Experimental Thermal and Fluid Science 57 (2014) 11–19
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Experimental Thermal and Fluid Science
journal homepage: www.elsevier.com/locate/etfs
Comparison of pressure drops through different bends
in dense-phase pneumatic conveying system at high pressure
Liang Cai ⇑, Shen Liu, Xu Pan, Xu Guiling, Yuan Gaoyang, Chen Xiaoping, Zhao Changsui
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China
a r t i c l e
i n f o
Article history:
Received 23 January 2014
Received in revised form 28 March 2014
Accepted 28 March 2014
Available online 5 April 2014
Keywords:
Pneumatic conveying
Dense-phase
High pressure
Bend
Pressure drop
a b s t r a c t
In order to investigate the effect material property, bend geometry and location on pressure drop through
the bend, experiments of dense-phase pneumatic conveying are carried out at conveying facility with the
pressure up to 4.0 MPa. Petroleum coke and anthracite powders with different particle sizes are applied
to examine flow characteristics. The empirical correlations of pressure drop through the bend are
obtained using Barth’s additional pressure theory and multi-variable linear regression. Results show that
pressure drop through vertical downward bend is the least, followed by pressure drop through horizontal
bend, pressure drop through vertical upward bend is the largest. Powders with larger size need consume
more energy than that with smaller size at the same solid loading ratio and conveying velocity as gas–
solid mixture flows across the same radius bend. Flow characteristics of petroleum coke and anthracite
are analyzed and compared. Pressure drop through the bend with the long radius is greater than that with
the short radius. While to unit length, pressure drop of long radius bend is less than that of short radius
bend. The empirical correlations of pressure drop through the bend are derived and predicted results
agree well with the experimental results. The flow characteristics of the bend offer the theoretical support for design, control and operation of dense-phase pneumatic conveying at high pressure.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
Pneumatic conveying is an important process in the chemical,
energy and pharmaceutical industry for transportation of granular
particles [1–4]. The aim of these transport systems is to transfer
particulate material between storage locations, or to feed different
kinds of reactors. One of the advantages of using pneumatic conveying system to transport bulk particulate material, compared
to other systems, is the flexibility in routing the pipeline. This often
results in transport pipes with many bends, which considerably increase the difficulty in predicting the performance of the system.
Bends and elbows play vital roles in giving pneumatic conveying
systems considerable flexibility by allowing routing and distribution. Dense-phase pneumatic conveying at high pressure is one
of the key technologies in gasification. Because of low velocity,
high pressure and high solid concentration in transportation,
gas–solid mixture across bend is very unsteady and complicated
[5–14]. Pressure drop through bend is seriously affected by bend
geometry, location, material property, etc.
⇑ Corresponding author. Tel./fax: +86 25 83795652.
E-mail address: Liangc@seu.edu.cn (L. Cai).
http://dx.doi.org/10.1016/j.expthermflusci.2014.03.016
0894-1777/Ó 2014 Elsevier Inc. All rights reserved.
