Influence of Chemical Accommodation on Re-Entry Heating and Plasma Wind Tunnel Experiments

Influence of Chemical Accommodation on Re-entry

Heating and Plasma Wind Tunnel Experiments

Markus Fertig∗, Sven Schäff†, Georg Herdrich‡ and Monika Auweter-Kurtz§

Institut für Raumfahrtsysteme, Universität Stuttgart, Pfaffenwaldring 31, D-70550 Stuttgart, Germany

The influence of chemical accommodation coefficients on the forebody heating of the

ballistic MIRKA re-entry capsule was investigated numerically with the nonequilibrium

Navier-Stokes code URANUS. Chemical accommodation coefficients have been modeled

physically consistent for all possible catalytic recombination mechanisms at the surface, i.e.

direct recombination, the Eley-Rideal reaction type as well as the Langmuir-Hinshelwood

reaction type. The difference in surface temperature was below 6 K when comparing the

thermal nonequilibrium simulations, taking into account chemical accommodation coeffi-

cients below unity, with the thermal equilibrium simulations. In a second step, typical

plasma wind tunnel conditions with stagnation pressures of 800 Pa and a total specific en-

thalpy of 20MJkg

used to determine recombination coefficients in the plasma wind tunnel

at IRS have been addressed. Given the chemical accommodation coefficient, only little

uncertainty in determining recombination coefficients has been found.

Nomenclature

A molar activation energy(

kJmole

)

~c molecular velocity(

ms

)

D molar dissociation energy(

kJmole

)

d diameter (m)E mean molar energy of reacting particles

(

Jmole

)

ε energy (J)

~E , ~F inviscid flux vectors(

kgm2 s , . . . ,

Nsm2 s , . . . ,

Jm2 s

)

~Ev, ~Fv viscous flux vectors(

kgm2 s , . . . ,

Nsm2 s , . . . ,

Jm2 s

)

E molar energy(

kJmole

)

e mass specific energy(

Jkg

)

~Fn, ~Fs flux vectors in the surface normal coordinate system(

kgm2 s , . . . ,

Nm2 , . . . ,

Jm2 s

)

HCIRA density equivalent altitude according to CIRA standard atmosphere 1972 (km)

h0 mass specific formation enthalpy(

Jkg

)

Lint mean molar energy function(

Jmol

)

M Mach number (−)

M molar mass(

kgkmole

)

m particle mass (kg)

N(−)

n , N(+)

n species molar particle flux to and from the surface(

molem2 s

)

n particle density(

1m3

)

PSter steric factor (−)

∗Head of Plasma Modelling and Simulation Group, Space Transportation Technology, AIAA Member†Student‡Scientist, AIAA Member§Professor, Head of department Space Transportation Technology, AIAA Associate Editor

1 of 17

American Institute of Aeronautics and Astronautics

p pressure (Pa)

~Q conservation vector(

kgm3 , . . . ,

N sm3 , . . . ,

Jm3

)

~R right hand side vector(

kgs , . . . ,N, . . . ,

Wm3

)

~S source term vector(

kgm3 , . . . ,

N sm3 , . . . ,

Jm3

)

T , T ∗, T , TQ pseudo temperatures (K)T temperature (K)TW surface temperature (K)t time (s)~V thermal velocity

(

ms

)

V volume(

m3)

~v flow velocity(

ms

)

x, y coordinate directions (m)zr number of reactions (−)zmol number of molecules (−)zs number of species (−)Greek Symbols

α parameter to consider vibrational energy contribution on activation energy (−)β chemical accommodation coefficient (−)δ parameter to consider rotational energy contribution on activation energy (−)εrot(J) rotational energy in quantum state J (J)εvib(υ) vibrational energy in quantum state υ (J)ν ′ stoichiometric coefficient of reactantsν ′′ stoichiometric coefficient of productsψ molar fraction (−)

ρ density(

kgm3

)

θ characteristic temperature of internal excitation (K)ωW molar reaction rate at the surface

(

molem2 s

)

Constants

k Boltzmann’s constant(

1.380662 · 10−23 JK

)

me electron mass(

9.10956 · 10−31 kg)

NA Avogadro number(

6.0221358 · 1023 1mol

)

ℜ universal gas constant(

8.31441 Jmol K

)

Indices

0 ground stateapp appearing, i.e. productsch chemistryER Eley-Rideal reaction mechanisme electron∞ freeflowJ rotational quantum numberLH Langmuir-Hinshelwood reaction mechanismmax maximum valuemin minimum valuemol moleculen, s, t components of the surface normal vectorR surface boundary of the continuum simulationr reaction indexrot rotational excitationtot total valuetr translational energyυ vibrational quantum numberυJ value coupled with rotational and vibrational excitationtr translational

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va vanishing, i.e. reactantsvib vibrational excitationW surfaceAbbreviations

CIRA COSPAR International Reference AtmosphereCOSPAR Committee on Space ResearchCVCV Coupled Vibration-Chemistry-VibrationHLRS Höchstleistungsrechenzentrum Stuttgart (High Performance Computing Center Stuttgart)IRS Institut für Raumfahrtsysteme (Institute of Space Systems)MIRKA Mikro Rückkehr Kapsel (Micro Return Capsule)PWK Plasma Wind Kanal (plasma wind tunnel)SFB Sonderforschungsbereich (Collaborative Research Center)TPS thermal protection systemURANUS Upwind Relaxation Algorithm for Nonequilibrium Flows of the University of Stuttgart

I. Introduction

During re-entry of a space vehicle into the atmosphere, high thermal and mechanical loads arise. Typicalre-entry velocities from low earth orbit are about 8 km

s , leading to a specific gas enthalpy of 32 MJkg . Due

to the high velocity, a compression shock forms in front of the vehicle. As a consequence, temperature andpressure increase by factors of approximately 250 and 1000, respectively. With rising atmospheric density,the speed of the vehicle decreases. For ballistic vehicles, the thermal loads reach a maximum at an altitudeof about 60 km, where the velocity is about 6000 m

s . Under these conditions, pressure increases to about20000 Pa and translational temperature to about 20000 K across the normal shock in front of the vehicle.Downstream of the shock, relaxation of internal degrees of freedom such as rotation and vibration as well aschemistry occurs. Roughly 2

3 of the energy of the high-enthalpy gas flow is converted into chemical enthalpyby chemical reactions such that the temperature is reduced to about 6000 K. Within the boundary layer thetemperature is reduced to about 2000 K at the surface. Under peak heating conditions, the reaction velocityin the boundary layer is not high enough to obtain chemical equilibrium. Hence, the gas flow reaching thesurface of the vehicle may be mainly composed of highly reactive atoms. Recombination of atoms at thethermal protection system (TPS) may release the energy stored in the chemical composition of the gas,hereby increasing the thermal loads of the surface. When restricting the investigation to non-ablative TPSmaterials, convective heat transfer and catalyzed reactions at the surface dominate the heat flux.

