Internal energy excitation and dissociation of molecular nitrogen in a compressing flow

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1 Center for Turbulence Research Annual Research Brefs Internal energy exctaton and dssocaton of molecular ntrogen n a compressng flow By T. E. Magn, M. Panes, A. Bourdon, R. Jaffe AND D. Schwenke 1. Motvaton and objectve Predcton of the radatve heat-flux to the surface of a spacecraft enterng a planetary atmosphere strongly depends on the completeness and accuracy of the physcal model used to descrbe the non-equlbrum phenomena n the flow. Durng an atmospherc entry, the translatonal energy of the flud partcles drastcally rses through a shock. Dependng on the ntensty of the shock, dfferent physco-chemcal processes may take place, such as exctaton of the nternal energy modes, dssocaton of the molecules, onzaton of the atoms and molecules. These non-equlbrum phenomena are strongly coupled to each other. For re-entry veloctes >1 km/s, a sgnfcant porton of the heatng experenced by the heat sheld can be due to radaton and s hghly nfluenced by the shape of the nternal energy dstrbuton functon. Understandng thermo-chemcal non-equlbrum effects s also mportant for a correct nterpretaton of expermental measurements n flght and n ground wnd-tunnels. Concentraton of the gas speces and dstrbuton of ther nternal energy level populatons can be estmated by means of ether mult-temperature models (Park 199 or collsonal radatve (CR models (Laux 22; Bultel et al. 26; Magn et al. 26; Panes et al. 29. In mult-temperature models, the physco-chemcal propertes of the ar flow are obtaned by assumng that, for all the speces, the populaton of each nternal energy mode follows a Boltzmann dstrbuton at ts own temperature (T r rotatonal, T v vbratonal or T e electronc temperature, respectvely. These models have been developed based on expermental data obtaned n flght and also n hgh-enthalpy facltes representatve of specfc flght condtons, such as n arc-jet and shock-tube wnd-tunnels (Appleton et al The problem wth ths approach s that the models may contan many uncertantes that can be extremely dffcult to quantfy. Moreover, there s no detaled nformaton about the specfc state of the gas snce these data are hghly averaged (e.g., stagnaton pont heat-flux measurement. Park (26 has worked extensvely on mult-temperature models for ar and has also shown that the use of these models, even f very effcent from a computatonal pont of vew, can be justfed only when the departure from the Boltzmann populaton s small,.e., for low-velocty and hgh-pressure re-entry condtons. Collsonal radatve models take nto account all relevant collsonal and radatve mechansms between the nternal energy levels of the dfferent speces n the flow. They consttute a vald alternatve to the mult-temperature models snce they exhbt a wder range of applcablty. By ncreasng order of complexty and computatonal tme, three knds of CR models can be dstngushed for ar: electronc, vbratonal and rovbratonal. In electronc CR models, transtons between the electronc states are consdered and the rovbratonal levels of the molecules are populated accordng to Boltzmann dstrbutons Insttute for Computatonal Engneerng and Scences, The Unversty of Texas at Austn Laboratore EM2C UPR 288 CNRS, Ecole Centrale Pars, France

