover required for the air supply system as any losses in the air compression
system have not been taken into account. In an economic study of a pneumatic conveying
system Marcus et al, (1986), Reed indicated that the Thames Polytechnic use a value of twice
this isothermal power requirement.
8
202
Conveying Limits - Power Consumption Distribution
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
• m s (kg/s)
-Total Power Consumption (kW)
Figure 8.2.5 Power Consumption Requirements: Bulk Solid: PF3,
Pipeline Arrangement, constant pressure drop: Ap, = 60kPa: T R 2 , Model P C I
Figures 8.2.6 and 8.2.7 are based on a constant air mass flow rate analysis and appear to show
slightly different trends from those previously presented. These Figures, however, do not
traverse the dense to lean phase conveying regions, but rather examine a vertical slice through
the conveying characteristics plot. At the air mass flow rate selected (0.06kg/s) it is clear from
Figure 8.2.1 that a lean phase conveying regime is in place. Consequently the dominance of the
bendfrictioncomponent across the range is to be expected. The specific transport energy curve
apparently shows the opposite trend to that demonstrated in the previous Figures. This is in fact
a result of the configuration chosen for this Figure, and the result confirms the previous Figures
in that increasing the solids loading results in a reduced specific transport energy requirement.
70.00
KE
PE
Solids Friction
Bend Friction
X—Air
Component
# — S p e c . Energy
"3
©
60.00
mm
mm
O
X
50.00
•mm
s
40.00
ii
E
.m
30.00
5
IX
20.00
&
10.00
0.00
200
Total Pressure D r o p (kPa)
Figure 8.2.6 Power Consumption Requirements
Bulk Solid: PF3; Pipeline Arrangement: T R 2 ; Model P C I ;
Constant air mass flow rate: mf= 0.06kg/s; [note: the specific energy curve denotes
the energy required to transport 1kg of material l m (units: J k g 1 m 1 ) ]
8
20
Conveying Limits - Power Consumption Distribution
40
60
80
100
120
140
160
203
180
200
Total Pressure Drop (kPa)
-ms(kg/s)
-Total Power Consumption (kW)
Figure 8.2.7
P o w e r Consumption Requirements: Bulk Solid: P F 3 ,
Pipeline Arrangement, constant air mass flow rate: mf— 0.06 kg/s: T R 2 , M o d e l P C I
The shape of the ms curve in Figure 8.2.7 is of interest in that it indicates the trend in spacing
solids mass flow rate contours in the conveying characteristics plot. The basically linear
relationship indicates an overall even spacing (as shown in Figure 8.2.1 at this air mass flow
rate), however the shape at the beginning and end of the curve suggests a tendency for contours
to be closer together at low ms values, and tendency for them to become more widely spaced at
higher values. N o major importance is attached to this result, though trends suggested appear
reasonable, as it is only one case of m a n y that could be analysed for this bulk sohd/pipeline
combination. However it does suggest an additional means of analysing the performance of
pneumatic conveying models.
The results presented in Figure 8.2.2 to 8.2.7 utilise the power based model (version PCI in this
case) to illustrate trends in the pneumatic conveying process as interpreted by this modelling
approach. T o be able to carry out this analysis, it isfirstnecessary to process a set of
experimental data and obtain a solids friction factor correlation and hence determine the required
model parameters. It is quite possible at this point to apply the same analysis to the original set
of experimental data points and determine the share in power consumption among the same set
of components for each of these data points. This procedure w a s carried out during the
development of the power based model, and in fact prompted the development of analysis tools
to allow the generation of data of the type shown in Figures 8.2.2 to 8.2.7. Using the model to
develop these trend graphs allows full control of the operating conditions being examined (for
example holding the solids mass flow rate constant and varying the air mass flow rate in even
steps) and also eliminates the scatter that would be expected if the analysis is applied to
experimental data. Hence it is possible to obtain clear relationships. Nevertheless it is useful to
observe the results obtained w h e n the same analysis is applied to raw experimental data. The
following Figures 8.2.8 to 8.2.12 demonstrate this for a pulverised coal (PFl) conveyed over
two different pipeline configurations (TRl and T R 2 ) . The solidsfrictionfactor correlation has
been applied to the combined data. Thefirstfour figures examine the percentage of total power
consumption absorbed in overcoming solids friction (straight pipe and bend components again
considered separately). Figure 8.2.12 shows the variation in specific transport energy indicated
for these data. Model P C 6 has been used in this analysis.
8 Conveying Limits - Power Consumption Distribution
204
90 y
80 ••
*
70 -•
i
m
ii
* 60 ••
e
* 509m
» 40-mm
*A
•S"
*
® 30 ••
20 ••
10 -•
+
0.01
0.03
0.02
0.04
0.05
0.06
0.07
Air M a s s Flow Rate (kg/s)
•Test Rig TR1 BTestRigTR2 ]
Figure 8.2.8 Power Consumption Requirement for Solid Friction (straight pipe
component): Bulk Solid: PFl; Pipeline Arrangement: T R l and TR2; Model P C 6
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Air Mass Flow Rate (kg/s)
•Test Rig TR1 BTest Rig TR2
Figure 8.2.9
Power Consumption Requirement for Bend Friction Component;
Bulk Solid: PFl; Pipeline Arrangement: T R l and TR2; Model P C 6
demon
conmonfnK i ? i
strate that the significant role identified for the solids friction
P
S
exZZS, A ,
? T u FlgUreS 8'22 t0 8-2-7' is * ° evident when apphed to the
p S o n o f Sri
** *" ^
* • * ™ ° b s e r v e d P * * * "frictionIncreasing in
r6duCed
tteTTi^
wSfohoLtT
8 t0
l^tc^A^/r^^
SetS
T ** **
^
^
* flows> bend ^
°
fdata from
follo
™S
«» opposite
** ^ ° P i P ^ arrangements,
<fi8tinctfromone
« « * « • ™ . is not surprising
(TR 1} ha§&d S m , w of2 5 4 m m
Wri*^2 S 52 I t f T ? * u
> approximately half mat of
r e at
S3me
massflowrate
S i would he I n •' ? w °
*•
^
** superficial air velocity in
t m W 8
fTR2 hence
te o ^ toTSTT^ ^
** °
'
****** P M c d interactions would
expected to apply. To explore the relevance of velocity in this context, the following two
8
205
Conveying Limits - Power Consumption Distribution
Figures 8.2.10 and 8.2.11, have been prepared. For these Figures the data from Figures 8.2.8
and 8.2.9 have been replotted with the X axis n o w being the actual superficial air velocity (ie the
superficial air velocity calculations include allowance for the presence of solids).
6
8
10
12
14
16
18
20
Actual Superficial Air Velocity (m/s)
•Test Rig TR1 •TestRigTR2
Figure 8.2.10 Power Consumption Requirement for Solid Friction (straight pipe
component): Bulk Solid: PFl; Pipeline Arrangement: TRl and
TR2; Model PC6
6
8
10
12
14
16
18
20
Actual Superficial Air Velocity (m/s)
•Test Rig TR1 BTest Rig TR2
Figure 8.2.11
Power Consumption Requirement for Bend Friction Component;
Bulk Solid: P F l ; Pipeline Arrangement: T R l and T R 2 ; M o d e l P C 6
Figures 8.2.10 and 8.2.11 show that w h e n this aspect of conveying performance is compared on
the basis of velocity rather than air mass flow rate, there is considerable similarity between the
results obtained on both testrigs.A s afinalexarnination along this line, Figure 8.2.12 shows the
variation of specific transport energy for the previous set of experimental data from both test rigs
plotted against the actual superficial air velocity.
206
8
Conveying Limits - Power Consumption Distribution
6
8
10
12
14
Actual Superficial Air Velocity (m/s)
•Test Rig TR1
•Tesst Rig TR2
Figure 8.2.12 Variation of Specific Transport Energy with Actual Superficial Air
Velocity; Bulk Solid: P F l ; Pipeline Arrangement: T R l and T R 2 ; M o d e l P C 6
The similarity in behaviour between both sets of data is again evident, though the results for TRl
are tending to diverge from T R 2 at higher velocities. It is important to be aware of the
differences between the pipeline arrangements for the two rigs w h e n considering these results,
and in this instance it should be noted that there are nine 90° bends in T R l compared to four in
T R 2 . Given the increasing influence of bends at higher velocities, the higher rate increase in
specific transport energy indicated for T R l at higher velocities, is to be expected. This factor
also affects the data presented in the preceding two Figures. A further point to note in regard to
Figure 8.2.12 is that the results presented are not influenced or modified in any w a y by the
assumptions (other than the VSR value selected) m a d e in applying this power based pneumatic
conveying model, or by the solidsfrictionfactor correlation process, but follow simply from the
conveying parameters recorded for each test point (pressure drop, air and solids mass flow rates)
and the application of the isothermal power equation.
8.3 Conclusions
The results presented in this Chapter identify a number of valuable insights that have emerged in
the development and application of the power based model. B y considering possible physical
processes for the treatment of the air and solids phases, the existence of a basic constraint on the
viable conveying range for a bulk solid can be identified. Application of this constraint, even in
the simplified form developed here, indicates conveying limit boundaries that relate well to
experimental experience.
The ability to split power consumption in the pneumatic conveying process into a number of
distinct areas, and observe the variation in these components across a range of conveying
conditions, provides a considerable additional insight into the behaviour of this model and, it is
suggested, into the pneumatic conveying phenomena as a whole. T h e power consumed in
overcoming frictional forces whilst moving the solids component (through the straight pipe
sections as well as the bends) is shown to dominate, as would be expected. The balance between
the power absorbed in the straight pipe sections and around bends varies considerably with the
air mass flow rate (or more particularly the air velocity), highUghting that potential impact bends
can have on a pneumatic conveying system under lean phase conditions.
The specific transport energy requirement for conveying a bulk solid can be readily deterrnined.
This parameter is of major importance in assessing the viability of pneumatic conveying systems
8
Conveying Limits - Power Consumption Distribution
207
as it is directly related to the overall power requirements, capital costs, operating costs,
maintenance costs and the likely product degradation and contamination (with eroded pipeline
material). The results presented in this Chapter support moving the operating point of a
conveying system as far as possible into the dense phase region, consistent with reliable
conveying performance. Graphs showing the variation of the specific transport energy across the
conveying spectrum would assist in selecting an appropriate compromise between reliable and
economical conveying.
9. Scaling, Long Distance Conveying, Stepped Pipelines
Chapter 9
SCALING
LONG DISTANCE CONVEYING
STEPPED PIPELINES
The development of pneumatic conveying models provides an opportunity for exploring the
physical interactions involved in this m o d e of transport, as demonstrated in several areas in the
previous Chapter. The application of these models in practical design tasks is one area where
tangible benefit m a y be obtained from the insights gained in this process. In most cases the
desired outcome in applying a model is the accurate extrapolation of a set of experimental data
obtained on a particular pipeline to alternative pipeline configurations. A n alternative pipeline
m a y have a different total length, a different diameter/s, different numbers of bends and so on.
The transfer of these results from the source pipeline configuration to alternative configurations
is called scaling.
Since it is less expensive to build a smaller test rig, generally (though not exclusively) this
scaling task will involve estimating the performance of larger pipeline systems based on test data
obtained on the smaller test rig. W h e r e the required pipeline reaches substantial lengths, issues
associated with long distance conveying need to be considered. These include the rehability of
using m e a n pipeline conditions to indicate the performance of the entire pipeline as the range of
conditions experienced along a long pipeline increases. T h e compressibility of the driving
medium in this m o d e of transport also means that economical conveying systems frequently
require the use of stepped pipelines in longer distance applications. L o n g distance conveying
systems require an increased total pressure drop to transport the bulk solid. W h e r e a single
diameter pipeline is used for such a system, it is likely to result in unacceptably high velocities at
the exit end of the system. This would be accompanied b y accelerated wear at impact points
along the pipeline, excessive power consumption around bends, and increased overall power
consumption and operating costs. A stepped pipeline, with the pipeline diameter increasing at
appropriate points along the pipeline, allows a measure of control over this velocity. Therefore
ability to estimate the performance of stepped pipeline configurations is an important feature of
pneumatic conveying models.
This Chapter examines a number of aspects relating to the performance of the power based
pneumatic conveying model in these areas.
9.1 Scaling Characteristics
The first thing to note in relation to the scaling procedures adopted in the power based pneumatic
conveying model is that, in the sense of there being an explicit set of procedures for this
9. Scaling, Long Distance Conveying, Stepped Pipelines
209
purpose, there are none. There is no requirement to apply a set of formulae, such as those
suggested by Mills et al (1982) or W y p y c h and Arnold (1987), to transform a set of
experimental data so that it m a y represent the performance of an alternative pipeline
configuration. Rather the treatment of the physical processes (determination of mass of solids,
available area, velocity calculations, solids friction factor values etc) when applying the model to
different pipeline configurations, in effect carries out the scaling function. The assumptions that
have been applied in developing the model include consideration of pipeline features (length,
diameter, number of bends, extent of vertical pipework), subsequently lead to the scaling
performance of the model w h e n it is applied to an alternative pipeline configuration.
The selection of which physical processes are significant, and the manner in which they are
modelled, will have the major impact on the scaling performance of the model. The trends
inherent in the power based model were examined in some detail in Chapter 7, in particular the
variation of system pressure drop with respect to pipe diameter, length and numbers of bends. A
particular focus of that analysis was to assess the influence of the dimensionless parameters
selected for use in the solids friction factor correlation, resulting in the recognition that there can
be major differences in the behaviour of the model depending on this selection - with two
versions (PCI and P C 6 ) consequently being preferred. That trend analysis indicated the scaling
characteristics of the model, and expectations with regard to the anticipated performance of the
pneumatic conveying system under different pipeline configurations; these analyses were used
in assessing the various model versions. Whilst those results are noted, the analysis, outcomes
and conclusions will not be repeated here. However the behaviour of the model in comparison
to some additional suggestions in the literature with regard to scaling predictions, will be
examined.
Wypych and Arnold (1987) discussed scale up guidelines suggested by Mills et al (1982) and
from a consideration of these suggestions, some theoretical work and experimental
investigations developed a modified set of relationships for scaling experimental data with
respect to pipe diameter and length. The suggested scaling relationships (Mills and W y p y c h ) are
presented in Table 9.1.1.
Table 9.1.1:
Scaling relationships proposed for solids mass flow rate
(Mills et al and W y p y c h et al)
Scaling with respect to Length
ms2 = mA
ms2 = msl
m
si = msl
V
UJ
V
V
( N B for constant m^
Mills et al.
and W y p y c h et al.
2
Mills et al.
2.8
W y p y c h et al.
.AIn Table 9.1.1, subscript 1 refers to the source data (solids mass flow rate and pipeline data from
the experimentalrig),subscript 2 refers to the proposed pipeline configuration. For the above
equations Aps, or the pressure drop attributed to the solids component, is held constant for both
the experimental data and the scaling predictions. The air mass flow rate (length case) or the air
mass flux (diameter case) is also held to a constant value. It m a y be noted that W y p y c h uses the
same relationship for scaling with respect to length as Mills, but suggests a significantly higher
power in the diameter relationship. For coarser materials such as grains and plastic pellets,
9. Scaling, Long Distance Conveying, Stepped Pipelines
210
W y p y c h (1989) has suggested a smaller power for the diameter relationship, ms2
A
= m
12.2
sl
LA J
while retaining the earlier expression for fine products.
T o obtain an indication of the similarity between the corresponding trends predicted by the
power based model and the above equations, the model has been used to determine the variation
in solids mass flow rate w h e n subjected to the same general constraints. O n e area of difference
in the analysis is that a fixed value for the total system pressure drop has been used, rather than
Aps. This is because the power based model uses specifically developed air-alone pressure drop
calculations, and hence there is no direct equivalent term to subtract from the total system
pressure drop to obtain the corresponding Aps term. Consequently a corresponding Aps term is
not available. Secondly, even if an approximately equivalent Aps term w a s constructed, the
computerised procedure used in deterrnining the conveying behaviour of the model does not
conveniently allow for solution on the basis of a fixed Aps value. Therefore the following
analysis is on the basis of the total systems pressure drop rather than Aps.
Since different physical processes dominate the model in lean and dense phase regimes, the
analysis has considered trends under both these modes of conveying. T o illustrate the trends,
Figures 9.1.2 to 9.1.6 are presented, based on experimental data from a fly ash (FA2). Figure
9.1.1 shows the conveying characteristics curves obtained for this material.
350
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Air Mass Flow Rate (kg/s)
Figure 9.1.1
Conveying Characteristics Plot: Bulk Solid: F A 2 ,
Pipeline Arrangement: T R 1 2 , M o d e l P C I
In each of the following figures, with the exception of Figure 9.1.6, a power function has been
fitted to the data calculated by the model. It should be noted that thefittingof a power function
is purely to allow a comparison with the Mills and W y p y c h results. It is not suggested that the
resulting function should or would be used in place of the modelling procedures that have been
previously described. Nevertheless, as is indicated in the following figures, in m a n y cases the
power functionfitsthe data well. However in some instances, particularly for solids mass flow
rate variation with pipe length under lean phase conditions, the fit is likely to be poor and
alternative functions such as the log function applied in Figure 9.1.6 or even a linear equation
would be a preferable choice.
9. Scaling, Long Distance Conveying, Stepped Pipelines
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
211
0.1600
Pipe Diameter (m)
- power function
m s calc.
Figure 9.1.2
0.0400
Scaling Solids Mass Flow Rates with respect to Pipe Diameter:
Bulk Solid: F A 2 , Pipeline Arrangement: TR12, Model P C I ;
Dense Phase Conveying Regime (constant Ap, and W/D" 2 )
0.0600
0.0800
0.1000
0.1200
0.1400
Pipe Diameter (m)
m s calc.
Figure 9.1.3
•power function
Scaling Solids Mass Flow Rates with respect to Pipe Diameter:
Bulk Solid: F A 2 , Pipeline Arrangement: TR12, Model P C I ;
,-2\
Lean Phase Conveying Regime (constant Ap, and «/Z>")
0.1600
9. Scaling, Long Distance Conveying, Stepped Pipelines
212
o
180.00
130.00
80 00
280.00
230.00
Pipe Length (m)
•
Figure 9.1.4
IP •
!
«
w
a
•power function
ms calc.
Scaling Solids M a s s Flow Rates with respect to Pipe Length:
Bulk Solid: F A 2 , Pipeline Arrangement: T R 1 2 , M o d e l P C I ;
Dense Phase Conveying Regime (constant Ap, and m/D' 2 )
*
•
12 •
•mm
ml 10 •
•s.
1
••
01
3 6-
•
•\
^s>
smm 4 •
S 2060.00
•
•
^ - ^
h110.00
1
160.00
1
1
5_i—
210.00
260.00
310.00
Pipe Length (m)
•
Figure 9.1.5
ms calc.
•power function
Scaling Solids M a s s Flow Rates with respect to Pipe Length:
Bulk Solid: F A 2 , Pipeline Arrangement: T R 1 2 , M o d e l P C I ;
Lean Phase Conveying Regime (constant Ap, and mfD'2)
9. Scaling, Long Distance Conveying, Stepped Pipelines
213
I
a
mm
mm
E
01
«
01
-a
©
SZ3
110.00
160.00
210.00
260.00
310.00
Pipe Length (m)
•
Figure 9.1.6
m s calc.
• log function
Scaling Solids M a s s Flow Rates with respect to Pipe Length:
Bulk Solid: F A 2 , Pipeline Arrangement: T R 1 2 , M o d e l P C I ;
L e a n Phase Conveying Regime (constant Ap, and mfD'2)
9.2 Scaling Characteristics - Diameter Variation
Experimental data from a number of bulk sohd/pipeline configurations have been analysed in
the same manner as illustrated in Figures 9.1.2 to 9.1.6, using both models P C I and P C 6 . A
power function has beenfitted,via a least squared error minimisation process, to the resulting
predictions of solids mass flow rate variation with pipe diameter. The power function is of the
same form as that used by Mills and Wypych, ie ms2 = KxD
constant term accounting for the base data component (ie, K =
for diameter, with K being a
m s\
AJ
CC
53
where the subscript 1
denotes the base data condition and " n " is the relevant power. Performance under both dense
and lean phase conditions has again been considered separately.
In this analysis, pipe length has been held constant, at the length of the pipeline used in obtain
the source data in each case.
There is also the question of whether the bend influence should be removed so that the trends
specifically associated with pipe diameter are isolated, by using zero bends in the application of
the models. O n the other hand the relative significance of bends in conveying processes,
particularly under lean phase conditions as demonstrated in Chapter 8, suggests that this m a y
unreahstically distort the behaviour of the model. Hence both options have been explored, ie
zero bends and assuming the same number of bends as existed in the base data pipeline
configuration.
An additional aspect that is also explored in the following tables is the value of carrying out th
solids friction factor correlation using experimental data taken from a range of pipeline length
and diameter configurations, in comparison to using data from a single pipe diameter/length
arrangement. It is relatively easy to vary conveying conditions when carrying out a set of
experiments so that the correlation analysis takes account of variations in system pressure and air
and solids mass flow rates. However the generality of the subsequent analysis m a y be limited
since the data will generally be drawn from a single pipeline length and diameter configuration.
It is more likely that a representative solids friction factor correlation will be achieved if the
9. Scaling, Long Distance Conveying, Stepped Pipelines
214
experimental data are drawn from a range of pipeline configurations, though this entails
additional expense, time and effort in providing for a range of pipeline configurations. The
following tables exarnine both the single and multiple pipeline configurations options by
appropriate selection of the experimental data
The analysis results in a substantial amount of data, which is presented in Tables 9.2.1
The key result is the value obtained for the power "«" in each case. If the model conforms to the
trend predicted by Mills et al, this value should be in the region of 2. Wypych et al suggest the
value should be in the region of 2.2 to 2.8. The first table in this series, Table 9.2.1, presents a
summary of the average results obtained for the range of cases considered to allow an initial
overview. It is again noted that the analysis used to determine the values presented in the
following Tables has been carried out on the basis of a constant total pressure drop, and a
constant air mass flux.
Table 9.2.1: Summary of scaling relationships with respect to diameter; Average values
obtained for the power "n" in assumed function: ms~
Dense Phase
Source Data
Single Dia./Len.
Multiple Dia/Len.
Bends
without
with
without
with
PCI
2.015
2.012
2.037
2.035
PC6
2.013
2.010
2.036
2.033
K-D
Lean Phase
PCI
2.248
2.075
3.511
2.710
PC6
2.245
2.063
3.440
2.508
A number of interesting points are suggested by Table 9.2.1. The trends in system performance
predicted by models PCI and PC6 under dense phase conditions, are very similar for the range
of cases considered (both models, with and without bends and when the correlation is carried out
on experimental data obtained from a variety ofpipe length and diameter conditions). It may
also be noted that, under dense phase conditions, the value of "«" deteimined in the analysis is
close to that predicted by Mills et al.
