[Elearnica.ir]-Transient Modeling of Non-Isothermal Dispersed Two-phase Flow in Natural g

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Transient modeling of non-isothermal, dispersed two-phase flow

in natural gas pipelines

Mohammad Abbaspour a,*, Kirby S. Chapman a, Larry A. Glasgow b

a National Gas Machinery Laboratory, KSU, United Statesb Department of Chemical Engineering, KSU, United States

a r t i c l e i n f o

Article history:

Received 31 January 2008

Received in revised form 10 May 2009

Accepted 1 June 2009

Available online 16 June 2009

Keywords:

Homogeneous

Two-phase flow

Fully implicit method

Non-isothermal

Vaporliquid equilibrium

Natural gas pipeline

a b s t r a c t

Unsteady-state or transient two-phase flow, caused by any change in rates, pressures or

temperature at any location in a two-phase flow line, may last from a few seconds to sev-

eral hours. In general, these changes are an order of magnitude longer than the transient

encountered during single-phase flow. The primary reason for this phenomenon is that

the velocity of wave propagation in a two-phase mixture is significantly slower. Interfacial

transfer of mass, momentum and energy further complicate the problem. It is primarily

due to the numerical difficulties anticipated in accurately modeling transient two-phase

flow that the state of the art in this important area is restricted to a handful of studies with

direct applicability to petroleum and gas engineering. A limited amount of information on

the subject of two-phase transport phenomena is available in the petroleum engineering

literature. Most of the publications for two-phase flow of gas assume that temperature is

constant over the entire length of the pipeline.

This study is the first effort to simulate the non-isothermal, one-dimensional, transient

homogenous two-phase flow gas pipeline system using two-fluid conservation equations.

The modified PengRobinson equation of state is used to calculate the vaporliquid equilib-

rium in multi-component natural gas to find the vapor and liquid compressibility factors.

Mass transfer betweenthe gas and theliquid phases is treated rigorously through flashcalcu-

lation, making the algorithm capable of handling retrograde condensation. The liquid drop-

lets are assumed to be spheres of uniform size, evenly dispersed throughout the gas phase.

The method of solution is the fully implicit finite difference method. This method is stable

for gas pipeline simulations when using a large time step and therefore minimizes the com-

putation time. The algorithm used to solve the non-linear finite difference thermo-fluid

equations for two-phase flow through a pipe is based on the NewtonRaphson method.

The results show that the liquid condensate holdup is a strong function of temperature,

pressure, mass flow rate, and mixture composition. Also, the fully implicit method has

advantages, such as the guaranteed stability for large time step, whichis very useful for sim-

ulating long-term transients in natural gas pipeline systems.

2009 Elsevier Inc. All rights reserved.

1. Introduction

Homogeneous two-phase flows are frequently encountered in a variety processes in the petroleum and gas industries.

In natural gas pipelines, liquid condensation occurs due to the thermodynamic and hydrodynamic imperatives. During

0307-904X/$ - see front matter 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2009.06.023

* Corresponding author. Present address: Weatherford International Inc., 15995 Barkers Landing, Suite 275, Houston, TX 77079, United States. Tel.: +1

832 201 4282; fax: +1 832 201 4300.

E-mail address: [email protected] (M. Abbaspour).

Applied Mathematical Modelling 34 (2010) 495507

Contents lists available at ScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te / a p m

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horizontal, concurrent gasliquid flow in pipes, a variety of flow patterns can exist. Each pattern results from the particular

manner by which the liquid and gas distribute in the pipe.

The accompanying liquids affect the transportation efficiency of the system. Most gathering pipelines (which typically

have liquid loads up to 100 barrels per million cubic feet of gas (bbls/MMSCF)) transport fluids as multiphase components.

Most of the pipeline companies typically use dry gas models for transmission pipelines, where the liquid entertainment is

usually less than 10 bbls/MMSCF of gas[1].

Many researchers have tried to model the transient two-phase flow behavior of gasliquid in pipelines for different flow

regimes that are divided to two major areas, isothermal and non-isothermal.

1.1. Isothermal

Isothermal condition is the simplified version of non-isothermal condition that the effect of energy equation is neglected.

