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Piping System Design
Technical Report · February 2000
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Piping
System
Design
IL Greeff
W Skinner
Department of Chemical Engineering
University of Pretoria
February 2000
TABLE OF CONTENTS
1.
INTRODUCTION ..............................................................................................................1
2.
PHYSICAL PROPERTIES AND UNITS............................................................................ 7
3.
BASIC FLUID DYNAMICS FOR PIPE FLOW................................................................. 10
4.
PRACTICAL APPLICATION OF THE ENERGY BALANCE ON PIPING SYSTEMS ........ 24
5.
COMPRESSIBLE FLOW................................................................................................ 39
6.
NON-NEWTONIAN FLOW. ............................................................................................58
7.
MULTIPHASE FLOW .................................................................................................66
8.
OTHER TYPES OF FLOW........................................................................................71
9.
PIPING SYSTEM DESIGN .........................................................................................77
10.
CONTROL VALVES ..................................................................................................89
11.
FLUID MOVERS........................................................................................................101
APPENDIX A 1 : Dimensional standards for plain steel pipes (unscrewed)
APPENDIX A2 : Dimensional standards for general purpose tubes
APPENDIX A3 : Dimensions for polypropylene and high density polyethylene tubes
APPENDIX B : Conversion factors
APPENDIX C1 : Absolute roughness for various pipe materials
APPENDIX C2 : Moody diagram
APPENDIX C3 : Equivalent lengths for various components
APPENDIX C4 : Resistance coefficient data for piping components
APPENDIX C5 : Resistance coefficient data for \wo-K method
APPENDIX C6 : Diagram for prediction of friction pressure loss
APPENDIX D1 : Calculation of tiP1K for isothermal compressible flow
APPENDIX D2 : Calculation of W for isothermal compressible flow
APPENDIX D3 : Calculation of &'jlc for adiabatic compressible flow
APPENDIX 04 : Calculation of W for adiabatic compressible flow
APPENDIX E : Generalised rheological constants for various fluids
APPENDIX F : Relation between X and
f]j for
two-phase flow
APPENDIX G1 : L!P 1 m and LIP100ft for mild steel systems
APPENDIX G2 : Ludwig criteria for linear velocity
APPENDIX H : C"' values
~-
-.-·---
·-:
·-
LIST OF SYMBOLS
a
A
c
c
c,
C,
d
D
F
Fx
r
g
h
H
k
k'
J(
K,
IC
ke
KE
L
I
m
M
Ma
MEB
II
N
NPSH
N,
p,
p
pb
P,
pe
PE
q
Q
r
acceleration, m/s2
area, m2
sonic velocity, m/s
constant of integration
corrosion allowance
flow coefficient
specific heat capacity at constant pressure, kJ/kmol. K
specific heat capacity at constant volume, kJ/kmol. K
pipe diameter, mm
pipe diameter, m
energy loss in the system due to friction, J/kg
safety factor
pulse factor
force in direction x, N
Darcy friction factor
acceleration due to gravity, m/s 2
specific enthalpy, J/kg
enthalpy, J
specific heat ratio
generalised rheological constant
resistance coefficient
constant for 2-K method
constant for 2-K method
specific kinetic energy, J/kg
kinetic energy
length, m
mixing length, m
mass, kg
non-isothermal exponent
molecular mass .
Mach number
mechanical energy balance
mol or kmol (number)
rheological constant
exponent
generalised rheological constant
revolutions per minute
net positive suction head
turbulent exponent
velocity head
pressure
back pressure
vapour pressure
specific potential energy, J/kg
potential energy, J
heat, J/kg
heat, J
general radius, m
proportionality constant
- ---... · - - . -. - ---........,4"
R
radius of pipe, m
ideal gas constant, J/mol. K
Reynolds number, dimensionless
specific entropy, J/kg.K
entropy, JIK
entropy production, JIK
specific mass
time, s
temperature
linear point velocity in the x direction, mls
average linear velocity, mls
maximum linear velocity, mls
velocity fluctuation components
Re
s
s
Sp
SG
t
T
temporal average of the product of v' and u'
u
internal energy, Jlkg
v
specific volume, m3/kg
v
volumetric flow rate, m3/s
V'
Volume
w
Mass flow rate
work energy, J/kg
W'
work energy, J
XJ'
coordinates
X
linear dimension, which is significant in the flow pattern, m
mass fraction
distance from the centre of the pipe, m
y
y
Martinelli parameter
z
height, m
compressibility factor
head, m
z
kinetic energy correction factor, dimensionless
thickness of the viscous sublayer, m
£
absolute roughness, m
!L
dynamic viscosity, Pa.s
v
kinematic viscosity, v = p./p , m2/s
p
density, kglm 3
r
shear stress, Pa
wall shear stress, Pa
non-isothermal correction factor
e
angle
angular velocity
efficiency
1J
•-r
•-.;-~-
--
--
-
--,,,-;
'
-
·-~:
--:•-
•
--~(">''_;;
.-;-;-•
SUBSCRIPTS
a
fluid mover, absolute, available, smaller diameter
b
larger diameter
B
beginning
c
calculated
CV
e
critical
control valve
equivalent
E
EL
elevation
EP
end point
EQ
equipment
f
friction
g
gauge
I
I
inside, component, first reference point
j
KE
end
in
second reference point
kinetic energy
L
m
n
o
liquid
0
out
p
r
pipe, pulse
R
s
STV
TV
TP
v
required
gas, vapour, volume
w
wall
momentary, maximum
normal
outside
pipe components I restrictions
sonic, piston, cylinder
sub total variable
total variable
two phase
. ···:
..
--~--~~-
--~---::
1.
INTRODUCTION
1.1
CHEMICAL ENGINEERING DESIGN
Chemical engineering is the science of the industry's manufacturing processes. Raw materials
are transformed into more valuable products by means of chemical, physical thermal,
biochemical and mechanical processing. It is performed by companies which are collectively
referred to as the process industry and in equipment collectively referred to as a plant. Planning
and evaluations associated with a specific envisaged manufacturing process are collectively
known as chemical engineering design or simply process design.
1.1.1
•
DESIGN STEPS
PROCESS DEVELOPMENT
Buy an existing process.
Develop an own process - laboratory and pilot plant investigations; ideal stage for
investigating different construction materials.
•
MATERIAL AND ENERGY BALANCES
Material balances - correlate feed and product flow rates of processing units; render
flow rates necessary for dimensional design of equipment; also composition of fluids.
Energy balances- render temperatures at different stages of manufacturing; needed for
evaluation of physical properties of fluids.
Compositions and temperatures -selection of suitable construction materials.
•
PLANT DESIGN
Selection and writing of specifications for construction materials.
Calculation of dimensions, making of sketches, writing of specifications for process
equipment (reactors, columns, heat exchangers, instrumentation) and components of
process equipment (tubes in heat exchangers, plates or packing in distillation columns,
pipe sections, fluid movers, control valves in piping systems).
•
INVESTMENT EVALUATION
After completion of the plant design, information is available to do a proper cost analysis.
This will determine whether the project is economically viable and whether the company
will proceed with the building of the plant.
1.1.2
DIAGRAMS
Several types of diagrams can be found for example architectural diagrams, instrumentation
diagrams, plant diagrams, diagrams of subsurface constructions, services diagrams (water, gas,
steam), pipe diagrams, pipe and instrumentation diagrams (P & ID) and flow diagrams.
1
Abbreviations, symbols for processing units and so called equipment tables are used to provide
maximum information on flow diagrams.
1.1.3
REPORTS
The basic principles of report writing are applicable. A typical layout of the main division is the
following:
•
INTRODUCTION.
•
PROCESS SELECTION.
•
MATERIAL AND ENERGY BALANCES.
appendix).
Motivation and description.
Selection and motivation.
Tables and figures (Bulk calculations in the
•
EQUIPMENT.
•
Selection and specification of construction materials.
Make proper use of subdivisions.
•
Selection, design principles, dimensions, sketches of columns, reactors, piping systems.
(Bulk calculations in the appendix).
•
WASTES.
•
OPERATING PROCEDURES. Commissioning, operation, decommissioning.
•
SAFETY.
gases).
•
APPENDIX.
Quantities, properties, treatments.
Dangers associated with operation and properties of materials (solids, liquids,
Bulk calculations and documentary material.
PIPING SYSTEM DESIGN
1.2
The design of piping systems can involve various engineering disciplines. It is based on a sound
knowledge of fluid dynamics combined with practical design guidelines and procedures.
Chemical engineers are usually involved in dimensional design and specification as well as the
selection of suitable construction materials. This course will focus on dimensional design. The
methods will either be applied to design new systems or to analyse existing systems. Analysis of
existing systems is necessary if unknown parameters need to be determined or if modifications
need to be implemented.
1.2.1
COMPONENTS OF PIPING SYSTEMS
Components include pipe sections, tubes, valves, elbows and T-pieces, equipment like pumps,
compressors and flow measuring instruments.
•
PIPES AND TUBES
•
The main distinctions between pipes and tubes are in the methods of fabrication, finishing
2
and in their codes of standards. The surface roughness of pipes varies to such an extent
that differences must be taken into consideration in the calculation of friction factors.
Tubes are taken as smooth with minimal surface roughness.
•
VALVES
•
Gale valve- The closing element operates at right angles to the fluid flow, the flow is
straight through. The gate wedges into the body. With the gate in the fully open position
there is little resistance to the fluid flow. Not suitable for flow rate control, used only as a
stop valve. May be difficult to open should the downstream pressure fall with the valve in
the closed position.
•
Globe valve- The closing element is parallel to the fluid flow, the body has a globular
shape. The disk can have various profiles to provide different controlled flow
characteristics. The head loss across these valves are large. Due to relatively large
disks the valves are limited to smaller sizes.
•
Plug valve - The closing element is a quarter turn plug, straight or tapered, with a
rectangular opening through its centre. The flow is straight through and there is little
resistance to flow or head loss. The valve is suitable for coarse throttling. The valve can
be subject to sticking if not lubricated suitably.
•
Ball valve- The closing element is in the shape of a ball with a hole through its centre.
The valve opens and closes through an angle of 90 degrees. Straight through flow
provides minimal resistance to flow. The ball can be manufactured with a contoured
cavity to give some degree of controlled flow characteristics.
•
Butterfly valve- The valve disk opens and closes through a 90 degree angle. Often used
for the control of gas and vapour flows. May be used as a stop valve except under
severe unbalanced pressure conditions. Should not be used as a terminal valve except
at very low pressures.
•
Check valve -Also known as a non-return valve. Used to prevent backflow of fluids in
process lines, closure effected by zero or reverse flow. Available in various patterns such
as lift disk or swing check. Produces resistance to flow, can cause water hammer if not
well designed and positioned.
•
Diaphragm valve -The closing element is a diaphragm clamped between the body and
cover of the valve and separating the fluid from the operating mechanism. Little
resistance to flow. Diaphragm can be manufactured of rubber or various elastomers.
Was invented in 1929 by a SA engineer- PK Saunders.
•
PIPING COMPONENTS
•
Elbows differ in their angle and also in the ratio bend radius : pipe diameter.
•
T-pieces include the following types- soft T (no flow in branched leg), hard T (no flow in
one main leg), reduced T (three flow streams are relevant and the diameters may differ).
3
•
•
Various types of reducers and enlargers are used. Changes in diameter can be gradual
(a typical example is the ASA type) or sudden.
EQUIPMENT
Strainers are used to remove solid particles larger than certain dimensions from fluids.
Flow measuring instruments are used to measure flow. They also provide signals for
flow regulating equipment. A typical example is the orifice plate. (Control valves are
sometimes also classified as equipment).
Fluid movers are used to move fluids from one process unit on a plant to anotherpumps for liquids and compressors for vapours or gases. To limit shut downs due to
faulty fluid movers, standby units are often installed in parallel.
Equipment also includes heat exchangers, reactors, columns etc.
1.2.2
STANDARDS
Standards are used for specification purposes. Important standards for piping systems are
standards for dimensions, material composition and material properties. Various standard
systems are in use. Examples are ASA, ASTM, API, AWWA, AISI, DIN, CABRA and BSJ. Each
system is subdivided into sections for pipes and tubes of different construction materials and also
for the various types of piping components and fittings. The general format for pipe system
specification is MATERIAL STANDARD, DIMENSIONAL STANDARD.
The material standard is a reference for information relating to the composition and properties of
the relevant construction material. Several of the given standards are used. In 1974 the UNS
(unified numbering system) was introduced and will hopefully become the only system to be
used. The latest edition of Perry's Chemical Engineers' Handbook is still using different systems.
Examples for mild steel are AISI1 020 and UNS G1 0200.
The dimensional standard is a reference for information relating to the diameter and wall
thickness of the pipe, tube, component or fitting.
The most important dimensional information for pipe and tube sections is that of nominal
diameter, outside diameter and wall thickness. The inside diameter can be calculated as
(outside diameter- 2 x wall thickness). Different formats are in use for pipes (variables are
material of construction and units of fabrication) and tubes (the same variables as for pipes).
Examples for mild steel pipes are 6" nom sch 40 and 150 mm nom 5 mm wall. Standards for
British pipes are in Perry. The standard for mild steel Sl pipes is given in appendix A1. In the
case of metal tubes formats are independent of the metal type; examples for metal tubes are 2"
nom BWG 14 and 50 mm nom 2 mm wall. The standard for British metal tubes is in Perry and
for Sl tubes in appendix A2.
Examples of dimensional standards for fittings and other pipe components are available in Perry.
The format is nominal diameter and class number. The class number refers to the maximum
allowable operating pressure (psig) and relates to the wall thickness. Plastic piping are also
4
specified according to class numbers. Dimensions for polypropylene and polyethylene pipes are
given in appendix A3.
1.2.3
DIAGRAMS
Piping systems are shown in various degrees of detail on design diagrams. Examples are pipe
diagrams, pipe and instrumentation diagrams and engineering flow diagrams. Pipe related
diagrams may be presented in orthographic and isometric formats. Use is made of various
abbreviations and symbols for process units. Process streams may also be abbreviated, e.g. 0
for oil and S for steam.
The Piping and Instrumentation Diagram should include the following:
1.
2.
3.
4.
5.
6.
All process equipment identified by equipment numbers.
All pipes identified by line numbers, size and construction material should be shown.
All valves and control valves with identification numbers, type and size should be shown.
All fluid machines (pumps and compressors), identified by suitable codes or numbers.
Ancillary fittings that are part of the system, such as strainers and steam traps; with an
identification number.
All control loops and instruments, suitably identified.
1.2.4
SYMBOLS
Various symbols are used to show piping components or process units on a diagram. Some
of the symbols to be used in this course are shown below.
-D--
Reducer
-E_G
-------C}---
Enlarger
------@--
----1XJ-
Gate valve
~~
Globe valve
-l/1---
Check valve
~
-II
-)(i~-
-Q---
Centrifugal pump
----s?--
Reciprocating pump
5
[_]
Orifice
Strainer
Blind flange
Heat exchanger
Column
1.3
LITERATURE
1.
Walas, S M, Rules of thumb - selecting and designing equipment, Chern Eng, 75,
March 16, 1987.
2.
Spitzgo, C R, Guidelines for overall chemical plant layout, Chern Eng, 103, Sept 27,
1976.
3.
Bruckman, c G and Mandersloot, W G B,
CENG 191.
4.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, val. 6, 2 ed:
Pergamon, 1993.
5.
R. H. Perry, Green, W.G., Perry's Chemical Engineers' Handbook, 7th ed: McGraw-Hill,
1997.
6
Writing informative reports, CSIR report
2.
PHYSICAL PROPERTIES AND UNITS
The most important physical properties for flow applications are density and viscosity. In
compressible flow applications thermodynamic properties also become important. Physical
properties are one of the most important aspects in chemical engineering design and the
literature is vast. Perry's Chemical Engineers' Handbook is a useful source for physical
properties. Another source is the databanks of flowsheeting simulators. If the physical
properties of a component is not known various estimation methods can be used to find the
properties e.g. group contribution methods. In the case of mixtures of liquids or gases properties
are estimated with thermodynamic methods. Modern ftowsheeting simulators are also very
useful in this regard.
2.1
UNITS
Sl units will mainly be used. Pipes and tubes of certain construction materials are however only
available in British units and some data in reference books are given in British units. Useful
conversion factors relevant to the course material are given in appendix B.
2.2
DENSITY
LIQUIDS
For design purposes liquids are considered to be incompressible and densities are functions of
composition and temperature but not of pressure. The density of most organic liquids, other than
those containing a "heavy atom" usually lies between 800 and 1000 kg/m 3 . Density can also be
calculated from specific mass if the latter is known:
PL = 1000 SG kg/m3
where
SG=SG (tWCJ
GASES AND VAPOURS
For general engineering purposes it is sufficient to consider gases and vapours as ideal. Density
is then calculated using the ideal gas law:
P'V=nRT
MP
p= RT kglm'
where Tis always in Kelvin.
If greater accuracy is needed the compressibility factor can be included:
7
P'V=znRT
Pv=zRT
MP k I 3
P =-- g m
zRT
The compressibility factor can be estimated using an equation of state for real gases such as the
Pang-Robinson equation or the Redlich-Kwong equation. A generalised compressibility plot,
which gives z as a function of reduced pressure and temperature can also be used. For mixtures
of gases the pseudo critical properties ofthe mixture should be used to obtain the compressibility
factor:
P~.~m=
Pc.a Ya + Pc.bYb + ···
T,,m =Tc.a Ya + T,;, y, + ...
where Pc and Tc are critical pressure and temperature, y is mol fraction, m refers to mixture and a
and b to the components.
2.3
VISCOSITY
Units for viscosity are cP for absolute/dynamic viscosity and eSt for kinematic viscosity. The
relation between the two viscosities is:
p=pv
The Sl units for viscosity are as follows:
1 cP = 1x 10'3 Pa.s
1 eSt = 1x10'6 m2/s
Viscosity of liquids vary with temperature and pressure but the pressure effect is not significant
except at very high pressures. Viscosity of liquids tend to decrease with an increase in
temperature whereas the opposite effect is found in the case of gases.
2.4
SPECIFIC HEAT CAPACITY
Specific heat capacities are required to find the specific heat ratio for gases which is used in
k=Cp
C,.
compressible flow calculations:
8
For a gas in the ideal state the specific heat capacity at constant pressure is given by:
CP
=a+ bT + cT 2 + d T 3
Values for the constants in the equation are available in handbooks. Several group contribution
methods are also available for estimation of these constants.
2.5
VAPOUR PRESSURE
The three-term Antoine equation can be used to determine vapour pressure for a pure
B
h1p.=A--,
T+C
component at a certain temperature:
where the constants can be found in literature, and the units will depend on the units of the
constants. T is usually in Kelvin. Knowledge of vapour pressure is important in cavitation
calculations.
2.6
LITERATURE
1.
J. Winnick, Chemical Engineering Thermodynamics. New York: Wiley, 1997.
2.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, val. 6, 2 ed:
Pergamon, 1993.
9
3.
BASIC FLUID DYNAMICS FOR PIPE FLOW
Fluid dynamics is the branch of fluid mechanics that is concerned with the motion of fluids.
Previously fluid dynamics existed as two separate disciplines namely hydrodynamics and
hydraulics. Hydrodynamics is a mathematical science based on the equations of motion of an
imaginary ideal fluid. Results of hydrodynamic studies are of limited practical value. For this
reason empirical formulae were developed from experimental studies on fluid flow. When
dealing with liquids this subject is called hydraulics. In fluid mechanics today, the basic
principles of hydrodynamics are combined with experimental data which satisfy the need for
a broader treatment.
3.1
IDEAL FLUIDS VERSUS REAL FLUIDS
In an ideal fluid the effects of viscosity are completely neglected. When such a fluid flows in
a pipe, the shear stresses are absent and only the pressure and inertia forces are considered.
There is also no velocity variation in the direction perpendicular to the pipe axis and as a
consequence the fluid must slip past the solid boundary of the pipe wall. Ideal fluid flow is
frictionless, without any losses and is also referred to as the reversible flow of
thermodynamics.
When considering real fluid flow, the tangential stresses due to shear as well as the normal
stresses due to pressure are taken into account. It is especially in the region near a solid
boundary that real fluid flow differs from ideal fluid flow. A real fluid adheres to the solid
boundary and does not slip. This no slip condition at the solid boundary causes the velocities
of the different fluid layers to vary across the cross-section of a pipe between a zero velocity
at the wall and a maximum velocity in the centre.
3.2
FUNDAMENTAL EQUATIONS DESCRIBING PIPE FLOW
The three-dimensional motion of any fluid can be described by the fundamental laws of fluid
dynamics and thermodynamics. These laws are mathematically formulated by the continuity
equation, the momentum equation and the energy equation. In addition to these laws certain
other relations are also employed in describing a fluid, for example the ideal gas law and
Newton's viscosity relation.
When the above laws are applied to flow in a straight pipe a one-dimensional approach works
very well since there is no curvature in the streamlines. The complete one-dimensional
differential equations describing flow in a pipe, applied in the x-direction, are the following:
3.2.1
CONTINUITY EQUATION
ap
u-
at
a
ax
+ -(pu)
=
0
In the case of steady flow, equation (3.1) becomes:
10
..... (3.1)
a
- (pu)
ax
~
0
..... (3.2)
or more simply
W
~
pAu
..... (3.3)
which describes the conservation of mass in a system through which a fluid flows.
3.2.2
MOMENTUM EQUATION
In the momentum equation
2
1 aP
az
- - - g - + ~ a u ~ au +
p ax
ax
p ax 2
at
du
dt
au
ax
!l-
..... (3.4)
the first term represents the normal stresses due to pressure and the third term the tangential
shear stresses due to viscosity. The second term represents the gravitational forces, which
are zero in the x-direction if the pipe is horizontal. For an ideal fluid the viscosity is zero and
equation (3.4) reduces to the well known Euler equation:
1 ap
az
ax
- - - g-
pax
au
+
at
~
au
11-
..... (3.5)
ax
The momentum equations in three-dimensional form are also known as the Navier-Stokes
equations.
3.2.3 ENERGY EQUATION
The one-dimensional energy equation for fluid flow is derived here using the basic
thermodynamic balances.