The lean phase results show greater variability. The value of the power "«" is markedly
indicating that under these conditions, increasing the pipe diameter considerably increases the
potential solids mass flow rate. Inclusion of bends in the pipeline configuration being modelled
has the effect of lowering the power "«" significantly (lean phase case), which is consistent with
the greater impact that bends have under lean phase conditions restricting the increase in solids
mass flow rate for a fixed total system pressure drop. For the case where the solids friction
factor correlation has been obtained on a single pipeline length/diameter configuration, inclusion
of bends under lean phase conditions results in similar results to those found for dense phase.
Where the experimental data have been drawn from a range of pipe lengths and diameters, the
value of "«" is somewhat higher for both with and without bends cases. In the case where bends
have been included the values are similar to those indicated by Wypych et al.
A possible reconciliation of these results with the relationships proposed by Wypych and
that the Mills analysis may have predominantly been based on a set of dense phase data,
compared to a greater emphasis on lean phase data in the case of Wypych.
The following Tables 9.2.2 to 9.2.9 present the detailed results of this analysis. Averag
for "n" as well as the range of values found for each of the cases examined are given. These
Tables also list the correlation (R) between the power function and the results predicted by the
model to indicate the closeness of thefit.Each of the cases in Tables 9.2.2 to 9.2.5 is based on
single diameter and length pipeline configurations, hence the solids friction factor correlations
do not mclude consideration of experimental data with a variance in these parameters.
9. Scaling, Long Distance Conveying, Stepped Pipelines 215
Table 9.2.2:
Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
m,D" using model PCI assumed function: ms - K- D Dense Phase Conveying Regime
Data
File
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3 / TRl
FA1/TR3
FA2/TR12
Diameter
Range(m)
kg/s
kz/s
0
48
.012
.035-.1
3
0
65
0.5
.003
.015 - .075
0
48
3
.035-.1
.015
0
65
0.4
.015 - .075
.004
0
.01
2
.035-.1
48
0
.003
0.5
65
.015 - .075
0
15
168.5
.07
.07-.2
0
4
172.2
.045-.15
.05
average
maximum
minimum
Number of bends mate1 the source data pipeline configuration
mf
ms
No. of Length
Diameter
Data
Range(m)
Bends
(m)
File
kz/s
4
48
.035-.1
PF1/TR2
3
.012
65
.015 - .075
0.5
9
.003
PFl/TRl
.035-.1
4
3
PF2/TR2
48
.015
.015 - .075
9
65
0.4
PF2/TR1
.004
.035-.1
4
48
.01
2
PF3/TR2
.015 - .075
9
65
0.5
PF3/TR1
.003
.07 - .2
168.5
15
5
FA1/TR3
.07
.045-.15
172.2
5
4
FA2
/ .05
TR12
average
maximum
minimum
mf
ms
No. of Length
Bends
(m)
power
R
2.006
2.014
2.012
2.027
2.008
2.016
2.010
2.027
2.015
2.027
2.006
power
2.004
2.01
2.009
2.020
2.006
2.012
2.009
2.023
2.012
2.023
2.004
j
R
216
9. Scaling, Long Distance Conveying, Stepped Pipelines
Table 9.2.3: Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
mf Dousing model PC6 assumed function: ms = K-D" Dense Phase Conveying Regime
Diameter
Range(m)
kg/s
kz/s
.035-.1
3
0
48
.012
0
65
0.5
.015 - .075
.003
48
3
.035-.1
0
.015
65
0.4
0
.015-.075
.004
.01
2
0
48
.035-.1
.003
0.5
0
65
.015 - .075
.07
15
168.5
0
.07 - .2
.05
4
172.2
0
.045-.15
average
maximum
minimum
Number of bends mateh the sour
ce data pipeline configuration
mf
Data
ms
No. of Length
Diameter
File
Bends
(m)
Range(m)
kg/s
kg/s
PF1/TR2
.012
3
4
48
.035-.1
PFl/TRl
0.5
.003
9
65
.017- .075
PF2/TR2
.015
3
4
48
.035-.1
PF2/TR1
.004
0.4
9
65
.015 - .075
PF3/TR2
.01
2
4
48
.035-.1
PF3/TR1
.003
0.5
9
65
.015 - .075
FA1/TR3
.07
15
5
168.5
.07 - .2
FA2
/ .05
4
5
172.2
.045-.15
TR12
Data
File
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3/TR1
FA1/TR3
FA2/TR12
L
mf
ms
No. of Length
Bends
(m)
average
maximum
minimum
power
V
2.005
2.013
2.014
2.030
2.007
2.013
2.009
2.016
2.013
2.030
2.005
R
power
R
2.004
2.009
2.010
2.022
2.006
2.010
2.008
2.013
2.010
2.022
2.004
9. Scaling, Long Distance Conveying, Stepped Pipelines
217
The above two Tables indicate very consistent and predictable system performance with respect
to changes in pipe diameter in the dense phase region. The average value of'V is close to that
suggested by Mills, and varies within a narrow band about this value. It is also clearfromthe
correlation term that the power function provides an excellentfitto the predicted data.
The following two Tables show the results of the corresponding analysis on the same sets of
experimental data, under lean phase conveying conditions. The power function again provides a
goodfitto the data. The major difference with this set of data however is the increase in the
value of "«" for the case where zero bends have been assumed. There is also a greater
variability in the value of "«" for this case. Similar results are predicted by both models P C I
and PC6.
Table 9.2.4: Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
mfD"2using model P C I assumed function:ms = K-D Lean Phase Conveying Regime
Data
File
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3/TR1
FA1/TR3
FA2/TR12
mf
ms
No. of Length
Bends
(m)
Diameter
Range(m)
kg/s
kg/s
.05
3
0
.035-.1
48
0
.015-.075
.01
65
0.5
0
.05
3
48
.035-.1
0
.012
0.4
65
.015 - .075
.05
0
2
.035-.1
48
0
.009
0.5
.015 - .075
65
.25
15
.07 - .2
0
168.5
.25
0
4
.045-.15
172.2
average
maximum
minimum
ition
Number of bends mateI the source data pipeline configurj
ms
Diameter
mf
No. of Length
Data
Range (m)
FUe
Bends
(m)
ke/s
kg/s
.035-.1
.05
4
48
3
PF1/TR2
9
65
.015 - .075
0.5
PFl/TRl
.01
4
48
.035-.1
PF2 / TR2 .05
3
.015 - .075
PF2/TR1
.012
9
65
0.4
.035-.1
4
48
.05
2
PF3/TR2
.015 - .075
.009
PF3/TR1
9
65
0.5
.07 - .2
.25
168.5
5
FA1/TR3
15
.045-.15
4
172.2
5
FA2/TR12 .25
average
maximum
minimum
power
R
2.124
2.176
2.142
2.278
2.241
2.169
2.133
2.722
2.248
2.722
2.124
power
2.034
2.062
2.040
2.091
2.066
2.052
2.049
2.205
2.075
2.205
2.034
R
9. Scaling, Long Distance Conveying, Stepped Pipelines
218
Table 9.2.5: Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
mfD2using model PC6 assumed function: ms = K-D" Lean Phase Conveying Regime
Diameter
Range(m)
kg/s
.035-.1
48
0
3
.05
.015 .075
65
0
0.5
.01
.035-.1
48
0
3
.05
.015-.075
65
0
0.4
.012
.035-.1
48
0
2
.05
.015 - .075
65
0
0.5
.009
.07-.2
168.5
0
15
.25
.045-.15
172.2
0
4
.25
average
maximum
minimum
Number of bends matel the source data pipeline configuration
Diameter
ms
No. of Length
mf
Data
Range(m)
Bends
(m)
FUe
kg/s ,
kg/s
48
4
.035-.1
.05
3
PF1/TR2
65
.017-.075
9
0.5
PFl/TRl
.01
4
.05
48
.035-.1
3
PF2/TR2
9
.012
0.4
.015 - .075
65
PF2/TR1
4
48
.05
2
.035-.1
PF3/TR2
.009
9
65
0.5
PF3/TR1
.015 - .075
5
.25
15
FA1/TR3
168.5
.07 - .2
5
4
FA2/TR12 .25
.045-.15
172.2
average
maximum
minimum
Data
FUe
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3/TR1
FA1/TR3
FA2/TR12
mf
ms
No. of Length
(m)
Bends
power
R
2.124
2.195
2.189
2.333
2.274
2.162
2.139
2.546
2.245
2.546
2.124
power
R
2.035
2.009
2.048
2.107
2.066
2.047
2.050
2.144
2.063
2.144
2.009
The following four Tables 9.2.6 to 9.2.9 repeat the analysis presented in the previous four tables,
with the key difference being that the solidsfrictionfactor correlation analysis in this case is
based on test data drawn from a number of different testrigsfor each material. It would be
expected that by including some variation in pipe dimensions in this phase of the model
development a more reliable result should be achieved.
The dense phase results (Tables 9.2.6 and 9.2.7) support the previous comments, ie the power
functionfitsthe data weU and the value of' V is in the vicinity of 2 throughout. Inclusion or
exclusion of bends in the analysis has little effect.
Table 9.2.6: Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
mfD' using model PCI, Multiple Length and Diameter Source Data assumed function:
ms = K-D
Data
FUe
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
Dense Phase Conveying Regime
Mf
ms
kg/s
kg/s
.007
.008
.007
2
2
2
1
.04
No. of Length
Bends
(m)
0
55
0
0
0
55
55
558
Diameter
Range(m)
J .025 - .08
.03 - .08
.03 - .08
.04-.12
power
V
2.011
2.016
2.011
2.128
R
1
1
1
1
9. Scaling, Long Distance Conveying, Stepped Pipelines
219
Table 9.2.6: (cont)
Data
File
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
No. of Length
Bends
(m)
0
381.2
0
233.66
0
563.4
0
381.2
0
252.7
0
146.1
0
390.3
0
261.1
Diameter
Range(m)
kg/s
kg/s
.04
1
.055-.12
.05
1
.07-.175
.04
1.5
.04-.12
.05
1.5
.06-.15
.06
1.5
.08-.2
2
.01
.04-.1
.015
2
.05-.15
2
.015
.07-.175
0
.02
150
2
.08 - .2
average
maximum
minimum
Number of bends mate l the source data pipeline configuration
PF1/TR1&2 .007
6
2
55
.025 - .08
PF2/TR1&2 .008
2
6
55
.03 - .08
Data
mf
No. of Length
Diameter
ms
FUe
Bends
Range(m)
(m)
kg/s
kg/s
PF4/TR16
.04
16
1
558
.04-.12
PF4/TR16
.04
1
12
381.2
.055-.12
PF4/TR16
.05
1
8
233.66
.07-.175
PF5/TR20
17
.04
1.5
563.4
.04-.12
PF5/TR20
1.5
12
.05
381.2
.06-.15
PF5/TR20
8
.08 - .2
1.5
.06
252.7
PF6/TR24
4
146.1
.04-.1
2
.01
PF6/TR24
390.3
.05-.15
2
.015
12
PF6/TR24
.07-.175
8
261.1
.015
2
PF6/TR24
.08
- .2
5
.02
150
2
average
maximum
minimum
mf
ms
power
R
2.079
2.045
2.061
2.061
2.030
2.023
2.012
2.007
2.003
2.037
2.120
2.003
2.009
2.012
power
1
1
R
2.126
2.079
2.045
2.053
2.056
2.029
2.022
2.011
2.007
2.003
2.035
2.126
2.003
1
1
;
1
1
1
1
1
1
1
1
Table 9.2.7: Scaling relationships; Variation of m, with Pipe Diameter constant Ap, and
mf D'2 using model P C 6 Multiple Length and Diameter Source Data assumed function:
ms = K-D
Data
FUe
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
Dense Phase Conveying Regime
kg/s
kg/s
No. of Length
Bends
(m)
.007
.008
.007
2
2
2
1
1
1
1.5
1.5
1.5
0
0
0
0
0
0
0
0
0
mf
.04
.04
.05
.05
.05
.05
55
55
55
558
381.2
233.66
563.4
381.2
252.7
Diameter
Range (m)
.03 - .08
.03 - .08
.03 - .08
.04-.12
.06-.12
.08 - .2
.06-.12
.07-.15
.08 - .2
power
2.010
2.019
2.012
2.104
2.066
2.036
2.077
2.052
2.018
R
!
9. Scaling, Long Distance Conveying, Stepped Pipelines
220
Table 9.2.7: (cont)
Diameter
Range(m)
kg/s
.05-.1
146.1
0
2
.01
.06-.12
390.3
0
2
.02
.07-.15
261.1
0
2
.03
.08 - .2
150
0
2
.04
average
maximum
minimum
Number of bends mateh the source data pipeline configuration
Diameter
No. of Length
ms
mf
Data
Range(m)
(m)
Bends
FUe
kg/s
kg/s
.03 - .08
55
6
PF1/TR1&2 .007
2
.03 - .08
55
6
PF2/TR1&2 .008
2
- .08
.03
55
PF3/TR1&2 .007
6
2
558
.04-.12
16
PF4/TR16
.04
1
.06-.12
381.2
PF4/TR16
12
1
.04
.08 - .2
233.66
PF4/TR16
8
1
.05
PF5/TR20
.06-.12
563.4
17
1.5
.05
PF5/TR20
.07-.15
381.2
1.5
12
.05
PF5/TR20
.08 - .2
8
252.7
1.5
.05
PF6/TR24
146.1
.05-.1
4
2
.01
PF6/TR24
390.3
.06-.12
.02
2
12
PF6/TR24
8
261.1
2
.07-.15
.03
PF6/TR24
5
150
.08 - .2
.04
2
average
maximum
minimum
Data
FUe
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
mf
ms
No. of Length
(m)
Bends
power
"n"
2.014
2.018
2.031
2.015
2.036
2.104
2.010
R
power
R
1
1
1
1
2.008
2.014
2.009
2.096
2.064
2.036
2.061
2.047
2.018
2.014
2.017
2.029
2.014
2.033
2.096
2.008
Thefinaltwo tables in this section (Tables 9.2.8 and 9.2.9) show the trends found under lean
phase conditions. There is considerably more variability in the values obtained for "«", and
extracting bends from the calculations has a major impact. The value of the power is in the
region suggested by Wypych et al (1987) for the case where bends are included (it should be
noted that the experimental data used by Wypych included bends in the pipeline).
Table 9.2.8: Scaling relationships; Variation of m5 with Pipe Diameter constant Ap, and
mf D' using model PCI, Multiple Length and Diameter Source Data assumed function:
ms = K-D
Data
File
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
Lean Phase Conveying Regime
Mf
ms
kg/s
kg/s
No. of Length
Bends
(m)
.05
.05
.05
.2
.2
.3
.2
.2
2
2
2
1
1
1
1.5
1.5
0
0
0
0
0
0
0
0
55
55
55
558
381.2
233.66
563.4
381.2
Diameter
Range(m)
.025 - .08
.03 - .08
.03 - .08
.055-.12
.055-.12
.07-.175
.055-.12
.06-.15
power
R
2.475
2.432
2.403
4.636
5.561
5.262
3.663
3.407
.998
.999
.999
.98
.959
.954
.992
.993
9. Scaling, Long Distance Conveying, Stepped Pipelines
221
Table 9.2.8: (cont)
Data
FUe
PF5/TR20
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
Mf
No. of Length
Bends
(m)
0
252.7
Diameter
Range(m)
kg/s
kg/s
.3
1.5
.08 - .2
0
.12
2
146.1
.04-.1
0
2
390.3
.12
.06-.15
0
2
261.1
.15
.07-.175
0
2
150
.08 - .2
.25
average
maximum
minimum
Number of bends mate l the source data pipeline configuration
ms
Mf
Diameter
Data
No. of Length
FUe
Range(m)
Bends
(m)
kg/s
kg/s
PF1/TR1&2 .05
2
6
55
.025 - .08
PF2/TR1&2 .05
2
6
55
.03 - .08
PF3/TR1&2 .05
2
55
6
.03 - .08
PF4/TR16
1
.2
16
558
.055-.12
PF4/TR16
1
.2
12
381.2
.055-.12
PF4/TR16
1
.3
8
233.66
.07-.175
PF5/TR20
1.5
17
563.4
.2
.055-.12
PF5/TR20
12
1.5
.2
381.2
.06-.15
PF5/TR20
8
252.7
.08 - .2
1.5
.3
PF6/TR24
4
2
.04-.1
.12
146.1
PF6/TR24
2
390.3
12
.12
.06-.15
PF6/TR24
2
8
.07-.175
261.1
.15
PF6/TR24
2
5
150
.08 - .2
.25
average
maximum
minimum
ms
power
R
3.281
4.102
2.792
2.718
2.911
3.511
5.561
2.403
.994
.975
.997
.998
.997
power
"n"
2.055
2.054
2.044
3.618
4.234
4.218
2.478
2.582
2.648
2.473
2.221
2.256
2.35
2.710
4.234
2.044
R
1
1
1
.992
.984
.978
.999
.999
.998
.999
1
1
1
Table 9.2.9: Scaling relationships; Variation of ms with Pipe Diameter constant Ap, and
mf IX2 using model PC6, Multiple Length and Diameter Source Data assumed function:
ms-K-D
Data
File
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
PF6/TR24
PF6/TR24
Lean Phase Conveying Regime
mf
ms
kg/s
kg/s
No. of Length
Bends
(m)
.05
.05
.05
.2
.2
.3
.15
.17
.35
.12
.12
.16
2
2
2
1
1
1
1.5
1.5
1.5
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
55
55
55
558
381.2
233.66
563.4
381.2
252.7
146.1
390.3
261.1
Diameter
Range(m)
.03 - .08
.03 - .08
.03 - .08
.055-.12
.06-.12
.08 - .2
.06-.12
.07-.15
.08 - .2
.05-.1
.06-.12
.07-.15
power
R
2.463
2.573
2.479
5.104
5.66
4.561
2.922
2.85
3.801
3.358
2.784
2.894
.999
.999
.999
.956
.958
.968
.998
.998
.987
.995
.999
.998
9. Scaling, Long Distance Conveying, Stepped Pipelines
222
Table 9.2.9: (cont)
Diameter
Range(m)
kg/s
.08 - .2
150
0
PF6/TR24
2
.3
average
maximum
minimum
Number of bends mateh the source data pipeline configuration
Diameter
No. of Length
ms
mf
Data
Range(m)
Bends
(m)
File
kg/s
kg/s
.03 - .08
PF1/TR1&2 .05
6
55
2
.03 - .08
PF2/TR1&2 .05
6
55
2
PF3/TR1&2 .05
.03 - .08
55
6
2
PF4/TR16
558
.055-.12
16
1
.2
PF4/TR16
381.2
.06-.12
1
.2
12
PF4/TR16
233.66
.08 - .2
1
8
.3
PF5/TR20
.06-.12
563.4
.15
1.5
17
PF5/TR20
.17
1.5
381.2
.07-.15
12
PF5/TR20
8
252.7
.08 - .2
1.5
.35
PF6/TR24
.12
4
146.1
.05-.1
2
PF6/TR24
.12
12
390.3
2
.06-.12
PF6/TR24
.16
2
8
261.1
.07-.15
PF6/TR24
.3
5
150
2
.08 - .2
average
maximum
minimum
Data
FUe
mf
ms
No. of Length
(m)
Bends
power
R
3.268
3.440
5.066
5.436
.995
power
R
2.048
2.062
2.041
3.013
3.367
3.18
2.335
2.395
2.729
2.423
2.243
2.316
2.448
2.508
3.367
2.041
1
1
1
.997
.995
.994
1
1
.998
1
1
1
.999
It may also be noted that the fit of the power function to the data has deteriorated to some
extent, particularly for the case of zero bends, and that this is associated with the very high
values of "n" that are reported in some instances. The following line of data from Table 9.2.9
is a case in point. It m a y appear that the correlation is still quite reasonable at 0.956, however
the associated value for the power at 5.104 appears excessive.
Data
FUe
PF4AR16
m
f
ms
kg/s
kg/s
No. of Length
Bends
(m)
.2
1
0
558
Diameter
Range(m)
.055-.12
power
R
5.104
.956
Figure 9.2.1 shows the model data and the power function for this set of data, and it is apparent
that in this instance a different function, for example a straight line would provide a better fit,
and that the power term is not a particularly useful characterising parameter. Other instances in
the data where correlation values of this order have been found associated with high power
values indicate a simUar inappropriateness offitfor the power function rather than a dubious
trend in model performance. It is again noted that the selection of a power function in this
analysis is purely to allow a comparison with other scaling data in the literature.
223
9. Scaling, Long Distance Conveying, Stepped Pipelines
10
!
a
mm
/
&
I
6
o
2 ••
•
•
4 -•
^ -mi 1 1 1 J
0.05
0.04
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Pipe Diameter (m)
• ms calc. • power function
Figure 9.2.1
Scaling Solids M a s s Flow Rates with respect to Pipe Diameter:
Bulk Solid: P F 4 , Pipeline Arrangement: T R l 6 , M o d e l P C 6 ;
Lean Phase Conveying Regime (constant Apt and mfD"2)
9.2.1 Summary of Scaling Trends with respect to Diameter
In summary, it m a y be concluded that the scaling trends with respect to diameter produced by
the power based pneumatic conveying model, conform to the general relationships suggested by
both MUls and Wypych. The variation of system performance under lean and dense phase
conditions provides a possible explanation for the differences between the proposals of those
two authors.
In seeking a reason for the difference in performance of conveying systems in this respect under
dense and lean phase conditions, the improved efficiency of the pneumatic conveying process
under dense phase is worth consideration. The dominant outcome of increasing pipe diameter is
to increase the avaUable cross-sectional area for flow, at a rate proportional to the diameter
squared. If the means of increasing the available area for flow was to duplicate the existing pipe
then the solids mass flow rate would necessarily increase in direct proportion to the increase in
area, since aU other parameters of the conveying process would remain identical. This would
lead to the trend suggested by Mills, with the variation in solids mass flow rate being directly
proportional to D2.
If other features of the conveying process are changing in concert with the pipe diameter
however, a variation from this diameter squared relationship m a y be expected. In particular, if
the base conveying point around which this variation is being examined is in the lean phase
region, and if increasing the diameter (wlulst holding the total system pressure drop and the air
mass flux constant) results in an increase in the solids to air mass flow ratio, the improved
efficiency of a higher density conveying regime wUl also impact on the scaling trends. The
improved efficiency wUl allow a higher solids mass flow rate than would be expected simply on
the diameter squared basis, leading to a power greater than 2 for the diameter scaling term. The
foUowing example taken from Table 9.2.9 has been used to Ulustrate this point.
