Energy Accommodation Coefficient Calculation Methodology
Using State-to-State Catalysis Applied to Hypersonic Flows
Georgios Bellas-Chatzigeorgis∗
von Karman Institute for Fluid Dynamics, 1640 Rhode-Saint-Genèse, Belgium Paolo F. Barbante†
Polytechnic University of Milan, 20133 Milan, Italy and
Thierry E. Magin‡
von Karman Institute for Fluid Dynamics, 1640 Rhode-Saint-Genèse, Belgium https://doi.org/10.2514/1.J058543
The interplay of a gas–surface interaction and thermal nonequilibrium is still an open problem in aerothermodynamics. In the case of reusable thermal protection systems, it is unclear how much of the recombination energy is stored internally in the molecules produced by surface catalytic reactions, potentially leading to nonequilibrium between their translational and internal energy modes. A methodology is developed to calculate the energy accommodation coefficient using a rovibrational state-to-state chemical mechanism for a nitrogen mixture coupled with a generalized form of the catalytic recombination coefficient. The flow around a spherical body is simulated in hypersonic conditions, allowing study of the amount of energy deposited on the surface and stored in the recombining molecules. Internal energy quenching into translational energy is found, which is a phenomenon also observed experimentally, keeping the total energy transferred to the surface overall constant. The methodology developed for the application of a state-to-state model in the computational fluid dynamics framework coupled with catalysis is generic and applicable to a variety of other similar mechanisms.
I. Introduction
C
OUPLED multiphysics phenomena occur in the flow surrounding atmospheric entry vehicles, adding uncertainties to the heat flux prediction used for the development of thermal protection systems. Two important phenomena are the strong deviation of the gas from local thermodynamic equilibrium and the interaction of the chemically reacting boundary layer with the vehicle’s surface. When the pressure in the shock layer is too low to reach local thermodynamic equilibrium (in particular, for the part of the trajectory before peak heating), the gas thermodynamic state must be described by means of nonequilibrium models ranging from multitemperature to state-to-state approaches [1–3]. In multitemperature models, the populations of internal energy levels follow Boltzmann distributions at distinct temperatures, which are characteristic of each thermal bath considered per energy mode. For instance, in the two-temperature model of Park [4], the rotational energy of molecules is grouped with the translational energy of atoms and molecules into a common energy mode; whereas the electronic energy of these species, as well as the vibrational energy of molecules, is grouped with the translational energy of free electrons into a second mode [1]. In state-to-state models, the populations are obtained directly from a detailed chemical mechanism, gathering all relevant collisional processes between the internal energy levels of the different species present in the flow. When radiative processes are taken into account, one refers to collisional-radiative models.Examples of nonequilibrium shock-layer flow simulations abound in the literature [5–8].
Thermal equilibrium is often imposed to the gas as boundary condition at the wall, where it interacts (with the wall itself) through catalytic processes in the case of reusable thermal protection systems [9,10]. Highly reactive atoms produced across the shock layer reach the thermal protection material; a fraction of them use the surface as a catalyst for recombination into molecules [11]. This is an exothermic process, with part of the recombination energy (equal in modulus to the dissociation energy of the formed molecule) directly released to the material [12]. The assumption of thermal equilibrium at the wall is questionable. It is unclear how much of the recombination energy is stored internally by the molecules produced, potentially leading to nonequilibrium between the translational and internal energy modes. For instance, quantum-mechanical calculations fail to fully explain experimental results on the vibrational excitation of molecules due to catalytic reactions [13]. A limited effort has been spent on assessing the coupled influence of catalytic and thermal nonequilibrium effects on heat fluxes. The chemical energy accommodation coefficientβ is a parameter to determine the ratio between the energy directly transferred to the wall and the recombination energy [14–16]. Contrary to experimental evidence [17], in most simulations, it is generally assumed that all of the recombination energy is deposited on the surface, provided that the molecule internal energy follows a Boltzmann distribution at the wall temperature, i.e., the value ofβ 1.
In hypersonic flow simulations, the elementary processes occurring at the surface are usually not described in detail, adopting a macroscopic point of view based on global recombination reactions [18]. Hence, to characterize the material catalytic properties, only one single coefficientγ needs to be defined for the catalytic re-combination of two atoms into a molecule. The simplicity of this approach has made the interpretation of complicated experimental results significantly easier. It proved an invaluable tool for the sizing of thermal protection systems for reentry vehicles, especially in the early years of mankind into space [19], as well as for the space shuttle [20]. Nonetheless, the model has some drawbacks. Even though the γ parameter should depend only on the material properties (e.g., roughness) and its surface temperature [21], it has been observed experimentally that it depends explicitly on pressure [22] and the degree of chemical nonequilibrium in the boundary [23,24].
Received 3 April 2019; revision received 26 July 2019; accepted for publication 17 September 2019; published online 12 November 2019. Copyright © 2019 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-385X to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp.
*Postdoctoral Researcher, Aeronautics and Aerospace Department, Chaussée de Waterloo, 72; also Politecnico di Milano, MOX, Department of Mathematics, Piazza Leonardo da Vinci 32, 20133 Milan, Italy; bellas@vki .ac.be.
†Researcher, MOX, Department of Mathematics, Piazza Leonardo da Vinci
32; [email protected].
‡Professor, Aeronautics and Aerospace Department, Chaussée de
Waterloo, 72; [email protected].
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Vol. 58, No. 1, January 2020
These effects limit the predictive capabilities of theγ model. Recent approaches to infer the catalytic properties of materials from ex-perimental results account for model and exex-perimental uncertainties. They represent a possible path to extend the predictive capabilities of theγ model [25].
Unlike the global macroscopic approaches, finite-rate surface-chemistry models have been developed by several researchers to provide a more detailed description of catalytic elementary processes [26–29]. These are expressed as separate chemical reactions, enriching the physics of the model but requiring more data for both the reaction rate coefficients of the processes involved and of the surface properties. The values of the basic data used for closure are often obtained from semiempirical models, or calibrated using experimental results and other simulations [30]. Finite-rate surface-chemistry models exhibit enhanced extrapolation capabilities as compared to theγ model and give a more thorough understanding of gas–surface interaction, but the data required are often unavailable or with large uncertainties.
First principle calculations yield model parameters based on physics, permitting us to extrapolate their validity range as compared to parameters deduced by fitting experimental data. Higher-fidelity methods allow us to retrieve the macroscopic properties of surfaces. They range from quantum chemistry [13,31], through molecular dynamics [32–35], to direct simulation Monte Carlo methods [36,37]. These approaches can be particularly efficient when data cannot be effectively measured through experiments. Despite their detailed physical description, they are often computationally prohibitive for macroscopic simulations.
