Applied Mathematical Modelling 34 (2010) 495507

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal h omepage: www.else vier.com/locate/apm

Transient modeling of non-isothermal, dispersed two-phase ow 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 2008Received in revised form 10 May 2009Accepted 1 June 2009Available online 16 June 2009

Keywords: Homogeneous Two-phase owFully implicit method Non-isothermal Vaporliquid equilibrium Natural gas pipeline

a b s t r a c t

Unsteady-state or transient two-phase ow, caused by any change in rates, pressures or temperature at any location in a two-phase ow 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 ow. The primary reason for this phenomenon is that the velocity of wave propagation in a two-phase mixture is signicantly slower. Interfacial transfer of mass, momentum and energy further complicate the problem. It is primarily due to the numerical difculties anticipated in accurately modeling transient two-phase ow 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 ow of gas assume that temperature is constant over the entire length of the pipeline.This study is the rst effort to simulate the non-isothermal, one-dimensional, transient homogenous two-phase ow gas pipeline system using two-uid conservation equations. The modied PengRobinson equation of state is used to calculate the vaporliquid equilib- rium in multi-component natural gas to nd the vapor and liquid compressibility factors. Mass transfer between the gas and the liquid phases is treated rigorously through ash calcu- 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 nite 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 nite difference thermo-uid equations for two-phase ow 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 ow rate, and mixture composition. Also, the fully implicit method has advantages, such as the guaranteed stability for large time step, which is very useful for sim- ulating long-term transients in natural gas pipeline systems. 2009 Elsevier Inc. All rights reserved.

1. Introduction

Homogeneous two-phase ows 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

* Corresponding author. Present address: Weatherford International Inc., 15995 Barkers Landing, Suite 275, Houston, TX 77079, United States. Tel.: +1832 201 4282; fax: +1 832 201 4300.E-mail address: m.abbaspour@yahoo.com (M. Abbaspour).

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

Nomenclature

CD drag coefcientCP specic heat at constant pressureD pipe diameterdmax maximum droplet diameterd p average droplet diameterf Darcy friction factorFd interfacial drag forceFmlg interfacial mass transfer forceFwg wall friction force for gas phaseFwl wall friction force for liquid phaseg gravitational accelerationh specic enthalpykij binary interaction coefcientn 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 fraction e pipe roughness q densityX heat owD differenceh angle of inclination of pipe to the horizontal clg mass rate of phase change from liquid to gas l viscosityx acentric factorrm surface tension for mixture

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

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

horizontal, concurrent gasliquid ow in pipes, a variety of ow 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 efciency of the system. Most gathering pipelines (which typically have liquid loads up to 100 barrels per million cubic feet of gas (bbls/MMSCF)) transport uids 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 ow behavior of gasliquid in pipelines for different ow regimes that are divided to two major areas, isothermal and non-isothermal.

1.1. Isothermal

M. Abbaspour et al. / Applied Mathematical Modelling 34 (2010) 495507497Isothermal condition is the simplied 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

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 ow pattern transition during a transient two-phase ow. This model was based on separate conservation equations for mass and momentum for each phase. The assumption of isothermal ow was also imposed and the results were presented for an airwater system.Sharma [4] presented an improved formulation for stratied two-phase ow 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 ow for homogeneous, stratied and annular ow 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 ow rates.Zhou and Adewumi [6] implemented an isothermal compositional multiphase hydrodynamic model for dispersed tran- sient gas/condensate two-phase ow in pipelines. They used the well-posed modied 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 ow, multi-uid and drift- ux models. Multi-uid model provides a general framework for the mathematical description of multiphase ows. However the multi-uid equations are not a complete description of a multiphase ow 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-ux 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 ow of gas assume that temperature is constant over the entire length of the pipe- line. In some cases, a temperature-versus-distance prole is specied to reect 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 ow rates and pressures, the uid will attain thermal equilibrium with the pipe wall very rapidly. The other important factor inuencing the temperature distribution is the JouleThompson coefcient, which can be important for variety of mass ow rates and gas compositions through the pipeline.Doster [12] derived the conservation equations that described mass, momentum and energy transport in a multiphase ow system from the classic NavierStokes equations for single-phase ows. He simplied the six-equation model to three separate models including, ve-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 ve-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 rst effort to develop a model for non-isothermal transient homogenous two-phase ow 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 nd the vapor and liquid compressibility factors and other properties. Furthermore ash calculations are used to determine the vapor and liquid mole fractions that are used to nd the mass transfer between the phases. The result of the ash calculation is compared with Vincent and Adewumi [13] to conrm the accuracy of our ash calculation used for two-phase ow simulation.