The total energy balance for an open system is:
(U
+
PE + KE)E ~ (U + PE + KE,)B +
- La (H
+
LJ (H
+
PE + KE)
PE + KE) + CiQ + 6W 1
..... (3.6)
where E = End and B = Beginning and refer to the system, I= In and 0= Out and refer to the
11
mass streams entering and leaving the system. Note that the sign convention for work is taken
as positive when done on the system and negative when done by the system. For steady flow
the energy of the system does not change and the balance reduces to:
La (H
+ PE + KE) -
L
1
(H + PE + KE) = oQ + oW'
or
11H + 11PE + ME
= oQ
+ oW 1
In terms of specific properties:
11h + 11pe + 11ke
=
q + w1
..... (3.7)
Potential energy and kinetic energy are calculated with the following well-known equations:
11pe
=
Me
=
g/1z
Substitution of the above into equation (3. 7) gives
1'1112
11h +g/1z + - -
=
2
q +w 1
..... (3.8)
The entropy balance over the system, in terms of specific properties, is:
..... (3.9)
where Lis is the change in entropy of the mass streams and Sp is the entropy production. The
following thermodynamic relation is now used:
dh
=
Tds + vdP
12
..... (3.10)
which on integration from inlet to outlet under the assumption of constant temperature yields:
Outlet
/l..h
T/l..s +
J vdP
..... (3.11)
Inlet
Noting that
1b ~ T,
equation (3.11) rearranges to
q ~T/I..s-TS
b p
..... (3.12)
Substituting equations (3.11) and (3.12) into (3.8) and noting that v
~
1/p :
Outlet dP
J-
+
g/l..z
..... (3.13)
+
p
Inlet
where TJ:fp is the frictional loss of energy of the flowing fluid and is now substituted by F. For
incompressible flow p is constant and equation (3.13) reduces to:
dP
+ gdz + udu +
p
oF ~
OW
..... (3.14)
which has the units of J/kg.
For an ideal fluid (i5F =D) without work being done on the fluid, equation (3.14) reduces to the
well-known Bernoulli equation:
dP
+gdz+udu~o
p
..... (3.15)
Since it is inconvenient to deal explicitly with the variations in the flow and fluid properties that
occur at a pipe cross-section, average flow quantities need to be defined. In this regard it is
only the linear velocity that is considered to vary significantly over the pipe cross-section for
a real fluid. It is obvious that the average linear velocity is dependent on the velocity profile.
In order to express equation (3.14) in terms of the average linear velocity, a kinetic energy
correction factor aKE• is introduced:
dP
p
+ gdz + aKE u du +
13
oF ~ oW
..... (3.16)
The kinetic energy equation can be determined with the following equation:
R
l/. 3
(~) rdr
J
..... (3.17)
11
0
where u1 is the linear velocity at a point and u is the average linear velocity.
Provided that density is constant, equation (3.16) can be integrated:
M
p
!<.u 2
. 2
+ gf'..z + u.K1<.-- + F =
w
..... (3.18)
Multiplication of each term with density transforms equation (3.18) to a pressure balance in
units of Pascals:
M
!<.u2
+ pgf'..z + pu.KE--
2
+ pF = pw
..... (3.19)
Equation (3.19) is now expressed using the following symbols and this is referred to as the
Mechanical Energy Balance (MEB):
..... (3.20)
Equation (3.19) can also be expressed in units of metres when divided by the term p g, such
an equation is known as a head balance (HB) :
..... (3.21)
The MEB form of the energy equation will mainly be used in this course. The first three terms
of the MEB deal with pressure energy, potential energy and kinetic energy respectively. The
frictional pressure loss term (AP1 ') deals with losses due to components in a pipeline as well
as losses in straight pipe sections due to internal fluid friction and friction between the fluid and
the pipe wall due to viscosity. LIP" is the energy added to or removed from the system due to
a fluid machine of some kind.
3.3
LAMINAR AND TURBULENT FLOW
A variety of flow features, including energy losses and velocity profiles, are affected by
14
whether the flow is laminar or turbulent. The Reynolds number is the criterion used to
distinguish between different flow regimes namely laminar, transitional and turbulent. This
dimensionless number is named after Osborne Reynolds, who was first to describe the
existence of laminar and turbulent flow quantitatively in 1883. The Reynolds number is the
ratio of the inertia forces to the viscous forces which are the only significant forces affecting
the flow pattern. Gravity and capillary forces do not have any effect on the flow pattern of a
fluid in a completely filled pipe. The general formula for the Reynolds number is the following:
Re
xup
=
~~
where x is the linear dimension which is significant in the flow pattern.
In the case of a pipe flowing full, the pipe diameter, D, is used as the linear dimension:
Re = Dup
..... (3.22)
fl
For flow in a straight circular pipe laminar flow exists for Reynolds numbers below 2000. It is
however subject to slight variations. In laminar flow, movement of the fluid appears as the
sliding of thin laminations over adjacent layers. The particles move in definite paths or
streamlines with relative motion occurring at a molecular scale.
The shear between the adjacent layers in laminar flow is expressed by
't
=
i5u
~~~
oy
..... (3.23)
which is also known as Newton's equation of viscosity.
When laminar flow is developing in a pipe, a laminar boundary layer, or annular outer zone,
starts to grow against the pipe wall until the boundary layers from opposite sides meet at the
pipe axis. At this point, the flow is termed fully developed. Theoretically an infinite distance
is required for the flow to become fully developed, although it has been established that the
maximum velocity in the centre of the pipe will reach 99% of its ultimate value in the distance
L = 0.058ReD.
Between Reynolds numbers of 2000 and 4000 there exists a transition region in which the flow
can be laminar or turbulent. The value of 4000 is also known as the upper critical Reynolds
number, above which flow is normally turbulent. This value is indeterminate, laminar flow has
been maintained in circular pipes for values of Reynolds number up to 50 000. The value of
2000 is much more definite and that is why it can be defined as the true critical Reynolds
number.
15
In turbulent flow the velocity at a point in the flow field fluctuates in direction and magnitude.
These fluctuations are caused by a multitude of small eddies created by viscous shear
between adjacent particles. When turbulent flow is developing in a pipe, a laminar boundary
layer starts to grow at the pipe wall up to a point when transition occurs and the boundary layer
becomes turbulent. This turbulent boundary layer increases in thickness much more rapidly
than the growth of the laminar boundary layer until the layers from opposite sides meet at the
pipe axis to result in fully developed turbulent flow. At a smooth pipe wall, the fluctuation in
velocity in the direction of the wall must be zero causing the turbulence to be inhibited. This
results in a laminar-like sublayer next to the wall. This is not a true laminar layer since it is
momentarily disrupted by the adjacent turbulent flow. Because shear in this layer is
predominantly due to viscosity alone, it is called the viscous sub/ayer. Fully developed
turbulent flow will be found at about 50 pipe diameters from the pipe entrance for a pipe with
no special disturbance at the entrance; otherwise, the flow will be fully developed within a
shorter distance.
In turbulent flow the shear stress is made up of two components: the viscous stress and the
Reynolds or inertia stress due to the turbulent fluctuations:
'
=
ou
11- - p
oy
11
"v
..... (3.24)
The concept of a boundary layer, within which viscosity is important, was advanced by Ludwig
Prandtl in 1904. According to Prandtl's hypothesis, the viscous effects of fluid friction at high
Reynolds numbers are limited to the boundary layer. In all the flow outside the boundary layer
(the core flow) the viscous effects can be ignored at high Reynolds numbers. Thus the core
flow can be considered ideal and is well described by the ideal Bernoulli or Euler equations.
3.4
EMPIRICAL EQUATIONS FOR FRICTION LOSS
For incompressible, laminar flow in a pipe the well-known Hagen Poiseuil/e Jaw can be used
to compute the frictional pressure drop through a straight pipe section:
..... (3.25)
Equation (3.25) can be derived from basic principles and is not empirical. The empirical
equation for the prediction of the frictional energy loss which is valid for both laminar and
turbulent flows is known as the Darcy-Weisbach equation:
- f'L pu2
M--f
D
2
..... (3.26)
The equation expresses the loss in terms of an empirical friction factor, j', known as the Darcy
16
friction factor, which corresponds to fully developed flow.
3.5
RELATION BETWEEN THE FRICTION FACTOR AND SHEAR STRESS
It will now be shown how the friction factor is related to the shear stress. It is valid for laminar
and turbulent flow. Consider a fluid element and the various forces acting on it in figure 3.1.
~~~-Pipe
~-,____
wall
{
~~
k---
Surface Area, A,
--T-~-----~----
r
Area ofplane, AP
--
Figure 3.1
Force Balance on a Fluid Element
If acceleration forces can be considered neglible, a simple force balance results in:
'L,FX = ma = 0
..... (3.27)
PA-PA
-TA s =0
I p
~.._p
..... (3.28)
or
..... (3.29)
T =
f'...Pr
2L
17
..... (3.30)
Combining equation (3.26) with equation (3.30) and simplifying, yields and expression for the
friction factor in terms of the shear stress :
..... (3.31)
3.6
VELOCITY DISTRIBUTION AND axE FOR LAMINAR FLOW
Due to viscosity the different layers of a fluid travel at different velocities resulting in a velocity
distribution over the cross section of a pipe. In the case of fully developed laminar flow it can
be shown that the velocity profile is a perfect parabola:
..... (3.32)
where u 1 is the velocity at one point and urn"" is the maximum velocity in the centre of the pipe.
This profile is always valid for laminar flow and is independent of the condition of the pipe wall
i.e. whether the wall is smooth or rough. The average linear velocity in laminar flow can be
shown to be precisely one half of the maximum centerline linear velocity :
U
=
0.5
llmax
..... (3.33)
The kinetic energy correction factor can now be calculated for laminar flow by inserting
equations (3.32) and (3.33) into equation (3.17) and integrating. This renders a value for aKE
of exactly 2.
3. 7
FRICTION FACTOR FOR LAMINAR FLOW
Since both the Darcy-Weisbach equation and the Hagen Poiseuille law are valid for laminar
flow, a simple friction factor relation can be derived by equating these equations, solving for
j' and substituting for the Reynolds number:
j'
=
64
Re
..... (3.34)
This result shows that for laminar flow the friction factor is dependent only on the Reynolds
number. In the critical region, between laminar and turbulent flow, values for the friction factor
are uncertain.
18
3.8
VELOCITY DISTRIBUTION AND
3.8.1
SMOOTH VERSUS ROUGH PIPE FLOW
aKe
FOR TURBULENT FLOW
In turbulent flow the condition of the pipe wall influences the velocity profile. The roughness
of commercial pipe walls is described by the absolute roughness e, which is an indication of
the size of the projections on the pipe wall. It has been found experimentally that a pipe with
a given wall roughness will sometimes behave as a smooth pipe and other times as a rough
pipe depending on the Reynolds number. The velocity profiles encountered are different.
Whether a pipe is smooth or rough is determined by the size of the absolute roughness (height
of the projections) with respect to the thickness of the viscous sublayer (ii), which decreases
with an increase in Reynolds number. If the roughness does not project through the viscous
sublayer (e < ii) in turbulent flow, the surface is said to be hydraulically smooth. It is then the
viscous shear alone that determines the flow resistance, and roughness has no effect on the
flow. In rough pipe flow the roughness projections protrude through the layer (e > ii) causing
a broken up viscous sublayer. It is now the form drag of the protrusions that determines the
flow resistance.
3.8.2
THE POWER LAW
The oldest representation of a smooth pipe velocity profile is the so-called power Jaw:
~'
= (
11.
Umax
'')~
1-R
I
..... {3.35)
where N, is the turbulent exponent. This law is strictly empirical.
3.8.3
THE SMOOTH LAW OF THE WALL
A newer approach, which is mostly used for smooth pipe velocity profiles, is the Jaw of the wall
which is semi-empirical. This law states that there is a wall layer where most of the velocity
variation occurs. Three regions are apparent in this layer namely the viscous sublayer which
is laminar, a transition zone (also referred to as the buffer layer) and a turbulent core where
the velocity profile is described by a logarithmic function. Since the equations describing the
velocity profile is very complex, the profile is only described qualitatively with the aid of figure
3.2. The vertical scale is exaggerated so that the three zones are clearly visible.
19
Turbulent zone
~----------
y
Logarithmic velocity profile
/
/
/
///~
/
-
Laminar velocity profile
Transition zone
I
~~/_/_/_/
-- ----~+~Viscous sublayer
Figure 3.2 Velocity profile for turbulent flow
The Viscous Sublayer
In the viscous sublayer the flow is assumed to be laminar and the Reynolds stress due to
turbulent fluctuations is neglible. The following expression for the thickness of the viscous
sublayer can be derived:
0
SuD
= -'-----
Re
J-c
0
/p
..... (3.36)
The relation between.{ and r is now used to find 15 as a function off:
0
SD
=
Re
~~
..... (3.37)
The Logarithmic Layer
The Prandtl mixing length theory is used to predict the logarithmic function describing the
velocity profile in this region analytically. In this region it is assumed that the viscous shear can
be neglected over most of the flow area and that the Reynolds stress dominates.
20
The Transition Region
The intersection of the velocity profiles for the viscous sublayer and the logarithmic layer
introduces an abrupt change in the velocity profile which is not realistic. The one curve must
rather merge gradually into the other, introducing a transition region where the viscous and
turbulent stresses are of the same magnitude.
3.8.4
THE ROUGH LAW OF THE WALL
In the case of rough pipe flow, the classification can be broken down into fully rough and
transitionally rough. In the first case the roughness protrudes right through the viscous
sublayer and the transition region to render a totally broken up wall layer. The resulting flow
is then completely turbulent. In transitionally rough flow the roughness projections protrude
only partially into the transition region. It is then both viscous shear and form drag determining
the flow resistance.
When < < o the roughness protrusions are contained within the viscous sublayer and the
roughness has no effect on the flow. When<> 14o the protrusions extend into the turbulent
layer and foro<<< 14o there is a transition region where flow resistance is a result of viscous
shear and form drag.
Since the development of the law of the wall by Prandlt, other authors have attempted to
describe the complete velocity distribution in a single equation. Experimental data have also
shown that the law of the wall equation for the logarithmic layer deviates from the real profile
near the centerline.
3.8.5
aKF:
FOR TURBULENT FLOW
When the law of the wall for the logarithmic layer is assumed to be valid over the entire crosssection of a pipe the following approximate relation can be derived for the kinetic energy
correction factor for turbulent flow:
aKE
3.9
~ I + 2.7f1
..... (3.38)
FRICTION FACTOR FOR TURBULENT FLOW
In laminar flow the Hagen-Poiseuille law can be used to determine the relation for the friction
factor. In turbulent flow no such law exists and use is made of the velocity profiles to predict
the forms of the equations for the friction factor.
For smooth pipe flow this was first done by Prandtl in 1933 and is known as the theoretical law
of friction. The experimental work of Nikuradse contributed to the exact values of the constant
21
giving an equation known as Prandtl's smooth pipe equation for the friction factor:
_I
= 2
ll
Jog (
Re /l ) - 0.8
..... (3.39)
By similar analyses von Karman developed the following equation for turbulent flow in fully
rough pipes:
_I_ = 2 log (
ll
.!2)
28
+ 1.74
..... (3.40)
In 1939 Colebrook combined equations (3.39) and (3.40) to yield an equation that is
applicable over the entire range, smooth, transitionally rough and fully rough:
_I_ = -2 log ( c/D +
/l
3.7
~
Re/l
)
..... (3.41)
Equation (3.41) provides a good approximation for conditions in the intermediate range and
reduces to the smooth pipe equation for c = 0 and the rough pipe equation for large Reynolds
numbers.
The problem with using more recent equations for the velocity distribution to derive equations
for the friction factor, is that they are complex and have to be numerically integrated. Churchill
and Chan derived an improved theoretically based expression for the friction factor but, as
stated in their article : "the net numerical corrections to the friction factor are too small to be
of practical interest".
3.10
NIKURADSE'S SAND ROUGHNESS SCALE
The measuring and specifying of the roughness of commercial pipes remains a problem. The
roughness projections vary in size, shape and distribution and cannot be quantified in terms
of a single number. The experimental work done by Nikuradse entailed the coating of different
sizes of pipe with sand grains that had been sieved to ensure uniform diameters.
diameters of the sand grains are represented by
c,
The
the absolute roughness. The correlations
for the friction factor are all based on Nikuradse's sand roughness scale. This means that in
order to use these correlations, one is bound to express any pipe roughness data on this
scale. Nikuradse did experimental work with values of the relative roughness (c/O) ranging
form 0.000985 to 0.0333.
3.11
LITERATURE
1.
R. P. Benedict, Fundamentals at Pipe Flow: John Wiley & Sons, Inc., 1980.
22
Fluid Mechanics
Hill, 1997.
with
Engineering
Applications,
ninth
ed.
2.
J. B. Franzini,
Belfast: McGraw-
3.
J. R. Welty, Fundamentals of Momentum, Heat, and Mass Transfer, third ed. New York:
John Wiley and Sons, Inc., 1984.
4.
A. J. Ward-Smith, Internal Fluid Flow, 1 st ed. New York: Oxford University Press,
1980.
5.
0. Reynolds, "An experimental investigation of the circumstances which determine
whether the motion of water will be direct or sinuous, and the laws of resistance in
parallel channels," Philosophical Transactions, pp. 935-982, 1883.
6.
L. Prandlt, "Uber Flussigkeitsbewegung bei sehr kleiner Reibung," Verhandl. Ill Int.
Math. Kongr., Heidelburg, 1904.
7.
L. Prandtl, "Uber die Ausgebildete Turbulenz," Proc. II Int. Congr. Appl. Mech., Zurich,
p. 62, 1926.
8.
L. Prandtl, "Neuere ergebnisse der turbulenzforschung," Z. VDI, vol. 77, p. 105,
1933.
9.
T. von Karman, "Uber laminare und turbulente reibung," Z. Angew. Math. Mech., vol.
1, p. 233, 1921.
10.
T. Von Karman, "Aspects of turbulence problems," Proc. IV Int. Congr. Appl. Mech.,
Cambrigde, England, 1934.
11.
J. Nikuradse, "Laws of turbulent flow in smooth pipes," Forsch. - Arb. lng. - Wesen,
val. 356, 1932.
12.
J. Nikuradse, "Laws of flow in rough pipes," Forsch.-Arb. lng.-Wesen, vol. 361, 1933.
13.
S. W. Churchill, "Friction factor equation spans all fluid-flow regimes," Chemical
Engineering, pp. 91-92, 1977.
14.
S. W. Churchill, "Improved correlating equations for the friction factor for fully
turbulent flow in round tubes and between identical parallel plates, both smooth
and naturally rough," Industrial and Engineering Chemistry Research, val. 33, pp. 20162019, 1994.
15.
C. F. Colebrook, "Turbulent flow in pipes, with particular reference to the transition
region between the smooth and rough pipe laws," Journal of the Jnsliluteof Civil
Engineering, London, val. 11, pp. 133-156,1938-1939.
23
4.
PRACTICAL APPLICATION OF THE ENERGY BALANCE ON PIPING SYSTEMS
4.1
MEB APPLICATION ON TYPICAL PIPING SYSTEM
The various terms of the MEB are now looked at in more detail:
..... (3.20)
APJ is subdivided as follows:
..... (4.1)
where
AP1 = friction pressure loss in pipe sections, components such as valves (except control
valves) and fittings such as elbows.
APcv = friction pressure loss in control valves.
APEQ = friction pressure loss in equipment such as reactors, columns, filters and flow
measuring
instruments.
The MEB can be applied between any two reference points in a piping system. Such
reference points are referred to as terminal points. Consider the system shown in figure 4.1.
The pressures P 1 and P 2 will normally be fixed and independent of flow rate. A MEB between
points 1 and 2 will contain all the terms in the given balance. A typical application of such a
balance
will be to establish the required APa for the system.
3
4
Figure 4.1
Typical piping system
24
When the balance is applied to the reference points 3 and 4, the AP. term become irrelevant:
A typical application of such a balance will be to calculate P 4 if P 3 is known from a pressure
gauge:
The pressuresP3 and P4 are not fixed and will be a function of flow rate. Such reference points
are referred to as intermediate points.
For the three integral terms (APE, LiPEP• APKJ, AX= XJa'""''"ampaint - X),P'"'"mpai"'· Their values can
be positive or negative. The three friction terms (L1P1, LiPcv. APEQ) are calculated by means of
specially developed methods. Apart from rare exceptions their values are positive.
For design purposes the terms L1P8 r and LIPEL between two terminal reference points are fixed
and values are independent of flow rate. In certain applications it is handy to group those
terms, that do vary with flow rate:
..... (4.2}
where TV= Total Varying. Due to the control valve flow regulating mechanism, LIPcv increases
with decreasing flow rate. The other three terms decrease with decreasing flow rate. In certain
applications these three terms are grouped:
..... (4.3}
where STV = Subtotal Varying. The MEB forms the basis for most piping system evaluations.
Different methods have been developed to calculate the different MEB terms for different flow
systems. These will be discussed in the following sections.
25
4.2.1
STRAIGHT PIPE SECTIONS
Frictional pressure loss equations
Frictional pressure loss in straight pipe sections are calculated using the Darcy equation:
Pa
..... (3.26)
The following forms of the Darcy equation are also very handy:
;:.,p ~ f'L
puz
D
2000
f
~ 62544f'LW 2
M
f
~ 62544f'LV"p
M
f
kPa
with all the parameters in Sl units
with W in kg/h and d in mm
kPa
kPa
with Vin m3/h and din mm
Reynolds number equations
The friction factors are a function of Reynolds number. Reynolds number also indicates
laminar or turbulent flow. The following equations are handy for the calculation of Reynolds
number:
Re ~ Dup
J.l
all parameters in Sl units
Re ~ 354W
dJ,I
with Win kg/h, din mm and f1 in cP
354Vp
du
with Vin m3/h, din mm and f1 in cP
Re
Any other set of units rendering a dimensionless number can be used.
26
Laminar I Turbulent flow
Laminar flow:
Critical flow:
Turbulent flow:
Re < 2000
2000 < Re < 4000
Re > 4000
The critical zone is avoided in design since the flow is unstable and friction factors are
uncertain.
Absolute pipe roughness
The difficulty in calculating accurate friction losses lies in the selection of a value for the
absolute pipe roughness. Pipe roughness generally increases with age due to corrosion,
pitting, etching or deposition of sediment or attachment of algal or bacterial slime to pipe walls.
Typical values for absolute roughness for various pipe materials are given in appendix C1.