The importance of the bend in the design of pneumatic transport
systems has been noted by many researchers. Mason et al. [15] apply polyethylene pellets and cement to examine the performance
envelope of the pneumatic conveying system and observe the flow
regimes of the gas–solid mixture in the pipe. An analytical model is
developed to determine the contribution of the solids to overall
pressure drop in both straight pipes and bends. Levy and Mason
[16] investigate the effect of bend on the distribution of particle
in pipe cross section and segregation in pneumatic conveying system using three-dimension numerical simulation. Results show
that the presence of a bend causes particles to concentrate around
the pipe wall downstream of the bend. The effect of pipe diameter,
bend radius ratio and different material properties on flow parameters is examined. Das and Meloy [17] study pressure drop in a
close-coupled double bend in pneumatic conveying of fly ash. Pressure drop across two close-coupled 90-degree bends is compared to
the pressure drop in an isolated single 90-degree bend. Resulting
bend pressure drops are correlated to the corresponding phase density and superficial air velocity at the bend inlet. Under similar flow
conditions, the pressure drop in a close-coupled double bend is less
than double of that in a single bend. This shows that pressure drop
through a close-coupled double bend conveying solid material is
not equivalent to the cumulative effect of two single bends. The
power correlation relating the pressure drop with superficial air
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L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
Nomenclature
D
dp
e
Fr
G
g
L
R
P2
Dp
Dpg
Dps
Dpsa
Dpsf
internal diameter of pipe, m
particle mean diameter, m
the natural constant, e = 2.718
Froude number
mass flow rate, kg/h
acceleration of gravity, m/s2
pipe length, m
bend radius, m
receiving tank pressure, MPa
pressure drop through the bend, kPa
pressure drop due to gas friction, kPa
additional pressure drop caused by solids, kPa
pressure drop due to accelerating the solids, kPa
pressure drop due to solid friction and impact, kPa
velocity and solid loading ratio shows a reasonable agreement with
test data. Lee et al. [18] apply polypropylene beads and glass beads
to study solid concentration and velocity distribution determination through 90° bend in the pneumatic conveying system. The
experimental results show a constant frequency pulsating flow
for polypropylene beads in the dense-phase flow regime. For dilute-phase flow regime, both polypropylene and glass beads show
a continuous annulus flow structure. Numerical simulation using
the Euler–Euler method is also conducted using computational
fluid dynamics and the fluid and particle flow characteristics are
compared with the experimental data. Wadke et al. [19] use the
numerical simulation and experiment to describe variation of aerodynamic forces and particle motion in a dilute horizontal pipe. It is
found that the bend causes an increase in mean particle velocity
compared with a horizontal pipe. Results show that the number
of impacts in the bend decreases as the velocity of the particle increases. The results from the simulation agree closely with the
experimental time-of-flight measurements. Mcglinchey et al. [20]
develop a model to predict the pressure drop across a 90° bend both
in a horizontal plane and in a vertical plane for an extended range of
conveying conditions. The model results presented are compared
with experimental data gathered from an industrial-scale pneumatic conveying test system. Broad qualitative agreement in trends
and flow patterns are found. Chu and Yu [21] use three-dimensional
combined continuum and discrete model to investigate flow characteristics in bend. The applicability of the approach is first qualitatively verified by comparing the simulated results with the
observations in the literature in terms of typical flow features in
bends such as roping, particle segregation, particle velocity reduction, particle recirculation, and pressure fluctuation. The gas–solid,
particle–particle and particle–wall interaction forces are then analyzed to understand their roles in governing the complicated flow.
Zhou et al. [22] carry out dense-phase pneumatic conveying experiments to reveal pressure drop through the bend. Effects of operating parameters on the pressure drops are examined in dense-phase
pneumatic conveying at high pressure. Hanley et al. [23] use macroscale model to examine the breakage of particles at a 90° bend during dilute phase pneumatic transport. Breakage results if the impact
force between the particle and pipe bend exceeds the intrinsic
strength of the particle. The latter is taken to be distributed according to the Weibull distribution. Impact force depends on impact
velocity and this relationship is obtained by a two-phase structural
model of the particle, based on the widely used Kelvin-Voigt model.
