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ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018

1

Towards a multicomponent real-fluid fully compressible two-phase flow model

Songzhi Yang1,2, Chaouki Habchi1,2*, Ping Yi1,2, Rafael Lugo1

1IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France

2Institut Carnot IFPEN Transports Energie

Abstract

A fully compressible two-phase flow with four-equation model coupled with a real-fluid multi-component

phase equilibrium solver is proposed. It is composed of two mass conservation equations, one momentum equa-

tion and one energy equation under the assumptions of mechanical and thermal equilibrium. The phase equilib-

rium solver is developed based on the Peng-Robinson equation of state. To be more specific, this solver not only

includes the entire processes of vapor-liquid equilibrium covering phase stability analysis which is based on

tangent plane distance (TPD) method [24] and isothermal-isobaric flash (TPn) [25], but also involves an isoen-

ergetic-isochoric (UVn) flash [27] for the computation of equilibrium temperature, pressure and phase composi-

tions. This model named as 4EQ-PR has been implemented into an in-house IFP-C3D software [8]. Compari-

sons between the predictions of the 4EQ-PR model with available numerical results in literature have been car-

ried out. The compared results demonstrated that the 4EQ-PR model can well reproduce the multicomponent

real-fluid behaviors in subcritical and transcritical conditions with phase change, and strong shock wave in su-

percritical conditions. Finally, the proposed 4EQ-PR model has been applied to typical 3D injection modeling

including the classical supercritical and transcritical injection regimes. It has been demonstrated that the simula-

tion results can be correlated with relevant theories and experiments very well. In particular, analysis of the

phase state (TPD) contour in different thermodynamic regimes has led to better understanding for the phase

change process around the liquid core in transcritical conditions.

Keywords: Two-phase flow, four-equation model, phase equilibrium model, real fluid, Peng-Robinson EOS

(*) Corresponding author: Chaouki.HABCHI@ifpen.fr

1. Introduction

Accurate and robust modeling of compressible two-phase flow is crucial for many engineering applications,

such as the fuel injectors, nuclear reactors, rocket motors, as well as gas turbines and heat pumps [23]. The

involved two-phase flow may be subcritical, transcritical or supercritical depending on the pressure and

temperature but also the composition operating conditions. Many physical and numerical models have been

developed for the simulation of two-phase flow. Based on the numbers of transport equations and initial

equilibrium assumptions, the models can be divided into three to seven equation system. The most general two-

phase flow model is the fully non-equilibrium seven-equation model, in which each phase has its own pressure,

velocity and temperature, and is governed by its own set of fluid equations. More precisely, it is based on

compressible model composed of three balance equations for gas phase and three balance equations for liquid

phase, together with a transport equation for the volume fraction for one of the phases. The relaxation method at

the interface generally are comprised of finite characteristic time [4][14] or the stiff (i.e. instantaneous)

relaxation approaches [32][16, 36] for the velocity, pressure, temperature and chemical potential. The seven-

equation model shows great capabilities in describing the complex wave patterns and correctly capturing the

wave propagation in liquid and gas phases [3]. However, the complexity of implementation has limited its

extensive use. The reduced five-equation models, in which mechanical and thermal equilibrium are assumed,

have been proposed extensively [1, 26, 30]. Kapila et al. [17] have constructed the most popular formulation

with two mass conservation equations for liquid and vapor, one mixture momentum equation, one mixture

energy equation, and together with the transport equation for the liquid volume fraction. It has been

demonstrated that this diffuse interface model shows excellent resolution of interfaces between two

compressible fluids [26, 28]. Besides, more simplified four-equation models are also widely used [13, 18, 35].

These models are usually composed of three conservation laws for mixture quantities (mass, momentum, energy)

along with partial density transport equations. Four-equation model has been proven to show high

computational efficiency [6]. Recently, Saurel et al. [33] proposed a new four-equation model, in which each

phase is compressible and the two phases share common pressure, velocity, temperature, and Gibbs free energy.

Moreover, they developed a specific phase equilibrium solver by applying the Noble-Abel-Stiffened Gas

equation of state (NASG-EoS) [19] for each phase. Indeed, this fitted-parameters NASG-EoS [19] is adopted to

simplify the thermodynamic computations. Many advantages of this four-equation model with the phase

equilibrium solver has been shown in terms of the computational efficiency and numerical stability. It is worth

noting that, this phase equilibrium solver with NASG-EoS is currently limited to the liquid phase merely with

single-component (i.e., gas solubility in liquid phase is neglected). In reality however, substantial amount of gas

dissolved in liquid phase under high pressure conditions plays primary influence on phase change. Therefore,

the multi-component real-fluid behaviors in liquid phase need to be considered.