2. Governing equations

Homogeneous two-phase ow occurs when either of the two phases owing simultaneously in the pipeline is completely dispersed in the other. In horizontal pipelines this takes place at very high liquid rates (dispersed bubble ow) or at very high gas rates coupled with low liquid loading (mist ow). The following assumptions are used in this paper:

Transient conditions. One-dimensional ow. Non-isothermal condition. Homogeneous two-phase ow. Phase behavior is described by ash calculation using the PengRobinson equation of state.

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@ @@t al ql @x al ql v l clg 1Gas phase

@ @@t ag qg @x ag qg v g clg 2

2.2. Momentum equation

Liquid phase

@ @ @P@t al ql v l @x al ql v l v l al @x Fwl Fd F mlg al ql g sin h 3Gas phase

@ @ @P@t ag qg v g @x ag qg v g v g ag @x Fwg Fd Fmlg ag qg g sin h 4

2.3. Mixture energy equation

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

@T @T

@P "

( T @qg )

( T @ql )#

g@x ag qg Cp;g vg al ql Cp;l v l @t ag qg Cp;g al ql Cp;l @x ag vg q

q@T p

al v ll

@T p@P "

( T @qg )

( T @ql )

# @v g

2 @v g

@v l

2 @v l

g @t ag q

@T p

al

ql

@T p

1 @x ag qg v g

@t ag qg v g @x al ql vl @t al ql vl

gX v 2

v 2 !

l A ag qg v g al ql vl g sin h clg hg hl 2 2

5

where

@q

q T @Z 1 6

TP@T

Z @T P

2.4. Conservation of phases

ag al 1 7

One of the difculties of two-phase ow analysis in pipelines is to dene appropriate constitutive equations for relating some relevant forces such as the drag force Fd, wall/uid interaction force Fwg, and interfacial momentum transfer Fmlg tothe primary measurable variables, such as vg, vl, ag, and al. The following equations present the constitutive equations forthis paper:

2.5. Wall friction forces

ag fg qg v g jv g jFwg

2D 8

and

Fwl a ql v jv jl fl l l2D 9

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:

1

( e

5:0452

" 1 e 1:1098

5:8506#)

fp 2 logg

3:7065D

Reg

log

2:8257 D

Re0:8981

10

and

1

( e

5:0452

" 1 e 1:1098

gl5:8506#)

fp 2 logl

3:7065D

Rel

log

2:8257 D

Re0:8981

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]:3C D qg ag al v g v l jv g v l jFd

2d p

12

where CD depends on the relative velocity between the gas and condensate, and the interfacial area over which drag is acting depends on the ow regime and, d p is the droplet diameter in dispersed ow.The drag coefcient CD is found by Cliff et al. [16]:

24

0:687

0:42 CD

Regl

1 0:15Regl

13

gl1 42500Re 1:16

where

Regl

qg jv g v l jd plg

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:

We

g l ;q v g v 2 dp maxrm

14

The term rm is 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 stabilitynumber l2 =q dp;max rm . Hinze [18] gave an expression for Weber number which included the stability number as:l l2 0:36 !

We 12 1 ll ql dp;max rm

15

Moeck [19] recommended that the Weber number be assumed as 13, which is adopted in this paper. To nd the average droplet diameter to use in Eq. (12), Ulke [17] suggested:

d p 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 ow 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]:v g qg yg xDx qg yg x clg

Dx 17The parameter yg is the equilibrium gas volume fraction which one can nd from vaporliquid equilibrium using ash calculation.

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:Fmlg clg vg vl 18

2.8. PengRobinson equation of state (PR)

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

where

RT

VP b

aV V b bV b

19

Nb X xi bii

RT cibi 0:077796

PciN Na X X xi xj ai aj 0:5 1 kij i jai aci ai

RT ci 2aci 0:457235

Pcia0:5

0:5 i 1 mi 1 T ri

If x < 0.5

imi 0:37646 1:54226xi 0:26992x2

Otherwise

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

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 nite difference approximations for the partial derivatives. Fig. 1 shows a mesh used in this transformation. The pipe has N nodes and n time levels.The partial derivatives with respect to time are approximated by:

n1

n1 n n @F@t

F i1 F i F i1 F i 2Dt 20The spatial partial derivatives are written as:

@F Fn1

n1 i1 F i @x

Dx 21

F is a generic variable that represents:

F P; T ; vg ; v l ; ag ; al

x

xn+1tn1 2 3 i-1 i i+1 N

Fig. 1. Mesh of the solution.