Data
File
PF5/TR20
mf
ms
kg/s
kg/s
.17
1.5
No. of Length
(m)
Bends
12
381.2
Diameter
Range(m)
.07-.15
power
R
2.395
1
9. Scaling, Long Distance Conveying, Stepped Pipelines
224
Figure 9.2.2 has been generated from the calculations carried out w h e n deterrnining the above
trend, and shows the variation in solids mass flow rate with pipe diameter.
12.00
o
11.00 ••
o
mm
01
01
«
%
5m.
10.00 ••
9.00 ••
<
O
01
=3
8.00 ••
7.00
0.0600
0.0700
0.0800
0.0900
0.1000
0.1100
0.1200
0.1300
0.1400
0.1600
0.1600
Pipe Diameter (m)
Figure 9.2.2: Scaling Trends with respect to Pipe Diameter - Variation of m*
Bulk Solid: P F 5 , Pipeline Arrangement: T R 2 0 , M o d e l P C 6 ; Lean Phase
Conveying Regime (constant Ap, and mfD'2)
The increase in m* depicted in Figure 9.2.2 is close to 40% across the range of diameters
considered, which contributes to the increase in the power of "«" (in this case a value of 2.395)
over the nominal value of 2.
9.3
Scaling Characteristics - Length Variation
This Section repeats the work of Section 9.2, with length replacing diameter as the pipeline
parameter being examined. A power function has again beenfittedto the data deterrnined by
application of models P C I and P C 6 , of the form: ms2 = KxL",
accounting for the base data component (ie, K =
mVtl
LA
with K being a constant term
where the subscript "i" denotes the base
data condition and " n" is the relevant power). Performance under both dense and lean phase
conditions has again been considered separately.
Both Mills and Wypych suggest that for the case of a fixed air mass flux and a constant pressure
drop (due to the solids component), the solids mass flow rate is inversely proportional to pipe
length. This would lead to a value of -1 being found for V in the power function.
For this analysis, pipe diameter has been held constant at the diameter of the pipeline used in
obtaining the source data in each case. The two options of applying the models with and without
bends have again been explored. The same experimental data arrangements (single and multiple
pipeline configurations) have been applied.
The first Table in this series, Table 9.3.1, presents a summary of the average results obtained
the range of cases considered to a U o w an initial overview. It is again noted that the analysis used
to determine the values presented in the following Tables has been carried out on the basis of a
constant total pressure drop, and a constant air mass flux.
9. Scaling, Long Distance Conveying, Stepped Pipelines 225
Table 9.3.1:
Summary of scaling relationships with respect to pipe length Average
values obtained for the power **«" in assumed function: m5 = K'L
Conveying Regime
Lean Phase
Dense Phase
Bends
without
with
Multiple Dia./Len. without
with
Source Data
Single Dia./Len.
PCI
PC6
-1.857
-1.595
-2.037
-1.909
-1.930
-1.646
-2.122
-2.007
Dense Phase
PCI
-2.072
-.965
-4.919
-2.113
PC6
-2.349
-1.058
-4.844
-1.656
Several aspects are apparent from Table 9.3.1. Firstly there is a considerable degree of
variabUity in the values of "«" found whenfittingthe power function to this set of data. In
addition, the magnitude of the power is somewhat higher than the value of -1 suggested by Mills
and Wypych, with the exception of the lean phase data for the single length/diameter case, with
bends included. The effect of the power being a larger negative value is that the estimated solids
massflowrate reduces at a faster rate with increasing pipe length. The inclusion of bends in the
model reduces the magnitude of the power, though the extent of the reduction is relatively minor
for the dense phase cases.
As noted in Section 9.1, the significance of the value of "«" is dhninished if t
function to the data is not good. In that case a misleading trend may be indicated, therefore
some attention needs to be paid to the correlation values listed in the foUowing tables. A general
comment would be that the power function does not provide as good afitto the data produced
by the models for length variation as it did for the diameter case.
The foUowing four Tables 9.3.2 to 9.3.5, show the results obtained from the anal
experimental datafromsingle length/diameter pipeline configurations. The performance in both
dense and lean phase conveying regimes is examined. It can be seen that within each table there
is a greater variability in the values of "«" found than there is in the corresponding tables for
diameter in Section 9.2.
Table 9.3.2: Scaling relationships; Variation of ms with Pipe Length constant Ap, and m/
using model PCI assumed function: ms-K-L
Data
FUe
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3 / TRl
FAl / TR3
FA2/TR12
m
f
ms
kg/s
kg/s
No. of Diameter
(mm)
Bends
.012
.003
.015
.004
3
0.5
3
0.4
2
0.5
15
4
0
0
0
0
0
0
0
0
.01
.003
.07
.05
52.5
25.4
52.5
25.4
52.5
25.4
105
69
Dense Phase Conveying Regime
Length
Range(m)
30 -100
50-130
30 -100
45-130
25 -100
45 -130
100 - 300
105 - 300
average
maximum
minimum
power
R
-1.793
-2.223
-1.790
-2.230
-1.754
-2.211
-1.640
-1.801
-1.930
•1.640
-2.230
.997
.998
.989
.998
.983
.995
.978
.997
9. Scaling, Long Distance Conveying, Stepped Pipelines
226
Table 9.3.2:
(cont.)
Number of bends mateh the source data pipeline configuration
Length
No. of Diameter
ms
ntf
Data
Range(m)
(mm)
Bends
File
kg/s
kg/s
30 -100
52.5
4
3
.012
PF1/TR2
50 -130
25.4
9
0.5
.003
PFl/TRl
30 -100
52.5
4
3
.015
PF2/TR2
45 -130
25.4
9
0.4
.004
PF2 / TRl
25 -100
52.5
4
2
.01
PF3/TR2
45 -130
25.4
9
0.5
.003
PF3/TR1
100 - 300
105
5
.07
15
FA1/TR3
105 - 300
69
5
4
FA2/TR12 .05
average
maximum
minimum
power
R
-1.279
-1.962
-1.453
-1.942
-1.564
-1.849
-1.484
-1.633
-1.646
-1.962
-1.279
.99
.996
.986
.995
.982
.992
.978
.996
Table 9.3.3: Scaling relationships; Variation of ms with Pipe Length constant Ap, and mf
using model PC6 assumed function: ms = K- ZT Dense Phase Conveying Regime
Data
FUe
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3 / TR2
PF3/TR1
FAl / TR3
FA2/TR12
mf
No. of Diameter
(mm)
Bends
Length
Range(m)
kg/s
kg/s
.012
3
0
52.5
30 -100
.003
0
0.5
25.4
50 -130
3
.015
0
52.5
30 - 100
0.4
0
.004
25.4
45 -130
.01
2
0
52.5
25 -100
.003
0.5
0
25.4
45 -130
.07
15
0
105
110-300
.05
4
0
69
80 - 300
average
maximum
minimum
Number of bends mateh the source data pipeline configuration
Data
mf
ms
No. of Diameter
Length
FUe
Bends
kg/s
(mm)
Range(m)
kg/s
PF1/TR2
.012
3
4
52.5
30 -100
PFl/TRl
.003
0.5
9
25.4
50-130
PF2/TR2
.015
3
4
52.5
30 -100
PF2 / TRl
.004
0.4
9
25.4
45 -130
PF3/TR2
.01
2
4
52.5
25 -100
PF3/TR1
.003
0.5
9
25.4
45 -130
FA1/TR3
.07
15
5
105
110-300
FA2/TR12 .05
4
5
69
80 - 300
average
maximum
minimum
ms
power
V
-1.367
-2.196
-2.200
-2.686
-1.736
-1.954
-1.569
-1.153
-1.857
-1.153
-2.686
R
power
R
-1.175
-1.922
-1.798
-2.305
-1.545
-1.630
-1.415
-.973
-1.595
-.973
-2.305
.99
.993
.995
.976
.994
.978
.995
.981
.998
.993
.975
.989
.977
.992
.982
.996
an be seen that within each Table there is a considerably greater variability in the values of
found than there is in the corresponding Tables for diameter in Section 9.2. For example, in
dense phase cases examined in Tables 9.3.2 and 9.3.3 the maximum to niinimum range is in
227
9. Scaling, Long Distance Conveying, Stepped Pipelines
the order of 6 0 % of the average value of "n", compared to a 1 % variation in the diameter case.
The following two Tables list results for the lean phase, where it m a y be noted that this
variability is increased further.
Table 9.3.4: Scaling relationships; Variation of ms with Pipe Length constant Ap, and w/
using model P C I assumed function: ms = K- L Lean Phase Conveying Regime
Length
Range (m)
30 -100
45 -130
30 -100
45 -130
20 -100
45 -130
100 - 300
80 - 300
power
R
-1.25
-2.427
-1.916
-2.647
-2.012
-2.503
-1.776
-2.043
.987
.996
.985
.996
average
maximum
minimum
Number of bends matel the source data pipeline configunition
mf
ms
No. of Diameter
Length
Data
FUe
Bends
(mm)
Range(m)
ke/s
.Ws
.05
3
4
52.5
30 -100
PF1/TR2
9
0.5
45 -130
PFl/TRl
25.4
.01
30 -100
4
3
PF2/TR2
52.5
.05
45 -130
9
.012
0.4
PF2/TR1
25.4
20 -100
4
.05
2
52.5
PF3/TR2
45 -130
9
.009
0.5
PF3/TR1
25.4
100 - 300
5
.25
105
15
FA1/TR3
80 - 300
5
69
4
FA2/TR12 .25
average
maximum
minimum
-2.072
-1.250
-2.647
Data
File
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3/TR1
FA1/TR3
FA2
/
TR12
mf
ms
kg/s
kg/s
No. of Diameter
(mm)
Bends
.05
.01
.05
3
0.5
3
0.4
2
0.5
15
4
0
0
0
0
0
0
0
0
.012
.05
.009
.25
.25
52.5
25.4
52.5
25.4
52.5
25.4
105
69
.97
.994
.974
.97
power
R
-.630
-1.315
-.769
-1.365
-.729
-1.198
-.904
-.81
-.965
-.63
-1.365
.986
.988
.983
.988
.969
.986
.983
.979
Table 9.3.5: Scaling relationships; Variation of ms with Pipe Length constant Ap, and mf
using model PC6 assumed function: ms = K- L Lean Phase Conveying Regime
Data
File
PF1/TR2
PFl/TRl
PF2/TR2
PF2/TR1
PF3/TR2
PF3/TR1
FA1/TR3
FA2/ TR12
mf
ms
kg/s
kg/s
No. of Diameter
(mm)
Bends
.05
.01
.05
3
0.5
3
0.4
2
0.5
15
4
0
0
0
0
0
0
0
0
.012
.05
.009
.25
.25
52.5
25.4
52.5
25.4
52.5
25.4
105
69
Length
Range(m)
30 -100
45 -130
30 -100
45 -130
20 -100
45 -130
110-300
80 - 300
average
maximum
minimum
power
V
-1.564
-2.791
-2.967
-3.505
-2.276
-2.402
-2.002
-1.289
-2.349
-1.289
-3.505
R
.988
.992
.968
.991
.958
.993
.972
.973
9. Scaling, Long Distance Conveying, Stepped Pipelines
228
Table 9..5: ;cont)
Number of bends match the source data pipeline configuration
Length
No. of Diameter
mf
ms
Data
(mm)
Range (m)
Bends
FUe
kg/s
kg/s
30 -100
52.5
4
3
PFl / TR2 .05
45 -130
9
25.4
0.5
.01
PFl/TRl
30 -100
52.5
4
3
.05
PF2/TR2
25.4
45 -130
0.4
9
.012
PF2/TR1
20 -100
52.5
4
.
0
5
2
PF3/TR2
25.4
45 -130
9
.009
0.5
PF3/TR1
105
110-300
5
.25
15
FAl/TR3
69
80 - 300
5
4
FA2/TR12 .25
average
maximum
minimum
power
"n"
-.618
-1.52
-1.243
-1.909
-.791
-1.065
-.958
-.357
-1.058
-.357
-1.909
R
.986
.986
.968
.981
.964
.986
.984
.982
One possibUity for the variability in the value of the power term found in this correlation process
may be the inadequacy of the power function tofitthe data reliably. The correlation term "R",
wlnlst stiU reasonable, is somewhat reduced from the typical values being achieved in the
corresponding diameter variation tables in Section 9.2. The following two rows of data are
extracted from Tables 9.3.3 and 9.3.5 respectively, and show the results obtained for one of the
pulverised coal samples using model PC6.
Data
Mf
FUe
kg/s
PF2/TR1
.004
dense phase
PF2/TR1
.012
lean phase
kg/s
No. of Diameter
Bends
(mm)
0.4
9
0.4
9
ms
power
R
25.4
Length
Range(m)
45 -130
-2.305
.989
25.4
45 -130
-1.909
.981
Figures 9.3.1 and 9.3.2 show the corresponding model data and the resulting power functions to
indicate the fit characteristics.
50
70
80
90
100
110
120
130
140
Pipe Diameter (m)
•
Figure 9.3.1
ms calc.
-power function
Scaling Solids M a s s Flow Rates with respect to Pipe Length:
BuUx Solid: P F 2 , Pipeline Arrangement: T R l , M o d e l P C 6 ; Dense Phase
Conveying Regime (constant Ap, and mfD2), n = -2.305
9. Scaling, Long Distance Conveying, Stepped Pipelines 229
01
0.8 •
*
0.7 •
Wii
'
' •
0.6 -
•V
OS
o
0.5 •
E 0.4 •
SB
^ V ^
2sn 0.3 •
•o 0.2 -
O
C« 0.1 •
0 •
-1
h-
1
1
1
1
1
1
1
40
50
60
70
80
90
100
110
120
130
140
Pipe Diameter (m)
m s calc.
Figure 9.3.2
• power function
Scaling Solids Mass Flow Rates with respect to Pipe Length:
Bulk Solid: PF2, Pipeline Arrangement: TRl, Model PC6; Lean Phase
Conveying Regime (constant Ap, and mfD"2), n = -1.909
Figures 9.3.1 and 9.3.2 indicate that whUst the fit of the power function to the
massflowrate values is quite good, the curvature of the resulting function is higher than the data
points it is intended to fit. The higher curvature is consistent with an increased magnitude for the
power "«". This result is typical of thefitof the power function to the length trend data,
suggesting that the power function is not an ideal choice and that the values listed for this
parameter in the foUowing tables err on the high side.
The foUowing four Tables 9.3.6 to 9.3.9 list corresponding results obtained usin
data obtained from multiple pipeline length/diameter configurations for each bulk solid. Since
the solids friction factor correlations in these cases is able to some extent to take variation in
pipeline length and diameter into account, it would be expected that the trends indicated would
be more reliable.
Table 9.3.6:
Scaling relationships; Variation of m, with Pipe Length constant Ap, and mousing model PCI, Multiple Length and Diameter Source Data assumed
function: ms = K- L Dense Phase Conveying Regime
Data
File
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
Length
Range(m)
power
R
kg/s
No. of Diameter
(mm)
Bends
2
2
2
1
1
1
1.5
1.5
1.5
2
0
0
0
0
0
0
0
0
0
0
40 - 100
40 -100
40 -100
300 - 800
300 - 800
200 - 500
400 - 800
300 - 800
200 - 500
100 - 300
-1.265
-1.507
-1.253
-2.044
-2.086
-2.075
-1.976
-2.042
-1.865
-2.993
.994
mf
ms
kg/s
.007
.008
.007
.04
.04
.05
.04
.05
.06
.02
40
40
40
69
81
105
69
81
105
60
.99
.99
1
1
1
1
1
1
.999
9. Scaling, Long Distance Conveying, Stepped Pipelines
230
Table 9.3.6:
(cont.)
69
81
105
350 - 700
350 - 700
100 - 400
.02
average
maximum
minimum
Number of bends mateh the source data pipeline configuration
Length
No. of Diameter
mf
ms
Data
Range(m)
(mm)
Bends
FUe
kg/s
kg/s
40 -100
40
6
PF1/TR1&2 .007
2
40 -100
40
6
PF2/TR1&2 .008
2
40 -100
40
6
2
PF3/TR1&2 .007
300 - 800
69
16
PF4/TR16
I
.04
300 - 800
81
12
PF4/TR16
.04
1
200 - 500
105
8
PF4/TR16
1
.05
400
- 800
69
17
PF5/TR20
1.5
.04
300 - 800
81
PF5/TR20
12
.05
1.5
200 - 500
105
PF5/TR20
8
.06
1.5
100 - 300
PF6/TR24
60
4
2
.02
350
- 700
PF6/TR24
69
12
.015
2
PF6/TR24
350 - 700
8
.015
81
2
PF6/TR24
105
100 - 400
5
.02
2
average
maximum
minimum
PF6/TR24
PF6/TR24
PF6/TR24
Table 9.3.7:
.015
.015
2
2
2
0
0
0
1
1
.991
power
R
-1.097
-1.204
-1.034
-1.983
-2.077
-2.075
-1.814
-1.998
-1.835
-2.697
-3.043
-3.147
-2.161
-2.007
-1.034
-3.147
.992
.989
.99
.997
1
.991
Scaling relationships; Variation of ms with Pipe Length constant Ap, and ntf
using model PC6, Multiple Length and Diameter Source Data assumed
function: ms = K- L
Data
FUe
PF1/TR1&2
PF2/TR1&2
Data
File
PF3/TR1&2
PF4/TR16
Data
File
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
PF6/TR24
-3.124
-3.182
-2.18
-2.122
-1.253
-3.182
Dense Phase Conveying Regime
ntf
ms
kg/s
kg/s
No. of Diameter
Bends
(mm)
.007
.008
mf
2
2
0
0
ms
kg/s
kg/s
No. of Diameter
Bends
(mm)
.007
.04
2
1
0
0
40
40
40
69
mf
ms
kg/s
kg/s
No. of Diameter
Bends
(mm)
.04
.05
.05
.05
.05
.01
.02
1
1
1.5
1.5
1.5
2
2
0
0
0
0
0
0
0
81
105
69
81
105
60
69
Length
Range(m)
40 -100
40 -100
Length
Range(m)
40 -100
400 - 800
Length
Range(m)
300 - 700
200 - 500
400 - 800
300 - 700
200 - 500
130 - 300
300 - 700
power
R
-.872
-1.538
power
V
-1.211
-1.773
power
.995
.986
-1.796
-1.803
-1.888
-1.947
-1.771
-2.915
-2.948
R
.988
1
R
.998
9. Scaling, Long Distance Conveying, Stepped Pipelines
Table 9.3.7:
231
(cont.)
Length
Range(m)
power
R
200 - 500
100 - 400
average
maximum
minimum
Number of bends mateI the source data pipeline configuration
ms
mf
No. of Diameter
Length
Data
Bends
Range(m)
FUe
(mm)
kg/s
kg/s
PF1/TR1&2 .007
2
6
40
40 -100
PF2/TR1&2 .008
2
6
40
40 -100
PF3/TR1&2 .007
2
6
40
40 -100
PF4/TR16
1
16
69
.04
400 - 800
PF4/TR16
.04
12
300 - 700
81
1
PF4/TR16
8
200 - 500
1
.05
105
PF5/TR20
1.5
17
69
400 - 800
.05
PF5/TR20
12
81
300 - 700
1
.
5
.05
PF5/TR20
.05
8
200 - 500
1.5
105
PF6/TR24
130 - 300
4
2
60
.01
PF6/TR24
300 - 700
2
69
12
.02
PF6/TR24
8
200 - 500
81
2
.03
PF6/TR24
5
100 - 400
105
.04
2
average
maximum
minimum
-3.277
-2.748
-2.037
-.872
-3.277
.999
.991
power
V
-.619
-1.271
-1.020
-1.676
-1.761
-1.796
-1.634
-1.833
-1.753
-2.861
-2.81
-3.103
-2.679
-1.909
-.619
-3.103
R
Data
File
PF6/TR24
PF6/TR24
mf
ms
kg/s
kg/s
No. of Diameter
Bends
(mm)
.03
.04
2
2
0
0
81
105
.991
.986
.988
1
1
1
.988 !
1
1
i
.998
.997
.998 i
.991
The results for the dense phase conveying case in Tables 9.3.6 and 9.3.7 above actuaUy show an
improved fit for the power function, however the range in values for the power has increased, to
1 1 1 % (from 6 0 % for the single pipeline length/diameter case). The final two tables in this series
show the results obtained for the lean phase conveying case. Thefitof the power function has
deteriorated, and the variability of the power values has again increased.
Table 9.3.8:
Scaling relationships; Variation of ms with Pipe Length constant Ap, and mf
using model P C I , Multiple Length and Diameter Source Data assumed
function: ms = K- Z," Lean Phase Conveying Regime
Data
FUe
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
mf
ms
kg/s
kg/s
No. of Diameter
(mm)
Bends
.05
.05
.05
.2
.2
.3
.2
.2
.3
2
2
2
1
1
1
1.5
1.5
1.5
0
0
0
0
0
0
0
0
0
40
40
40
69
81
105
69
81
105
Length
Range(m)
40 -100
40 -100
40 -100
300 - 800
300 - 600
200 - 350
400 - 800
300 - 700
200 - 500
power
V
-1.586
-1.993
-1.639
-5.410
-6.233
-5.884
-4.041
-5.442
-6.208
R
.981
.975
.974
.895
.96
.983
.98
.959
.933
9. Scaling, Long Distance Conveying, Stepped Pipelines
232
Table 9.3.8:
(cant)
60
69
81
105
100 - 260
350 - 700
350 - 700
100 - 400
average
maximum
minimum
Number of bends matel the source data pipeline configuration
Length
No. of Diameter
ms
mf
Data
Range(m)
(mm)
Bends
FUe
kg/s
kg/s
40
40 -100
PF1/TR1&2 .05
6
2
40 -100
PF2/TR1&2 .05
40
2
6
40 -100
PF3/TR1&2 .05
40
6
2
300 - 800
PF4/TR16
16
69
1
.2
PF4/TR16
81
300 - 600
1
12
.2
PF4/TR16
105
200 - 350
1
8
.3
PF5/TR20
17
400 - 800
1.5
69
.2
PF5/TR20
300 - 700
81
.2
1.5
12
PF5/TR20
8
105
.3
200 - 500
1.5
PF6/TR24
.12
2
4
60
100 - 260
PF6/TR24
2
12
.12
69
350 - 700
PF6/TR24
.15
2
8
81
350 - 700
PF6/TR24
2
.25
5
105
100 - 400
average
maximum
minimum
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
.12
.12
.15
.25
Table 9.3.9:
2
2
2
2
0
0
0
0
.905
.993
.983
.961
power
V
-.295
-.363
-.271
-2.924
-4.184
-4.414
-1.269
-2.076
-2.614
-1.198
-1.666
-3.563
-2.263
-2.113
-.271
-4.414
R
.991
.99
.991
.961
.976
.985
.993
.986
.98
.981
.993
.989
.961
Scaling relationships; Variation of ms with Pipe Length constant Ap, and mf
using model PC6, Multiple Length and Diameter Source Data assumed
function: ms = K-H
Data
FUe
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
-6.163
-4.636
-8.748
-5.962
-4.919
-1.586
-8.748
Lean Phase Conveying Regime
mf
ms
k
?/s
kg/s
No. of Diameter
Bends
(mm)
.05
.05
.05
.2
.2
.3
.15
.17
.35
.12
.12
.16
.3
2
2
2
1
1
1
1.5
1.5
1.5
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
_.
