Different turbulent models.
Abstract
When the natural gas with vapor is flowing in production pipeline, condensation occurs and leads to serious problems such as condensed liquid accumulation, pressure and flow rate fluctuations, and pipeline blockage. This chapter aims at studying phase change of vapor and liquidlevel change during the condensing process of waterbearing natural gas characterized by coupled hydrothermal transition and phase change process. A hydrothermal mass transfer coupling model is established. The bipolar coordinate system is utilized to obtain a rectangular calculation domain. An adaptive meshing method is developed to automatically refine the grid near the gasliquid interface. During phase change process, the temperature drop along the pipe leads to the reduction of gas mass flow rate and the rise of liquid level, which results in further pressure drop. Latent heat is released during the vapor condensing process which slows down the temperature drop. Larger temperature drop results in bigger liquid holdup while larger pressure drop causes smaller liquid holdup. The value of velocity with phase change is smaller than that without phase change while the temperature with phase change is bigger. The highest temperature locates in gas phase. But near the pipe wall the temperature of liquid region is higher than gas region.
Keywords
 hydrocarbons pipeline
 vapor/condensationstratified flow
 heat transfer
 phase change
 multicomponent
1. Introduction
Condensation occurs when the natural gas with vapor is flowing in production pipeline and leads to serious problems such as condensed liquid accumulation, pressure and flow rate fluctuations, and pipeline blockage or corrosion. The pipe flow together with phase change is commonly encountered in various heat and mass transfer processes over the past four decades, for instance, in petroleum and chemical processing industry, steamgenerating equipment, nuclear reactors, geothermal fields, heat exchangers, cooling systems, and solar energy system [1, 2, 3, 4]. In petroleum transportation, twophase flow characterization is a very common and economic technique, where vaporliquid twophase stratified flow is often observed in horizontal or slightly inclined systems [5, 6].
There exist several problems in the pipeline network system that the saturated vapor in gas would condense due to pressure and temperature drop [7, 8]. The condensate would attach to the pipe wall as a form of film or droplet [9, 10]. The condensation will decrease the effective crosssectional area and cause the increase of pressure drop which may lead to system shutdown [11, 16]. Generally, the condensed water accumulates at the lower parts of the pipeline due to the hilly pipeline route topography, which results in a continuous change of liquid holdup along the pipeline [12, 13, 14]. The changing liquid holdup and flow area are bounded to affect the flow patterns which inevitably influence the operating pressure and temperature inversely. Thus, the flow of condensed water and waterbearing gas in production pipelines is a complex process with coupling of hydraulic, thermal, and phase change phenomena [14, 15, 16, 17]. Researchers have investigated the gasliquid twophase pipe flow system by experiments or hydrodynamic and thermodynamic models.
It has been observed experimentally that when phase change occurs during the saturated vapor pipeline transportation process, the thermal gradients are created in the wall of the pipeline that lead to severe liquid condensation and stratified vaporliquid twophase flow [17, 18]. The fundamental engineering parameters are the pressure drop, liquid holdup, phase fraction, phase flow rate, temperature, thermophysical properties of the fluids, and pipeline geometry [19].
Not limited to hydraulic parameters, more recent attention has turned to nonisothermal flow in a pipe or plane channel where some numerical studies are also found [20, 21, 22, 23, 24, 25, 26, 27]. The detailed characteristic of heat transfer is taken into consideration in these mentioned models instead of an average value being represented for the temperature profiles in a pipe [28, 29]. According to their studies, the wall temperature distribution is different from the assumption of fully developed isothermal state [30]. The energy transfer model has been taken into the flow progress for the optimization of transportation, estimation of corrosion, or prediction of wax deposition [31, 32]. Concretely, the twodimensional (2D) momentum and threedimensional (3D) energy equations for both phases have been established for dynamic and thermal numerical simulation [8, 27, 30, 33]. The smooth or wavy interface between phases was obtained in a different range of flow rates [17]. For such a twophase nonisothermal stratified flow, analytical and numerical heat transfer solutions limited to laminar flow and without phase change have been obtained for fully developed stratified flow under different thermal boundary conditions [27]. Then, solutions which are more applicable to fully developed turbulent gasliquid smooth stratified flow have been obtained through the use of high Reynolds model [30]. Recently, the steadystate axial momentum and energy equations coupled with a low Reynolds model were established and solved [34]. The pressure drop, liquid height, and temperature field which are included in the solutions could match well with the experimental data. However, although equation of state (EOS) was utilized in previous onedimensional (1D) models to calculate the phase fraction [10, 12, 16, 22, 34, 41, 44], the flow rate, temperature, and pressure were not coupled with the varying liquid level.