Table 1Approximation of the individual terms at an interface between the nodes.

Pn1 Pn1

gZn ZnP i1 i

i1 i 2 Zg

ZlTn1 T n1 n

g2

lZn

2 Zl T i1 i

i1 i 2v n1

n1

Cn Cnv g

gi1 v gi

Cpg2

pgi1 pgi 2vn1 v n1

Cn Cnv l

li1 li

C2 pl

pli1 pli 2an1

n1

ln nag

gi1 agi

l2 g

gi1 lgi 2an1 an1

ln lnal

li1 li

l2 l

li1 li 2

Table 2Composition and component critical properties of gas mixture [13].

CompoundMWMol.%Tc (K) Pc (kPa) x

Methane16.04375.57190.734604.320.0115

Ethane30.07011.22305.614880.110.0908

Propane44.0977.78370.004249.240.1454

n-Butane58.1241.71425.343796.940.1928

Iso-butane58.1240.78408.313648.020.1756

n-Pentane72.1510.31469.783368.780.2510

Iso-pentane72.1510.28460.573381.190.2273

Hype176.20.41622.222141.920.26

Hype2112.40.33666.672106.420.28

Hype3133.00.28707.782073.460.295

Nitrogen28.0131.01126.273399.11 0.223

Carbon dioxide44.0100.32304.377384.280.2250

Table 3Composition and properties of pseudo-ternary mixture.

CompoundMWMol.%Tc (K) Pc (kPa) x

Pseudo-116.20176.58189.884588.4260.008

Pseudo-239.36822.40350.444548.8920.14

Pseudo-3130.001.02722.221585.7920.408

Finally, Table 1 shows how the individual terms were approximated at an interface between the nodes.Substituting Eqs. (20) and (21) and Table 1 into Eqs. (1)(5) results in ve sets of equations for each element and there will be (5N 5) equations for a pipe. But from Eq. (7), there are N equations for N nodes, 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 6N unknowns and 6N equations. These equations are completely non-linear and the Newton Raphson method can be applied to solve these equations for two-phase, non-isothermal transient ows through a pipe.

3. Computational results and comparisons

The result of the ash calculation is compared with Vincent and Adewumi [13] to conrm the accuracy of our ash cal- culation used for two-phase ow simulation. The composition and critical properties of the natural gas are tabulated in Table2.To reduce the computation time these composition are broken down into a pseudo-ternary system. Table 3 shows the pseudo-components for natural gas that appears in Table 2 [13].Fig. 2 shows the ash curves for the natural gas with composition shown in Table 3 for various temperatures using thePengRobinson equation of state.To validate the accuracy of the result for ash calculation, Fig. 3 shows 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 is1600 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 in Table 3.

15000

Pressure (kPa)10000

5000

0

236 K250 K264 K278 K292 K306 K319 K0 5 10 15 20 25Equilibrium Liquid Volume Percent

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

15000

Pressure (kPa)10000

T=277.78 K

5000

Present workVincent and Adewumi (1990)

00 2 4 6 8 10Equilibrium Liquid Volume Percent

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

a 28

24

Gas Mass Flow Rate (kg/s)20

X/L=0.0

X/L=0.5

X/L=0.6

b 28

24

20

X/L=0.0

16 16

X/L=0.812 12

Gas Mass Flow Rate (kg/s)8 8

4 4

X/L=1.00 0

X/L=0.5

X/L=0.6

X/L=0.8

X/L=1.0

510152005101520Time (sec)Time (sec)-4 -40

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

Initially steady-state ow is assumed along the pipeline with a mixture gas ow 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 ow 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 xed 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

a 2 b 2

1.5

Liquid Mass Flow Rate (kg/s)1.5

X/L=0.0

1

Liquid Mass Flow Rate (kg/s)0.5

0

X/L=0.0X/L=0.5 X/L=0.6

X/L=0.8

X/L=1.0

1

0.5

0

X/L=0.5

X/L=0.8

X/L=1.0

X/L=0.6

0510152005101520Time (sec)Time (sec)-0.5

-0.5

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

a 1.85

b 1.85

1.8

1.8

Liquid Condensate Holdup (%)1.75

1.75

1.7

051015200 5101520Time (sec)Time (sec)1.65

X/L=0.0X/L=0.5X/L=0.6X/L=0.8X/L=1.0

Liquid Condensate Holdup (%)1.7

1.65

X/L=0.0X/L=0.5X/L=0.6X/L=0.8X/L=1.0

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

a 3 b 3

Liquid Condensate Holdup (%)2.5

2.5

Liquid Condensate Holdup (%)X/L=0.02 2

050100150200050100150200Time (sec)Time (sec)1.5

1.5

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

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 signicant role in natural gas condensation is the pipeline

