Ref.1: Brill & Beggs, Two Phase Flow in Pipes, 6th Edition, 1991.
Chapter 1.
Ref.2: Menon, Gas Pipeline Hydraulic, Taylor & Francis, 2005,
Chapter 2.
General Flow Equation

U1 : InternalEnergy
P1 V1 : Expansionor Compression Energy
m v12
: KineticEnergy
2 gc
m g Z1
: P otentialEnergy
gc
Energy balance at steady state:

mv12 mgZ1
mv22 mgZ2


U1  P1V1 

 q  Ws  U 2  P2V2 

2 gc
gc
2 gc
gc

Where q  Heat added to the fluid and Ws  Work done on the fluid

U2
P2 V2
m v22
2 gc
m g Z2
gc
General Flow Equation
Dividing by m and writing in differential form:
 P  v dv g dZ
dU  d  

 dq  dWs  0
gc
   gc
By using the enthalpy and entropy definition:
P
dh  dU  d ,

dP
dh  TdS 
dP

v dv g dZ
 TdS 


 dq  dWs  0

gc
gc
General Flow Equation
For irreversible process TdS  dq  d(losses) therefore:
dP
No Work
v dv g dZ


 d(losses)  dWs  0

gc
gc
For an inclined pipe, dZ  dL sin  therefore:
 dP   v dv g  sin   d(losses)




gc
dL
 dL  gc dL
For Up Flow :   0
For Down Flow :   0
 dP 


 dL  friction
General Flow Equation
Fanning friction factor ( f ):
w
f 
  v2 


 2gc 
Wall shear stress:
P
P+dP
4 w 2  v 2 f
 d2 
 dP 

   w ( d ) dL   
 
P  ( P  dP)
d
gc d
 dL  f
 4 
Darcy or Moody friction factor (fm):
fm  v2
 dP 
fm  4 f     
2 gc d
 dL  f
General Flow Equation
Pressure gradient in pipe:
2
 dP   v dv g  sin  f m  v




gc
2 gc d
 dL  gc dL
 dP 
 dP 
 dP 
 dP 
 
 

  



 dL total  dL  acceleration  dL elevation  dL  friction
Usually negligible
Zero for horizontal pipe
Single Phase Gas Flow
Reynolds Number
Reynolds Number in Gas Pipeline:
N Re
d (ft) v(ft/sec)  (lbm /ft3 )
 1488
 (cp)
Mass flow rate v A  g  qg sc  g sc  v 
 N Re
 4 q g sc 0.0764 g
d
2


d
g

 1488

qg sc  g sc
A g

 g

20.14 q g sc (Mscfd)  g


 (cp) d (in )
Single Phase Gas Flow
Friction Factor
Laminar Flow (NRe < 2100):
64
fm 
N Re
Turbulent Flow (NRe > 2100): Moody Diagram
Smooth Wall Pipe:
0.32
f m  0.0056 0.5 NRe
for 3103  NRe  3106
Rough Wall Pipe:
  21.25 
1
 1.14  2 log10   0.9  , Commonly :   0.0006 in
fm
 d N Re 
Single Phase Gas Flow
General Equation
 v dv g  sin  f m  v 2
 dP 

  
 dL  gc dL
gc

2 gc d
, v
4  g sc qg sc
 d 2 g

P M
 4q g sc g
sc
 PM g  
RTsc

fm 
 z RT  
 PM g 
PM g
 g
  d 2

 g sin 



z g RT
 dP   z g RT 




gc
2 gc d
 dL 
 dP 


 dL 
PM g g sin 
g c z g RT








2
8 Psc2 M g q g2sc z g T f m
 2 g cTsc2 d 5 P R
Single Phase Gas Flow
General Equation
C2
If T and zg are constant (T=Tav and zg=zav):
2 2
2
2


8
P
q
z
T
M
g
sin



dP 

sc g sc av av f m
g
2

  P  2 2 5
 P
  
dL   g c zav RTav  
 gTsc d sin  


P1
P2
M g g L sin  S
P dP


2
2
P C
g c zavTav R
2


 P12  C 2 
2
S 2
2 S


 ln 2
 S  P1  e P2  C e 1
2 
 P2  C 
Single Phase Gas Flow
General Equation
P e P 
2
1
S
2
2
16Psc2 qg2sc zavTav f m L M g
 T d gc S R
2
2
sc
5
e
S

 1 Where S 
0.0375 g Z (ft)
zavTav (o R )
5
o
2
2
.
527

10

z
f
T
(
R)
q
eS 1 
g av m av
g sc ( Mscfd) 
2
S 2
 L(ft)