According to the description of the physical process of gasliquid twophase pipe flow, the model can be divided into isothermal model, gasliquid twophase pipe flow model coupled with heat and gasliquid twophase pipe flow model coupled with heat and mass transfer. In the gasliquid twophase pipe flow model under the isothermal condition, it is assumed that all phases are in thermodynamic equilibrium state without considering the heat transfer process between pipe flow and environment, and the physical parameters of gasliquid twophase are just the single value function of pressure. In the gasliquid twophase flow model coupled with heat, the heat transfer of gasliquid and surrounding environment is considered. In the gasliquid twophase flow model coupled with heat and mass transfer, the coupling effect of flow, heat transfer, and mass transfer are considered simultaneously.
The twodimensional (2D) or threedimensional (3D) stratified gasliquid twophase model including mass conversation equation, momentum conversation equation, turbulence model equation, boundary conditions, and related auxiliary equations for model closure were applied to describe the flow in pipelines. Differences among them mainly existed in two aspects. On the one hand, different turbulence models were built including SpalartAllmaras Model (SAM),
Recently, attempts have been made to introduce energy equation into the improved model and the detailed solutions about temperature distribution have also been worked out by considering potential energy, kinetic energy, heat transfer, and JouleThomson effect [17, 35, 36, 37, 38]. The phase change was ignored in these models. Although equation of state (EOS) was utilized in previous studies to calculate gas condensation of gascondensate flow in pipelines [12, 13, 14, 15, 17, 18, 19, 20, 39], the flow rate, temperature, pressure were not coupled with liquid level. Turbulent flow is not considered, which would lead to different numerical results in their onedimensional (1D) model. Moreover, 1D model could not present the detailed distribution of hydraulic and thermal parameters at pipe crosssection.
This chapter mainly introduces the different turbulence models, the interface shape model, and the phase transition model in the process of gasliquid twophase stratified flow in the horizontal pipeline. The turbulence model mainly includes the
2. Numerical modeling of stratified gasliquid wavy pipe flow with phase change
Since pressure and temperature drop along the pipeline, the saturated vapor of hydrocarbons would condenses gradually, as shown in Figure 1. The condensed liquid would attach to the pipe wall as liquid film or accumulate at the lower part of the pipeline, which could result in continuous change of the liquid level [17, 29, 30].
Thus, some models of stratified vaporliquid twophase flow coupled with phase change were proposed. Assumptions could be made as follows: (1) precipitation of condensed liquid is a flash evaporation equilibrium process which occurs in a short moment; (2) regardless of the attachment to inner wall of the pipe, all the condensed liquid accumulates at the bottom of the pipe; (3) twophase flow in vaporliquid pipeline has a stratified flow pattern as well as a stable developing flow area in every calculated pipeline segment; (4) the smooth vaporliquid interface model is adopted to describe the interface shape; (5) the heavy components of hydrocarbon are simplified as pseudocomponent
Several kinds of flow patterns are likely to form for vapor/condensation flowing in pipeline: stratified flow, slug flow, annular flow, and stratifieddispersed flow. It is hard to calculate the liquid level, the exact position of liquid film, and the migration patterns of condensed liquid (as shown in Figure 1) via onedimensional model which could not present the detailed distribution of hydraulic and thermal parameters at pipe cross section. Moreover, the 1D model does not consider the turbulent flow which would lead to different numerical results. Hence, the control volume in threedimensional coordinates is adopted to discretize the calculating area, as shown in Figure 2.
In this section, the governing equations based on physical conservation are chosen and established in threedimensional coordinates, which include mass conservation equation, momentum conservation equation, energy conservation equation, turbulent flow model, and phase change model.
2.1. Mass conservation equation
The total mass flow rate
The mass transfer rate of phasechange within pipe segment has been taken into consideration due to the vapor phase gradually condensing or the liquid evaporating during the flow process. The change of vapor mass flow rate
The liquid volume fraction
2.2. Momentum conservation equation
The law of momentum conservation is a universal principle for any flow system that the varying rate of momentum is equal to the sum of the forces imposed on the control volume. Considering the compressibility of vapor and liquid, the equation of momentum is as follows:
here
According to the theory of CFD, the NS equation is applicable to any kind of flow. A direct calculation of NS equation requires a high computer capacity which is not practical in engineering. Hence, an item of
where
2.3. Turbulent flow model
Following a similar approach as Xiao et al. [36] and Reboux et al. [37, 38] for twophase turbulent flow, the eddy viscosity is modeled using the large eddy simulation (LES) turbulence model based on the assumption of nonisotropic turbulence. Meanwhile, changes are made to take account of the progressive attenuation of turbulence close to the wall. LES model is applicable to the flow with different
The
The filter width
And
The choice of turbulence model is crucial for this sort of study, due to the presence of these secondary flows. Nallasamy and Rodi explained that the wellknown
Formula name  expression 

Duan et al. [5]  Low Reynolds number model: 
Jiang et al. [40]  LES model: 
2.4. Heat transfer model
It is of great significance to study turbulent flow and heattransfer mechanism due to the frequent occurrence in many industrial applications, such as heat exchangers, vapor turbines, cooling systems, and nuclear reactors [27]. In this study, a quasisteady state of temperature profile is calculated where axial thermal conduction is neglected. Since the operating temperature is influenced by ambient temperature, fluid properties, and hydraulic parameters such as flow rate and liquid level; a heattransfer model for vaporliquid flow is established.
