NON-EQUILIBRIUM REACTION RATES IN THE MACROSCOPIC CHEMISTRY METHOD FOR DSMC CALCULATIONS. March 19, 2007

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1 NON-EQUILIBRIUM REACTION RATES IN THE MACROSCOPIC CHEMISTRY METHOD FOR DSMC CALCULATIONS M.J. GOLDSWORTHY, M.N. MACROSSAN & M.M ABDEL-JAWAD March 9, 7 The Direct Simulation Monte Carlo (DSMC) method is used to simulate the low o rareied gases. In the Macroscopic Chemistry Method (MCM) or DSMC, chemical reaction rates calculated rom local macroscopic low properties are enorced in each cell. Unlike the standard total collision energy (TCE) chemistry model or DSMC, the new method is not restricted to an Arrhenius orm o the reaction rate coeicient, nor is it restricted to a collision cross-section which yields a simple power-law viscosity. For reaction rates o interest in aerospace applications, chemically reacting collisions are generally inrequent events and, as such, local equilibrium conditions are established beore a signiicant number o chemical reactions occur. Hence, the reaction rates which have been used in MCM have been calculated rom the reaction rate data which are expected to be correct only or conditions o thermal equilibrium. Here we consider artiicially high reaction rates so that the raction o reacting collisions is not small and propose a simple method o estimating the rates o chemical reactions which can be used in the Macroscopic Chemistry Method in both equilibrium and non-equilibrium conditions. Two tests are presented: () The dissociation rates under conditions o thermal non-equilibrium are determined rom a zero-dimensional Monte-Carlo sampling procedure which simulates intra-modal non-equilibrium; that is, equilibrium distributions in each o the translational, rotational and vibrational modes but with dierent temperatures or each mode; () The -D hypersonic low o molecular oxygen over a vertical plate at Mach 3 is calculated. In both cases the new method produces results in close agreement with those given by the standard TCE model in the same highly nonequilibrium conditions. We conclude that the general method o estimating the non-equilibrium reaction rate is a simple means by which inormation contained within non-equilibrium distribution unctions predicted by the DSMC method can be included in the Macroscopic Chemistry Method. Centre or Hypersonics, The University o Queensland, Brisbane, 7, Australia ARC Centre or Functional Nanomaterials, The University o Queensland, Brisbane, 7, Australia

2 I. MODELLING OF CHEMICAL REACTIONS IN DSMC Inclusion o chemical reactions in the DSMC method has been considered by many authors and recently summarized by Boyd. Traditionally, these conventional models are collision based; when two particles are chosen to collide, a reaction probability is computed i there is suicient energy or the particles to reach the product state. Since there is limited inormation on state-speciic reaction probabilities or the reactions o interest in aerodynamics, probability unctions have generally been developed on a phenomenological basis and the models are calibrated to reproduce experimental reaction rates under conditions o thermal equilibrium. Validation o these models under conditions o thermal non-equilibrium has, with recent exception, generally consisted o comparisons with other DSMC simulations, rather than with experimental results -5. The standard chemistry method or DSMC is the total collision energy (TCE) method o Bird. The TCE method is ormulated or collision probabilities matching the variable hard sphere (VHS) total collision cross-section, and or equilibrium chemical reaction rate coeicients in the Arrhenius orm. A dierent collision cross-section, which might be required to yield a certain viscosity µ = µ ( T ), or a multi-temperature reaction rate model cannot be used. In response to these diiculties, the macroscopic chemistry method (MCM) was proposed by Lilley and Macrossan 7. In this method, chemical reactions are not processed on a collision pair basis. Instead a reaction rate is calculated rom the state inormation in each cell to determine the total number o reaction events required at any time step. The number o molecules or atoms in the cell is then adjusted to conorm with the required number o reaction events. The reaction rate in the MCM may be derived rom any experimental or theoretical source, and it may be based on any macroscopic parameters. In previous applications o MCM 7- the reaction rates have been calculated () rom the kinetic temperature in an Arrhenius rate equation, () rom the density-dependent reaction rates given by Gupta et al., and (3) rom the two-temperature model o Park. In the irst two examples, the reaction rate was taken as the value expected or conditions o thermal equilibrium. It was argued that chemically reacting collisions are inrequent events and, as such, local equilibrium conditions are established beore a signiicant number o chemical reactions occur. I the raction o collisions which result in a chemical reaction is not small, then it would be necessary to account or the non-equilibrium distribution o energy in collisions. In order to capture some o the inormation contained within the non-equilibrium distribution unctions predicted by the DSMC method and which is not included in thermally averaged experimental rate data, we propose a simple reaction rate adjustment. The adjustment is such that i the rate o suiciently energetic collisions is ound to dier rom the theoretically expected rate at the local equilibrium condition, the reaction rate is modiied in proportion to the dierence. II. CORRECTION FOR NON-EQUILIBIRUM RATES The problems or which the DSMC method is commonly applied involve predominately bimolecular chemical reactions. These elementary reactions require two-body collisions, a common example being the dissociation o a molecule through collisions with a second molecule or atom. I the reaction is given by: A + B products, then the rate o depletion o a reactant species can be expressed as: dn A = k NA NB [] dt

