Fluid Flow Design Calculation
Equation 12
Metric :
P1 V12 P V2
Z e1 1000 Z e2 1000 2 2 H f
1 2g 2 2g
Customary :
144P1 V12 144P2 V22
Z e1 Z e2 Hf
1 2g 2 2g
where :
Z e vertical elevation rise of pipe, m ft
P pressure, kPa psia
density of liquid, kg / m 3 lb / ft 3
V average velocity, m / sec ft / sec
g acceleration of gravity, 9.81m / sec 2 32.2ft / sec 2
H f head loss, m ft
Velocity, as used herein, refers to the average velocity of a fluid at a given cross section,
and is determined by the steady state flow equation:
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Equation 13
Metric :
Q W
V s
3600A A
Customary :
Q Ws
V
A A
where :
V average velocity, m / sec ft / sec
Q rate of flow, m 3 / sec ft 3 / sec
A cross sectional area of pipe, m 2 ft 2
Ws rate of flow, kg / sec lb / sec
density of liquid, kg / m 3 lb / ft 3
2.4. Flow Regimes
Experiments have demonstrated that there are two basic types of flow in pipe, laminar and
turbulent. In laminar flow, fluid particles flow in a straight line, while in turbulent flow the
fluid particles move in random patterns transverse to the main flow.
At low velocities, fluid flow is laminar. As the velocity increases, a "critical" point is reached
at which the flow regime changes to turbulent flow. This "critical" point varies depending
upon the pipe diameter, the fluid density and viscosity, and the velocity of flow.
Reynolds showed that the flow regime can be defined by a dimensionless combination of
four variables. This number is referred to as the Reynolds Number (Re), and is given by:
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Equation 14
Metric :
DV
Re
Customary :
D V 1488DV 124dV 7738SG dV
Re
'
where :
Re Reynolds number, dimensionless
density, kg / m 3 lb / ft 3
d pipe ID, mm in
D pipe ID, m ft
V average velocity, m / sec ft / sec
' viscosity, lb / ft - sec cp 0.000672
absolute viscosity, Pa - sec cp
SG specific gravity of liquid relative to water
At Re < 2000 the flow shall be laminar, and when Re > 4000 the flow shall be turbulent. In
the "critical" or "transition" zone, (2000 < Re < 4000), the flow is unstable and could be
either laminar or turbulent.
The Reynolds number can be expressed in more convenient terms. For liquids, Equation
(14) can be shown to be:
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Equation 15
Metric :
Rel 353.13
SG Ql
d
Customary :
Rel 92.1
SG Ql
d
where :
absolute viscosity, Pa - sec cp
d pipe ID, mm in
SG specific gravity of liquid relative to water
Ql liquid flow rate, m 3 / hr BPD
The Reynolds number for gas flow can be shown to be:
Equation 16
Metric :
Qg S
Reg 0.428
d
Customary :
Qg S
Reg 20,000
d
where :
Qg gas flow rate, std m 3 / hr MMSCFD
S specific gravity of gas relative to air
d pipe ID, mm in
absolute viscosity, Pa - sec cp
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2.5. Darcy-Weisbach Equation for Pressure Drop
The head loss due to friction is given by the Darcy-Weisbach equation as follows:
Equation 17
fLV 2
Hf
D2g
where :
L length of pipe, m ft
D pipe ID, m ft
f Moody friction factor
V average velocity, m / sec ft / sec
g acceleration of gravity, 9.81 m / sec 2 32.2 ft / sec 2
H f pipe friction head loss, m ft
Equations 12 and 17 can be used to calculate the pressure at any point in a piping system
if the pressure, average flow velocity, pipe inside diameter, and elevation are known at any
other point. Conversely, if the pressures, pipe inside diameter, and elevations are known
at two points, the flow velocity can be calculated. Neglecting the head differences due to
elevation and velocity changes between two points, Equation 12 can be reduced to:
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Equation 18
Metric :
P1 - P2 P 9.81 10 -3 H f
Customary :
P1 - P2 P Hf
144
where :
P pressure drop, kPa psi
H f pipe friction head loss, m ft
density of liquid, kg / m 3 lb / ft 3
Substituting Equation 17 into Equation 18 and expressing pipe inside diameter in inches:
Equation 19
Metric :
fLV 2
P 0.5
d
Customary :
fLV 2
P 0.0013
d
where :
d pipe ID, mm in
f Moody friction factor
density of liquid, kg / m 3 lb / ft 3
L length of pipe, m ft
V average velocity, m / sec ft / sec
P pressure drop, kPa psi
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2.6. Moody Friction Factor
The Darcy-Weisbach equation can be derived rationally by dimensional analysis, except for
the friction factor (f), which shall be determined experimentally. Considerable research has
been done in reference to pipe roughness and friction factors. The Moody friction factor is
generally accepted and used in pressure drop calculations.