2 6 T. E. Magn et al. at temperatures T r and T v. In vbratonal CR models, transtons between the vbratonal states of the molecules are also consdered and only a rotatonal temperature T r s defned. Fnally, n rovbratonal CR models, no temperature s requred to descrbe the nternal energy. The qualty of the results obtaned wth a CR model reles manly on the accuracy of the rate coeffcents for elementary processes between energy levels. Dfferent theoretcal models have been developed n the lterature to determne elementary rate coeffcents: for nstance, the quasclasscal trajectory method usng a potental energy surface whch s a ft to ab nto electronc structure calculatons (Schwenke 199, the same method based on approxmate energy surfaces (Esposto et al. 26 and analytcal models such as the forced harmonc oscllator (Macheret & Adamovch 2. Recently, the computatonal chemsts at NASA Ames Research Center have embarked on the characterzaton of non-equlbrum ar chemstry from frst prncples (Chaban et al. 28; Jaffe et al. 28, 29. So far, the N 2 + N system has been studed to yeld rate coeffcents and cross-sectons for rovbratonal exctaton and dssocaton of molecular ntrogen n the ground electronc state. Frst prncple quantum chemstry calculatons are used to generate realstc nuclear nteracton potentals. The quasclasscal trajectory method s then used to yeld the fundamental data requred for a rgorous treatment of non-equlbrum chemstry. The present work s at the nterface between computatonal chemstry and computatonal flud dynamcs and ams at developng new models based on mcroscopc theory and applyng them to macroscopc scale. The database recently developed at NASA Ames Research Center was used to derve vbratonal dssocaton and exctaton rate coeffcents for molecular ntrogen (Bourdon et al. 28. In ths work, we propose to develop a 1D vbratonal CR model to smulate a shock n a ntrogen flow. Ths model, so far only collsonal, s the frst component of a larger model for ar that wll eventually nclude radaton. It s mportant to menton that mult-quantum jumps are taken nto account for the exctaton mechansm. The free stream condtons are carefully selected to stay n the valdty range of a smplfed mechansm for the N 2 + N system comprsng dssocaton and vbratonal-translatonal (VT relaxaton. Fnally, we compare the results obtaned by means of the vbratonal CR model to a mult-temperature model often used n the aerospace communty. 2. Physco-chemcal model 2.1. Energy levels The NASA Ames database (Schwenke 28 comprses 939 (v, J rovbratonal levels for the electronc ground-state of ntrogen, where ndex v stands for the vbratonal quantum number, and ndex J, the rotatonal quantum number. These levels can also be denoted by means of a global ndex. The relaton between the and (v, J notatons s expressed as follows = (v, J, v =,..., v max, J =,..., J max (v. Conversely, the relaton between the (v, J and notatons s gven by the relatons v = v(, J = J(, I BP, where I BP s the set of global ndces for the ntrogen energy levels. Most of these levels are truly bound,.e., ther energy s lower than the dssocaton energy relatve to the level (v =, J =, equal to 9.75 ev for the electronc ground-state of ntrogen, whle some of the energy levels are predssocated,.e., ther energy s hgher than the dssocaton

3 Exctaton and dssocaton of molecular ntrogen 61 energy relatve to the level (v =, J =. The bound energy levels are denoted by the set I B, and the predssocated energy levels, by the set I P. The set for all the levels s then gven by I BP = I B I P. The total wave functon for the ntrogen molecule must be symmetrc wth respect to exchangng the nucle (Bose-Ensten statstcs, and thus the degeneracy of the energy levels s gven by the expresson where the nuclear spn degeneracy s g = (2J( + 1g NS, I BP, g NS = { 6 : even J(, 3 : odd J(. The database for the N 2 + N system comprses more than 23 mllon reactons, for dssocaton of truly bound states and predssocated states, k Df N 2 ( + N N + N + N, I BP, k Db for predssocaton of the predssocated states, and for exctaton between all states, k P f N 2 ( N + N, I P, k