Dpsh
Dpsh1
Dpsh2
Re
U
pressure drop due to raising and suspending the solids,
kPa
pressure drop due to suspending solid particles, kPa
pressure drop due to raising solid particles, kPa
Reynolds number
superficial gas velocity, m/s
Greek letters
kg
resistance coefficient of gas phase
ks
resistance coefficient of additional solid phase
l
solid loading ratio
qg
gas density, kg/m3
qs
solid density, kg/m3
The impact velocity is distributed as a result of a distribution in particle velocity and in impact angle, though the variability in the latter
is shown to be the significant component. Rinoshika [24] studies
the effect of the dune model and soft fins in horizontal pneumatic
conveying involving a 90° bend. At the upstream of bend and in
the bend, the particle velocity of using the dune model is evidently
higher than that of the conventional pneumatic conveying and
using soft fins. However, the effect of soft fins and dune model on
the particle velocity is maintained downstream of the bend. Hidayat and Rasmuson [25] investigate the effects of particle diameter,
particle density, particle volume fraction, gas velocity and bend radius ratio on relevant quantities in engineering applications in a Ubend. A small bend radius ratio will produce a faster dispersion of
particles, which benefits drying, but on the other hand, will increase
the total pressure drop. Thus, optimizing gas velocity and bend radius ratio is important in reducing energy consumption. Laín and
Sommerfeld [26] apply Euler/Lagrange approach in connection
with the k–e turbulence model accounting for full two-way coupling to simulate the pneumatic conveying. The structure of the
secondary flow developing in the bend is investigated and the influence of the particles as well as inter-particle collisions on the secondary flow structure and intensity is addressed. A detailed
analysis of the segregation phenomena occurring in the bend and
the influence of the particle phase on the flow structure can be performed by the calculations. Despite numerous studies, both experimental and numerical, have been conducted on different
pneumatic conveying systems to characterize the flow behaviors
of the solids in bend, most of those researches are mainly carried
out to dilute-phase pneumatic conveying at low pressure. Effect
of material property, bend geometry and location on the pressure
drop through the bend in dense-phase pneumatic conveying is
not fully understood at high pressure.
This paper presents a comprehensive study of effect of material
property, bend geometry and location on pressure drop through the
bend in dense-phase pneumatic conveying at high pressure. A series
of cases with powders of different particle sizes and material categories are performed at different gas velocities. Pressure drops through
horizontal bend, vertical upward bend and vertical downward bend
are examined and compared. Empirical correlations of pressure drop
through the bend are derived and analyzed. The findings should be
useful not only for establishing a comprehensive understand about
the effect of material property, bend geometry and location but also
for designing and controlling pneumatic conveying systems.
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
13
metal tube variable-area flow meters. Weight of bottom-discharge
blow tank is measured by load cells sintered on the surface of bottom discharge blow tank to obtain mass flow rate in the experiments. The signals of pressures, differential pressures, weight
and flow rates are sent to the data acquisition and control system
composed of a computer and an A/D converter.
2.2. Material properties
Fig. 1. Schematic diagram of dense-phase pneumatic conveying. 1 – Nitrogen
cylinder. 2 – Buffer tank. 3 – Supplementary gas. 4 – Fluidizing gas. 5 – Pressurizing
gas. 6 – Top-discharge blow tank. 7 – Bottom-discharge blow tank. 8 – Load cell. 9 –
Electrostatic charge sensor. 10 – Data acquisition system. 11 – Observation window.
12 – Horizontal pipe. 13 – Vertical pipe. 14 – Inclined pipe.
2. Experimental system and material properties
2.1. Dense-phase pneumatic conveying system at high pressure
The schematic diagram of the pneumatic conveying experimental system is shown in Fig. 1. The pneumatic conveying system
consists of material storage system, gas supply system, conveying
pipeline, measurement system, data acquisition and control system. The material storage system is composed of two tanks, one
top-discharge blow tank and the other bottom-discharge blow
tank with a capacity of about 0.6 m3. The top discharge blow tank
adopts bottom-fluidization and top-discharge arrangement. The
bottom-discharge blow tank adopts bottom-fluidization and bottom-discharge arrangement. In the experiments, particles are
transported from feeding tank to receiving tank through conveying
pipeline, driven by the total differential pressure and carrier gas
from gas supply system. Each of the blow tanks can serve as feeding or receiving tank, which can be converted by controlling the
valves. In the gas supply system, high-pressure nitrogen with
gauge pressure up to 12 MPa from nitrogen cylinders is injected
into a buffer tank and then divided into fluidizing gas, pressurizing
gas and supplementary gas. During the experiment process, fluidizing gas from buffer tank is injected into feeding tank through a