Energy equation means the increase of energy, which is equal to the result of heat flux entering the representativeelement volume deducting the work from internal force. Ignoring axial heat conduction and viscous dispersive item, the equation of energy conservation is as follows:
2.5. Phase change model
The pressure and temperature drop along pipelines, which incurs the change of ratio of gas mass volume and liquid mass volume. It is prone to evoke P–T flash evaporation in each component. The mass percentage would change accordingly, influencing the parameters like molar mass, density, viscosity, heat capacity and thermal conductivity. The vapor enthalpy would change during the process of phase change and dissipate into the vapor or liquid in the form of phase change heat which results in variation of fluid temperature [22, 32, 45].
Cong Guo et al. considered a modification of the original model, in which the rate of conductive heat through the tube wall due to temperature difference can be calculated [43]. Peneloux et al. proposed the concept of volume translation. They argue that the volume obtained using SRKEOS is a “pseudovolume” and proposed a method of calculating the “pseudo volume” [44]. Sadegh et al. proposed an equilibrium criterion for the PengRobinson equation of state (PREOS) based on the volume translated PengRobinson equation of state (VTPREOS) and a translated functional relationship is used based on the theory of Peneloux et al. to discuss the volume translation technique [45]. The study of Li Zhang et al. shows that the condensation heat transfer coefficient reduces with the increase of wall subcooling from around 2 to 14°C. With the rise in the wall subcooling, the heat flux increases, resulting in an increasing rate of steam condensation, which brings forth a thicker condensate film on the tube surface. The thicker condensate film around the tube offers a higher thermal resistance to steam condensation and in turn reduces the condensation heat transfer coefficient [22]. Considering the volume addition of LSI phase due to the coalescence, Kai Yan et al. used the additional velocity and considered all the conditions when some portion of SSI phase can come into LSI phase [46]. Bonizzi proposed a model for calculating the atomization flux and the bulk concentration based on the recommendation by Williams et al. and Pan and Hanratty [47, 48, 49, 50, 51]. Vinesh et al. showed the physical model, considered for phase change which corresponds to hydrodynamically as well as thermally developing vaporliquid stratified flow in a plane channel, with heating from the top and cooling from the bottom wall [52] (Table 2).
Formula name  Expression 

Cong Guo et al. [43]  The rate of conductive heat through the tube wall: The temperature of the condensate around circumferential wall: 
Peneloux et al. [44]  The “pseudovolume” obtained using SRKEOS: The definition of “pseudo volume”: 
Sadegh et al. [45]  The “pseudo partial volume” can be defined as: With this definition “pseudo fugacity coefficients” can be defined as: 
Kai Yan et al. [46]  The expression of the additional velocity

Williams et al. & Pan and Hanratty [47, 48]  The atomization flux: In which 
Bonizzi et al. [51]  The flux of droplet deposition is usually expressed as: 
The equation of state in P–T Flash involves PengRobinson (PR) equation, which was proposed by PengRobinson in 1976. It can predict the molar volume more accurately than SRK equation and can be applied for polar compounds. Apart from that, the equation is applicable to vapor and liquid at the same time and widely used in the calculation of phase equilibrium.
During the P–T flash calculation process, the required parameters are the component of light hydrocarbon, gasification rate, density, molar mass, and enthalpy. The relation between input and output is explained in Figures 3 and 4.