Scoggins[2]developed a three-equation isothermal model formulation based on individual mass conservation equation for

Nomenclature

CD drag coefficientCP specific heat at constant pressureD pipe diameterdmax maximum droplet diameterdp average droplet diameter

f Darcy friction factorFd interfacial drag forceFmlg interfacial mass transfer forceFwg wall friction force for gas phaseFwl wall friction force for liquid phaseg gravitational accelerationh specific enthalpykij binary interaction coefficientn time levelN number of nodesP pressureRe Reynolds numbert timeT temperaturev velocityWe Weber numberx distance along the pipeyg equilibrium gas volume fractionZ compressibility factor

Greek charactersa volume fractione pipe roughnessq densityX heat flowD differenceh angle of inclination of pipe to the horizontal

clg mass rate of phase change from liquid to gasl viscosityx acentric factorrm surface tension for mixture

Subscripts

g gasi,i+ 1 number of node in discretizationl liquidc crirical

Superscriptn,n + 1 nth and (n+ 1)th time levels respectively

496 M. Abbaspour et al. / Applied Mathematical Modelling 34 (2010) 495507


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two phases, and a mixture momentum equation for the two-phase mixture. Mass transfer across the gasliquid interface

was accounted for through the black oil model. Taitel et al. [3]presented a model for predicting flow pattern transition

during a transient two-phase flow. This model was based on separate conservation equations for mass and momentum

for each phase. The assumption of isothermal flow was also imposed and the results were presented for an airwater system.

Sharma[4]presented an improved formulation for stratified two-phase flow by including an interfacial pressure term.

Roy[5]improved the formulations, which were developed by Sharma [4]and developed a model for transient phenomena

in two-phase horizontal flow for homogeneous, stratified and annular flow patterns under isothermal conditions. However,

the iterative scheme used for evaluating the in situ liquid volume fraction and the no-slip liquid holdup exhibited poor con-

vergence for rapidly changing flow rates.

Zhou and Adewumi[6]implemented an isothermal compositional multiphase hydrodynamic model for dispersed tran-

sient gas/condensate two-phase flow in pipelines. They used the well-posed modified Soos [7] partial pressure model in con-

servative form, which serves as the transient multiphase hydrodynamic model, and the phase behavior model for natural gas

compositional mixture. Adewumi[8]presented a new class of high-resolution hybrid higher-order schemes to solve a sys-

tem of four non-linear hyperbolic partial differential equations (PDE) for gas/condensate problems.

Mahgerefteh et al. [9,10] used Method of Characteristic to simulate Full Bore Rupture of long pipeline containing condens-

able or two-phase hydrocarbon mixture. They modeled the pertinent conservation equation in conjunction with an equation

of state.

Lezeau and Thompson[11]described two common mathematical formulations of two-phase flow, multi-fluid and drift-

flux models. Multi-fluid model provides a general framework for the mathematical description of multiphase flows. However

the multi-fluid equations are not a complete description of a multiphase flow because they need to be supplemented by suit-

able constitutive relationships which govern the way the phases interact with each other (microscopic level). The drift-flux

model is characterized by the fact that the momentum equations applying to each phase are combined to form a total

momentum equation, which must be supplemented with constitutive relationships giving the so-called slip velocity be-

tween the phases.

1.2. Non-isothermal

Most of the publications for two-phase flow of gas assume that temperature is constant over the entire length of the pipe-

line. In some cases, a temperature-versus-distance profile is specified to reflect changes in the surrounding environment.

Transient temperature effects are neglected in these models on the basis that for a long pipeline subject to rather gradual

time variations in flow rates and pressures, the fluid will attain thermal equilibrium with the pipe wall very rapidly. The

other important factor influencing the temperature distribution is the JouleThompson coefficient, which can be important

for variety of mass flow rates and gas compositions through the pipeline.

Doster[12]derived the conservation equations that described mass, momentum and energy transport in a multiphaseflow system from the classic NavierStokes equations for single-phase flows. He simplified the six-equation model to three

separate models including, five-equation models, four-equation models and homogeneous equilibrium mixture (HEM) mod-

els. Five-equation models employ four phasic and one mixture equation to describe the two-phase system. Typically the en-

ergy equation is considered as a mixture equation in five-equation model. The four-equation models are based on the

combination of two phasic and two mixture equations. The homogenous model or three-equation models are based solely

on the mixture equation and require both phases to be at saturation.

However, Doster[12]developed the non-isothermal equations, but the results and solution methods were not provided.