3 Here N i is the number density o species i, and k is a rate coeicient with units o a reaction volume per unit time. Following Vincenti and Kruger 3, the rate o reactant depletion can be expressed in the orm o: dn A = Z AB F S [] dt Here Z is the bimolecular collision rate or the species o interest, F is the raction o all such collision pairs which have energy above a threshold value and S is the raction o collision pairs satisying the energy criteria which actually react (the steric actor). This expression is convenient in that it is physically intuitive and can be used to derive an approximate theoretical expression or the reaction rate. The rate coeicient or the reaction can thus be expressed as: Z AB k = F S = Zc F S [3] NANB Here Zc is the collision constant, analogous to the rate constant, with units o collision volume per unit time. Each o the terms on the RHS may be interpreted as averaged quantities where the averaging is over an ensemble o collisions. Here we assume that the steric actor under thermal non-equilibrium conditions is equal to the steric actor at a nearby equilibrium condition. Hence, we propose that the reaction rate in conditions o thermal equilibrium or non-equilibrium may be approximated as Z AB F k = k [] Z AB F In this equation () denotes equilibrium conditions and the equation reduces to k = k at equilibrium. The actual non-equilibrium collision rate Z AB, and non-equilibrium raction o colliding particle pairs with energy greater than the threshold F, are monitored during the collision selection procedure. We could equally well consider the ratio o the rate o non-equilibrium to equilibrium high energy collisions. I, during the collision calculations or any cell, there are ewer high energy collisions than expected or equilibrium conditions, then the chemical reaction rate k given by Eq. is proportionally less than k. III. IMPLEMENTATION The above rate correction which accounts or the actual collision rate and high energy portions o the distribution may be implemented in the MCM DSMC routine with little computational cost. The implementation discussed here reers to a reaction rate model without biasing or a particular energy mode. Although such a model is known to be insuicient, given or example the inluence o coupled vibrational motion and dissociation, our purpose is to demonstrate the general procedure or correcting equilibrium rates or thermal non-equilibrium conditions. Corrections including the preerential treatment o speciic energy modes are a topic or later investigation. At the start o the simulation, reaction rates are computed using the equilibrium expressions. As the collisions are computed, the total number o collisions and the number with energy rom the speciied degrees o reedom in excess o the threshold energy are counted. The expected equilibrium collision rate and high energy raction are computed rom the time-averaged macroscopic cell parameters.