P1  e P2 
5
d (in)
S 

Le
qg sc


 P e P d
 198.94 
  g zavTav f m Le
2
1
s
2
2
5



0.5
 Tsc
 5.6354
 Psc


  P e P d
 
   g zavTav f m Le
2
1
s
2
2
eS 1
For HorizontalPipe: lim
 1  Le  L
S 0
S
5



0.5
Single Phase Gas Flow
Average Pressure
P e P 
2
1
S
2
2
16Psc2 qg2sc zavTav f m L M g
 T d gc S R
2
2
sc
5
e
S

1
Px2  P22  K L(1  x)
P12  Px2  K L x Where 0  x  1

P12  Px2 Px2  P22

 Px  P12  x( P12  P22 )
x
1 x
1
Pav  
0

0.5
2
P22  2  P13  P23 
   2

Px dx  Pav   P1 
2 
3
P1  P2  3  P1  P2 
Single Phase Gas Flow
Erosional Velocity
Higher velocities will cause erosion of the pipe interior
over a long period of time. The upper limit of the gas
velocity is usually calculated approximately from the
following equation:
vmax (ft/s) 
100
 g (lbm/ft )
3
Usually, an acceptable operational velocity is 50% of the above.
Single Phase Gas Flow
Pipeline Efficiency
In Practice, even for single-phase gas flow, some water or
condensate may be present. Some solids may be also
present. Therefore the gas flow rate must be multiply by
an efficiency factor (E).
A pipeline with E greater than 0.9 is usually considered
“clean” .
Single Phase Gas Flow
Non-Iterative Equations
Several equations for gas flow have been derived from General
Equation. These equations differ only in friction factor relation
assumed:
Gas Transmission Pipline
1. AGA equation
2. Weymouth equation
3. Panhandle A equation
4. Panhandle B equation
Gas Distribution Pipeline
1. IGT equation
2. Spitzglass equation
3. Mueller equation
4. Fritzsche equation
Single Phase Gas Flow
AGA Equation
The transmission factor is defined as:
2
F
fm
First, F is calculated for the fully turbulent zone. Next, F is
calculated based on the smooth pipe law. Finally, the smaller of
the two values of the transmission factor is used.

d

F

4
log
3
.
7
 Fully Turbulent
10 




Min 
 F  4 log10  N Re , Ft  4 log10  N Re   0.6 Sm oothPipe
 1.4125F 
 F 

t 

 t 
Single Phase Gas Flow
Weymouth Equation
The Weymouth equation is used for high pressure, high flow
rate, and large diameter gas gathering systems.
The Weymouth friction factor is:
0.032
f m  1/ 3
d
Single Phase Gas Flow
Panhandle A Equation
The Panhandle A Equation was developed for use in large
diameter natural gas pipelines, incorporating an efficiency factor
for Reynolds numbers in the range of 5 to 11 million. In this
equation, the pipe roughness is not used.
The Panhandle A friction factor is:
0.0768
f m  0.1461
N Re
Single Phase Gas Flow
Panhandle B Equation
The Panhandle B Equation is most applicable to large diameter,
high pressure transmission lines. In fully turbulent flow, it is
found to be accurate for values of Reynolds number in the range
of 4 to 40 million.
The Panhandle B friction factor is:
0.00359
f m  0.03922
N Re
Single Phase Gas Flow
Gas Transmission Equations
A. Comparison of the calculated Output Pressure by AGA,
Colebrook, Weymouth and Panhandle equations: Figure 2.5
B. Comparison of the calculated Flow rate by AGA, Colebrook,
Weymouth and Panhandle equations: Figure 2.6
We therefore conclude that the most conservative flow equation
that predicts the highest pressure drop is the Weymouth equation
and the least conservative flow equation is Panhandle A.
Single Phase Gas Flow
IGT Equation
The IGT equation proposed by the Institute of Gas Technology is
also known as the IGT distribution equation:
qg sc
 Tsc
 35.861
 Psc
 P  e P 

 


   T  Le 
2
1
0.8
g
av
s
2
2
0.2
0.555
d 2.667 ,   cp
Single Phase Gas Flow
Spitzglass Equation
The Spitzglass equation originally was used in fuel gas piping
calculations. This equation has two version
A. Low pressure (less than 1 psig):
0.5



 Tsc  
P1  P2
 d 2.5
q g  278.956   
 Psc    g Tav Le (1  3.6  0.03d ) 
d


sc
B. High pressure (more than 1 psig):
0.5
q g sc



 Tsc  
P12  e S P22
 d 2.5
 53.016   
 Psc    g Tav z av Le (1  3.6  0.03 d ) 
d


Single Phase Gas Flow
Mueller and Fritzsche Equation
The Mueller equation is:
qg sc
 Tsc
 35.4509
 Psc


P

e
P

 

T 
Le 
 
2
1
0.7391
g
av
s
2
2
0.2609
0.575
d 2.725 ,   cp
The Fritzsche formula, developed in Germany in 1908, has found
extensive use in compressed air and gas piping:
qg sc
 Tsc
 41.28 
 Psc
 P  e P
 

T L

2
s 2
1
2
0.8587
g
av e




0.538
d 2.69
16 in., 100 MMSCFD, 80°F
roughness of 700 μ in. for AGA and Colebrook,
pipeline efficiency of 0.95 in Panhandle and Weymouth
30 in., 100 miles, 80°F, output pressure of 800 psig
roughness of 700 μ in. for AGA and Colebrook,
pipeline efficiency of 0.95 in Panhandle and Weymouth