______________________________ * Corresponding Author: maria-magdalena.poschner@unibw.de Proceedings of the European Combustion Meeting 2009

CFD-Simulation of supercritical LOX/GH2 combustion considering consistent real gas thermodynamics

M. Poschner, M. Pfitzner

Thermodynamics Institute, Faculty for Aerospace Engineering,

University of the Federal German Armed Forces Munich Abstract

Due to the high pressures and very low injection temperatures of the propellants in modern rocket combustors real gas effects play an important role in rocket combustion simulation. These have to be accurately modelled in combustion CFD simulations to enable an accurate prediction of the performance of the rocket combustion chamber. This work presents the implementation of a thermodynamically consistent real gas mixing equation of state into the commercially available CFD-code ANSYS CFX and its validation using experimental data from the Mascotte test rig (V03) operated at ONERA [4],[5].

Introduction

Recently, some very detailed investigations of real gas effects on mixing and combustion processes at high pressures using LES and DNS CFD simulations were published [11]-[17]. These detailed investigations are very important for the understanding of the complex fundamental processes occurring in rocket combustion chambers. However, these methods are still computationally too expensive to be used in industrial applications, where more conventional RANS methods are applied to support the design process.

In former publications [1] the implementation of the real gas thermodynamics into the commercial CFD-solver ANSYS CFX has been described and validated against the high pressure version of the Mascotte single-injector test rig V03. It has been demonstrated that besides the real gas treatment of the pure components also the significant real gas effects on the mixture properties have to be taken into account. Therefore a consistent real gas formulation of trans- and supercritical mixtures based on a volume-corrected Peng-Robinson equation of state established by Harstad et al. [2] has been developed and implemented into the commercial code ANSYS CFX. Specific Objectives / Theoretical Formulation Experiment & simplified CFD-Model

The experiment used for the validation of the code is the test case RCM-3 presented on the 2nd IWRCM [4]. In this test case the single element combustor Mascotte V03 is fed with liquid oxygen at 85 K and gaseous hydrogen at 287 K.

Conditions H2 O2 Pressure [MPa] 6 6 Mass flow rate [g/s] 70 100 Temperature [K] 287 85 Density [kg/m3] 5.51 1177.8 Velocity [m/s] 236 4.35

Table 1. Conditions for Mascotte test case RCM-3 [4].

Pressures up to 100 bar can be achieved in the combustion chamber [6], representing supercritical

conditions for hydrogen and transcritical conditions for oxygen (T

2

Figure 2. 2-sector of a rot.-sym. combustion chamber. Governing equations

The commercial CFD solver ANSYS CFX is a pressure based fully coupled flow solver of a Favre averaged system of governing equations (1-3). In CFX the momentum and enthalpy equations shown below are solved. Instead of a continuity equation (1) a pressure equation is solved. In a first step, the coupled system of pressure and momentum equations is solved. The enthalpy is solved in a second step.

(1) ( ) 0ii

ut x

+ =

(2) ( ) ( ) k( )" " ,j ii ij i j M ij j j

u uu p u u St x x x

+ = + +

23

ji kij ij

j i k

uu ux x x

= +

(3) ( ) ( )tot j totj

h u hpt t x

+ =

k ( )" "j tot j ij Ej j i

T u h u Sx x x

+ +

Here is the average mixture density and u is the Favre-averaged velocity field. is the stress tensor, accounting for the molecular viscosity of a fluid. In the enthalpy equation htot is the total enthalpy and is the heat conductivity.

MS and ES are source terms in momentum and energy. Radiation effects have been neglected in the work presented below.

The Reynolds flux term k" "j totu h is modelled using

an eddy diffusivity hypothesis [ k ( )" "j tot t tot ju h h x = ] and the eddy viscosity hypothesis applied to model the Reynolds stress terms:

(4) k" " 2 2( )3 3

ji ki j t ij ij

j i k

uu uu u kx x x

= +

The turbulent diffusion coefficient t is modelled as t = t / Prt with the turbulent Prandtl number Prt (CFX default = 0.9). The system can be closed by choosing a model for the turbulent viscosity t. In the present work the standard k- turbulence model has been chosen, solving two additional transport equations for the turbulent kinetic energy k and the eddy dissipations rate . t is then calculated as 2( )t C k = [7].

In the cold flow simulations presented below a transport equation (5) is solved for the mass fraction of n-1 of the n components occurring in the flow. The remaining mass fraction is calculated as the deviation of the sum of those n-1 mass fractions from 1.