If roughness values for a given pipe material are not available in literature it will have to be
determined experimentally. The roughness projections cannot be measured physically and
an experiment to obtain the value will involve a flow test to generate values of the friction
factors for various Reynolds numbers. Roughness can then be calculated using the Colebrook
equation rendering a value that corresponds to the Nikuradse sand roughness scale.
Calculation of friction factors
In laminar flow the friction factor is only a function of Reynolds number:
j'
=
~
..... (3.34)
Re
In turbulent flow the relations for friction factor is much more complicated. The Colebrook
equation has long been accepted as the formula to use in design for turbulent flow:
E/D
2.51
+ - )
3.7
Re/1
1 -- -2 Iog ( -
/1
..... (3.41)
The only disadvantage of the equation is the implicitness of the friction factor,}. A trial and
error method is required to calculate] from given values of Re and e/D. In order to overcome
this difficulty many explicit equations have been proposed to use in the place of the Colebrook
equation. One such an equation that seems to be widely accepted was proposed by Haaland
in 1983:
s/D)
- 1 -_ -1.8 log ( ( IJ
3.7
27
1.1!
+ -6.9 )
Re
..... (4.4)
In the past numerical values for the friction factor were read from a chart prepared by Moody
in 1944, also referred to as the Moody diagram. This diagram has been plotted with the aid
of equations (3.39) and (3.40) and is shown in appendix C2. The beauty of the Moody
diagram is that the different zones of importance are clearly visible, namely the laminar, critical,
transitional and fully rough zone. The chart has been plotted to cover the range of roughness
values and Reynolds numbers that would typically be encountered in practical situations,
rendering values for the friction factor varying between 0.008 and 0.1. These correspond to
values for the kinetic energy correction factor for turbulent flow ranging between 1.0216 and
1.27, according to equation (3.38):
I + 2.7f1
aKE =
..... (3.38)
In practice a value of 1.0 is used for all turbulent pipe flows.
The dotted line separating the transition region from the fully rough region was suggested by
R.J.S. Pigott and the equation for this line is:
Re
3500
=
..... (4.5)
E/D
In 1977 Churchill introduced a friction factor equation that is valid for both rough and smooth
pipes as well as for the full range of laminar, transitional and fully turbulent flow regimes :
j'-8
-
[ (
8
Re )
12
]
1
1
12
..... (4.6)
+ (A +B)u
where A and 8 are computed as
A
= (
-2.457 In [ (
;e
r
9
+
0.27 (;)
lr
and
The Churchill equation eliminates the need to test for the flow regime and is very suitable for
use in computer programming codes.
28
4.2.2
PIPING SYSTEMS
Piping systems consist of pipe sections (often with more than one diameter), fittings and other
components. The Darcy equation can not directly be applied for the evaluation of friction
pressure losses in fittings and other components -
the contribution of form friction is larger
than in pipe sections and length and diameter dimensions are complex. Other methods were
developed for such evaluations. They are based on equivalent lengths and resistance
coefficients.
Equivalent length method
The equivalent length of a fitting or component is that length of pipe section which will render
the same friction pressure loss as the restriction.
Equivalent lengths are determined
experimentally and are tabulated in literature as the number of equivalent pipe diameters (LID).
The equivalent length of the restriction (Lr) can be calculated as follows:
Lr~LrxD
..... (4.7)
D
The inside diameter of the pipe in which the unit will be mounted must be used for the
evaluation.
The friction pressure drop through the restriction can be calculated using the Darcy equation:
;.,pi ~
f'
Lr pu 2
D 2000
kPa
..... (4.8)
In the calculation of the friction pressure loss of a pipe system it is standard practice to first
calculate a global equivalent length of the system. In the case of a single diameter system:
Le ~ '£Lp + '£Lr
..... (4.9)
Then
~
f'
2
Le pu
D 2000
kPa
..... (4.10)
Literature data for LID values are subject to inaccuracies. In reality the value for a restriction
is a function of certain dimensions as well as Reynolds number. These dependencies are
often omitted and only average values are given. The data given in appendix C3 are valid for
Re > 1000. When used for calculations where Re < 1000 the results should be treated as rough
estimations. This is sometimes acceptable in piping system evaluations.
29
Resistance coefficient method
It is also known as the velocity head method. The following format of the friction pressure drop
equation was used in the development:
f'L
pu2
D.P = - ·
f
D
2000
kPa
which is now written as:
..... (4.11)
J'L
with K
D
Pv
=
pu2
2000
=
resistance coefficient
..... (4.12)
velocity head
..... (4.13)
The friction pressure drop through a restriction can be calculated with any of the given
equations. In the case of a pipe system with only one diameter the global resistance
coefficient can be calculated as:
Ke = LKP + L:Kr
..... (4.14)
where
_ f'Lp
Kp - D
and
..... (4.15)
Literature data forK values are subject to inaccuracies for the same reasons as those of LID
values. Depending on the type of restriction, values for K may be more dependent on
30
Reynolds number than on LID values, or less dependent; note thatK ~ (LID)f'. The Kvalues
given in appendix C4 are valid for Re > 2000. For Re < 2000 the K values become highly
inaccurate; for rough estimations it may be assumed that the LID values would be
independent of Reynolds numbers; this enables the evaluation of K values at low Reynolds
numbers by prorating:
Kr(Re < 2000)
~
Kr (II.terature) x
f'
at Re
at Re ~ 2000
--,--"-----..:.:_:_::_:c:___
f'
..... (4.16)
For more accurate calculations the two-K method can be used. The two-K method takes the
dependency of K-values on Reynolds number and exact geometry of the fitting into account
in the following equation:
K~
Kt
Re
I
+K(l+-)
o•
d
..... (4.17)
where
K 1 = K for the fitting at Re = 1
K. = K for a large fitting at Re = =
d = inner diameter of attached pipe in inches
Values for K 1 and K. are given in appendix C5.
In the cases where restrictions are associated with changes in diameter (e.g. reducers and
enlargers) and/or changes in flow rates (e.g. T pieces) different possibilities for the calculation
of velocity head exist. The data source must specify which velocity head should be combined
with the given K value for the calculation of friction pressure drop.
Combination of equivalent length and resistance coefficient methods
Various approaches can be followed. The following method is in general use:
•
First consider those restrictions to be treated with the K method or two-K method and
calculate EKr
•
•
Transfer EKr to 2:Lr!Dsflbtoraz
Add all the LID values of those restrictions to be treated with the LID method to find
•
LIDTotal
Calculate ELr and then Le ~ ELp + ELr for calculation of the friction pressure loss in the
usual way.
31
Systems with more than one diameter
Various approaches can be followed. Each diameter can be evaluated on its own. The friction pressure
Joss of the system is obtained by summation:
Mf
="
L..J !'J.P!;
In certain applications it is more convenient to combine the different diameter systems into a single system,
based on one diameter with a single Le or Ke. The friction pressure drop of the system can then be
evaluated in a single calculation with any of the given equations.
Prorating is used in combining the different diameter systems. Consider a two diameter system for
illustration purposes:
Now using the resistance coefficient method:
!JP! ~ Kea. Pv,a + Keb · Pv,b
Pv
0:
J/d'
.: Pv,a ~ {d/d.J'. Pv,b
.. !1P1 ~ (Keb + (d/d) 4Ke) p,.,b
The term (Keb + (d/d) 4 Kej is known as the resistance coefficient of the system based on db. Similarly an
equation can be derived for calculations based on da :
Using the equivalent length method:
kPa
32
The term (Le• + Le.(djd,J' (f'jf'~) is known as the equivalent length of the system based on diameter "b".
Similarly with base diameter "a":
In practice the diameters normally do not differ substantially,.('.~ f'• and the friction factor ratio term is
omitted in the correlations.
Division of Flow
Division by a T-piece renders two flow systems between two sets of reference points. Each must be
analysed individually. Calculation of friction pressure loss can be performed by summation or by using
combined equations. In prorating for combined equations it is necessary to also take into account the
difference in flow rate in the relevant sections. Examples of combined equations (for division of flow with
aT-piece) where the main line has a diameter "b" and the branched line a diameter "a", are
62544(/W
· b
2
b
kPa
5
pdb
Individual restrictions
In the case of restrictions in which division of flow occurs and/or where there is a change in diameter, the
literature source forK-data will specify the base for the relevant p,. calculation. In certain applications it
may be convenient to make use of the other p,. value; such evaluations require transformation of Kvalues.
The principle is prorating; examples are:
Reducer:
LlP! ~ K,Pv,a ~ (d/d,,J' K,p,.,b
T-piece:
L1PJ.3. 1 ~ K3.1 p,., 3 ~ (d/d;I 4(W/WY K3. 1 p,., 1
33
Diagrams
In older literature diagrams are often used to do friction pressure drop calculations. Evaluations are based
on certain equivalent lengths like 1 m (for AP, ...) and 100ft (for LiP 100}. The friction pressure loss of a
system can then be calculated as follows:
:::::;
Le
MlOO X-~
100
An example of a diagram is given in appendix C6.
Safety factors
Calculations of friction pressure loss are subject to various inaccuracies such as approximated correlations
for friction factors, average values for LID and K and also changes in relative roughness with time.
Calculated values may be either too low or too high. Applications where these inaccuracies may cause
operating problems, e.g. the specification of a fluid mover which will not be able to deliver the required
flow rate, must be identified; the use of suitable safety factors is recommended for such systems. A
typical safety factor for non-complex flow systems is F = 1.2 :
L1P1 (TO BE
4.3
USED)~
1.2 L1P1 (CALCULATED)
CONTROL VALVES
Special equations are available to calculate the friction pressure loss through control valves (LiPcvl· This
will be dealt with in section 10.
4.4
EQUIPMENT
In practice AP,Q for various types of equipment are reported as part of design results and will be
associated with a certain flow rate. It is often necessary to calculate LiPEQ at other flow rates.
34
This is done by prorating:
2
.dPEQ a W
.. L1PEQ,2 ~
(W/WJ 2 .;JPEQ,J
Reported L1PEQ values will already include exit and entrance pressure drops. Exit and entrance pressure
drops of such equipment must not again be incorporated into L1P1 calculations!
4.5
ELEVATION PRESSURE DIFFERENCE
Elevations on a plant are normally given relative to a common reference elevation. In equipment where
fluid levels may vary, the most conservative level is used in calculations. Equations for calculation of L1P,"
are the following :
..... (4.18)
kPa
..... (4.19)
L1PEL for gases and vapours are normally considered as small (low density) and omitted in evaluations.
L1PEL for liquids can be large; if L1PEL is large and negative, it may be possible to obtain the desired flow
rate by means of gravity flow- no pump is needed.
4.6
ENDPOINT PRESSURE DIFFERENCE
..... (4.20)
In situations where one of the two reference pressures is not known, it may be calculated by application
of the MEB. In the case of gases or vapours, the calculation of an unknown P 1 will require a trial and error
approach. If L1PEP (terminal) is large and negative, it may be possible to obtain the desired flow rate
without the use of a fluid mover.
In principle, if for a given flow system
35
it will be theoretically possible to obtain the associated flow rate in the absence of a fluid mover; it may
however require impractically large pipe diameters.
4.7
KINETIC PRESSURE DIFFERENCE
The following equation is used:
kPa
..... (4.21)
with the following values for the kinetic energy correction factor:
Re < 2000 :
Re > 2000 :
o.KE
o.KE
~
~
2
I
Liquids are incompressible and densities are fixed. In the cases of gases and vapours for systems which
can be approximated as incompressible, calculations may be based on p 1 for both end points. In the case
of compressible flow the density in the energy equation:
dP
-
p
+ gdz + u.KE 11 du +
oF = oW
..... (3.16)
should be integrated using an appropriate equation of state to describe the relation between density and
pressure. Special equations will be derived for compressible flow in section 5.
LIPKE is often relatively small in relation to other terms in the MEB; in some applications it is convenient
to approximate LiPKE as zero in initial calculations;
this simplifies trial and error calculations;
the
assumption is checked in later calculations. Linear velocities of fluids in processing units with relatively
large diameters are taken as approximately zero; if such a unit is one of the reference points, the
associated kinetic energy is zero.
It is important to realise that LiPKE is applicable between the two
reference points only; all kinetic energy changes in between are irrelevant.
Two types of pipe exits are encountered in the case of liquids liquid level:
36
below the liquid level and above the
Below:
kPa since u2
=
0
Above:
4.8
FLUID MOVER PRESSURE DIFFERENCE
The energy supplied to a system by a fluid mover is represented by the LIP. term in the MEB. The LIP.
of a fluid mover is a function of flow rate.
The relation between LIP. and flow rate is a unique
characteristic of a fluid machine and this information is supplied by the manufacturer in the form of a
graph, usually as LIZ. plotted against volumetric flow rate, V in m3/h.
In the design of new systems, the required LIP. is calculated using the MEB:
..... (3.20)
The terms on the left hand side of the MEB can be plotted against flow rate and such a curve is called the
system curve. The intersection of the system curve and characteristic pump curve is known as the
operating point and indicates the flow rate that will achieved. This is illustrated in figure 4.2.
37
----J~ump
-~--
Pressure head
LIPa
curve
~perating point
- - - - - - - - - - - - - -- - - - - - - -- - - - -
System
~
curve~~~
~
AP
Ll
STV
--~-~c-c-c-~~~:~--- _______ _'Jj_ __ ---
ljl__
Flow rate
Figure 4.2
Pump curve and system curve
4_9
LITERATURE
1_
J. B. Franzini, Fluid Mechanics with Engineering Applications, ninth ed. Belfast: McGraw-Hill,
1997_
2_
L F_ Moody, "Friction Factors for Pipe Flow," Transactions of the American Society of
Mechanical Engineers, vol. 66, PP- 671-684, 1944.
3.
S. E. Haaland, "Simple and explicit formulas for the friction factor in turbulent pipe flow,"
Journal of Fluids Engineering, vol. 105, 1983.
4.
Hooper, W B, "The two-K method predicts head losses in pipe fittings", Chemical Engineering,
96, Aug 24, 1981.
5.
S. W. Churchill, "Friction factor equation spans all fluid-flow regimes," Chemical
Engineering, pp. 91-92, 1977.
38
5.
COMPRESSIBLE FLOW
5.1
INTRODUCTION
One-dimensional gas flows through nozzles, orifices and in pipelines are the most important applications
of compressible flows in chemical processing. With compressible fluids density and hence velocity may
vary considerably in a pipeline. In engineering applications liquids are considered to be incompressible.
Although the flow of gases and vapours are always compressible, for design purposes it is considered
incompressible if
..... (5.1)
where LIPsrv is calculated with the fluid density at the upstream reference point. Otherwise gas and
vapour flow in piping systems must be treated as compressible.
Isothermal or adiabatic conditions are assumed in the calculation of compressible pipe flows. Most real
situations are polytropic which increases the complexity of calculations tremendously. The isothermal
and adiabatic models for pipe flow fortunately provide bounds for the range of real behaviour and in
many cases the two models provide similar results.
In the case of flow through nozzles the flow is assumed to be adiabatic and reversible, or isentropic.
Ideal gas behaviour is assumed in the derivation of all the models. To consider non-ideal gases the
compressibility factor, z, can be included when deriving the equations.
In older literature graphical solutions for compressible flows were given, examples are the method of
Lobo, Friend and Skaperdas for isothermal flow and the Crane method for adiabatic flow. Graphical
solutions were very handy since the models are complicated to solve by hand. With the development
of computer software the emphasis in literature has now shifted to algorithms that can easily be
translated to computer code or implemented on a spreadsheet. The subsequent sections will therefore
concentrate on the fundamental equations and the necessary algorithms to solve them.
5.2
THERMODYNAMIC CONSIDERATIONS
The necessary thermodynamic principles used in the derivation of the various models are only
summarised here:
•
Thermodynamic properties of a gas :
cp
I
R ) CP> Cv' k=C > P > T' v='v
p
39
•
Ideal gas:
~
Pv
MP
p
~-
Pv n
•
~
0,
~
~
~
0,
11 ~
k
11
> k for compression
Definition of enthalpy:
~ 11
h
h
•
0
Adiabatic irreversible process:
n < k for expansion,
•
1
Adiabatic and reversible process (isentropic) :
11s
•
11 ~
Adiabatic process :
q
•
RT
constant
Isothermal process :
!1T
•
nRT
~ 11 +
+ pv
RT for ideal gas
Definition of specific heats :
l
cP ~
cv ~ ~;) v
dh
~
CpdT
d11
~
CiT
40
l
dh)
dT
p
for ideal gas
•
Relations between specific heats for ideal gases:
cp - c"
=
R
C = kR
k-i
p
c
"
5.3
R
=-
k-i
THE MACH NUMBER AND SONIC FLOW
The Mach number is a dimensionless number named after Ernst Mach:
II
Ma = -
..... (5.2)
c
where c is the sonic velocity.
This is the velocity at which a pressure wave will travel through a
compressible fluid. The sonic velocity is calculated with the following equations for an ideal gas:
..... (5.3)
ForMa< 1 the flow is termed subsonic, sonic flow occurs if Ma = 1 and supersonic flow if Ma > 1. Sonic
flow causes shock waves and vibrations in a system and should be avoided. Sections of possible sonic
flow should be identified and tested. Such sections include small diameters and areas of low density,
since u ~ 1/pd.
5.4
ISOTHERMAL PIPE FLOW MODEL
The isothermal model is used for flow in long, uninsulated pipelines. Consider the energy balance
applied between two points within a single diameter pipeline without a fluid mover:
dP
-
p
+
gdz +
UKE II
d11 +
oF
= 0
The following substitutions are now made:
•
•
•
1 for turbulent flow
the elevation term is neglected for gas or vapour flow
F is replaced with the Darcy equation
(J.KE =
41
..... (5.4)
Equation (5.4) now becomes:
dP
j'dL u 2
udu
=
p
..... (5.5)
+ ----
D
2
At normal pressures the viscosity of a gas or vapour is only a function of temperature and therefore the
Reynolds number as well as the friction factor can be considered constant in isothermal flow.
From the ideal gas law:
p
=
MP
RT
..... (5.6)
and the continuity equation:
II =
w
..... (5.7)
pA
the following expression for linear velocity is derived:
WRT
PAM
II---
..... (5.8)
2
_ 2
f 1dL
- - dP - -d11 + - pu2
11
D
..... (5.9)
Equation (5.5) is now multiplied by 21u 2 :
Substitution of (5.6) and (5.8) into (5.9) gives:
3.du +
u
f' dL
..... (5.10)
D
Equation (5.1 0) is now integrated between two points in the pipe:
p2
2
f PdP f !;d11
=
P1
L,
11 2
u1
42
+
r~
L,
dL
..... (5.11)
..... (5.12)
From continuity, for a constant diameter pipe:
Substitution of the ideal gas law for density into the above equation yields:
MP 1
RT
MP 2
--II
I
= - - 1 12
RT
=
Substitution of the above result into equation (5.12) gives the isothermal model:
L
5.6
----
ADIABATIC PIPE FLOW MODEL
The adiabatic model is most appropriate for shorter, insulated pipelines. The derivation starts with
equation (3.8):
..... (3.8)
The equation is applied between two points in a pipeline without the incorporation of a fluid mover. The
elevation term is neglected since the fluid is a gas and q~O since the flow is adiabatic:
43
II 2
l
..... (5.14)
+-
2
u2 2
-
ut 2
2c
2(h t - h2) = M p
(Tt - T2 )
=
..... (5.15)
where the right hand side of the equation is divided by molar mass to convert the units of Cp from
kJ/kmol. K to kJ/kg. K. Substitution of
T
=
PvM
R
and
cp
=
kR
(k-1)
into equation (5.15) gives:
u2
2
-
u2
l
=
2k (P v - P2v2)
k-1 l l
..... (5.16)
Substitution of
w
11 = -
pA
into equation (5.16) gives:
which can be rewritten as
2k
(k-1)
+ --(Pv) = C
2 2
..... (5.17)
where C is a constant evaluated from known conditions at point 1. From the above relation the
following can be derived:
44
P =
2
k-1(
W
)
2k Cp - pA
2
Multiplication by p and integration yields:
Now substitute for C :
2pdP
!
1
=
k- 1
2k
[ __!!'____
2 ( P22 - P2)
1
p2A2
1
2
1 + 21n p2
p1
)
..... (5.18)
Equation (5.5) is now used:
dP
p
fdL u 2
D 2
+ udu + - - - = 0
..... (5.5)
Multiply by p2 and replace pu with WIA:
2
w)
pdP+ (A du+
!'DA
( w)2 dL=O
2
45
..... (5.19)
Since pAu =constant from continuity, it can be shown thatpdu
(5.19) gives:
(w) -dp
2
pdP -
A
w) dL A
2
!' ( -
+ -
p
2/J
-udp. Substitution of this into equation
=
_
0
..... (5.20)
Integrating and assuming that f' is constant (this is not really true for adiabatic flow) :
2
J
pdP = (
2
2
2
W)
L
I
(
W)
p
A In"P;" - f D 2 A
1
..... (5.21)
I
Now set equation (5.18) =equation (5.21) and simplify:
(Aw) \/2p
1
_ !'.!:.. !._
lJ 2
( w)
2
=
2
!:.::i(
w)
( p;
4k A
_
p~
A
2( w) \n .':.!_ - !:.::i( w) P; - I
2 (
A
p2
2k
P~
A
- k- 1 ( p; - I + 2In p2 ]
2k
p;
p1
+ k-1. (1-p;]-
2k
.. j'!:... = k+ lin P2
D
..
!'.!:...
D
k
p,
= k+ 1 In P2
k
Pt
k-1
2kp2
2
Pt
2
2
p,
p~ ~ p;
46
l
r
P, ( A
w
p,
+ --2 (pl -p2) + -
+ (
2
p1(~)
( 2- 2)
W
p2
Pt
2
2
(p2 - p,)
( k-1
( A
2k + PtPt W
r)
Which leads to the adiabatic pipe flow model:
[
5.7
:. .fL
--D
=
k+1 lnP2 + ( 1
k
p1
FLOW CHOKING
In both isothermal and adiabatic flow in a pipe of constant diameter there is a limiting pipe length at
which flow choking takes place. This happens because as P 2 decreases along the pipe, p2 decreases
and since pu = constant, u increases. Since it is physically impossible for P 2 to drop to zero, there is a
choking of the flow that limits the mass flow rate. For isothermal flow this occurs at
Ma
=
1
Iii
and for adiabatic flow at
Ma
= 1
It is possible through successive calculations to plot a curve such as shown in Figure 2 for any assumed
flow and initial conditions where P 2 represents any pressure along the pipe at any distance x2 . In the
case of isothermal flow equation (5.13) is only valid for
1
Ma<-
/1(
whereas equation (5.22) for adiabatic flow apply equally well for supersonic flow.