metal gas distribution plate located at the bottom of feeding tank
and particles in the feeding tank are fluidized by fluidizing gas
and driven into conveying pipeline by differential pressure. Supplementary gas is added at the outlet of feeding tank to enhance
the conveying stability. Pressurizing gas is injected into feeding
tank at the upper-half of feeding tank in order to regulate feeding
pressure. And the pressure of receiving tank is regulated by the gas
exhaust valve. Conveying pipeline, composed of straight pipes and
bends, is made of a smooth stainless pipe with an inside diameter
of 10 mm and a total length of 35 m. Every section of the straight
pipe for measuring pressure drop is 1 m in length. Inclination angle
of inclined straight section can be adjusted to 30°, 45° or 60°,
respectively. The bend radius of horizontal bend can be adjusted
to 120 mm, 200 mm and 300 mm. Pressures of feeding tank,
receiving tank and buffer tank are measured by pressure transducers. Pressure drop of every section of pipeline is measured by differential pressure transducers. Volume flow rates of pressurizing
gas, fluidizing gas and supplementary gas are measured by three
Anthracite and petroleum coke are transported in the experiments and material properties are shown in Table 1. The particle
sizes of the experimental materials are measured by a laser particle
analyzer (LS, Beckman Coulter Inc., USA). Four experimental materials all cover a wide range of particle size and their size distributions are shown in Fig. 2. Fig. 3 illustrates the SEM (scanning
electron microscope) micrographs of experimental materials. It
can be seen from Fig. 3 that four kinds of experimental materials
have poor sphericity and rough surfaces. Particle size distributions
of anthracite #2 and petroleum coke #2 are similar and their volume mean particle diameters are nearly the same. Thus, these two
kinds of materials are treated as particles with the same particle
size in the analysis of experimental results. Moisture contents of
materials are measured according to the National Standards of
the People’s Republic of China. External moisture contents and total moisture contents of experimental materials are very small,
which means the influence of moisture content on flow characteristics and resistance properties can be ignored. Real densities of
experimental materials are measured by mercury intrusion analysis and real density of anthracite is larger than petroleum coke for
the same particle size.
3. Results and discussion
3.1. Comparison of pressure drops through different location bends
In pneumatic conveying system, three different location 90°
bends with 200 mm radius are investigated using petroleum coke
and anthracite. In the conveying process, the pressure in receiving
tank is maintained at 3.0 MPa and other operation parameters are
same except the conveying velocity. Superficial gas velocity, which
will be referred to simply as conveying velocity in the paper, can be
calculated at gas temperature and average pressure in conveying
pipeline. Figs. 4 and 5 show the pressure drops through horizontal
bend, vertical upward bend and vertical downward bend. With the
increase in conveying velocity, pressure drops through different
bends rise slowly at first and then increase rapidly [15,22]. Forces
between gas and particles, between particles, and between particles and wall are the key factors responsible for the features of
gas–solid two-phase flow in a bend. As gas–solid two-phase flow
enters the bend, particle–wall collision can lead to particle–particle
collision in regions close to a bend wall. It is because some particles, after colliding with the bend wall, will change their velocities
and directions, and collide with incoming particles, forming a
shielding layer. Strong particle–wall interactions exist in the outer
wall of a bend, which can lead to strong particle–particle interactions. Particle–wall and particle–particle interactions both contribute to pressure drop. Then gas–solid two-phase mixture flows out
from the bend and particles are accelerated by the gas. Particle
velocity decreases considerably at a bend and the afterward acceleration causes additional pressure lose. The pressure drop through
the bend depends on not only the particle velocity but also on the
particle concentration. When the conveying velocity is low, solid
loading ratio decreases with the increase in conveying velocity because of constant mass flow rate. Pressure drop contributed by
those factors almost rises slowly as shown in Figs. 4 and 5. As
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L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
Table 1
Material properties.
Material
Mean particle diameter (lm)
External moisture content (wt.%)
Total moisture content (wt.%)
Real density (kg/m3)
Bulk density (kg/m3)
Petroleum coke #1
Petroleum coke #2
Anthracite #1
Anthracite #2
163.0
56.69
139.9
52.78
1.05
0.59
1.84
2.35
1.19
0.72
2.63
2.99
1103
1103
1490
1490
616
475
736
588
5
4
4
Petroleum coke, dp =163 μm
Anthracite, dp =139.9μ m
Petroleum coke, dp =56.69 μm
Anthracite , dp =52.78μ m
Volume (%)
Volume (%)
3
3
2
1
1
0
2
10
100
1000
0
1
10
100
Particle Diameter (μm)
Particle Diameter (μm)
(a) Petroleum coke
(b) Anthracite
1000
Fig. 2. Particle size distributions of experimental materials.