18

Liquid Condensate Holdup (%)1620 sec14 80 sec12 140 sec200 sec10

8

6

4

2

00 0.25 0.5 0.75 1X/L

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

16

Liquid Condensate Holdup (%)14

12

10

8

6X/L=0.454 X/L=0.4

2X/L=0.30

X/L=0.350 50 100 150 200Time (sec)

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

a 330

325

320

315

X/L=0.3

X/L=0.35

X/L=0.25

Temperature (K)X/L=0.2

b 310

305

X/L=0.0X/L=0.5X/L=0.6X/L=0.8X/L=1.0

Temperature (K)310

305

300

X/L=0.15

X/L=0.1

X/L=0.05

X/L=0.0

300

295

290

5010015020005101520Time (sec)Time (sec)0

295

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

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 ow in the pipes.Fig. 4 shows the variation of gas mass ow rate with respect to time for different locations along the pipe. As shown in this gure, the wave propagation period for non-isothermal condition is about 3.2 s which is almost half of the wave propagation

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 reects and propagates downstream again. The time reection depends on length of pipe and the wave speed which is function of pressure, temperature and properties of liquid and gas.

3Isothermal Condition (X/L = 0.25)

3Non-isothermal Condition (X/L = 0.25)

2.5

2.5

Velocity (m/s)Velocity (m/s)2 2

1.5

1.5

1

0.5

GasLiquid

1

0.5

GasLiquid

0501001500 50100150200Time (sec)Time (sec)0 0200

3Isothermal Condition (X/L = 0.5)

3Non-isothermal Condition (X/L = 0.5)

2.5

2.5

Velocity (m/s)2

1.5

2

Velocity (m/s)1.5

GasLiquid

1

0.5

GasLiquid

1

0.5

050100150200050100150200Time (sec)Time (sec)0 0

3Isothermal Condition (X/L = 0.75)

3Non-isothermal Condition (X/L = 0.75)

2.5

2.5

Velocity (m/s)Velocity (m/s)2 2

1.5

1.5

1 Gas 1Liquid

GasLiquid

0.5

0.5

050100150200050100150200Time (sec)Time (sec)0 0

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

Same procedure applies for liquid mass ow rate. As shown in Fig. 5, the liquid mass ow rate uctuates 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 ow rate, especially for the nodes in downstream of pipe, is different with isothermal condition.As shown in this gure, after rst wave reection (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 uid. Therefore, with changing the condition to non-isothermal causes a signicant change in the liquid holdup.Fig. 6 illustrates the calculated liquid condensate holdup (%) history at the locations explained in Fig. 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 gure, the liquid holdup for non-isothermal increases similar to isothermal condition but with different wave propagation time for X/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 rst wave reection reach to the upstream end. As illustrated in Fig. 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 for X/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 gure as time increases the wave penetrates and the liquid condensate holdup increases in different locations. Fig. 9 shows the variation of liquid holdup for intermediate points between X/L = 0 and X/L = 0.5 where the non-isothermal effect starts appearing in pipeline.Fig. 10 illustrates 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 ow has temperatures that are directly proportional to the pressure. How- ever, for a two-phase ow the temperature is not only a function of pressure but also a function of liquid holdup, which is elucidated in the Fig. 9. It is clear from both the gures that the instant at which the temperature stops increasing is the instant where the liquid holdup suddenly starts increasing.As an example, at X/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 inuence 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, and X/L = 0.75 for isothermal and non-isothermal conditions. As shown in this gure, 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 signicant 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 gure, at X/L = 0.25 gas and liquid speeds have signicant differences, but for X/L = 0.5 and X/L = 0.75 the difference is less.As already explained in Fig. 7, the variation of liquid holdup for X/L = 0.5, X/L = 0.6, X/L = 0.8 and X/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 in Fig. 11 for X/L = 0.5 and X/L = 0.75.