40
40
40
69
81
105
69
81
105
60
69
81
105
Length
Range(m)
40 -100
40 -100
40 -100
400 - 750
300 - 550
200 - 325
400 - 800
300 - 700
200 - 400
130 - 230
300 - 700
200 - 500
100 - 300
average
maximum
minimum
power
"n"
-1.994
-3.220
-2.178
-6.982
-6.552
-6.605
-2.966
-3.835
-5.866
-6.502
-4.729
-5.476
-6.072
-4.844
-1.994
-6.982
R
.972
.949
.958
.888
.946
.972
.994
.987
.961
.964
.988
.986
.970
233
9. Scaling, Long Distance Conveying, Stepped Pipelines
Table 9.3.9:
(cont.)
Number of bends mate I the source data pipeline configuration
Data
mf
Length
No. of Diameter
ms
File
kg/s
kg/s
Bends
(mm)
PF1/TR1&2
PF2/TR1&2
PF3/TR1&2
PF4/TR16
PF4/TR16
PF4/TR16
PF5/TR20
PF5/TR20
PF5/TR20
PF6/TR24
PF6/TR24
PF6/TR24
PF6/TR24
.05
.05
.05
.2
.2
.3
.15
.17
.35
.12
.12
.16
.3
2
2
2
1
1
1
1.5
1.5
1.5
2
2
2
2
6
6
6
16
12
8
17
12
8
4
12
8
5
40
40
40
69
81
105
69
81
105
60
69
81
105
Range(m)
40 -100
40 -100
40 -100
400 - 750
300 - 550
200 - 325
400 - 800
300 - 700
200 - 400
130 - 230
300 - 700
200 - 500
100 - 300
average
maximum
minimum
power
R
-.369
-.596
-.355
-1.931
-2.452
-2.792
-1.246
-1.826
-2.052
-1.436
-1.782
-2.231
-2.454
-1.656
-.355
-2.792
.99
.986
.99
.988 !
.988
.991 !
.995
.991
.987
.992
.988
.98
.962
There are a number of general conclusions that m a y drawn from these Tables. Firstly the power
function is not ideaUy suited to model the solids mass flow rate versus pipeline length trends
under the conditions that have been specified. Where a power function is used there is a
tendency for the magnitude of the power value to err on the high side. Nevertheless, when
compared with the relationship suggested by MUls and later Wypych, the power based model in
the form it has been developed here results in a higher rate of decrease in solids mass flow rate
with respect to increasing pipe length. This statement should be qualified however by noting
that there are some differences in the basis of calculating these trends. The impact of these
differences is discussed in more detail below.
Inclusion of bends in the pipeline configuration consistently leads to a reduction in the pow
term obtained. In many cases the extent of this reduction is substantial and converts an
apparently unreasonable power term to one that could be regarded as quite feasible.
Why should the inclusion of bends result in this outcome?
The main effect of including bends would be to reduce the achievable solids mass flow rate fo
given air mass flow rate and total system pressure drop. Under lean phase conditions where
bend power consumption (and hence share of total pressure drop) is likely to be dominant, this
effect would be particularly noticeable. The power "«" however is not an indication of the
amount of solids that can be transported, but an indication of the rate of change in the solids
mass flow rate as the pipe length as varied.
The basis for deternmiing the contribution of bends to overall system performance in the powe
based model, means that for afixedtotal pressure drop for the system, the bend component is
not directly affected by pipe length. Solids velocity around the bend is the key parameter in
determining the power consumption at the bends and solids velocity is based on the air mass
flow rate, the assumed pressure at the bend and the VSR value selected for the bulk solid. The
bend component wUl change in response to changing solids mass flow rates as a result of
variations in mass of solids in the bends and changes to the solids friction factor; however these
are second order effects in comparison to the velocity squared term.
234
9. Scaling, Long Distance Conveying, Stepped Pipelim
Therefore inclusion of bends introduces a relatively stable component into the total pressure
drop calculation that is not directly affected by changes in length. Presence of bends will reduce
the solids mass flow rate overall for a given total system pressure drop, but also wUl reduce the
extent to which the solids mass flow faUs in response to increasing bend length. This wtil tend
to restrict the variation to an extent, hence reducing the magnitude of "«". It would be expected
that under lean phase conditions this effect would be more noticeable. Table 9.3.1 confirms this
outcome.
It was previously noted in this Section that differences in the basis of calculating solids mass
flow rate variation with respect to pipe length between the power based model and the
calculations used by Mills and W y p y c h also contribute to the difference in the power term
finahy obtained. This difference is principaUy associated with the air-only pressure drop
component. Mills and W y p y c h isolate the pressure drop due to the solids component by
estimating the pressure drop due to air-only flow and subtracting this from the total pressure
drop. Conveying conditions, primarily ms in this case since mf is held constant, are then adjusted
so that a constant pressure drop for the solids component is maintained. W h e r e an equation such
as the Darcy-Weisbach Equation is used to calculate the air-only pressure drop, this component
is proportional to pipe length. Hence doubling the pipe length wUl result in a doubling of the
air-only pressure drop, and a U o w an equivalentrisein the total system pressure drop in order to
maintain a constant solids pressure drop component. This would a U o w a higher solids mass
flow rate than would be the case if the total system pressure drop was fixed.
The treatment of the air-only component in the power based model has been complicated by
seeking to incorporate the presence of the solids, to consider an interstitial air flow, non-circular
flow channels and the interaction between the solids and air velocities. Consequently there is no
direct correspondence between the approach of Mtils and W y p y c h and that adopted here. For
the purposes of this comparison it was elected to maintain a constant total pressure drop, rather
than basing the analysis on an estimated pressure drop attributable to the solids component. If
there is a significant variation in the pressure drop associated with the air component across the
pipe lengths being considered, is wUl influence the outcome of this scaling analysis and is likely
to contribute to differences between the results reported here and those indicated by Mtil and
Wypych.
Fixing the total system pressure drop means that as the air-only component increases with
increasing pipe length, the power avaUable to transport the solids component shrinks, resulting in
a more rapid rate of decline in ms: hence the greater is the magnitude for the power term "«".
This effect would be more pronounced under lean phase conditions due to the increased impact
of the air component which corresponds to the high values of "«" listed in Table 9.3.1.
9.3.1 Summary of Scaling Trends with respect to Length
In summary, it m a y be concluded that the scaling trends with respect to length produced by the
power base model, suggest that the solids mass flow rate is likely to decrease at a faster rate than
proposed by MUls and Wypych. The model also suggests, however, that a simple power
function provides an inadequate representation of this trend and m a y magnify apparent
differences. The different treatment of the air component between the M U l s and W y p y c h
approach and that adopted in the power based model also contributes to the differences between
the outcomes.
Exclusion of bends from the pipeline configuration for the purposes of isolating the length
effect would not appear to be a useful approach.
9. Scaling, Long Distance Conveying, Stepped Pipelines 235
9.4
Some Scaling Examples
The preceding sections considered specific characteristics of scaling behaviour separately, which
is important because in combination, inappropriate trends can cancel out in some cases, leading
to a false level of confidence in the overall performance of the model. Therefore analysing the
contribution of the individual components allows the development of a more stable foundation.
In practice, however, variation of all the pipeline configuration parameters (le
number of bends, vertical sections) should be expected. To assess effectively the scaling
performance of the model under these circumstances, ideally requires an examination of the
accuracy of the model in reproducing experimental results across a wide range of conveying
conditions. One means of doing this is to use experimental datafromone pipeline configuration
to carry out a solidsfrictionfactor correlation and establish the required parameters for the
model. Then, using these parameters, use the model to produce a set of conveying
characteristics curves of a different pipeline configuration, for which there are also experimental
data avaUable. Plotting this second set of experimental data points on the conveying
characteristics curves aUows some assessment of the closeness of the scaling process, as well as
any variation in thisfitover different conveying regimes. Figure 9.4.1 shows one example of
this process, using datafromthe pulverised coal PF2 obtained on pipeline configuration TR2.
200. N
M l |
Pipe Dotal Is
Dia.
Length
l mini (nil
25 65
Tot. i
Q_
65
Q_
O
DC
•
LU
DC
Ci
CD
CO
LU
100,
CC
Q_
I
J _ J L_J_
I
000
0-002
I I I I
0-004
0.006
0-008
PIR MASS FLOH RATE
CONTOURS:
Figure 9.4.1
I II
I I
I
I I J L
0-010
0.012
(KQ/S)
11 Soli ds Mass Flowrate
(kg/s)
Scaling Results: Bulk Solid: PF2, Base Data from TR2, Scaled to Pipeline
Configuration: TRl, Model PC6
Data in the form presented in Figure 9.4.1 give an indication of the success of
exercise by comparing the location of the experimental data points (indicated by the decimal
point in each case) with the surrounding conveying characteristics curves. In this way the
accuracy of the calculations over a range of conveying conditions can be viewed. Though
useful, this assessment is largely subjective.
Some objective measures can be introduced by using the air mass flow rate and pressure drop
coordinates of each experimental data point and interpolating between the conveying
9. Scaling, Long Distance Conveying, Stepped Pipelines
236
characteristics contours to determine the corresponding estimated conveying value. In this w a y a
set of experimentaUy deterrnined and scaled solids mass flow rates can be assembled and
compared. Figure 9.4.2 shows the result of this process, for the data presented in Figure 9.4.1.
0.9 -1
u
m*
OS
0.8 -
•
0.7 -
• •
©
m.
01
01
3 *
o
Vm
ii
•m
•
• ^ ^
0.5 -
IS*
,
0.4 -
•
%
^
• • ^
•^
0.3 -
m>
V
• X"
0.6 •
0.2 -
'•m
i)
m
0.
0.1 •
0 - mt-C-
1
1
1-
1
1
_H
1
_|
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Experimental Solids Mass Flow Rate (kg/s)
Figure 9.4.2
Scaling Performance - Comparison of Predicted and Experimental ms
values, for data in Figure 9.4.1
It is clear from Figure 9.4.2 that the solids mass flow rate values are generally over predicted
this instance, though the correspondence is quite reasonable. A cross-correlation between these
two sets of data gives a correlation coefficient of 0.897. Expressing the error as the difference
between the predicted and experimental data points, as an average of the experimental data point
value, the average error is 1 8 % .
If experimental data from more than one pipeline configuration are available, the combined data
can be incorporated into the solidsfrictionfactor correlation, which should lead to improved
scaling performances. Figure 9.4.3 provides an example of this procedure, again using the bulk
solid P F 2 but combining the results obtained on test rigs T R l and T R 2 in the correlation
analysis.
9. Scaling, Long Distance Conveying, Stepped Pipelines
L50.
1
237
f ' i i 'i ' ' j 1 i ' ' ' ' i
Fl pe Det.ol I s
Ola. Length
l mini (hi)
52 48
Tot.
100,
4B
o
DC
a
LU
DC
=>
en
en
50.
"0797
LU
0-86
cc
Q_
-I
I I 1_1
• 00
I.J -J-.J
. ..I . .
0.06
I 1 „l, :f.:,,l:,.:\ ,
0.02
0. 04
R I R M R S 5 FLOW R A T E
0.08
(Kg/s)
CQNT0UR5: fll Solids Mass Flovrole (kg/s)
Figure 9.4.3 Scaling Results: Bulk Solid: PF2, Base Data from TRl and TR2, Scaled to
Pipeline Configuration: T R 2 , Model P C 6
A visual examination of the results shown in Figure 9.4.3 suggests that a better fit to the
experimental data has been obtained. This is more clearly illustrated in Figure 9.4.4, comparing
the experimentaUy deterrnined and predicted values of ms. The correlation coefficient between
these two sets of numbers has improved to 0.959, and the average error between the predicted
and experimentally determined values of ms is 3.1%.
0.5
1
1.5
2
2.5
3
3.5
4
Experimental Solids Mass Flow Rate (kg/s)
Figure 9.4.4
Scaling Performance - Comparison of Predicted and Experimental ms
values, for data in Figure 9.4.3
9. Scaling, Long Distance Conveying, Stepped Pipelines
238
An alternative approach for evaluating the predictive or scaling performance of the models is
shown in Figures 9.4.5 to 9.4.8. In this case a constant solids massflowrate curve is determined
for a specific pipeline configuration, based on the analysis of experimental data obtained on that
pipeline. It has previously been demonstrated (Chapter 7, Section 1) that the power based model
is very effective at reproducing conveying characteristics for the pipeline on which the
experimental data are obtained. Using experimental data obtained on a second pipeline, but
retaining the same bulk solid, the model is used to reproduce the same massflowrate. IdeaUy
these curves would be collinear throughout their range. Displacements between the curves,
differences in slope, curvature and location of the pressure minimum point etc can provide a
good indication of the scaling characteristics of the model across a range of conveying
conditions. This form of presentation has the potential of highlighting conveying condition
regions where the model performance could benefitfromsome improvement. Results for both
models PCI and PC6 are combined on these four figures.
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Air Mass Flow Rate
•base data-PC1
—
• base data - PC6
-
scaled data - PC1
X
scaled data - P C 6
Figure 9.4.5: Scaling Results, Lines of Constant Solids Mass Flow Rate (ms = 0.4kg/s)
Bulk Solid: PFl, Pipeline Arrangement: TRl, Scaling from TR2, Model
PClandPC6
239
9. Scaling, Long Distance Conveying, Stepped Pipelines
^-s 120 -
^
OS
wa
*X^
w 100 •
Oo
s
X
^&L2&^
80 -
S
60 -
s
40J
AH
•a
o
H
20 0 -—
i
0.01
1
1
1
1
1
1
1
0.02
0.03
0.04
0.05
0.06
0.07
0.08
1
0.09
—
0.1
Air Mass Flow Rate
• base data - PC1
Figure 9.4.6
• base data -PC6
-
scaled data - PC1
X
scaled data - PC6
Scaling Results, Lines of Constant Solids M a s s F l o w Rate (ms = 2kg/s):
B u l k Solid: P F l , Pipeline A r r a n g e m e n t : T R 2 , Scaling f r o m T R l , M o d e l
PCI and PC6
300
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Air Mass Flow Rate
• base data -PC1
Figure 9.4.7
-base data -PC6
-
scaled data - PC1
X
scaled data - PC6
Scaling Results, Lines of Constant Solids M a s s F l o w Rate (ms = 0.4kg/s)
B u l k Solid: P F 3 , Pipeline A r r a n g e m e n t : T R l , Scaling f r o m T R 2 , M o d e l
PCI and PC6
9. Scaling, Long Distance Conveying, Stepped Pipelines
240
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Air Mass Flow Rate
•base data -PC1
Figure 9.4.8
• base data - PC6
scaled data - PC1
X
scaled data - PC6
Scaling Results, Lines of Constant Solids Mass Flow Rate (ms = 2kg/s) :
Bulk Solid: PF3, Pipeline Arrangement: T R 2 , Scaling from T R l , Model
P C I and P C 6
The contours listed as "base data" in Figures 9.4.5 to 9.4.8 are the "accurate" lines of constant
solids mass flow rate for the respective pipeline configurations, as the model parameters used in
generating them were obtained on the same pipeline. It can be seen that "base data" contours
produced by models PCI and P C 6 are very similar in most cases, the exception being the
PF3/TR2 combination on model P C 6 (Figure 9.4.8) where there is some divergence between
PCI and P C 6 curves as the flow moves further into the lean phase region. The data points listed
on each figure indicate the scaling performance of the respective models, with the model
parameters being generated from the alternative pipeline experimental data.
The differences between the two pipelines (TRl and TR2) may be summarised as:
TR2toTRl
3 5 % increase in length
Length:
T R 1 to T R 2
1 0 0 % increase in diameter
Diameter:
Bends:
TR2toTRl
approximate doubling in number of bends (4 to 9)
With the exception of the results generated using model PF3/TR2 combination, the shape of
scaled data is similar to the "accurate" ms lines in aU cases. In particular the slope, and the
displacement between the "accurate" and scaled data, in the lean phase region, indicate good
performance from the model in that area. The transition from dense to lean phase conveying
regimes is also weU modeUed, with the change in curvature and slope into the dense phase region
being weU matched. However the general trend is for the displacement between the "accurate"
and scaled data to increase in that region.
In reflecting on the performance of the models as illustrated by this set of figures it is
considering the differences within the model that m a y result in a disparity between the
"accurate" and scaled ms curves. The basic modelling of the physical processes is the same in all
cases. The key differences are in the velocity relationship selected (the VSR value) and in the
parameters determined in the solids friction factor correlation. These parameters are interrelated
since the VSR value is chosen on the basis that it provides the best solidsfrictionfactor
correlation for the set of experimental data being analysed.
9. Scaling, Long Distance Conveying, Stepped Pipelines
241
In seeking to refine the scaling performance, and hence predictive value of the power based
model, the options are to modify either the treatment of the basic physical processes in some
way, or to enhance the solids velocity and friction factor analysis. The general agreement
between the shape of the "accurate" and scaled data suggests that the modelling of the basic
physical processes (and the selection of the set of physical processes that are significant) is
satisfactory at this point. However it is the author's view that there would be some merit in
extending the sophistication of the relationship used for the solids velocity.
Of necessity the relationship chosen in the development of this model has been simple to allow
the development of the remaining elements to a stage suitable for evaluation of the overall
performance. The capacity of the power based model to match established conveying behaviour
suggests that further refinement is warranted. This refinement wtil not be without difficulty, as
any n e w relationship will need to be incorporated into the existing interrelationships (mass of
solids in the pipeline, air velocity, interstitial air component etc) within the model.
There is also the possibility of finding a better set of dimensionless parameters for the solids
friction factor correlation. The stable and reliable performance of the simple m* correlation
recommends itself as the preferable platform at this stage, despite the efforts at establishing a
more responsive set of parameters.
9.5 Long Distance Conveying - Stepped Pipelines
With the rising levels of understanding and confidence in positive pressure pneumatic conveying
systems, the possibilities and potential of this m o d e of bulk solid transport over longer distances
has been increasingly exploited. A largely inescapable feature of these systems is the substantial
pressure drop (say 400 to 600 kPag) required to overcome the resistance to flow though the pipe.
This can in turn lead to very high conveying velocities towards the exit end of the system as the
air expands to ambient pressure.
Conveying velocities higher than required to move the bulk solid are undesirable because they
are associated with high specific power consumption, high levels of pipeline degradation and
particle fragmentation. This characteristic of using compressible media for the transport of bulk
materials m a y be overcome by increasing the diameter of the pipe from inlet to outlet. This
would ideally be accomplished by a continuously tapered pipe, designed to maintain conveying
conditions in some optimum state, however it is difficult to envisage a circumstance where such
a solution would be economicaUy justifiable. However, using a number discrete steps in pipe
diameter along the pipeline is a simple and effective alternative.
The capacity to predict the performance of a long distance and stepped pipeline is obviously a
valuable feature of a pneumatic conveying model, as the alternative of constructing a trial
pipeline of similar dimensions is an expensive exercise. O n the other hand, the availabUity of a
long distance stepped pipeline to obtain test data, even if it does not precisely match the
envisaged system, would be very useful. A n issue that arises in this latter case is h o w to
incorporate the resulting data into the pneumatic conveying model.
It is not possible with the power based model to take total system pressure drop data across a
range of pipe diameters and corresponding lengths, and use this in the correlation analysis to
determine appropriate model parameters (predominantly the solids friction factor correlation and
the solids velocity relationship). Virtually all aspects of the model development, from the
estimation of the mass of solids in the pipe to the evaluation of air flow, treatment of bends and
calculation of dimensionless parameters for the solidsfrictionfactor equation, require specific
values ofpipe length and diameter. Therefore several different lengths and diameters cannot be
lumped together in this process. However, the presence of several distinct sections along the
pipeline can be used to advantage by allowing multiple sets of experimental data to be taken
during each test run, one for each segment measured. Providing sufficient pressure tapping
9. Scaling, Long Distance Conveying, Stepped Pipelines
242
points are set up to aUow the pressure drop across each pipe section to be measured, each pipe
segment can be treated as a quasi-independent system. To utilise these data, the model needs to
be designed to handle exit end pressures above ambient, and the same steady state solids mass
flow rate needs to be assumed for each section. The pressure tapping points should also be far
enough away from flow disruption zones such a bends or diameter changes to avoid acceleration
zones, which may otherwise distort the data to some extent.
Where such a test facility is avaUable it is possible to build up relatively quickly,
of data on the behaviour of a bulk solid over a range of pipeline configurations. The avaflabUity
of data for a range of lengths and diameters should assist in the determination of reliable
conveying model parameters. Generation of the conveying characteristics of the total system
then requires the reassembly of the pipe network, working from the known exit conditions
(typicaUy ambient pressure). The inlet parameters deterrnined for the last or exit pipe length
then form the known exit condition for the upstream pipe section, and the analysis proceeds up
through the segments. The process within each pipe section is identical to the treatment of single
diameter systems, with the additional requirement of successively recording and accumulating
the overaU effect. Figure 9.5.1 Ulustrates the outcome of this process for a 1.2km pipeline
containing three pipe sections conveying a pulverised briquette material.
600Prpe D«t.ai | s
Dro.
Length
500.
563
a
-i
381
400.
253
Q_
O
tr
Tot.! 1197
300.
a
LU
cc
en
en
200,
LU
DC
100.
-i—i—i
00
i
i_i
i
i_
0- 10
j—i—i—i—1_
0-20
AIR MAS5 FLOW R R T E
0.30
0.40
(Kg/s)
CONTOURS: fll Solids Moss F l o w r a t e (kg/s)
Figure 9.5.1
Conveying Characteristics for 1.2km Pipeline: Bulk Solid: PF5,
Pipeline Configuration: T R 2 0 (incorporating 69,81 and 1 0 5 m m
diameter pipe sections), Model P C 6
It is interesting to note that a transition from lean to dense phase is stiU identified
aU the data points are clearly in the lean phase region. The correlation coefficient between the
experimental data and the equivalent model generated points is 0.99, and the mean error is 1.6%,
a good overall result.
foUowing Figure 9.5.2, shows the result of a simUar set of calculations for a pulverised
m coal usmg model P C 1. Four pipeline diameters are in use for this testrig.Thefitis again
9. Scaling, Long Distance Conveying, Stepped Pipelines
243
good, with a correlation coefficient of 0.98 and an average difference between the experimental
and model derived values of 3 . 7 % .