(a) Petroleum coke, 163μm
(b) Petroleum coke, 56.69μm
(c) Anthracite, 139.9µm
(d) Anthracite, 52.78μm
Fig. 3. SEM micrographs of experimental materials.
conveying velocity continues to rise, gas-phase friction increases.
Particle collision frequency and intensity become fierce highly,
which results in more energy consumption. All of those lead to increase of pressure drop through bend with the increase in conveying velocity. In addition, pressure drop through the vertical
downward bend is the lowest, followed by pressure drop through
the horizontal bend, pressure drop through vertical upward bend
is the largest at the same conveying velocity and solid loading
ratio. The pressure drop through the pipe can be described using
Eq. (1)
DP ¼ DPg þ DPsf þ DPsh þ DPsa
ð1Þ
DPg is the pressure drop due to the gas friction, DPsf is the pressure
drop due to solid friction and impact, DPsh is the pressure drop due
to raising and suspending the solids, DPsa is the pressure drop due
to accelerating the solids. To the three different location bends with
15
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
26
28
Horizontal bend
Vertical upward bend
Vertical downward bend
G=900kg/h
P2=3.0MPa
24
24
Δ P (kpa)
Δ P (kpa)
26
Horizontal bend
Vertical upward bend
Vertical downward bend
G=760kg/h
P2=3.0MPa
22
22
20
18
20
16
18
4
6
8
10
12
4
6
8
10
U (m/s)
U (m/s)
(a) Coarse petroleum coke
(b) Fine petroleum coke
12
Fig. 4. Pressure drops through different location bends of petroleum coke.
28
Δ P (kpa)
26
24
30
Horizontal bend
Vertical upward bend
Vertical downward bend
G=830kg/h
P2=3.0MPa
27
Δ P (kpa)
30
22
20
21
18
18
16
24
Horizontal bend
Vertical upward bend
Vertical downward bend
G=900kg/h
P2=3.0MPa
6
8
10
12
14
15
6
8
10
U (m/s)
U (m/s)
(a) Coarse anthracite
(b) Fine anthracite
12
Fig. 5. Pressure drops through different location bends of anthracite.
the same geometry in an experimental case, pressure drops due to
the gas friction, solid friction and acceleration are essentially equal.
Pressure drop of raising and suspending the solids can be divided
into two parts: one part is used to suspended solid particle in conveying gas. Another part is used to change gravitational potential
energy because of change of solid particles height. Pressure drop
of raising and suspending the solids can be expressed by
DPsh ¼ DP sh1 þ DPsh2
ð2Þ
DPsh1 is the pressure drop due to suspending particles. DPsh2 is the
pressure drop due to raising solid particles. In dense-phase pneumatic conveying, because suspended velocity is very less and particles across horizontal bend flow at the same height level, pressure
drop of suspending and raising particles can be ignored in the horizontal bend. To the vertical bend, it is different to the pressure drop
of suspending and raising particles. As the gas–solid mixture flows
through the vertical upward bend, directions of gravity and flow are
opposite in the vertical part of bend. Height of solid particles is continuously elevated and gravitational potential energy rises. In order
to overcome effect of gravity, conveying gas must consume more
energy to change particles height. While directions of gravity and
flow are same in vertical part of vertical downward bend, gravitational potential energy of particles can be used to accelerate solid
particles and overcome flow resistance in conveying process. Thus
the conveying gas saves some energy because of gravity and pressure drop decrease as gas–solid mixture flows in vertical downward
bend. Therefore pressure drop through vertical upward bend is the
largest, followed by pressure drop through horizontal bend,
pressure drop through vertical downward bend is the lowest at
the same conveying velocity as shown in Figs. 4 and 5.
