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

Conference Paper

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

Abstract and Figures

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 equation and one energy equation under the assumptions of mechanical and thermal equilibrium. The phase equilibrium 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 compositions. This model named as 4EQ-PR has been implemented into an in-house IFP-C3D software [8]. Comparisons between the predictions of the 4EQ-PR model with available numerical results in literature have been carried 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 simulation 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.
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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
3.7e-3
3.1.2
60
6
158
222
0.999999
1e-7
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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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.
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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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)
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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)
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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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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Case 2.1
Case 2.2
Figure 6. 3D transcritical modeling with the 4EQ-PR model. (K,M,O,Q) show the phase state (gas phase
TPD=0, liquid phase TPD=1 and two-phase mixture TPD=2) for four cases at different time instants. (L, N, P,
R) show the mass fraction of vaporous n-dodecane for cases 2.1 and 2.2 at 46 and 70.
(K)
(M)
(L)
(N)
(O)
(Q)
(P)
(R)
ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 22-26, 2018
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... One example is that, in diesel engines when the liquid fuel is injected into the ambient gas at a pressure higher than its critical value, the fuel jet will be heated to supercritical temperature before combustion takes place. This process is often refered to as transcritical injection [1]. Intensive fluid-dynamical structures and thermodynamical processes may happen in the typical complex two-phase flow. ...
... Simulation of such complicated unsteady, multi-scale two-phase flows with multiple fluids of thermodynamic properties is very challenging. Accurate flow solver with the capability of capturing high-resolution interface and realfluid thermodynamic solver which can precisely describe phase and component states under varying temperature and pressure conditions are greatly desired [1,2,3]. Regarding the flow solver, diffuse-interface methods are widely used with various numerical schemes. ...
... Once the convergence criterion is reached, the equilibrium temperature T and pressure p are obtained; see details of the UVn-flash we consider in the present study in [1,3] and other relevant solvers, e.g. in [20]. One critical difficulty of the present UVn-flash (which is based on the deterministic gradient-dependent quasi-Newton algorithm) lies in the convergence of iterative root-finding for equilibrium T and p due to the complex non-linearity of PR-EoS and VLE constraints. ...
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In this study, a mass-fraction based fully conservative multi-component two-phase flow solver is considered using characteristic-wise finite-difference (CWFD) discretization with the 5th-order WENO scheme, in order to reduce numerical interface smearing and oscillations. Real-fluid thermodynamic properties are accounted for by a vapor-liquid equilibrium (VLE) model according to the local total density, internal energy and composition of the homogeneous mixture, with each phase being separatedly described by its Peng-Robinson equation of state (PR-EoS). A multicomponent Roe averaging for the cell-face eigen-system in CWFD methods has been developed with pressure derivatives resulted from the VLE model. Several 1D testing examples, e.g. interface advection, shock-tube problems and double-expansion cavitation, have been examined to demonstrate the low-oscillation, low-dissipation and robust performance of the present solver, in comparison with finite-volume schemes. A 2D transcritical injection process has also been simulated. It has been shown that high-order numerical schemes, such as the current CWFD method may be the way to reduce the smearing in diffuse interface modelling.
... Inspired by these works, some researchers proposed a hypothesis that the spray appears to produce supercritical fluid from the nozzle outlet when injected into the most elevated temperature and pressure conditions, and the classical atomization theory used in the subcritical condition is unavailable for the spray simulation under the supercritical condition. However, Yi et al. [19] proposed a compressible two-phase flow model with the real-gas equation of state to unravel the droplet evolutions under the same ambient conditions, and they found that the n-dodecane vapor is still generated by the evaporation rather than diffusion, and the droplet is still in two-phase regime even though the droplet surface is apparently deformed as observed in the experiment. Therefore, it is a necessary to investigate the thermodynamic evolutions of dense sprays under the supercritical conditions. ...