Adiabatic flow with friction is also termed Fan no Flow. If a gas entering a duct is flowing at subsonic
velocity, friction will have the effect of accelerating the flow so that sonic velocity is approached;
likewise, if the flow at the entrance is supersonic, the gas will be decelerated, also approaching Mach
1. In each case, when Mach 1 is reached choking of the flow occurs.
Figure 1 shows the conditions along the pipe length for isothermal as well as adiabatic flows.
47
p
u
T
I
I
I
1
Distance along pipe
Figure 1
Conditions along pipeline for compressible flows
5.8
PRACTICAL APPLICATION OF COMPRESSIBLE PIPE FLOW MODELS TO PIPING SYSTEMS
5.8.1
THE MEB AND COMPRESSIBLE FLOW
The effect of variation in density on the MEB is as follows:
•
L1PEL is not relevant because it is zero for gases and vapours
•
L1PEP ~ P2 - P 1 as for incompressible flow
•
Calculation of L1Pcv is based on a compressible flow model (see Control Valve design)
•
L1PEQ calculation must take into account changes in density. The prorating equation becomes
L1PEQ
•
X
W2/p
Evaluation of L1P1 requires special calculation methods. Variations in density with changes in
pressure can not be ignored. Density variations also lead to variations in linear velocities for flow
in single diameter piping systems; they are the cause of additional pressure changes. Friction
pressure losses and these additional kinetic energy related pressure losses are combined as a
MEB term L1PJ.K· L1P1K is calculated with equations (5.13) or (5.22), this will be discussed in the
following section.
48
5.8.2
APPLICATION OF THE COMPRESSIBLE PIPE FLOW MODELS
The equations are only applicable between two reference points inside a single diameter pipe. In the
case of piping systems, evaluations must be done incrementally for different diameters. Frictional losses
due to pipeline fittings which do not significantly reduce the pipe cross-sectional area may be added to
the velocity head term, ]LID, otherwise incremental evaluation of the sections upstream and
downstream of the restriction is necessary.
The MEB is applied between two points within a single diameter pipeline:
/';.PEP + Mf,K
p2 - PI
+
=
0
Mf,K = 0
:. Mf,K = PI - p2
Either P 1 or P2 will be known, the unknown will be calculated using one of the models.
Consider for illustration purposes the calculation of the storage tank pressure (1'2) for the flow system
shown in Figure 2. 1'1 is known.
FIGURE 2
Compressible Flow
The first step will be calculation of 111}; 1 _2 and then checking for compressible flow according to !JPsr,IP1
> 0, 1. Assume the control indicates compressible flow. Incremental evaluation is necessary:
Pipe entrance :
-iJPEP
.. -(P,-PJ
:. P,
A MEB over the pipe entrance will give P,
ilP1 + iJPKE
ilP1 + ilPKE
P1 - ilPr ilPKE
49
Normally, for the pipe entranceLIPSTI/1'1 « 0,1 and LIP1andLIPKE may be calculated as for incompressible
flow; confirm after the evaluation.
Pipeline and fittings :
A MEB over the pipe will give Pi
LJPJ,K
P;- LJPJ.K
It should be noted that in this evaluation LIPSTI/P, may be < 0,1; calculation of Pj will then not require a
compressible analysis.
Pipe exit :
A MEB over the pipe exit will render P 2
-LJPEP
Because L1P1
p2
All components over which changes in diameter and/or flow rate occur must be analysed incrementally!
5.8.3
ALGORITHMS FOR SOLUTION OF COMPRESSIBLE FLOW MODELS
ISOTHERMAL MODEL
..... (5.13)
•
Calculation of APJ.K
Either P1 or P2 will be known, the unknown will be calculated using a trial and error procedure or the
"solver" function of a spreadsheet. The term on the right hand side of the equation can be neglected as
a first approximation. A typical algorithm for this calculation is given in appendix D1.
•
Calculation of Flow Rate
In the case of a flow rate calculation a trial and error procedure is necessary to find the friction factor.
The von Karman equation can be used as an initial estimate for the friction factor. An algorithm for this
calculation is given in appendix D2.
50
AD/ABA TIC MODEL
..... (5.22)
•
Calculation of APr,K
Either P 1 or P2 will be known, the equation is in terms of p 1 and P2 and therefore the ideal gas law must
be used to calculate the relevant pressures. Since temperature is not constant, T2 must also be
calculated to find P2 from p 2 . An algorithm is given in appendix 03.
•
Calculation of Flow Rate
In the case of a flow rate calculation a trial and error procedure is necessary to find the friction factor.
The von Karman equation can be used as an initial estimate for the friction factor. An algorithm for this
calculation is given in appendix 04.
5.9
COMPRESSIBLE FLOW THROUGH NOZZLES
This theory is highly relevant to the design of relief valves or bursting discs which are often incorporated
into pressurised systems in order to protect equipment and personnel from the dangers which may arise
if equipment is subjected to pressures in excess to design values.
5.9.1
CONVERGING NOZZLES
-------
I
------------------------
Throat
Inlet
Figure 3 Converging nozzle
51
Isentropic conditions is assumed since frictional loss is neglected (reversible) and the area of heat
transfer is small (adiabatic).
Equation (5.16), derived previously, is used:
u
2 2
u 2 ~ 2k (P v I
k-J
P2v2)
..... (5.16)
I I
The following relation is also used:
from which the following is derived:
v :::: v
2
I(
p
- 2
p
)-Ilk
..... (5.23)
I
Substitution of (5.23) into (5.16) and expressing in terms of densities :
u2-u2
2
I
1/ 2
2
112
2
-
ut 2
~
-p I -2k
-( I - ( -p 2 ) (k-IYk)
Pt k-1
PI
..... (5.24)
The following version of equation (5.24) is also valid:
..... (5.25)
The velocity of approach (u 1) can be considered neglible compared to the outlet velocity:
52
..... (5.26)
Using equation (5.26) together with the following relation:
A2
~
w
makes it possible to find the required nozzle throat area for the pressure to be reduced to P 2•
Noting that
equation (5.26) can be expressed in terms of the Mach number:
(
112) 2--Ma 2-- 2 ((pl)(k-l)lk1)
c2
k-1
..... (5.27)
P2
The velocity at the throat, u2 , depends on the ration P/P2- If there is a large enough pressure differential
betweenP1 and the back pressure Pb, sonic velocity will occur in the nozzle throat. With further increase
in the pressure differential, the flow rate will increase {due to the density increase) but the velocity at the
throat will remain sonic.
During subsonic flow, P2 ~ P• ~ back pressure, but if the flow in the throat is sonic 1'2 :> P•.
Assuming sonic flow, letMa = 1 in equation (5.27):
(;~r-1)/k
k+l
2
53
Equation (5.28) is the critical back pressure for sonic flow:
p
crit
2
p1
..... (5.28)
For subsonic flow
..... (5.29)
P2 1 1'1 can never be smaller than P2 / P/nt
Calculation of Flow Rate for converging nozzles
A converging nozzle is a very handy device for the measurement of flow rate.
calculated as follows:
_3/!_(( P
P2
p2 k-1
W
=
A2
2_.
kp
k-1
1
1) (k- )'k _
p2
1)lk- 1)
((p1)(k2p2 p
2
and because
p 1v1k
=
it can be shown that
54
p 2 v2k
Flow rate can be
1)
..... (5.30)
Substitution of the above expression into equation (5.30) yields:
W
=
2_.kp
A2
k-1
1
((p2)2/k(p2)k;
)
1p1
p
p
I
..... (5.31)
1
The maximum flow rate occurs at sonic flow in the throat, therefore substitution of equation (5.28) into
(5.31) yields:
2
W
max
=A
2
Pp
1 1
or
-
)(k+1)1(k-l)
k+
. l
'
W
mox
=
A2P1
{f;
kM( -.2) (k+1)1(k-1)
R
..... (5.32)
k+ 1
Note that the square root expression on the right hand side of equation (5.32) only depends on the
properties of the gas. By measuring the pressure and temperature in the tank, the flow rate through
the nozzle can be calculated.
At the point where 1'2 I P 1 reaches the value of PjP 1 ,·rir, the flow in the nozzle throat is sonic. As P 1 is
increased beyond the threshold point, P jl' 1 maintains the value of P JP
However, W increases directly with 1'1 , as shown in equation (5.32).
5.9.2
1 crir
and u remains sonic.
CONVERGING-DIVERGING NOZZLES
The flow through a converging-diverging nozzle is shown in figure 4. If a diverging section is placed
after a converging nozzle, it is possible to attain supersonic velocities in the diverging section if sonic
flow exists in the throat. The gas will continue to expand in the diverging section to lower pressures and
55
the velocity will continue to increase. If the velocity at the throat is not sonic, the gas will behave in the
same manner as a liquid : it will accelerate in the region up to the throat and decelerate in the diverging
region. This is shown by the dashed lines ABO in figure 4.
Suppose the back pressure P, in figure 4 is reduced gradually while P 1 remains constant. Then P1 = P,
and the pressure at the throat decreases while the velocity at the throat increases until the limiting sonic
velocity is reached. The pressure plot is ACE. If the back pressure is now further reduced to H, the
pressure plot is ACFGH; the jump from F toG is a pressure shock, or a normal shock wave (normal to
the approaching flow), which is analogous to the hydraulic jump, or standing wave, often seen in open
channels conveying water. Through the shock wave the velocity is reduced abruptly from supersonic
to subsonic, while at the same time the pressure jumps as shown by the lines FG, F'G' and F"G". The
flow through the shock wave is not isentropic, since part of the kinetic energy is converted to heat.
Further reduction of the back pressure causes the shock wave to move further downstream until at some
given value H"' the shock wave is located at the downstream end of the nozzle. If P, is lowered below
the level of H"' the shock wave occurs in the flow field downstream of the nozzle exit. Such flow fields
are either two or three dimensional and cannot be described by the foregoing one-dimensional
equations.
Figure 4 Flow through a converging-diverging nozzle
56
If the back pressure is lowered to H"", the flow will proceed isentropically to supersonic throughout the
entire region downstream from the throat, the velocity will increase continuously from 1 to its maximum
value at 3 and the pressure will drop continuously from 1 to 3. As long asP, is above H"" then P 3 ~ P•
; but if P, drops below H"" then P
3
> P • and supersonic flow occurs through the entire length of the
divergent portion of the nozzle.
If the back pressure is above E, the flow rate through the nozzle is given by equation (5.31). In this
instance theP2 of equation (5.31) must be replaced by the P , of Figure 3. If the back pressure is below
E, critical pressure, as defined by equation (5.29), will occur at the throat and the flow rate will be given
by equation (5.32).
If P 1 is increased, the sonic velocity may be shown to remain unaltered, but since the density of the gas
is increased, the rate of discharge will be greater. The converging nozzle and the converging-diverging
nozzle are alike insofar as discharge capacity is concerned.
The only difference is that with the
converging-diverging nozzle, a supersonic velocity may be attained at discharge from the device, while
with the converging nozzle, the sonic velocity is the maximum value possible.
5.10
LITERATURE
1.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, vol. 6, 2 ed: Pergamon,
1993.
2.
J. B. Franzini, Fluid Mechanics with Engineering Applications, ninth ed. Belfast: McGrawHill, 1997.
3.
R. H. Perry, Green, W.G., Perry's Chemical Engineers' Handbook, 7th ed: McGraw-Hill, 1997.
4.
Winnick J., Chemical Engineering Thermodynamics. New York: Wiley, 1997.
57
6.
NON-NEWTONIAN FLOW
Gases and simple low molecular weight liquids are all Newtonian and viscosity may be treated
as constant unless there are significant variations in pressure and temperature.
Fluids which do not adhere to Newton's viscosity law are classified as non-Newtonian. Examples
are colloidal suspensions, emulsions like certain paints, sewerage sludge, melted polymers and
melted metals. Non-Newtonian fluids deviate in different ways from Newton's viscosity law.
Viscosities are functions, not only of temperature, but also of shear stress and shear rate. In
evaluations use is made of so called apparent viscosities.
Non-Newtonian flow are much more likely to be laminar due to high apparent viscosities
compared to the viscosities of simple Newtonian fluids. In order to predict the transition from
laminar to turbulent flow it is necessary to define a modified Reynolds number. The transition
from laminar to turbulent flow is not always sharp as in the case of Newtonian flow.
The terms LiP,, LiPEv L!PEP, L1Pc"" L!PEQ and LiPKE of the MEB are calculated in the usual way.
Special correlations were developed for the calculation of L1P1
6.1
RHEOLOGY
Rheology is the science concerned with the flow of both Newtonian and non-Newtonian fluids.
A Newtonian fluid at a given temperature and pressure has constant viscosity which does not
depend on shear rate and obeys Newton's viscosity equation :
T =
ou
J.t-
0)'
..... (6.1)
The apparent viscosities of non-Newtonian fluids may depend on the rate they are sheared and
on their previous shear history. At any position or time in the fluid the apparent viscosity is
defined as the ratio of the shear stress to the shear rate at that point:
T
J.l
=-a
8u/8y
..... (6.2)
When the apparent viscosity is a function of the shear rate, the behaviour is said to be shear-
dependent; when it is a function of the duration of shearing at any time it is said to be limedependent. Any shear-dependent fluid must to some extent be time-dependent because the
apparent viscosity does not change instantaneously, in many cases the effect of timedependence is negligible.
Typical forms of curve of shear stress versus shear rate are shown in figure 6.1. Such a plot is
known as a rheogram since it represents the rheological properties of a fluid.
58
/
Newtonian
~""""""Bingham-plastic
Shear-thickenin
_"/shear-thinning
';;:..-:..----
,#/>;;;:::>' '
ouloy
Figure 6.1
Shear stress versus shear rate
A general plot of apparent viscosity versus shear rate is shown in figure 6.2. This plot describes
the behaviour of dispersions, emulsions, polymer solutions or slurries in general. At low enough
shear rates the viscosity is constant and relatively high (Newtonian behaviour). As the shear rate
increases the viscosity begins to fall (shear-thinning). Eventually the curve becomes a straight
line when plotted on log-log axes (power-law region). At even higher shear rates the viscosity
usually begins levelling out, falling towards a constant level.
f.l,
Cross model
Sisko model
Power law model
~
107
7
-,~~~
105
1000
~
10
0.1
0.001
L _ _ o L _ _ _ l_
10"'
Figure 6.2
10~
_ _ l_ _
L _
_ I __
_ L _ _ L_
0.01
~
_L__
_L
100
Typical flow curve for non-Newtonian fluid
59
~
------'---
104
ou!oy
Two exceptions to the general behaviour described by figure 6.2 are possible. First the existence
of a yield stress (Bingham plastic fluids) and secondly the appearance of shear thickening
(dilatancy) at the high end of the curve, both shown in figure 6.3.
Time dependency is another exception to the general viscosity curve. The behaviour described
so far relates to steady state behaviour. Some materials take a long time to achieve steady state,
and during the unsteady state period they can either show a continual decrease (thixotropy) or
an increase (rheopexy) in viscosity when sheared at constant shear rate/stress.
Yield stress
107
10'
1000
10
0.1
_j _
10
Figure 6.3
__l__
_)___
_[__
1000
100
10'
'&u/'&y
Yield stress and shear thickening behaviour
Apart from typical viscous behaviour described above some liquids also show the elastic
response usually associated with solids. Materials which behave like this include concentrated
solutions of high molecular weight polymers, shower gels, shampoos and polymer melts.
The ideal elastic solid obeys Hooke's law in which the relation between distortion and stress is:
T =
Gdx
dy
..... (6.3)
where G is Young's modulus and dxldy is the ratio of the shear displacement of two elements to
their distance apart. Materials that exhibit some properties of both a solid and a liquid are termed
viscoelastic.
60
The viscosity function shown in figure 6.2 is well described by the Cross model:
..... (6.4)
The Cross model can characterise the complete flow curve if the fluid does not show a yield
stress of shear thickening. For the higher shear rates of more interest to the chemical engineer,
equation (6.4) simplifies to:
..... (6.5)
A simple redefinition of some of the terms in equation (6.5) allows a rearrangement to give the
Sisko model:
..... (6.6)
where m
~
l-11 and k
~flo
I
K"'.
If the extrapolated viscosity at infinite shear rate is negligible compared to the viscosity at the
shear rate of interest, the Sisko model reduces to the well-known power-law model:
f.l
a
=
du n-1
kdy
..... (6.7)
du"
kdy
..... (6.8)
which can also be expressed as:
1 =
In equation (6. 7) when
n> I,
f.la
n<l,
f.la
n~1, f.la
increases with increase in shear rate and shear thickening behaviour is described
decreases with increase in shear rate and shear thinning behaviour is described
is constant and equal to the Newton's viscosity of the fluid.
The dimensions fork are ML'1T"·2 .
A further simplification to the Sisko model is when n=O:
..... (6.9)
which can also be written as
61
..... (6.10)
where T0 is the yield stress and !lP is the plastic viscosity. This equation is known as the Bingham
equation.
Some materials give more complex behaviour and the plot of shear stress against shear rate
approximates to a curve, rather than a straight line with an intercept T0 on the shear stress axis.
The following equation, known as the generalised Bingham equation or Herschei-Bulkleyequation
can then be used:
..... (6.11)
Bingham shear thickening or Bingham shear thinning behaviour can be described using equation
(6.11).
6.2
LAMINAR FLOW CORRELATIONS FOR L1P1
A number of simple situations can be described mathematically, usually for shear-thinning
behaviour. In the case of laminar flow the correlations can be derived from first principles using
the flow curve equation (relation between shear stress and shear rate) and the definition for shear
rate. Through substitution and integration the relation between flow rate and frictional pressure
drop can be found. If the flow curve equation is too complicated it might be necessary for a
numerical integration. For comparison the Newtonian flow correlation is first shown.
NEWTONIAN FLUID
Hagen-Poiseulle law:
..... (6.12)
In order to make use of the Darcy-Weisbach equation, which is always valid, the following relation
for the Darcy friction factor is derived:
f'
·
=
64
..... (6.13)
Re
POWER-LAW FLUID
In this case the flow curve equation is more complicated, but still simple enough to derive
analytical equations. Relation between flow rate and pressure drop :
62
APf
~
( 6nn+2)" 4 k L u" JY<
n+ll
..... (6.14)
The above equation reduces to the Hagen-Poiseulle equation for n ~ 1 and k ~ p.
Using the Darcy-Weisbach equation the following relation for the friction factor is derived:
!'
6n+2) "ku n-lD -n
~ 8 --
(
p
11
..... (6.15)
For a Newtonian fluid the friction factor is a function of Reynolds number. In the case of nonNewtonian flow the Reynolds number changes with shear rate since it is a function of the
viscosity. It is therefore difficult to define an appropriate Reynolds number. Metzner and Reed
(1955) defined a Reynolds number Re,m for a power-law fluid so that it is related to the friction
factor in the same way as for a Newtonian fluid. It is derived by substituting equation (6.15) into
equation (6.13). The following expression is then found after simplification:
Re 1111
'
_
-
8
(
2
-11-) "pu -"D"
6n+2
k
..... (6.17)
The transition value is approximately the same as for Newtonian flow, although streamline flow
may in some cases persist to somewhat higher values. Setting n~l in equation (6.17) leads to
the standard definition of the Reynolds number.
The effect of the power law index on the velocity profile is that it is flatter for a shear thinning fluid
(n < 1) compared to the Newtonian parabolic profile and sharper for a shear thickening fluid (n
> 1).
BINGHAM PLASTIC FLUID
In the case of a Bingham-plastic fluid the cross-section of flow in a pipe can be considered in two
parts:
1)
A central unsheared core in which the fluid is all travelling at the centre-line velocity.
2)
An annular region separating the core from the wall over which the whole of the velocity
profile is concentrated.
63
The relation between flow rate and pressure drop is derived by considering the two parts of the
flow mentioned above separately and then adding them. The relation is much more complicated
than for a power-law fluid:
with
..... (6.18)
For a Newtonian fluid X and ~0 is zero. Equation (6.18) is sometimes referred to as Buckingham's
equation.
GENERALISED EQUA T/ONS
Fluids whose behaviour can be approximated by the power-law or Bingham plastic models are
special cases. The rheology are frequently very complex and simple algebraic equations cannot
be fitted to the flow curves. A general method for time-independent fluids in fully developed flow
is given here. A general model with parameters that can be measured for any fluid is used.
The following general relation between pressure drop and flow rate can be derived:
l!.P
!
~
4k'L (
D
8u)
n'
D
..... (6.19)
where k' and n' are generalised rheological parameters. These parameters are widely used in the
literature on rheology. Values for the generalised parameters for various fluids are given in
appendix E.
It can be shown that for a power-law fluid:
..... (6.20)
An equation for the friction factor can be found, similar to equation (6.15):
f'
.
~
1
1
8k
pu2
(
!"_)
"
D
..... (6.21)
The Metzner and Reed Reynolds number can also be generalised:
ReMR
~
pu 2 -n 'nn I
gn 1-lk I
64
..... (6.22)
6.3
TURBULENT FLOW CORRELATIONS FORLIP1
In the case of turbulent flow the relations for pressure drop can not be derived from first
principles, empirical relations are used. Many correlations have been published for different types
of non-Newtonian fluids.
The following relation proposed by Yoo (1974) gives values for the friction factor accurate to
within ±1 0%. The friction factor is expressed in terms of the Metzner and Reed generalised
Reynolds number and the power law index, n:
..... (6.23)
The above equation should be used with caution, particularly if the fluid exhibits any plastic
properties.
Large safety factors are recommended (F ~ 1,5).
6.4
LITERATURE
1.
H. Barnes, "Rheology for the chemical engineer," The Chemical Engineer, June 24, 1993.
2.
D. C. H. Cheng, "Pipeline design for non-Newtonian flow," The Chemical Engineer, vol.
525, 1975.
3.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, vol. 6, 2 ed:
Pergamon, 1993.
65
7.