FT pe Det.al I s
Din.
Length
f»)
146
390
261
150
Q_
D
CC
O
LU
QC
m
en
en
LU
az
94B
AIR M A 5 5 FLOW RATE
(Kg/s)
C O N T O U R S : 111 S o l i d s M o s s F l o w r a t e (kg/s)
Figure 9.5.2
Conveying Characteristics for 948 m Pipeline: Bulk Solid: P F 6 ,
Pipeline Configuration: T R 2 4 (incorporating 60,69,81 and 1 0 5 m m
diameter pipe sections), M o d e l P C I
The procedure for developing conveying characteristics diagrams for long distance stepped
pipeline systems is identical whether the source data are obtained from a single pipe
length/diameter configuration or from a multiple length/diameter arrangement. The only
variation is in the generation of the model parameters (solidsfrictionfactor correlation and VSR
value). If the source data were obtained on a single length/diameter pipeline, it would be
expected that the accuracy of a prediction for a stepped pipeline system would suffer since the
solids friction factor correlation did not include consideration of a range of pipe lengths and
diameters. This is generafly the case, though in most cases the correlation is quite good. Figure
9.5.3 shows conveying characteristics generated for a pulverised brown coal (PF4) over a 1.2km
pipeline. The conveying characteristics are developed for a three pipe section configureation,
however the results presented are based only on the experimental data obtained for the smallest
diameter ( 6 9 m m ) section. The agreement between the predicted and experimental ms values is
good, with a correlation coefficient of 0.95 and an average error of 9.8%.
9. Scaling, Long Distance Conveying, Stepped Pipelines
244
400.
P T p e D o t a l Is
Or a. Length
(mini
fm)
69
577
61
391
-al
105
234
Q_
Tot.
1202
300- -
O
CC
200.
o
LU
CC
r>
en
tn
L00. -
LU
CC
Q_
0.20
RIR MflSS FLOW RATE
0.30
0-40
(KQ/S)
CONTOURS: fll Solids Moss Flowrate (kg/s)
Figure 9.5.3 Conveying Characteristics for 1202m Pipeline: Bulk Solid: PF4,
Pipeline Configuration: TR16 (incorporating 69,81 and 105mm diameter
pipe sections), Model Correlation based on TRl 7 (69mm diameter)
experimental data, Model PCI
Similarly, Figure 9.5.4 shows conveying characteristics generated for a fly as
through a 945m stepped pipeline. The data for estabhshing the model parameters were obtained
on a single diameter pipeline (TRl4), with the model subsequently used to predict the
performance of the stepped pipeline configuration (TRl 5). The data points indicated on Figure
9.5.4 are the experimental results obtained on the stepped pipeline and allow some assessment
of the scaling process. In this case the correlation between the scaled and experimental values is
0.93, with an average error of 8.5%.
9. Scaling, Long Distance Conveying, Stepped Pipelines
245
500.
P T p e D o t a l 1s
400.
-
D r a-
Length
fin in)
69
81
fml
562
382
Tot.
945
jat
Q_
300.
-
o
cc
o
LU
200.
cc
=J
tn
cn
LU
L00.
a:
Q_
• 30
AIR MH55 FLOW RATE
(KQ/S)
CONTOURS: fll Solids Moss Flowrate
Figure 9.5.4
Ug/s)
Conveying Characteristics for 9 4 5 m Pipeline: Bulk Solid: F A 2 ,
Pipeline Configuration: T R T R 1 5 (incorporating 69 and 8 1 m m diameter
pipe sections), M o d e l Correlation based on T R l 4 ( 6 9 m m diameter)
experimental data, M o d e l P C I
A key area where a difference can occur when developing model parameters from a subset of
the experimental data, is in the selection of the VSR term. The value of this term directly effects
the solids velocity used in the model, and subsequently affects other calculated parameters such
as total mass of solids in the pipeline. Optimising the solids friction factor correlation is the
basis for selecting a particular value for VSR. The value chosen is the one which produces the
highest correlation coefficient w h e n deterrnining the solidsfrictionfactor relationship. It would
be better if this selection is based on thefinalassessment of model performance, ie the ability of
the model to reproduce accurately the experimental data points. However the application of this
criterion is extremely arduous from a computational perspective, and is not adopted routinely.
Figure 9.5.5 is a third example, using the same basic data set as in Figure 9.5.3 (only the 6 9 m m
pipe section data), but carrying out the analysis with model PC6. The optimum VSR value based
on the solidsfrictionfactor correlation in this case is 0.35 (a low value), compared to a value of
0.7 if the fuU data set is included in the analysis.
9. Scaling, Long Distance Conveying, Stepped Pipelines
246
400.
Pipe Dotal I e
300. 0
Dia.
Length
l mini
(ml
69
577
81
391
105
234
Tot.
1202
CD
cc
200. -
Q
LU
cc
:n
en
en
LU
az
D_
100. -
0. L0
0.20
0.30
0. 40
AIR MASS FLOW RATE (Kg/s)
CONTOURS: fU Solids Moss Flowrate (kg/s)
Figure 9.5.5
Conveying Characteristics for 1 2 0 2 m Pipeline: Bulk Solid: P F 4 , Pipeline
Configuration: T R 1 6 (incorporating 69,81 and 1 0 5 m m diameter pipe
sections), M o d e l Correlation based o n T R 1 7 ( 6 9 m m diameter)
experimental data, M o d e l P C 6
A comparison between the calculated contours in Figure 9.5.5 and the original experimental data
indicates a good correlation (0.95), however the average error is 19.5%. T h e overall result,
nevertheless, is very sirnUar to that presented in Figure 9.5.3 .
If a VSR value of 0.7 is specified in the above case, overriding the automatic optimising
procedures within the model, there is a marginal improvement to the correlation coefficient (to
0.96) but a significant reduction in the average error to 7.3%. Figure 9.5.6 shows the conveying
characteristics obtained if this value of VSR (0.7) is adopted.
9. Scaling, Long Distance Conveying, Stepped Pipelines
247
400.
P i p e D o t a l Is
-
300- -
Q_
CD
CC
Q
D f a.
Length
t mini
fm]
69
577
61
391
105
234
Tot.
1202
200. -
LU
CC
CJ
en
en
LU
100. -
QC
Q_
0-20
AIR MRSS FLOW RATE
0.30
0.40
(Kg/si
C O N T O U R S : fll Solids Mass Flowrate (kg/s)
Figure 9.5.6
Conveying Characteristics for 1202m Pipeline: Bulk Solid: PF4, Pipeline
Configuration: T R 1 6 (incorporating 69,81 and 1 0 5 m m diameter pipe
sections), Model Correlation based on T R l 7 (69mm diameter)
experimental data, Model PC6, VSR set to 0.7
This outcome, where an improvement in the overall model performance is achieved at a VSR
value different from that suggested by optimising the solidsfrictionfactor correlation, is also
noted in a number of other cases. It would appear, therefore, that there is scope to refine the
model performance in this area.
9.6 Conclusions
Results have been presented in this chapter indicating that the power based model is able to
predict the performance of pipeline configurations other than the configuration used in
evaluating the model parameters. The scaling trends demonstrated by models PCI and PC6 for
the range of fly ash and pulverised coals examined appear reasonable, and generally match
results suggested elsewhere. There is scope for further refinement and improvement in accuracy
however, and the treatment of the solids velocity is one area that is identified as having potential
for development.
The capacity of the model to be applied to stepped pipeline situations has also been
demonstrated. The inherent tendency of the model to identify a transition to dense phase
conveying conditions, even where there is no specific data in this region to "force" arisein the
solids mass flow rate contours, has been highhghted.
A potential of the model that has not been explored at this stage is the possibility of a
"pipe fuU" criteria to determine the appropriate time to step the pipe diameter. For a given
diameter and air mass flow, working upstream from the pipe outlet, results in increasing pipeline
pressure which is accompanied by reduced air and solids velocities. The falling solids velocity
means that an increasing proportion of the pipe cross-section is occupied by the solids
component. Increasing the velocity by reducing the pipe diameter as the cross-sectional area
occupied by the solids component approaches some threshold value, would provide a rational
basis for deciding the location of steps in the pipeline.
248
/0
Power Based Model - Review and Further Work
Chapter 10
POWER BASED MODEL
REVIEW AND FURTHER WORK
The pneumatic conveying model based on the power requirements of a set of physical processes
that has been proposed and developed over the preceding Chapters, has demonstrated a number
of positive features. It differs significantly from pressure drop type models and models derived
from an extension of single phase flow theory, and as such is not faced with the task of trying to
coerce a model developed for one set of conditions to produce trends that are fundamentally
different.
An intuitive understanding of the pneumatic conveying process readtiy results in the type of
concave upwards solids mass flow rate contours that result from a graphical presentation of
experimental data, as adopted in the mf, Apt format in this thesis. Increasing the solids loading in
the pipe is bound, at some point, to generate higher pressure drops and potential blockages as the
solids component comes to dominate the pipe cross-section. Hence, foUowing a line of constant
solids mass flow rate in the direction of reducing air mass flow rate, will at s o m e point m o v e
into a region of increasing pressure drop with reducing air flow. This quite reasonable outcome
is not so easy to impress on a model that is tied to a trend of increasing pressure drop with
increasing fluid flow rates. There is a transition between the dominant physical processes in
different phases of the conveying regime, and it is likely that a model that is able to reflect this
characteristic wUl be more successful.
The power based model demonstrates a capacity to generate conveying characteristics of the
form required, and successfuUy modeUed all the experimental data presented to it. These data
were obtained on a range of pulverised coals and fly ashes. It should be noted that these
materials were aU "good conveying" type materials that could straddle the lean to dense phase
transition. If a different style of conveying is to be analysed, such as low velocity granular flow,
it is suggested that the basic approach promoted in the power based model would continue to
have merit, however differences in the physical processes to be m o d e U e d m a y well require
significant modifications to the arrangement and detaUed content of the model.
A notable feature of this model is that the solids friction factor takes on values that can
conceivably be identified with the solidsfrictionphenomenon. This is in contrast to the solids
friction factor values resulting from the application of s o m e of the alternative approaches to
modeUing this process. This outcome supports the selection of physical mechanisms identified
as being significant for this process, and the manner of treating these mechanisms, as adopted in
this model. The generaUy good correlation between the experimentally deduced and analyticaUy
determined solids friction factors also supports this conclusion.
10
Power Based Model - Review and Further Work
249
It has been shown that the selection of the dimensionless parameters for the solidsfrictionfactor
correlation is important. Inappropriate parameters can result in erroneous trends being generated
by the model. The simplest arrangement proposed (Xs being a power function of m*) proved to
be the most reliable of the set that is examined. However the possibility that a different selection
could improve the performance of the model is not discounted.
An additional benefit flowing from the application of this model is the insights into the
conveying process that become avaUable. The power based approach focuses attention on the
power consumption of this m o d e of transport and facilitates the selection of suitable conveying
conditions based on mininnsing the specific power consumption requirement. The marked
difference in this respect of lean and dense phase conveying options is apparent.
It is of interest to note the contributions of the major pipeline features (length, diameter, ben
to the overall power requirement of the transport system. The major role indicated for bends
under lean phase conditions is a particular feature. Recognition of these factors has the potential
of buUding on the strengths of the pneumatic conveying system and minimising the influence of
the weaknesses at the design stage.
A simple criteria for identifying the "no go" zone for reliable conveying on the conveying
characteristics diagram is a further outcome of the application of this model. The zones
identified in this manner accord well with experimental experience and expectations. It is also
noted that with some additional development work there m a y be the scope to apply this criteria
at other locations along the pipe. From a design perspective, this has the potential to check and
modify designs that might otherwise result in pipeline configurations that promote the incidence
of flow instabUities and blockages. The impact of specific elements of the pipeline, such as a
bend at a particular location, could be explored and alternatives evaluated.
An important aspect of pneumatic conveying design is the prediction of the performance of
pipeline systems based on experimental data obtained on a testrig.This m a y involve conversion
or scaling of the test data so as to replicate the expected performance of the n e w pipeline. This
procedure is already an integral part of the power based model in that the treatment of the
physical processes sets in place the manner in which the performance of different pipeline
configurations will be modeUed. Providing the treatment of these physical processes is realistic
the outcome should be stable with reliable performance predictions. O f particular interest in this
respect is the variation in scaling trends deterrnined for lean and dense phase conveying
conditions. A n intuitive understanding of the pneumatic conveying process would suggest that
this is quite a reasonable outcome. A s different physical processes are dominating in these two
phases of transport, it is likely that the trends in performance with varying pipeline parameters
would be different. The power based model allows differences of this nature to be incorporated
in a w a y that is not possible if scaling is based on a set of equations such as those proposed by
Mills et al (1982) or W y p y c h et al (1987).
It is perhaps appropriate at this point, in view of the number of alternative options that have b
explored in the preceding chapters, to collect the preferred set of equations for implementing the
power based model. The original equation numbers are retained to assist the reader in referring
to the relevant section of the thesis. The fundamental relationship on which this model is built is:
Power = m^AKEf + APEfj + ms(AKEs + APES) + (air frictional power loss)
+ (solids frictional power loss)
E q 6.2.1
/0
250
Power Based Model - Review and Further Work
where the power is deterrnined from the isothermal expansion of the air, i.e.
P
power, isothermal expansion... Eq 6.3.1
mfRTIn
\Pe)
There is some interdependence in the remaining terms, in particular with respect to the solids
velocity as this affects the cross-sectional area occupied by the solids and subsequently the
average actual air velocity (Vfa). The foUowing expression has been adopted in relating the solids
velocity to the superficial air velocity:
Vs = VSRxVfa
Eq 6.5.14
The VSR term is a constant factor between zero and one, and is deterrnined from an analysis of
experimental data on the bulk solid being conveyed. In this thesis the criteria adopted in
selecting a value for VSR is that value that gives the best result in the solidsfrictionfactor
correlation.
The Vfa term at any point along the pipeline, using the appropriate air density and pipe area
values, can then be found from:
m
1
m,
f ,
V
—
Eq 6.5.7
»—A
vs* P.• +Pf)
Power =
With equations 6.5.14 and 6.5.7 it is possible to determine air and solids velocities at any point
along the pipeline. Values of these parameters at the inlet and exit ends allow an estimate to be
made of the power associated with change in kinetic energy along the pipeline, (for both the air
and solids components of the flow):
f
2
\
POKE = ms x VSR • +m•f ] ( 2
2\
The power associated with a change in potential energy is deterrnined from:
POPE = (ms + mf)xgx
(change in height)
Eq 6.7.1
The power loss associated with air only flow is estimated from an application of the is
expansion equation, applied at the mean pressure of the pipe section under consideration.
Po DTI \Pi +Pe+ APf )
Pof = mfRT\v\
Eq 6.4.5
The Apf term in equation 6.4.5 is the pressure loss attributed to the air only flow, and
calculated based on consideration of that component of the air flow lhat may be considered to be
passing above a moving fluidised bed of solids. SimUarly, the W/term is also restricted to that
component of the air flow, and is not the total air mass flow rate. Detennination of the required
terms andfinalevaluation of equation 6.4.5 is a somewhat elaborate procedure, and the reader is
referred to Section 6.4.5 of the text where a worked example details thefifteensteps involved. It
is noted, however, that the following equation is proposed for the determination of the interstitial
air velocity:
V
M = Vs+~f(vfa~Vs) Eq 6.4.12
and that the power determined from Equation 6.4.5 is further reduced by restricting it
between the air and the pipe waU, i.e.
Pn
Dx(n-9)
f =
Dx(n-0 + sm(0))XP°f
(from Eq 6.4.14)
The power loss associated with frictional interactions between the pipe wall and the so
considered in two parts. The contribution from the straight pipe sections is determined from:
p
°s = \xgxVsmx Ms
Eq 6.5.34
10
Power Based Model - Review and Further Work
251
where the mass of solids in the pipeline (Ms) is calculated from:
LxpsxAxm*xp
M
s=—1
TT^
Eq 6.5.29
m*xPfin+VSRxps
If a stepped pipeline is being analysed, the contributionfromeach diameter section is calculated
separately using Equations 6.5.34 and 6.5.29, and the combined value found.
The second solids friction component is that associated with conveying around bends.
contribution of gravity to the solids/pipe wall interface force is ignored at the bends, this power
loss component is calculated from:
PoB=Xsx(BVFxVsi)
xmsx(j)B
Eq6.6.30
where V& is the solids velocity at the inlet to the bend, ^ is the angle of the bend (typicaUy 90°)
and BVF is found from:
BVF = 0.5x i+e
Eq 6.6.32
V
If the contribution of gravitational force is included, the foUowing equations apply:
For bends lying in a horizontal plane:
0.5
Po„ = l x
Rl
•+?
xmsxRBx<f>B
Eq 6.6.33
For bends oriented in a vertical plane:
i. where the exit velocity is upwards:
PoB = m , x R B x
A x
(BVFxVsif g
RB +V2
+ JL
Eq 6.6.34
n. where the exit velocity is down:
PoB=msxRBx0Bx\Zsx
(BVFxV,)2 _ g j _
RB & V2
g
Eq 6.6.35
The remaining term to be evaluated is the solids friction factor (Xs), a key parameter in all the
solids friction equations. Values for this parameters are obtained from the analysis of
experimental data, and are extracted from:
Isothermal Power - \m.+mf)x(AKE + APE)+ Air Friction Loss
X =
[solids velocity x normal force in straight pipe + solids velocity x normal force in bends]
Eq 6.9.1
A power relationship using dimensionless parameters relevant to the flow regime isfittedto the
solidsfrictionfactor data. A range of formulations for this relationship were considered,
however it is suggested that the simplest:
be applied.
Xs = const. xm%
Despite the merits of the power based pneumatic conveying model outlined above, there are a
number of aspects of the model that would benefitfromsome further development.
Perhaps the major feature in this respect is the estimation of the solids velocity. A
step in the process of evaluating the power based model parameters is the establishment of a
suitable relationship for the solids velocity. Once an expression for this term has been adopted,
it is then possible to evaluate the contribution of the components of the overaU conveying
252
10
Power Based Model - Review and Further Work
system. This extends from calculating various aspects for the air velocity ("actual" superficial air
velocity, interstitial and top air velocity), to estimating the total mass of solids in the pipeline,
determining velocities around bends and so on.
The path chosen at this stage of the development of the model has been a simple proportional
relationship (VSR value) between the actual superficial air velocity and the solids velocity. The
selection of the most appropriate proportional constant for a specific bulk solid has been based
on the subsequent solids friction factor correlation. The VSR value which resulted in the best
correlation between the experimental and calculated solids friction factors is chosen as the
appropriate one, and is then considered as a derived bulk solid characteristic. T o a large extent
this is an expedient rather than an ideal solution. T o facUitate the development of this model it is
necessary to proceed from the basis of a simple initial assumption in this area. Even with this
assumption, a substantial amount of iteration and computation remains to determine model
parameters. The feasibility of adopting this simplifying assumption has been explored, and it is
concluded that it is an acceptable compromise for this stage of the model's development.
The difficulty with such an assumption is that whtist it allows the solids velocity to vary in a
reasonable fashion with air velocity, this same ratio applies regardless of whether the conveying
conditions are in the lean or dense phase, and is carried through to different pipe configurations
(diameters and lengths). It is possible that the solidsfrictionfactor correlation is able to adjust
the overaU performance of the model to counter the simplicity of this restrictive solids velocity
assumption to some extent; however n o w that the power based approach has demonstrated merit
in its capacity to model the pneumatic conveying mechanism it would appear justifiable to
explore more complex solids velocity relationships. Preferably this would include the coUection
of experimental data on solids and air velocities, and m a y involve parameters such as m* and
flow turbulence.
A number of novel approaches are suggested in this model for calculating the contribution of
some of the components of the pneumatic conveying process. T h e treatment of the air
component and the power consumption of transporting the solids around bends are cases in
point. The implications of the approaches adopted have been explored in s o m e detaU, and there
are reasonable grounds for considering these to have been successful. Nevertheless a useful area
of further work would be an attempt to collect experimental data that m a y a U o w the refinement
of the analysis in these areas.
A further refinement that could be incorporated would be a more precise identification of the
location of pipeline features such as bends. This is likely to have a relatively minor impact on
the overall system performance, and at this stage is not considered worth the additional
complexity, however it is an area where the accuracy of the model m a y be further tightened.
Finally, one of the core tasks in setting the model parameters is the determination of the solids
friction factor correlation. Considerable effort has been expended on developing and evaluating
a number of alternatives in this area. It has been clearly established that the dimensionless
parameters selected for use in this relationship can have a crucial impact on the performance of
the model. In particular, inappropriate parameter selection can lead to totaUy incorrect trends
being predicted by the model. Given the range of alternatives that have been explored in this
thesis, it is perhaps ironic that the simplest approach, and one that has frequently been
incorporated in other quite different models, has proved the most stable and reliable.
Nevertheless the key influence the solidsfrictionfactor relationship has on the performance of
the model warrants additional investigation in this area, and it m a y well be that a preferable set
of dimensionless parameters is waiting to be identified for this purpose.
In conclusion, whtist it is noted that this work has opened up significant additional areas to be
explored in further refuting this model, it is suggested that this approach has demonstrated
/0
Power Based Model - Review and Further Work
253
substantial merit. The power based model has been successfuUy applied to experimental data
obtained on a range of fly ash and pulverised coal bulk solids, conveyed through a diverse set of
pipeline configurations. The basic structure of the model allows different aspects of the
conveying process to dominate in the lean and dense phase regions, faciUtating the generation of
realistic conveying characteristics plots, in particular reproducing the lean to dense phase
transition behaviour. This approach to modelling the pneumatic conveying process includes
scaling capabilities based on core pipeline geometry parameters, and provides n e w insights into
the contributions of various key components to the overaU conveying load. Consequently it has
the potential to improve the understanding and application of pneumatic conveying technology.
254
11 References
Chapter 11
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Sutton, H.M., and Richmond, R.A., 1973. "How to improve Powder Storage and Discharge in
Hoppers by Aeration", Process Engineering. Sept.
Swamee, P.K. and Jain, A.K., 1976. "Explicit Equations for Pipe-Flow Problems", J. Hydr. Div.,
Proc. A S C E , M a y 1976, pp657-664.
System 2600 Instruction Manual, Malvern Instruments Ltd, M 026, Issue 2.1, April 1991, pp
5.11-5.15.
Szikszay, G., 1988. "Friction Factor for Dilute Phase Pneumatic Conveying", Bulk Solids
Handling. Vol 8, No. 4, 1988, pp 395-399.