The current paper is the first effort to develop a model for non-isothermal transient homogenous two-phase flow in gas pipe-

line systems using microscopic and macroscopic conservation equations. This model includes vaporliquid equilibrium in a

multi-component natural gas and the PengRobinson equation of state is used to calculate the vaporliquid equilibrium in

multi-component natural gases to find the vapor and liquid compressibility factors and other properties. Furthermore flash

calculations are used to determine the vapor and liquid mole fractions that are used to find the mass transfer between the

phases. The result of the flash calculation is compared with Vincent and Adewumi [13]to confirm the accuracy of our flashcalculation used for two-phase flow simulation.

2. Governing equations

Homogeneous two-phase flow occurs when either of the two phases flowing simultaneously in the pipeline is completely

dispersed in the other. In horizontal pipelines this takes place at very high liquid rates (dispersed bubble flow) or at very high

gas rates coupled with low liquid loading (mist flow). The following assumptions are used in this paper:

Transient conditions.

One-dimensional flow.

Non-isothermal condition.

Homogeneous two-phase flow.

Phase behavior is described by flash calculation using the PengRobinson equation of state.

M. Abbaspour et al. / Applied Mathematical Modelling 34 (2010) 495507 497


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Liquid droplets are uniformly dispersed in a continuous gas phase.

Liquid droplets are spherical and of uniform size.

The following equations are the continuity, momentum [6], and energy equations for liquid and gas phase [14,21]:

2.1. Continuity equation

Liquid phase

@

@talql

@

@xalqlvl clg 1

Gas phase

@

@tagqg

@

@xagqgvg clg 2

2.2. Momentum equation

Liquid phase

@

@t

alqlvl @

@x

alqlvlvl al@P

@x

FwlFdFmlg alqlgsin h 3

Gas phase

@

@tagqgvg

@

@xagqgvgvg ag

@P

@xFwg FdFmlg agqggsin h 4

2.3. Mixture energy equation

The following equation is the modified version of the energy equation developed by Doster[12]:

@T

@xagqgCp;gvg alqlCp;lvl

@T

@tagqgCp;g alqlCp;l

@P

@x agvg

T

qg

@qg@T

p

( ) alvl

T

ql

@ql@T

p

( )" #

@P

@t agT

qg

@qg@T

p

( ) al

T

ql

@ql@T

p

( ) 1

" #

@vg

@x agqgv2g

@vg

@t agqgvg @vl

@x alqlv2l

@vl

@t alqlvl

X

A agqgvg alqlvlgsin h clg hghl

v2g

2

v2l

2

! 5

where

@q@T

P

qT

1 T

Z

@Z

@T

P

6

2.4. Conservation of phases

ag al 1 7One of the difficulties of two-phase flow analysis in pipelines is to define appropriate constitutive equations for relating

some relevant forces such as the drag force Fd, wall/fluid interaction force Fwg, and interfacial momentum transfer Fmlg to

the primary measurable variables, such as vg, vl, ag, and a l. The following equations present the constitutive equations for

this paper:

2.5. Wall friction forces

Fwgagfgqgvgjvgj

2D 8

and

Fwlalflqlvljvlj

2D 9



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There exist different friction factor equations used for the wall friction force equations for gas and liquid phases. In this

study, the Chens[15]equation is used (functions of Reynolds number and pipe roughness) as follows:

1ffiffiffiffifg

p 2log e3:7065D

5:0452

Reglog

1

2:8257

eD

1:1098

5:8506

Re0:8981g

" #( ) 10

and

1ffiffiffifl

p 2log e3:7065D

5:0452

Rellog

1

2:8257

eD

1:1098

5:8506

Re0:8981l

" #( ) 11

2.6. Interfacial drag force

In the gascondensate system, the gas is considered the continuous phase. Therefore, the interfacial drag force per unit

volume is[13]:

Fd 3CDqgagalvg vljvg vlj

2dp12

where CDdepends on the relative velocity between the gas and condensate, and the interfacial area over which drag is acting

depends on the flow regime and, dp is the droplet diameter in dispersed flow.The drag coefficientCD is found by Cliff et al.[16]:

CD 24

Regl

1 0:15Re0:687gl

0:42

1 42500Re1:16gl

13where

Regl qgjvg vlj

dp

lg

The Weber number, We (ratio of inertia and surface tension force), is of great importance in the determination of the stability

of a single droplet and is evaluated from:

Weqgvg vl

2dp;max

rm14

The term rmis the mixture surface tension. The suggested values of the critical Weber number that give the maximum stable

droplet diameter range from 8 to 20 [17]. However the liquid viscosity has a stabilizing effect that is shown as the stability

number l2l =qldp;maxrm. Hinze[18]gave an expression for Weber number which included the stability number as:

We 12 1 l2lqldp;maxrm

0:36 ! 15

Moeck[19]recommended that the Weber number be assumed as 13, which is adopted in this paper. To find the average

droplet diameter to use in Eq. (12),Ulke[17] suggested:

dp 0:06147dp;max 16

2.7. Interfacial momentum transfer

The interface mass transfer rate cannot be determined a priori but must be calculated simultaneously with the dependent

variables. In this study, we use the equilibrium cell method to evaluate the mass transfer rate. The mass transfer rate is the

difference in the equilibrium gas mass flow rate between the inlet and outlet. The velocity of the gas at outlet is assumed

equal to the velocity at the inlet, therefore the equation for the mass rate of phase transfer per unit volume is [13]:

clgvgqgygxDx qgygx

Dx 17

The parameter yg

is the equilibrium gas volume fraction which one can find from vaporliquid equilibrium using flash

calculation.



6/13

If liquid evaporates and enters the gas stream, the momentum force that would be transferred depends on the mass rate

of phase change and the relative velocity of the gas and condensate:

Fmlgclgvg vl 18

2.8. PengRobinson equation of state (PR)

Peng and Robinson[20]presented an equation of state of the form:

P RT

V b

a

VV b bV b 19

where

bXN

i

xibi

bi 0:077796RTciPci

aXN

i

XNj

xixjaiaj0:5

1 kij

ai aciai

aci 0:457235RTci

2

Pci

a0:5i 1 mi 1 T0:5ri

Ifx < 0.5

mi 0:37646 1:54226xi 0:26992x2i

Otherwise

mi 0:379642 1:48503 0:164423 1:1016666xixixi

2.9. Numerical formulation using the fully implicit method

The fully implicit method consists of transforming Eqs.(1)(5)from partial differential equations to algebraic equations

by using finite difference approximations for the partial derivatives.Fig. 1shows a mesh used in this transformation. The

pipe hasNnodes and n time levels.

The partial derivatives with respect to time are approximated by:

@F

@t

Fn1i1 Fn1i F

ni1F

ni

2Dt

20

The spatial partial derivatives are written as:

@F

@x

Fn1i1 Fn1i

Dx 21

F is a generic variable that represents:

F P; T;vg;vl;ag;al

x

1 2 3 i-1 i i+1 N

x

t

n+1

n

Fig. 1. Mesh of the solution.



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Finally,Table 1shows how the individual terms were approximated at an interface between the nodes.

Substituting Eqs.(20) and (21)andTable 1into Eqs.(1)(5)results in five sets of equations for each element and there

will be (5N 5) equations for a pipe. But from Eq.(7), there are Nequations forNnodes, therefore the total number of equa-

tions become (6N 5). The number of unknown values at time level n+ 1, which consists of pressure, temperature, gas

velocity, liquid velocity, gas fraction, and liquid fraction at each node, is 6N. Five equations will be obtained from boundary

conditions, and then there are 6Nunknowns and 6Nequations. These equations are completely non-linear and the Newton

Raphson method can be applied to solve these equations for two-phase, non-isothermal transient flows through a pipe.

3. Computational results and comparisons

The result of the flash calculation is compared with Vincent and Adewumi [13]to confirm the accuracy of our flash cal-

culation used for two-phase flow simulation. The composition and critical properties of the natural gas are tabulated in Table

2.

To reduce the computation time these composition are broken down into a pseudo-ternary system. Table 3shows the

pseudo-components for natural gas that appears in Table 2[13].

Fig. 2shows the flash curves for the natural gas with composition shown in Table 3for various temperatures using the

PengRobinson equation of state.

To validate the accuracy of the result for flash calculation,Fig. 3shows the comparison between the Vincent and Adew-

umi [13] and the present work. There is a slight difference between the results which created from the compressibility factor

calculation for vapor and liquid phases.

In this paper a simple horizontal pipe with the instantaneous closure of the downstream valve is considered. A pipe is

1600 m long and 304.8 mm in diameter has a natural gas mixture with three pseudo-ternary components. The composition

of the natural gas is shown inTable 3.

Table 1

Approximation of the individual terms at an interface between the nodes.