The fluid in the pipe includes
Therefore, the latent heat may result in the temperature change of the vaporliquid system. The enthalpy difference in the process of vaporization or condensation can be obtained by virtue of P–T flash calculation. The latent heat helps to retard the temperature change. The process can be calculated as follows:
The parameters of flow state and related dimensionless parameters are given by:
The properties of fluids are calculated by:
Both gas and liquid phases are not ideal fluids. Vapor phase is a mixture of multicomponent light hydrocarbons. Thus, the property of vapor phase is a combination of their quality weighting. The PREOS equation is currently acknowledged as the most accurate formula to calculate the density of mixed vapor. If the pressure, temperature, and relative density of light hydrocarbon are known, the viscosity can be calculated using experimental formulas. Both the thermal conductivity and the heat capacity at constant pressure for vapor mixture are related to temperature and pressure, which also can be calculated using the experimental formulas. For the liquid phase, the density is calculated by PR EOS. The other physical properties including viscosity, thermal conductivity, and specific heat capacity are respectively a function of temperature, density, and other critical properties of components.
The relationship between liquid level
The value of
As shown in Figure 6, the evaporation fraction, density, and molar weight of vapor and liquid phase can be obtained with the known mole fraction of each component since the temperature and pressure are the same as inlet data in the first cross section. The liquid holdup can be calculated as follows:
The temperature, pressure, and mole fraction vary in the second and other subsequent cross section due to the influence of thermal conduction, pressure drop, and phase change but can be derived from the former section. Hence, it requires the calculation of the evaporation fraction, density, and molar weight in each grid cell. The liquid holdup can be calculated as follows:
Then, the liquid level
2.6. Wavy gasliquid interface model
Dimensionless shift at wavy gasliquid interface is defined as:
It can be calculated by (Cebeci and Smith, 1974):
The abovementioned correlation is based on the data fit, the lower limit corresponds to a smooth surface, where
The
where
The flow geometry of stratified flow in circular receiver is very complex. In order to simplify the model, many researchers use different models in their researches. The following are some models.
2.7. Boundary conditions
The boundary conditions included in vaporliquid twophase stratified pipe flow with heat transfer and phase change involve vaporliquid interface condition, wall boundary condition, symmetrical boundary condition and inlet boundary condition. As for the vaporliquid interface, in order to illustrate the mutual influences among flow, heat transfer and phase change, the equal interfacial shear stress has been prescribed as the vaporliquid interface condition which is related to the fluid properties and velocity distribution and can be calculated as:
The temperature and heat flux of two phases are respectively equal at vapor liquid interface (Table 3).
At the pipe wall, the nonslip condition is applied for velocity in both two phases.
The temperature boundary condition at pipe wall of vapor and liquid phases are convective heat transfer and its coefficient of pipeline outer wall remains constant.
The gradients of velocity and temperature are zero at the symmetrical boundary.
At the inlet of pipeline, the velocity and temperature filed of two phases are respectively equal to the pipe inlet values.
3. Results and discussion
Based on the theory of flow and heat transfer, turbulent flow and phase equilibrium, the model is solved by multiphysical field coupling numerical simulation. The noncircular liquid and vapor domains in stratified pipe flow can be simply modeled with the bipolar coordinate system, which is helpful in solving the problem caused by the inhomogeneity of boundaries. Bipolar cylindrical coordinate is composed of two orthogonal circles in rectangular coordinate. As the flow field in both phases is bounded by a circular pipe wall and a plane interface, the calculation domain has been converted to rectangle form from the anomalous physical domain by adopting the bipolar coordinate system.
With the increase of axial distance, the liquid level in pipeline changes constantly and leads to the change of flow area in both phases. The grid size changes adaptively along with the flow area, where the flow area is determined by the height of gasliquid interface. Variablesize grid has advantages in calculating the changing interface. The grid number remains unchangeable. The location of the gasliquid interface is obtained by the secant method, and the convergence condition is that the conservation of the mass flow rate of the gasliquid phase and the total mass flow rate equal to the inlet mass flow rate. In this way, the interface is detected.
Vaporliquid two phase flow and heat transfer coupled with phase change have been simulated in this section. The simulated pipeline is with inner diameter of 100 mm and total length of 6000 m. The superficial velocities of vapor and liquid are respectively
Compound  CH_{4}  C_{2}H_{6}  C_{3}H_{8}  iC_{4}H_{10}  nC_{4}H_{10}  iC_{5}H_{12}  nC_{5}H_{12}  nC_{6}H_{14} 


Mole percent (%)  78.03  4.73  5.98  3.05  3.54  2.85  0.54  0.69  0.59 
3.1. Mole fraction, density distribution, and liquid level along the pipeline
The mole fractions of each component, density distribution, temperature distribution, and liquid level along the pipeline are obtained in the condition of condensation production.