Weber, M., 1981. "Principles of Hydraulic and Pneumatic Conveying in Pipes", Bulk Solids
Handling. Vol 1, No. 1, 1981, pp 57-63.
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Handling. Vol 2, No. 2, 1982, pp 231-233.
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257
258
11 References
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Zenz, A.F. and Othmer, 1960. "Fluidization and Fluid-Particle Systems", Reinhold Publ
Corporation, N e w York, pp315-319.
Zenz, F.A. and Othmer, D.F. 1960. "Fluidization and Fluid-Particle Systems", Reinhold
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Zenzs, F.A., 1984. "Conveyability and Powder Classification", Proceedings of the Tech
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1984, pp396-407.
258
Appendix A
APPENDIX
259
A
Experiment Data
Materials
Pipeline Configurations
Steady State Conveying Data
Appendix A
260
Table A.1:
Test Materials - Power Based Pneumatic Conveying Model
2000
800
Associated
Test Rigs (see
Table A.2)
TRl, TR2
2080
800
TR1,TR2
2170
1500
750
520
Pulverised Briquettes
1571
482
PF6
Pulverised Brown Coal
1495
520
FAl
Fly Ash
2230
874
FA2
Fly Ash
2197
634
TR1,TR2
TR16, TR17,
TR18, TR19
TR20, TR21,
TR22, TR23
TR24, TR25,
TR26, TR27,
TR28
TR3toTRll
incl.
TR12, TR13,
TR14, TR15
Designation
Material
A
PIP
3
kgm"
PFl
PF3
PF4
Pulverised Coal, sieved to
7 0 % < 75 u m
Pulverised Coal, sieved to
8 4 % < 75 u m
Pulverised Coal
Pulverised Brown Coal
PF5
PF2
kgm
Appendix A
Table A.2:
Test Rigs - Pipeline Configurations - Power Based Pneumatic
Conveying Model
Designation
TRl
TR2
TR3
TR4
TR5
TR6
TR7
TR8
TR9
TRIO
TR11
TR12
TR13
TR14
TR15
261
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
a section of TR3
stepped pipeline
TR16
stepped pipeline
TR17
TRl 8
TR19
TR20
a section of TRl 6
a section of TRl6
a section of TRl6
stepped pipeline
TR21
TR22
TR23
TR24
a section of TR20
a section of TR20
a section of TR20
stepped pipeline
TR25
TR26
TR27
TR28
a section of TR24
a section of TR24
a section of TR24
a section of TR24
L
D
(m)
65
48
(mm)
168.5
110.5
43.7
46.65
20.15
48.85
28.7
95.5
139.2
172.1
382.2
553.5
562.4
382.4
558
391.2
233.66
105
105
105
105
105
105
105
105
105
69
81
69
558
391.2
233.66
563.4
381.2
252.7
563.4
381.2
252.7
146.1
390.3
261.1
150
146.1
390.3
261.1
150
(m)
3.9
3.6
7
0
0
0
0
7
7
7
7
Number
of Bends
6.3
9
4
5
2
0
2
0
2
2
4
4
5
12
17
69
81
69
81
105
0
6.75
0
0
0
17
12
16
12
8
69
81
105
0
0
0
16
12
8
69
81
105
0
0
7
17
12
8
69
81
105
0
0
7
17
12
8
60
69
81
105
0
0
0
7
4
12
8
5
60
69
81
105
0
0
0
7
4
12
8
5
25.4
52.5
6.65
6.75
Bend
Radius (m)
0.254
0.254
Appendix A
262
Table A.3:
Steady State Conveying Data - PFl / TRl
Test Rig: TRl
Material: PF I
J
Loose Poured Density: 800 kg m J
Particle Density-: 2000 kg m
Ambient Temperature: 15 ° C
Ambient Pressure: 101 kPa
No. of Bends Bend Radius
Lift
Pipe Length
Pipe ID
No. data
(m)
(m)
(m)
(mm)
points
.254
9
3.9
65
25.4
21
Exit Pressure
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
1
1
(kPa,)
Rate (kgs )
(kPa)
(kgs" )
0
0.231
81
0.0097
0
0.243
80
0.0089
0
0.262
79
0.0076
0
0.283
80
0.0061
0
0.262
79
0.0035
0
0.206
82
0.0022
0
0.405
117
0.0099
0
0.44
116
0.0084
0
0.438
115
0.0075
0
115
0.45
0.0056
0
0.423
113
0.004
0
0.367
118
0.0024
0
0.571
150
0.0099
0
0.608
148
0.0085
0
0.633
0.0068
149
0
0.0057
144
0.622
0.0035
152
0
0.508
0.0098
188
0
0.801
0.0083
184
0.832
0
0.0063
187
0.84
0
0.0049
184
0.782
0
Table A.4:
Steady State Conveying Data - PFl / T R 2
Material: PFl
Test Rig: TR2
Particle Density: 2000 kg m 3
Loose Poured Density: 800 kg m
Ambient Pressure: 101 kPa
Ambient Temperature: 15 ° C
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
SSSmmm.
52.5
30
0.254
48
3.5
Air Mass Flow Rate
Pressure Drop
Exit Pressure
Solids Mass Flow
0.0061
0.0117
(kPa)
Rate (kgs-1)
46
26
0.709
0.562
o_
o
Appendix A
263
Table A.4: (cont.)
Air Mass Flow Rate
(kg s 1 )
0.0232
0.0348
0.0592
0.0072
0.0135
0.0245
0.0364
0.0602
0.0086
0.0121
0.0233
0.0331
0.0613
0.0104
0.014
0.0242
0.0346
0.0639
0.0135
0.0172
0.0228
0.0444
0.0641
0.0149
0.0186
0.0352
0.0453
0.0677
Pressure Drop
(kPa)
23
24
29
45
42
36
37
45
55
47
48
47
60
63
65
58
64
74
71
69
70
75
86
90
87
94
101
109
Solids Mass Flow
Rate (kgs"1)
0.584
0.578
0.479
0.829
1.054
1.092
1.109
0.879
1.308
1.498
1.661
1.62
1.303
1.825
2.535
2.311
2.285
1.8
2.749
2.66
2.927
2.618
2.213
4.051
3.873
3.782
3.674
2.915
Exit Pressure
(kPa*)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
264
Table A.5:
Appendix A
Steady State Conveying Data - P F 2 / T R 1
Test Rig: TRl
Material: PF2
Loose Poured Density: 800 kg m 3
Particle Density: 2080 kg m *
Ambient Temperature: 15 ° C
Ambient Pressure: 101 kPa
No. of Bends Bend Radius
Lift
Pipe Length
Pipe ID
No. data
(m)
(m)
(m)
(mm)
points
0.254
9
3.9
65
25.4
20
Exit Pressure
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
1
1
(kPa,)
Rate (kgs )
(kPa)
(kg s" )
0
0.189
81
0.0102
0
0.213
80
0.0089
0
0.25
81
0.0078
0
0.28
79
0.006
0
0.265
81
0.0035
0
0.36
116
0.0103
0
0.409
116
0.009
0
0.417
112
0.0079
0
0.439
112
0.0063
0.39
0
111
0.0039
0.355
118
0
0.003
0
0.0106
0.532
148
148
0.591
0.0092
0
0.0084
147
0.611
0
0.0066
0.608
143
0
147
i
0.0041
0.537
0
0.0034
152
0.499
0
0.0101
175
0.716
0
0.0073
185
0.834
0
0.0041
196
0.738
0
Table A.6:
Steady State Conveying Data - P F 2 / T R 2
Material: PF2
Test Rig: TR2
3
Particle Density: 2080 kg m
Loose Poured Density: 800 kg m
Ambient Pressure: 101 kPa
Ambient Temperature: 15 ° C
No. data points
Pipe ID
Pipe Length
Bend Radius
Lift
No. of Bends
(mm)
_(m)_
(EL
fa)
52.5
31
0.254
48
3.5
Air Mass Flow Rate
Pressure Drop
Exit Pressure
Solids Mass Flow
(kg s"1)
(kPa)
Rate (kgs"1)
0.0322
66
2.547
_0_
0.0218
63
0
2.659
0.0133
33
0.987
0.0068
41
0.8
0.0242
31
0
0.971
Appendix A
Table A.6:
265
(cont.)
Air Mass Flow Rate
(kg s"1)
0.0367
0.0483
0.0089
0.0124
0.0236
0.0334
0.05
0.0106
0.0234
0.0228
0.0351
0.0516
0.0148
0.0196
0.0278
0.0373
0.0649
0.0692
0.014
0.017
0.0221
0.0392
0.066
0.0187
0.0244
0.0412
Pressure Drop
(kPa)
32
38
52
49
48
50
56
57
54
63
58
69
47
60
52
56
86
105
72
66
70
70
81
87
93
92
Solids Mass Flow
Rate (kgs"1)
0.856
0.802
1.877
1.691
1.784
1.727
1.54
1.961
2.171
2.651
1.971
2.045
1.682
2.446
1.892
1.914
2.275
2.817
3.045
2.749
3.036
2.566
2.103
3.824
4.275
3.355
Exit Pressure
(kPa,)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
!
266
Table A.7:
Appendix A
Steady State Conveying Data - P F 3 / T R 1
Test Rig: TRl
Material: PF3
3
Loose Poured Density: 750 kg m 3
Particle Density: 2170 kg m
Ambient Temperature: 15 ° C
Ambient Pressure: 101 kPa
No. of Bends Bend Radius
Lift
Pipe Length
Pipe ID
No. data
(m)
(m)
(m)
(mm)
points
0.254
9
3.9
65
25.4
21
Exit Pressure
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
1
1
Rate (kgs" )
(kPa.)
(kPa)
(kg s")
0
0.238
82
0.01
0
0.265
82
0.0088
0
0.301
80
0.0071
0
0.309
79
0.0061
0
0.333
81
0.0039
0.276
0
81
0.0021
0
116
0.418
0.0095
116
0
0.443
0.0086
0
0.489
115
0.0072
0.0059
0.505
115
0
116
0.0037
0.537
0
0.0023
115
0
0.395
0.009
156
0.668
0
0.0083
156
0.688
0
0.0067
155
0.724
0
0.0053
151
0.747
0
0.0037
155
0.741
0
0.01
190
0.82
0
0.0079
186
0.875
0
0.0067
183
0.902
0
0.0059
184
0.896
0
Appendix A
Table A.8:
261
Steady State Conveying Data - P F 3 / T R 2
Material: PF3
Test Rig: T R 2
Particle Density: 2170 kg m 3
Loose Poured Density: 750 kg m 3
Ambient Pressure: 101 kPa
Am jient Temperature: 15 ° C
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
(m)
21
25.4
65
3.9
9
0.254
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kg s"1)
(kPa)
Rate (kgs"1)
(kPa,)
0.01
82
0.238
0
0.0088
82
0.265
0
0.0071
80
0.301
0
0.0061
79
0.309
0
0.0039
81
0.333
0
0.0021
81
0.276
0
116
0.0095
0.418
0
0.0086
116
0.443
0
0.0072
115
0.489
0
0.0059
115
0.505
0
0.0037
116
0.537
0
115
0.395
0.0023
0
156
0.668
0.009
0
156
0.688
0.0083
0
155
0.0067
0.724
0
151
0.747
0
0.0053
155
0.741
0
0.0037
0.82
190
0.01
o
0
0.875
186
0.0079
0
0.902
183
0.0067
0
0.896
184
0.0059
Appendix A
Table A.9:
Steady State Conveying Data - P F 4 / T R 1 6
Material: PF4
Particle Density: 1500 kg m 3
Ambient Pressure: 101 kPa
Pipe ID
No. data
(mm)
points
17
69
Test Rig: TRl6 (3 pipe sections)
Loose Poured Density: 520 kg m 3
Ambient Temperature: 15 ° C
No. of Bends
Lift
Pipe Length
(m)
(m)
558
0
16
Bend Radius
(m)
1
N B this pipe section on its own also constitutes test rig TR17 (PF4 / TR17 combination)
Exit Pressure
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
1
1
(kPa.)
Rate (kgs" )
(kPa)
(kg s")
74.46
1.069
102.16
0.081
69.96
0.92
106.63
0.082
86.5
0.78
135.1
0.135
69.6
0.5
102.6
0.129
77.4
0.52
117.4
0.196
98.7
0.9
146
0.195
129.9
1.49
181.1
0.199
1.68
141.3
195.3
0.206
153.4
212.6
1.91
0.209
154.1
1.58
211.6
0.242
142.7
0.241
199.8
1.29
0.246
163.4
0.87
116.9
146
0.246
0.62
102.1
0.246
121
0.34
86.2
0.291
114
153
0.59
0.283
192.9
141.3
1.1
0.29
202.9
1.16
145.1
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
(m)
17
81
391.2
0
12
1
N B this pipe section on its own also constitutes testrisTRl 8 IPF4 / TRl 8 combination)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
1
(kg s"1)
(kPa)
(kPa,)
Rate (kgs" )
0.081
56.62
1.069
17.84
0.082
54.37
15.58
0.92
0.135
63.1
0.78
23.4
0.129
50.6
19
0.5
0.196
58.2
19
0.52
0.195
71.2
25.5
0.9
0.199
94.2
34.6
1.49
0.206
102.9
36.7
1.68
0.209
111.2
40.9
1.91
0.242
112.9
39.8
1.58
0.241
104.2
37.1
1.29
Appendix A
Table A.9:
269
(cont.)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
1
(kg s"1)
(kPa)
Rate (kgs" )
(kPa,)
0.246
86.2
0.87
29.9
0.246
76.6
0.62
24.7
0.246
67
0.34
19
0.291
86
0.59
27.5
0.283
104.9
1.1
35.7
0.29
107.7
1.16
36.4
Pipe ID
No. data
Pipe Length
Lift
No. of Bends Bend Radius
(mm)
points
(m)
(m)
(m)
105
233.66
17
0
8
1
N B this pipe section on its o w n also constitutes test rig T R 1 9 (PF4 / T R 1 9 combination)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
1
1
(kgs" )
(kPa)
Rate (kgs" )
(kPa,)
0.081
13.16
1.069
4.68
0.082
10.74
0.92
4.85
0.135
20
3.4
0.78
3.4
15.6
0.129
0.5
3.9
15.1
0.196
0.52
4.7
0.9
20.8
0.195
6.2
1.49
28.4
0.199
6.9
1.68
0.206
29.8
8.3
1.91
0.209
32.6
7.6
1.58
32.2
0.242
7.3
1.29
29.8
0.241
6
0.87
23.9
0.246
5.5
0.62
19.2
0.246
4
.7
0.34
14.3
0.246
6.2
0.59
21.3
0.291
7.6
1.1
28.1
0.283
7.6
1.16
28.7
0.29
270
Table A.10:
Appendix A
Steady State Conveying Data - P F 5 / T R 2 0
Test Rig: TR20 (3 pipe sections)
Loose Poured Density: 482 kg m 3
Ambient Temperature: 20 ° C
No. of Bends Bend Radius
Pipe Length
Lift
(m)
(m)
(m)
563.4
26
69
0
17
1
N B this pipe section on its own also constitutes test rig T R 2 1 (PF5 / T R 2 1 combination)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kgs"1)
(kPa)
Rate (kgs'1)
(kPa,)
0.214
215
1.571
136
0.218
214
1.577
143
0.146
216
2.066
137
0.144
223
2.041
138
0.192
234
2.048
163
0.195
227
1.923
149
0.185
252
2.29
175
0.184
168
1.121
113
0.147
170
1.277
114
0.146
170
1.353
117
0.169
109
0.413
71
0.238
134
0.276
82
0.274
154
0.332
101
0.331
182
0.331
124
0.351
187
0.284
131
0.369
217
0.56
157
0.251
211
1.327
145
0.332
214
0.809
147
0.323
217
0.87
150
0.29
265
1.979
167
0.325
256
1.629
180
0.325
280
1.955
190
0.35
280
1.699
200
0.179
285
3.077
177
0.292
290
2.372
202
0.165
215
1.721
143
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
(m)
26
81
381.2
0
12
1
N B this pipe section on its own also constitutestestHff n m (PIT* / T W ? ? rftmhinarinn>
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kgs"1)
1
(kPa)
Rate (kgs" )
(kPa*)
0.214
106
1.571
35
0.218
101
1.577
Material: PF5
Particle Density: 15 74 kg m
Ambient Pressure: 101 kPa
Pipe ID
No. data
(mm)
points
37
Appendix A
271
Table A.10;
(cont.)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kg s"1)
(kPa)
Rate (kgs"1)
(kPa,)
0.146
100
2.066
36
0.144
114
2.041
38
0.192
107
2.048
49
0.195
107
1.923
42
0.185
127
2.29
48
0.184
81
1.121
32
0.147
81
1.277
33
0.146
81
1.353
36
0.169
52
0.413
19
0.238
63
0.276
19
0.274
77
0.322
24
0.331
94
0.331
30
0.351
99
0.284
32
0.369
119
0.56
38
0.251
104
1.327
41
0.332
110
0.809
37
0.323
110
0.87
40
0.29
122
1.979
45
0.325
130
1.629
50
0.325
135
1.955
55
144
0.35
1.699
56
0.179
129
3.077
48
0.292
144
2.372
58
0.165
1.721
101
42
Lift
No. data
Pipe ID
Pipe Length
No. of Bends Bend Radius
points
(m)
(mm)
(m)
(m)
7
8
1
252.7
26
105
N B this pipe section on its o w n also constitutes testrigT R 2 3 (PF5 / T R 2 3 combination)
Exit Pressure
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
1
1
(kPa,)
Rate (kgs' )
(kPa)
(kg s" )
0
1.571
0.214
35
0
1.577
37
0.218
0
2.066
0.146
36
0
2.041
38
0.144
0
2.048
49
0.192
0
1.923
0.195
42
0
2.29
0.185
48
0
1.121
0.184
32
0
1.277
0.147
33
0
1.353
0.146
36
0
0.413
0.169
19
0
0.276
0.238
19
0
0.274
0.322
24
0
0.331
0.331
30
Appendix A
Table A.10:
(cont.)
Air Mass Flow Rate
(kgs"1)
Pressure Drop
(kPa)
0.351
0.369
0.251
0.332
0.323
0.29
0.325
0.325
0.35
0.179
0.292
0.165
32
38
41
37
40
45
50
55
56
48
58
42
Table A.11:
Solids Mass Flow
Rate (kgs1)
0.284
0.56
1.327
0.809
0.87
1.979
1.629
1.955
1.699
3.077
2.372
1.721
Exit Pressure
(kPa,)
0
0
0
0
0
0
0
0
0
0
0
0
Steady State Conveying Data - PF6/TR24
Test Rig: TR24 (4 pipe sections)
Loose Poured Density: 520 kg m 3
Ambient Temperature: 20 ° C
Pipe Length
No. of Bends Bend Radius
Lift
(m)
(m)
(m)
1
146.1
9
60
0
4
N B this pipe section on its own also constitutes testrigTR25 (PF6 / TR25 combination)
Air Mass Flow Rate
Pressure Drop
Exit Pressure
Solids Mass Flow
1
1
(kPa)
Rate (kg^s" )
(kgs" )
(kPa,)
0.12
91
2.5
275
0.127
96
2.94
298
0.12
72
1.8
245
0.096
76
2.08
242
0.087
75
2.14
250
0.125
76
1.67
230
0.18
76
1.43
244
0.2
85
1.79
295
0.2
110
355
2.94
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
(m)
9
69
390.3
0
1
12
N B this pipe section on its own also constitutes testrigTR26 (fPF6 / TR26 combination)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kg s"1)
(kPa)
Rate (kgs"1)
(kPa2)
0.12
165
2.5
110
0.127
175
123
2.94
0.12
147
98
1.8
Material: PF6
Particle Density: 1495 kg m 3
Ambient Pressure: 101 kPa
No. data
Pipe ID
points
(mm)
Appendix A
Table A.ll:
273
(cont.)
Air Mass Flow Rate
(kgs"1)
0.096
0.087
0.125
0.18
Pressure Drop
(kPa)
143
150
138
145
166
209
0.2
0.2
No. data
points
Pipe ID
9
105
Solids Mass Flow
Exit Pressure
1
Rate (kgs" )
(kPa^)
2.08
99
2.14
100
1.67
92
1.43
99
1.79
129
2.94
146
Lift
No. of Bends Bend Radius
(m)
(m)
Pipe Length
(m)
9
261.1
0
8
1
N B this pipe section on its own also constitutes test rig T R 2 7 (PF6 / T R 2 7 combination)
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
1
(kgs"1)
(kPa)
Rate (kgs' )
(kPa,)
81
0.12
2.5
29
0.127
91
2.94
32
0.12
73
1.8
25
0.096
72
2.08
27
0.087
72
2.14
28
0.125
69
1.67
23
0.18
72
1.43
27
0.2
97
1.79
32
0.2
108
2.94
38
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(mm)
(m)
(m)
(m)
(mm)
81
150
7
5
1
N B this pipe section on its own also constitutes testrigT R 2 8 (PF6 / T R 2 8 combination)
Exit Pressure
Solids Mass Flow
Air Mass Flow Rate
Pressure Drop
1
1
(kPa,)
Rate (kgs' )
(kPa)
(kgs" )
0.12
0.127
0.12
0.096
0.087
0.125
0.18
0.2
0.2
29
32
25
27
28
23
27
32
38
2.5
2.94
1.8
2.08
2.14
1.67
1.43
1.79
2.94
0
0
0
0
0
0
0
0
0
Appendix A
274
Table A.12:
Steady State Conveying Data - F A l / T R 3
Test Rig: T R 3
Loose Poured Density: 874 kg m 3
Am jient Temperature: 20 ° C
No. of Bends Bend Radius
Lift
Pipe Length
(m)
(m)
(m)
5
1
7
168.5
Exit Pressure
Solids Mass Flow
Pressure Drop
1
(kPa«)
Rate (kgs' )
(kPa)
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data
(mm)
points
18
105
Air Mass Flow Rate
(kgs"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Table A. 13:
74.43
67.18
65.98
67.29
64.45
79.36
98.91
91.14
84.9
113.27
110.99
184.65
172.48
163.53
155
217.66
222.23
182.92
Air Mass Flow Rate
(kg s"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Steady State Conveying Data - F A l / T R 4
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
No. data
Pipe ID
points
(mm)
18
4.81
4.48
5.44
6.28
6.07
6.47
7.14
7.52
7.54
12.99
13.33
15.96
18.52
19.05
22.99
23.81
22.99
27.4
105
Test Rig: T R 4
Loose Poured Density: 874 kg m 3
Ambient Temperature: 20 ° C
Pipe Length
Lift
No. of Bends Bend Radius
(m)
(m)
(m)
110.5
1
0
2
7
Pressure Drop
Solids Mass 1low
Exit Pressure
(kPa)
Rate (kg sf1)
(kPa.)