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The trans-critical transition of multi-component sprays under the ECN Spray-A conditions has attracted extensive interest. However, the drastic change of critical points of multi-components makes this controversial. In this study, the trans-critical transition of sprays before and after the end of injection is evaluated by numerical methods, and new findings have been obtained. First, the robust and accurate thermodynamic equilibrium solver and multi-component droplet evaporation model were developed and implemented into the OpenFOAM. These models have been validated against the measured phase change diagram and isolated droplet evaporation rate. Then, the mixture critical temperatures of n-dodecane/nitrogen and multi-component diesel/nitrogen were calculated under a wide range of pressures based on the thermodynamic equilibrium solver. The mixture critical temperature decreases almost linearly with the increase of the pressure. Following that, extensive sprays of n-dodecane and multi-component diesel under high temperature and pressure conditions were simulated, and the predicted spray liquid penetrations were validated against the experimental data. The results found that the dilute sprays after the end of injection are more prone to transition to the supercritical mixing regime, while the trans-critical possibility of dense sprays decreases due to the cooling effect of extensive evaporation. Finally, the trans-critical transitions of multi-component diesel sprays before and after the end of injection were evaluated and compared with those of n-dodecane sprays. Due to the low mixture critical temperature and evaporation rate of diesel, its sprays are more likely to transition to the supercritical regime. Therefore, it can conclude that the dilute sprays after the end of injection cannot represent the thermodynamic state of dense sprays. Moreover, the multi-component diesel and n-dodecane sprays follow a different trans-critical transition pathway.
... Several comprehensive reviews have been focused on the phenomenological differences in the mixing of droplets under sub-and supercritical conditions [26][27][28][29].The traditional droplet evaporation theory, CFD simulations and droplet experiments have been widely used to study the transition behavior of droplets under supercritical conditions in detail [4,9], although these methods are unable to resolve the nanoscale liquid-gas interfaces. In recent years, molecular dynamics (MD) has been increasingly applied to the study of phase transitions of fuel droplets [4,[30][31][32][33][34][35][36][37][38][39]. MD simulations are very suitable for investigating the interfacial behaviors of droplets at atomic scales under both subcritical and supercritical conditions, thus offering a possibility for revealing the mechanisms behind mixing mode transition [30,40]. ...
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For a multi-component hydrocarbon mixture under supercritical conditions, especially for fuels injected into compression ignition engines, the mechanism for the transition of the dominant mixing mode from evaporation to diffusion is not well established. In this paper, evaporation processes of a six-component hydrocarbon fuel (13.16 mol% toluene, 13.81 mol% n-decane, 22.30 mol% n-dodecane, 24.60 mol% n-tetradecane, 14.66 mol% n-hexadecane and 11.47 mol% n-octadecane) droplet in nitrogen environments were studied using molecular dynamics (MD) simulations, in comparison with those of three-component and single-component fuel droplets. The ambient pressure ranged from 2 MPa to 16 MPa and the ambient temperature ranged from 750 K to 1350 K. Results indicated that the transition characteristics of the mixed fuel were not the linearly weighted average of the physical properties of individual components in the mixture based on their mole fractions. The reason why there is a limitation on the maximum transition temperature when diffusion dominates the fuel-ambient gas mixing process under high pressures has been discussed. The average resultant force on a fuel atom of an individual component increases with increasing pressure or decreasing temperature at the supercritical temperature, and diffusion will gradually dominate the mixing process of the fuel. The clustering behavior of fuels under supercritical conditions has also been discussed.
... While most commercial solvers today rely on the assumption of incompressible multiphase flows, high temperatures and/or high pressures can be encountered which violate this hypothesis. These conditions can be found in various engineering applications such as fuel injectors, nuclear reactors, rocket motors or gas turbines and heat pumps [1], for which accurate modeling and simulation are crucial. Another important field in which compressible multiphase models are required concerns the prediction of the hydrodynamics of melting materials, such as metallic alloys or glassy materials, surrounded by an external gas. ...
... Following this, a further reduced 4-Equation model with the assumption of mechanical and thermal equilibrium has been proposed and combined with the real fluid phase equilibrium solver [20]. This 4-Equation model has been successfully applied to 2D supercritical and transcritical injection modelling in [24], and achieved a good agreement with experimental results in the 1D flash boiling conditions, as reported in [20]. Based on these previous studies, the real fluid DIM solver will be further utilized to explore its potential in solving multi-scale injection problems at HTHP conditions. ...
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A fully compressible two-phase flow model consisting of four balance equations including two mass, one momentum, and one internal energy equation, formulated with the mechanical and thermal equilibrium assumptions is developed in this article. This model is closed with a real fluid equation of state (EoS) and has been applied to the simulation of different 1D academic cases, in addition to the 3D Large-Eddy Simulation (LES) of the Engine Combustion Network (ECN) Spray A injector including the needle to target part with and without the phase change (i.e. frozen) assumptions. The obtained numerical results from the model with phase change have proven to be able to accurately predict the liquid, vapor penetrations and rate of injection compared to experimental data. However, the frozen model has presented some uncertainties and deviations in predicting the penetration length as with different measure criterions, even though an excellent agreement can be achieved in the estimation of rate of injection, near-nozzle mass and velocity distribution. Several conclusions are drawn from the simulations: (1) the initial in-nozzle flow has a strong effect on the early jet development; (2) considering phase change is still essential in the high temperature, high pressure (HTHP) injection modelling since it strongly affects the temperature distribution, turbulence intensity and thereby the jet development; (3) significant variations of liquid compressibility factor and density, as well as the cooling effect through the nozzle are highlighted. Overall, the detailed analysis of the numerical results reported in this article may complement the Engine Combustion Network (ECN) experimental database.