The study here has proposed a fully compressible multicomponent real-fluid four-equation model coupled

with a phase equilibrium solver. Since the cubic EoS are known to be able of predicting the real-fluid behaviors

ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018

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for two-phase flow, in particular, the Peng-Robinson (PR) EoS, as a good trade-off between computational

efficiency and accuracy, has been chosen in the current research. A multicomponent phase equlibrium solver has

been developed based on PR-EoS. This solver is able to deal with the mass and heat transfer at interfaces for

each computational time step with the assumption of instantaneous equilibrium between liquid and gas phases.

Similar research can be found in the work of Matheis et al. [22] but with different transport equations of species.

Qiu et al. [29] also developed a consistent and efficient phase equilibrium solver using PR-EOS based on the

Lagrange-Eulerian framework of KIVA-3 [2].

The structure of this paper is as follows: Section 2 introduces the partial differential equations and

thermodynamic models, referred to below as 4EQ-PR. Section 3 provides the comparisons between predictions

of 4EQ-PR model and previously published numerical results. Then, this model thus checked has been applied

for the simulation of several typical 3D injection conditions, including the classical supercritical and transcritical

regimes. Finally, the conclusions are summarized in Section 4.

2. Numerical Model

The fully compressible multicomponent two-phase flow four-equation model has assumed the liquid phase

and vapor phase in mechanical and thermal equilibrium. As shown in the following Eqs. (1-4), the four-equation

model includes the balance equations for different species in gas and liquid phases, mixture momentum, and

mixture specific internal energy, respectively:

(1)

(2)

(3)

(4)

in which, index and denote the liquid and gas phase respectively; represents different species; the

RHS terms of are the phase transition mass of liquid and gas phases, respectively, following

+ = 0;

is the shear stress tensor covering the laminar (L) and turbulent (T) contributions, which

can be written as

with for turbulent flows. As described in our previous work [16], a

standard Boussinesq approximation is used for the modeling of

using a subgrid-scale turbulent viscosity

given by the Smogorinsky LES model. However, the laminar viscosity is computed from Chung’s correlation

[11]. In Equation (4), represents the specific internal energy;

is the heat conduction flux and usually

modelled as

based on Fourier’s law. The heat conduction coefficient has also a laminar and

turbulent contributions. The laminar contribution is also computed from Chung’s correlation [11] and the turbu-

lent one is estimated using a given Prandtl number, . denotes the volume fraction of phase p. It is

computed in the phase equilibrium solver along with . Considering the limited space in this article, the

details of the phase equilibrium solver are not described here. However, they can be found in [27, 31] and also

the work of Matheis et al. [22] and Qiu et al. [29]. The main difference lies in the mass conservative equation in

which the overall partial densities of species in the mixture is transported instead of each phase. The final model

coupled with the thermo-solver is referred to below as 4EQ-PR. Its first validation is performed with a series of

test cases and compared with available literature references, as described below in Section 3.

3. Results and Discussion

To prove the validity and accuracy of the models, two one-dimensional (1D) shock tube test cases at differ-

ent operating conditions ranging from subcritical to supercritical are performed. Then, the 4EQ-PR model is

used to simulate three dimensional (3D) typical injection conditions including classical supercritical and tran-

scritical regimes.

3.1 1D Shock tube test cases

Table 1 Initial conditions for test cases; L and R denotes shock tube left and right sides, respectively.

Section

No.

/MPa

/MPa

/K

/K

3.1.1

0.2

0.1

293

293

1e-5

1e-5

3.7e-3

3.1.2

60

6

158

222

0.999999

0.999999

1e-7

ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018

3

3.1.1 Subcritical shock tube

The shock tube is one meter long and with the initial discontinuity at 0.5 m. The pressure ratio is 2 (2 bar in

the left and 1 bar in the right) and the temperature is 293K throughout the tube. The whole tube is filled with

more than 99% liquid water. The volume fraction of gas in the liquid is around 4e-3. These initial condi-

tions (see Table 1) are the same as in Chiapolino et al. [10]. The results from 4EQ-PR model are compared with