Mole fractions of each component at pipe inlet in both phases are shown in Table 5. In vapor phase, the mole fraction of methane is larger than all the other light hydrocarbons (
Compound  CH_{4}  C_{2}H_{6}  C_{3}H_{8}  iC_{4}H_{10}  nC_{4}H_{10}  iC_{5}H_{12}  nC_{5}H_{12}  nC_{6}H_{14} 

Total 

Mole fraction in vapor phase (%)  80.54  4.69  5.61  2.71  3.05  2.25  0.41  0.44  0.30  100 
Mole fraction in liquid phase (%)  43.31  5.35  11.05  7.82  10.30  11.21  2.32  4.10  4.54  100 
Mole fractions of each component in both vapor and liquid phase are shown in Figure 5. The content of methane in vapor phase increases when flowing in the pipe while the content of the other light hydrocarbons become less and less in vapor phase. During the condensing process which is dominated by temperature drop, the methane keeps evaporating; while during the evaporating process which is dominated by pressure drop, the other light hydrocarbons keep condensing. Meanwhile, the bigger the molar mass is, the faster the condensing rate is.
Six pipe cross sections located at every 1000 m along the pipeline are selected to illustrate the change of density and temperature distribution, as shown in Figures 6 and 7. It is exactly because the pressure on the cross section is the same, so the uneven distribution of the temperature leads to the uneven distribution of the fluid density. In the two phases, the density distribution is opposite to the temperature distribution. The high temperature means low density. The temperature value distributed at every single cross section can be ranked in descending order: the interior of liquid phase, most parts in vapor phase, vapor phase near the top wall. However, the descending rank of density is liquid phase, vapor phase near the top wall, and the interior of vapor phase. Along the pipeline, the temperature in the area referenced above decreases gradually while the density increases gradually. Therefore, the temperature of the sequential cross sections tends to be the same and the density distribution within the two phases gradually becomes uniform.
Figure 8 illustrates the selected 12 pipe cross sections located at every 500 m along the pipeline, where the varying trend of liquid level are presented in threedimensional coordinate system. The minimum liquid level is 7.82 mm at pipeline inlet. The liquid level at outlet is about 16.18 mm and keeps declining trend, which can also be found in Figure 9(b).
3.2. Pressure gradient, liquid level, fluid mass flow rate, and temperature along the pipeline
The pressure gradient, liquid level, fluid mass flow rate, and temperature along the pipeline obtained in the condition of condensation production have been compared with that found in literature.
When the phase change behavior is considered along pipe flow, the vapor in gas phase starts to condense to liquid which begins from the pipe inlet due to the significant temperature drop at pipe wall. The pressure drop fit well with the simulated results presented by Sadegh, as shown in Figure 9(a) [16].
During the condensing process, vapor mass flow rate gradually reduces, as shown in Figure 9(c), and the liquid mass flow rate increases due to the constant total mass flow rate. The increase of liquid mass flow rate leads to further rise of liquid level, as shown in Figure 9(b).
The liquid holdup firstly increases until it reaches a maximum value and then gradually decreases. The reason behind this is as follows: The increase of liquid holdup results from the liquid precipitation caused by dominant temperature drop. Due to the large difference between fluid temperature and ambient temperature, the amount of liquid precipitation is greater than liquid evaporation. On the contrary, the decrease of liquid holdup is leaded by liquid evaporation due to dominant pressure drop. Being same to liquid holdup, the liquid mass flow rate maintains the same trend, that is, gradually increasing to reach a maximum value and then gradually reduced, which is depicted in Figure 9(c). As the total mass flow is constant, the mass flow rate of the vapor phase decreases first and then increases. When compared with the process of evaporation, the precipitation process caused by temperature drop is transient and intense, which is related to the temperature difference between the inside and outside of the pipeline and to the convective heat transfer coefficient.
The tendency of temperature drop is similar to that in the existing research [16]. But there also exists difference between vapor bulk temperature, liquid bulk temperature, and the total bulk temperature, which cannot be revealed by onedimensional model. The liquid bulk temperature is always higher than the vapor bulk temperature while the vapor bulk temperature is almost always equal to the total bulk temperature. Latent heat is revealed during the vapor condensing process which slows down the temperature drop, as shown in Figure 9(d).
3.3. Velocity and temperature distribution at pipe length of 3000 m
Through solving the model, with phase change happening, it can be obtained that the pressure gradient is 21.29 Pa/m and the liquid level is 16.02 mm when axial distance reaches 3000 m.
Figure 10(a) shows that the velocity of vapor phase slows down while approaching either the pipe wall or the vaporliquid interface because of the hindering effects and fluid viscosity. The velocity of liquid phase keeps increasing from pipe wall to the interface. The maximum velocity at the pipe cross section occurs within the vapor phase.