4 Although the method may be applied generally to any DSMC collision routine, expressions or the variable hard sphere (VHS) model with continuously distributed internal energy modes are given here: v v.5 σ re ggre Γ( v) m% Γ( ν + ς int, Eo / kt ) Zc = F = [5] π kt Γ( ν + ς int ) Here σ re and gre are reerence values o the collision cross-section and relative speed, ( ν +.5) is the temperature exponent in the power law viscosity relation, m% is the reduced mass, ζ int is the number o internal degrees o reedom, k is Boltzmann s constant and Eo is the threshold energy. The rate correction compares the actual collision rate and high energy raction with that expected at a nearby equilibrium condition. The act that the collision rate is proportional to the translational temperature implies that the translational temperature be used to evaluate the equilibrium condition. On the other hand, since the high energy raction F is proportional to all the energy available in collisions, it may well be more appropriate to evaluate the equilibrium values o Z c and F using an overall temperature deined by: 3Tt + ζ rtr + ζ vtv Tov = [] 3+ ζ r + ζ v However, since the actual collision rate and high energy raction are always accounted or by the method, the dierences due to the temperature chosen arise only in the orm o the departure o the actual steric actor rom the equilibrium one. In this study, Tov is used to evaluate the reaction rate. The computational time or evaluation o Z c and F is negligible since they are evaluated only when the macroscopic cell parameters are updated using the accumulated sample collected in each cell. IV. RESULTS Since we are considering the case where there is no biasing or a particular energy mode, we compare the non-equilibrium MCM rates with those predicted by the TCE model. The correct non-equilibrium chemical reaction rates are not known, but the proposed rate calculation is inspired by the physical modelling o the TCE method. Agreement o our model with the TCE model does not indicate that MCM yields correct reaction rates, but only that it can be used to obtain plausible non-equilibrium rates. Wadsworth and Wysong 5 argue that comparisons o the non-equilibrium rate coeicients are a poor test o DSMC chemistry models. The temperature dependence o the rate is primarily due to the exponential term which accounts or the threshold energy and the chemistry models essentially dier only in their interpretation o the steric actor. They say that it is more accurate to compare the energy distributions o those pairs selected or reaction. However, such detailed comparisons are neither necessary nor inormative in this case. The reason is that in the MCM, the selection procedure or reactant particles is completely decoupled rom the number o reaction events which occur; it is possible in MCM to adjust independently the number o reaction events and the energy distributions o post reaction particles through the use o dierent energy disposal models. Since we consider here only a rate correction, comparisons o the non-equilibrium rate coeicients predicted by the MCM are suicient to investigate the ability o the method to resolve chemical reaction rates in regions o thermal non-equilibrium.

5 We consider two methods o comparing reaction rates in thermal non-equilibrium. A zero-dimensional Monte-Carlo collision sampling program is used to investigate a special case o non-equilibrium which we reer to as intra-modal non-equilibrium. By this, we mean that even though the kinetic temperatures o each energy mode are dierent, the energy within each mode conorms to the Boltzmann distribution. The rate correction is aimed at accounting or the dierent distributions o energy in collision pairs owing to a non-equilibrium partitioning o energy between the modes. Although the total collision energy is constant and there is no preerence or energy in a particular mode, dierent reaction rates can arise due to varying collision rates, dierent amounts o energy absorbed in the motion o the centre o mass o collision pairs and the varying quantity o energy stored in the vibrational mode. In the second method, comparisons are made based on a highly non-equilibrium D low simulation. This is the most general comparison since chemical reactions may arise rom regions which contain any realistic energy distributions. A. MONTE CARLO SAMPLING A Monte-Carlo sampling procedure has been used to sample translational, rotational and vibrational energy or potential collision pairs rom the actual equilibrium distribution unctions where each distribution is considered at a dierent temperature. The simulation is zero-dimensional in that molecules are not moved. Potential collision pairs are selected using the acceptance-rejectance NTC collision procedure with the VHS collision model. The maximum value o the product o the collision cross-section and relative speed is set as a constant, large value. Although computationally ineicient, this method allowed or varying temperatures in the individual translational components. The TCE reaction probability is then computed and the number o successul reaction events is recorded. For the non-equilibrium MCM rate determination, the number o collisions and high energy raction are accumulated. The actual TCE rate coeicient, collision rate and high energy raction are computed rom: Nreact N N collisions energy k = ( σg), Z ( ), max c = σg F = [7] max N N N O pairs pairs collisions In this study the dissociation reaction o molecular oxygen is considered. Results are presented or dissociation through O + O collisions. Similar results were ound or dissociation through + Ocollisions. Species speciic collision cross-section data o Gupta et al. have been used to it the ollowing power law viscosity relations at a reerence temperaturetre = K: 5 O + O : µ r =.9 kg/m/s, υ=.9 5 O + O: µ r =.5 kg/m/s, υ=.5 5 O+ O: µ r =.7 kg/m/s, υ=. The reaction rate data o Park have been used or the equilibrium reaction rate: O + O :.5 k =. T / θ exp θ / T 3 m / s ( ) ( ).5 ( θ ) ( θ ) D D 5 3 O + O: k =. T / D exp D / T m / s Whereθ D = 595K is the dissociation temperature o oxygen. The characteristic vibrational temperature o oxygen or the harmonic oscillator model was taken as 5K.