The goal of this paper is to develop a methodology to calculate the energy accommodation coefficient β using state-to-state catalysis models. The method is applied to hypersonic flows aiming at the study the coupled effects of the gas thermal nonequilibrium and surface catalysis. This is achieved by simulating the flow around a spherical body in hypersonic conditions using an ab initio state-to-state model for the N2 N system (where N denotes nitrogen) [38] concerning the
gas phase, coupled with a generalized form of theγ model for catalysis. This approach allows us to study the amount of energy deposited on the surface and stored in the recombining molecules. A similar model for catalysis was proposed by Armenise et al. [39] and Kustova et al. [40] to study oxygen recombination inside hypersonic boundary layers, however, without any emphasis on the calculation of the heat flux and energy accommodation at the wall. We simulate simultaneously both the shock- and boundary-layer flows by solving stagnation line conservation equations resulting from a Chapman–Enskog perturba-tive solution to the Boltzmann equation [41]. We derive optimal expressions for the multicomponent transport properties obtained via a state-to-state approach, provided that the transport collision integrals are identical for all the internal energy levels of the species. Analyzing the simulations, we focus on the effects of thermal nonequilibrium to the heat fluxes experienced by the reentry body: in particular, the energy quenching of the internally excited molecules. The thermal nonequilibrium behavior analysis can be used for the development of multitemperature models, such as the widely used two-temperature model of Park [4].
This paper is split into the following sections. After this introduction, the necessary tools to study catalysis in thermal nonequilibrium are presented in Sec. II, followed by the governing equations and thermochemical models for a state-to-state approach in Sec. III. The simulations performed are discussed in Sec. IV, with the conclusions of the paper given in Sec. V.
II. Modeling Gas–Surface Interaction
Heterogeneous catalysis describes the recombination of dissociated atoms at the wall using, in reentry flows, the thermal protection material as a catalyst. The adjective heterogeneous indicates that the recombining species and the catalyst are in different phases: respectively, gas and solid. Without being consumed, the catalyst increases the rate of chemical reactions by offering an alternative energetically favored reaction path, without altering the chemical equilibrium composition, because it affects both forward and backward reaction rates equally [42].
In hypersonic applications, surface recombination reactions are usually exothermic, releasing energy to the wall, which can contribute to a substantial fraction of the total heat flux experienced by space vehicles. An accurate modeling of surface processes directly impacts the reentry heat load prediction, and thus the proper sizing of thermal protection systems based on reusable materials, which are developed to achieve an optimal reradiation of the heat while keeping the catalytic properties of the material low. All of the recombination energy is not deposited to the surface. A part of it can be stored in the internal energy modes of the molecules produced [14,35], resulting in thermal nonequilibrium of the gas at the wall and inside the boundary layer. This can potentially modify the total heat flux experienced by the vehicle.
Conventional chemistry models usually couple a multitemperature description of the gas phase to a global recombination surface-catalysis approach. Thermal nonequilibrium effects at the wall directly result from the boundary condition imposed for the gas translational and internal temperatures [1]. Thermal equilibrium is assumed in absence of any sound theory. Instead, we propose to use state-to-state models for the gas phase because they do not require any hypothesis on the temperature to describe the internal energy level populations. These models treat each distinct internal energy level as a discrete species, with their population being computed based on a mass continuity equation. Assumptions on the catalytic properties of the surface allow us to impose recombination to specific energy levels: a procedure that can describe thermal nonequilibrium in a straightforward way.
A no-slip catalytic surface is imposed as follows: for each (pseudo) species i, the conservation of mass states that the normal diffusive flux ji⋅ n is equal to the chemical production rate _ωcati due to
catalysis, or
ji⋅ n _ωcati ; i ∈ S (1)
whereS stands for the set containing all species in the mixture, and n is the unit vector normal to the surface considered. Diffusion is modeled using the multicomponent Stefan–Maxwell equations described in Sec. III.B, whereas the chemical term is based on theγ model. By solving Eq. (1) for all the species, the surface composition can be obtained and then provided to the fluid solver as boundary condition. Because the set of Eq. (1) is linearly dependent, an additional mass constraint is imposed on the chemical composition for the closure of the system, i.e.,
X
i∈S
yi 1
where yiis the species mass fraction. An algorithm to solve the system,
based on the Newton–Raphson method, has been implemented in the open-source Mutation++ library so that it can be easily used by any computational fluid dynamics code.
A. Multitemperature Gamma Recombination Coefficient
Theγ model, introduced by Goulard [18], is the conventional way to treat catalysis for aerothermodynamic applications. Considering a binary mixture composed of homonuclear molecules A2and atoms
A, in multitemperature models, catalytic reactions are described macroscopically as nonelementary processes of the form
A A !γAA2 (2)
To determine the chemical production term for this reaction, a recombination probability is defined for the recombining atom A as
γAF rec A F↓ A (3)
where F↓Ais the flux of atoms impinging the surface, and Frec A is the
flux of atoms actually recombining there. When the distribution function of species translational energy at the wall is well
approximated by a Maxwellian distribution, in absence of temperature slip, the impinging flux is equal to
F↓ A nA kBTw 2πmA s (4)
according to the kinetic theory of gases. Symbol Twstands for the wall
temperature, and kBis the Boltzmann constant. Quantity mAis the mass,
and nA ρA∕mAis the number density of species A. Corrections to the
flux for deviations of the impinging species distribution function from the Maxwellian one can be found in Refs. [9,43].
The input parameterγAfor the model, introduced as a probability,
can take values between zero and one. WhenγAis equal to zero, no
reaction takes place, corresponding to a noncatalytic or chemically inert wall. When γA is equal to one, all the particles of species
A impinging the surface recombine at the wall, corresponding to a fully catalytic wall. Any values ofγA between these two bounds
correspond to a partially catalytic wall, which is the case for most of the surfaces. Once the probability γA is determined, the surface
chemical source terms of Eq. (1) are given by the expressions
_ ωcat
A mAγAF↓A and _ωcatA2 −mA2
γAF↓A
2 (5)
B. State-to-State Gamma Recombination Coefficient
In the approach described previously, recombination does not take into account the internal energy level to which the molecules are recombining. Implicitly, recombination is assumed to occur in thermal equilibrium, with the internal energy levels populations following a Boltzmann distribution at the wall temperature. This assumption can be assessed by using state-specific recombination reactions instead of Eq. (2):
A A!γA;lA2l (6)
where index l∈ L describes a specific internal energy level of the molecule. Assuming atoms A in their ground electronic state and molecules A2with internal energy levels, one hasS L ∪ fAg. The
set of energy levels for A2defined asL fA2ljl 1; : : : ; nLg
depends on the choice of thermodynamic model [39,40].
The expressions for the recombination probability given in Eq. (3) should be adapted to distinguish the internal energy level l as
γA;l
Frec A;l
F↓A
(7)
where l∈ L. The expression for the impinging flux F↓A given in
Eq. (4) remains valid in state-to-state models, and Frec
A;lis the flux of
atoms actually recombining at the wall into level l. A constraint that should be imposed on the state-specific recombination coefficient is that they sum up to the global recombination coefficient, i.e.,
X
l∈L
γA;l γA
The coefficientsγA;lcan be computed based on quantum-mechanical
or molecular dynamics approaches [32,33,35].