The recombination of the atoms may be catalyzed by the TPS. The borderline cases for heterogeneouscatalysis are a fully catalytic surface, where all impinging atoms recombine, and a non-catalytic surface,where no recombination occurs. Due to the energy release caused by recombination, the heat load of theTPS may increase by a factor of about three. Up to now, TPS design has been based on the fully catalyticdesign assumption. Hence, the TPS especially in the stagnation area may be over-sized. An optimal TPSdesign will become possible as soon as accuracy and reliability of heat flux predictions based on finiterate catalysis models can be proved. The development of such models requires knowledge of the catalyticproperties of the TPS materials used. Due to the high surface temperatures during re-entry, the necessaryinvestigations require high enthalpy ground testing facilities, e.g. plasma wind tunnels. A common methodfor determining of the recombination probability of atoms at surfaces is based on the comparison of surfacetemperature or heat flux onto a material of known catalytic properties to the heat flux onto the materialunder investigation.1, 2

The factor of three comparing fully catalytic to non-catalytic surfaces implies that the reaction enthalpyreleased by recombination is completely transferred to the vehicle TPS. Typically, vibrational and rotationaldegrees of freedom of molecules formed during recombination are highly excited. Hence, the molecules donot completely accommodate to the surface conditions. Therefore, part of the recombination energy remainsstored in the molecules, which may reduce the surface heat load. Making use of detailed catalysis modelsaccounting for direct recombination, Eley-Rideal type reactions and Langmuir-Hinshelwood type reactions,3

the energy accommodation coefficient has been determined physically consistent,4 which will be explained insection III. A on page 8. Although, the accommodation coefficient was significantly lower than unity, onlyvery little influence on re-entry heat load was found, which will be shown in section IV. A on page 12.

The influence of chemical accommodation coefficients on surface heat load differs significantly when

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comparing ground tests and flight. During flight, the dissociation takes place in the area between bow shockand boundary layer edge. In rarefied regimes, i.e. at high altitudes, the number of collisions between thegas molecules is too low to obtain a significant degree of dissociation in front of the vehicle. Hence, surfacecatalysis and chemical accommodation coefficient have only little influence on surface heat load. Numericalinvestigations show that with decreasing altitude, where catalysis becomes important, the influence of thechemical accommodation on surface heat flux is rather low since collisions between gas molecules close tothe surface tend to reduce thermal nonequilibrium significantly. In plasma wind tunnel facilities, however,the dissociation of the gas molecules arises in the combustion chamber of the plasma generator.5 Hence, thedissociation degree of the gas flow is high even if the pressure at the surface is low. Therefore, a significantinfluence of chemical accommodation coefficients on surface heat flux during catalysis experiments may occur.The influence of chemical accommodation coefficients on surface heating for typical operating conditionsof the inductively heated plasma wind tunnel at IRS was quantified employing different recombinationcoefficients and accommodation coefficients, which will be discussed in sections B on page 11 and sectionIV. B on page 14.

In the following section the axisymmetric nonequilibrium Navier-Stokes code URANUS (Upwind RelaxationAlgorithm for Nonequilibrium Flows of the University of Stuttgart) used for the numerical studies will bebriefly described.

II. URANUS

In order to simulate the thermal and mechanical loads during re-entry, the URANUS nonequilibriumcode has been developed at the Institute of Space Systems (IRS) of the Universität Stuttgart in coopera-tion with the High Performance Computing Center Stuttgart (HLRS) within SFB 259a. In the URANUSnonequilibrium Navier-Stokes code, the governing equations in finite volume formulation are solved fullycoupled.6 The Navier-Stokes equations

∂ ~Q

∂t+∂[

~E − ~Ev

]

∂x+∂[

~F − ~Fv

]

∂y= ~S , (1)

where

~Q =

[

ρi, ρ vx, ρ vy, ρ etot, ρj evib,j ,

zmol∑

k=1

ρk erot,k, ρe ee

]T

(2)

is the conservation vector of the 2D/axisymmetric code, consist of ten species continuity equations for N2,O2, NO, N, O, N+

2 , O+2 , NO+, N+ and O+, two momentum equations, the total energy equation, three

vibrational energy equations for the molecular species N2, O2 and NO, a rotational energy equation for themolecular species and an electron energy equation. Assuming charge neutrality, no continuity equation forthe electrons is solved. ~E , ~F are the inviscid flux vectors in x- and y-direction; ~Ev, ~Fv are the viscous fluxvectors. ~S denotes the source terms vector for chemical reactions and energy exchange.

The discretization of the inviscid fluxes of the governing equations is performed in the physical space bya Godunov-type upwind scheme employing Roe/Abgrall’s approximate Riemann solver.7 Second order accu-racy is achieved by employing TVD or WENO limited extrapolation.8, 9 The viscous fluxes are discretized inthe transformed computational space by central differences on structured grids using formulas of second orderaccuracy. Thermochemical relaxation processes in the gas phase are accounted for by the advanced multipletemperature Coupled Vibration-Chemistry-Vibration (CVCV) model.10 The influence of vibrational as wellas rotational excitation on chemistry and the influence of chemical reactions on vibration and rotation aremodeled consistently in the source terms of the conservation equations. The vibrational and rotational exci-tation is taken into account not only in dissociation but also in exchange and associative ionization reactions.In dissociated and partly ionized re-entry flows, strong gradients are observed in densities, temperatures andvelocities. To describe the exchange of mass, momentum and energy under these conditions, Chapman-Cowling’s approximations for the transport coefficients translational thermal conductivity of heavy particlesas well as electrons, viscosity and multi-component diffusion coefficients were implemented.11 A diffusionapproach is used to determine the thermal conductivity due to internal excitation of the molecules.

aSonderforschungsbereich 259, Collaborative Research Center 259: “High Temperature Problems of Reusable Space Trans-portation Systems”

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To calculate the steady state solution of the finite volume Navier-Stokes equations

V(

∂ ~Q/∂t)

= ~R( ~Q) (3)

with the volumes V and the sum of the inviscid and viscous fluxes and the source terms ~R( ~Q), the implicit

Euler time differencing with the usual Taylor series linearization for ~R is applied. The resulting linear system

[

V/∆t−(

∂ ~R/∂ ~Q)]

∆ ~Q = ~R( ~Q) (4)

with ~Qn+1 = ~Qn + ∆ ~Q has to be solved for each time step. For ∆t → ∞ the scheme is exactly Newton’smethod. All of the advanced complex surface boundary conditions as well as all other boundary conditionsare implemented implicitly.