MULTIPHASE FLOW
The complexity of multiphase flow is so great that design methods depend on an analysis of the
behaviour of such systems in practice and, only to a limited extent, on theoretical predictions.
Some of the more important systems are:
Mixtures of liquids with gas or vapour
Liquids mixed with solid particles (hydraulic transport)
Gases carrying solid particles (pneumatic transport)
Multiphase systems containing solids, liquids and gases
For all multiphase flows it is important to understand the nature of the interactions between
phases and how these influence the flow patterns- the ways that the two phases are distributed
over the cross-section of the pipe. Pressure drop will depend on the flow pattern as well as the
relative velocity of the phases - slip velocity. This slip velocity will influence the hold-up, the
fraction of the pipe volume which is occupied by a particular phase.
In the flow of a two-
component mixture, the hold-up of a component will differ from that in the mixture discharged
at the end of the pipe because, as a result of slip of the phases relative to one another, their
residence times in the pipe will not be the same.
Only liquid-liquid and gas(vapour)-liquid multiphase flows will be considered in this course.
7.1
LIQUID-LIQUID FLOW
Two phase liquid-liquid flow is encountered when two liquids which are relatively immiscible flow
in the same piping system. The method of Woods and Dukler is suitable for liquid-liquid flow.
It is less accurate for vapour-liquid flow but is however convenient for certain estimations in the
latter case. The same equations and methods as for single phase flow are used but with
weighted average values for physical properties, volume flow rate and linear velocity:
I
=
Prp
PrP
L
xi
pi
I:~
=
I
f1rp
'LWi /pi
=
Lxi
lli
66
..... (7.1)
..... (7.2)
..... (7.3)
~w
vTP ~ --'
p
..... (7.4)
TP
u
TP
~
~wi
p A
--
..... (7.5)
TP
..... (7.6)
7.2
VAPOUR(GAS)-LIQUID FLOW
Two phase vapour-liquid flow is encountered in steam-condensate lines, reboiler lines and lines
from partial condensers. Three phase flow, with two liquid phases and a vapour phase, is
relatively common in the petroleum industry.
The MEB in its general format is applicable, special methods are used to predict AP1 . A fluid
mover may be part of a liquid-liquid two phase system but will not be mounted in a vapour-liquid
line; if a fluid mover is required for such a system, it will be placed in a line section where the
flow is still single phase. Similarly a control valve will also not be used in the two phase flow
section of a pipe system.
7.2.1
FLOW REGIMES AND FLOW PATTERNS
Vertical and horizontal flow patterns differ, in the case of vertical flow axial symmetry exists.
Principal characteristics of the flow patterns are described in figure 7.1.
The regions over which the different types of flow occur are conveniently shown on a flow
pattern map in which a function of the gas flowrate is plotted against a function of the liquid
flowrate and boundary lines are drawn. The distinction between the flow patterns are not clear
cut and several workers have produced their own flow maps. Figure 7.2 shows a flow pattern
map prepared by Chhabra and Richardson.
67
c ...
~
Bubble flow
Churn/Froth flow
Bubble flow
Plug flow
Stratified flow - wavy flow
~~d
Slug flow
Slug flow
t====l]
Annular flow
Annular flow
Upward vertical flow
Mist/spray flow
Horizontal
Description
Typical velocities
mls
flow regimes
Liquid
Vapour
Bubble flow
Gas bubbles dispersed throughout the liquid
1.5-5
0.3-3
Plug flow
Plugs of gas in liquid phase
0.6
<1.0
Stratified flow
Layer of liquid with layer of gas above
<0.15
0.6-3
Wavy flow
As stratified but with a wavy interface due to
higher velocities
<0.3
>5
Slug flow
Slug of gas in liquid phase
Wide range
Annular flow
Liquid film on inside walls, gas in core
>6
Mist flow
Liquid droplets dispersed in gas
>60
Figure 7.1
Flow patterns in two-phase flow
68
1.._,:
Slug flow causes unstable flow conditions with vibrations and is highly undesirable. In the design
of piping systems for two phase vapour-liquid flow, a check for possible slug flow must always
be done. If slug flow prevails, other pipe diameters must be considered. It is desirable to design
so that annular flow still persists at loadings down to 50% of the normal flow rates. Mist flow
should also be avoided since once mist flow is reached there is virtually no way of returning to
any other flow regime.
7.2.2
PRACTICAL METHODS TO EVALUATE AP1
The most widely used method is that proposed by Lockhart and Martinelli and later modified by
Chisholm.
The method requires the evaluation of friction pressure drop for the liquid phase as if it were the
only fluid in the system (LIPt.Ll and also the evaluation of friction pressure drop for the vapour
phase as if it were the only fluid in the system (LIPt.al· The two-phase pressure drop (APt.rP) is
taken as LIPJ.L or LIPr.a multiplied by some factor <D0 2 or <DL2 where
69
f>,.pf,TP
<1>2
G
..... (7.7)
MG
Mf,TP
D.PL
2
= <l>L
..... (7.8)
<I>G2 and <I>~.2 are given graphically as functions of the hold-up parameter X, where:
..... (7.9)
The graph is given in appendix F.
Reynolds numbers for distinguishing between turbulent and viscous flow are calculated as if
only the fluid of the case in question flowed in the pipe.
Strictly speaking the method is only applicable for isothermal, non-flashing, incompressible flow
in horizontal pipe. It was shown that the method renders relatively low values for annular flow
and large values for stratified, wave and slug flow.
Chisholm has developed the following relation between <l>1. and X:
c
+- +
x
1
x2
..... (7.10)
where c has a value of 20 for turbulent/turbulent flow, 10 for turbulent liquid/laminar gas, 12 for
laminar liquid/turbulent gas and 5 for laminar/laminar flow.
7.3
LITERATURE
1.
R. Kern, Piping design for two-phase flow, Chem Eng, 145, June 25, 1975.
2.
W.W. Blackwell, Calculating two-phase pressure drop, Chem Eng, 121, Sept 7, 1981.
3.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, vol. 6, 2 ed:
Pergamon, 1993.
70
8.
OTHER TYPES OF FLOW
8.1
NON-ISOTHERMAL FLOW
Changes in temperature over a piping system may be gradual or stepwise.
8.1.1
GRADUALAT
Gradual temperature changes occur when hot or cold pipelines are not effectively insulated and
in tubes of shell and tube heat exchangers. The normal MEB is used with the following
modifications:
FRICTION PRESSURE LOSS
A nonisothermal correction factor is incorporated:
2
62544fL W q> kPa
pds
..... (8.1)
Physical properties are evaluated at the mass temperature:
rp ~ (fliflJ"'
For laminar flow m = 0,25 and for turbulent flow m = 0.14. Subscript)w refers to wall conditions.
The evaluation of mass and wall temperatures are dealt with in heat transfer literature.
OTHER TERMS OF THE MEB
LiP,, AP8 Q. LiPEP and APcv are calculated as for isothermal systems. Strictly speaking LiPEL must
be calculated by integration. In most applications variation in density is ignored and the
calculation is based on the density at the upstream reference point. LIPKE is calculated as A(X)
with properties and other variables evaluated at the conditions of the two relevant reference
points.
8.1.2
STEPWISE AT
Sudden changes in fluid temperature are normally caused by heat exchangers which form part
of the piping system. LIP,, LiPEpLiPcv and LiPKE are calculated as discussed in 2.4.1.2. In
calculations of LiP!!Q for heat exchangers non-isothermal flow is taken into consideration. Strictly
speaking, in prorating for other flow rates, densities should not be taken as constant; constant
densities are however often acceptable for design calculations. For the evaluation of L1P1 and
APEL the system is subdivided into sections of isothermal flow; then
71
iJP1
iJPEL
8.1.3
1.
~
I:iJP!.,
~ 1L1PEL,i
LITERATURE
K.N. Murty, Assessing effects of temperature and flow rate of an incompressible fluid,
Chem Eng, 101, Jul25, 1983.
8.2
PULSATING FLOW
Pulsating flow is the result of fluid being moved by reciprocating fluid movers. See figure 8.1.
w
Figure 8.1
Reciprocating fluid mover
Many variations of cylinder, piston, valve and drive mechanisms have been designed. In fluid
dynamics it is necessary to distinguish between single acting (single piston or plunger with only
one delivery per cycle), double acting (single piston or plunger with two deliveries per cycle);
simplex (one cylinder), duplex (two cylinders), etc. From a flow dynamic point of view double
acting simplex = single acting duplex because both render two deliveries per cycle.
72
Cylinder movement in the case of crankshaft movers is simple harmonic:
ll
s,m
Ol
11s,max
us,min
=
=
Oll'
cos
e
=
Oll'
cos rot
2 1tN rad!s
60
Oll'
= 0
The continuity equation gives a correlation between linear velocity of the cylinder and linear
velocity of an incompressible fluid in the pipeline; for simplification the subscript)p for pipeline
is ignored:
The concept of average linear velocity still applies and is correlated with the production rate:
W = pV = puA
puA = Pllvll ,A,
8.2.1
FRICTION PRESSURE LOSS FOR CRANKSHAFT MOVERS
Because 11,,, varies between zero and u '·"""' 11, and .dP f.m also vary between zero and their
maximum values. Average values are however required for application in the MEB.
GASES AND VAPOURS : For purposes of piping system design calculations it is assumed
that the compressibility of gases and vapours to a large degree absorbs the pulsating action.
Friction pressure losses are calculated as for non-pulsating systems. Where relevant, a safety
factor of 1.3 is recommended.
LIQUIDS: No absorption of the pulsating action takes place and the approximation used for
gases and vapours is not valid. An equation can be developed to correlate the average friction
pressure drop of pulsating flow with the equivalent friction pressure drop of non-pulsating flow.
Consider the fluid mover shown in figure 8.1:
73
us,m
..
oh 2cos28
2
us,m =
:.
X
2
=
=
corcos8
:::::
ro 2r 2(1-sin28)
lf/(11,~,)
=
x2
ro2r\l-.:...)
r
=
lfl(u,;,) = lfl(M'f,m)
The correlation is parabolic as shown in figure 8.2.
-
-
-
-
~- -~~----
-
-=-=--.:::::-___ -
-~-
-
-
-
-
/
r
Figure 8.2
Correlation between LiP;: m and X
area beneath parabokt = 3. ( area
2r
3
=
2r
3_( !<.Pf,max X 2r)
3
L1P1 .,~
~f rectangle)
2r
can be calculated with one of the forms of the Darcy equation:
!<.P[,max
f'L . _P) ( J]vrorA,)2 kPa
( D
2000
A
74
X
=
=
1
fL . ~ .
D
2000
wrA
2
/I.Pf,ass
where Fp
1l,
)
2
'l wrA
"uA '
)
2
'l wrA
3(
uA
-
=
kPa
'
uA
(
non-puls (
2
"
kPa
2
'
=
)
..... (8.2)
pulsating factor
Where relevant the normal safety factor of 1.2 is used. Calculation of pulsating factors is
dependent on the type of fluid mover.
MOVERS WITH ONE DELIVERY STROKE PER CYCLE
u = 'l,ll ,A ,I A
us =
F = .3_ ( 'l,wrA,IA)
P
3
II
2
2rA/f - 2rN
--60A,
60
2
2(
2
= .3_ ( 'lvwrA,IA) =
rur) - -2
3
Tj.,II,A,/A
3 u,
3
2nN
--·r
2
60
2rN
-60
.. F =
p
3.3
n2
=
75
6.58
..... (8.3)
MOVERS WITH TWO DELIVERY STROKES PER CYCLE
11
=
'
2 ( 2rA,N) = 4rN
60A s
60
..... (8.4)
It follows that for the same flow rate friction pressure loss is much less for movers with two
delivery strokes per cycle than for movers with only one delivery stroke per cycle.
Air chambers may be included in delivery lines to reduce friction pressure losses. The use of
air chambers do not eliminate pulsating actions.
The following pulsating factors are
recommended in the presence of air chambers.
ONE DELIVERY STROKE :
TWO DELIVERY STROKES:
8.2.2
FP = 1.3
FP = 1.1
FRICTION PRESSURE LOSS FOR OTHER MOVER TYPES
Steam driven movers are in general use. Steam buffers the pulsating action. Pulsating factors
as for air chamber systems are recommended.
8.2.3
OTHER TERMS IN THE MEB
LiPEL is calculated as for non-pulsating flow.
LiPEP may pulsate; normally based on average values.
LiPcvand LiPEQ are normally not relevant.
LiPKE is calculated as Fyl(X). Li(X) is calculated with the same equations as for non-pulsating
flow. Pulsating factors are the same as for friction pressure losses.
LiPa is calculated with the MEB.
8.2.4
LITERATURE
1.
T.L. Henshaw, Positive displacement pumps, Chern Eng, Sept 21, 1981.
2.
J.D. Ekstrum, Sizing pulsation dampeners for reciprocating pumps, Chern Eng, 111,
Jan 12, 1981.
76
9.
PIPING SYSTEM DESIGN
It embraces selection of construction materials and dimensional design of pipelines, fittings,
components, fluid movers, instruments for measuring and control of flow rates. It enables the
writing of specifications for buying and construction.
Different engineering disciplines are
normally involved; this course concentrates on the responsibilities of chemical engineers.
9.1
PIPELINES, FITTINGS AND COMPONENTS
For pipelines it embraces design of wall thickness and optimal economic diameter. Design of
wall thickness is based on mechanical engineering principles; designs for wall thickness and
diameter are however interrelated and are normally performed by the same engineer. A trialand-error approach is required. A typical procedure is an estimation of diameter, selection of
an industrially available pipe (nominal and outside diameter known), design of wall thickness and
check with optimum diameter criteria.
Nominal diameters of fittings and components are taken the same as that of the associated
designed pipe section; the maximum pressure which may occur determines the class type which
fixes dimensions like wall thickness. Design of a check valve diameter however requires an own
procedure.
9.1.1
WALL THICKNESS
The wall thickness must be sufficient to prevent failures which will result if stresses in the system
exceed the yield strength of the construction material. It is normal practice to allow for a certain
thickness of material loss due to corrosion. The system is designed to withstand the maximum
pressure which may be encountered.
The maximum pressure can be calculated with the MEB; at the beginning of the design, pipe
diameters necessary for MEB analyses are not available and a trial-and-error approach is
required. Any operation which may cause momentary pressure increases, like pulsating flow
and the occurrence of water hammer must also be taken into consideration. Water hammer is
only relevant with liquids and can be caused by the sudden closing of a valve or failing of a
pump; momentary pressure increases are functions of fluid momentum.
Momentary pressure increases due to water hammer may be calculated from basic principles:
AZ = af::..u
-Ill
g
..... (9.1)
D.
a
=
velocity of a pressure shock wave
=
d )
1
p(+ -)
(
K
tE
-0,5
77
K is the compressibility modulus of the liquid (N/m 2 ) and E is the elasticity modulus of the
construction material of the pipe (N/m2 ).
The momentary pressure increases may also be obtained from tables in literature. See table 9.1.
Table 9.1 Water hammer allowances
for above ground cast-iron pipe
Pipe diameter in
Water Hammer
inches
allowance, psi
4 to 10
120
12 to 14
110
16 to 18
100
20
90
24
85
30
80
36
75
42 to 60
70
The maximum pressure to be used for wall thickness design is the sum of the maximum pressure
according to the MEB and any momentary pressure increase which may occur.
The method of wall thickness design is also a function of the type of construction material. In the
case of brittle materials, tables are available which correlate maximum system pressure and wall
thickness.
In the case of ductile materials wall thicknesses are calculated with the following mechanical
design equation
t.=M (
nun
p
fmin
M
s
E
y
c
=
=
=
=
=
=
=
Pdo
2(SE + PY)
+C)
..... (9.2)
maximum system pressure (gauge pressure)
minimum wall thickness
fabrication tolerance factor; typical value for mild steel pipes is 1.125
allowable stress (Perry); _ 0,5 YS
joint quality factor (Perry)
factor which deals with toughness of material (Perry)
sum of allowances for corrosion, erosion and any thread or groove depth
(experience)
78
The equation is derived from basic principles; thus units of the variables in the right hand side
determine the unit for tmiw After evaluation of tmin an industrially available pipe is selected for
Which I
:>c I min·
9.1.2
DIAMETER
The aim is to establish the optimum economic diameter of a piping system for the required flow
rate. Under favourable conditions for LIFE/' and LIPEt it may be possible to obtain the desired
flowrate in the absence of a fluid mover. It is most important to distinguish between systems
without fluid movers and systems with fluid movers.
WITHOUT A FLUID MOVER
Such systems are possible if (LIPEP + LIPEJ is negative and relatively large. It must be large
enough to render the relevant flowrate in a piping system for which the total cost will be equal
to or less than that of a piping system with a smaller diameter but which requires a fluid mover.
The choice of a system therefore actually requires a proper economic evaluation. A shortcut
method which is often acceptable is based on linear velocities. If, for Newtonian fluids, -(LIPEP+
LIPEJ is large enough to render a pipe diameter for which the linear flow velocity is larger than
1 m/s for liquids or 5 m/s for gases or vapours, the system without the fluid mover is considered
to be more economical; if the linear velocity is less than the given values, it is indicative of the
obtained diameter being too large and that a smaller diameter system, combined with a fluid
mover, may be more economical. For liquids with high viscosities and also for non-Newtonian
fluids this rule of thumb does not apply; systems without fluid movers with u « 1 may still be more
economical.
A suitable design method is the following. Determine LIFt,,..""''' by application of the MEB.
Estimate values for j' and Le and calculate d with the Darcy equation. Base calculations on the
maximum flowrate which may need to be processed. Choose an industrially available pipeline,
design for wall thickness and perform check calculations (Re, .f, Le, APr, 11). Prorate for other
diameters - the optimum economic diameter is the smallest diameter for which
LJPf S
L1Pf,availabk
Linear velocities play important roles, in design of piping systems. Low velocities may be
indicative of an uneconomic system; it may also cause undesirable sedimentation of suspended
solids and problems with crevice corrosion. High velocities may cause problems with erosion
corrosion, vibrations and the development of static charges. Rough rules of thumb for systems
without fluid movers are the following.
LIQUIDS :
GASES AND VAPOURS:
1 < u < 5 m/s
5 < 11 < 100 m/s
WITH A FLUID MOVER
Piping cost increases with pipe diameter and power cost decreases. The diameter with minimum
system cost is the optimum economic diameter. Different variables are important in the
79
evaluation of the optimum economic diameter. Various correlations have been developed for
its determination (see references). A simple but reliable method is based on recommended
values for API"' (AP 100 rJ combined with criteria for linear velocities. The following design
procedure is recommended:
•
•
•
Obtain the relevant API, from the literature and calculate by means of trial and error the
associated diameter with the Darcy equation. Base calculations on the flowrate
associated with the planned production rate.
Choose an industrially available pipe diameter and design for wall thickness.
Do check calculations (Re J, APim, u). The optimum economic diameter is that diameter
which renders values for APim and u which correlate best with relevant criteria. Prorating
simplifies calculations.
Cavitation occurs when, due to pressure drop in liquid flow systems a fraction of liquid
evaporates and a two phase system results. At higher pressure zones the vapour suddenly
condenses and shock waves are formed. They cause problems with vibrations and erosion
corrosion. In the diameter design of piping systems with pumps it is necessary to distinguish
between systems where cavitation is a threat and systems where it is not.
•
SYSTEMS WITHOUT CAVITATION
They are all gas and vapour systems, all delivery lines where liquids are pumped as well as
suction lines for which the operating temperature is much lower than the bubble point
temperature, provided that the liquid is not saturated with dissolved gases.
Criteria for APim and AP100ft for mild steel systems are given in appendix G1. If the piping system
cost deviates substantially from that of mild steel, the criteria must be adjusted; for instance in
the case of UNS 30400 (austenitic stainless steel often used in the chemical industry)APvalues
= 2 xAP values for mild steel are recommended. The criteria can be used for all fluid flow types.
Ludwig criteria are used for linear velocities. Examples for different systems are given in
appendix G2. In most cases a designed pipe which complies with the relevant AP criterion will
also comply to the relevant linear velocity criterion. As a general rule, if a discrepancy does
occur, the AP criterion should be considered dominant; exceptions are steam (linear velocity
criteria are better developed), certain construction material-corrosive medium systems (e.g. for
mild steel and concentrated sulphuric acid for which u must be lower than 1.5 m/s to limit erosion
corrosion) and certain two-phase vapour-liquid systems (for vertical lines urp > 6 m/s to prevent
slug forming; slug flow in horizontal lines is also not acceptable but a different criterion is used).
For non-Newtonian flow and also for liquids with high viscosities the criterion of linear velocity
loses its relevancy.
The criterion APim- 0.000164 P"P"'"m may lead to two different sizes for the suction and delivery
lines for gas and vapour systems with AP" relatively large.
80
•
SYSTEMS WITH CAVITATION
When a liquid is pumped at a temperature which is close to that of its bubble point at the
prevailing system pressure, a problem with cavitation may occur in the pump. Typical pressure
changes in a centrifugal pump is shown in figure 9.1. If the pressure drops to a value lower than
that of the bubble point at the prevailing system temperature, cavitation will occur. In the case
of pumps, cavitation not only causes problems with vibrations and erosion corrosion but also with
decreasing delivery head. According to the Hydraulic Institute a limited amount of cavitation is
allowable- the decrease in delivery head must however not be more than 3%. See figure 9.2.
Entrance loss
P
----~-__F_r~ction
______
_::_-::~Increasing P
I
-~
::::::::::~-:._:_----..--..._
I
I
~:::~~
3%
Decreasin~LIP due ;~~
due to im7elle
Turbulence, friction,
to cavitation
1
_~
entrance loss at van
--~-~·------'-----
Point of lowest P where
vaporization starts
Flow rate
Points along liquid path
Figure 9.1
Figure 9.2
When the liquid is nearly saturated with dissolved gases a less harmful type of cavitation may
occur at operating temperatures much less than bubble point temperatures. At zones of reduced
pressure two phase systems also form; however when pressure increases at the delivery side,
the return to a single phase system is gradual and less serious.
The concept known as the nett positive suction head (NPSH) is used in the analyses of
cavitation problems. It relates to the pressure at the suction flange of a pump. It is necessary
to distinguish between the available NPSH and the required NPSH.