PPn1i1 P

n1i

2 Zg

Zngi1Zngi

2

TTn1

i1 Tn1i

2 Zl

Znli1Znli

2

vgv

n1gi1

vn1gi2

CpgCnpgi1

Cnpgi2

vl v

n1li1

vn1li

2 Cpl

Cnpli1Cnpli

2

agan1gi1 an1gi

2 lg

lngi1 lngi2

al a

n1li1

an1li

2 ll

lnli1

lnli

2

Table 2

Composition and component critical properties of gas mixture[13].

Compound MW Mol.% Tc(K) Pc(kPa) x

Methane 16.043 75.57 190.73 4604.32 0.0115

Ethane 30.070 11.22 305.61 4880.11 0.0908

Propane 44.097 7.78 370.00 4249.24 0.1454

n-Butane 58.124 1.71 425.34 3796.94 0.1928

Iso-butane 58.124 0.78 408.31 3648.02 0.1756

n-Pentane 72.151 0.31 469.78 3368.78 0.2510Iso-pentane 72.151 0.28 460.57 3381.19 0.2273

Hype1 76.2 0.41 622.22 2141.92 0.26

Hype2 112.4 0.33 666.67 2106.42 0.28

Hype3 133.0 0.28 707.78 2073.46 0.295

Nitrogen 28.013 1.01 126.27 3399.11 0.223

Carbon dioxide 44.010 0.32 304.37 7384.28 0.2250

Table 3

Composition and properties of pseudo-ternary mixture.

Compound MW Mol.% Tc(K) Pc(kPa) x

Pseudo-1 16.201 76.58 189.88 4588.426 0.008

Pseudo-2 39.368 22.40 350.44 4548.892 0.14Pseudo-3 130.00 1.02 722.22 1585.792 0.408



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Initially steady-state flow is assumed along the pipeline with a mixture gas flow rate of 26 kg/s and inlet pressure 12 MPa.

At time zero the downstream valve is closed instantaneously, while the supply of gas mixture flow rate at the upstream end

is kept constant at the initial value. The absolute pipe roughness is 0.127 mm and the isothermal temperature is 300 K.

In this study, the fully implicit method is used to discretize the conservation equations. The time step is 0.2 s and the

number of nodes is 50 for this example. A very small step time and large number of nodes leads to more computation time

in fixed period of simulation time. For this example, the fully implicit method can accept a large step time and small number

of node comparing to method of characteristic to get the same result. The grid independency shows that 50 nodes are enough

Equilibrium Liquid Volume Percent

Pressure(kPa)

0 5 10 15 20 250

5000

10000

15000

236 K250 K264 K278 K292 K306 K319 K

Fig. 2. Flash curve for pseudo-ternary mixture.

Equilibrium Liquid Volume Percent

Pressure(kPa)

0 2 4 6 8 100

5000

10000

15000

Present work

Vincent and Adewumi (1990)

T=277.78 K

Fig. 3. Comparison of flash curve at T= 277.78 K.

Time (sec) Time (sec)

0 5 10 15 20-4

0

4

8

12

16

20

24

28

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0GasMassFlowRate(kg/s)

GasMassFlowRate(kg/s)

0 5 10 15 20-4

0

4

8

12

16

20

24

28

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

a b

Fig. 4. Comparison of gas mass flow rate history at five points for (a) isothermal and (b) non-isothermal condition in 20 s of operation.



9/13

for this simulation using fully implicit method. Also the results show that the time steps smaller than 0.2 s have same behav-

iors as 0.2 s.

The long pipelines are often exposed to varying temperature conditions as a result of regional differences as well as

temporal climatic variations. One of the factors that play a significant role in natural gas condensation is the pipeline

LiquidMassF

lowRate(kg/s)

LiquidMassF

lowRate(kg/s)

0 5 10 15 20-0.5

0

0.5

1

1.5

2

X/L=0.0

X/L=0.5 X/L=0.6

X/L=0.8

X/L=1.0

0 5 10 15 20-0.5

0

0.5

1

1.5

2

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

Time (sec) Time (sec)

a b

Fig. 5. Comparison of liquid mass flow rate history at five points for (a) isothermal and (b) non-isothermal condition in 20 s of operation.

Time (sec)0 5 10 15 20

1.65

1.7

1.75

1.8

1.85

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

Time (sec)

LiquidCondensateHoldup(%)

LiquidCondensateHoldup(%)

0 5 10 15 20

1.65

1.7

1.75

1.8

1.85

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

a b

Fig. 6. Comparison of liquid condensate holdup history at five points for (a) isothermal and (b) non-isothermal condition in 20 s of operation.