6 The overall temperature and the raction o the total energy in the translational mode were varied between 5-K and.-. respectively. The temperatures o the rotational and vibrational modes were equal. These cases covered the range corresponding to the low which may be ound behind a shock wave where the translational temperature is above that o the rotational and vibrational temperatures, or in an expansion where the internal modes may be rozen at a temperature above that o the translational temperature. Separate adjustment o the components o translational temperature or a given total translation energy was ound to have negligible inluence and is not included here. The sampling continued until at least 3 reactions and at least collisions had been computed or each condition. In the TCE model, the probability o a reacting collision is a unction o the total collision energy o the colliding particles. However, the probability unction also includes a term corresponding to the average number o internal degrees o reedom. Since vibration is rarely ully excited, it might be expected that the number o vibrational degrees o reedom in the TCE probability unction should also vary. However, according to Haas, the variation o the average number o internal degrees o reedom over the chemically reacting region o a general low-ield is small and thus may be taken approximately as a constant value characteristic o the overall low-ield. In this study, the overall temperature has been used to deine the constant number o vibrational degrees o reedom in the TCE model. The expression or the raction o high-energy collisions F in equation [5], which is used in the rate equation [] corresponds to the continuously distributed harmonic oscillator model or the vibrational energy levels. However, the quantized version o the vibrational energy distribution was used in the collision calculations, since this is generally considered more accurate 5. Thus the reaction rate given by equation [] diers slightly rom the equilibrium rate, even or equilibrium conditions. As noted by Gimelshein et al. 5, the TCE chemistry model is also derived or continuously distributed vibrational levels and a similar small dierence between the theoretical equilibrium rate and the actual TCE rate at equilibrium is expected. The let plot in Fig. shows the overall reaction rates as computed by the TCE method and rate corrected MCM. The overall temperature is K and the translational and vibrational temperatures vary between -3K and -K respectively to give a range o non-equilibrium conditions, as expressed by the raction o the total energy in the translational modes. The right plot in Fig. shows how the collision rate and high energy raction vary or these dierent conditions. It can be seen that when the majority o the energy is in the translational modes, the reaction rate decreases signiicantly. This is because all o the internal energy contributes to the reaction whilst only the relative components o the translational energy contribute. The reaction rate decreases aster as the raction o energy in the translational mode approaches unity. Since it is the overall temperature which is constant rather than the total energy, the overall energy (and hence the rate) decreases due to the reducing number o vibrational degrees o reedom. A reduced collision rate causes the reaction rate to decrease when the majority o the energy is in the internal modes. The rate corrected method closely matches the TCE model. The small variation present is due to the variation o the distribution o high energy particles between the non-equilibrium and nearby equilibrium condition.