3.2. Effect of material category and particle size on pressure drop
through the bend
In order to obtain effect of material category and particle size on
pressure drop through the bend, pressure drop through the horizontal 90° bend with 200 mm radius is applied to investigate the
flow characteristics. Effect of particle size on pressure drop through
the horizontal bend is shown in Fig. 6. It can be seen that pressure
drop through horizontal bend with larger size particles is greater
than that with smaller size particles for the same kind of material.
As solid particles have larger size in pneumatic conveying process,
single particle is weightier and it is difficult to wholly suspend
those particles in the conveying gas. So particles with larger size
are very easy to sink to the bottom of conveying pipe. Particle layer
formed by the sunken particles slides at the bottom of conveying
pipe and solid friction rises. In addition, as larger size particles collide with the bend wall, the particles have larger inertia force and
can easily be crashed. This inelastic collision will consume a lot of
energy of particles. Formed fine particles because of collision will
fly at different directions. In order to change those fine particles
flow directions to mainstream direction, conveying gas needs to
expand more energy to accelerate those particles. While to the
smaller size particles, they are light and easy to be suspended in
the conveying gas. Energy loss to the suspended flow is less than
that to the stratified flow at the same conveying velocity.
16
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
26
26
dp=56.69μm, G=760kg/h
24
P2=3.0MPa
Δ P (kpa)
Δ P (kpa)
28
dp=163μm, G=700kg/h
22
20
24
dp=139.90μm, G=830kg/h
dp=52.78μm, G=900kg/h
P2=3.0MPa
22
20
18
18
4
6
8
10
12
16
6
8
10
12
U (m/s)
U (m/s)
(a) Petroleum coke
(b) Anthracite
14
Fig. 6. Effect of particle size on pressure drop through the horizontal bend.
28
Δ P (kpa)
26
24
parameters and material properties. But mass flow rates of petroleum coke and anthracite are about 760 kg/h and 900 kg/h, respectively. So if the two kinds of powders flow at the same solid loading
ratio and conveying velocity, pressure drop though the bend for
petroleum coke is larger than that for anthracite. Because petroleum coke particles adhered at the pipe wall form the shell around
the conveying pipe as shown in Fig. 8, friction coefficient greatly
increases. Then at the same solid loading ratio and conveying
velocity, pressure drop of petroleum coke is larger than that of
anthracite.
Petroleum coke, dp =56.69μm, G=760kg/h
Anthracite, dp =52.78μm G=900kg/h
P2=3.0MPa
22
20
18
3.3. Pressure drop through the bend with different radiuses
16
6
8
10
12
U (m/s)
Fig. 7. Effect of material category on pressure drop through the horizontal bend.
Moreover, as fine particles collide with bend pipe wall, probability
of particles breakage is low and energy loss is small. Solid particles
have a good tracking ability in the conveying gas. More particles
have same flow direction with mainstream, which save more energy to change particle flow direction and accelerate them.
Although mass flow rate of powders with the smaller size is
slightly larger than that with the larger size, pressure drop of powder with larger size is higher than that with smaller size at the
same conveying velocity in Fig. 6. Fig. 7 shows the influence of
material category on pressure drop through the bend. Result indicates that pressure drops are almost same at the similar operation
In order to investigate effect of bend radius on pressure drop
through bend, 90° bends with radius of 120 mm, 200 mm and
300 mm are used to examine flow characteristics of coarse anthracite in dense-phase pneumatic conveying at high pressure. In
experimental process, pressure in receiving tank and mass flow
rate keep stable. From the Fig. 9, results show that the short radius
bend gives an overall pressure drop less than the long radius bend.
As gas–solid mixture enters the bend, particles collide with particles and pipe wall by inertia force and form powder layer in the
bend. Gas–solid mixture distribution in bend is very non-uniform
in conveying pipe cross-section. After gas–solid mixture flows
out from the bend, gas–solid mixture becomes uniform slowly in
pipe cross-section. Apparently the short radius bend takes the
abrupt losses quickly and then returns to a steady state while
the long radius carries an unsteady condition over a long distance.
