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

Energy balance at steady state:

Energy Potential :

Energy Kinetic :2

Energyn Compressioor Expansion :Energy Internal:

1

21

11

1

c

c

g

Zgmg

vm

VPU

c

c

g

Zgmg

vm

VPU

2

22

22

2

2

ccs

cc g

mgZ

g

mvVPUWq

g

mgZ

g

mvVPU 2

22

2221

21

111 22

fluid on the done Work and fluid the toaddedHeat Where sWq

General Flow Equation

Dividing by m and writing in differential form:

By using the enthalpy and entropy definition:

0dddd

dd

s

cc

Wqg

Zg

g

vvPU

P

SThP

Uhd

dd,ddd

0ddddd

d scc

Wqg

Zg

g

vvPST

General Flow Equation

For irreversible process therefore:

For an inclined pipe, therefore:

0d)d(ddd

scc

Wlossesg

Zg

g

vvP

)(ddd lossesqST

No Work

sindd LZ

L

losses

g

g

Lg

vv

L

P

cc d

)(dsin

d

d

d

d

0 : FlowDown For 0 :Flow For Up

frictionL

P

d

d

General Flow EquationFanning friction factor ( f ):

Wall shear stress:

Darcy or Moody friction factor (fm):

c

w

gv

f

2

2

P P+dP

Ldd

PPP w d)(4

)d(2

dg

fv

dL

P

c

w

f

224

d

d

dg

vf

L

Pff

c

m

fm 2d

d4

2

dg

vf

g

g

Lg

vv

L

P

c

m

cc 2

sin

d

d

d

d 2

General Flow Equation

Pressure gradient in pipe:

frictionelevationonacceleratitotal L

P

L

P

L

P

L

P

d

d

d

d

d

d

d

d

Usually negligible Zero for horizontal pipe

Single Phase Gas FlowReynolds Number

Reynolds Number in Gas Pipeline:

)cp(

)/ftlb()ft/sec()f(1488

3m

Re vtd

N

g

ggggg A

qvqAv scsc

scsc

rate flow Mass

)in()cp(

)Mscfd(14.20

0764.04

14882

Re d

qd

qd

N gg

gg

gg

sc

sc

Single Phase Gas FlowFriction Factor

Laminar Flow (NRe < 2100):

Turbulent Flow (NRe > 2100): Moody Diagram

Smooth Wall Pipe:

Rough Wall Pipe:

Re

64

Nfm

6Re

332.0Re 1031035.00056.0 NforNfm

in0006.0:,25.21

log214.11

9.0Re

10

CommonlyNdfm

Single Phase Gas FlowGeneral Equation

g

gg

c

m

cc d

qv

dg

vf

g

g

Lg

vv

L

Pscsc

2

2 4,

2

sin

d

d

d

d

dg

RTz

PMd

RT

MPq

RTz

PMf

g

gRTz

PM

L

P

c

g

g

sc

gscg

g

gm

c

g

g

sc

2

4

sin

d

d

2

2

RPdTg

fTzqMP

RTzg

gPM

L

P

scc

mgggsc

gc

g sc

522

228sin

d

d

Single Phase Gas Flow General Equation

sin

8sin

d

d522

22222

dgT

fTzqPP

RTzg

gM

L

PP

sc

mavavgsc

avavc

g sc

If T and zg are constant (T=Tav and zg=zav):

2

sind1

222

S

RTzg

LgM

CP

PPP

Pavavc

g

SCP

CP

22

2

221ln 122

22

1 SS eCPeP

C2

5.052

22

1

5.052

22

1 6354.594.198

emavavg

s

sc

sc

emavavg

s

g LfTz

dPeP

P

T

LfTz

dPePq

sc

S

eL

d

qTfzPeP

SgavmavgS sc

1(ft)

in)(

Mscfd)(R)(10527.25

2o52

22

1

Single Phase Gas Flow General Equation

116

522

222

22

1 S

csc

gmavavgscS eRSgdT

MLfTzqPPeP sc

Le

)R(

)ft(0375.0o

avav

g

Tz

ZSWhere

LLS

ePipeHorizontalFor e

S

S

11

lim:0

Single Phase Gas Flow Average Pressure

10221 xWherexLKPP x )1(2

22 xLKPPx

116

522

222

22

1 S

csc

gmavavgscS eRSgdT

MLfTzqPPeP sc

5.022

21

21

22

2221 )(

1PPxPP

x

PP

x

PPx

xx

22

21

32

31

21

22

1

1

0 3

2

3

2d

PP

PP

PP

PPPxPP avxav

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:

)lbm/ft(

100ft/s)(

3max

g

v

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 Pipline1. AGA equation2. Weymouth equation3. Panhandle A equation4. Panhandle B equation

Gas Distribution Pipeline1. IGT equation2. Spitzglass equation3. Mueller equation4. Fritzsche equation

Single Phase Gas Flow AGA Equation

The transmission factor is defined as:

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.

mfF

2

PipeSmoothF

NF

F

NF

TurbulentFullyd

F

Min

tt

t

6.0log4,4125.1

log4

7.3log4

Re10

Re10

10

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:

3/1

032.0

dfm

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:

1461.0Re

0768.0

Nfm

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:

03922.0Re

00359.0

Nfm

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:

cp,861.35 667.2

555.0

2.08.0

22

21

d

LT

PeP

P

Tq

eavg

s

sc

scg sc

Single Phase Gas Flow Spitzglass Equation

The Spitzglass equation originally was used in fuel gas piping calculations. This equation has two versionA. Low pressure (less than 1 psig):

B. High pressure (more than 1 psig):

5.2

5.0

21

)03.06.3

1(956.278 d

dd

LT

PP

P

Tq

eavgsc

scg sc

5.2

5.0

22

21

)03.06.3

1(016.53 d

dd

LzT

PeP

P

Tq

eavavg

S

sc

scgsc

Single Phase Gas Flow Mueller and Fritzsche Equation

The Mueller equation is:

The Fritzsche formula, developed in Germany in 1908, has found

extensive use in compressed air and gas piping:

cp,4509.35 725.2

575.0

2609.07391.0

22

21

d

LT

PeP

P

Tq

eavg

s

sc

scg sc

69.2

538.0

8587.0

22

2128.41 d

LT

PeP

P

Tq

eavg

s

sc

scg sc

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