Some texts including API RP 14E utilize the "Fanning friction factor," which is one fourth
(1/4) the value of the Moody friction factor, restating the Darcy-Weisbach equations
accordingly, where:
f fanning 1 / 4f
This has been a continual source of confusion in basic engineering fluid analysis. This
Tutorial uses the Moody friction factor throughout. The reader is strongly cautioned always
to note which friction factor (Moody or Fanning) is used in the applicable equations and
which friction factor diagram is used as a source when calculating pressure drops.
The friction factor for fluids in laminar flow is directly related to the Reynolds Number (Re <
2000), and is expressed:
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Equation 20
Metric :
f 64
dV
Customary :
64
f 0.52
Re dV
where :
f Moody friction factor
Re Reynolds number
absolute viscosity, Pa - sec cp
d pipe ID, mm in
V average velocity, m / sec ft / sec
density of fluid, kg / m 3 lb / ft 3
If this quantity is substituted into Equation 19, pressure drop in pounds per square inch for
fluids in laminar flow becomes:
Equation 21
Metric :
LV
P 32
d2
Customary :
LV
P 0.000676
d2
The friction factor for fluids in turbulent flow (Re > 4000) depends on the Reynolds number
and the relative roughness of the pipe. Relative roughness is the ratio of pipe absolute
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roughness, , to pipe inside diameter. Roughness is a measure of the smoothness of the
pipe's inner surface. Table 1 shows the absolute roughness, , for various types of new,
clean pipe. For pipe which has been in service for some time it is often recommended that
the absolute roughness to be used for calculations shall be up to four times as much as the
values shown in Table 1.
Table 1: Pipe Roughness
Absolute Roughness ()
Type of Pipe
(New, clean condition) (mm) (ft) (in)
Unlined Concrete 0.30 0.001-0.01 0.012-0.12
Cast Iron - Uncoated 0.26 0.00085 0.0102
Galvanized Iron 0.15 0.0005 0.006
Carbon Steel 0.046 0.00015 0.0018
Fiberglass Epoxy 0.0076 0.000025 0.0003
Drawn Tubing 0.0015 0.000005 0.00006
The friction factor, f, can be determined from the Moody diagram, Figure 5, or from the
Colebrook equation:
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Fluid Flow Design Calculation
Equation 22
1 2.51
- 2 log 10
1 3.7D 1
f 2 Re f 2
where :
f Moody friction factor
D pipe ID, m ft
Re Reynolds number
absolute roughness, m ft
The pressure drop between any two points in a piping system can be determined from
Equation 21 for laminar flow, or Equation 19 for turbulent flow using the friction factor from
Figure 5 or Equation 22.
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Figure 5: Friction Factor as a Function of Reynolds Number and Pipe Roughness
(Courtesy of API)
2.7. Effect of Elevation Changes
In single phase gas or liquid flow the pressure change between two points in the line shall
be affected by the relative elevations of those points but not by intermediate elevation
changes. This is because the density of the flowing fluid is nearly constant and the
pressure increase caused by any decrease in elevation is balanced by the pressure
decrease caused by an identical increase in elevation.