In Figure 10(b) shows that the temperatures of both vapor and liquid phase drop while approaching the pipe wall because of the lowest ambient temperature and the convective heat transfer effects. The temperature of liquid phase keeps increasing from pipe wall to the interface. The maximum temperature at the pipe cross section exists within the liquid phase near the interface.
Figure 10(c) shows that the temperature at pipe wall of vapor phase is lower than that of liquid phase when convective heat transfer exists due to the smaller heat carried by the vapor phase than liquid phase. Thus, lower specific heat capacity results in bigger temperature drop at pipe wall of vapor phase. The thermal conductivity of the liquid phase is greater than vapor phase, hence, the temperature gradient in liquid phase is smaller than that in vapor phase, and the bulk average temperature of the liquid phase is higher than the vapor phase. Heat is transferring from liquid phase to vapor phase through the interface, which makes the temperature drop of the liquid phase and reduces the temperature difference between the two phases.
Figure 10(d) reveals the velocity distribution at the centerline of the pipe. By the dragging force of the interface, the velocity of liquid phase reaches the maximum value at the interface while the velocity of vapor phase reaches the maximum value at the location between the interface and its bulk center. The liquid phase slows down while approaching the pipe wall because of the hindering of the pipe wall and its high viscosity.
4. Conclusion
The vaporliquid twophase pipe flow and heat transfer are studied by virtue of numerical simulation in light hydrocarbon transportation pipeline coupled with hydraulics, thermodynamics, and phase change. A threedimensional nonisothermal vaporliquid stratified flow model including phase change model in bipolar coordinate system has been established, where LES turbulence model is utilized to simulate the turbulence flow and the wall attenuation function is used to describe the inadequacy performance of vaporliquid interface. The vapor phase and the liquid phase are both considered to be compressible and the PR equation of state is chosen for the vaporliquid equilibrium calculation where the multicomponent hydrocarbon flash calculation is used to evaluate the physical properties, gasification rate, and enthalpy departure of the phases. The P–T flash calculation has been applied to predict the varying liquid level and the multicomponent mass fraction in each phase during the process of vapor/liquid stratified pipe flow. The axial pressure gradient, liquid holdup, velocity, and temperature fields have been presented. The fluid mass flow rate, mole fraction, density distribution, and liquid level along the pipeline are also given out.
The simulation results indicate that the influence of pressure and temperature on liquid holdup is different. During the light hydrocarbon transportation process in pipeline, the temperature drop leads to the reduction of vapor mass flow rate and the rise of liquid level as well as mass flow rate. Larger temperature drop results in bigger liquid holdup while larger pressure drop causes smaller liquid holdup due to the change of physical properties and phase equilibrium. After the increase of liquid holdup caused by dominant temperature drop reaching the maximum value, then the decrease of liquid holdup maintains its trend till the pipe outlet, which results from liquid evaporation due to dominant pressure drop.
The highest velocity locates in vapor phase while the highest temperature locates in liquid phase. The liquid bulk temperature is always higher than the vapor bulk temperature. The vapor bulk temperature is almost always equal to the total bulk temperature, which cannot be revealed by onedimensional model. Latent heat is revealed during the vapor condensing process which slows down the temperature drop. The average velocity of liquid is lower than that of vapor, but the temperature of liquid is higher than vapor.
When the fluid flows in the pipeline, the content of methane in vapor phase increases all the time while the content of the other light hydrocarbons (
Thus, models in this chapter can be utilized to accurately predict pressure gradient, velocity, temperature field, liquid holdup, fluid physical properties, and mole fraction, which are essential to the determination of pipe size, design of downstream equipment, and guarantee of flow assurance.
Acknowledgments
This work was supported by the National Natural Science Foundation of China [Grant number 51474228]; and the Beijing Scientific Research and Graduate Joint Training Program [Grant number ZX20150440].
References
 1.
Akansu SO. Heat transfers and pressure drops for porousring turbulators in a circular pipe. Applied Energy. 2006; 83 :280298  2.
Siavashi M, Bahrami HRT, Saffari H. Numerical investigation of flow characteristics, heat transfer and entropy generation of nanofluid flow inside an annular pipe partially or completely filled with porous media using twophase mixture model. Energy. 2015; 93 :24512466  3.
Revellin R, Lips S, Khandekar S. Local entropy generation for saturated twophase flow. Energy. 2009; 34 (9):11131121  4.
Ferreira RB, Falcão DS, Oliveira VB. Numerical simulations of twophase flow in an anode gas channel of a proton exchange membrane fuel cell. Energy. 2015; 82 :619628  5.