Time (sec)0 50 100 150 200

1.5

2

2.5

3

X/L=

0.0,0

.5,0.6

,0.8

and1

.0

Time (sec)

LiquidCond

ensateHoldup(%)

LiquidCond

ensateHoldup(%)

0 50 100 150 2001.5

2

2.5

3

X/L=0.0

X/L=0

.5

X/L=

0.6,

0.8an

d1.0

a b

Fig. 7. Comparison of liquid condensate holdup history at five points for (a) isothermal and (b) non-isothermal condition in 200 s of operation.



10/13

temperature. Lower temperature generally illustrates more liquid condensate holdup. Apart from the season-imposed lower

temperature, declining temperature occurs as a result of JouleThompson cooling effect due to restricted flow in the pipes.

Fig. 4 shows the variation of gas mass flow rate with respect to time for different locations along the pipe. As shown in this

figure, the wave propagation period for non-isothermal condition is about 3.2 s which is almost half of the wave propagation

X/L

LiquidConde

nsateHoldup(%)

0 0.25 0.5 0.75 10

2

4

6

8

10

12

14

16

18

20 sec

80 sec

140 sec

200 sec

Fig. 8. Variation of liquid condensate holdup along the pipe for various time.

Time (sec)

LiquidCondensateHoldup(%

)

0 50 100 150 2000

2

4

6

8

10

12

14

16

X/L=0

.0

X/L=0

.05

X/L=0

.1

X/L=0

.15

X/L=0

.2

X/L=0

.25

X/L=0.3

X/L=0.35

X/L=0.4

X/L=0.45

Fig. 9. Liquid condensate holdup histories for non-isothermal condition for 200 s simulation for intermediate points betweenX/L= 0 andX/L= 0.45.

Time (sec)

Temperature(K)

0 50 100 150 200290

295

300

305

310

315

320

325

330

X/L=0.0

X/L=0.05

X/L=0.1

X/L=0.15

X/L=0.2

X/L=0.25

X/L=0.3

X/L=0.35

Time (sec)

T

emperature(K)

0 5 10 15 20295

300

305

310

X/L=0.0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

a b

Fig. 10. Temperature history at different points for 200 s and 20 s of operation.



11/13

time for the isothermal condition (6 s). This is because the system is treated non-isothermally where the response time is

lesser than the time for isothermal simulation in a linepack problem.

With the instantaneous closure of the downstream valve, the pipeline outlet pressure increases very sharply, then the

pressure wave propagates upstream at about 6 s and 3.2 s for isothermal and non-isothermal conditions respectively. At this

time, the wave reaches the upstream end, then reflects and propagates downstream again. The time reflection depends on

length of pipe and the wave speed which is function of pressure, temperature and properties of liquid and gas.

Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

GasLiquid

Isothermal Condition (X/L = 0.25)

Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Gas

Liquid

Non-isothermal Condition (X/L = 0.25)

Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Gas

Liquid


Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Gas

Liquid


Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Gas

Liquid


Time (sec)

Velocity(m/s)

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Gas

Liquid


Fig. 11. Variation of gas and liquid speed in different location for isothermal and non-isothermal condition.



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Same procedure applies for liquid mass flow rate. As shown in Fig. 5, the liquid mass flow rate fluctuates till reaches to

stable condition. As liquid holdup in non-isothermal condition is function of temperature and pressure, therefore the vari-

ation of liquid mass flow rate, especially for the nodes in downstream of pipe, is different with isothermal condition.

As shown in this figure, after first wave reflection (3.2 s) in upstream end, the liquid start to be appeared in pipe and liquid

appearance will move to the other locations of pipe as time increases. The condensate holdup occurring in a pipeline may not

necessarily follow the equilibrium liquid condensate curve (Fig. 2) and strongly depends on hydrodynamic behaviors, which

determine the pressure, temperature and liquid holdup of fluid. Therefore, with changing the condition to non-isothermal

causes a significant change in the liquid holdup.

Fig. 6illustrates the calculated liquid condensate holdup (%) history at the locations explained inFig. 4. Because the tem-

perature along the pipeline is assumed a constant value for isothermal condition, the liquid condensate holdup is only a

function of pressure history. With increasing pressure, the liquid condensate holdup increases simultaneously.