When no data are available for the state-to-state model, one can extract information about the influence of thermal nonequilibrium on catalysis by studying some characteristic, ad hoc cases. Because, in our case, we do not have data for the recombination probabilities for each internal energy level, we consider four distinct cases for the values of the γA;l coefficients. In the lower-catalytic case, one
assumes that all atoms recombine to the lower internal energy level l 1; i.e., γA;l δ1lγA, with the Kronecker symbolδml 1, when
m l, and δml 0, otherwise. The maximum amount of energy is
then transferred to the wall, whereas the molecules produced are in
thermal nonequilibrium, with a population depleted as compared to a Boltzmann distribution. In the upper-catalytic case, one assumes that all atoms recombine to the highest energy level l nL; i.e., γA;l δlnLγA. When the energy of this level is equal to the
dissociation energy, the chemical heat flux to the wall is equal to zero. When its value is higher, then the recombination reaction becomes endothermic and the wall is cooled via catalysis. Lastly, an equal catalytic case is studied, where all levels share the same probability, i.e., γA;l γA∕nL. Finally, it is often useful to assume that the
recombination at the wall occurs with the internal energy levels being in thermal equilibrium at the wall temperature. In the thermal equilibrium catalytic case, the surface chemical production rate is supposed to follow a Boltzmann distribution; the state-specific catalytic recombination probabilities read as follows:
γA;l γA
glexp−eIl∕RlTw
P
m∈Lgmexp−eIm∕RmTw
(8) Symbol glstands for the degeneracy; eIlis the internal energy of
level l in joules per kilogram, as will be presented in the next section; and Rl is the specific gas constant of the species levels that l
represents: in this case, A2.
Once the probabilities γA and γA;l are determined, the surface
chemical source terms of Eq. (1) are given by the expressions _
ωcat
A mAγAF↓A; ω_catA2l −mA2
γA;lF↓A
2 (9)
for state-to-state models.
C. Beta Energy Accommodation Coefficient
The recombination of atoms at the surface releases chemical energy due to the creation of chemical bonds, and a part of this energy is transferred to the surface. The remaining fraction may be stored within the created molecule as internal (rotational, vibrational, or electronic) energy. State-to-state models for catalysis allow us to determine how much of this energy is released to the wall and how much is stored internally by the molecules. The chemical energy accommodation coefficientβ was introduced in Ref. [15] as
β qdif _ ωcat A2ΔH dis A2Tw (10)
where quantity ΔHdis
A2Tw is the dissociation energy of the A2
molecule, and
qdifX i∈S
hiji⋅ n
is the diffusive heat flux normal to the surface, which is equivalently defined as
qdifX i∈S
hiω_cati
considering Eq. (1). Also, _ωcat
A2 is the total catalytic rate, and it is
simply equal to X l∈L _ ωcat A2l
An equivalent formulation of the accommodation coefficient for the state-to-state catalytic recombination is
β P qdif
l∈Lω_catA2lΔH
dis
A2lTw
(11)
where quantityΔHdis
A2lTw is the dissociation energy of level l of the
A2molecule, and _ωcatA2lis the rate of the thermal equilibrium catalytic
case at the wall temperature. Using Eq. (8), it can be shown that both expressions forβ given in Eqs. (10) and (11) are equivalent, provided that the dissociation energy is consistent with its state-to-state counterpart, i.e.,
ΔHdis
A2Tw
P
l∈LΔHdisA2lTwglexp−e
I
l∕RlTw
P
m∈Lgmexp−eIm∕RmTw
(12) Theβ coefficient is nondimensional. Avalue of β 0 means that no fraction of energy released during catalytic recombination is directly transferred to the wall. A value ofβ 1 means that all the energy released is fully transferred to the surface, whereas the populations of internal energy levels of molecules produced by surface catalysis follow Boltzmann distribution at the wall temperature. To characterize the catalytic efficiency derived from experiments [44], the product of theβ and γAcoefficients is often used as an overall effective catalytic
recombination probability ofγeff
A βγA.
Note that a single global energy accommodation coefficient has been defined. In some cases it would be more practical to define one coefficient for each gas–surface recombination reaction or even to distinguish in which internal mode the energy is stored, giving better understanding of the heat transfer process.
III. State-to-State Fluid Model
In a state-to-state fluid model, the evolution of the species mass, mixture momentum, and total energy is governed by the Navier– Stokes equations: ∂ρi ∂t ∇ ⋅ ρiu ji _ωi; i ∈ S (13) ∂ρu ∂t ∇ ⋅ ρu ⊗ u pI τ 0 (14) ∂ρE ∂t ∇ ⋅ ρuH τ ⋅ u q 0 (15) whereρiis the partial density of (real or pseudo-) species i∈ S. We
recall that, in this work (without loss of generality), the set of species is composed of the molecule internal energy levels (setL) and a single atom (A), forming the full mixtureS L ∪ fAg. Mixture properties are introduced as usual; quantity ρ is the density, u represents the velocity, E e u ⋅ u∕2 is the total energy, e is the thermal energy, H E p∕ρ is the total enthalpy, and p is the pressure of the mixture. The identity matrix is denoted by symbol I. Quantityτ is the Newton viscous stress tensor, and q is the heat flux vector. The chemical source term _ωi describes the production or
destruction of species i due to inelastic collisions in the gas phase, including excitation and deexcitation reactions. When Eq. (13) is summed over all the species in the mixture, the total continuity equation is retrieved; i.e.,∂ρ∕∂t ∇ ⋅ ρu 0. To close the system, Eqs. (13–15) are solved together with the perfect gas law p ρRT, with the mixture gas constant
R X
i∈S
yiRi
where Ri is the gas constant of species i, and yi ρi∕ρ its mass
fraction, as well as a constitutive law for the energy. The mixture energy per unit mass is computed by summing the species specific energy eiweighted by the mass fractions, i.e.,
e X
i∈S
yiei
For multitemperature models, an additional energy equation is solved for each mode considered, assuming a Boltzmann population of the energy levels at the bath temperature [1]. State-to-state models
treat the distinct energy levels of atoms and molecules as chemical species without any a priori assumption. The evolution of their population is governed by the species continuity Eq. (13). The rovibrational state-to-state model that will be presented in Sec. III.C is only for the gas phase. The gas–surface interaction is instead based on the gamma model presented in Sec. II.B. The internal energy levels along with their degeneracy and the chemical reaction rate coefficients are thus the necessary inputs for the closure of the governing equations.
A. Thermodynamic Properties
Thermodynamic properties of the full mixture can be retrieved from the populations of the species energy levels. The thermal energy of each species is given as the sum of the translational, internal, and formation energy contributions: eiT eTiT eIi eFi, i∈ S.
The translational energy population is assumed to follow a Boltzmann distribution at the translational temperature T: eTi 3∕2RiT. The
internal energy eIiis associated with the internal degrees of freedom of species i. This quantity is constant for state-to-state models. The formation energy eFi for the ground energy level is due to the energy stored in chemical bonds, which is common to all the levels of a given molecule.
Gibbs free energy is also an important thermodynamic property of a mixture because it is necessary for computing the equilibrium constant of a chemical reaction. For each species i∈ S, Gibbs free energy is given by gip; T −RiT ln gikBpT 2πmikBT h2 P 3∕2 eI i eFi (16)
where symbol hPstands for Planck’s constant.
In state-to-state models, the molecule internal energy is defined as X
l∈L
ρleIl
No internal temperature is required; nevertheless, this quantity can be computed a posteriori to describe macroscopically the thermal nonequilibrium state of the flow. The internal temperature TIis then
calculated by solving the following nonlinear equation: P l∈LρleIl P m∈Lρm P l∈LmleIlglexp−eIl∕RlTI P m∈Lmmgmexp−eIm∕RmTI (17)
This quantity represents the temperature that the internal energy bath would have if the populations of internal energy levels of the species would follow a Boltzmann distribution function.