(5) ( ) k( )" "j ii ii i j ij j j j

u YY Y Y u St x x x x

+ = +

In equation 5 the term k" "i jY u is modelled as

(6) k" " t ii j

t j

YY uSc x =

Here i and i iY = are the Favre-averaged density and mass fraction of component i, respectively. Si is the source term for component i.

Sct is the turbulent Schmidt number, which is identical to the turbulent Prandtl number in order to treat the turbulent diffusion of heat and species mass fractions identically.

The turbulent scalar fluxes k" "i jY u are modelled

using an eddy dissipation assumption. Molecular diffusion due to temperature gradients (Soret effect) and heat diffusion due to concentration gradients (Dufour effect) are neglected. The molecular diffusion coefficient i is calculated form the bulk viscosity.

In the case of a combusting flow field a flamelet model developed by Peters [9, 10] is applied. This model is available in CFX as a tool called CFX-RIF. Here the mean mass fractions of the single components in the flow and the resulting combustion temperatures are tabulated depending on the mixture fraction, its variance and the scalar dissipation rate. For those three properties transport equations are solved during the simulation. The transport of the enthalpy and the single component mass fractions is not necessary here. Unfortunately real gas effects could not be accounted for in the combustion modelling up to now.

Thermodynamic properties and EOS

All thermodynamic properties are calculated as the sum of an ideal reference value and a departure function accounting for real gas effects which is calculated using a real gas equation of state. To close the system of governing equations described above enthalpy, entropy and constant-pressure specific heat have to be provided in CFX. These are defined the as (7) ( ) ( )0,

mm

V

m mVV

ph T V h T p T dVT

= +

(8) ( ) ( )0,mm

V

mV mV

p nRs T V s T dpT V

= + +

(9) ( ) ( ) 2, ,m

p m V mV m T

p pc T V c T V TT V

=

with ( ),m

V mV

uc T VT

= where

(10) ( ) ( )0,mm

V

m mVV

pu T V u T p T dVT

= +

3

Here the subscript 0 refers to the ideal reference state at low pressure. The departure functions on the right hand side have to be determined using an appropriate equation of state. In the work presented here the Peng-Robinson (PR) equation (eq. 11) has been applied. This equation takes the form

(11) 2 2

( )( ) 2m m m

RT a TpV b V V b b

= +

where Vm is the molar Volume and R is the universal gas constant with a value of R = 8.314472 J/molK. The constants a(T) and b are calculated from empirical relations. a(T) accounts for attractive forces between the molecules in the fluid and is calculated from the relation

(12) ( ) ( )0a T a T=

The universal constant a0 is calculated from the relation a0 = 0.457235R2Tc2/pc and the temperature dependent function (T) is given by

(13) ( ) ( )( )21 1 cT T T = +

Where = 0.37464 + 1.54226 0.269922 is a function of the acentric factor . b = 0.077796RTc/pc takes into account the effects of the reduction of free volume by the particular volume of the molecules.

In the standard version of ANSYS CFX it is possible to choose the ideal gas equation as well as two different real gas equations of state (Redlich-Kwong (RK) and Peng-Robinson) for the calculation of the pure component density. For the mixing process an ideal approach is used even in the case of the application of real gas equations of state for the pure component properties. The volume of the mixture is assumed to be equal to the sum of the volumes of the single components of the mixture at a certain pressure. For a real mixture this is usually not accurate.

As shown in former publications [1] this leads to a significant misprediction of the density of the real gas mixture. There are also important effects on the mixture enthalpy, entropy and specific heat capacity. In the work presented here a consistent real gas formulation on the basis of the PR-equation of state has been implemented into the solver. The extended corresponding states principle is applied, where the multi-component mixture is assumed to behave like a single component but with coefficients a, b in the EOS modified appropriately through mixing rules. The mixture properties are also calculated using the PR equation of state with parameters calculated form real gas mixing rules. For the results shown below the following simple van der Waals mixing rules (14-16) have been applied:

(14)

i j iji j

a x x a= (15)

i ii

b x b= (16)

i ii

x = As the Peng-Robinson equation of state is known to

be not very accurate in predicting the density in transcritical regions a volume-corrected Peng-Robinson

equation of state established by Harstad et al. [2] has been chosen for the final implementation.