A. Surface Boundary Conditions

At altitudes above approximately 85 km, the temperature and velocity jumps at the surface have to beconsidered for vehicles with a characteristic dimension of L = 1 m in order to determine surface loads. Forsimple chemistry and accommodation models it is possible to determine slip velocity and slip temperatureanalytically.12 As shown by Daiß, who developed the predecessor of the model used for the simulationspresented in this paper, it is not necessary to evaluate the slip values.13, 14 For explanation, a virtual volume

D s

DnC e n t e r o fs u r f a c ee l e m e n t

C e l l c e n t e r( i , j = 1 )( i , j = 2 )

n

( i - 1 , j = 2 )( i - 1 , j = 1 ) ( i + 1 , j = 1 )

( i + 1 , j = 2 )

sB o u d a r y l i n e R

B o u d a r y l i n e W

Figure 1. Grid geometry at the surface with virtual surface volume constituted by the edges ∆s and ∆n.14

The cell indices are given in parentheses.

element shown in Fig. 1 with edge lengths ∆s and ∆n is introduced at the surface. For ∆n→ 0 the residualvector ~R only contains the fluxes across the boundary lines ’R’ and ’W’ indicated in Fig. 1, leading to thediscretized equation

~R( ~Q) =(

~Fn,W − ~Fn,R

)

∆s, (5)

where ~Fn is the flux vector component perpendicular to the surface; ~n and ~s are the surface normal andtangential vectors, respectively. The volume of the virtual surface element becomes zero and the numericalscheme given by Eq. (4) reduces to

−(

∂ ~R/∂ ~Q)

∆Q = ~R( ~Q) (6)

with Qt+1 = Qt + ∆Q, which differs from Eq. (4) just by the absence of the time step.

The fluxes at the gas-phase interface ~Fn,R are computed with the usual continuum approach as in the

flow field.6 The flux at the surface interface is split into fluxes due to particles approaching the surface ~F(−)n

and fluxes ~F(+)n due to particles leaving.14 The flux vector ~F

(+)n is determined dependent on ~F

(−)n and non-

reactive as well as reactive models for the gas–surface interface. The main advantage of flux based boundaryconditions is the possibility of introducing sophisticated models for surface chemistry under consideration ofincomplete energy accommodation at the surface.3, 15

The flux vector ~F(−)n is determined by evaluation of

F(−)n,i =

∫

cn,i<0

Ψi cn,i fi d~ci (7)

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where ~ci is the molecular velocity, Ψi ∈{

mi,mi~ci,12mic

2i ,mih0,i, εvib,j(υ), εrot,k(J)

}

is the vector of thetransported quantities and fi is the velocity distribution function. For an accurate determination of thefluxes to the surface a perturbed distribution function is employed in Eq. (7).14

III. Modeling of Surface Reaction Rates

In order to account for chemical reactions at the surface, the species molar particle flux

N(−)

n,i = −F

(−)n,m,i

Mi

(8)

is introduced by division of the species mass flux to the surface F(−)

n,m,i by the species molar mass Mi. Theparticle fluxes of particles vanishing or appearing at the surface are determined by

N(−)

n,ch,i =

zr,W∑

r=1

νi,r′ ωW,r (9)

N(+)

n,ch,i =

zr,W∑

r=1

νi,r′′ ωW,r, (10)

where zr,W is the number of reactions considered at the surface, νi,r′, νi,r

′′ are the stoichiometric coefficientsof the reactants and the products, respectively, and ωW,r is the molar reaction rate. Assuming that only thekinetic energy component normal to the surface of the particles

εn,tr =1

2mc2n (11)

contributes to the reaction rates and that the reaction probability of all particles whose energy exceeds theactivation energy A

NAhave identical reaction probability, the reaction rate becomes

ωW,max =1

NA

∫

cn≥

r

2AM

cn f d~c (12)

at most. By introduction of a steric factor PSter , which is intended to account for collision geometry andother unknown factors independent of temperature, the reaction rate

ωW = PSter ωW,max (13)

is found. Evaluating Eq. (13) employing an unperturbed Maxwell distribution function

f[0]i = ni

(

mi

2πkTi

)

3

2e−miV

2i

2kTi , (14)

where ~V is the thermal velocity and Ti is the species translational temperature, one obtains

ωW =PSter

NA

n(−)i c

(

T(−)

i

)

4e

−A

ℜT(−)

i (15)

for the reaction rate, where

ci =

√

8 k Ti

πmi

(16)

is the mean velocity and ℜ is the universal gas constant. The temperature

T(−)

i =mi

2 k

F(−)

n,tr,i

F(−)

n,m,i

(17)

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and the particle density

n(−)i = −F

(−)n,m,i

4

mi ci

(

T(−)

i

) (18)

of the particles approaching the surface is found from the species mass and translational energy fluxesdetermined according to Eq. (7) by comparison with the unperturbed Maxwell solutions

F(−)

n,m,i,Maxwell (vn = 0) = −ρi ci4

(19)

and

F(−)

n,tr,i,Maxwell (vn = 0) = −ρi ci4

2 k T(−)

i

mi

. (20)

In order to account for the influence of the internal energy states of reacting molecules, the CVCV-model of Kanne10 was extended to surface reactions. Extending Eqs. (12) and (13) towards rotational andvibrational excitation leads to

ωW,υJ = N−1A

Jmax∫

J=0

frot

υmax∑

υ=0

fvib

∞∫

cn,min

∞∫

−∞

∞∫

−∞

cn f d~cdJ. (21)

The distributions of vibrational energy εvib ≈ υ k θvib

fvib =1

Zvib

e−

εvib(υ)

k Tvib (22)

and of rotational energy εrot = J(J + 1) k θrot

frot =1

Zrot

(2J + 1)e−εrot(J)k Trot (23)

were modeled using the rigid rotator and the harmonic oscillator models. It must be noted that the vibrationalground level is not considered in determining the distribution function. Since the introduction of this groundlevel simply leads to an additional activation energy term, the activation energies were increased by ℜθvib/2if not yet included in measured activation energies.