B. Transport Properties
Closure of the mass, momentum, and energy transport fluxes also requires the computation of the transport properties: diffusion coefficients, viscosity, and thermal conductivity. These transport properties are computed by solving linear transport systems of dimension proportional to the number of species in the mixture, which result from the Chapman– Enskog solution to the Boltzmann equation [45]. Because levels are treated as conventional species in state-to-state models, no modifications are required for the linear systems, but the dimension of the corresponding matrices increases drastically. In this work, we assume that all energy levels interact with other species using the same intermolecular potential to compute the transport collision integrals [46]. In this case, a substantial reduction of the linear systems can be achieved by lumping together all the pseudospecies with the same collision integrals when calculating the viscosity and thermal conductivity properties, as discussed in the Appendix.
The correct modeling of the mass diffusion fluxes is crucial when gas–surface interaction phenomena become important because the diffusion of species in the boundary layer is the main process feeding the surface with chemically active species. It has been observed that
using Fick’s law instead of a more rigorous multicomponent diffusion formalism can lead to substantial errors in the predicted heat fluxes [47]. The mass diffusion fluxes can be obtained by solving the multicomponent Stefan–Maxwell system as an alternative to the transport systems for the multicomponent diffusion coefficients [41]:
X j∈S j≠i xixj ρjDijjj− X j∈S j≠i xixj ρiDijji ∇xi; ∀ i ∈ S (18)
where quantity xiis the mole fraction of species i, andDijis the
binary diffusion coefficients for the i, j species pair. Stefan–Maxwell equations are linearly dependent. A constraint is imposed to respect mass conservation:
X
i∈S
ji 0
The right-hand sides of the equations consider as driving forces only mole fraction gradients, with the pressure, temperature, and volume forces contributions neglected. Unfortunately, the reduction of the Stefan–Maxwell system by lumping the levels is not possible because the mass diffusion fluxes are proportional to the species diffusion forces.
The viscous stress tensorτ expresses the variation of momentum when velocity gradients are present in the flowfield. It is defined as
τ −μ ∇u ∇uT−2 3∇ ⋅ uI (19) where μ is the shear viscosity coefficient of the flow, which is obtained by solving the reduced linear system presented in the Appendix. Notice that the bulk viscosity does not appear in a state-to-state approach.
The heat flux appearing in Eq. (15) expresses the energy transferred by conduction and diffusion:
q qcon qdif; qcon −λ∇T and qdifX i∈S
hiji (20)
The first term represents Fourier’s law with the translational temperature gradient force, whereas the second term is the transfer of species enthalpy hiT eiT RiT by mass diffusion. We recall
that the diffusion flux of species i (ji) is computed based on the
Stefan–Maxwell equations [Eq. (18)], whereas the thermal con-ductivityλ is obtained by solving the linear transport system shown in the Appendix. It is important to note that, within a state-to-state formalism, there is no need to consider Eucken’s correction to the thermal conductivity [45]. The energy transfer of the internal energy levels is taken into account through chemical reactions, and is therefore part of the diffusive heat flux.
C. Mixture Description and Chemical Mechanism
A variety of detailed chemistry models are available in the literature [2,48–51]. Most of them are hybrid models combining a state-to-state approach, for part of the internal energy levels populations, together with a multitemperature approach, assuming that the rest of the energy level populations follow Boltzmann distributions at their relevant temperature.
In this work, we use a state-to-state approach for the N2 N
system developed by the computational chemistry group at NASA Ames Research Center [52,53]. The full model comprises 9390 distinct rovibrational energy levels for the ground electronic state of the N2molecule, which are denoted by the set of indicesF. By
ordering these internal levels with increasing energy, the first 7421 levels (setFb) are truly bound; i.e., their energy is lower than the
dissociation energy (9.75 eV) of the N2molecule as compared to the
ground rovibrational level. They can dissociate when external energy is provided by collisions. The upper 1969 levels (setFp) are
quasi-bound or predissociated levels. Their internal energy being higher
than the dissociation energy, they are easier to dissociate than the truly bound levels through collisional dissociation or predissociation reactions. This distinction is important because the elementary processes governing the behavior of these two types of levels are different. We haveF Fb∪ Fp and Fb∩ Fp ∅. Following
Ref. [54], we consider the predissociation process negligible. The mixture is completed by adding the ground electronic state of N, with its degeneracy equal to gN 12, accounting for the nuclear spin
degeneracy.
Two categories of collisional reactions by atom impact in the gas phase are selected here:
1) Dissociation/recombination: N2l N ⇄ kf;dis l kb;dis l N N N 2) Excitation/deexcitation: N2l N ⇄ kf;exc l→l 0 kb;exc l→l 0 N2l0 N
where l; l0∈ F with l0> l. The total number of reactions is on the order of 23 million, with rate coefficients calculated from first principle quantum-chemistry calculations and fitted into Arrhenius form. The large dimensionality of the aforementioned state-to-state model is a stumbling block for practical applications. For this reason, order reduction methods are essential. They can be performed by means of mathematical approaches [55] or using physical arguments [54,56– 58]. Even though the former can be more effective, the latter approach gives a more intuitive representation of the flow physics. In this work, the uniform rovibrational collisional (URVC) reduction method [54] is used, considering its adequacy when developing models for transport phenomena. Contrary to the original formulation, where a uniform energy grid was applied for the binning operation, a variable-space energy grid was adopted in this work to improve the accuracy of the result, as previously recommended by Torres and Magin [59] for direct simulation Monte Carlo applications.
Starting with the lower boundE0, the upper bound for bin k of the
variable-spaced energy grid is defined as follows:
Ek 8 < : eI nFb− eI1 k nBb ξ k eI 1; k ∈ Bb; eI nF− e I nFb1 k nBp ξ k eI nFb1; k ∈ Bp (21)
where nFb #Fband nF #F. The set of bins is introduced as Bb
for the bound levels andBpfor the predissociated levels. These sets
are disjoint to avoid mixing elementary processes:Bb∩ Bp ∅ and
B Bb∪ Bp. One has nBb #Bband nBp #Bp. Quantityξkis a
stretching factor for the energy grid. Chosen to be greater than unity, it provides a good approximation of the full model at low temperatures encountered in the freestream of atmospheric entry flows because the energy grid is finer close to the low-energy levels. One of the main assumptions of the URVC model is that the population of each energy level inside bin k∈ B is uniform. The total population of energy levels in bin k is denoted by symbol nk, whereas the population of
each level l within bin k is given by nl glnk∕gk, l∈ Fk, where the
set of levels in bin k is denoted by symbolFk. The degeneracy of the
bin gkis obtained by summing the degeneracies of each level:
gk
X
l∈Fk
gl
The energy of bin k can be computed a priori as eI k X l∈Fk gleIl∕gk
because the population is assumed to be uniformly distributed in each bin.
For the reduced mechanism, the two reaction categories of the full model are kept at the bin level.
1) Dissociation/recombination: N2k N ⇄ kf;dis k kb;dis k N N N 2) Excitation/deexcitation: N2k N ⇄ kf;exc k→k 0 kb;exc k→k 0 N2k0 N
where k; k0∈ B, with k0> k. The bin averaged rate coefficients for dissociation and excitation reactions are
kf;dis k 1 gk X l∈Fk glkf;disl (22) and kf;exc k→k0 1 gk X l∈Fk X l0∈F k 0 glkf;excl→l0 (23) respectively.