In the approach given in [2] the departure functions are determined from the PR equation of state. For the reference state values at a low pressure (p0 = 1bar) empirical correlations are given for all thermodynamic properties. The coefficients needed for these correlations are provided for a number of different substances, but not for every species occurring at the combustion of H2 and O2. For substances where no coefficients were available, NASA polynomials have been applied to determine a reference value.

Every pure component in the flow field has been modelled as a real gas, although pressure effects on the radicals occurring only in the hot flame region are probably negligible. For the critical points of all substances values published by Ribert et al. [19] have been applied. These values are summarized in Table 2.

H2 O2 H O

Tc [K] 33.2 154.6 404.3 367.4 pc [bar] 13 50.4 88.2 76

OH HO2 H2O H2O2 Tc [K] 443.7 487.3 647.3 544.3

pc [bar] 85.4 82.8 221.2 93.5 Table 2. Critical points of all species occurring at the combustion of oxygen and hydrogen [19].

As CFX is a pressure-based solver, the equation of

state has to be solved for the density. In two-phase regions possibly occurring in the flow field this may lead to problems as the equation of state is a polynomial of 3rd degree in density providing up to three different solutions for a given pressure and temperature.

Figure 3. p,V-diagram for oxygen demonstrating the problem of calculating the density from the EOS at subcritical conditions.

Here one of the three possible values has to be chosen. This is done by minimizing the Gibbs free enthalpy. (17) ( ) ( ) ( )0

1, ln

N

m k k kk

g T V g T RT x x p=

= + k is the fugacity coefficient which can be calculated from the equation of state: (18) ( ) , ,ln j

m

V

i i m mT v xV

RT p x RT V dV =

4

The density with the lowest Gibbs enthalpy is the most stable state and therefore its value is chosen as the solution.

Transport properties

Molecular viscosity and thermal conductivity at elevated pressures are calculated using a formalism established by Chung et. al. [3].

Boundary conditions and numerical method

As inlet boundary conditions the mass flow rates provided by the test case RCM-3 (Table 1) were set. At the outlet a pressure of 60 bar was prescribed. The walls of the chamber were assumed to be smooth and adiabatic.

The problem has been solved using a so called high resolution scheme, blending between first and second order accuracy in order to avoid oscillations in the flow field at an improved accuracy compared to a simple first order scheme.

Results and Discussion

With the focus on implementing fully consistent real gas thermodynamics into the solver ANSYS CFX real gas equations of state of very different kinds have been validated against pure component properties provided by the NIST [8] thermophysical properties database (Figure 4). A detailed validation of real mixing at experimental data could not be performed due to a lack of data for H2/O2 mixtures at elevated pressures.

Figure 4. Validation of pure component density and constant pressure specific heat capacity at a pressure of 60 bar for different real gas equations at NIST data.

As a first step the effects of the real gas mixing process as compared to the ideal mixing rule described above has been investigated for a simple hydrogen/oxygen mixture. In the standard solver only the Peng-Robinson equation of state is available, therefore this comparison has been carried out without volume correction. Results for varying compositions are shown in figures 5 to 8.

Comparing the mixture density resulting from a consistent real gas formulation with the mixture density resulting from the approach applied by the standard version of ANSYS CFX significant deviations can be found. Figure 5 compares the densities of a mixture of H2 and O2 for different compositions at a pressure of 60 bar which is a supercritical value for both components.

The temperature ranges from 100 K to 200 K. This is a realistic range in the mixing region of LOx and

GH2 in the Mascotte single injector application, where the injection temperatures of oxygen and hydrogen are 85 K and 287 K, respectively.

Figure 5. Comparison of mixture densities of a H2/O2 mixture at a pressure of 60 bar

Figure 5 shows that the mixture density is highly

underestimated for oxygen rich mixtures at subcritical temperatures. This has been expected to affect the momentum flux predicted for the jet entering the combustion chamber leading to a stretched flame.

Figures 6 and 7 show that there are also remarkable effects on the mixture enthalpy and entropy. Both are highly underestimated by the assumption of an ideal mixing process. For the entropy even negative values arise form the ideal calculation which is unphysical.

Figure 6. Comparison of the mixture enthalpy of a H2/O2 mixture at a pressure of 60 bar

Figure 7. Comparison of the mixture entropy of a H2/O2 mixture at a pressure of 60 bar

The assumption of an ideal or a real gas mixing rule

also affects the mixture constant-pressure heat capacity. cp is known to have a maximum at the pseudo boiling temperature for supercritical pressures, which is the temperature value in the (p,T)-diagram which results for a certain pressure from the extrapolation of the boiling line.