It is assumed that vibrational energy may contribute to the dissociation energy of a molecule up tothe fraction αD of the dissociation energy.16 The parameter U used by Knab16 to describe the influenceof vibrational excitation on reaction cross sections was skipped since a molecule either hits or leaves thesurface. Kanne et al.17 showed that the dependence of the dissociation energy on the rotational excitationmay be described by a linear equation

D(J) = D0 −NAδDεrot(J). (24)

Based on these assumptions, the coupled rovibrational distribution function

fυJ(υ, J) =1

ZD0

υJ

(2J + 1) e−εrot(J)

k Trot e−εvib(υ)

k Tvib (25)

is found, where the index D0 of the coupled partition function which will be given below indicates that themaximum of the rotational and vibrational energies of a molecule

εvib,D,max = εvib(υD,max(J)) =D(J)

NA

(26)

depends on dissociation energy as well as on the parameter δD.Extending the approach to all reactions of molecules, one finds the minimum normal velocity of a reacting

molecule to be

cn,min =

√

2

mmax {(1 − αA)A(J), A(J) −NAεvib}. (27)

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Employing Eqs. (14) and replacing (22) and (23) in Eq. (21) by Eq. (25) one obtains

ωW,υJ = PSter

n (−)c

4NA

e−A0ℜ T

Zαch,AA0

υJ

(

T ∗, T)

+ eαch,AA0

ℜ T

[

ZD0

υJ

(

Tvib, T)

− Zαch,AA0

υJ

(

Tvib, T)

]

ZD0

υJ (Tvib, Trot)(28)

where D is the molar dissociation energy. In order to simplify Eq. (28) the abbreviation of the coupledrovibrational partition function

ZE0

υJ (Tvib, Trot) =1

θrot

1

1 − e−

θvibTvib

{

Trot

(

1 − e−εrot,max

k Trot

)

− TQ

(

1 − e−

εrot,maxk TQ

)

e−

E0ℜ Tvib

}

(29)

was used. Furthermore, the rotational energy at the maximum rotational quantum level

εrot,max(E) =E0

NA δE(30)

and the pseudo temperatures

1

T=

1

Trot

−δ

T,

1

T ∗=

1

Tvib

−1

T,

1

T=

1

Trot

−(1 − α)δ

T,

1

TQ

=1

Trot

−δ

Tvib

(31)

have been used. Note that to determine the reaction rate (28) the parameter of rotational influence onactivation energy

δA = δDαAA0

D0(32)

is connected to the parameter of vibrational influence and dissociation modeling.

Eq. (28) contains three major contributions. The first one, n (−) c4 NA

, is the molar effusion flux to the

surface. The second one, exp(− A0

ℜT), is the Arrhenius type activation energy at the ground state. The third

contribution,

ΛαAA0

υJ (T, Tvib, Trot) =ZαAA0

υJ (T ∗, T ) + eαAA0ℜT

[

ZD0

υJ (Tvib, T ) − ZαAA0

υJ (Tvib, T )]

ZD0

υJ (Tvib, Trot), (33)

describes the influence of rotational and vibrational nonequilibrium on the reaction rate.Note that the nonequilibrium partition function in Eq. (29) can not be solved numerically in thermal

equilibrium because TQ becomes infinite. In our computations, the reduced equilibrium expression is usedif the equation is evaluated at surface temperature. In the nonequilibrium case, |TQ| is limited to 1010 andthe resulting value is corrected making use of the derivative of the partition function with respect to TQ.

A. Chemical Accommodation of Elementary Surface Reactions

In order to account for thermal nonequilibrium at the surface, the energy fluxes of impinging and departingparticles are balanced with the continuum flux.14 Therefore, the mean energy flux of reacting particles has tobe determined. Restricting the approach for atoms first, where no internal energy distribution is considered,the translational energy fluxes

F(−)

n,tr,ch,i = −

zr,W∑

r=1

νi,r′ ωW,r Etr,va,i,r , F

(+)n,tr,ch,i =

zr,W∑

r=1

νi,r′′ ωW,r Etr,app,i,r (34)

have to be determined, where E represents the mean molar energy of reacting particles and app, va standfor appearing and vanishing, respectively. The mean molar translational energy of reacting atoms

EAtr,i

(

T(−)

i

)

=PSter

ωW

∫

cn≥

r

2AMi

mi

2c2 cn f

[0]i d~c dJ = A+ 2ℜT

(−)i (35)

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is found by integrating the translational energy 12 mi ci

2 employing the uperturbed Maxwell distributionfunction (14) and division through the reaction rate.

For the flux balance equations of internal degrees of freedom the fluxes

F(−)

n,int,ch,i = −

zr,W∑

r=1

νi,r′ ωW,r Eint,va,i,r , F

(+)n,int,ch,i =

zr,W∑

r=1

νi,r′′ ωW,r Eint,app,i,r (36)

are determined for int ∈ {rot, vib} corresponding to Eqs. (34). Hence, similarly to Eq. (35) the equations

EA0

tr,vJ =PSter

ωW,υJ

JD,max∫

J=0

υD,max(J)∑

υ=0

fυJ

∫

cn≥cn,min,A

m

2c2 cn f

[0] d~c dJ (37)

for the mean molar translational energy and

EA0

int,vJ =PSter

ωW,υJ

JD,max∫

J=0

υD,max(J)∑

υ=0

εint fυJ

∫

cn≥cn,min,A

cn f[0] d~c dJ (38)

for the mean molar internal energies have to be evaluated to obtain the energy distribution of reactingmolecules. Since, similar to the molecular reaction rates, quite a number of integrals of the rovibrationaldistribution function have to be evaluated, it is advantageous to introduce the mean molar rotational energy

LD0rot(Tvib, Trot) = NA

JD,max∫

J=0

εrot(J)

υD,max(J)∑

υ=0

fυJ dJ

=1

ZD0

υJ (Tvib, Trot)

ℜ

θrot

(

1 − e−

θvibTvib

)