7 These tests o the rate correction are not particularly severe since we have assumed that equilibrium distributions exist in each mode. However, they do demonstrate that a simple correction can be used to account or the actual number o collision pairs with suicient energy to react. B. FLOW SIMULATION To test the non-equilibrium rate equation [] urther, we require a low-ield where signiicant chemical reactions occur in regions o thermal non-equilibrium. To this end, we consider a lat plate aligned perpendicularly to a high Mach number rareied low o diatomic oxygen. A two-dimensional simulation domain is used so that the plate is ininitely wide. The ree-stream conditions and computational parameters are given in Table. Although the density is very low, the stagnation temperature behind the shock is o the order o the dissociation temperature o oxygen. As a result, although the collision rate is low, the raction o potential reacting collisions is high. The vertical plate has a height o approximately nominal upstream mean ree path. Here the nominal mean ree path is deined as: λ nom = µ /( ρ kt / πm ) The plate is modelled as a diuse relector with complete accommodation in the normal and translational components o thermal energy and the rotational energy. The vibrational accommodation coeicient is zero and no surace chemical reactions are modelled. The downstream boundary is modelled as a vacuum boundary; the normal component o Mach number is supersonic everywhere along this boundary. Particles entering the domain through the top boundary have properties characteristic o the ree-stream. This is expected to introduce a small error due to the slightly higher temperatures along this boundary. A simulation was run with the top boundary location twice as ar rom the plate and the resulting low-ield dierences were negligible. The collision rate data given in the previous section is used with the VHS collision model. The serial application o the Borgnakke-Larsen (BL) energy exchange model is implemented with constant rotational and vibrational exchange probabilities o. and. respectively. The continuously distributed model o rotational energy and the discrete unbound harmonic oscillator model o vibrational energy are used. The low-ield is sampled at intervals o 5 t with the samples taken beore and ater the collision routine alternately as suggested by Rebrov and Skovorodko. The macroscopic parameters are calculated at intervals o t with the sample reset at these intervals during the approach to steady state. Steady state is assumed to have occurred at a non-dimensional time o 5 corresponding to the ree-stream low traversing the computational domain ive times. The sampling is continued until a non-dimensional time o is reached. The domain is divided into regions or the purpose o calculating the cell-structure. The cell size in each direction is varied using geometric progressions to concentrate cells near the plate. The macroscopic chemistry method is implemented according to Lilley where particles are selected at random or reaction and the dissociation energy is removed rom the translational mode only. Chemical reactions are considered at each time-step using the modiied Arrhenius rate data given in the previous section and the overall cell temperature. Reverse reactions were included in an initial simulation using the equilibrium constant data o Prabhu and Erickson 7. Since no recombination events were ound to occur, recombination reactions were subsequently excluded in all cases. For case, the rate correction