Because gas friction almost holds constant at a certain length as
conveying parameters keep stable, pressure drop of gas friction
rises with the increase in bend length. As conveying velocity and
solid loading ratio maintain constant in conveying experiments,
30
Δ P/L (kpa)
27
24
R=300mm
R=200mm
R=120mm
G=860kg/h
P2=3.0MPa
21
18
15
6
8
10
12
14
U (m/s)
Fig. 8. Petroleum coke of the adhered pipe wall.
Fig. 9. Pressure drop through horizontal bend with different radiuses.
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
the number of particles collision with particles and pipe wall is
great when mixture flows across long radius bend, which consumes more kinetic energy of particles. Then kinetic energy which
uses to accelerate the solid particles in long radius bend is larger
than that in short bend radius. To the pressure drop of solid friction, although friction coefficient decreases slightly with the increase in bend radius, effect of bend length on pressure drop of
solid friction is stronger. So the pressure drop of friction rises with
the increase in bend radius. Pressure drop of raising and suspending the solids can be ignored because of horizontal bend. So pressure drop of long radius bend is greater than that of short radius
bend at the similar operation parameters and solid loading ratio
as shown in Fig. 9. Pressure drop per unit length through the bend
is shown in Fig. 10. It indicates that pressure drop per unit length
decreases with the increase in bend radius. As gas–solid mixture
flows in short radius bend, gas–solid two-phase flow changes more
in direction per unit length. Collision frequency and intensity in the
short radius bend are greater than that in the long radius bend. So
mixture needs to consume more energy to overcome solid friction
and accelerate particles per unit length in short bend as shown in
Fig. 10.
3.4. Empirical correlations of pressure drops through different bends
DP s ¼ k s l
pR qg U 2
2D
17
ð6Þ
2
where solid friction coefficient is expressed as a function of
velocity, solid loading ratio, particle size and bend radius for the
bend with a certain diameter.
ks ¼ f ðU; l; R; dp Þ
ð7Þ
According to the dimensional analysis, the equation for ks is:
ks ¼ alb Fr c
m n
dp
R
D
D
ð8Þ
pffiffiffiffiffiffi
where Fr ¼ U= gD, dp is the average particle size. The a, b, c, m
and n are the constants. According to the experimental data,
empirical correlation of solid friction coefficient using multi-variable linear regression can be given as follows.
The solid friction coefficients of bend with 200 mm radius for
petroleum coke are given as follows.
Solid friction coefficient of horizontal bend:
ks ¼ e0:046 l1:14 Fr 1:931
0:133
dp
D
ð9Þ
Solid friction coefficient of vertical upward bend:
In order to predict pressure drop through the bend, empirical
correlation of pressure drop through bend, based on Barth’s additional pressure theory, is analyzed and derived. The pressure drop
is considered as the sum of gas and solid pressure drop components. Total pressure drop can be derived as follows:
DP ¼ DP g þ DP s
ð3Þ
where Dpg is pressure drop due to gas friction and Dps is additional
pressure drop due to solids. The gas pressure drop component is
evaluated by assuming that only gas is flowing in the bend. Dpg
can be calculated according to the equation below:
DPg ¼ kg
Lqg U 2
2D
ð4Þ
When 300 > ReðD=2RÞ2 > 0:034, gas friction coefficient can be given
[27]:
0:25
0:029 þ 0:304½ReðD=2RÞ2 kg ¼
ð2R=DÞ
ks ¼ e0:183 l1:131 Fr1:969
ð5Þ
ks ¼ e1:129 l1:129 Fr 1:856
ð11Þ
The solid friction coefficients of bend with 200 mm radius for
anthracite are given as follows.
Solid friction coefficient of horizontal bend:
ks ¼ e1:334 l1:086 Fr 1:688
0:042
dp
D
ð12Þ
Solid friction coefficient of vertical upward bend:
0:038
dp
D
ks ¼ e1:794 l1:085 Fr 1:61
0:012
dp
D
ð13Þ
ks ¼ e0:126 l0:961 Fr 1:537
100
80
R=120mm
R=200mm
R=300mm
G=860kg/h
P2=3.0MPa
60
40
8
10
12
14
U (m/s)
Fig. 10. Pressure drop per unit length through horizontal bend with different
radius.