[10] in which a thermodynamic solver using the Stiffened Gas (SG) EoS combined with a four-equation model

has been adopted. However, in Chiapolino et al. [10] model, only single component in liquid phase has been

considered, which means the dissolved gas is neglected. In the following discussions and Figures, the Chiapo-

lino et al. [10] model is referred to as 4EQ-SG. It is worth to note that binary interaction parameter (BIP) for the

water and nitrogen mixture is set with 0.4788 based on ref [9]. The results presented are at 1.5ms. As shown in

Figure 1(c), the density from SG-EoS is larger than from PR-EoS. Because the initial condition in 4EQ-PR

model is at real equilibrium state, this has induced different water-vapor mass fraction (see Figure 1(e)) with

that of 4EQ-SG model. Affected by the expansion and compression waves, obvious phase transition can be seen

in Figure 1(e). Indeed, the influence of the dissolved nitrogen on phase change, in this test case, is small due to

the negligible amount of nitrogen in the mixture (see Figure 1(f)). This explains well the almost equivalent vap-

orous water level by the 4EQ-SG and 4EQ-PR models, as shown in Figure 1(e). In addition, the pressure and

temperature variations for two models have presented similar trend (see Figures 1(a) (b)). But, the expanding

width from 4EQ-PR is slightly narrower than from 4EQ-SG. The reason may come from the different cell num-

bers,100 for 4EQ-SG vs. 1000 with 4EQ-PR. Overall, this shock tube test case has proved that 4EQ-PR model

compare well with the Chiapolino et al. [10] numerical results and can capture the wave transmission and phase

transition accurately using a cubic EoS in subcritical conditions.

Figure 1 1D subcritical shock tube at t=1.5ms.The numerical results (solid line with symbols) from 4EQ-PR are

compared with the numerical results from Chiapolino et al. [10] (dashed bold line) from 4EQ-SG. The dot-

dashed lines are the initial conditions. The simulations with 4EQ-PR are carried out with 1000 cells and a CFL

number equals to 0.2.

4

3.1.2 Supercritical Shock Tube

In this test case, a supercritical shock tube problem is simulated and the results are compared with the nu-

merical results from Ma et al. [20]. In the latter model, referred to below as 4EQ-PR-REF, a fully conservative

solver combined with Peng-Robinson EoS, but without considering phase change is used. The detailed condi-

tions of this test case can be found in Table 1. Particularly, an almost pure nitrogen is used throughout the com-

putational domain. Based on tangent plane distance (TPD) test [24, 27], the initial mixture of nitrogen with the

trifle amount of n-dodecane (Yn-c12h26=1e-6) is still in single phase state and the mixture critical point is almost

the same as pure nitrogen, as shown in Figure 5(J). The mesh and boundary conditions are the same as the pre-

vious subcritical shock tube case. The initial discontinuity is also in the middle of the tube, at x=0.5 m. But, in

this case, the pressure and temperature are above the critical point of nitrogen for both sides. The numerical re-

sults from 4EQ-PR model are compared with the results from Ma et al. [20] in Figure 2. Good agreements at

0.05ms can be seen in Figure 2. Noteworthy, the 4EQ-PR model can capture the shock wave correctly although

we are using a quasi-conservative formulation for energy balance (Eq. 4). In addition, there are no spurious os-

cillations and visible diffusion showing up at the interface (i.e. contact discontinuity) which can prove the ro-

bustness of the current numerical solver (see also in [27, 31]).

Figure 2 1D supercritical shock tube at t=0.05ms. The computational results (solid line with symbols from

4EQ-PR), are compared with results from Ma. [20]. (dashed bold line from 4EQ-PR-REF). The computations

were conducted with 1000 cells and a CFL number equals to 0.4.

3.3 3D supercritical and transcritical injection modelling

To further test the capability of the newly developed 4EQ-PR model in dealing with supercritical and tran-

scritical conditions, a series of 3D simulations are carried out in this section. The computational configuration is

a typical injector which consists of a single-hole (Length = 1 mm and Diameter=100 µm) fitted to a hexahedral

chamber, as shown in Figure 3(A). The total number of cells are 1504800 with a minimum mesh size of 10

The boundary conditions are set with pressure inlet and outlet in the left and right side of the geometry, respec-

tively, as shown in the cross-section of the grid presented in Figure 3(B). The injection pressure is set with 70

bar, which is above n-dodecane pressure critical point (18.2 bar). No larger injection pressure is necessary for

the present investigations of supercritical and transcritical regimes.