8 is implemented using the method described above assuming continuously distributed internal modes. The TCE method is implemented in the collision routine beore the serial BL procedure. The proportional energy exchange method is used or consistency with the MCM energy exchange method. The number o vibrational degrees o reedom in the TCE probability unction is set as a constant value characteristic o a low-ield at a temperature o 5K. Only.% o the total CPU time was devoted to computing the reaction rate correction. Simulations using the TCE, MCM and new rate corrected MCM (NEQ-MCM) chemistry models have been run or the ree-stream conditions list in Table. Contours o the atomic oxygen mole raction and translational temperature or the TCE and MCM methods are shown in Fig.. The TCE model has resulted in more dissociation with the chemically reacting region extending urther out rom the plate. Contours o the translational temperature show that the shock is marginally thicker in the TCE case and that the translational temperatures in the wake region behind the plate are similar. The translational, rotational and vibrational temperatures and the atomic oxygen mole raction along the stagnation streamline are shown in Figs. 3 and. Although the low-ields are qualitatively similar, some quantitative dierences are present. In the MCM case, the peak translational temperature within the shock is approximately5t lower, and behind the plate approximately5t higher than the TCE result. The rotational and vibrational temperatures are signiicantly lower in the TCE case. The MCM simulation predicts between 5 and % less dissociation in the region behind the plate and almost 5% less dissociation in the region directly in ront o the plate. Properties along the stagnation streamline or the rate corrected MCM simulation are shown in Figs. 3 and. The rate correction has reduced the dierences between the MCM and TCE solutions. The peak translational temperatures in the shock or the two methods agree and the dierence between the local maximums o the translational temperature behind the plate has been reduced. The non-equilibrium reaction rate has led to a generally increased dissociation rate in ront o and behind the plate. There is a marked dierence between the rotational and vibrational temperatures or TCE and MCM, or both orms o reaction rate. Coeicients o drag, shear and heat transer are shown or each case in Table. The variation o the total drag and shear orces on the plate is negligible over all the cases, including the non-reacting lowield. However the heat transer coeicient does vary considerably. Less heat transer occurs or reacting low-ields since energy has been absorbed in the dissociation processes within and behind the shock. The MCM predicts a heat transer coeicient % higher than the TCE method while the NEQ- MCM predicts the same heat transer coeicient as the TCE case. The overall raction o reacting collisions per particle or the TCE, MCM and NEQ-MCM cases are shown in Fig. 5. Within the shock, up to % (TCE), % (NEQMCM) and % (MCM) o the collisions are reacting. In the region behind the plate where the densities and hence the collision rates are much lower, less than % o the collisions result in a reaction or the TCE and NEQ-MCM cases. For the MCM case, proportionally more reactive collisions occur urther downstream. In the uncorrected MCM, reactions are less likely to occur until some energy has been transerred to the rotational and vibrational modes. V. DISCUSSION

9 It is useul to consider again how we estimate the non-equilibrium reaction rate in the MCM simulations. To calculate the required number o reaction events, we have used the actual collision rate and the actual number o particles with suicient energy to react, as ound in the simulations. We have assumed, in eect, a steric actor equal to the steric actor at equilibrium, as determined by the speciied equilibrium reaction rates. The equilibrium steric actor incorporates inormation regarding the orientation o particles, their relative vibrational requencies and phases, the amount o total energy they have above the threshold energy and other unknown actors. We have not speciied how the energy is distributed amongst the product particles or which particles we will choose or the reaction (the distribution o reactant particles). In the low-ield simulation considered, the raction o reacting collisions was not small and the predicted heat transer coeicient rom the modiied and unmodiied macroscopic chemistry methods varied signiicantly. Thus or this simulation, the use o equilibrium reaction rates in the MCM resulted in some probable inaccuracies. The simple rate correction produced results closer to the TCE method with the predicted heat transer to the plate being exactly the same. The discrepancies between the rotational and vibrational temperatures may be attributed to the mechanics o accounting or the dissociation energy in the MCM method, rather than the reaction rate speciically. In the MCM method, the dissociation energy was removed rom the translational mode only, whereas in the TCE method, energy may be removed rom all modes. It is expected that this may have led to the higher than expected internal temperatures in the MCM simulation cases. The MCM is not restricted to the speciic energy disposal methods used in this study. The simulation considered here has both a very high speed low where the kinetic energy is on the order o the dissociation energy and a set o reaction rates which are ast in comparison to other available experimental data. The reaction rate data used in this study is based on the two-temperature model o Park. Park states that the model is applicable up to translational temperatures o 5K, but only to a geometric temperature o K. In this study, the overall cell temperature was used when evaluating the reaction rate. Thus, the dissociation rates observed in this study were above those which are recommended by Park or these conditions. Lilley and Macrossan 9 have considered a similar low-ield and made comparisons between the macroscopic and total collision energy methods or the low o oxygen over a blunt-ended cylinder at Mach 5. In that study, the slower reaction rate data o Gupta et al. was used. The resulting dierences between the translational and rotational temperatures and the mole raction o atomic oxygen along the stagnation streamline between the MCM and TCE predictions were negligible. A small dierence in vibrational temperature was noted which may be due to the method o energy disposal previously discussed. It is thus apparent that the use o a very ast reaction rate in the current study is the reason or the signiicant departure o the reaction rate rom the equilibrium rate. VI. CONCLUSIONS The advantages o the macroscopic chemistry method are its lexibility and generality. It would be possible, or example, to use detailed vibrational state dependent reaction rates and an-harmonic oscillator vibrational levels. In previous work with MCM the reaction rate in any cell was taken as the equilibrium reaction rate, regardless o the degree o non-equilibrium in the low. This worked well because reacting collisions were rare events and thermal equilibrium was established beore any signiicant number o reaction events took place. It was not known how the method would perorm i a signiicant number o reaction events took place in conditions o non-equilibrium. We have addressed