ð14Þ
Solid friction coefficient of horizontal bend with different radius
is given
120
Δ P (kpa/m)
0:104
dp
D
Solid friction coefficient of vertical downward bend:
The total pressure drop through the bend is measured in the
experiment and pressure drop of gas phase is calculated using
Eq. (4). Then the correlation is derived for the solid pressure drop
component. Addition pressure drop Dps is given as follows:
6
ð10Þ
Solid friction coefficient of vertical downward bend:
ks ¼ e0:8 l1:102 Fr 1:801
1=2
0:121
dp
D
0:072 0:634
dp
R
D
D
ð15Þ
Then empirical formulas of pressure drops through the different
bends for petroleum coke and anthracite are derived in densephase pneumatic conveying at high pressure. In order to verify
the accuracy of the empirical formulas, comparisons between predicted results based on the above correlation and experimental
data of pressure drops through different bends are shown in Figs.
11–13. The relative errors between the experimental results and
predicted values are mostly less than 8%, which means the predicted results agree well with the experimental data. So empirical
formulas of pressure drops through different bends in dense-phase
pneumatic conveying at high pressure are obtained and can be
used to design, control and operation of dense-phase pneumatic
conveying at high pressure.
18
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
27
Experimental value
Predicted value
Experimental value
Predicted value
Predicted Δ P (kpa)
Predicted Δ P (kpa)
28
24
%
+5
%
-5
20
16
16
20
24
24
%
+5
%
21
-5
18
18
28
21
24
Experimental Δ P (kpa)
Experimental Δ P (kpa)
(a) Horizontal bend
(b) Vertical upward bend
27
27
Predicted Δ P (kpa)
Experimental value
Predicted value
24
%
+5
21
%
-5
18
15
15
18
21
24
27
Experimental Δ P (kpa)
(c) Vertical downward bend
Fig. 11. Comparison of predicted and experimental pressure drop through bend for petroleum coke.
30
Experimental value
Predicted value
Predicted Δ P (kpa)
Experimental value
Predicted value
24
+5
21
%
-5
%
18
15
15
18
21
24
+5
24
%
-5
18
18
27
%
24
Experimental Δ P (kpa)
Experimental Δ P (kpa)
(a) Horizontal bend
(b) Vertical upward bend
27
Predicted Δ P (kpa)
Predicted Δ P (kpa)
27
24
21
Experimental value
Predicted value
+5
-5
18
15
15
%
18
%
21
24
27
Experimental Δ P (kpa)
(c) Vertical downward bend
Fig. 12. Comparison of predicted and experimental pressure drop through bend for anthracite.
30
L. Cai et al. / Experimental Thermal and Fluid Science 57 (2014) 11–19
Predicted Δ P (kpa)
30
Experimental value
Predicted value
24
%
+7
%
-8
18
12
12
18
24
30
Experimental Δ P (kpa)
Fig. 13. Comparison of predicted and experimental pressure drop through different
radius bends for coarse anthracite.
4. Conclusions
The influence of material property, bend geometry and location
on pressure drop through the bend in dense-phase pneumatic conveying at high pressure is investigated. According to Barth’s additional pressure theory, the empirical formulas are achieved using
the dimensional analysis and multi-variable linear regression.
With the increase in conveying velocity, pressure drops through
the bends rise slowly at first and then increase rapidly. Pressure
drop through the vertical downward bend is the lowest, followed
by pressure drop through the horizontal bend, pressure drop
through the vertical upward bend is the largest at the similar solid
loading ratio and operation parameters. Pressure drop through the
horizontal bend with larger size particles is greater than that with
smaller size particles for the same kind of material. At the same solid loading ratio and conveying velocity, pressure drop of petroleum coke is larger than that of anthracite. The short radius bend
give an overall pressure drop less than the long radius bend. While
pressure drop per unit length decreases with the increase in bend
radius. Pressure drops through different bends are analyzed and
empirical correlations are derived. The predicted results agree well
with the experimental results and the relative errors between the
experimental results and predicted values are mostly less than 8%.
Acknowledgment
Supported by the Special Funds of National Key Basic Research
and Development Program of China (2010CB227002).
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