Duan J, Liu H, Gong J, Jiao G: Heat transfer for fully developed stratified wavy gas–liquid twophase flow in a circular crosssection receiver. Solar Energy. 2015; 118 :338349  6.
Lun I, Calay RK, Holdo AE. Modeling twophase flows using CFD. Applied Energy. 1996; 53 :299314  7.
Oliemans RVA. Modeling of gascondensate flow in horizontal and inclined pipes. In: Proc. of the ASME Pipeline Engineering SymposiumETCE Dallas; 1987  8.
Adewumi MA, NorAzlan N, Tian S. Design approach accounts for condensate in gas pipelines. SPE Eastern Regional Meeting, Society of Petroleum Engineers. 1993  9.
Zhou J, Adewumi MA. Transients in gascondensate natural gas pipelines. Journal of Energy Resource Technology. 1998; 120 :3240  10.
Schouten JA, Janssenvan Rosmalen R, Michels JPJ. Condensation in gas transmission pipelines. International Journal of Hydrogen Energy. 2005; 30 :661668  11.
Jin T. Network modeling and prediction of retrograde gas behavior in natural gas pipeline systems [thesis]. Doctoral dissertation. The Pennsylvania State University; 2013  12.
Deng D, Gong J. Prediction of transient behaviors of gascondensate twophase flow in pipelines with low liquid loading. In: 2006 International Pipeline Conference. American Society of Mechanical Engineers; 2006  13.
Vincent PA, Adewumi MA. Engineering design of gascondensate pipelines with a compositional hydrodynamic model. SPE Production Engineers. 1990:53815386  14.
Mucharam L, Adewumi MA, Watson RW. Study of gas condensation in transmission pipelines with a hydrodynamic model. SPE Production Engineers. 1990:52365242  15.
Sadegh AA, Adewumi MA. Temperature distribution in natural gas/condensate pipelines using a hydrodynamic model. SPE Eastern Regional Meeting; Society of Petroleum Engineers. 2005  16.
Abbaspour M, Chapman KS, Glasgow LA. Transient modeling of nonisothermal, dispersed twophase flow in natural gas pipelines. Applied Mathematical Modelling. 2010; 34 :495507  17.
Dukhovnaya Y, Adewumi MA. Simulation of nonisothermal transients in gas/condensate pipelines using TVD scheme. Powder Technology. 2000; 112 :163171  18.
Almanza R, Lentz A, Jimeenez G. Receiver behavior in direct steam generation with parabolic troughs. Solar Energy. 1997; 61 :275278  19.
Newton CH, Behnia M. A numerical model of stratified wavy gas–liquid pipe flow. Chemical Engineering Science. 2001; 56 :68516861  20.
Newton CH, Behnia M. Numerical calculation of turbulent stratified gas–liquid pipe flows. International Journal of Multiphase Flow. 2000; 26 :327337  21.
Berthelsen PA, Ytrehus T. Numerical modeling of stratified turbulent two and threephase pipe flow with arbitrary shaped interfaces. In: The 5th International Conference on Multiphase Flow; 30 May–4 June, 2004; Yokohama, Japan  22.
Zhang L, Yang S, Xu H: Experimental study on condensation heat transfer characteristics of steam on horizontal twisted elliptical tubes. Applied Energy. 2012; 97 :881887  23.
Gada VH, Datta D, Sharm A. Analytical and numerical study for twophase stratifiedflow in a plane channel subjected to different thermal boundary conditions. International Journal of Thermal Science. 2013; 71 :88102  24.
Manabe R. A comprehensive mechanic heat transfer model for twophase flow with high pressure flow pattern validation [thesis]. Department of Petroleum Engineering: University of Tulsa; 2001  25.
Fontoura VR, Matos EM, Nunhez JR. A threedimensional twophase flow model with phase change inside a tube of petrochemical preheaters. Fuel. 2013; 110 :196203  26.
Mansoori Z, Yoosefabadi ZT, SaffarAvval M. Two dimensional hydro dynamic and thermal modeling of a turbulent two phase stratified gas–liquid pipe flow. In: ASME 2009 Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers; 2009. p. 753758  27.
Singh AK, Goerke UJ, Kolditz O. Numerical simulation of nonisothermal compositional gas flow: application to carbon dioxide injection into gas reservoirs. Energy. 2011; 36 :34463458  28.
Ebadian MA, Vafai K, Lavine A. Single and multiphase convective heat transfer. Applied Energy. 1992; 43 :291292  29.
Gong G, Chen F, Su H, Zhou J. Thermodynamic simulation of condensation heat recovery characteristics of a single stage centrifugal chiller in a hotel. Applied Energy. 2012; 91 :326333  30.