10 that issue here by assuming reactions rates or oxygen dissociation much greater than those recommended in the literature. We have demonstrated a simple procedure or calculating the reaction rate in the macroscopic chemistry method in conditions o strong non-equilibrium. The procedure takes into account the nonequilibrium collision rate and the raction o high-energy collision pairs which arise rom the energy distribution in the cell and the collision cross-section (collision probability) used in the DSMC collision procedure. This same inormation is captured by the standard TCE method. The MCM is more general in that it is not tied to the VHS collision cross-section; it could be used with a cross-section or a more realistic inter-molecular potential which includes attractive as well as repulsive orces. It is plausible that the non-equilibrium high-energy collision rate arising rom such a realistic potential should be relected in the chemical reaction rate. The remaining actor which determines the collision rate is the steric actor, the raction o suiciently energetic collisions which result in a reaction event. In MCM we assume, in eect, that the steric actor is the same or equilibrium conditions as or non-equilibrium conditions. In TCE a reaction probability as a unction o collision energy is speciied and adjusted so that, when averaged over all high-energy collisions at equilibrium, it yields the equilibrium steric actor. In both methods we know the equilibrium steric actor rom the speciied equilibrium reaction rate. In the TCE method the average o the reaction probability or non-equilibrium conditions, the eective steric actor, varies rom its equilibrium value, which is why the MCM reaction rate diers slightly rom the TCE rate in nonequilibrium conditions. Considering the scarcity o experimental data, it is diicult to say whether the particular variation o the steric actor arising rom the TCE model is more realistic than the MCM approximation. We regard the Macroscopic Chemistry Method, with the reaction rate determined by equation [], as a reasonable approach to modelling chemical reactions in the DSMC method in both near equilibrium and non-equilibrium lows. REFERENCES. Boyd, I., Nonequilibrium chemistry modelling in rareied hypersonic lows, in Chemical Dynamics in Extreme Environments, edited by R. A Dressler, World Scientiic, Singapore,, pp Bose, D. and Candler, G., Kinetics o the N+O->NO+N reaction under thermodynamic nonequilibrium, J. Thermodynamics and Heat Transer, Vol., No., 99, pp Boyd, I., Bose, D. and Candler, G., Monte carlo modelling o nitric oxide ormation based on quasi-classical trajectory calculations, Physics o Fluids, Vol. 9, No., 99, pp.-7.. Wysong, I., Boyd, I., Direct Simulation Monte Carlo dissociation model evaluation: comparison to measured cross sections, J. Thermodynamics and Heat Transer, Vol., No.,, pp Wadsworth, W. and Wysong, I., Vibrational avouring eect in DSMC dissociation models, Physics o Fluids, Vol. 9, No., 997, pp Bird, G.A, Molecular gas dynamics and the direct simulation o gas lows, Clarendon Press, Oxord, Lilley, C. R. and Macrossan, M. N. A macroscopic chemistry method or the direct simulation o gas lows, Physics o Fluids A, Vol., No.,, pp Lilley, C.R, A macroscopic chemistry method or the direct simulation o non-equilibrium gas lows, PhD thesis, The University o Queensland, 5.