E j k P b N 2 ( + N N 2 (j + N,, j I BP, j >. k E j The drect reacton rate coeffcents k Df, k P f and kj E, j > are obtaned based on the reacton cross-sectons for nne values of the gas translatonal temperature (T =75; 1,; 12,5; 15,; 2,; 25,; 3,; 4,; and 5, K. The reverse reacton rate coeffcents k Db, k P b and kj E, j < are computed based on mcroreversblty, only usng endothermc rate coeffcents, although some exothermc rates were also avalable: k Db k P b (T = k Df (T = k P f (T g Q t N 2 (T exp( (E 2EN k BT [g N Q t N (T ]2, I BP, (T g Q t N 2 (T exp( (E 2EN k BT [g N Q t N (T ]2, I P, j >, k E j(t = k E j(t g exp( (E Ej k BT g j,, j I BP, j >, where the energy of level s defned as E and the translatonal partton functons as ( 3/2 ( 3/2 Q t 2πkB m N2 T N 2 (T =, Q t 2πkB m N T N(T =, h 2 P where symbol k B stands for Boltzmann s constant, and h P, Planck s constant. The ntrogen atom degeneracy s gven by g N = 12 (nuclear and electronc spn. h 2 P

4 62 T. E. Magn et al Vbratonal collsonal radatve model A 1-D vbratonal CR model was developed to descrbe the energy relaxaton and dssocaton processes n a ntrogen flow. The 61 vbratonal energy levels for N 2 of the NASA Ames database are taken nto account n the CR model, as well as all the averaged vbratonal rate coeffcents for dssocaton and exctaton for the N 2 + N system. The energy for the rovbratonal level I BP s defned based on the vbratonal contrbuton and the relatve rotatonal contrbuton wth respect to the vbratonal energy level E = Ẽv + Ẽ(v, J, v =,, v max, J =,, J max (v, wth Ẽv = E (v,. The rotatonal energy level populatons are assumed to follow a Boltzmann dstrbuton at the translatonal temperature ( n = 1 ñ v Q v (T g Ẽ(v, J exp, I BP, k B T where the number densty, ñ v = J max(v J= n, and the rotatonal partton functon, Q v (T = J max(v J= g exp[ Ẽ(v, J/(k BT ], are ntroduced for the vbratonal energy levels v =,..., v max. The average vbratonal rate coeffcents for dssocaton and exctaton are gven by the expressons k v Df (T = 1 J max(v ( Ẽ(v, J g exp Q v (T k B T J= k vv E (T = 1 J max(v ( Ẽ(v, J g exp Q v (T k B T J= k Df (T, v =,... v max, Jmax(v J = k E j(t, v, v =,... v max, where j = j(v, J. Predssocaton s not accounted for. Reverse rate coeffcents are computed by means of the followng relatons k Db v (T (T = Qt k Df v N 2 (T Q v (T exp( (Ẽv 2EN k BT [g N Q t N (T ]2, v =,... v max, k v E v (T Qv k vv E (T = (T exp( (Ẽv Ẽv k BT, v, v Q =,... v max, v > v. v (T In secton 3, we wll choose sutable free stream condtons for whch ths smplfed mechansm s domnant, neglectng onzaton, VV relaxaton and electronc exctaton. Post-shock condtons are derved from the Rankne-Hugonot jump relatons assumng frozen gas composton and vbratonal energy modes. The rotatonal mode s assumed to be n equlbrum wth the translatonal mode. The downstream flowfeld s obtaned by solvng the followng steady Euler conservaton equatons of mass for the N atoms and

5 Exctaton and dssocaton of molecular ntrogen 63 for the vbratonal energy levels of N 2 molecule, global momentum and global energy d dx (n Nu = ω N, (2.1 d dx (ñ vu = ω v, v =,..., v max, (2.2 d dx (ρu2 + p =, (2.3 d (ρuh =. dx (2.4 The chemcal producton rates are gven by the expressons v max ω N = 2 k v Df (T v= ω v = k Df v (T v max v =v+1 [ [ ñ v n N k E vv (T [ ñ v n N Db k v (T k v Df (T (n N 3 Db k v (T k v Df (T (n N 3 ] ], v 1 v = ñ v n N k E v v (T k E vv (T ñv n N ] k E v v(t, [ ] ke vv (T k v E v (T ñvn N ñ v n N for v =,..., v max. The total enthalpy s gven by the expresson H = e nt +e t u2 +p/ρ, wth the translatonal energy e t = 3 2 (ñ N 2 + n N k B T/ρ, the mass densty, ρ = ñ N2 m N2 + n N m N, the number densty for N 2, ñ N2 = v max v= ñv, and the mxture