Figure 3. (A) 3D geometry and mesh configuration

(B) Central cross-section of the grid

The details of initial boundary conditions are listed in Table 2. For all test cases, the working fluid is a mix-

ture of n-dodecane and nitrogen. The global scenario is liquid n-dodecane (at 363 K) is injected into higher tem-

x

5

perature gas (at 900 K) in the chamber. In the first two cases (Cases 1.1 and 1.2), an almost pure liquid n-

dodecane (at 363 K lower than n-dodecane critical point, 658 K) is injected into the chamber filled with an al-

most pure vaporous n-dodecane (at temperature 900 K and pressure, 40 or 60 bar, higher than n-dodecane pres-

sure critical point, 18.2 bar). On the other hand, Cases 2.1 and 2.2 are aimed at modelling a more realistic condi-

tions of diesel injection in transcritical conditions. In these cases, the chamber is filled with an almost pure gas-

eous nitrogen (at temperature 900 K and pressure 40 or 60 bar, higher than nitrogen pressure critical point, 33.1

bar). In fact, the chamber conditions are above the critical point of any single component in this system regard-

less of pressure or temperature.

Table 2 Initial conditions for the 3D injection simulations

Case

No

1.1

7

6

6

900

363

0.9999

0.9999

1.2

7

4

4

900

363

0.9999

0.9999

2.1

7

6

6

900

363

0.9999

0.0001

2.2

7

4

4

900

363

0.9999

0.0001

denotes the mass fraction of n-dodecane.

Firstly, as for Case1.1 and 1.2, the pressure in the whole injection process is above the critical point of fuel, but

the temperature has been through the transition from subcritical to supercritical in the chamber. No phase transi-

tion is observed in whole injection process. In this mixing regime, the liquid jet behaves like a gaseous jet, as

depicted by the fluid phase state TPD in Figure 4(C, E) at 70 . The phase state is identified by the tangent

plane distance (TPD) criterion [24]. In current study, for the post-processing convenience, we have employed

different TPD values to show the flow state. TPD = 0, 1 or 2 correspond to single gas phase state, single liquid

state or two-phase state, respectively. In current research, the model used to differentiate between single liquid

(or liquid-like) and single gas (or gas-like) is based on the relation of feed and equilibrium constant (i.e. species

composition and equilibrium constant ):

(pure vapor) and (pure liquid) [25]. There-

fore, it is easy to verify from Figure 4(C, E) that the overall flow is in single fluid state (TPD=0 and 1) and, the

fluid is still in supercritical regime in all the cells of the computational domain. To be more specific, the flow

has transited from pure liquid (TPD = 1) in the liquid core to higher temperature gas state (TPD=0) directly

without crossing the two-phase envelop (i.e. only cross the pseudoboiling-line but not the co-existence line here,

as shown in Figure 5(G) for supercritical conditions). Hence, the whole injection process is in supercritical re-

gime. Moreover, the boundary of white isosurface marked with TPD = 1 in Figure 4(D, F) is very close to the

maximum value of heat capacity (Cp), which corresponds to the pseudoboiling-line, depicted in Figure 5(G). To

better understand these intricate phenomena of supercritical injection regime, reduced pressure (Pr) and tempera-

ture (Tr) radial profiles are plotted at 70 in Figure 5(H), in a section located at a distance of 0.5 mm from the

nozzle outlet, as depicted by the arrow in Figure 3(B). The temperature has seen a gradual increase from liquid

core to the out layer of the jet. However, the pressure is approximately constant and the same as chamber pres-

sure along this section. The mixing layer exhibits a similar behavior as a gaseous jet. When plotting the varia-

tion of heat capacity (Cp) at this section versus temperature, a maximum value can be observed for both cases, as

shown in Figure 5(I). These maximum points in the Cp curves belongs to the pseudoboiling-line depicted in Fig-

ure 5(G), that separates liquid-like from gas-like supercritical fluids [5]. In addition, the collected Cp values are

compared with the data from Coolfluid thermal properties library [7] and a good agreement has been achieved.

It can also be seen in Figure 5(I) that the maximum value becomes higher when the ambient pressure is closer to

the critical point. Indeed, the variation of Cp at lower chamber pressure (Pr = 2.2, case 1.2) has presented strong-

er non-linearity than higher chamber pressure (Pr = 3.3, case 1.1). The Cp distribution also widens and flattens at

higher pressure as confirmed by Banuti [5]. The case with lower chamber pressure (Case 1.2) also corresponds

to lower pseudoboiling temperature which can be proved in the path 2 and path 3 of Figure 5(G).