11 9. Lilley, C.R and Macrossan, M.N., Applying the Macroscopic Chemistry Method to Dissociating Oxygen, 5th International Symposium on Rareied Gas Dynamics, - July,, St. Petersburg, Russia, Proceedings, ed. M. S. Ivanov and A. K. Rebrov (Siberian Branch o the Russian Academy o Sciences, 7), pp Lilley, C.R. and Macrossan, M.N., Modeling Vibrational-Dissociation Coupling with the Macroscopic Chemistry Method, In th International Symposium on Rareied Gas Dynamics, - July,. AIP Conerence Proceedings, Vol. 7 (Ed Capetilli, M) pp Gupta, R., Yos. J., Thompson, R. and Lee, K. A review o reaction rates and thermodynamic and transport properties or an -species air model or chemical and thermal nonequilibrium calculations to 3K, NASA, Washington, 99, NASA reerence publication 3.. Park, C., Nonequilibrium Hypersonic Aerothermodynamics, Wiley, New York, Vincenti, W. and Kruger, C., Introduction to physical gas dynamics, Wiley, New York, Haas, B., Models o energy exchange mechanics applicable to a particle simulation o reactive low, J. Thermodynamics and Heat Transer, Vol., No., 99, pp Gimelshein, S., Gimelshein, N. Levin, D., Ivanov, M. and Wysong, I., On the use o chemical reaction rates with discrete internal energies in the direct simulation Monte Carlo method, Physics o Fluids, Vol., No. 7, pp A. K. Rebrov and P. A. Skovorodko, An Improved Sampling Procedure in DSMC Method, Proceedings o the th International Symposium on Rareied Gas Dynamics (ed. Ching Shen), Peking University Press, Beijing, China, 997, pp Prabhu, R. and Erickson, W., A rapid method or the computation o equilibrium chemical composition o air to 5K, NASA, Washington, 9, NASA reerence publication 79.

12 TABLES Free-stream conditions: Computational parameters Cases M :3 Plate height :.m : Non-reacting T : 3K Cells : 5 : Standard TCE - 3 3: Standard MCM ρ : kg / m Simulators : : Corrected MCM 9 λnom :.9m Sample size : 3 7 X O : t = 5 s Boundary conditions: Simulation details Diuse relection φr =. Tw = 3K φv =. a = a = a =. a =. n tan r v Table Simulation details C D C S C H C H normalized Case Non-reacting TCE MCM NEQ-MCM Table Plate orce and heat transer coeicients

13 FIGURES. NEQ-MCM TCE.3. Z/Z F/F..9 k /k..7 Ratio E t /E tot E t /E tot Figure Non-equilibrium rate comparisons at T ov = K. Ratio o equilibrium to non-equilibrium rate (let) or MCM and TCE. The MCM rate correction accounts or the non-equilibrium high energy raction and non-equilibrium collision rates shown in the right igure. Equilibrium conditions correspond to E / E.7. t tot 5 5 W/λ W/λ W/λ.. W/λ Figure Contours o mole raction o atomic oxygen (let) and translational temperature (right) rom MCM with k = k (top) and TCE (bottom) simulations.

14 .9. MCM NEQ-MCM TCE X o Figure 3 Mole raction o atomic oxygen along the stagnation streamline or MCM ( k = k ), NEQ- MCM ( k rom []) and TCE cases. A logarithmic scale is used. The let and right plots correspond to the regions let and right o the plate respectively. T t /T MCM NEQ-MCM TCE T r /T / MCM NEQ-MCM TCE T v /T MCM NEQ-MCM TCE X o MCM NEQ-MCM TCE Figure Properties along the stagnation streamline. translational temperature, rotational temperature, vibrational temperature and mole raction o atomic oxygen.

15 W/λ W/λ W/λ Figure 5 Fraction o reacting collisions per particle or TCE (top), MCM (middle) and NEQ-MCM (bottom) cases. The TCE and NEQ-MCM solutions show similar characteristics. The MCM solution shows a reaction incubation period where the internal temperatures increase such that the overall temperature is suiciently high or reactions to take place..