{

T 2rot

[

1 −

(

εrot,max

k Trot

+ 1

)

e−εrot,max

k Trot

]

−T 2Q,D

[

1 −

(

εrot,max

k TQ,D

+ 1

)

e−

εrot,maxk TQ,D

]

e−

D0ℜ Tvib

}

(39)

and the mean molar vibrational energy

LD0

vib(Tvib, Trot) = NA

JD,max∫

J=0

υD,max(J)∑

υ=0

εvib(υ) fυJ dJ

=1

ZD0

υJ (Tvib, Trot)

ℜ

θrot

(

1 − e−

θvibTvib

)

{

Trot

(

1 − e−εrot,max

k Trot

) θvib

eθvibTvib − 1

− TQ,D

[ (

1 − e−

εrot,maxk TQ,D

)

(

θvib

eθvibTvib − 1

+D0

ℜ− δDTQ,D

)

+ δDεrot,max

ke−

εrot,maxk TQ,D

]

e−

D0ℜ Tvib

}

(40)

of impinging particles first. Eq. (38) may then be expressed by

EA0

int,vJ (T, Tvib, Trot) =Z αAA0

υJ

(

T ∗, T)

LαAA0

int,υJ

(

T ∗, T)

ZD0

υJ (Tvib, Trot) ΛαAA0

υJ (T, Tvib, Trot)

+e

αA A0ℜ T

(

ZD0

υJ

(

Tvib, T)

LD0

int,υJ

(

Tvib, T)

− Z αAA0

υJ

(

Tvib, T)

LαAA0

int,υJ

(

Tvib, T)

)

ZD0


υJ (T, Tvib, Trot). (41)

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Finally, the evaluation of the mean molar translational energy of reacting molecules from Eq. (37) leads to

EA0

tr,vJ (T, Tvib, Trot) = 2ℜT +A0

−δAZ

αAA0

υJ

(

T ∗, T)

LαAA0

rot,υJ

(

T ∗, T)

+ Z αAA0

υJ

(

T ∗, T)

LαAA0

vib,υJ

(

T ∗, T)

ZD0


υJ (T, Tvib, Trot)

−αAA0

(

ZD0

υJ

(

Tvib, T)

− Z αAA0

υJ

(

Tvib, T)

)

ZD0


υJ (T, Tvib, Trot)e

αAA0ℜ T

−ZD0

υJ

(

Tvib, T)

LD0

rot,υJ

(

Tvib, T)

− Z αAA0

υJ

(

Tvib, T)

LαAA0

rot,υJ

(

Tvib, T)

ZD0


υJ (T, Tvib, Trot)· δA (1 − αA) e

αAA0ℜ T . (42)

To determine all types of dissociation processes at the surface, the energy states of the molecules are

obtained from Eqs. (41) and (42) using the temperatures T (−), T(−)rot,i and T

(−)vib,i of the impinging molecules

derived from the fluxes to the surface similar to the kinetic temperature given in Eq. (17).14 The evaluationof the effective activation energy for the different dissociation processes, i.e. non catalyzed dissociation aswell as dissociative, has been described in a previous paper.3

In the final modeling step, the mean molar energy of product molecules formed in surface reactions isobtained by detailed balancing, i.e.

Etr,app,i = EA0

tr,υJ,i (TW , TW , TW ) and Eint,app,i = EA0

int,υJ,i (TW , TW , TW ) , (43)

where TW is the surface temperature. In the modeling of the surface reactions, it was assumed that theactivation energy of the dissociation processes is carried by the reacting molecules. Hence, molecules whichdissociate directly, i.e. where the surface only acts as a collision partner, must contain more than the dis-sociation energy in total. Correspondingly, a molecule formed in the reverse process will contain in totalmore than the dissociation energy as well (Eapp,i ≥ D). Similar arguments hold for a rough estimationof the energy content of molecules formed by Eley-Rideal or Langmuir-Hinshelwood reaction mechanisms.Therefore, a rough estimate of the energy of the particles formed may be given without actually evaluatingthe Eqs. (43). Molecules formed within the Eley-Rideal mechanism, where a gas-phase atom recombines withan atom adsorbed at the surface, contain in total more that the dissociation energy minus the adsorptionenergy (Eapp,i ≥ D − EAd). The Langmuir-Hinshelwood mechanism finally states that two atoms adsorbedat the surface recombine. Since one of the adsorbed atoms must be mobile at the surface, the energy of themolecules formed depends not only on adsorption energy but also on the energy required for mobilization.Nevertheless, for the total energy of the molecule formed Eapp,i ≥ D − 2EAd may be estimated.

For a detailed investigation, the amount of energy stored in the formed molecules may be characterisedby the chemical accommodation coefficient β. Once the energy of particles appearing or vanishing withinsurface reactions is determined, the chemical accommodation coefficient for any surface reaction can beexpressed by

βr =

zs∑

i=1

ν′

i,rEva,i −zs∑

j=1

ν′′

j,rEapp,j

zs∑

k=1

ν′

k,rEva,k −zs∑

l=1

ν′′

l,rE0l (TW , TW , TW )

(44)

which is the difference in total energy of the reacting particles divided by the energy difference if the moleculeis formed in equilibrium with the surface temperature. The mean molar total energies of the particles aredetermined by

Eva,i = EA0

tr,vJ,i

(

T(−)i , T

(−)vib,i, T

(−)rot,i

)

+ EA0

rot,vJ,i

(

T(−)i , T

(−)vib,i, T

(−)rot,i

)

+ EA0

vib,vJ,i

(

T(−)i , T

(−)vib,i, T

(−)rot,i

)

+h0,i

Mi

, (45)

Eapp,i = EA0

tr,vJ,i (TW , TW , TW ) + EA0

rot,vJ,i (TW , TW , TW ) + EA0

vib,vJ,i (TW , TW , TW ) +h0,i

Mi

(46)

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and

E 0l (TW , TW , TW ) = 2ℜTW + LD0

rot,i(TW , TW ) + LD0

vib,i(TW , TW ) +h0,i

Mi

, (47)

where the contribution of internal energy is zero in the case of atoms. If the energy contribution as afunction of surface temperature is low compared to the dissociation energy, it follows from the estimationof the energy of molecules formed at the surface that β ≈ 0 for direct recombination, β ≈ EAd

Dfor the

Eley-Rideal reaction mechanism and β ≈ 2EAd

Dfor the Langmuir-Hinshelwood reaction mechanism. Even if

the chemical accommodation coefficient is low, an influence on surface heat flux only occurs, if

1. the gas temperature differs significantly from the surface temperature and

2. the recombination probability of the reaction mechanism is high.