In Figure 6, case A, the elevation head increases by H from point 1 to point 2. Neglecting
pressure loss due to friction, the pressure drop due to elevation change is given by:
P1 - P2 H
144
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Figure 6: Effect of Elevation Changes on Head
In case B, the elevation head decreases by H from point 1 to point A. Neglecting pressure
loss due to friction, the pressure increase from 1 to A is determined from Equation 12:
PA - PB - H
144
Similarly, elevation pressure changes due to other segments are:
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PA - PB 0
PB - P2 2H
144
The overall pressure change in the pipe due to elevation is obtained by adding the
changes for the individual segments:
P1 - P2 P1 - PA PA - PB PB - P2
P1 - P2 0 - H 0 2H
144
P1 - P2 H
144
Thus, for single-phase flow, the pressure drop due to elevation changes is determined
solely by the elevation change of the end points. Equation 12 can be rewritten as:
Equation 23
Metric :
PZ 9.79 SG Z
Customary :
PZ 0.433 SG Z
where :
PZ pressure drop due to elevation increase, kPa psi
Z total increase in elevation, m ft
SG specific gravity of liquid relative to water
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Fluid Flow Design Calculation
In two-phase flow, the density of the fluids in the uphill runs is higher than the density of the
fluids in the downhill runs. In downhill lines flow is stratified with liquid flowing faster than
gas. The depth of the liquid layer adjusts to the depth where the static head advantage is
equal to the pressure drop due to friction, and thus the average density of the mixture
approaches that of the gas phase.
The uphill segments at low gas rates are liquid full, and the density of the mixture
approaches that of the liquid phase. As a worst case condition, it can be assumed that the
downhill segments are filled with gas and the uphill segments are filled with liquid.
Referring to Figure 6, case A, assuming fluid flow from left to right and neglecting pressure
loss due to friction:
P1 - P2 H l
144
For Case B:
g
P1 - PA - H
144
PA - PB 0
PB - P2 2H l
144
Thus:
P1 - P2 l 2 l - g
144
Since l g
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Fluid Flow Design Calculation
P1 - P2 2H l
144
Thus, one would expect a higher pressure drop due to elevation change for case B than for
case A even though the net change in elevation from point 1 to point 2 is the same in both
cases.
So, neglecting pressure changes due to any elevation drops, the maximum pressure drop
due to elevation changes in two-phase lines can be estimated from:
Equation 24
Metric :
PZ 9.79 SG Z e
Customary :
PZ 0.433 SG Z e
where :
Z e sum of vertical elevation rises only, m ft
PZ pressure drop due to elevation changes, kPa psi
SG specific gravity of liquid relative to water
With increasing gas flow, the total pressure drop may decrease as liquid is removed from
uphill segments. More accurate prediction of the pressure drop due to elevation changes
requires complete two-phase flow models that are beyond the scope of this manual. There
are a number of proprietary computer programs available that take into account fluid
property changes and liquid holdup in small line segments; they model pressure drop due
to elevation changes in two-phase flow more accurately.
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3. Pressure Drop in Piping
3.1. Liquid Flow (General Equation)
For flowing liquids in facility piping, the density is constant throughout the pipe length.
Equation 19 can be rewritten to solve for either pressure drop or flow rate for a given length
and diameter of pipe as follows:
Equation 25
Metric :
2
fL Ql SG
7
P 6.266 10
d5
Customary :
2
fL Ql SG
P 1.15 10 -5
d5
Equation 26
Metric :
1
P d 5 2
Ql 1.265 10 -4
fL SG
Customary :
1
P d 5 2
Ql 295
fL SG
The most common use of Equations 25 and 26 is to determine a pipe diameter for a given
flow rate and allowable pressure drop. First, however, the Reynolds number shall be
calculated to determine the friction factor. Since the Reynolds number depends on the
pipe diameter, the equation cannot be solved directly. One method to overcome this
disadvantage is to assume a typical friction factor of 0.025, solve for diameter, compute a
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Reynolds number, and then compare the assumed friction factor to one read from Figure
5. If the two are not sufficiently close, iterate the solution until they converge.
Figure 2.2 in API RP 14E can be used to approximate pressure drop or required pipe
diameter. It is based on an assumed friction factor relationship which can be adjusted to
some extent for liquid viscosity.