58.63
24.78
4.81
52.73
19.37
4.48
51.04
15
5.44
53.41
13.77
6.28
50.38
15.55
6.07
63.66
24.18
6.47
76.64
33.7
7.14
Appendix A
Table A.13:
(cont.)
Air Mass Flow Rate
(kg s-1)
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Table A.14:
Pressure Drop
_(kPa)
73.24
66.73
91.83
90.9
149.7
139.81
133.88
127.7
179.73
182.43
152.18
Air Mass Flow Rate
(kgs"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Exit Pressure
(kPaJ
27.61
18.03
32.33
31.36
60.69
55.08
53.86
52.21
79.18
79.1
66.09
Steady State Conveying Data - F A l / T R 5
Material: F A l
Particle Density: 223 0 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data points
(mm)
18
Solids Mass Flow
Rate (kgs-1)
7.52
7.54
12.99
13.33
15.96
18.52
19.05
22.99
23.81
22.99
27.4
275
105
Test Rig: T R 5
Loose Poured Density: 874 kg m 3
Am jient Temperature: 20 ° C
Pipe Length
Lift
No. of Bends
Bend Radius
(m)
Cm)
(m)
43.7
0
0
1
Pressure Drop
Solids Mass Flow Rate
Exit Pressure
(kgs"1)
(kPa)
(kPa,)
58.63
4.81
43.96
52.73
38.14
4.48
51.04
5.44
35.51
35.99
53.41
6.28
50.38
34.97
6.07
44.07
6.47
63.66
58.96
7.14
76.64
54.55
7.52
73.24
46.5
7.54
66.73
12.99
64
91.83
61.97
13.33
90.9
113.06
15.96
149.7
103.3
18.52
139.81
97.3
19.05
133.88
91.34
22.99
127.7
135.72
23.81
179.73
138.59
22.99
182.43
110.49
27.4
152.18
276
Appendix A
Table A.15:
Steady State Conveying Data - F A l / T R 6
Test Rig: TR6
Loose Poured Density: 874 kg m 3
Ambient Temperature: 20 ° C
No. of Bends Bend Radius
Lift
Pipe Length
(m)
(m)
(m)
1
2
0
46.65
Exit Pressure
Solids Mass Flow
Pressure Drop
1
Rate (kgs' )
(kPaJ
(kPa)
30.24
4.81
43.96
25.06
4.48
38.14
20.43
5.44
35.51
22.87
6.28
35.99
22.44
6.07
34.97
30.37
6.47
44.07
41.59
7.14
58.96
36.24
54.55
7.52
7.54
27.71
46.5
64
43.7
12.99
61.97
42
13.33
80.22
113.06
15.96
73.06
103.3
18.52
97.3
70.24
19.05
91.34
67.14
22.99
135.72
100.63
23.81
138.59
101.96
22.99
110.49
27.4
83.31
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data
points
Xmm)
18
105
Air Mass Flow Rate
(kg s'1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Table A.16:
Steady State Conveying Data - FAl / T R 7
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
No. data
Pipe ID
points
(mm)
18
Air Mass Flow Rate
(kg s"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
105
Test Rig: TR7
Loose Poured Density: 874 kg m 3
Ambient Temperature: 20 ° C
Pipe Length
Lift
No. of Bends
(m)
(m)
20.15
0
()
Pressure Drop
(kPa)
30.24
25.06
20.43
22.87
22.44
30.37
41.59
Solids Mass Flow
Rate (kgs'1)
4.81
4.48
5.44
6.28
6.07
6.47
7.14
Bend Radius
(m)
1
Exit Pressure
(kPaJ
24.78
19.37
15
13.77
15.55
24.18
33.7
Appendix A
Table A.16:
277
(cont.)
Air Mass Flow Rate
(kgs'1)
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Table A.17:
Pressure Drop
(kPa)
36.24
27.71
43.7
42
80.22
73.06
70.24
67.14
100.63
101.96
83.31
Air Mass Flow Rate
(kg s"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Exit Pressure
(kPa.)
27.61
18.03
32.33
31.36
60.69
55.08
53.86
52.21
79.18
79.1
66.09
Steady State Conveying Data - F A l / T R 8
Test Rig: T R 8
Loose Poured Density: 874 kg m 3
Ambient Temperature: 20 ° C
Lift
Pipe Length
No. of Bends Bend Radius
(m)
(m)
(m)
1
48.85
7
2
Exit Pressure
Solids Mass Flow
Pressure Drop
(kPa^)
Rate (kgs"1)
(kPa)
0
4.81
30.24
0
4.48
25.06
0
5.44
20.43
0
6.28
22.87
0
6.07
22.44
0
6.47
30.37
0
7.14
41.59
7.52
36.24
o
7.54
27.71
o
0
12.99
43.7
0
13.33
42
0
15.96
80.22
0
18.52
73.06
0
19.05
70.24
0
22.99
67.14
0
23.81
100.63
22.99
101.96
o
0
27.4
83.31
Material: FAl
Panicle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data
points
(mm)
18
Solids Mass Flow
Rate (kgs'1)
7.52
7.54
12.99
13.33
15.96
18.52
19.05
22.99
23.81
22.99
27.4
105
Appendix A
278
Table A.18:
Steady State Conveying Data - F A l / T R 9
Test Rig: TR9
Loose Poured Density: 874 kg m 3
Am Dient Temperature: 20 ° C
No. of Bends Bend Radius
Lift
Pipe Length
(m)
(m)
(m)
1
2
7
28.7
Exit Pressure
Solids Mass Flow
Pressure Drop
1
(kPaJ
Rate (kgs' )
(kPa)
0
4.81
24.78
0
4.48
19.37
0
5.44
15
0
6.28
13.77
0
6.07
15.55
0
6.47
24.18
7.14
0
33.7
0
27.61
7.52
18.03
7.54
0
12.99
32.33
0
13.33
31.36
0
15.96
60.69
0
55.08
18.52
0
53.86
19.05
0
52.21
22.99
0
79.18
23.81
0
79.1
22.99
0
66.09
27.4
0
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data
(mm)
points
18
105
Air Mass Flow Rate
(kg s-1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
Table A.19:
Steady State Conveying Data - F A l / T R I O
Material: FAl
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
No. data
Pipe ID
points
(mm)
18
Air Mass Flow Rate
(kg s-1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
It)5
Test Rig: TRIO
Loose Poured Density: 874 kg m 3
Am nent Temperature: 20 ° C
Pipe Length
Lift
No. of Bends
(m)
(m)
L
95.5
7
Pressure Drop
(kPa)
43.96
38.14
35.51
35.99
34.97
44.07
58.96
54.55
Solids Mass Flow
Rate (kgs"1)
4.81
4.48
5.44
6.28
6.07
6.47
7.14
7.52
Bend Radius
(m)
1
Ex it Pressure
(kPa.)
0
0
0
0
0
0
0
0
Appendix A
279
Table A.19: (cont.)
Air Mass Flow Rate
(kg s-1)
Pressure Drop
(kPa)
Solids Mass Flow
Rate (kgs"1)
Exit Pressure
(kPa,)
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
46.5
7.54
12.99
13.33
15.96
18.52
19.05
22.99
23.81
22.99
27.4
0
0
0
0
0
0
0
0
0
0
Table A.20:
64
61.97
113.06
103.3
97.3
91.34
135.72
138.59
110.49
Steady State Conveying Data - F A l / T R l 1
Material: F A l
Particle Density: 2230 kg m 3
Ambient Pressure: 103.5 kPa
Pipe ID
No. data
points
(mm)
18
Air Mass Flow Rate
(kg s"1)
0.273
0.234
0.155
0.103
0.089
0.053
0.275
0.224
0.155
0.108
0.083
0.286
0.207
0.163
0.108
0.237
0.268
0.144
105
Test Rig: TR11
Loose Poured Density: 874 kg m 3
Am jient Temperature: 20 ° C
Pipe Length
Lift
No. of Bends
(m)
(m)
L
139.2
7
Pressure Drop
(kPa)
58.63
52.73
51.04
53.41
50.38
63.66
76.64
73.24
66.73
91.83
90.9
149.7
139.81
133.88
127.7
179.73
182.43
152.18
Solids Mass Flow
Rate (kgs'1)
4.81
4.48
5.44
6.28
6.07
6.47
7.14
7.52
7.54
12.99
13.33
15.96
18.52
19.05
22.99
23.81
22.99
27.4
Bend Radius
(m)
1
Exit Pressure
(kPa,)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
280
Appendix A
Table A.21:
Steady State Conveying Data - F A 2 / T R 1 2
Test Rig: T R 1 2
Loose Poured Density: 634 kg m 3
Ami>ient Temperature: 15 ° C
No. of Bends Bend Radius
Lift
Pipe Length
(m)
(m)
(m)
1
5
6.65
172.1
Exit Pressure
Solids Mass Flow
Pressure Drop
1
(kPa,)
Rate
(kgs"
)
(kPa)
0
3
136.12
0
4.26
164.71
0
4.2
190.07
0
3.35
172.46
0
7.3
239.51
0
4.35
199.69
1
62.84
0
0
1.1
69.89
116.27
2.45
0
155.77
4.75
0
4.75
158.86
0
92.77
1.5
0
81.98
0.9
0
84.05
0.8
0
75.53
0.6
0
142.06
2.1
0
187.2
2.15
0
232.34
5.7
0
119.73
2.15
0
101.71
2
0
161.98
3.87
0
169.05
3.11
0
207.69
4.7
0
248.81
5.05
0
91.28
1
0
88.65
1
0
152.83
2.5
0
196.4
2.9
0
239.69
4.3
0
270.05
5.35
0
93.61
1.1
0
160.13
2.9
0
191.1
4.1
0
223.9
5.1
0
262.88
6.77
0
283.97
8.5
0
103.03
1.6
0
Material: F A 2
Particle Density: 2197 kg m 3
Ambient Pressure: 101 kPa
Pipe ID
No. data
(mm)
points
47
Air Mass Flow Rate
(kg s"1)
0.1372
0.1224
0.1894
0.1868
0.1511
0.1905
0.1254
0.1472
0.1299
0.0971
0.0622
0.1631
0.1963
0.2149
0.2271
0.2189
0.3156
0.1863
0.1527
0.1482
0.1655
0.2123
0.1953
0.2557
0.2056
0.2082
0.2118
0.275
0.2691
0.2752
0.2045
0.2075
0.1968
0.2066
0.2032
0.1946
0.1641
69
Appendix A
Table A.21:
(cont.)
Air Mass Flow Rate
(kg s 1 )
0.1658
0.1717
0.1485
0.1491
0.0562
0.1334
0.1033
0.0568
0.2505
0.2438
Table A.22:
Pressure Drop
(kPa)
117.11
155.09
133.12
128.36
169.28
71.99
69.23
75.65
219.74
134.82
Air Mass Flow Rate
(kg s-1)
0.1253
0.1681
0.2291
0.1695
0.13
0.1234
0.2149
0.1622
0.1132
0.0727
0.1035
0.1058
0.0799
Exit Pressure
(kPa,)
2.1
3.3
3
2.8
0
0
0
0
0
0
0
0
0
0
4.55
1.1
1.3
1.7
3.9
1.7
Steady State Conveying Data - F A 2 / T R 1 3
Material: F A 2
Particle Density: 2197 kg m 3
Ambient Pressure: 101 kPa
Pipe ID
No. data
points
(mm)
13
Solids Mass Flow
Rate (kgs"1)
81
Test Rig: TR13
Loose Poured Density: 634 kg m 3
Am Dient Temperature: 15 ° C
Pipe Length
Lift
No. of Bends Bend Radius
(m)
(m)
(m)
6.75
382.2
12
1
7
Pressure Drop
Solids Mass ]low
Exit Pressure
1
(kPa)
Rate (kgsi )
(kPa.)
0.79
0
100.51
1.4
141.52
0
1.35
0
134.02
0.9
0
103.88
0
1.8
142.24
0
1.12
95.62
0
0.17
51.1
0
0.44
61.16
0
0.36
48.02
0
0.96
62.48
0
1.11
68.9
0
1.08
74.1
1.16
64.6
o
Appendix A
282
Table A.23:
Steady State Conveying Data - FA2 / TR14
Test Rig: TR14
Loose Poured Density: 634 kg m 3
Ambient Temperature: 15 ° C
No. of Bends Bend Radius
Lift
Pipe Length
(m)
(m)
(m)
17
1
6.3
553.5
Exit Pressure
Solids Mass Flow
Pressure Drop
1
(kPa,)
Rate (kgs' )
(kPa)
Material: FA2
Particle Density: 2197 kg m 3
Ambient Pressure: 101 kPa
Pipe ID
No. data
(mm)
points
14
Air Mass Flow Rate
(kg s"1)
0.1678
0.1715
0.1667
0.1618
0.2025
0.2073
0.2095
0.2389
0.2858
0.1191
0.0821
0.0386
0.0914
0.1256
69
247.43
247.17
353.75
180.94
169.88
255.1
338.46
348.84
342.81
204.24
175.43
176.76
247.76
358.43
1.43
1.61
2.79
0.79
0.65
1.43
2.42
2.42
2.02
1.17
1.2
0.85
1.57
2.81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Appendix A
Table A.24:
283
Steady State Conveying Data - F A 2 / T R l 5
Material: F A 2
Particle Density': 2197 k g m 3
Ambient Pressure: 101 kPa
No. data
Pipe ID
points
(mm)
Test Rig: TRl5 (2 pipe sections)
Loose Poured Density: 634 kg m 3
Am ?ient Temperature: 15 ° C
Pipe Length
Lift
No. of Bends Bend Radius
(m)
(m)
(m)
13
69
562.4
0
17
1
Air Mass Flow Rate
Pressure Drop
Solids Mass Flow
Exit Pressure
(kg s"1)
(kPa)
Rate (kgs"1)
(kPaJ
0.1253
147.47
0.79
100.51
0.1681
227.74
1.4
141.52
0.2291
231.74
1.35
134.02
0.1695
182.49
0.9
103.88
0.13
234.05
1.8
142.24
0.1234
192.48
1.12
95.62
0.2149
95.25
0.17
51.1
0.1622
107.42
0.44
61.16
0.1132
120.85
0.36
48.02
0.0727
148.85
0.96
62.48
193.79
0.1035
1.11
68.9
0.1058
181.38
1.08
74.1
0.0799
1.16
64.6
211.23
No. data
Pipe ID
Pipe Length
Lift
No. of Bends Bend Radius
points
(m)
(mm)
(m)
(m)
382.2
13
12
1
6.75
81
Solids Mass Flow
Pressure Drop
Air Mass Flow Rate
Exit Pressure
1
1
(kPa,)
(kPa)
Rate (kgs' )
(kg s" )
0.79
0
100.51
0.1253
0
!
1.4
141.52
0.1681
0
1.35
134.02
0.2291
0
0.9
103.88
0.1695
0
1.8
142.24
0.13
0
1.12
95.62
0.1234
0
0.17
51.1
0.2149
0
0.44
61.16
0.1622
0
0.36
48.02
0.1132
0
0.96
0.0727
62.48
0
1.11
0.1035
68.9
0
1.08
74.1
0.1058
0
1.16
0.0799
64.6
284
Appendix B
APPENDIX B
TRENDS IN MODEL PERFORMANCE
This Appendix presents comparisons between the performance of models P C I , P C I A , P C 2 ,
PC3, P C 4 , P C 5 and PC6, based on solids friction factor correlations from eight different sets of
experimental data.
The trends (in predicted total pipeline pressure drop) in each of these models with respect to
changes in number of bends, pipeline length and pipe diameter are examined. Model P C I uses a
solids friction factor correlation based on m* only - Model P C I A is an extension of this model,
using two parameters (m* and Frm) in the correlation relationship.
Model performance has been examined under both "dense phase" and "lean phase" conditions.
This has been achieved by selecting ms and W / values in the appropriate regions of the conveying
characteristics plots for each material. In fact, this categorisation only accurately reflects the
conveying regimes of the bulk solids at the respective pipeline configurations at which the data
was collected. A s the pipe length or pipe diameter is varied from the original setup it is quite
possible that the operating point will m o v e into a different conveying regime.
B.l Trends in Model Performance - Pipe Length
This section shows the relationship between system pressure drop and pipe length for both dens
and lean phase conveying regimes. Pipe diameter is held constant (at the diameter of the base
data pipeline configuration in each case), and zero bends are assumed in the application of each
model.
Appendix B
B.l.l
Variation with Length - Dense Phase Conveying Conditions
300
50
100
150
200
Pipe Length (m)
Figure B.l.1.1 Base data: Bulk Solid: PFl, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: /«/= 0.012 kg/s, ms - 3 kg/s
(dense phase for base data)
20
40
60
80
Pipe Length (m)
Figure B.l.1.2 Base data: Bulk Solid: PFl, Test Rig: TRl (L = 65 m )
Conveying Conditions: m/= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
Appendix B
286
350
50
100
150
200
Pipe Length (m)
Figure B.l.1.3 Base data: Bulk Solid: PF2, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: /»/= 0.015 kg/s, ms = 3 kg/s
(dense phase for base data)
700
50
100
150
Pipe Length (m)
Figure B.l.1.4 Base data: Bulk Solid: PF2, Test Rig: T R l (L = 65 m )
Conveying Conditions: mf= 0.004 kg/s, ms = 0.4 kg/s
(dense phase for base data)
200
Appendix B
300
100
150
200
Pipe Length (m)
Figure B.l.1.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: m / = 0.01 kg/s, ms = 2 kg/s
(dense phase for base data)
400
350 -
^
300 -•
» 250 +
o.
Q 200 f
£ 150
100 •50 -•
0
50
100
150
Pipe Length (m)
Figure B.l.1.6 Base data: Bulk Solid: PF3, Test Rig: T R l (L = 65 m )
Conveying Conditions: mf= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
200
Appendix B
288
50
100
150
200
250
Pipe Length (m)
Figure B.l.1.7 Base data: Bulk Solid: FAl, Test Rig: T R 3 (L = 168.5 m )
Conveying Conditions: #«/= .07 kg/s, ms = 15 kg/s
(dense phase for base data)
600
100
150
200
250
300
350
400
Pipe Length (m)
Figure B.l.1.8 Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (L = 172.1 m )
Conveying Conditions: mf= 0.05 kg/s, ms = 4 kg/s
(dense phase for base data)
Appendix B
B.1.2
289
Variation with Length - Lean Phase Conveying Conditions
Diameter held constant (at the diameter of the base data pipeline configuration in each case)
zero bends are assumed in the application of each model.
50
100
Pipe Length (m)
Figure B.l.2.1 Base data: Bulk Solid: PFl, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: mf= 0.05 kg/s, ms = 3 kg/s
(lean phase for base data)
50
100
Pipe Length (m)
Figure B.l.2.2 Base data: Bulk Solid: PFl, Test Rig: T R l (L = 65 m )
Conveying Conditions: mf = 0.01 kg/s, ms = 0.5 kg/s
(lean phase for base data)
Appendix B
290
50
100
150
200
Pipe Length (m)
Figure B.l.2.3 Base data: BuJk Solid: PF2, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: /M/= 0.05 kg/s, ms = 3 kg/s
(lean phase for base data)
50
100
150
Pipe Length (m)
Figure B.l.2.4 Base data: Bulk Solid: PF2, Test Rig: T R l (L = 65 m )
Conveying Conditions: mf= 0.012 kg/s, ms = 0.4 kg/s
(lean phase for base data)
200
Appendix B
291
300
100
150
200
Pipe Length (m)
Figure B.l.2.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (L = 48 m )
Conveying Conditions: m / = 0.05 kg/s, m , = 2 kg/s
(lean phase for base data)
700 •
600 mm
%
500 -
a
o
400 ii
m
s 300 -
01
m^9*^
tm m***^^r
_X
01
ii
m.
6. 200 V^^k^"•dfc^^T^^ff
V
J^~—^"—^
100 -
fcfcfaJ^gajj
0-
L+_
1
1
-J
50
100
150
200
Pipe Length (m)
Figure B.l.2.6 Base data: Bulk Solid: PF3, Test Rig: T R l (L = 65 m )
Conveying Conditions: « / = 0.009 kg/s, ms = 0.5 kg/s
(lean phase for base data)
292
Appendix B
50
100
150
200
250
300
350
400
Pipe Length (m)
Figure B.l.2.7 Base data: Bulk Solid: FAl, Test Rig: T R 3 (L = 168.5 m )
Conveying Conditions: mf= .25 kg/s, ms = 15 kg/s
(lean phase for base data)
400
350 ••
^ 300 ••
w 250 -a,
a 200
a
3 150 -
s
X X X X X X X X ^ C
&< too -•
50 -•
0
50
100
4-
4-
4-
150
200
250
300
350
400
Pipe Length (m)
Figure B.l.2.8 Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (L = 172.1 m )
Conveying Conditions: mf = 0.25 kg/s, ms = 4 kg/s
(lean phase for base data)
Appendix B
B.2
293
Trends in Model Performance - Pipe Diameter
The following section shows the relationship between system pressure drop and pipe diameter
for both dense and lean phase conveying regimes. Pipe length is held constant (at the length of
the base data pipeline configuration in each case), and zero bends are assumed in the application
of each model.
B.2.1
Variation with Diameter - Dense Phase Conveying Conditions
140
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.1.1 Base data: Bulk Solid: PFl, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: mf= 0.012 kg/s, ms = 3 kg/s
(dense phase for base data)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.1.2 Base data: Bulk Solid: PFl, Test Rig: T R l (D = 25.4 m m )
Conveying Conditions: mf= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
Appendix B
294
14U -
100 -|
§
""x1^—--3
m
^""^VfwJlF^^
~+»-~.«j«. j ^ ^ ^
S
Pressure Drop
1?
-"-f~~J
20 00.03
1
1
0.04
0.05
—i
0.06
f
1
1
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure B.2.1.3 Base data: Bulk Solid: PF2, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: /«/= 0.015 kg/s, ms = 3 kg/s
(dense phase for base data)
200
180 -•
160 -•
«140-
a 120
a 100 i
£
5
80 +
01
0i
£
60 +
m,
40 -•
20 -•
0
0.01
0.02
0.03
4-
4-
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.1.4 Base data: Bulk Solid: PF2, Test Rig: T R l (D - 25.4 m m )
Conveying Conditions: mf = 0.004 kg/s, ms = 0.4 kg/s
(dense phase for base data)
Appendix B
120
100 -•
«
a
2
a
£
9
0i
01
Z
ft.