... supercritical and transcritical injection modelling in [108], and achieved a good agreement with experimental results in the 1D flash boiling conditions, as reported in [97]. Based on these previous studies, the real fluid DIM solver will be further utilized to explore its potential in solving multi-scale injection problems at HTHP conditions. ...
Thesis
To satisfy latest stringent emission regulations, important progress is still be expected from internal combustion engines. In addition, improving engine efficiency to reduce the emission and fuel consumption has become more essential than before. But many complex phenomena remain poorly understood in this field, such as the fuel injection process. Numerous software programs for computational fluid dynamics (CFD) considering phase change (such as cavitation) and injection modelling, have been developed and used successfully in the injection process. Nevertheless, there are few CFD codes able to simulate correctly transcritical conditions starting from a subcritical fuel temperature condition towards a supercritical mixture in the combustion chamber. Indeed, most of the existing models can simulate either single-phase flows possibly in supercritical condition or two-phase flows in subcritical condition; lacking therefore, a comprehensive model which can deal with transcritical condition including possible phase transition from subcritical to supercritical regimes, or from single-phase to two-phase flows, dynamically. This thesis aims at dealing with this challenge. For that, real fluid compressible two-phase flow models based on Eulerian-Eulerian approach with the consideration of phase equilibrium have been developed and discussed in the present work.More precisely, a fully compressible 6-equation model including liquid and gas phases balance equations solved separately; and a 4-equation model which solves the liquid and gas balance equations in mechanical and thermal equilibrium, are proposed in this manuscript. The Peng-Robinson equation of state (EoS) is selected to close both systems and to deal with the eventual phase change or phase transition. Particularly, a phase equilibrium solver has been developed and validated. Then, a series of 1D academic tests involving the evaporation and condensation phenomena performed under subcritical and supercritical conditions have been simulated and compared with available literature data and analytical results. Then the fully compressible two-phase flow models (6-Equation and 4-Equation systems) have been employed to simulate the cavitation phenomena in a real size 3D nozzle to investigate the effect of dissolved N2 on the inception and developing of cavitation. The good agreement with experimental data proves the solver can handle the complex phase change behavior in subcritical condition. Finally, the capability of the solver in dealing with the transcritical injection at high pressure and temperature conditions has been further validated through the successful modelling of the engine combustion network (ECN) Spray A injector.
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A fully compressible four-equation model for multicomponent two-phase flow coupled with a real-fluid phase equilibrium-solver is suggested. It is composed of two mass, one momentum, and one energy balance equations under the mechanical and thermal equilibrium assumptions. The multicomponent characteristics in both liquid and gas phases are considered. The thermodynamic properties are computed using a composite equation of state (EoS), in which each phase follows its own Peng-Robinson (PR) EoS in its range of convexity, and the two-phase mixtures are connected with a set of algebraic equilibrium constraints. The drawback of complex speed of the sound region for the two-phase mixture is avoided using this composite EoS. The phase change is computed using a phase equilibrium-solver, in which the phase stability is examined by the Tangent Plane Distance approach; an isoenergetic-isochoric flash including an isothermal-isobaric flash is applied to determine the phase change. This four-equation model has been implemented into an in-house IFP-C3D software. Extensive comparisons between the four-equation model predictions, experimental measurements in flash boiling cases, and available numerical results were carried out, and good agreements have been obtained. The results demonstrated that this four-equation model can simulate the phase change and capture most real-fluid behaviors for multicomponent two-phase flows. Finally, this validated model was applied to investigate the behaviors of n-dodecane/nitrogen mixtures in one-dimensional shock and double-expansion tubes. The complex wave patterns were unraveled, and the effects of dissolved nitrogen and the volume translation in PR EoS on the wave evolutions were revealed. A three-dimensional transcritical fuel injection is finally simulated to highlight the performance of the proposed four-equation model for multidimensional flows.
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