4. Conclusion

In the area of homogenous two-phase, natural gas ow, most of the research done considers the steady state condition assuming a temperature prole along the pipe or isothermal transient condition. However, as the results in this study show the non-isothermal condition has a very signicant impact on the solution especially on liquid holdup, it becomes imperative that any two-phase ow analyses incorporate the ndings of this study.In this paper, the non-isothermal, transient homogenous two-phase ow gas pipeline model is developed using fully im- plicit nite 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 deection dampen with increasing time in linepack problems. The effect of non-isothermal conditions is very signicant on results in which the variation of liquid holdup is strongly a function of pressure and temperature.

The non-isothermal condition reaches stability faster than the isothermal condition in linepack problem. There is a signicant difference between gas and liquid speed for non-isothermal condition.

References

[1] B. Asante, Two-phase ow: accounting for the presence of liquids in gas pipeline simulation, in: 34th Annual Meeting Pipeline Simulation InterestGroup (PSIG), Portland, Oregon, 2325 October, 2002.[2] M.W. Scoggins, A Numerical Simulation Model For Transient Two-Phase Flow in a Pipeline, Ph.D. Thesis, University of Tulsa, 1977.[3] Y. Taitel, N. Lee, A.E. Dukler, Transient gasliquid ow in horizontal pipes: modeling the ow pattern transitions, AIChE Journal 24 (5) (1978) 920934. [4] Y. Sharma, Modeling Transient Two-Phase Flow in a Stratied Flow Pattern. MS. Thesis, University of Tulsa, 1983.[5] K.D. Roy, Transient Phenomena in Two-phase Horizontal Flowing for the Homogeneous, Stratied and Annular Flow Patterns, Ph.D. Thesis, Universityof Tulsa, 1984.[6] J. Zhou, M.A. Adewumi, 1998. Transient in gascondensate natural gas pipelines. Energy Source Technology Conference & Exhibition, ASME, ETCE98-4660.[7] S.L. Soo, Equation of multiphase-multidomain mechanics, Multiphase Transport: Fundamentals, Reactor Safety Applications, vol. 1, HemispherePublishing Co., Washington, D.C., 1980, pp. 291305.[8] M.A. Adewumi, J. Zhou, N. Nor-Azlan, Transients in Wet Gas Pipelines, in: ASME Fluid Engineering Division Conference, vol. 1, FED-Vol. 236, 1996, pp.695-700.[9] H. Mahgerefteh, P. Saha, I.G. Economou, Fast numerical simulation for full bore rupture of pressurised pipelines, AIChE Journal 45 (6) (1999) 11911201.[10] H. Mahgerefteh, P. Saha, I.G. Economou, Modelling uid phase transition effects on the dynamic behaviour of ESDV, AIChE Journal 46 (5) (2000) 9971006.[11] P. Lezeau, C.P. Thompson, Numerical Simulation of One-Dimensional Transient Multiphase Flow, Applied Mathematics and Computing Group, Craneld University, 1999. pp. 133.[12] J.M. Doster, Conservation equations in multiphase ows, in: Simulation VI: Proceeding of the SCS Multiconference on Simulators VI, March 2831,1989, pp. 5158.[13] P.A. Vincent, M.A. Adewumi, Engineering Design of GasCondensate Pipelines with a Compositional Hydrodynamic Model. SPE ProductionEngineering, November, 1990, pp. 381386.[14] M. Abbaspour, Simulation and Optimization of Non-isothermal One-dimensional Single/Two-phase Flow in Natural Gas Pipeline, Ph.D. Dissertation, Kansas State University, 2005.[15] N.H. Chen, An explicit equation for friction factor in pipe, Ind. & Eng. Chem. Fund. 18 (3) (1979) 296297.[16] R. Cliff, J.R. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press Inc., Washington, DC, 1978. p. 112.[17] A. Ulke, A study of a two-component, two-phase ow system in one dimension, in: Multi-Phase Flow and Heat Transfer III., Part A Fundamentals, Elsevier Science Publishers, 1984, pp. 5977.[18] J.O. Hinze, Critical speeds and sizes of liquid globules, Appl. Sci. Res. A1 (1948) 275288.[19] E.D. Moeck, Annular dispersed two-phase ow and critical heat ux, AECL Report 3650, Alberta, 1970. [20] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fund. (1976) 5964.[21] M. Abbaspour, K.S. Chapman, L.A. Glasgow, Z.C. Zheng, Dynamic simulation of gasliquid homogeneous ow in natural gas pipeline using two-uidconservation equations, in: 2005 ASME Fluids Engineering Division Summer Meeting and Exhibition, 9th International Symposium on GasLiquidTwo-Phase Flow, FEDSM2005-77136, Houston, TX, June 1923, 2005.