Item 1 applies under rarefied conditions if pressure is low. If pressure is high, collisions in the gas-phase leadto a quick relaxation and the heat flux is determined by thermal conductivity in the gas. Only if item 2is fulfilled, is a significant amount of energy stored in dissociation released and available for an increase ofsurface heating. Typically, the recombination coefficient of direct recombination is of the order γ < 10−5.Therefore, the influence of direct recombination is negligible. Remaining candidates for an influence of βon surface heating are Eley-Rideal and Langmuir-Hinshelwood reaction mechanisms. For the applicationtowards re-entry heating, the main interest are surface temperatures TW > 1000 K. Since recombinationof the above mentioned recombination mechanisms depends on surface coverage with adsorbed atoms, highrecombination probabilities in this high-temperature range only arise if adsorption energy exceeds 100 kJ

mole .

Hence, chemisorption is another necessary prerequisite. For SiC and SiO2 chemisorption energies of 180 kJmole

for oxygen adsorption and 250 kJmole for nitrogen adsorption have been found.3 With the estimations given

above, recombination coefficients of βER,O ≈ 0.36, βLH,O ≈ 0.73, βER,N ≈ 0.27 and βLH,N ≈ 0.53 arise forthe recombination of oxygen and nitrogen due to Eley-Rideal and Langmuir-Hinshelwood reaction types.

B. Simplified Catalysis Model

The detailed model described above allows for a determination of the surface heat flux on materials withknown properties. Experimental data is necessary for the calibration of the model, i.e. it can not be usedto determine the catalytic properties. Several plasma wind tunnels (PWK) are in operation at IRS forthe qualification of TPS materials and other re-entry related investigations.18 In recent years, one of thePWKs was equipped with a newly developed inductively heated plasma source, which allows the catalyticproperties of TPS materials to be investigated. Due to the electrodeless operation of the plasma source,no contamination of the gas flow occurs, even if highly reactive gases such as pure oxygen are used forthe experiments. To determine recombination coefficients by means of plasma wind tunnel experiments,the interaction of a plasma flow with a material probe is characterized by the heat flux onto the material.The recombination coefficients are then determined by comparison of the heat flux onto the material underinvestigation with the heat flux onto a material with known catalytic properties. Usually, it is assumedthat the chemical accommodation coefficient equals unity.2, 19 In order to rate the influence of the chemicalaccommodation coefficient on the experimental results, a numerical study was performed by Schäff employingtemperature independent recombination coefficients.20 It follows from Eq. (44) that

zs∑

j=1

ν′′

j,rEapp,j = (1 − βr)

zs∑

i=1

ν′

i,rEva,i + βr

zs∑

l=1

ν′′

l,rE0l (TW , TW , TW ). (48)

In order to estimate the upper borderline for the difference in heat flux due to chemical accommodationcoefficients, it was assumed that complete accommodation arises for translational and rotational degrees offreedom. Since the relaxation time of vibrational excitation is higher than for translation or rotation, highlyvibrationally excited molecules may carry a significant amount of the energy released due to recombinationaway from the surface into the gas flow.

As an example the model described above is applied to the reaction

O + O → O2. (49)

11 of 17


Making use of

Eva,O = 2ℜT (−) +h0

O

MO, (50)

E 0O2

(TW , TW , TW ) = 2ℜTW + LD0,O2

rot,O2(TW , TW ) + L

D0,O2

vib,O2(TW , TW ) +

h0O2

MO2

, (51)

Eapp,tr,O2 = 2ℜTW and (52)

Eapp,rot,O2 = LD0,O2

rot,O2(TW , TW ) (53)

one obtains

Eapp,vib,O2 = 2 (1 − β)

(

2ℜT (−) +h0

O

MO

)

+ (β − 1)(

2ℜTW + LD0,O2

rot,O2(TW , TW )

)

+ β LD0,O2

vib,O2(TW , TW ) (54)

for the vibrational energy of the oxygen molecules formed.

IV. Results

In order to determine the influence of chemical accommodation on re-entry heating, the forebody flow ofthe MIRKA capsule was investigated for three flight conditions along the trajectory with different detailedcatalysis models.3, 4 MIRKA is a German experimental re-entry capsule, with a spherical shape of 1 m indiameter, which was successfully flown in October 1997. Two weeks after the ascent on top of a RussianFOTON capsule, a ballistic descent took place for both reentry vehicles separately. To protect MIRKAagainst thermal loads during re-entry, the surface was covered by a C/SiC heat shield.

A. MIRKA Re-entry Flight

The inflow conditions for the MIRKA simulations are summarized in Table 1 on the following page. Sincethe influence of accommodation on re-entry heating lessens with decreasing altitude, only the results fort = 1273 s, i.e. at an altitude of approximately 70 km, will be shown, where the influence of catalyticrecombination on re-entry heating is strong.21

Figure 2 shows the structured, shock adapted finite volume mesh with 60 × 52 volume cells. The mesh

x (m)

y(m

)

0 0.2 0.40

0.2

0.4

0.6

0.8

Figure 2. Shock adapted MIRKA fore-body grid with 60 × 52 volume cells fort = 1273 s, i.e. at an altitude of about70 km.

n (m)

T(K

)

00.010.020.030.040.050.060

5000

10000

15000

20000

25000Ttr (K)Tvib,N2 (K)Tvib,O2 (K)Tvib,NO (K)Trot (K)Te (K)

Figure 3. Temperature distribution along MIRKA’s stagnationline for t = 1273 s determined with a detailed SiO2 catalysis model.

was adapted to the flow such that the resolution at the surface as well as in the shock is of the order of the

12 of 17


t (s) v∞(

ms

)

ρ∞

(

kgm3

)

HCIRA (km) T∞ (K) p∞ (Pa) ψN2,∞ (−) ψO2,∞ (−)