Hu H, Zhang C. A modified k–e turbulence model for the simulation of twophase flow and heat transfer in condensers. International Journal of Heat and Mass Transfer. 2007; 50 :16411648  31.
Sarica C, Panacharoensawad E. Review of paraffin deposition research under multiphase flow conditions. Energy Fuel. 2012; 26 :39683978  32.
Duan J, Gong J, Yao H. Numerical modeling for stratified gas–liquid flow and heat transfer in pipeline. Applied Energy. 2014; 115 :8394  33.
Ullmann A, Brauner N. Closure relations for twofluid models for twophase stratified smooth and stratified wavy flows. International Journal of Multiphase Flow. 2006; 32 (1):82105  34.
Vincent PA, Adewumi MA. Engineering design of gascondensate pipelines with a compositional hydrodynamic model. SPE Production Engineers. 1990; 5 (04):381386  35.
Haaland SE. Simple and explicit formulas for the friction factor in turbulent pipe flow. Fluids Engineering. 1983; 105 :8990  36.
Helgans B, Richter DH. Turbulent latent and sensible heat flux in the presence of evaporative droplets. International Journal of Multiphase Flow. 2016; 78 :111  37.
Vargaftik NB, editor. Handbook of Physical Properties of Liquids and Gasespure Substances and Mixtures. Hemisphere Pub; 1975  38.
DenHerder T. Design and simulation of PV super system using simulink [thesis]. San Luis Obispo: California Polytechnic State University; 2006  39.
Zaghloul JS. Multiphase analysis of threephase (gascondensatewater) flow in pipes (Doctoral dissertation [thesis]). Pennsylvania State University; 2006  40.
Jiang X, Siamas GA, Jagus K, et al. Physical modeling and advanced simulations of gas–liquid twophase jet flows in atomization and sprays. Progress in Energy & Combustion Science. 2010; 36 (2):131167  41.
Jones WP, Launder BE. The calculation of lowReynoldsnumber phenomena with a twoequation model of turbulence. International Journal of Heat and Mass Transfer. 1973; 16 (6):11191130  42.
Hishida M, Nagano Y, Tagawa M. Transport processes of heat and momentum in the wall region of turbulent pipe flow. In: Proceedings of the 8th International Heat Transfer Conference. Hemisphere Publishing Corp; 1986;Washington, DC; 1986. p. 925‐930  43.
Guo C, Wang T, Hu X, et al. Experimental investigation of the effects of heat transport pipeline configurations on the performance of a passive phasechange cooling system. Experimental Thermal and Fluid Science. 2014; 55 :2128  44.
Péneloux A, Rauzy E, Fréze R. A consistent correction for RedlichKwongSoave volumes. Fluid Phase Equilibria. 1982; 8 (1):723  45.
Sadegh A A, Adewumi M A. Temperature distribution in natural gas/condensate pipelines using a hydrodynamic model. In: SPE Eastern Regional Meeting. Society of Petroleum Engineers; January, 2005  46.
Yan K, Zhe D. A coupled model for simulation of the gas–liquid twophase flow with complex flow patterns. International Journal of Multiphase Flow. 2010; 36 (4):333348  47.
Williams LR, Dykhno LA, Hanratty TJ. Droplet flux distributions and entrainment in horizontal gas–liquid flows. International journal of multiphase flow. 1996; 22 (1):118  48.
Pan L, Hanratty TJ. Correlation of entrainment for annular flow in horizontal pipes. International Journal of Multiphase Flow. 2002; 28 (3):385408  49.
Laurinat JE, Hanratty TJ, Jepson WP. Film thickness distribution for gas–liquid annular flow in a horizontal pipe. PhysicoChemical Hydrodynamics. 1985; 6 (1):79195  50.
Pitton E, Ciandri P, Margarone M, Andreussi P: An experimental study of stratified–dispersed flow in horizontal pipes. International Journal of Multiphase Flow. 2014; 6 :92103  51.
Bonizzi M, Andreussi P. Prediction of the liquid film distribution in stratifieddispersed gas–liquid flow. Chemical Engineering Science. 2016; 142 :165179  52.
Gada VH, Datta D, Sharma A. Analytical and numerical study for twophase stratifiedflow in a plane channel subjected to different thermal boundary conditions. International Journal of Thermal Sciences. 2013; 71 :88102  53.
Badie S, Hale CP, Lawrence CJ, et al. Pressure gradient and holdup in horizontal twophase gas–liquid flows with low liquid loading. International Journal of Multiphase Flow. 2000; 26 (9):15251543