As shown in this figure, the liquid holdup for non-isothermal increases similar to isothermal condition but with different

wave propagation time forX/L = 0.5, 0.6, 0.8, and 1.0. The non-isothermal effect starts appearing in pipe inlet and liquid hold-

up suddenly increases at time = 3.2 s when the first wave reflection reach to the upstream end. As illustrated inFig. 7, this

effect moves to other locations as time increases and reaches to X/L= 0.5 at time = 150 s. The interesting thing is that the

liquid holdup for X/L= 0.5, 0.6, 0.8, and 1.0 increases similar to the isothermal condition till 150 s beyond which the liquid

holdup starts to increase forX/L= 0.5. As time increases, the non-isothermal effect appears in entire pipeline. Fig. 8 illustrates

the variation of liquid condensate holdup along the pipe for various times. As shown in this figure as time increases the wave

penetrates and the liquid condensate holdup increases in different locations. Fig. 9shows the variation of liquid holdup for

intermediate points between X/L= 0 andX/L= 0.5 where the non-isothermal effect starts appearing in pipeline.

Fig. 10illustrates the temperature variation over a 20- and 200-s operation at various locations. For the case of the line-

pack example, a pipeline with single-phase gas flow has temperatures that are directly proportional to the pressure. How-

ever, for a two-phase flow the temperature is not only a function of pressure but also a function of liquid holdup, which is

elucidated in theFig. 9. It is clear from both the figures that the instant at which the temperature stops increasing is the

instant where the liquid holdup suddenly starts increasing.

As an example, atX/L= 0.05 the liquid hold up exhibits a sudden transition (increase) at 15 s which is also the time instant

at which the temperature stops increasing after which the temperature slowly attains a steady state. The phenomena can be

better understood by appreciating that an increase in the liquid holdup causes an increased liquid phase. In the liquid phase,

the temperature ceases to be a strong function of pressure. However, as part of mixture in the pipe is in a gas phase where

the temperature is a strong function of pressure and part of the mixture is in a liquid phase where the temperature is no

more a strong function of pressure the liquid holdup become critical in determining the temperature the mixture in the pipe-

line. Obviously as the liquid holdup increases the influence of the liquid phase overrides that of the gas phase and the total

temperature in the pipeline slowly attains a steady state.

If the liquid holdup suddenly changes, the variation of liquid speed is not related to gas speed anymore.Fig. 11 illustrates

the variation of gas and liquid speed at X/L= 0.25,X/L= 0.5, andX/L= 0.75 for isothermal and non-isothermal conditions. As

shown in this figure, gas and liquid speed are almost identical for isothermal condition due to the boundary conditions (gas

and liquid speed at inlet and outlet are considered to be equal), and similarly liquid holdup behavior for all locations in

pipeline.

But there is a significant difference between gas and liquid speed for non-isothermal condition. Because the liquid holdup

starts to increase right from pipe inlet (Fig. 9) and transmit to outlet of pipe with increase in time, the difference between gas

and liquid speed in entrance region is more than at pipe outlet. As illustrated in this figure, atX/L= 0.25 gas and liquid speeds

have significant differences, but for X/L= 0.5 andX/L= 0.75 the difference is less.

As already explained inFig. 7, the variation of liquid holdup forX/L= 0.5,X/L= 0.6,X/L= 0.8 andX/L= 1.0 are same as iso-

thermal condition, therefore the gas and liquid speed should be very close for these regions and are same as isothermal con-

dition, which is clearly shown inFig. 11for X/L= 0.5 andX/L= 0.75.

4. Conclusion

In the area of homogenous two-phase, natural gas flow, most of the research done considers the steady state condition

assuming a temperature profile along the pipe or isothermal transient condition. However, as the results in this study show

the non-isothermal condition has a very significant impact on the solution especially on liquid holdup, it becomes imperative

that any two-phase flow analyses incorporate the findings of this study.

In this paper, the non-isothermal, transient homogenous two-phase flow gas pipeline model is developed using fully im-

plicit finite difference technique. The conservation of continuity, momentum, and mixture energy equations are developed

and discretized for this study. The numerical results show that:

The liquid condensate holdup is a strong function of temperature, pressure and composition of mixture.

The wave propagation and deflection dampen with increasing time in linepack problems.

The effect of non-isothermal conditions is very significant on results in which the variation of liquid holdup is strongly a

function of pressure and temperature.



13/13

The non-isothermal condition reaches stability faster than the isothermal condition in linepack problem.

There is a significant difference between gas and liquid speed for non-isothermal condition.

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