For both fine- and coarse-grain models, the backward rate coefficients are computed in a way that the equilibrium composition is retained, respecting the second law of thermodynamics. This is achieved by applying the microreversibility principle. The backward rate is thus computed based on the equilibrium constant keq;rin the
following way:
kb;rT k
f;rT
keq;rT (24)
for each reaction r. The equilibrium constant can be computed based only on thermodynamic properties and in particular Gibbs free energy for any reaction as
ln keq;rT −X i∈S νr i gipref; T RiT − ln pref kBT (25)
whereνri νri0 0− νir0, with symbolsνri0 0andνri0as the stoichiometric coefficients of the reactants and products in reaction r; and prefis an
arbitrary reference pressure.
For the N N2model used in this work, the URVC coarse-grain
model allowed us to reduce the 9390 rovibrational levels of the N2
molecule to just 10 bins (nB 10). According to the analysis performed in Refs. [58,60], it was decided that nBb 9 truly bound bins and nBp 1 predissociated bins gave sufficient fidelity to the model. For the truly bound levels, a stretching factor ofξk 1.5,
k ∈ Bb, was chosen. No stretching was applied to the predissociated
levels because they were all lumped into a single energy bin:ξk 1,
k ∈ Bp. The resulting energy and degeneracy of the bins can be found
in Table 1.
During the previous section on catalysis (Sec. II), symbolL was used to indicate the set of generic levels of constant energy for a single species i∈ S. In this section, we differentiate between true levels for the N–N2 system F and levels resulting from the binning
procedureB. It can be shown that the formulas presented based on set L can be applied for both sets F and B without problem. In particular, because of the nature of the URVC bins, Eq. (1) can be reduced rigorously from the full system to bin one, assuming that all energy levels in each bin share the same collisional properties.
D. Numerical Methods
State-to-state approaches, even with the bin reduction, greatly increase the computational cost of a simulation compared to multitemperature approaches because an additional species continuity equation needs to be added for each distinct internal state (true levels or bins). To alleviate this problem, a special formulation of the Navier– Stokes equations was used. This approximation [3,61] allows us to simulate the flowfield on the stagnation line of a reentry vehicle. The full Navier–Stokes equations are written in spherical (r, θ, ϕ) coordinates. The flow is assumed to be axisymmetric (uϕ 0 and ∂∕∂ϕ 0). The
symmetry of the flow implies that ur, p, T, andρiare symmetric with
respect to the stagnation line axis (located atθ 0), whereas uθis antisymmetric with respect to it. Furthermore, the pressure distribution is computed with the Newtonian theory of hypersonic flows. Finally, by taking the limit forθ → 0, the dimensionally reduced Navier–Stokes equations are obtained. The stagnation line simulations were compared for validation with the axisymmetric computational fluid dynamics solver, Cosmic [9]. Both codes discretize the equations with the cell-centered finite volume method in space and perform the time iteration with an implicit backward Euler method. The AUSM up all speed flux vector splitting scheme, coupled with the van Albada limiter, was used for the convective fluxes [62], whereas the diffusive fluxes were discretized with a Gauss–Green central scheme in order to achieve second-order accuracy [9].
Both codes were coupled with the Mutation++ library [63], which provides thermodynamic and transport properties, chemical kinetics, and energy transfer terms needed for the closure of the Navier–Stokes equations. Mutation++ was also used in both codes to account for the modeling of gas–surface heterogeneous catalysis at the wall.
IV. Hypersonic Flow Simulations
We study the effect of thermal nonequilibrium induced by catalytic reactions on the heat flux for a reentry body. The freestream and wall boundary conditions for the three test cases considered are reviewed in Table 2: 1) strong nonequilibrium (NEQ), high freestream velocity, and medium catalysis; 2) medium NEQ, intermediate freestream velocity, and high catalysis; and 3) weak NEQ, low freestream velocity, and low catalysis. The freestream velocities and pressure are characteristic of atmospheric reentry conditions, for which thermal nonequilibrium effects can become significant, whereas the freestream temperature was fixed equal to T∞ 200 K. To induce the excitation and dissociation processes, the flow is seeded with a small amount of N atoms (mole fraction xN 0.028) because the gas-phase mechanism
does not consider N2 N2collisions responsible for creating the first
N atoms after the shock.
The reentry object studied is a reaction-cured glass (RCG) sphere of radius equal to 0.0254 m, which is small enough to promote nonequilibrium in the flow. Its surface temperature Twis set at three
distinct values typical of the reentry speeds. The choice of a reaction-cured glass surface was because this material is commonly used for thermal protection systems, with its global recombination pro-bability given as a function of the surface temperature predicted by formula [20]:
Table 1 Degeneracygkand energyeIkfor the 10-bin URVC model
Bin k gk eIk, MJ/kg 1 7,767 0.69375 2 36,426 2.5440 3 91,239 5.1406 4 182,634 8.2399 5 312,789 12.0715 6 498,210 16.2218 7 772,773 20.7992 8 1,176,771 25.7693 9 2,013,810 31.2041 10 3,109,059 38.7546
γN 8 > > > > > > < > > > > > > : 5⋅ 10−4 : Tw< 465 K; 2⋅ 10−5exp 1500 Tw : 465 K < Tw< 905 K; 10 exp −10;360 Tw : 905 K < Tw< 1575 K; 6.2⋅ 10−6exp 12;100 Tw : Tw> 1575 K (26)
In reality, Eq. (26) corresponds to the effective catalytic re-combination coefficientγeff
N because it is obtained from experiments.
Here, the values will be used for the actual catalytic recombination coefficient γN. Considering that the state-specific recombination
probabilitiesγN;l for the energy levels l are not available, the first
three catalytic cases described in Sec. II.B are simulated. For the variably spaced energy bin URVC model B, the following cases will be considered: lower-catalytic (LC), equal catalytic (EC), and upper-catalytic (UC) cases, together with the nonupper-catalytic (NC) case. The results will be compared to those obtained by means of the two-temperature model proposed by Park P for the following cases: fully catalytic (FC), partially catalytic (PC), and noncatalytic.
A. Verification Against Axisymmetric Solver Results
Figure 1 shows a comparison of the temperature and atomic nitrogen mole fraction profiles along the stagnation line between the Cosmic code and the stagnation line code results for the strong NEQ case, assuming a noncatalytic material behavior. The aim of this comparison is to verify that the stagnation line code, with its special formulation of the Navier–Stokes equations, gives sufficiently accurate results. The matching in both cases is good, especially for the atomic nitrogen mole fraction. Cosmic underpredicts the overshoot of the translational and internal temperatures in the compression region, which is an effect associated with the coarser mesh used for the two-dimensional simulations to save computational time. This discrepancy did not affect the predicted heat fluxes, which were equal to 3.39 MW∕m2for the
Cosmic code and 3.47 MW∕m2for the stagnation line code. The grid
refinement at the wall is sufficiently refined for both codes, and the difference between the predicted heat fluxes is around 2%. Therefore, it is valid to use an order-reduced formulation of the Navier–Stokes equations instead of the full axisymmetric code, which requires an increased computational cost.