(a)
AVAILABLE NPSH
NPSHA is a property of flow in the suction line and is defined as the available pressure difference
at the suction flange to limit cavitation:
=
(total pressure which, at the conditions of flow, will exist at the suction
flange) - (effective vapour pressure of the liquid at the operating
temperature) in units of liquid head.
81
Consider the pump suction line system shown in figure 9.3.
!\I
I
I
!J.Z
I
I
~----D---------
I
-··-·~-···-·---5-Q
Figure 9.3
Pump suction line
Total head
=
p2
+ (I.Jlv,2
A MEB between reference points 1 and 2 gives
D.PED + MEP + MKE + D.Pr=
o
NPSHA is calculated as follows:
"". (9. 3)
Use is made of "effective vapour pressure" to take into account the possible presence of
dissolved gases. If dissolved gas is not relevant the effective vapour pressure = vapour
pressure. If the liquid is at its bubble point at reference point 1, then P,!f= P 1• Various methods
are in use to calculate the effective vapour pressure if dissolved gases are relevant. A rough
method proposed by Whistler is
82
p_:_l_+_P_.:.:va:!::po:::"'--',.P::.:re:::"::::"r_:_e
pe/J = 2
(b)
..... (9.4)
REQUIRED NPSH
NPSHR is an intrinsic property of each pump and is defined as the required pressure difference
at the suction flange to limit cavitation to 3% loss in delivery head. It is determined
experimentally as
(total pressure required at the suction flange to limit cavitation to 3% loss
in delivery head)- (effective vapour pressure of the liquid at the operating
temperature) in units of liquid head.
NPSHR
NPSHR is a function of flowrate as well as the geometry and relative roughness of the pump's
suction side. The NPSHR characteristic of a pump can be obtained from commercial pump
suppliers. It is normally determined with water as fluid. Correlations exist for transforming
NPSHR, wATER to NPSH R, FLu 10; in most design applications NPSH R, WATER is also used for
other fluids; results are normally conservative.
(c)
DESIGN
To limit problems with cavitation, NPSHA for the suction line must be larger than NPSHR for the
pump.
The following criteria are used:
(1)
(2)
NPSHA ~ NPSH" + 0.5 m
For hydrocarbons:
For aqueous solutions:
NPSHA ~ 1.1 NPSHR
NPSHA ~ 1.2 NPSHR
That criterion which renders the largest NPSHA (in cases where the pump characteristic is fixed)
or the smallest NPSHR (in cases where the suction line characteristics are fixed) must be applied.
If the pressure at reference point 1 and the operating temperature are fixed the following three
variables play important roles:
(1)
NPSHR : If NPSHR is small, the associated NPSHA is relatively small. This implies that
!JP1 may be relatively large; thus relatively inexpensive small diameter suction pipe may
be specified; pumps with small NPSH" values are however relatively expensive.
(2)
Suction line characteristics : They determine !JP1 . Lines should always be as short as
possible to minimise !JP1 . Large diameters and large NPSHR go together; small
diameters and small NPSHR go together.
83
(3)
L1Z :
If L1Z is negative and large, large L1P1 values and NPSHR values can be
accommodated (thus inexpensive pipelines and pump). Large negativeL1Zvalues may
however require expensive constructions.
In the design of a suction line which is subject to cavitation problems a compromise is normally
made between pipe diameter and NPSHR. Diameters are designed according to criteria which
render larger diameters than criteria for pipe with no cavitation problems. See appendix G1.
Such designed lines are considered optimum economic for the relative costs of pipeline, pump
and elevation constructions. Linear velocities in these lines will not comply to Ludwig criteria.
It remains good practice to calculate linear velocities; with certain material-medium systems (e.g.
stainless steel with water containing dissolved chlorides) linear velocities « 1 m/s are not
acceptable because of sedimentation with crevice corrosion problems which may occur. After
obtaining the optimum economic suction line diameter, NPSHR can be calculated for pump
specification purposes.
9.1.3
CHECKVALVES
For effective operation the friction pressure drop across a check valve must be larger than a
certain minimum. Check valves are designed independently. Minimum pressure drop across
check valves are given in appendix C3.
9.1.4
LITERATURE
1.
R. K. Sinnott, Coulson and Richardson's Chemical Engineering, Design, vol. 6, 2 ed:
Pergamon, 1993.
2.
J. T. Petroskry, Determining economic pipe diameters, Plant Engineering, 114, June 24,
1976.
3.
G. R. Kent, Preliminary pipeline sizing, Chern Eng, 119, Sept 25, 1978.
4.
C. B. Nolte, Optimum pipe size selection, Trans Tech Publications, 1978.
5.
M.S. Peters and K. D. Timmerhaus, Plant design and economics for chemical engineers.
9.2
ORIFICES
Various instruments are available for the measuring of flowrates. Examples are orifices, venturi
meters, pilot tubes and rotameters. Orifices are in general use when pipe diameters ~ 50 mm;
pressure drop over an orifice plate is also often used as a signal for flowrate control.
An orifice is a plate through which a hole is drilled and which is mounted in a pipe between
flanges as shown in figure 9.4. The pressure differential is measured between two pressure
taps. A typical pressure profile in the vicinity of an orifice is shown.
84
Pressure along orific:e pipe·run
I
\L ---- _
_l,_T_C>r-P"
-----=M±fo=ss)=;:::=;=
·---- -- -~
~
\_
~ -~
--
Flow--
.---+1-
h~
_L_ __
FIGURE 9.4
Mounting of an orifice
A MEB between the two pressure taps is as follows:
AP1 could be calculated with the Darcy equation if values for Kr or Lr!D for the orifice were
known. Calculation of APKE would require information of the flow profile (see figure 9.4) and is
()
a function of the positions of the pressure taps. Different arrangements in use are corner taps,
radius taps, line taps, flange taps and vena-contracta taps. See figure 9.5. For flange taps
positions are one inch upstream and downstream of the plate. The point of minimum pressure
(maximum kinetic energy) is known as the vena contracta.
For analyses the terms AP1 and APKE are combined. Examples of equations in use are the
following:
V
0.000397p2 d 2C.jhJSG
=
W
= o.o125pWC.jphw
m 3!h
kglh
hw
d
= mm water
fJ
= diameter ratio = orifice diameter/pipe inside diameter
= diameter in mm
The basic correlation of APs-rv a
w> still applies.
85
..... (9.5)
l..--------M----->.'__:_N-~
I
I'
Vena Contracta Taps:
= 1 x pipe dia, N varies with d0
M
IJ.,
ratio (see chart}
Radius Taps:
M = 1 x pipe dia, N = 0.5 x pipe dia
Corner Taps
r-2Y.. pipe di.,,..t<c------- 8 pipe dia. --~------>
I
0
I
0
:.. ;_
·'·
..., ;:.)) ., .
._.. ·
..
"
~
11-
~
1f-
"""' "
~
~
'\
1f-
r\.
'\
1:
0.3 '
0.2
tine Taps
FIGURE 9,5
"
0.3
..
0.4
0.5
''
0.6
Hatio, d 0 /d 1
Arrangement of taps
86
[\.
..~
0.7
0.8
In practice 0.25 ~
p ~ 0. 75; p = 0. 7 is popular.
Cis known as the coefficient of discharge. It is
not a constant but a function of the positions of the pressure taps,
p and Re.
See figure 9.6.
Positions of the pressure taps and pare fixed per orifice installation. However, because C varies
significantly with Re < 30 000, orifices are less suitable for flow measurement at low Re;
calibration is required. For Re > 30 000, C is approximately constant and flowrate is a function
of hw as shown in the equations.
0.95·
r---r-r-r/--rri/n.....,..-/nynorn
··~
_L';'~"\"
v
0.90
lo.'
[.6:
0.80
"·~·"".'"
.
.i!
u
,
g
;;
Ji
~---
\
[%:::
~
1':-:- l'i
PS
I'f'
1~-<0 '."m :'"~
~
~ 6::::
~~
~~
'
~-·
..
~
'
10
4
10
'
4
'
10
'
10
•
'
10
Coefficient of discharge for square-edged circular orifices with
Downstream pressure top focolion
comer taps. [Tuve and S7Jrenklc, Instruments, 6, 201 (1933).]
in pipe diameters
FIGURE 9.6
Influence of taps, J3 and Re on C.
DESIGN
Various design approaches may be followed. One suitable approach is the following. Establish
Re for the designed pipeline. If Re > 30 000, the orifice may be mounted in it; if Re < 30 000 a
special section of smaller diameter pipeline must be incorporated for orifice mounting and for
which Re > 30 000. Certain lengths of pipe upstream and downstream of the orifice must be free
of restrictions. Perry gives more detail.
=
Decide on the following : Construction material, p (/3 0. 7 is popular), plate thickness (thickness
< 0.033 pipe diameters and < 0.125 orifice opening), type of pressure-taps (each has own
advantages and disadvantages, vena contracta is popular).
Calculate LiPEQ based on the normal flowrate (h
87
w
follows from the basic equation; it however
includes LiPKE which is recovered downstream;
calculated from the following relationship:
LiP EQ
,jpEQ, MEB / ,jpEQ, ORIFICE
to be used in the global MEB can be
~
j -
fJ 2
..... (9.6)
Calculate the maximum pressure drop which may be recorded by the recording instrument; this
is h., (may be specified in mm water or kPa) based on the maximum flow rate which may be
encountered; report as the maximum instrument reading.
LITERATURE
1.
J. Powers, Flow meter selection guide, Chem Proc, 79, Oct, 1979.
2.
A. Noor, Sizing orifice and venturi meters, Chem Eng, 97, Aug 22, 1983.
88
10.
CONTROL VALVES
10.1
INTRODUCTION
Control values are primarily used for the control of flow rates. Control of flow rates may indirectly
lead to the control of variables such as level and temperature. Flow rates may also be controlled
by regulating the rotational speed of fluid movers; this method of control is not dealt with in this
course. Various types of control valves are commercially available. See examples in figure 10.1.
Operating principles are the same- a signal from a flow rate measuring instrument (typically a
pressure drop across an orifice plate) is transmitted (pneumatic or electric) to the control valve
which then takes action. Closing causes an increase of LIPcvwith an equivalent decrease of LIPSTv
and flow rate; opening causes LIPcv to decrease and LIPSTv and flow rate to increase. This course
deals with size design of control valves; Masoneilen control valves are used for illustration
purposes.
A typical mounting of a control valve is shown in figure 10.2. Two gate valves are mounted on
either side of the control valve; it is necessary for maintenance purposes. For continuation of
operation during a maintenance period a loopline with globe valve is part of the system. The
diameter of the loopline is normally taken the same as that of the main line; it may also be
designed on its own; the principle for such a design would be LIP8 n~ loopu,, s LIPrv; m,1,u,, ucuon •
Diameters of control valves are often smaller than those of the pipelines; coupling of a control
valve directly to the gate valves establishes a sudden reducer and enlarger in the system; ASA
reducers and enlargers may also be used.
Figure 10.2
Mounting of control valves
89
AIR PRESSURE
DIAPHRAGM
SPRING
STEM
PARABOLIC
INNER VALVE
Diaphragm+Qpo:rated, doubk·pon control valve.
IaI
IaI
IcI
I bl
IbI
IcI
Typ1e21 double·pon«l·v1lve cnnbguntions for lot tiHOU!h aow. (b) blending. 1nd tel
1tre1m Jplithng.
Figure 10.1
Types of control valves
90
10.2
FLOW RATES
Three types of flow rates are used in piping system design:
NORMAL FLOW RATE (W,,) : It is the flow rate associated with the planned production rate. In
the absence of any additional requirements optimum economic diameters are designed with
normal flow rates.
MAXIMUM FLOW RATE (Wm): Some processings require flow rates larger than normal flow
rates for relatively short intervals. Examples are reflux lines of distillation columns and quench
lines for exothermic reactors. If such lines incorporate fluid movers, their optimum economic
diameters are still designed for w;,; the fluid movers are designed and specified to deal with the
occasional larger flow rates. If they are operated without fluid movers, the optimum economic
diameters are designed with Wm.
DESIGN FLOW RATE (WJ) : Control valves cannot control flow effectively when they are fully
open. For effective control of maximum flows, control valves must be " 95% open with maximum
flow. Design flow rates are approximately 5% larger than maximum flow rates and are used in
control valve design to enable the effective control of maximum flow rates.
Typical correlations among the flow rates are the following
BATCH LOADING:
NORMAL PROCESSING :
SPECIAL PROCESSING :
10.3
Wa = 1.05 Wm = 1.05 W,,
wd = 1.05 wm = 1.15 w;,
Wa = 1.05 W,, = 1.30 w;,
CONTROL VALVE SIZING EQUATIONS
Equations correlate flow rate and friction pressure loss over control valves and are based on the
Darcy equation. A control valve coefficient Ccv is used in stead of a Lr!D or Krvalue. Literature
values for Ccv are for fully open control valves. See appendix H. Ccv values are determined
experimentally as the volume flow rate (US gpm at 60 •F) through the fully open control valve
when the friction pressure drop over the valve is 1 psi (liPcv = 1 psi).
10.3.1 LIQUIDS
The design equation for non-flashing liquids is:
V = CCl' f(x)
..... (10.1)
where
91
X
=
=
f(x)
=
v
=
LIPcv
pressure loss across valve in psi
valve stem position
fraction of the total flow area of the valve (the curve of f(x) versus x is called the
inherent characteristic of the valve)
flow rate in gpm
In the fully open position, at the design flow rate f(x) = 1, after the valve has taken action,J(x)
changes.
10.3.2 GASES AND VAPOURS
Pressure drops over control valves are normally relatively large and flow is mostly compressible.
The criterion for compressibility is L1Pcv/P1 > 0.1. See figure 10.3.
2
1
Figure 10.3 Reference points for control valves
Two variations of the basic control valve equation are used to deal with compressible flow:
1)
The first equation is in terms of the average density across the valve:
Expansion of a gas through a valve is polytropic. For purposes of design calculations, it may be
approximated as isothermal. Strictly speaking the calculation of LIPcv requires a trial-and-error
approach. The main application of the equation in this format is for the estimation of f(x) from
LIPCl\ availab/, when an approximated value for pav is acceptable.
2)
More accurate evaluations require incremental solutions. Depending on the placing of the
compressor, either P1 or P2 can be accurately calculated from a MEB. Masoneilen derived an
92
equation which eliminates trial-and-error methods for the calculation of the other pressure (P2 or
P 1 ;L1Pcv~P 1 -P2):
SGv
Pav
=
with pav =
M(P 1 +P2)/2
Pair,60°F
!0.73ZT
520M
29ZT
:. Pav
=
520M
29ZT
X
(p +P2 )/2
I
10.73
29
X- =
520
0.0026 SG (P
v
I
Substitution in the control valve equation gives:
..... (10.3)
Evaluations of LiPcv with these equations can only be performed if the control valve is fully open,
that is for f{x) = 1.
CHECKS FOR SONIC VELOCITIES
If flow in pipelines is identified as compressible, it is necessary to check for sonic velocities in the
lines. This can be done using the critical pressure ratio for sonic flow. Refer to section 5.
10.4
CONTROL VALVE CHARACTERISTICS
By changing the shape of the plug and the seat of the valve, different relations betweenj(x) and
x can be obtained. Common flow characteristics used are linear trim valves and equalpercentage trim valves as shown in figure 10.4.
The reason for using different control valve trims is to keep the stability of the control loop fairly
constant over a wide range of flows. Linear trim valves are used, for example, when the pressure
drop over the control valve is fairly constant and a linear relationship exists between the controlled
variable and the flow rate of the manipulated variable. Equal percentage (increasing sensitivity)
control valves are often used when the pressure drop over the control valve is not constant. This
is best illustrated using pump and system curves. The pressure drop across the control valve is
not considered as part of the system curve, the pump and system curve is plotted and the
distance between the pump and system curve then represents the LiP"" The pressure loss across
the valve can then be determined very easily for a range of flow rates. This is shown in figure
10.5. In figure 10.6 the choice between linear and increasing sensitivity valve trims are shown.
93
Linear
f(x) =x
f(x}
Equal percentage (a=5D)
j{x)
= ax-1
/
~/
/
0
y--~--------~---
1
X
Figure 10.4
Control valve characteristics
Pressure head
Mev
Systelll~u~e
__ -----~-----/~
_______________
li\--~-c-c-c::CC
ljl_L____
L1Psrv
'V/__ ____ _
----------------
Flow rate
Figure 10.5
Pump curve, system curve and control valve pressure loss
94
n
r--.
~ ,·
'
"Tl
«5.
c
~
<1>
Head
9(J)
Hoad
Pump Charactorlatlc
~
Ooveloped
"
"
Aoqulred
0
Roqulred
H
::::r
Pump CharacktriaUc
Do!v•loped
H
0
cr
<1>
tr
<1>
~
<1>
<1>
::J
<
"'<1>
<
-
I
Static
.....
Delivery.
I
-
Syatem Curve
Syatom Curve
Volumetric Aowrate,
Desired flowrate range
:::!.
3(/)
Use linear trim
Volumetric Flowrite ,
Q
Desired flowrate
range
Desired flowrate range
Use linear trim
Use equal % trim
Desired flowrate
range
Use equal
(!)
~{,
trim
"'
H•od
Hoad
Pump Char.acterletlc
OovelopOO
.,
Oovoloped
"
Required
Required
Sysotem Curv•
H
H
Static
.....
Use equal% trim
I
B
~~
Syatom Curve
.
Stdc
Oo!lvery
,.,
Delivery
Rowrate range
Pump Ch.aractedatic
VolumeiTic Aowrite , 0
Flowrate range
Use linear trim
Volunw.ttrio Flowrito
I
a
Q
10.5
DESIGN
Standard optimum economic diameter design principles are applicable. Several operating
requirements are also relevant. Control valve rules were developed to take both economic and
operating requirements into account.
SYSTEMS WITH FLUID MOVERS
RULE 1(d):
j(>:)d
5{
1
This rule takes care of the requirement that the chosen control valve must be able to
accommodate design flow rates when it is fully open; thus it will be able to control maximum flow.
RULE 1(n): j(Y), 2 0.1
Too much control valve action is associated with less effective control. This rule prevents
operating in relatively closed positions with normal flow.
RULE 2(n):
0.5 L1Psrl~n
5{
L1Pc,~n
5{
1.5 L1Psn~n
Effective control also requires thatLJPcvmust be a substantial fraction of L1P7 v; the lower limit of
these two rules takes care of this requirement. Economic analyses require that LiPcv must not be
unnecessarily large relative to L1Prv; the upper limit of these two rules takes care of this
requirement.
Except for rule 1(d), the rules must not be applied rigidly. Exceeding a limit is an indication that
a better option most likely exists and that it should be investigated.
SYSTEMS WITHOUT FLUID MOVERS
Rules 1(d) and 1(n) are still applicable. In the absence of operating costs, the upper limits of rules
2(d) and 2(n) are less relevant. They do however, serve a handy purpose in dividing an available
L!Prv between L!Psrv (for pipeline design) and LiPcv (for control valve choice).
In the design of piping systems with control valves it is necessary to distinguish between different
systems. The main types are new systems with fluid movers, new systems without fluid movers,
new systems with fluid movers where the fluid mover supplies flow to two or more branched lines,
each with its own control valve and existing systems without control valves which must be
provided with control valves. Different design approaches may be followed. The following are
suitable:
96
10.5.1 NEW SYSTEMS WITH FLUID MOVERS
Design the pipeline on its own. Take as a first choice a control valve one size smaller. Test with
control valve rules. If the rules are satisfied, the system qualifies as optimum economic. If not,
the results will indicate whether a line size valve or a two diameters smaller valve should be
considered next; this system will most likely qualify; control valves more than two line sizes
smaller are rare exceptions; control valves larger than line size are never specified. If the initial
pipeline design indicated the possibility of a second line size which may also qualify as optimum
economic, it may also be investigated. Different procedures are recommended for the testing of
control valve rules for liquids and gases:
•
LIQUIDS
Control valves are mounted in delivery lines. Mounting in suction lines promotes cavitation.
Assume a fully opened control valve with design flow - the method of calculation and
specification of LIP.justifies this assumption; calculateLIPc,~dwith the control valve equation. This
implies thatj(.'C)d ~ 1 and the system conforms to rule 1(d).
Calculate LIPsn~ .. and prorate for L!Psn~d· Check for rule 2(d). Consider another control valve if
necessary. Calculate LIPa.d with the MEB. Assume LIP a,n ~ LIP a,d - the accuracy of this
approximation depends on the head-flow characteristic of the pump which normally, at this stage
of the design, is not available. Calculate APe,~ .. with the MEB and check for rule 2(n). Adjust the
system if necessary.
Calculate f(.'C), with the control valve equation and test with rule 1(n). Adjust the system if
necessary.
•
GASES OR VAPOURS
Control valves may be mounted in suction or delivery lines. Because flow over the valve is
normally compressible, it is necessary to divide the pipe system in two sections - upstream of
the valve and downstream of the valve. See figure 10.7.
97
SELDOM
Figure 10.7
OFTEN
Control valves in gas systems
Consider the case with the valve in the suction line. Calculate P1.a with a MEB. Assume a fully
opened valve with design flow (see liquids) and calculate P 2,a with the control valve equation.
APc,~a ~ 1'1,"-
PM (This evaluation for the valve in the delivery line requires a trial-and-error
approach). Again, this implies thatj(."x:)a = 1 and the system conforms to rule 1(d).
Calculate APsr'~" and AP sr'~"; remember that strictly speaking prorating is invalid for compressible
flow. Check for rule 2(d). Adjust the system if necessary. Calculate AP."with the MEB. Assume
AP., ~ AP.a (see liquids). Calculate P 1,, with a MEB. Calculate P2,, with a MEB. Calculate APcv.n
~ 1'!_,- P2,, and check with rule 2(n). Adjust the system if necessary.
Calculate JM, with the control valve equation and test with rule 1(n). Adjust the system if
necessary.
Check for sonic flow in the control valve. If analysis proves that flow in the lines is also
compressible, tests for sonic flow at relevant points in the pipeline must also be performed.
NOTE : If APcv is large, it should be established whether the criterion L1P1 " ' - 0, 000164 P does
not render two different diameters for the two pipe sections upstream and downstream of the
control valve.
98
10.5.2 NEW SYSTEMS WITHOUT FLUID MOVERS
-Calculate L1Pr 1 ~a.m·ailabl< with the MEB.