1273 7362, 2 1, 474 · 10−4 66, 18 228, 6 9, 708 0, 7883 0, 2117

1301 6626, 3 5, 043 · 10−4 56, 23 260, 7 37, 881 0, 7883 0, 2117

1343 3948, 0 2, 599 · 10−3 42, 94 263, 2 197, 123 0, 7883 0, 2117

Table 1. Inflow conditions of the trajectory points investigated for the MIRKA re-entry. The time given isthe internal time of the MIRKA systems. The altitude has been determined for the density derived duringpost-flight analysis22 from the CIRA standard atmosphere 1972.23

local mean free path. To obtain accurate results at the surface, the boundary layer was resolved with 28 cells.The temperature distribution determined at t = 1273 s employing the detailed catalysis model for SiO2

3 isshown in Fig. 3. Through the bow shock, which is approximately 3.4 cm away from the surface, translationaltemperature jumps to about 20000 K. Equilibrium between translational and rotational temperature isreached after roughly 1.0 cm. From the differences between vibrational and translational temperatures itfollows that thermal equilibrium does not arise along the stagnation line. Thermal nonequilibrium is mainlycaused by the coupling between chemistry and internal energy.17 Since the dissociation probability of excitedmolecules is higher than in ground state, dissociation processes tend to lower vibrational temperature. Theopposite phenomenon arises in the boundary layer. Due to heat conduction, energy is transferred to thesurface. Therefore, translational temperature in the boundary layer decreases and chemical equilibrium shiftstowards molecule formation. According to detailed balancing, the recombination of atoms generates highlyexcited molecules which results in an increase of the vibrational temperatures within the boundary layer.

The chemical composition along the stagnation line at t = 1273 s is shown in Fig. 4. The change of mole

n (m)

Ψ(-

)

00.010.020.030.040.050.0610-7

10-6

10-5

10-4

10-3

10-2

10-1

100

N2

O2 O NO

N

e-

NO+

O2+

N2+

O+

N+

Figure 4. Mole fraction distribution along the stag-nation line of MIRKA for t = 1273 s determined witha detailed SiO2 catalysis model.

q W(M

W/m

2 )

0

0.5

1

1.5

2

HEATIN experiment

s (m)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

non catalytic

fully catalytic

s (m)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

SiC

SiO2

Figure 5. Simulation of surface heat flux at t = 1273 s

applying different catalysis models in comparison tothe HEATIN flight experiment.24 Non-catalytic andfully catalytic surface assumptions are the borderlinecases for catalytic influence.

fractions behind the shock along the stagnation line indicates that chemical equilibrium does not arise inthe post shock area. Nevertheless, nitrogen as well as oxygen are significantly dissociated. Since the molefraction of molecular oxygen becomes lower than ψO2 ≤ 10−5, nearly complete dissociation arises. Under theassumption of a non-catalytic surface, where no recombination occurs, the mole fractions of atomic nitrogenand oxygen at the surface equal those at the boundary layer edge. Due to catalysis on SiO2, the mole fractionof the atomic species increases significantly within the boundary layer. Although the catalytic activity ofSiO2 is low, the release of the dissociation energy at the surface results in a significant increase of the heatflux to the surface as compared to the non-catalytic surface assumption. In Fig. 5, the heat fluxes computedwith the detailed catalysis models for SiO2 and SiC described in a previous publication4 are compared to theresults of the HEATIN experiment.24 In addition, simulated heat fluxes are shown for a hypothetical fullycatalytic surface where all impinging particles recombine to N2 and O2 as a borderline case for catalysis. The

13 of 17


rather good agreement between the SiO2 result and the experiment indicates that a SiO2 layer has formedat the C/SiC–gas interface.

Figures 6 and 7 compare results obtained with the detailed SiO2 catalysis model at t = 1273 s for

n (m)

T(K

)

00.00050.0010.00150.0020

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Ttr (K)Tvib,N2 (K)Tvib,O2 (K)Tvib,NO (K)Trot (K)Te (K)

n (m)

T(K

)

00.00050.0010.00150.0020

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Ttr (K)Tvib,N2 (K)Tvib,O2 (K)Tvib,NO (K)Trot (K)Te (K)

Figure 6. Comparison of temperature distribu-tion close to the surface for slip boundary conditions(black lines) and no-slip boundary conditions (bluelines).

s (m)

TW

(K)

∆TW

(K)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

250

500

750

1000

1250

1500

1750

2000

0

1

2

3

4

5

6

TW (K), slipTW (K), no-slip∆TW (K)

Figure 7. Surface temperature distribution along thevehicle contour for slip boundary conditions (blackline) and no-slip boundary conditions (blue line). Thered line shows the difference between no-slip bound-ary conditions and slip boundary conditions.

slip boundary conditions where chemical accommodation is considered with no-slip boundary conditions. InFig. 6, the temperature distribution along the stagnation line close to the surface is shown for slip and no-slipboundary conditions. Comparing the scales of the abscissas from Figs. 6 and 3 it becomes obvious that theinfluence of chemical accommodation is restricted to a small part of the boundary layer. Figure 7 comparesthe radiation equilibrium surface temperatures along the vehicle contour for slip and no-slip conditions.As can be seen from the red line, the temperature obtained with no-slip boundary conditions exceeds thevalue found for slip boundary conditions by less than 6 K. The scatter in the values up to s = 0.3 m arisessince accuracy of the temperature output does not exceed 1

10 K. The increase of the difference at s = 0.7 moriginates from a disturbance due to the outflow boundary condition. The difference of 6 K in radiationequilibrium temperature originates from a difference in heat flux of 8.7 kW

m2 . Since the heat flux in the

stagnation area is of the order of 1.2 MWm2 , this is below 1 %. Since up to now no higher differences have been

found for any forebody flow, the influence of chemical accommodation on re-entry heating seems negligible.

B. Plasma Wind Tunnel Conditions

The influence of chemical accommodation coefficients on the determination of recombination coefficients inplasma wind tunnel experiments was investigated for a flow condition typical for the catalysis experimentsconducted at IRS. Since the plasma composition and the internal temperature distribution of the flowfield inthe plasma wind tunnel is not known in detail, only the total values of ambient pressure p∞ = 40 Pa, Machnumber M = 4, total enthalpy htot,∞ = 20 MJ

kg and stagnation pressure ptot = 875 Pa have been used for thespecification of the inflow conditions. The remaining inflow parameters which are summarized in Table 2have been set assuming thermochemical equilibrium for the inflow.