3.2. Gas Flow
3.2.1. General Equation
1. The Darcy equation assumes constant density throughout the pipe section
between the inlet and outlet points. While this assumption is valid for liquids, it is
incorrect for gas pipelines, where density is a function of pressure and
temperature. As gas flows through the pipe the drop in pressure due to head
loss causes it to expand and, thus, to decrease in density. At the same time, if
heat is not added to the system, the gas will cool and tend to increase in density.
In control valves, where the change in pressure is nearly instantaneous, (and
thus no heat is added to the system), the expansion can be considered
adiabatic. In pipe flow, however, the pressure drop is gradual and there is
sufficient pipe surface area between the gas and the surrounding medium to add
heat to the gas and thus to keep it at constant temperature. In such a case the
gas can be considered to undergo an isothermal expansion.
2. On occasion, where the gas temperature is significantly different from ambient,
the assumption of isothermal (constant temperature) flow is not valid. In these
instances, greater accuracy can be obtained by breaking the line up into short
segments that correspond to small temperature changes.
3. The general isothermal equation for the expansion of gas can be given by:
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Fluid Flow Design Calculation
Equation 27
Metric :
2 2
g A2 P1 - P2
9
Ws 2 1.322 10
fL P1 P1
v 2 ln
D P2
Customary :
144g A2 P1 2 - P2 2
Ws 2
fL P P1
v 2 ln 1
D P2
where :
Ws rate of flow, kg / sec lb / sec
g acceleration of gravity, 9.81 m / sec 2 32.2 ft / sec 2
A cross - sectional area of pipe, m 2
ft
2
v specific volume of gas at upstream conditions, m 3 / kg ft 3 / lb
f Moody friction factor
l length of pipe, m ft
D pipe ID, m ft
P1 upstream pressure, kPa psia
P2 downstream pressure, kPa psia
4. This equation assumes that:
a) No work is performed between points 1 and 2; i.e., there are no compressors or
expanders, and no elevation changes.
b) The gas is flowing under steady state conditions; i.e., there are no acceleration
changes.
c) The Moody friction factor, f, is constant as a function of length. There is a small
change due to a change in Reynolds number, but this may be neglected.
5. For practical pipeline purposes,
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Fluid Flow Design Calculation
P1 fl
2 ln
P2 D
6. Making this assumption and substituting it into Equation 27, one can derive the
following equation:
Equation 28
Metric :
2 2 S Qg2 ZTfL
P1 - P2 52,430
d5
Customary :
S Qg2 ZTfL
P1 2 - P2 2 25.2
d5
7. The "Z" factor will change slightly from point 1 to point 2, but it is usually assumed to be
constant and is chosen for an "average" pressure.
8. Please note that
P12 - P22
2
in Equation 28 is not the same as ( P) . Rearranging Equation 28 and solving for Qg we
have:
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Fluid Flow Design Calculation
Equation 29
Metric :
d 5 P12 - P22
Qg 4.367 10 -3
Z TfL S
Customary :
d 5 P12 - P22
Qg 0.199
Z TfL S
9. As was the case for liquid flow, in order to determine a pipe diameter for a given flow
rate and pressure drop, it is first necessary to estimate the diameter and then to
compute a Reynolds number to determine the friction factor. Once the friction factor is
known, a pipe diameter is calculated and compared against the assumed number. If
the two are not sufficiently close, the process is iterated until they converge.
3.2.2. Small Pressure Drops
For small pressure drops, an approximation can be calculated. The following formula can
be derived from Equation 28 if P1 - P2 < 10 percent of P1 and it is assumed that
P12 - P22 2P1 P
Equation 30
Metric :
S Qg Z T f L
2
P 26,215
P1 d 5
Customary :
S Qg Z T f L
2
P 12.6
P1 d 5
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