-Divide this judiciously between LIP c'~'!"'"""''' and LIPSTv.d.amilabt' by using the limits set in rule 2(d):
E.g. LJPSTI~d, m•ailable ~ 0. 75 LJPTl~d, available
- Determine the associated LIP1,(m·ailab/, and design the pipeline.
-Make a first choice for control valve diameter (e.g. one line diameter smaller or make use of the
relation .dPCl~d,amilable . . . . . 0. 25L1PTJ~cl,amilabk).
-Check the system with the control valve rules; the economic limits of rules 2(d) and 2(n) are less
relevant. It should be noted that it is unlikely that the valve will be fully open with Wd and LIPc'~"
must be calculated from a MEB application.
- The final check requires conformation of the designed system to the MEB. Adjust the chosen
system if necessary.
10.5.3 EXISTING SYSTEMS
- Sufficient LIP must be available for the control valve.
-First of all calculate LIPc,~damilabt' with a MEB application.
-Check whether it conforms to the operating limit requirement of rule 2(d); if not, additional LIP
(new fluid mover or larger upstream pressure, etc) must be supplied. If it does, calculatej(':Ja with
the control valve equation and choose a valve for which Ccv >J(x) a Ccv·
- Check with the control valve rules and adjust the control valve if necessary.
10.5.4 BRANCHED PIPING SYSTEMS
A typical example of a branched system with control is the top piping system of distillation
columns. One pump is used to serve both the reflux and product lines; each may be provided
with a control valve. One suitable design approach is the following:
Do designs for the two piping systems up to the point where LIP,a is calculated with the MEB.
Establish which system requires the largest LIP.d and design it fully.
Test the second system with the control valve rules. The format for these tests is as discussed
in 10.5.2. LIPn~amilaht< is calculated with the MEB.
10.6
EFFECT OF HEAD CAPACITY CURVE
For design purposes it is assumed that LIP""~ LIP• .,. In reality LIP,,, will be> LIP,,a· See figure 10.8.
99
Pressure head
·-- -"-----"-,---fu~ curve
~~­
'"'I
I
I
1
l
I
/
I/
~~r1
System curve
~II
I
I
--= -...=- _ _ - _ _ _ _ _ _ _ _ _ _ _I ___I_ ~ _ _ _ _ _ _
71\ --LlJ>EP
+ dJ>I!L
'!_
---
--
1
:
:
:
L____ _ _ _ - - - _ _ _ __,___,I
I
Vn
Figure 10.8
II
VmVd
Flow rate
Effect of head capacity curve
10.7
LITERATURE
1.
R. Kern, Control valves in process plants, Chern Eng, 85, April14, 1975.
2.
M. Adams and D. Boyd, Control valves: time for review, Hydrocarbon Processing, 87,
May, 1984.
3.
J.R. Connell, Realistic control valve pressure drops, Chern Eng, 123, Sept 28, 1987.
4.
H.D. Baumann, Control valve versus variable speed pump, Chern Eng, 81, Jun 29, 1981.
5.
W.L. Luyben, Process modelling, simulation and control for chemical engineers, second
ed. McGraw-Hill, 1990.
100
11.
FLUID MOVERS
In general terms fluid movers for liquids are known as pumps and those for vapours and gases
are referred to as compressors.
11.1
TYPES
The main types are shown in figure 11.1.
KINETIC
POSITIVE DISPLACEMENT
Radial flow
Figure 11.1
Types of fluid movers
101
11.2
CHARACTERISTICS
The various fluid mover characteristics are mostly presented in diagram format and are
available from commercial suppliers. Most common is the head capacity curve. The
correlation differs substantially for kinetic and positive displacement movers. See figure 11.2.
Positive displacement
Kinetic
Head
r---~
Pump curve
Head
Pump curve
System curve
System curve
Flow rate
Figure 11.2
Flow rate
Head capacity curves
r',;,
"""'
The operating point for a piping system is obtained when the head capacity curve of the fluid
mover and the system curve according to the MEB are combined.
Other important characteristics which are obtainable in diagram format include NPSHR, power
and efficiency vs capacity; influence of parameters like rotational speed and impeller size;
information regarding the surge zone for compressors.
Examples of characteristic curves are shown in figure 11.3.
102
U.S.qpm
50
0
0
150
200
150
50
Lll
14
J.
400
350
I'
I I
73
..:......t.....'
50
''
I
.::,..1) " 1'l ·;.
''
II
rrao- I
I
I'
I:
II
t jl
I
I'
I
''
'I
'I
601
II
''
II
f1
30
701
I; 1
'I
l" 1
6
''
I
'
I I
4
l
!
I I
I I
,
'
40
50
60
''I'
I I''
'''
i!•i
'•''
i;l\
;o.:
<
90
80
70
''''
'
l'i 1
I
100
J
,-
••
I
''
II
NPSH
I
10
I
I
!
'
120
110
130
''''I
:-'~5H -'~<f<lOon ,"nd.''~·~--:•tfe; ;~o•h••:-~~~log _.on.Q.5 "'.~1~·tl~
--:--.
I I
''
,.,
<
''
''
! 1I
I
Q m3/h JO
10
','
''
I
''
''
0
20
I'
'
11! I
12
''
I I!
''
'
10
'
NPSH
m
f1
''
2
'
'
~
I II I
'
I
I':
'
' '
'II '
'! II ' II I
4
kW
I
I I
I
3
I
i
I I
'
'
I
I
I
I ! I
I
I I' I
'
I
II
I
I
I I II
i I
i
I I
0
0
'
I
I
I
1•:
'
I I
'
I
I
Ill I
I
II
I
'
I
I
';
'I ' '
' 'I
'
'
I
II
I
I
U/min · RPM
'
''
.....'
'
I
0
'
I
Characteristic curves
103
'
'
''
'
'
''
'
'
:
'I
''
I I
'
I
'
I
II
'
go
25
'
'
I
''
'
I I I I
I I
I
I
''
I
I
i
I
I
I I
I
I'
'
'
I
' I' II
'
I
I
I
I
100
'
I
I
:'
6
' 'I
:
'
I
I
I
110
I I•
I I
I I'
I I
'I
I
'I
'I I
I I
'
:
I' !
I
p
hp.
4
3
'
I
I
I
I I
'
I
80
:
II I
I
''
I
'""'':
·--~
I
I I
I I
I 'I
I
I
ll-
'
i
4
' '' '
I '
'
'I ' I I ' ' ' :' I • '
'
'
I I II I
' '...;........-.....-:-;;
I
70
I
''
' '
; !
I
'
' i'
'
'
' II
'•
'I !
I
I
I
20
Impeller
Width
I
I
'
I
' ·I
'' I
I
I I
i
,........,I '
I
'! 'I
15
I
'
I
I I
'
I I I
::I
U_l
60
'
I
r'l•'
''
I
'
'
I I
I
I
I
I
'
'
! '
:
'
I
'I
! I
'
' ' I' '
! '
'I
'
'
I
•
'
'
I
I
- ·-
0m¥h JO
40
Otis 10
5
'
I I I
I I •
I I I
' II I I
I
I
I
/1450/
'!
I I
I I
.....
'• ' I'
'
'
I
I
I
_,..,I
I
I I
I I I
I I
I
II
II
I
i
:
J..-r'
I
10
'
'
:'
'I
I
I
I
'
''I
:
I
I
..,.
'
'I
'
:
I
:
I
I
I
! '
'
I
'
I
I
r ' I'
_.. I
1~01·
I
I
I I I i
I
I I
I
190'
180'
I
i
I
209'
200'
I
''
I '· I I
I
'
I
I
'"
I I
''
,-,
I
'
II
'II'
I
2
Figure 11.3
':
TTl~
I
''
p
0
I
' '
5
6
I I
''
c
'''
!'
1
C'-•
H
·s; ; : ' ' :
'I
I
I I
'-·!J_ ++-~-~·fi-r:1
'
'
II
'.
'
·.~c
40
I I
1l I I
I' I i
o
(I I
8
':
I''
::::-,_...,,4'
.'
2
450
I'
, I
10
550
I I 1!
, r i
H .l:l,9.'
m
·I
500
ro thsH '
''
_2£.0~
JOO
2y0
450
4()0
r+-+
'209.
"
II
]50
I
I I
I I
II
JOO
250
100
2
'I
'
'
' I I:
I
I
!
'
I
I
IJO
120
JO
J5
209·170 mm@
23mm
0
11.3
SELECTION
Selection of the most suitable, cost effective fluid mover for chemical processing is normally the
responsibility of a fluid mover specialist with lots of experience. Variables that must be considered
include type, construction material, mechanical design, characteristics of the mover and
characteristics of the system. It is the responsibilities of the relevant engineering disciplines to
provide him with the necessary information. Information regarding the following is normally the
responsibility of the chemical engineer: fluid characteristics (temperature, type, viscosity, density),
flowrates (W•. W.n· Wa), terminal system pressures, LIP. (associated with w. and Wm in the case of
pumps), P at the suction flange and P at the delivery flange for both w. and Wm in the case of
compressors, NPSHA and NPSHR, construction material.
r.
',_
1
~·
Useful general guidelines are the following. Centrifugal movers are suitable for most ordinary
applications. Application examples where other types of fluid movers are better choices are the
following. • Positive displacement movers for high delivery pressures (roughly LIP. > 3000 kPa)
• Gear and screw rotary movers for liquids with high viscosities (roughly v > 250 eSt)
•
Diaphragm pumps for corrosive inorganic liquids e.g. certain acids • Positive displacement
movers where a fixed flowrate is of great importance (e.g. dosage pumps). Diagrams were
developed to simplify the selection of fluid movers. Examples are shown in figure 11.4, 11.5 and
11.6.
30
q'
~)
,·
•
~
"E
1111
_llllill
I Si
-'
Jmps
:Dr~~
10
I
6
'I
I flow
~
'0
.2
3
:n•
r...:::
c.
"'
0.6
f:::
0.3
r-N
'\il
1 ii"' t-Ht+
114
D, = DH 1.,(0,
=Speed, rpm
r-- ZD' :
0.1
Jpump_
-
~~.~: ::'"Is,_ m.e ter, ,_+-+f-tll~'
..
11
- ifT
"~T
Httt--J;-;;T_.Jj·._.L -Htttffii-~ ~"\"r
°
"
:mf~ i'1lll
i"
1111
li1
~~
I II
L__!_..L.LL!LillL..-.L....L.t...L:Ll.L!L--L--:':-.L!..:l:ll:-:-:--L:::::-.1....:':-::':!-'-!-'=---':::-::;:;~~
1=
0.1
0.3
0,6
1
3
6
10.
30
60 100
300
600 1,000
3,000
Specific speed, N1
Figure 11.4
Initial type selection of a single stage fluid mover
104
10,000
The diagram of figure 11.4 may be used for both liquids and gases. For liquids H = AH, as
calculated with the MEB. For gases and vapours H must be calculated as AH, (adiabatic).
[ (Pft' )fk - l!lk
L1Ha {adiabatic}
)
1
= suction flange,
)2
=
_
1]
= delivery flange,
T
= oR,
P
= psia,
k
..... (11.1)
ft
(k - 1)/k
Cp/Cv.
=
Application requires the calculation of N.; see equation in diagram; if N is not known, a typical
value must be assumed (in SA a typical value is 1 450 or 2 900 rpm). The type of fluid mover
follows from the diagram. If the mover is a kinetic type, the minimum impeller diameter associated
with maximum efficiency may also be calculated. With the type and approximate size known,
·;__
.
characteristic diagrams may be obtained from suppliers for final selection.
Figure 11.5 shows a second type of diagram which may be used for the initial type selection.
Head capacity curves for different sizes of centrifugal fluid movers may be combined in a single
diagram as shown in figure 11.6; it is convenient for size screening before final selections are
made.
' I
I
'
'
I I I
I
I
I
I . I I 'I ~I
Sp~Cf<lllh·g~·soeM
centnfu~ai-
I. .I.. II I I
'
..
l,OO 0
1
~'
'--•
"
l
"'0
~
~
so 0
400
300
I
II
I I I
I I
I
I
.
I !
'1 • -
11
I
I
J'll~
I~
' I I I '1
7
I II I .
0
aI
I I
'f
~1""11
/
I
I I
I I
' '
I I
I I II I I I ill
,In:-;.,~-++"-~"''?;;,·~-+~f---1---l----i----i--g-+I" 'r~~
lI ~
V.ltll
:!I
I I I I i/
20 0
1/1 I
';71%-,':1
I
'
. I
I ·1
-f
~
'%-i>-
I I I
I I
(~I
I
~#.•
I I
I
Ji;;
1
I I
I
~
1.
I
I I
l I II I
!
I
I
Ill
i
I ' I
1
I I iI I
10 0
70
10
1
!;---_j___l__l_!----1-1-f:
Y
___!_T-'' '1-'""I-bT_j_TlT:-'t~~_jlruJIJInt.4~~f¥~~~~~,t'~Et:Jo···_'''"_·1-L·~~::~~~~I_LI
0
tO
50
tOO
500
1.000
5.000 lO,OOO
CJoacity, gpm.
Figure 11.5
Diagram for initial type selection
105
SO.OCO 100.009
800
600
400
~
"
200
u
"0
c
•c
}100
~
-"
~
"'
20
40
20
60 80 100
'
200
400
600 800 1000
2000
GPM
r~
.
Range
Curve
Pump
No.
1
2
3
4
5
6
7
8
731
731
731
731
4x3x6
1.5x1x8 731
3x 1.5x8 731
3 x 1.5 x 8.5 E 731
3x2x8.5E 731
1.5x6E
3x1.5x6
3x2x6
Figure 11.6
Plus
Plus
Plus
Plus
Plus
Plus
Plus
Plus
A-8475
A-6982
A-8159
A-8551
A-8155
A-8155
A-8529
A-8506
Range
Curve
Pump
No.
4x3x8.5
731 Plus A-8969
9
6x4x8.5
731 Plus A-8547
10
2x1 x10E 731 Plus A-8496
11
3x1.5x11E 731 Plus A-8543
12
731 Plus A-8456
3x2x11
13
4x3x11
731 Plus A-7342
14
3x1.5x13E 731 Plus A-3492
15
3 X 2 X 13
731 Plus A-7338
16
Head capacity curves for centrifugal fluid movers
MAXIMUM PRESSURE AT THE DELIVERY FLANGE: It must be calculated for specifying of the
class type (often A, B or C) of the fluid mover.
For centrifugal machines :
pmax
=
pfeed tank, max -
LJPEL, suction line, max + LJPa, max
For positive displacement machines :
p mer<
=
p delivery tank, max + LJPEL. delivery line, max + LJP/. delil.'ery line, max
The maximum friction pressure drop must be calculated with the maximum linear velocity. Pmax is
the maximum pressure which will be encountered at the delivery flange under normal operation.
If however a delivery line valve in a positive displacement system is closed, the pressure will
increase until a mechanical failure (e.g. burst of pipewall) will occur. To prevent this a safety valve
with recycling facilities is normally used; a typical safety valve setting is 10% largerthanPm"""'"'"''
Class specification should be based on the pressure of the safety valve setting.
106
11.4
COUPLING IN SERIES AND PARALLEL
The capacity of an existing piping system may be increased by series or parallel coupling of
centrifugal fluid movers. See figure 11.7.
Pump I + Pump 2
/
r~
Flow
\ •.. :
Head
Pump I + Pump 2
Series
/
Q
o· 0 '
Q
Flow
Figure 11.7
Coupling of fluid movers
Evaluation of operating points requires construction of the combined head capacity curve of the
fluid mover system. This can be done by simple addition of the relevant variables- note that in
the case of series coupling, flowrate is common; for parallel coupling, delivery head is common.
11.5
WORK AND POWER
Hydraulic work is defined as the energy received by the fluid from the fluid mover. Real work is
the energy which must be provided to the fluid mover. Efficiency (tl) of a fluid mover is defined
as hydraulic work/real work. Power is work per unit time. Suitable equations for the calculation
of power are
Hydraulic power
=
L1Pa W/3600 p
kW; Win kg/h
In the British unit system real power is also known as brake horse power.
107
11.6
LITERATURE
1.
R.F. Neerken, Pump selection for the chemical process industries, Chern Eng, Feb 18,
1974.
2.
R.F. Neerken, Progress in pumps, Chern Eng., Sept 14, 1987.
.
C·•
·'
108
Piping
System
Design :
Appendix
APPENDIX A1 : Dimensional standards for plain steel pipes (unscrewed)
Nominal
diameter
(mm)
Outside
diameter
(mm)
Nominal
diameter
Outside
diameter
Nominal
diameter
Outside
diameter
10
17.2
125
139.7
800
813
15
21.3
150
165.1
900
914
20
26.9
200
219.1
1000
1016
25
33.7
250
273.0
1200
1220
32
42.4
300
323.9
1400
1420
40
48.3
350
355.6
1600
1620
50
60.3
400
406.4
1800
1820
65
76.1
500
508
2000
2020
80
88.9
600
610
2200
2220
100
114.3
700
711
Recommended wall thicknesses: 4, 4.5, 5, 6, 8, 10, 12, 14, 16, 20, 22, 25 mm
Normal stock lengths : 6 m
APPENDIX A2 : Dimensional standards for general purpose tubes
Outside
d
(mm)
Thickness (mm)
0.
5
0.
6
0.
8
1.
0
1.
2
1.
5
2.
0
2.
5
3.
0
3.
5
4.
0
5.
0
2
X
2.5
X
X
3
X
X
X
4
X
X
X
5
X
X
X
X
6
X
X
X
X
8
X
X
X
X
X
10
X
X
X
X
X
12
X
X
X
X
X
X
14
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
25
X
X
X
X
30
X
X
X
X
X
32
X
X
X
X
38
X
X
X
X
40
X
X
X
X
44.5
X
X
X
X
50
X
X
X
X
57
X
X
X
X
76.1
X
X
X
X
X
88.9
X
X
X
X
X
X
108
X
X
X
X
X
X
133
X
X
X
X
X
159
X
X
X
X
X
16
20
Sizes marked with an x are those most likely to be available, but you are advised to always consult
your suppliers' stock list.
APPENDIX A3 : Dimensions for polypropylene and high density polyethylene tubes
PP (Polypropylene)
SABS 1315 of 1981
Wall
Thlcl<ness
ODmm
"""'""'
Tokraoc:e
ClASS6,3
ClASS 10
ClASS 16
""'
-
-
""'
-
-
-
-
-
-
-
-
-
HomiMI
10
12
.03
-
16
03
20
Min Wall
Thkl<-
-
>Muf
NomfNI
lO
{kg)
-
Mou/
Min Wall
Thkl<·
-
-
15
23
0,13 0,19
022
031
0,37
o·s8c
()
Thkl<·
-
03
-
-
-
-
03
-
-
-
-
-
25
-
-
19
2,8
32
03
-
-
-
27
24
0,22
24
3,6
40
04
-
-
-
34
29
033
31
45
50
OS
45
24
035
42
37
053
38
56
63
06
57
3,0
056
75
07
68
36
0,80
90
110
09
81
4,3
113
10
99
s;2
1,67
125
12
113
60
2,18
140
126
6,7
2,73
160
13
t:5·-
145
7,6
3,53
630
57
568
300
55 5
710
64
640
33,8
705
53
64
76
93
106
119
136
153
170
190
211
237
266
300
338
381
423
474
533
·-
800
72
721
38,1
895
-
900
81
812
429
1133
-
180
17
163
86
4,49
200
18
181
95
5,51
225
21
203
10,7
6,97
250
23
226
11,9
280
26
253
133
10,7
315
29
285
15,0
13,6
355
32
320
16,9
17,7
400
361
19,0
224
450
36
41 .
406
21,4
283
500
45
451
238
35,0
560
51
505
26,7
439
1000
90
902
AJI dimensions in miltimetres
47,6
8,59
140,0
46
0,83
48
71
55
117
57
85
66
168
69
101
81
250
84
124
92
323
96
14,1
103
404
107
15,8
11 8
5,30
122
18 0
132
666
137
20,3
14 7
824
152
225
165
106
171
254
MM>/
""'
,....
MM>/
(le!)
-
20
006
-
11
1·f
22
O,CfJ·
27
015
18
34
23
44
049 ·-::28'c · - sse076
36
68
1,20
45.
86
1 71 . -53
103
244
64
12 3
365
79
151
4 72
89
17 1
592
99
192
7,86
113
219
995
·128
24 7
12 3
142
274
156
160
308
192
177
342
240
199
38,4
304
223
43,2
386
252
486
49,1
620
13 2
190
28,2
206
213
315
232
209
240
35,5
261
264
270
40,0
294
335
305
451
331
425
343
507
368
524
-
-
-
41 2
658
-
-
831
-
_,
-
-
Min Wall
Thld<·
.
8
16 5
-
I>
(kg)
184
463
ClASS20
""'
Min Wall
(kg)
'
_,.,
-
NomlMI
089
142
2 01
288
431
566
711
927
1175
145
183
226
284
359
455
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
APPENDIX B : Conversion factors
Sl units
Mass: kg
Length :m
Time: s
Temperature: K
Conversion factors
g = 9.807 m/s 2
1 US gallon= 0.8327 Imp gallon= 3.785 I
1 psi = 6.895 kPa
1 cP = 1 X 10'3 Pa.s
APPENDIX C1 : Absolute roughness for various pipe materials
New Pipe Material
Roughness
(mm)
Riveted steel
0.9-9.0
Concrete
0.3-3.0
Cast Iron
0.25
Galvanized iron
0.15
Commercial steel, wrought
iron, welded-steel pipe
0.046
Glass, plastic (smooth)
Drawn tubing (brass,
copper)
0.0
0.0015
PVC pipe
0.002
HOPE pipe
0.007
-----"I
, , , , ', II II!
,I , I
o.o9o . I.,
Lafro~aic;~~ceall-ltii-~-T-r·
0.1 oo
~·~:~
j\ _-- 'Tlr-~~ ~
s;
0.060
•
0.050
I
I
I
I
I"-,
I
I >-'5
0.030
I~
o.o2s
I
'
: -
u
-+-~~
~~
I
I
If.'-<
Ii
II
!
H-111---1-[~;~:~-1
0.015
-
~
+H-------- -Turbulent flow
~:-
li
-- -
II - ' \ ' - - I
··
·
-
· -
~
··
·-
·· -
---
i --
- --
-~
· ·
I
-:
'
---
,
'
-
11~--~-3500!(6/lJ)
"
-
1
1
'c
-_,+~ ~ -- =-
__
-
I
I
-
•
~·~~~
0.01~
--
0.008
-
1
.