Case 1.1

Case1.2

Figure 4. 3D supercritical modeling with the 4EQ-PR model. (C,E) show the phase state (gas phase TPD=0 and

liquid phase TPD=1) at different time instants. (D,F) show the heat capacity contours at 70 around the liquid

core represented by isosurface between TPD = 0 and 1.

(C)

(D)

(E)

(F)

6

Figure 5. 3D supercritical modeling with the 4EQ-PR model. (G) denotes different modeling regimes with re-

gards to the variation of Tr and Pr from ref [5]. (H) illustrates the evolution of pressure and temperature from

the middle of liquid core to the out layer of the jet in the radial section with a distance of 0.5mm from the outlet

of the nozzle. (I) plots the variation of heat capacity with temperature in the radial section at a distance of

0.5mm from the outlet of the nozzle for case1.1-1.2 (solid line with symbols). The dashed lines are data from

CoolProp open source library [7]. (J) illustrates the variation of mixture critical point (Pc, Tc) with the concen-

tration of N2 (solid line), numerical data source: Simulis Thermodynamics software package [34]. the experi-

mental data are from ref [15].

In contrast, Cases 2.1- 2.2 demonstrate the transcritical regime. The thermal condition of Case 2.1 is similar

to ECN Spray A [21] except the lower injection pressure of 70 bar that has been adopted in this study instead of

1500 bar. Different with Cases 1.1-1.2, the chamber is, in Cases 2.1- 2.2, filled with almost pure nitrogen (see

Table 2). During the simulation, there are a lot of N2 mixing with the n-dodecane jet, which has raised signifi-

cantly the mixture critical point. The mixture ‘true’ critical point values used in this study are from Simulis

Thermodynamics software [34]. A good agreement can be achieved with available experimental points, as

shown in Figure 5(J). In this Figure, one may see that very dilute mixtures of n-dodecane in nitrogen (Yn-

c12h26=1e-6) is still in single phase state and the mixture critical point is almost the same as pure nitrogen, as dis-

cussed above. In contrast, the mixture critical pressure has proved to exceed 3000 bars at low n-dodecane con-

centration (around 0.05). As the chamber pressure is usually much lower than such high mixture critical pres-

sure, some zones lying in subcritical regime surely exists around the liquid core, in a region where n-dodecane

concentration is low. To verify this hypothesis, the TPD fluid state is shown at 46 and 70 in Figure 6(K,

M, O, Q). A clear two-phase zone (TPD = 2: red color) enveloping the liquid jet can be observed for both cases.

This two-phase zone is the main feature of the transcritical regime making it distinct with the previous results of

Case 1.1-1.2 (compared to Figure 4(C, E)). These results corroborates the latest experimental findings from

Crua et al., [12], who have proved that there exists phase transition even when the operating conditions is under

supercritical condition with regards to the pure fuel critical point. Finally, it is worth noting that with smaller

pressure difference, Case 2.1 has shown wider two phase zone because of the lower velocity (Figure 6(K, M).

Therefore, the two-phase zone enveloping the liquid core should be thinner using higher injection pressure. But,

this work has proved that this subcritical zone exists and is the location of liquid evaporation and may be prima-

ry atomization. The contour of mass fraction of vaporous n-dodecane for these two cases are demonstrated in

Figure 6(L, N, P, R).

(G)

(H)

(I)

(J)

7

4. Summary and Conclusions

A fully compressible two-phase flow model combined with phase equilibrium solver with real-fluid equa-

tion of state has been developed in the current study. A series of one-dimensional shock tube cases covering

subcritical, transcritical and supercritical conditions are performed and checked with results from available liter-

ature. Then, the 4EQ-PR model has been used to simulate the classical isobaric supercritical and transcritical

transition regimes through several 3D liquid injection simulations. The simulation results have confirmed rele-

vant theory analysis from available literatures. In particular, the analysis of the phase state TPD contour in dif-

ferent thermodynamic regimes has led to better understanding of the phase change process around the liquid

core in transcritical conditions.

Acknowledgements

This project has received funding from the European Union Horizon 2020 Research and Innovation pro-

gram. Project IPPAD: Grant Agreement No 675528.

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