B. Comparison with Two-Temperature Model
The results of the NASA bin model for nitrogen are compared to the widely used two-temperature model proposed by Park [1], which is properly modified to enforce consistency between both models. More specifically, only the ground electronic states of atomic and molecular nitrogen are considered. Regarding the dissociation and internal-translational energy transfer processes, only N2 N
collisions are accounted for. In the two-temperature model, the rate coefficient for dissociation is computed based on the average temperaturepTTv, where Tvis the vibrational temperature, using a nonpreferential chemistry–energy coupling. The energy transfer is described by means of the Landau–Teller formula with the Millikan– White–Park relaxation time [1]. The thermodynamic properties of the nitrogen molecules are computed based on the rigid-rotor and harmonic-oscillator models using a steric factor equal to two. The ground state of the nitrogen atom is then assigned a degeneracy of four for the multitemperature approach, excluding the nuclear spin. Figure 2 shows the comparison of the stagnation line temperature profiles as predicted by the NASA bin and the two-temperature Park models for the three test cases, assuming a noncatalytic material behavior. As explained earlier, the internal temperature of the state-to-state model is a postprocessing result of the internal energy level distribution of the nitrogen molecule, which acts here as an indicator of the degree of thermal nonequilibrium between the translational and the internal energy modes. In all three cases, the shock standoff distance and the translational temperature peak are underpredicted by the Park model. Nonetheless, the equilibrium temperature reached in the shock layer is comparable for the two models, showing that the bin model accurately captures equilibrium. As the freestream velocity decreases and pressure increases, the temperature overshoot becomes less significant: par-ticularly for the weak NEQ case, where Park’s model completely fails to capture the overshoot of the translational temperature [1].
A better insight of the shock-layer features can be obtained by observing the mole fraction of atomic nitrogen in Fig. 3. For the strong NEQ case, despite the difference in the predicted shock standoff distance, a fully dissociated flow is found for both models. Park’s model slightly underpredicts nitrogen recombination the boundary layer [1] as compared to the bin model, resulting in a small difference in the mole fractions at the wall (xN, with Park 0.973;
and xN, with bin 0.956). For the medium NEQ case, the wall
nitrogen mole fractions are xN, Park 0.721 and xN, bin 0.690,
while for the weak NEQ case, the values are xN, with Park 0.367; Table 2 Freestream and wall boundary conditions for the three test casesa
u∞, m∕s p∞, Pa T∞, K ρN;∞, kg∕m3 ρN 2;∞, kg∕m 3 T w, K γN Strong NEQ 10,000 20 200 4.71⋅ 10−6 3.27⋅ 10−5 1,800 0.00515 Medium NEQ 8,000 50 200 1.18⋅ 10−5 8.18⋅ 10−4 1,600 0.0119 Weak NEQ 6,000 100 200 2.36⋅ 10−5 1.60⋅ 10−3 1,000 0.0003 aThe freestream N
2internal energy levels follow a Boltzmann distribution at this temperature.
a) Translational (black) and internal (red) temperatures b) Mole fraction of N
Fig. 1 Temperature and mole fraction profiles along the stagnation line for the strong NEQ case using a noncatalytic material: Cosmic code (dashed lines), and stagnation line code (solid lines).
and xN, with bin 0.358. The gas composition near the surface is
similar for both models, despite the differences in the gas-phase chemical mechanism. When studying catalytic properties of materials in the following section, one expects that the heat flux due to catalysis should not be affected by the gas mechanism because the number of atoms that reach the wall and can recombine is almost the same.
C. Catalytic Cases
The normalized populations (Nk nk∕gk) of the bins are presented
in Fig. 4 as a function of the energy of each bin for the strong NEQ test case and the four catalytic cases. The results are computed by means of the stagnation line code and are shown at different distances from the wall: 0, 1.5× 10−5, 3× 10−5, and 6× 10−5 m. In all four cases, the population at 6× 10−5 m is a straight line, meaning that it follows a Boltzmann distribution at the internal temperature. Moving closer to the wall, the slope of population distribution changes because of the boundary-layer cooling, which also favors recombination of the N atoms through gas-phase reactions. The lines seem to have three distinct regions: a behavior particularly apparent in the noncatalytic and lower-catalytic cases. This happens because gas-phase recombination occurs preferably in the upper bins, which are populated with a common temperature. This trend is not followed by the last bin (bin 10), which contains all the predissociated levels, because these levels follow a different chemical dynamics. The bin is populated with an internal temperature similar to the one of the first three bins. This behavior is driven by the gas-phase chemistry and is marginally affected by the catalytic model.
The effect of catalysis becomes clearer in Fig. 5, where the normalized populations at the wall are plotted for all four catalytic cases. The com-parison of the lower- and noncatalytic cases shows the overpopulation of the first bin because of the wall’s catalytic activity. The second and third bins are also overpopulated (this time due to gas-phase excitation reactions), whereas the minor depletion of the higher bins can be ex-plained by the fact that more nitrogen atoms are available in the noncatalytic case for gas-phase recombination, which is a process that occurs preferentially in the upper levels. The dynamics governing the equal (blue) and upper (magenta) catalytic cases are very similar. Smoothing of the populations is performed by gas-phase reactions pulling the system toward equilibrium. An overpopulation of the last four bins is noticeable for the upper-catalytic case when compared to the equal one. The additional internal energy added to the gas in the upper-catalytic case is transferred to the bins by gas-phase deexcitation reactions.
Figure 6 depicts the internal temperature profiles close to the wall with the same color code. Even in the noncatalytic case, the internal temperature at the surface is not equilibrated with the translational one imposed at the wall. From a microscopic point of view, this means that, when a molecule hits the surface, it does not accommodate any of its internal energy into it; there is only exchange of translational energy. Molecular dynamic simulations have shown that this might not correspond to physical reality because, by internal cooling [64], the internal temperature of molecules interacting with the surface can be lower than the translational one. Accommodation of molecules’ internal energy modes with the surface can be modeled by reactions of the kind:
N2l wall ↔ N2l0 wall (27)
with l; l0∈ L. However, rates for this kind of reaction are practically not available in the literature; and we decided to neglect them, with the aim of studying only the effect of catalysis.
The internal temperatures are substantially different in the catalytic cases only up to a small distance from the wall (1.5× 10−5 m), similar to the trend observed for the internal energy level populations. In the lower-catalytic case, catalysis forces a fraction of the available nitrogen atoms to recombine in the first level. Therefore, a maximum of energy is transferred from the gas to the wall, effectively cooling the gas phase.
In the equal and upper-catalytic cases, the internal temperature is significantly higher than the translational one. Again, this thermal nonequilibrium region extends only up to a small distance from the surface, meaning that these internally excited molecules disappear from the gas phase. In our model, we consider only two mechanisms that can explain this phenomenon: deexcitation and dissociation reactions. Both of them redistribute the internal energy into translational energy: a phenomenon often referred as internal energy quenching. The importance of this process will become clearer in Sec. IV.D, when the wall heat fluxes are presented.