-t ·1 I 11111·- ·· - -
--
-
-
·- - ·
1
~~~:
6 3
103
2
'r,
I
3 4 56 8
1Q5
2
I
'
--- --. -- --"-..---
o.oo1
"'"'
0.000. 3
J1)_
t--._
. ---
'
l-==t--
,
'
t--.
'
o.ooo 6 0
0.000 4
0.000 2
O.OCJ 1
o.oo·Jo1
3 4 56 8
106
2
3 4 56 8
10 7
2
3 4 56 8
Reynolds number Re
·----
"'0
~:::>
dD
3 4 56 8
104
<
(1)
o.oo 2
--.
2
~
iU
c.
(1)
_
~~f-.
I ~t:::---r-r-
1-- _, -~
_
0 050
0.040
0.03D
rr~l-nlllll1.l !,t~~~lb<' ~~~"
1=.--:- ~H-- ~.-TITIIIIII~~---. I~ltl
o.oo~oors~ J -1-1-1-H-1-8 '·"'"
~:~~~J
0.010
\
Hlll-·----1--1-ii-1-1111- :-1-11-·-·- ' ---- ~§;'±::
f'i::
1~1
ITI
- -- ---
,: --~- r--
draulically smooth pipes
I
L
...
_, __ . ·-
--! , -
- '; --'
1-
I }{
Complete _turbulence, fully rough p1pes -_
r\
I
~~----. ---- -~~~-I
o.o2o
I
t---=:t--.h
I .S
I t
' ·c
----
- §:::F:-h
Hill---+, I-
I'I' Ill! fll
--r-r Ill'1·-·-- -·- -------. ·: --- --- -[-- ~F
I - f - - - - - - --~
z~_ne ~ . t ~~-~-
t-1~ ---r-·f-:::~- i ~
I
'r-.
IHI~---~-~~- r---_
0040HH1
I
Transition
,_
----------------.
1QB
)>
"0
"0
m
z
0
x
0
N
;;:
0
0
0.
'<
0.
n:;·
"'ill
~
3
APPENDIX C3 : Equivalent lengths for various components
Valid for Re > 1000
Lr/D
Description
Globe
valves
Stern
perpendicular
to run
With no obstruction in flat, bevel or plug type seat
With wing or pin guided disc
Fully open
Fully open
340
450
Y-pattern
(no obstruction in flat, bevel or plug type seat)
Stem 60 degrees from run of pipe line
Stem 45 degrees from run of pipe line
Fully open
Fully open
175
145
With no obstruction in flat, bevel or plug type seat
With wing or pin guided disc
Fully open
Fully open
145
200
Angle valves
Gate
valves
Check
valves
Wedge, disc,
double disc or
plug disk
Fully
Three- quarters
One-half
One-quarter
open
open
open
open
13
35
160
900
Pulp stock
Fully open
Three- quarters open
One-half open
One-quarter open
17
50
260
1200
0.5§ .............. Fully open
Conventional swing
0.5§ .............. Fully open
Clearway swing
Globe lift or stop; stern perpendicular to run or Y-pattern
2.0§ .............. Fully open
Angle lift or stop
2.0§ .............. Fully open
In-line ball
2.0 vertical and 0.25 horizontai§ .............. Fully open
Butterfly valves 8 inch and larger
Cocks
Fittings
§
Fully open
Straightthrough
Rectangular plug port area equal to 100% of pipe area
Three-way
Rectangular plug port area equal to 80%
of pipe area (fully open)
Fully open
Flow straight through
Flow through branch
135
50
same as
globe
same as
angle
150
40
18
44
144
90 Degree standard elbow
45 Degree standard elbow
90 Degree long radius elbow
30
16
20
90 Degree Street Elbow
45 Degree Street Elbow
Square Corner Elbow
50
26
57
Standard Tee
20
60
With flow through run (soft Tee)
With flow through branch (hard Tee)
Minimum calculated pressure drop (psi) across valve to provide sufficient flow to lift disk fully
APPENDIX C4 : Resistance coefficient data for piping components
Valid for Re > 2000
PIPE ENTRANCES
L
0.5 < Kr < 1.0
L
L
Kr= 0.5
Kr = 0.04
PIPE EXITS
Kr = 1.0
STRAINERS
Nomd
mm
Kr
inch
Nomd
mm
Kr
inch
25
1
3.70
200
8
2.20
40
1%
3.25
250
10
2.15
50
2
3.00
300
12
2.09
65
2%
2.90
350
14
2.04
80
3
2.75
400
16
2.00
100
4
2.60
500
20
1.95
150
6
2.35
600
24
1.90
VERANDERING IN DIAMETER I CHANGE IN DIAMETER
: ... : .... : ......... ! ..•...... ;
.... -·····. ) •••. : ..•. 1.• '/.'; •• _l .... :. _:,.
I
--:~-~-
'\:./·
.
..:~--~
... ! __ _
.'' '.-· ''--l---!-.6,-3-f---'--+--'--i---:--T.-::--ct-:--',;:i-:\' '--,---;---c--f--,--.."".,--j---,-7."
...
-.·.i:
Kr-DRTA
0.1
6l
<>-)
Loss coefficients K
~
..:••
e.--·-· -
VIIi?.
~-
0.4
13
o-S
TsTUKK£
"'"'
0-"1
'""'"" ~,.,..~,. o,Jo.J
o..
t.o
o._,
Cor 90 degree sharp -edged combining tees
---·/----/----·' ___
, -
+
~
~
f---·· .... ··- -·+--1---/-~-.'-·--
"'
- ······· -· ·-· --l---l-l----/--~-
<0S
·
·· ·I ··-/·-"-!·· · I
• ~~
~ . ~~~~~t8~7-r~·tf~·t·l~-~--/~--~~--~
=! -f c-/
.:<..
·i·
-/
--
•>·~-~~---~-.-~+--.~~-~~+_-7+-~--1~-----~--~~--
.,t~f--~-j~~~-~~-~--~1--·~f§·-~·tf-~-~~~~~
·· .... C-f ·/ -f-f .
.. ------__ ~--------~..-,~~~--.I(E--J..
•• L..~~-~---~--,~---~~~_i~-L-~---~'~-~-·~_J
o-1
o·l
o.l
o•
oS
o-4
o.ll
o9
00
f'Low
Loss coefficients K
23
.u.,,.,
.,;'1
t-o
o.,j;. .3
for 90 degree sharp- edged combining tees
•••
.-~--- .: ..
!+·!+H-'
.
o.l
..,
o.ol:...lJU.Ll...l..L.i_.:...l_!__L:..lJU.Ll-'-.L.i_~-'--L-'-''--L-'-'-L..L--J
o.o
o,l
o.l.
o.J,
o.+
o-S
o . ..:.
o.;
o-8
o-~
l.o
fl.ow RA.r,o 0.1 /Gtl
Loss coefficients K
31
for' 90 degree sharp- edged dividing tees
/oO
d. •
(1.9
d. ' d,
0·8
0·7
~
'<
o.o.
"'.,
o,s
0
'
It~
\.{ o,q.
0.3
0·<
0
0.1
o.z
0-.l
04
c.s
o.~
PLOW <RTIO
0,]
oi0_J
0.3
d. -ct..
1,0
Gyc-'
CP.
-·1
·~tO
0$
0,9
~d.3
·o.:r
0.7
~
',/ -· 0..0
"',, .. o.s
0
'
'X
~
~
.It~
or;
...... 0.3
o.z..
0
0,/
Q,2
0.3
a.~
a.s
t="(.OW REI TIQ
o.;;
o;0
J
0,7
o.-<:
o,,
Ox-fo.J
0.9
/,0
APPENDIX C5 : Resistance coefficient data for two-K method
K=
K,
Re
1
+ Kjl +-d)
d
=
inner diameter of attached pipe in inches
Fitting type
K,
Standard (RiD= 1 ), screwed
Standard (RiD= 1 ), flanged/welded
Long-radius (RiD= 1.5), all types
Elbows
90 °
Tees
Valves
Mitered 1 Weld (90° angle)
elbows 2 Weld (45° angle)
3 Weld (30° angle)
(RiD= 1.5)
4 Weld (22.5° angle)
5 Weld (18° angle)
K.
800
800
800
0.40
0.25
0.20
1000
800
800
800
800
1.15
0.35
0.30
0.27
0.25
500
500
500
500
0.20
0.15
0.25
0.15
45 °
Standard (RiD = 1), all types
Long-radius (RiD= 1.5), all types
Mitered, 1 weld (45° angle)
Mitered, 2 weld (22.5° angle)
180 °
Standard (RiD = 1), screwed
Standard (RiD = 1), flanged/welded
Long-radius (RiD= 1.5), all types
1000
1000
1000
0.60
0.35
0.30
Flow
through
run
Standard , screwed
Long-radius , screwed
Standard, flanged/welded
Stub-in-type branch
500
800
800
1000
0.70
0.40
0.80
1.00
Flow
through
branch
Screwed
200
150
100
0.10
0.50
0.00
300
500
1000
0.10
0.15
0.25
Globe, standard
Globe, angle or Y-type
Diaphragm, dam type
Butterfly
1500
1000
1000
800
4.00
2.00
2.00
0.25
Check
2000
1500
1000
10.00
1.50
0.50
Gate,
ball, plug
Flanged/welded
Stub-in-type branch
Full line size, ~ = 1.0
Reduced trim, ~ = 0.9
Reduced trim, ~ = 0.8
Lift
Swing
Tilting-disk
Note: Use RiD= 1.5 values for RiD= 5 pipe bends, 45° to 180°.Use appropriate tee values for flow through
crosses.
SPECIAL CASES :
K,
K
K.
Pipe entrances
160
0.5
Pipe exit
0
1.0
Orifice
variable
(1-P ')((liP 'J-IJ
APPENDIX C6 : Diagram for prediction of friction pressure loss
INSIOEJ DIAMETER
QUANTITY OF WATER
HDPE/PP PIPE STEEL PIPE _
1 mm rustcoat
--~---
0,05-+-
3
~
-t'L..
j
-'--- 40
0,1
!
--T.
"
1
-L
0,1l
o2I
60
0,3
-t_
0,4
--!,...
"
,-
0,5 -~-
60 -
'
70
10
.:-
50
~
"1
i:
"'
50~
I
4
5
40 ......:::
20
J
j
JO
0,21
40
~
l
----1
~
~
80
2
90
90
100
l
100
-t
3 ---,-
4_]_
"
5 -t-
---
-
j
l '"
150-
i
20
J
J
400
500
-J
_)
40__s-
.r
2000
500
1,5
400
3000
'
:f-+-
OA
0,5
J
r-
j
1,5
2
"
4
~
5000
J-
2
5
'
3-i
'
l
10000
_J-
t
20000
30000
40000
:~
3
m /mio
==t
3i
4
j
10
5
15
-j
"
_i
><
0,5
3
I
2000_3:500
t
''l
0,2
0,3
'l
-(_ 4000
1
'
iI
_,
--:::.._ 50000
1000
----1-
~
,-.::~
300 --!...
300-;- 300
0,1
~
3000
-(_
400
0,2
-'_,
200--f
~
~0,15
1---i
}-
-!
~
--'
300
c
~
500
0.4
0,5
1000
30-+
100
"
400
200
0,1
0,05
--'
"
200 ~-- 200
i
'
-l-
50-:-
J
350
100
10 _;....
I
250
-+-
o.~t
j
0,3
}-
0,03
0,15 ~
50
1_1_
LOSS OF PRESSURE
m/100 m PIPELINE
HDPE/PP PIPE STEEL PIPE
-----~
-l-
35 ------4
~
m/sec
.f/sec I-f/min
·-----~----·---·----~----·~·----~----
j
VELOCITY
20
10
JO
10
100
40
200
300
15
20
20
1he nomogram is based on the Prantl-(oalbrook formula using a k factor of: k =
Factors "ffO)Iicable to other flow formulae ore:
Hazen Williams ...... _........ _.. c ='1 50
h\anning ..................... . n = 0,010
Darcy roughness factor ......... :o--= o,rxn rrm
opo7 rrm
50
APPENDIX 01 : Calculation of APr.K for isothermal compressible flow
Known : Pp T, M, 11, W, D, L, e
=
P!
MP 1
I
RT
w
111
I
p!A
I
Re
p1u 1D
=--
11
Since flow is turbulent for gas flow, solve Colebrook equation for j',
j'=0.02 can be used as first estimate :
e!D
3.7
+
2.51 )
Re/1
Solve isothermal model for P2 , the In-term can be neglected for first estimate:
p2 = , p; - (
:r (: ) (f'j; ;J )
+
21n (
APPENDIX 02: Calculation of Wfor isothermal compressible flow
Known : PI' P2 , T, M, f!, D, L, s
Find first estimate for f' using the von Karman equation for fully rough flow:
-
1
-
IJ:
~
2 log ( _!}__) + 1.74
2s
A ~ "-D2
4
Solve for W using isothermal model:
p2 - p2
I
w~A
\
I
I
( ::) (
2
~
+
2ln (
;J )
w
p,A
u,
--
I
I
Re
~
p u 1D
-1 -
I
fl
Since flow is turbulent for gas flow, solve Colebrook equation for ],
]=0.02 can be used as first estimate:
s/D + 2.51 )
3.7
Re/1
IFf' = /',,, STOP
ELS!O find new Wand repeat until convergence
APPENDIX 03 : Calculation of L1Pr.K for adiabatic compressible flow
Known : PI' Tl' M, 111' k, W, D, L, e
Pt
MP 1
---
RT1
I
Ill
w
--
p 1A
I
Find Re at upstream conditions, assume Re to be constant
throughout:
Since flow is usually turbulent for gas flow, solve Colebrook equation for ],
j'=0.02 can be used as first estimate :
1
11
-2 log (
s!D
3.7
+
2.51 )
Re/1
Solve adiabatic model for p 2
:
Find P2 from the following relation, T2 can then also be calculated:
w)
~ ( P0
2
+
2k (
k-1
Pzp2)
APPENDIX 04: Calculation of Wfor adiabatic compressible flow
Known : P" P2 , T" M, f.lp D, L, s
=
p,
MP 1
RT
I
Find first estimate for f' using the von Karman equation for fully rough flow:
J1
= 2 log (
~)
+ 1.74
1
Solve for Wand p2 simultaneously using the following two relations:
]L
D
=
~
lnp'p
k
+ (
1_(P,)') (k-1
p1
1
(pw)
A
1
2
2k(P~
1
+ k-1
+
2k
P,p,(~)')
W
+ 2_.
k (
)
k-1
r,)
Pz
Find Re at upstream conditions, assume Re to be constant
throughout:
p 1u 1D
Re = fl1
Since flow is usually turbulent for gas flow, solve Colebrook equation for ],
f=0.02 can be used as first estimate :
IFf
e!D
2.51 )
3.7 +
Re/1
~f,"
STOP
ELSE find new Wand repeat until convergence
APPENDIX E : Generalised rheological constants for various fluids
MEDIUM
0.67%
1.5%
3.0%
23%
33%
10%
4%
54.3%
14.3%
21.2%
25.0%
31.9%
36.8%
40.4%
Carboxymethylcellulose in water (CMC)
CMC
CMC
lime in water
lime in water
napalm in kerosene
paper pulp in water
fine cement rock in water
clay in water
clay in water
clay in water
clay in water
clay in water
clay in water
n'
K'
0.716
0.554
0.566
0.178
0.171
0.520
0.575
0.153
0.350
0.335
0.285
0.251
0.176
0.132
0.121
0.920
2.80
1.04
0.983
1.18
6.13
0.331
0.034
0.086
0.204
0.414
1.07
2.30
FLOW OF MULTIPHASE MIXTURES
1000
"
....
~
€<
Vise-Turb
ReL
>2000
<1000
>2000
<1000
m
ReG
>2000
>2000
<1000
<1000
x
Turb- turb
Vise- turb
Turb- vise
Vise- vise
100
€<
Turb-Turb
Reynolds No.
8
6
4
3
2
8
6
4
Turb-Vise
Vise-Vise
~
"tl
"tl
2
0
"T1
::0
CD
!!!.
a·
<t>L
::J
0"
i
CD
::J
N.B. If X 2: 1.0 use <I>L
If X<1.0 use <t>G
3
2
~
m
::J
a.
'El<
0'
~
4
3
2
t
1
!
Turb- turb
Vise- turb
<t>G~ Turb- vise
Vise- vise
1
2
1
'
"0
::J"
m
-~"'
CD
H3JHjr~-;c.-;--;}~
~I
6 8 0.012
6 8 0.1 2
6 81.0 2
6 81 0 2
Iii
4
4
4
X=
V -~PL/-~PG
4
1 J1iiJqil1
4 6 8100
APPENDIX G1 :L1P 1"' and L1P 100ft for mild steel systems
MILD STEEL AND NO CAVITATION
•
LIQUIDS
FLOWRATE
•
m3/h
lgpm
LIP,"' (kPa)
L1P1ooft (psi)
< 25
< 100
25- 125
125- 1250
> 1250
100- 500
500- 5 000
> 5000
0.35- 1.35
0.25-0.90
0.10-0.50
0.04-0.25
1.5-6.0
1.0-4.0
0.5- 2.0
0.2- 1.0
GASES AND VAPOURS
LIP1 m- 0.000164 x UPSTREAM PRESSURE kPa
L1P100ft- 0.005 x UPSTREAM PRESSURE psi
MILD STEEL WITH CAVITATION
•
LIQUIDS AT BUBBLE POINT
API m- 0.04 kPa
Lll'woft- 0.2 psi
APPENDIXG2
LUDWIG SNELHEDE I VELOCITIES
Suggested trial veloCity
Fluid
Acetylene
{Observe pressure
limitations)
.Air, . . . .!0-30psigl
lo
206,84
,,,
m/!.
66,67
20.32
Steel
66,67
20,32
Steel
6
100
6
1,83
30,1.8
\,83
Steel
Steel
Steel
'
I, 22
10,16
1,22
1,83
Glass
Gloss
Steel
Steel
Pipe
mot~riol
~o:.Pogl
Ammonia
Liquid
Go'
Benzene
Bromine
Liquid
Go'
Calcium ehlor ide
C.orbon tetrachloride
Chi or 1ne ldr yl
liquid
33,33
'6
s
33,33 - 83,33
Go<
\, s
2
\0,16 - 25,1.0
Steel, Sch.BO
Steel, Sch.80
Chloroform
6
Liquid
Gas
33,33
Ethylene gas
Ethylene dibromide
Ethylene dichloride
E\hylene glycol
100
'6
6
1,83
10,16
30,1.8
1,22
1,83
1,83
20,32
Copper & steel
Copper & steel
Steel
Glass
Steel
Steel
Steel
Hydrogen
66,67
Hydrochloric acid
Liquid
Gas
66,67
20,32
Rubber lined
Rvbber-lioed steel,
Saran, Ha"e9
6
100
6
1,83
20.32
30,48
1,83
Steel
Steel
Steel
Sleet
Ambient temperature
30 Mo:<.
9,11, Mcx.
low temperature
66,67
s
1.S2
Hethyl chloride
liquid
Gas
66,67
Natural gas
Oils, lubrico1ing
Oxygen
P erchlor e t/'ly len e
Propylene glycol
Sodium hydroxide
0-30%
30 -50%
so -73%
Sodium chloride solution
N~
s otids
With solids
6
5
6
s
's
7,5
( 6 !-lin.- IS Mox.l
20,32
1,83
l,S2
1,83
1,5 2
1,22
Steel (JOO p<>ig 1-lax.l
(2068,44 KPog Hox.
Type JOt, SS
Steel
Steel
Ste et
Steel
Steel
''
'
nicKel
oicl<el
nicl<e\
I, 52
Steel
2,29
Monel 0' oicl<el
{1,83-I.,S7)
{Min. -!-lox.!
Steam
Saturated:
to- 30 ::>Sigl
{0- 206,84 KPog}
Saturated or $LSf?Nheo\e{(
{30-\50 ;JS19]
(20C.,8t. i03t.,22 kPng]
Superhec1ed, (ISJ DS><J uoi
(103<., 12 kPo9 u;.J
Shor1 l1nes
Sulluric oc.id
88-93%
93·100%
S<.Jllur dioxide
Styrene
Tr ichlor e tnyle n e
'/my\
~htor1de
'linylident;> c.n!orioe
Wolt;>r
Pump suction U•t:>s
Averc9e SN"icc
Mo.~. ec.o,..om-cc.~ (v~ol}
SE:o cr>d brc<..ki~n w:ru.
NOTE:
=
66,67100.
100
20,3230,4S.
30,1.8
166.67.
\08.3 3 .
250.
2$0 ,'-.lox.
SO, 60.
33,02.
76,20.
76,20 Mo;o;.
-
''
66.b7
6
6
6
6
3 -8
3 -e
{0119. SJ
7- \0
S-8
tJ ~-"J
~- 12
{J !-lin)
-
1,22
I, 22
20,32
I ,&3
1,83
t,eJ
'\,83
0,914-2.1.4
0,9\t. -2,44
\Oil'}, \,'OJ)
2.13- J,OS
152-2,1.£.
Steel
Steel
Steel
SS-316 ieod
Cost oron i iteel, Scto.80
Steel
Steel
Steel
Steel
Steel
Steel
Sto:e I
i0.91 '-l:r.)
I,S2- 3,66.
tO.St
~1inJ
The V~!-\ocal~s e1re suqqeshve o<'~ly ond ctre to Pe u~>e<i t¢ appto)fimote tine ~iH
storlm<j point lor pas~rc- Qrop c.pfcvkttions. fhe linn! tin<! size. ~r.ould oe 5vch
9i11>? o."\1"1 ecomtnlC-O\ boSon(::e oetwe<'n pressure drop on.o re(l<;."On<fbte ve!oc:ty.
0'
0
o~
to
APPENDIX H : C., values
C., values
Size (nom)
View publication stats
inch
mm
single port
double port
0.75
20
5.4
8
1
25
9
12
1.25
32
14
18
1.5
40
21
28
2
50
36
48
2.5
65
54
72
3
80
75
110
4
100
124
195
6
150
270
450
8
200
480
750
10
250
750
1160
12
300
1080
1620
14
350
1470
2000
16
400
1920
2560