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure B.2.1.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: m / = 0.01 kg/s, ms = 2 kg/s
(dense phase for base data)
300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.1.6 Base data: Bulk Solid: PF3, Test Rig: T R l (D = 25.4 m m )
Conveying Conditions: mf= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
Appendix B
296
300
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
0.21
Pipe Diameter (m)
Figure B.2.1.7 Base data: Bulk Solid: FAl, Test Rig: T R 3 (D = 105 m m )
Conveying Conditions: /»/= .07 kg/s, ms = 15 kg/s
(dense phase for base data)
400
350 •
^
300 +
w 250
a
o 200 ••
m
a
150 -•
m
ts 100 --
Cm
50 -•
T-~""«4«,,„„ i
"*"h—~-4—~~4-.
0.04
0.06
4-
4-
4-
0.08
0.1
0.12
0.14
0.16
Pipe Diameter (m)
Figure B.2.1.8 Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (D = 69 m m )
Conveying Conditions: mf = 0.05 kg/s, ms = 4 kg/s
(dense phase for base data)
Appendix B
B.2.2
297
Variation with Diameter - Lean Phase Conveying Conditions
Length held constant (at the length of the base data pipeline configuration in each case), and zero
bends are assumed in the application of each model.
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure B.2.2.1 Base data: Bulk Solid: PFl, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: mf= 0.05 kg/s, ms = 3 kg/s
(lean phase for base data)
0.04
0.05
0.06
0.08
Pipe Diameter (m)
Figure B.2.2.2 Base data: Bulk Solid: PFl, Test Rig: T R l (D = 25.4 m m )
Conveying Conditions: /«/= 0.01 kg/s, ms = 0.5 kg/s
(lean phase for base data)
Appendix B
298
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure B.2.2.3 Base data: Bulk Solid: PF2, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: m / = 0.05 kg/s, ms-3 kg/s
(lean phase for base data)
400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.2.4 Base data: Bulk Solid: PF2, Test Rig: T R l (D = 25.4 m m )
Conveying Conditions: mf= 0.012 kg/s, ms = 0.4 kg/s
(lean phase for base data)
Appendix B
120
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Pipe Diameter (m)
Figure B.2.2.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (D = 52.5 m m )
Conveying Conditions: m / = 0.05 kg/s, ms = 2 kg/s
(lean phase for base data)
300
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Pipe Diameter (m)
Figure B.2.2.6 Base data: Bulk Solid: PF3, Test Rig: T R l (D = 25.4 m m )
Conveying Conditions: mf= 0.009 kg/s, ms = 0.5 kg/s
(lean phase for base data)
Appendix B
300
600
0.065
0.085
0.105
0.125
0.145
0.165
0.185
0.205
Pipe Diameter (m)
Figure B.2.2.7 Base data: Bulk Solid: FAl, Test Rig: T R 3 (D = 105 m m )
Conveying Conditions: mf= .25 kg/s, ms - 15 kg/s
(lean phase for base data)
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Pipe Diameter (m)
Figure B.2.2.8 Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (D = 69 m m )
Conveying Conditions: mf = 0.25 kg/s, ms = 4 kg/s
(lean phase for base data)
Appendix B
B.3
301
Trends in Model Performance - N u m b e r of Bends
The following section shows the relationship between system pressure drop and the number of
bends (90°) in the pipeline layout, for both dense and lean phase conveying regimes. Pipe l
and diameter is held constant (at the length of the base data pipeline configuration in eac
B.3.1
Variation with N u m b e r of Bends - Dense Phase Conveying Conditions
110 105 •
^^JBK^^J^^J^X*-*^*
1
<s
en
to
o
en
<n
mm^K^*m
S
Pressure Drop
S °° -
75 |
70-
I
-4
1
1
6
\—
8
1
1
1
10
12
14
16
Number of Bends
Figure B.3.1.1 Base data: Bulk Solid: PFl, Test Rig: T R 2 (4 Bends, RB = .254 m )
Conveying Conditions: m / = 0.012 kg/s, ms = 3 kg/s
(dense phase for base data)
6
8
10
12
Number of Bends
Figure B.3.1.2 Base data: Bulk Solid: PFl, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
Appendix B
302
6
8
10
Number of Bends
Figure B.3.1.3 Base data: BuUt Solid: PF2, Test Rig: T R 2 (4 Bends, RB .254 m )
Conveying Conditions: m / = 0.015 kg/s, ms - 3 kg/s
(dense phase for base data)
6
8
10
12
14
16
Number of Bends
Figure B.3.1.4 Base data: Bulk Solid: PF2, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf= 0.004 kg/s, ms = 0.4 kg/s
(dense phase for base data)
Appendix B
6
8
10
303
12
14
16
Number of Bends
Figure B.3.1.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (4 Bends, RB = .254 m )
Conveying Conditions: m / = 0.01 kg/s, ms = 2 kg/s
(dense phase for base data)
130
6
8
10
Number of Bends
Figure B.3.1.6 Base data: Bulk Solid: PF3, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf= 0.003 kg/s, ms = 0.5 kg/s
(dense phase for base data)
Appendix B
304
6
8
10
12
14
Number of Bends
Figure B.3.1.7 Base data: Budk Solid: FAl, Test Rig: T R 3 (5 Bends, RB =
Conveying Conditions: mf= .07 kg/s, ms = 15 kg/s
(dense phase for base data)
lm)
200
6
8
10
12
14
16
Number of Bends
Figure B.3.1.8 Base data: Bulk Solid: F A 2 , Test Rig: T R 1 2 (5 Bends,
Conveying Conditions: m / = 0.05 kg/s, m , = 4 kg/s
(dense phase for base data)
RB=lm)
Appendix B
B.3.2
305
Variation with N u m b e r of Bends - Lean Phase Conveying Conditions
300
6
8
10
12
14
16
Number of Bends
Figure B.3.2.1 Base data: BuUt Solid: PFl, Test Rig: T R 2 (4 Bends, RB = .254 m )
Conveying Conditions: mf= 0.05 kg/s, ms = 3 kg/s
(lean phase for base data)
6
8
10
12
Number of Bends
Figure B.3.2.2 Base data: Bulk Solid: PFl, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf= 0.01 kg/s, ms = 0.5 kg/s
(lean phase for base data)
Appendix B
306
350
6
8
10
12
14
16
Number of Bends
Figure B.3.2.3 Base data: BuBi Solid: PF2, Test Rig: T R 2 (4 Bends, RB = .254 m )
Conveying Conditions: mf= 0.05 kg/s, ms = 3 kg/s
(lean phase for base data)
6
8
10
12
14
16
Number of Bends
Figure B.3.2.4 Base data: Bulk Solid: PF2, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf= 0.012 kg/s, ms = 0.4 kg/s
(lean phase for base data)
Appendix B
6
8
10
307
12
14
16
Number of Bends
Figure B.3.2.5 Base data: Bulk Solid: PF3, Test Rig: T R 2 (4 Bends, RB = .254 m )
Conveying Conditions: m / = 0.05 kg/s, ms = 2 kg/s
(lean phase for base data)
6
8
10
Number of Bends
12
14
Figure B.3.2.6 Base data: Bulk Solid: PF3, Test Rig: T R l (9 Bends, RB = .254 m )
Conveying Conditions: mf = 0.009 kg/s, ms - 0.5 kg/s
(lean phase for base data)
Appendix B
308
350
6
8
10
Number of Bends
Figure B.3.2.7 Base data: BuUi Solid: FAl, Test Rig: T R 3 (5 Bends, RB =
Conveying Conditions: m / = .25 kg/s, ms = 15 kg/s
(lean phase for base data)
6
8
10
12
14
lm)
16
Number of Bends
Figure B.3.2.8 Base data: BuUi Solid: F A 2 , Test Rig: T R 1 2 (5 Bends, RB =
Conveying Conditions: mf = 0.25 kg/s, ms = 4 kg/s
(lean phase for base data)
lm)
Appendix C
APPENDIX C
CONVEYING CHARACTERISTICS PLOTS
309
Appendix C
310
200- | r
0 Pipe Dotali s
Dfa. Length
Imml fm)
^7 25 65
Tot.! 65
0' 4!
0.440.44
Q_
o
DC
Q
UJ
CC
L00,
0.28
0.26
0.240-2.3
m
tn
u
CC
D_
J__L.
0.0 10
0.005
• 000
AIR MASS FLOW RATE
(KQ/S)
CONTOURS: 111 Solids Moss Flowrate Ug/s)
Figure C.l Conveying Characteristics - BuUi SoUd: PFl, Test Rig: TRl, Model: PCI
200.
0 Pipe Dotal Is
0-84
Dfa. Length
(mini I nil
0.6i^~-&*• 57 ^5 65
0.63
Tot.s 65
D_
CD
CC
•
UJ
DC
0.44 0. 4 4
0--41
100
0.28
0.26
0.240-23
en
en
u
cn
_i
000
i_
-
i
—
i
i
•
i
0.005
FUR MASS FLOW RATE
•
i
i
0.010
(Kg/s)
CONTOURS: fl) Solids Moss Flowrate (kg/s)
Figure C.2 Conveying Characteristics - Bulk SoUd: PFl, Test Rig: TRl, Model: PC6
Appendix C
311
150.
Pipe D « t B l I s
Dia. Length
i mini (ml
52 48
Tot. !
100.
4S
Q_
O
CC
•
DC
ZJ
en
en
50. -
LJ
oc
0_
00
0.01
0-02
0-03
0-04
0.05
AIR MASS FLOW RATE
(K9/si
0.06
0.07
CONTOURS: 111 Solids Moss Flowrate (kg/s)
Figure C.3 Conveying Characteristics - Bulk Solid: PFl, Test Rig: TR2, Model: P
150.
Pipe Dotal Is
Ola. Length
I mini (ml
52 48
o
a.
.a:
Tot.!
100. ~
46
Q_
o
DC
•
LU
CC
CD
tn
50. -
cn
UJ
DC
Q_
3'. 00
0.01
0.02
0-03
0-04
0-05
0.06
0-07
AIR MASS FLOW RATE (Kg/s)
C0NT0UR5: fl) Solids Moss Flowrale (kg/s)
Figure C.4
Conveying Characteristics - Bulk SoUd: PFl, Test Rig: TR2, Model: PC6
Appendix C
312
200.
tZIj. 7 4 • ' ' i
Ftp© Dotal | s
Din- Length
(mini (ml
25 65
Tot. !
O
DC
•
LU
CC
65
L00,
28
0.25 0.21_0,
=J
v>
en
LU
DC
0_
- i — i — i _ i
-t_j
i. i
000
i I_J_
- > — ' — i — i — i
0-004
0-008
i
'
•
i
0.0 12
AIR MRS5 FLOW RATE (KQ/S)
CONTOURS: (1) Solids Moss Flowrate (kg/s)
Figure C.5 Conveying Characteristics - BuUi SoUd: PF2, Test Rig: TRl, Mod
200,
1
' ' ' • ' ' 0|- 7 4 ' N-Q'J i
T — i — i — i — r -
P f p e D o t a l Is
D (a. Length
(mini ihiJ
25 65
Q_
Tot.
B5
Q_
O
CC
•
UJ
100.
0.28
cc
m
in
UJ
cc
I)
0.25 0.2
0•20
Q_
•
000
•
'
' — ' — ' — I — i — i — i — i — I — i
0-004
i
.
,
I
,
,
,
0. 008
, i
0-012
AIR MRSS FLOW RATE (Kg/s)
CONTOURS: fl) Solids Moss Flovrate (kg/s)
Figure C.6 Conveying Characteristics - Bulk SoUd: PF2, Test Rig: TRl, Mode
Appendix C
150.
i|M
313
M | iM I i | i '
F r p e Dotal Is
Dfa. Length
(mini (ml
52 48
0_
•Tot. !
100,
48
O
CC
Q
LU
CC
n
in
50,
en
LU
-en-97
cc
I
00
I I
I
1I i ..I I I I i i i i I i i i i I i i i i . I i t i 'i1I i i i
0.01
0-02
0-03
0-04
0.05
0.05
0.07
AIR MRS5 FLOW RATE (Kg/s)
CONTOURS: (11 Solids Mass Flowrate (kg/s)
Figure C.7 Conveying Characteristics - BuUi SoUd: PF2, Test Rig: TR2, Model: PC
P r p e Dotal Is
Dia. Length
(mini 'nil
52 48
Tot. !
48
0O
DC
Q
LU
CC
CJ
cn
tn
LU
3.00
0.01
0-02
0-03
0-04
0-05
0-06
0.
AIR MASS FLOW RATE (Kg/s)
C0NT0UR5: fl) Solids Mass Flowrate (kg/s)
Figure C.8
Conveying Characteristics - Bulk SoUd: PF2, Test Rig: TR2, Model: PC6
Appendix C
314
200-
i
'
•
•
- i — | — i — r — r — i — | — i — r -
'
P i p e D o t a l Ie
0.9®.90 0'8^
Dia. Length
(mini (ml
25 65
o
Tot.!
Q_
-si
D
DC
O
UJ
CC
CJ
65
100. 8.28
0.33
0.310.30
0
"26 0'2f,
(11 0-?0_
en
in
LU
cc
o_
, I
0-004
• 000
0-012
0- 008
AIR MASS FLOW RATE
(Kg/si
CONTOURS: fll Solids Moss Flowrate (kg/s)
Figure C.9 Conveying Characteristics - Bulk SoUd: PF3, Test Rig: TRl, Model: PCI
200FIpe D o t a l I e
D f a. Length
(mini (ml
25 65
Tot.!
65
Q_
D
DC
L00.
a
33
LU
DC
=3
in
in
LU
0
0.31 B. 30
" 2 e 0 - 2 _l
(11 _§j
DC
o_
J
•000
i_
-J
0-004
1—1
0. 008
AIR MASS FLOW RATE
1
1
1
L...J...J
0.012
(Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure CIO Conveying Characteristics - BuUi SoUd: PF3, Test Rig: TRl, Model: PC6
Appendix C
100
-' I' I ' 11 ' M
315
' ' I i i i i I i r i i J&iyi i i i i i
Pipe Detal I s
90
D(a. Length
(mini I'm)
80
52 48
70
Q_
O
CC
Q
LU
CC
Tot. s
60
50
40
Cj
in
in
LU
cc
CL.
48
0.5
30
0
- 3 8 0.85
0.75
0.B7
20
10
i i i i I i i i i I i i i i I i i i i i i i i i I i i i i 1I i i i i
00
0-01
0-02
0-03
0-04
0.05
0-06
0-07
AIR MA55 FLOW RATE (Kg/s)
CONTOURS: d) 5o I ids Moss Flowrat-e (kg/s)
Figure C.l 1 Conveying Characteristics - Bulk SoUd: PF3, Test Rig: TR2, Model: PCI
100
Pipe D « t a l
90
80
Q_
-*L
0_
O
CC
n
70
U
Dia-
Length
(mini
52
(ml
48
Tot- !
4B
60
50
LU
cc
40
CD
in
r
n
UJ
az
M
CL
20
10
I I I I I I I I I I I I I I 1 I I I I I I I I I It
00
0.01
0-02
0-03
0-04
0-05
I I I I I I I1
0-06 0,
AIR MHS5 FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.12
Conveying Characteristics - Bulk SoUd: PF3, Test Rig: T R 2 , Model: P C 6
Appendix C
316
400.
PIpe Dotal I 6
tt Dfa. Length
300- -
Q_
o
cc
(mini
(ml
68
577
81
391
105
234
Tot. !
1202
200.
Q
LU
CC
CJ
tn
en
LU
100.
cc
j
00
i
i
-I
i
I
I
•
L_
0. 10
•
.
.
f
0-20
•
•
•
•
0.30
AIR MASS FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.13 Conveying Characteristics - BuUi SoUd: PF4, Test Rig: TRl6, M
400,
Pipe DotalIe
300-
Q_
D
DC
Q
LU
DC
CJ
DTa.
Length
(mini
(nil
69
577
81
391
105
234
Tot. !
1202
200.
cn
in
LU
DC
CL
100. -
- J — i — u _ 1 — i — i
0- 10
i
i
L
i.
i
i
i
l
i
0.20
0.30
AIR MASS FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrote (kg/s)
Figure C.14
Conveying Characteristics - Bulk SoUd: P F 4 , Test Rig: T R 1 6 , Model: P C 6
Appendix C
317
500
F f pe Dotal Is
Dfa.
Length
(mini
(ml
400. -
68
563
81
38J
105
253
300.
o
DC
Q
DC
CJ
in
in
LU
az
o_
Tot.! 1197
200.
100.
J...-L.-I—I i I — I
J, , I,, ,1,
I — U _ l — I — I — J — I
0-10
00
U-l, ,i,—1—'
1,
'
<t
I
*
>
0-20
•
•
1
•
•
•
•
<
•
f
0-30
AIR M A S S FLOW R A T E
•
•
0-40
(Kg/s)
C O N T O U R S : (11 S o l i d s M o s s F l o w r a t e
(kg/s)
Figure C.15 Conveying Characteristics - Bulk SoUd: PF5, Test Rig: TR20, Model: PCI
500P f p o Dotal Ie
Dia.
Length
Iinml
(ml
69
563
400.
81
381
105
253
Tot.t
1197
CL
300o
cc
o
LU
CC
CJ
in
in
LU
cc
CL
200.
100.
•
00
•
•
•
i
•
•
•
•
i
.. -i
0-10
1
•
• • •
l
'
•
•
•
'
•
'
A I R M A S S FLOW R A T E
i
•
0-30
0.20
i
•
i
0.40
(Kg/s)
C O N T O U R S : (11 S o l i d s M o s s F l o w r a t e
(kg/s)
Figure C.16 Conveying Characteristics - Bulk SoUd: PF5, Test Rig: TR20, Model: PC6
Appendix C
318
500,
Pf p o D o t a l Ie
Dia.
Length
(inlnl
(nil
60
146
69
390
81
261
105
150
Tot.
948
400.
0
CL
O
DC
D
LU
CC
300.
200.
-
-
CJ
in
in
LU
DC
L00. -
o_
00
0.05
0.10
0-15
0-20
0-25
AIR MASS FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.17 Conveying Characteristics - BuUi SoUd: PF6, Test Rig: TR24, M
500,
400.
i I i
i
Plpa
DotalIe
Dia.
Length
(ml
-
146
390
261
Q_
300.
150
o
DC
CJ
LU
CC
948
200.
CJ
in
in
LU
cc
100.
Q_
J — i — i — i
00
I
i
0-05
i
i
i
I J—i—i—i—I
0-10
i ''
0-15
i
-I—I—1
0.20
L
0.25
AIR MRS5 FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.18
Conveying Characteristics - BuUi SoUd: PF6, Test Rig: TR24, Model: PC6
Appendix C
250,
319
I' 1 ' I ' f ' ' 1
FIpe Dotal I s
Dfa. Length
(mini (in)
200,
105 169
CL
Tot.
0_
169
150,
o
DC
CJ
DC
CJ
in
in
LU
DC
100,
50,
I
00
0- H
-J
i
i
L
- i — i — i — i
i
0-20
i
i
,
,
0.30
AIR MASS FLOW RATE (Kg/s)
CONTOURS: fl) Solids Moss Flovrate (kg/s)
Figure C.19 Conveying Characteristics - Bulk SoUd: FAl, Test Rig: TR3, Mo
250.
P i p e D o t a l Ie
Dia. Length
200.
ilnln'l I hi J
-
105 169
CL
CL
O
CC
CD
LU
CC
=>
tn
in
UJ
DC
CL
Tot.
169
150. -
100. -
50. -
0-10
0-20
0.30
AIR MASS FLOW RATE (Kg/s)
CONTOURS: fl) Solids Moss Flowrate (kg/s)
Figure C.20
Conveying Characteristics - Bulk SoUd: FAl, Test Rig: TR3, Model: PC6
Appendix C
320
300
PIpe Dotal Is
Dia. Length
(mini (ml
69 175
CL
Tot.! 172
200,
Q_
O
CC
•
UJ
cc
C5
in
in
UJ
cc
100
Q_
•
0-10
00
1
•
•
-^1
j_L
0-30
0-20
AIR MRSS FLOW RATE
0-40
(Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.21 Conveying Characteristics - Bulk SoUd: FA2, Test Rig: TR12, Model:
300.
Prpe Dotal Ie
Dia. Length
(mini (ml
69 172
Tot.i 172
200.
CL
O
CC
CJ
LU
DC
CD
<n
m
100,
UJ
DC
CL
• i i
• 00
i
i
i
• • i I
0- 10
• • • • I
, . . . i ,
0-20
AIR MRSS FLOW RATE
1 •, ,
0-30
0-40
(Kg/s)
CONTOURS: (11 Solids Moss Flovrate (kg/s)
Figure C.22
Conveying Characteristics - BuUi SoUd: FA2, Test Rig: TR12, Model: PC6
321
Appendix C
400P i p e D o t a l Ie
Dia. Length
(mini i nil
69 554
300. -
Tot.
EL
554
Q_
o
cc
a
200.
UJ
cc
CJ
in
in
LU
DC
Q_
100. -
00
0.10
0-20
0. 40
0-30
AIR MRSS FLOW RATE (Kg/s)
CONTOURS: f 11 • Solids Moss Flowrate (kg/s)
Figure C.23 Conveying Characteristics - Bulk SoUd: FA2, Test Rig: TR14,
400-
P i p e D o t a l Is
Dra. Length
(mini (nil
69 554
300,
o
Tot.s 554
CL
-i-
o
cc
200.
•
LU
CC
CJ
in
in
UJ
100.
DC
0_
t • ...
.00
I *.'•'•' •
0.10
.
0-20
•
•
• i
i
•
• • 1
•
• . • i
0.30
i
'
J
0-40
AIR MRSS FLOW RATE (Kg/s)
CONTOURS: 111' Solids Moss Flowrate (kg/s)
Figure C.24
Conveying Characteristics - Bulk SoUd: FA2, Test Rig: TR14, Model: P C 6
Appendix C
322
Pipe Dotal I G
D(a. Length
(mini (nil
69 562
81 382
-
Tot.
945
CL
o
cc
•
LU
CC
CJ
in
in
UJ
DC
Q_
0. 10
0-20
0.30
AIR MASS FLOW RATE (Kg/s)
CONTOURS: (11 Solids Moss Flowrate (kg/s)
Figure C.25 Conveying Characteristics - Bulk SoUd: FA2, Test Rig: TR1S, M
400.
Pipe Dotal Is
D r a. Length
(mini (ml
a
69 562
300.
81 382
CL
Tot.
o
cc
CJ
LU
CC
C3
in
in
UJ
945
200.
100.
DC
CL
_i
00
•
•
'
•
I
'
0- 10
• ...J—i—L
0.20
0-30
AIR MRSS FLOW RATE (Kg/s)
CONTOURS: fl) Solids Moss Flowrate (kg/s)
Figure C.26 Conveying Characteristics - Bulk SoUd: FA2, Test Rig: TR15, M