The evolution of the normalized populations gives more information than an equivalent internal temperature in regions of strong non-Boltzmann behavior, such as in boundary layers and across shocks. Because the emphasis of the paper is on catalysis, we focus only on what happens in the boundary layer. Extensive discussions about what happens across shocks can be found in Ref. [3].
a) Strong NEQ b) Medium NEQ c) Weak NEQ
Fig. 2 Translational (black) and internal (red) temperatures profiles along the stagnation line using a noncatalytic material: bin model (solid lines), and Park two-temperature model (dotted lines).
a) Strong NEQ b) Medium NEQ c) Weak NEQ
Fig. 3 Mole fraction profiles of atomic nitrogen along the stagnation line using a noncatalytic material: bin model (solid lines), and Park two-temperature model (dotted lines).
D. Catalytic Thermal Nonequilibrium Effect on Heat Transfer
The heat fluxes for all the simulations performed for this work are reviewed in Table 3. The change of freestream conditions has an important effect on the total heat flux. For the noncatalytic cases, the total heat flux decreases with the freestream velocity. The phenomena
become more intricate when catalysis is considered. For all freestream conditions, and regardless of the catalytic model (LC, EC, or UC), the total heat flux remains almost unchanged; what changes is the energy mechanism transfer to the wall. In the lower-catalytic case, because only a small amount of the recombination energy is stored in
a) Noncatalytic case b) Lower-catalytic case
c) Equal catalytic case d) Upper-catalytic case
Fig. 4 Normalized internal energy populations as a function of the energy of each bin for the four catalytic cases. Distances from the surface of 0 m (solid lines);1.5 × 10−5m (dashed lines); 3 × 10−5m (dashed–dotted lines); and; 6 × 10−5m (dotted lines).
a) Strong NEQ b) Medium NEQ c) Weak NEQ
Fig. 5 Normalized internal energy populations for the three freestream conditions: NC (black); LC (red); EC (blue); and UC (magenta).
a) Strong NEQ Tw = 1800 K b) Medium NEQ Tw = 1600 K c) Weak NEQ Tw = 1000 K
Fig. 6 Internal temperature profiles along the stagnation line for the three freestream conditions: NC (black); LC (red); EC (blue); and UC (magenta).
the molecules created by catalysis, the diffusive heat flux has its highest value. It decreases for the equal catalytic case and becomes even negative for the upper one. As it was mentioned in Sec. III.C, the upper bin contains only predissociated levels. These levels have energy higher than the energy of the two dissociated atoms of nitrogen, making the recombination reaction an endothermic process. For this reason, the surface (which is considered isothermal) provides the necessary energy to the flowfield for recombination to occur. Even in this extreme case, the total heat flux at the wall reduces by a small amount because these highly unstable produced molecules, when they diffuse from the wall into the boundary layer, redistribute their internal energy into translational (the“quenching” discussed in Sec. IV.C). The increased translational energy returns to the wall under the form of a conductive heat fluxqcon, as can be seen
in Table 3. This conclusion is in agreement with the results obtained by examining the internal energy level population in Sec. IV.C.
The wall heat flux computed by means of the Park two-temperature model is also shown in Table 3. Except for the weak NEQ case, the Park model heat flux differs from the bin model predictions. This is especially true for the medium NEQ case, which has the higher global recombination probabilityγN. The discrepancy in heat fluxes
suggests that multitemperature models can be inaccurate when the interaction between the wall and the gas promotes thermodynamic nonequilibrium.
E. Energy Accommodation Coefficient Extraction
One of the main goals of this work was the extraction of the energy accommodation coefficientβ as presented in Table 3. In Sec. II.C, the β coefficient was shown to be an indicator of how much of the recombination energy is directly released to the wall surface and how much is stored in the produced molecules as internal energy. Because a state-to-state description of catalysis is performed, this coefficient can be directly extracted from the simulation.
Some important comments should be given about the values of the β coefficient presented in Table 3. They can be higher than one or less than zero; both of these cases are consequences of the model used. For the lower-catalytic case, the β coefficient is higher than one because the energy directly transferred to the wall is higher than the energy that would be transferred in Boltzmann equilibrium. We recall that the denominator of Eq. (10) is the energy released by the catalytic chemical reactions when the products are in thermodynamic
equilibrium at the wall temperature. Because recombination occurs only in the lowest-energy bin, the energy stored inside the molecules nN2Ek1is lower than the value
X
k∈B
n
kEk
it would have in thermal equilibrium (i.e., the population nk; k ∈ B is assumed to follow a Boltzmann distribution). As a consequence, more energy is transferred to the wall. Negative values of the β coefficient result from the fact that the upper energy bin is predissociated, making the recombination reaction an endothermic process. Whatβ indicates here is that the wall is not heated by the reaction, but it transfers energy to gas in the form of internal energy. This energy, though, is returned to the wall in the form of translational energy through quenching.
The fact that the total wall heat fluxes do not depend on theβ coefficient appears to be true for all the three different values ofγN
studied. In the weak NEQ case, the global recombination probability is really low, with minor effects on the diffusive heat fluxes. However, even in this extreme case, exactly the same behavior for the heat flux appears. This is the main reason why such low catalytic conditions were presented in this contribution.
Even though it seems that, for our bin model, the different β coefficients do not substantially affect the heat flux, this might not be the case for other systems; and its correct determination is still relevant for the determination of the catalytic recombination probability [65].
V. Conclusions
The interplay of the gas–surface interaction and thermal nonequilibrium is still an open problem in aerothermodynamics. The limited amount of relevant experiments along with the complex and partially incomplete physicochemical models make any numerical simulation trying to cope with this problem very hard to validate. Even though experiments are the ultimate way of proving or disproving an idea, studying the trends and the extreme cases numerically can still provide us with some understanding of the underlying physics. Such an approach was followed here, retaining some of the important features of thermal nonequilibrium during gas–surface interaction. By applying a sensitivity analysis on the amount of thermal nonequilibrium existing at the wall due to catalysis, interesting conclusions were drawn about how the heat fluxes are affected. The main idea was that there are cases where it is irrelevant how much energy is actually deposited on the surface or how much is stored in the internally excited molecules, considering the predicted heat fluxes. Internal energy quenching into translational energy is found, which is also a phenomenon observed experimentally; and it keeps the total energy transferred to the surface overall constant. This result gives more credibility to the popular assumption in the hypersonics community that all the energy is deposited on the surface. Nonetheless, more simulations and especially experiments are needed for validating this idea, whereas the problem of the degree of thermal nonequilibrium of a catalytic surface remains an open question.