M∞ (−) 4 v∞(

ms

)

5071.6 htot,∞

(

MJkg

)

20 p∞ (Pa) 40

T∞ (K) 3159 ρ∞

(

kgm3

)

3.6 · 10−5 ψ∞,N2 (−) 6.5 · 10−1 ψ∞,O2 (−) 1.4 · 10−3

ψ∞,NO (−) 4.4 · 10−3 ψ∞,N (−) 1.5 · 10−3 ψ∞,O (−) 3.4 · 10−1 ψ∞,N+2

(−) 10−12

ψ∞,O+2

(−) 4.9 · 10−11 ψ∞,NO+ (−) 8.5 · 10−7 ψ∞,N+ (−) 10−12 ψ∞,O+ (−) 1.7 · 10−11

Table 2. Inflow conditions for the simulation of the flow around the plasma probe.

14 of 17


The flowfield around a European standard probe with d = 50 mm in diameter was resolved with the grid

x (m)

y(m

)

0 0.05 0.10

0.02

0.04

0.06

0.08

0.1

0.12

Figure 8. Flow grid for the simulation of the flowaround the plasma probe

stagnation line (m)

T(K

)

00.010.020.030

5000

10000

15000

20000Ttr (K)Tvib,N2 (K)Tvib,O2 (K)Tvib,NO (K)Trot (K)Te (K)

β = 0,1

β = 0,2

β = 0,5

β = 1

Figure 9. Temperature distribution along the stag-nation line for temperature independent recombina-tion coefficients γ = 0.1 and for chemical accommoda-tion coefficients ranging from β = 0.1 to β = 1.

shown in Fig. 8, which consists of 84×60 finite volume cells. The grid resolution at the surface is of the order

stagnation line (m)

ψ(-

)

00.010.020.0310-3

10-2

10-1

100

ψN2

ψO2

ψNO

ψN

ψO

β = 1

β = 0,2

β = 0,5

β = 0,1

Figure 10. Mole fraction distribution of neutralspecies along the stagnation line for temperature in-dependent recombination coefficients γ = 0.1 and forchemical accommodation coefficients ranging fromβ = 0.1 to β = 1.

γ (-)

q W,n

(Wm

-2)

10-310-210-1100750000

800000

850000

900000

950000

1000000β = 1β = 0,5β = 0,2β = 0,1

.

Figure 11. Comparison of the heat flux to the surfaceat the stagnation point for different recombinationcoefficients and chemical accommodation coefficients.The lines connect the simulations with equal chemicalaccommodation coefficients.

of the local mean free path. The flow around the plasma probe was numerically simulated applying threedifferent temperature independent recombination coefficients γ ∈ {1, 0.1, 0.01} and four different chemicalaccommodation coefficients β ∈ {1, 0.5, 0.2, 0.1} with the model described in section III. B on page 11. Thetemperature distributions obtained for γ = 0.1 are shown in Fig. 9. The vibrational temperature of molec-ular oxygen increases significantly in the direction to the surface with decreasing chemical accommodationcoefficient. Nevertheless, the stagnation point heat flux decreases by not more than 4.6 % from 845 kW

m2 for

β = 1 to 806 kWm2 for β = 0.1.

Figure 10 shows the distribution of mole fractions along the stagnation line for γ = 0.1. For the air flowconsidered for the simulation, only little nitrogen dissociation arises for the total flow enthalpy of 20 MJ

kg .This explains why no difference for the vibrational temperature of molecular nitrogen was found. The figureshows that the mole fraction of molecular oxygen at the surface increases with rising chemical accommodationcoefficient. This is a result of the coupling of vibrational excitation with dissociation in the gas phase. Due

15 of 17


to incomplete energy accommodation at the surface, the molecules formed leave the surface highly excited.These highly excited molecules are characterized by a high dissociation probability. Therefore, a significantamount of the molecules formed at the surface dissociates in the boundary layer close to the surface. Figure 11shows numerically determined stagnation point heat fluxes for different chemical accommodation coefficientsversus recombination coefficients. With the help of this figure it is possible to determine errors originatingfrom chemical accommodation on the determination of recombination coefficients from PWK experiments.

• As a first example, a heat flux of 850 kWm2 is examined. Based on the typical assumption β = 1, a

recombination coefficient of γ = 0.1 would be determined. However, in case the chemical accommo-dation coefficient equals β = 0.1, a recombination coefficient of γ = 0.3 would be correct, assuming alogarithmic dependency of heat flux on the recombination coefficient indicated by the connecting linesin Fig. 11.

• As a second example, an uncertainty of ±5 % is assumed to determine the stagnation point heatflux. Therefore, the recombination coefficients for qW,n = 807.5 kW

m2 and qW,n = 892.5 kWm2 have to

be determined. For a chemical accommodation coefficient of β = 1 this results in recombinationcoefficients of γ = 0.03 and γ = 0.25, respectively.

From the detailed modeling explained in section III. A on page 8 with adsorption energies of about EAd ≈200 kJ

mole , it is to be expected that the chemical accommodation coefficient is lowest for nitrogen recombina-tion. Although the dissociation energy of nitrogen is high, the chemical accommodation coefficients of theEley-Rideal mechanism were found to exceed 0.25. Therefore, it is concluded that uncertainties in stagna-tion point heat flux determination have a much higher impact on the accuracy of recombination coefficientdetermination than chemical accommodation coefficients.

V. Summary and Conclusion

Previously published models for kinetic surface boundary conditions and a detailed description of hetero-geneous catalytic reactions on SiC and SiO2 have been briefly introduced. A new model has been presentedwhich allows for the simulation of the incomplete energy accommodation of molecules formed by direct re-combination, Eley-Rideal type and Langmuir-Hinshelwood type reaction mechanisms. Application of thisdetailed model on re-entry heating showed little influence of the chemical accommodation coefficients on theheat flux.

In a second step, a systematic, numerical investigation was performed to rate the influence of chemicalaccommodation coefficients on recombination measurements derived by the analysis of surface heat fluxmeasurements in plasma wind tunnel experiments. For this purpose, a simplified catalysis model withtemperature independent recombination coefficients and a variety of chemical accommodation coefficientswas applied to typical experimental conditions for the measurements at IRS. According to our results,uncertainties by a factor of two have to be expected due to incomplete chemical accommodation. Moresevere differences, however, might originate from uncertainties in surface heat flux determination.

VI. Acknowledgment

This work has been supported by the Deutsche Forschungsgemeinschaft in the framework of the Collabo-rative Research Center (SFB) 259 - Hochtemperaturprobleme rückkehrfähiger Raumtransportsysteme at theUniversität Stuttgart.

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