5

Figure 8. Mixture specific heat capacity at constant pressure of a H2/O2 mixture at a pressure of 60 bar By definition this temperature depends on the critical point of a fluid. As the critical temperature of a mixture changes depending on its composition, the maximum in the specific heat capacity at constant pressure has to change its position (Figure 8). This is the case for the assumption of real but not for ideal mixing, where the mixture value is calculated as a mass-fraction weighted average of the pure component values.

As a second step the test case RCM-3 has been investigated without combustion, in order to determine the effects of a real mixing process on the simple flow field in the single-injector combustor.

Figure 9. CFD-density field on the rotational symmetric model of the Mascotte burner (cold flow).

Figure 10 shows the effects of the application of a

consistent real gas formulation compared to the ideal mixing assumption used by default in CFX. In the case of a consistent real gas formulation the cold very dense oxygen jet breaks up further upstream compared to the case assuming an ideal mixing process. The mixing process itself is faster for the ideal case and therefore the density gradient is somewhat smoother for real mixing.

Figure 10. Density distribution on the axis (black line figure 9) resulting for a cold flow CFD-simulation applying ideal and real mixing.

The dense core shortens a little for the consistent real gas approach which was expected to lead to a shortened flame and thus to a better match with the experiment than the results published in [1].

Unfortunately this is not the case. Fig. 11 shows the axial density distribution for the RCM-3 test case with combustion. There are still effects of the real mixing on the resulting density distribution but compared to the cold flow case they are very small. The effects on the flame are also much smaller.

Figure 11. Axial density distribution resulting from a combustion simulation applying ideal and real mixing.

Figure 12 compares the simulated flames resulting from both approaches. The flame temperature is in general a little higher when using the real gas mixing rule. This is due to the enthalpy of the mixture which is significantly underestimated in the case of an ideal assumption. Due to these increased temperatures also the mass fraction of the OH radical predicted by the combustion model increases slightly (Fig. 11). Unfortunately there is no quantitative data available to determine which solution is more accurate.

Figure 12. Temperature and OH distribution resulting for ideal and real mixing using a simple PR-EOS.

Figure 13. Experimentally determined OH distribution compared to CFD results applying consistent real gas thermodynamics.

6

A comparison of the flame shape resulting from the

CFD with the experimental results shows that the general shape of the resulting flame seems to be realistic, but it is somewhat too long and spreads too quickly in radial direction in the simulations. Also the qualitative OH distribution in the flame shows some differences compared to the experimentally determined distribution. The OH concentration in the very thin flame front right downstream of the injector is much higher in the experiment. This could possibly be explained by the neglect of the species OH* resulting from molecular collisions in the simulations.

Conclusion

A consistent real gas formulation on basis of a volume-corrected Peng-Robinson equation of state has been implemented into the commercial solver ANSYS CFX and validated against experimental results published as test case RCM-3 on the 2nd International Workshop on Rocket Combustion Modelling [4].

Analysing H2/O2 mixtures at a pressure of 60 bar and varying compositions significant real gas mixing effects could be found of oxygen rich mixtures near and below the critical temperature for pure O2 in all thermodynamic properties.

In the cold flow simulation applying consistent real gas thermodynamics the dense oxygen jet breaks up a little further upstream compared to the assumption of an ideal mixing process. Effects of the volume correction by Harstad et al. can be seen in the resulting oxygen density. The flow field however almost doesnt change.

Investigating the combusting flow field the effects become much smaller. Here the flame resulting from the consistent approach is almost identical to the one resulting from using an ideal mixing rule. One effect that can be found is a slightly higher flame temperature which results form a higher mixture enthalpy predicted by the consistent real gas mixing rule. The effects of the volume-correction on the flame shape have turned out to be very small as well.

Comparing the OH distribution resulting from the CFD simulations to the experiments the general flame shape matches the experiment quite well but the flame in the CFD is still too long and spreads slightly too quickly in axial direction.

As a future task, the turbulence modelling which has been kept very simple up to now using a RANS approach with a simple k--model will be changed to an LES formulation.

Later real gas effects will also be accounted for in the combustion modelling.

Acknowledgements

This work was performed within the collaborative research center SFB-TR 40 sponsored by the Deutsche Forschungsgemeinschaft (DFG).

The support by the ANSYS team providing valuable information about CFX solver and modelling details is gratefully acknowledged.

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