The reentry of an RCG sphere with a radius of 0.0254 m was studied in three freestream conditions at different degrees of thermal nonequilibrium (strong, medium, and weak NEQs). In all cases, the freestream temperature was 200 K and the mole fraction of nitrogen was xN 0.028. The velocity and pressure were equal to
10;000 m∕s with 20 Pa, 8000 m∕s with 50 Pa, and 6000 m∕s with 100 Pa, with the surface temperatures being, respectively 1800, 1600, and 1000 K. To capture the thermal nonequilibrium effects, state-of-the-art models had to be used for both the gas phase and at the surface. Starting from the gas, an ab initio mechanism for molecular nitrogen developed at NASA Ames Research Center was reduced, based on a variably spaced energy bins approach applied for the first time to a continuum simulation. Using a stagnation line Navier–Stokes code, this state-to-state model was compared to the two-temperature model proposed by Park [1] for the noncatalytic case, showing a good matching for the gas chemical composition on the surface and the heat flux. For catalysis, an extension of the regular gamma model was
Table 3 Total, conductive, and diffusive heat fluxes (in megawatts per square meter); and energy accommodation coefficientβ and internal temperature
TIin kelvins for binB and Park P models
NEQ case q qcon qdif β TI
Strong B NC 3.47 3.47 — — — — 1949 B LC 5.00 3.28 1.72 1.023 1906 B EC 4.98 4.03 0.95 0.565 2172 B UC 4.93 5.12 −0.19 −0.113 2417 P NC 3.17 3.17 — — — — — — P PC 4.37 3.08 1.29 — — — — P FC 8.92 3.03 5.89 — — — — Medium B NC 3.08 3.08 — — — — 1674 B LC 5.70 2.47 3.23 1.018 1678 B EC 5.66 3.88 1.78 0.560 1863 B UC 5.61 5.94 −0.33 −0.103 2028 P NC 2.67 2.67 — — — — — — P PC 4.18 2.27 1.91 — — — — P FC 5.80 2.23 3.57 — — — — Weak B NC 2.56 2.56 — — — — 1085 B LC 2.64 2.53 0.11 1.010 1085 B EC 2.64 2.58 0.06 0.571 1096 B UC 2.64 2.65 −0.01 −0.095 1106 P NC 2.44 2.44 — — — — — — P PC 2.69 2.63 0.06 — — — — P FC 3.81 2.30 1.51 — — — —
used, which takes into account to which internally excited level (here, bin) the catalytic recombination will occur. Because data for these recombination probabilities were not available, three artificial test cases were chosen using extremeγA;lvalues. In the first one, all the
recombination was occurring in the lower-energy bin; in the second one, all the recombination was occurring in the highest-energy bin; and in the third one, an equal probability of recombination was chosen. All the tests cases showed the same trend; it was observed that, for the total heat flux, it was not important to which internally excited state the recombination was occurring. When recombination was forced on the highly energetic bins, this internal energy was very quickly quenched in the gas phase into the translational energy mode and was returned on the surface as translational heat flux, keeping the total heat flux constant. This observation is very encouraging because it shows that the popular assumption that all the energy is directly deposited on the surface will not, in the end, influence the predicted heat fluxes. The energy accommodation coefficient was also com-puted, observing that it can be higher than unity, or even negative.
The bin approach for catalysis presented in this paper was academic because no experiment can reproduced the physical conditions discussed. One parameter that restricted its applicability was the chemical model. At the time these simulations were performed, only data for the N2–N
collisions were computed; very recently, the data for the N2–N2collisions
became available. With the tools developed and the know how obtained, the addition of the rest of the reactions has been made trivial. As soon as the full model is available, pure nitrogen experiments could be performed for several catalytic materials; and the ideas discussed here could be validated by comparing to the experiments.
The catalytic model discussed for the bins could be improved by using several methods to obtainγ (e.g., molecular dynamics), but the method based on finite-rate chemistry models is more promising. By using the knowledge available on the physics included in the elementary reactions occurring on the surface, one could develop such a model and observe how it behaves. Of course the validity of such a model should be assessed by comparison with experiments.
Appendix: Transport Properties Linear Systems Reduction for State-to-State Models
Calculating transport properties as solutions to linear transport systems derived from a Chapman–Enskog method applied to the Boltzmann equation can be expensive for state-to-state models due to the large number of pseudospecies in the mixture. These properties are based on transport collision integrals that depend on the degree of internal excitation of the molecules [66,67]. In this work, we use the transport collision integrals of the ground rovibronic state of the nitrogen molecule [68], neglecting how these data are affected by internal energy excitation. This approach also has the advantage of significantly reducing the order of the linear transport systems for viscosity and thermal conductivity to the order of a binary mixture for atoms and molecules.
The mass transport is governed by the Stefan–Maxwell equations given in Eq. (18). The closure demands the definition of binary diffusion coefficientsDij, i≠ j, for the collision pair i − j. They depend only on
the thermodynamic state T; p asDij 3p2πkBT∕ mij∕16n Q1;1ij ,
with mijbeing the reduced mass of the two colliding species and Q1;1ij
being the diffusion transport collision integral function of the tra-nslational temperature.
Assuming again, for simplicity, the standard set of species used in this work ofS fL; Ag, the linear systems for viscosity and thermal conductivity take the following form:
2 4 Gκ A2l;A2l0 l;l0∈L Gκ A2l;A l∈L Gκ A;A2ll∈L G κ A;A 3 5 aκA aκ A2ll∈L xA xA2ll∈L (A1) with the symbolκ ∈ fμ; λg indicating either viscosity or thermal conductivity. For viscosity, the matrix Gμ is symmetric positive definite, with the diagonal terms given as
Gμ ii X j∈S j≠i kBT pDij xixj mi mj 1 3 5 mj mi Aij x2i μi (A2)
where the pure viscosity of species i∈ S is introduced as μi165
πmikBT
p
∕ Q2;2ii . The offdiagonal terms i; j∈ S with i ≠ j
are introduced as Gμ ij kBT pDij xixj mi mj −1 3 5Aij (A3)
with the transport collision integral ratio Aij Q2;2ij ∕ Q1;1ij . The
thermal conductivity matrixGλis also symmetric positive definite, with the diagonal terms given as
Gλ ii X j∈S j≠i kBT pDij xixj mi mjmij 16 25Aij− 12 25 mj mi Bij6 5 mi mj mj mi 4 15kB x2 imi μi (A4)
i ∈ S, with Bij 5 Q1;2ij − 4 Q1;3ij ∕ Q1;1ij . The offdiagonal
terms i; j∈ S with i ≠ j are introduced as Gλ ij kBT pDij xixj mi mjmij 16 25Aij 12 25Bij− 11 5 (A5)
The corresponding transport property is computed based on the inner product of the solution and mole fraction vectors:
κ aAxA
X
l∈L
aA2lxA2l
Returning to the system of Eq. (A1), it can be reduced to a matrix with dimensions equal to the size of the number of chemically different species (for this case, the setS⋆ fA2; Ag) under the assumption that
all the levels share the same collision properties. This means that all collision integrals Qp;qij , p; q∈ f1; 1; 1; 2; 1; 3; 2; 2g are identical for all the energy levels:
Qp;q A2lA2l0 Q p;q A2A2 Qp;q A2lA Q p;q AA2l Q p;q A2A Q p;q AA2 l;l 0∈ L
The same result holds for the expressions Aij, Bij, andDij. By
summing together all the lines containing different internal energy levels of the same chemical species and by taking into account that
xA2
X
l∈L
xA2l
the full system [Eq. (A1)] can be reduced to Gκ A;A GκA;A2 Gκ A2;A G κ A2;A2 aκ A aAκ2 xA xA2 (A6) where the matrixGκi;jhas the same structure as in Eqs. (A2–A5), with i; j ∈ S⋆. The transport property equals to κ xAaA x
A2aA2
because this procedure gives that xN2aN2
X
l∈L
xA2laA2l
The result is that, by solving the reduced system, we get the full viscosity and thermal conductivity of our full state-to-state mixture. For mass diffusion, such a reduction is not possible; and the full systemS should always be solved.