pressure, p = 2 3 ρe t. vmax The nternal energy reads ρe nt = m N2 v= ñv[ẽv + Ẽrot v (T ] + m N n N E N, wth the rotatonal energy of level v gven by Ẽrot v (T = k B T 2 (ln Q v / T k B T. The conservatve system of equatons (2.1-(2.4 s transformed nto a system of ordnary dfferental equatons (Magn et al. 26. No addtonal conservaton equaton for the vbratonal energy s consdered, snce ths quantty s drectly computed from the vbratonal energy populaton obtaned by means of the CR model e vb (x = 1 v max ñ v (xẽv. (2.5 ñ N2 One can also defne a vbratonal temperature based on the relatve populaton among the frst excted vbratonal energy level and the ground state, as follows: v= Ẽ 1 T v = (, (2.6 ñ k B ln ñ 1 where Ẽ1 s the energy of the frst excted vbratonal energy level. The post-shock ntal populaton of vbratonal energy levels s assumed to follow a Boltzmann dstrbuton ñ v = Q ( v (T ñ N2 Q(T, T v exp Ẽv, v =,..., v max, k B T v where the total partton functon s gven by the expresson ( v max Ẽv Q(T, T v = exp Q v (T, k B T v v =

6 64 T. E. Magn et al T [K] 4 4, p [Pa] u [km/s] x N [ ] Table 1. Translatonal temperature, pressure, velocty and ntrogen atom mole fracton for the free stream LTE condtons (1, post-shock non-equlbrum condtons at the shock locaton (2 and post-shock LTE condtons (3. wth the vbratonal temperature T v equal to the freestream translatonal temperature. 3. Results The N 2, N free stream gas mxture s assumed to be n Local Thermodynamc Equlbrum (LTE at p=4 Pa pressure and T =4 K translatonal temperature (.e., there s 8.6% mole fracton of N atoms. The free stream temperature does not correspond to flght condtons. A hgh value of temperature was chosen to have enough ntrogen atoms n the flow, snce only the mechansm for the N + N 2 system s consdered n ths work; the free stream and post-shock condtons based on the jump relatons are revewed n Table 1, together wth the post-shock LTE condtons. Fgure 1 compares the evoluton of the rotatonal-translatonal temperature and vbratonal temperature as a functon of the dstance from the shock for the mult-temperature model of Park (T = T r, T v = T ele = T e for a 5-speces mxture (N, N 2, N +, N + 2 and e wth standard mechansm and for a 2-speces mxture (N and N 2 wth smplfed mechansm. The standard mechansm comprses all the reactons of dssocaton, onzaton and VT relaxaton frequently used n mult-temperature models (Park The speces electronc energy levels s accounted for when computng the flow enthalpy. The smplfed mechansm comprses N 2 + N dssocaton and Landau-Teller VT relaxaton, based on a Mllkan-Whte-Park relaxaton tme computed from the N 2 N nteracton only (Park Both mult-temperature models gve very close results, the smplfed mechansm s thus domnant for the selected free stream condtons when usng a mult-temperature model. In both cases, an overshoot of the vbratonal temperature and a slow relaxaton to equlbrum are notced. In the followng, we wll compare the results obtaned by means of the collsonal radatve model to those based on the mult-temperature model for the 2-speces mxture wth smplfed mechansm (Appleton et al Fgure 2 compares the evoluton of the pressure and velocty for a flud partcle as a functon of the Lagrangan tme, startng at t = wth the shock, for the mult-temperature model (2-speces mxture wth smplfed mechansm, Park reacton rate coeffcents and for the vbratonal collsonal radatve model presented n secton 2. We notce that, as expected, the level of accuracy to descrbe the ntrogen dssocaton has a neglgble nfluence on the pressure of the flow. A more sgnfcant nfluence s observed on the velocty feld. Table 2 gves the Lagrangan tme for a flud partcle, and the correspondng dstance from the shock, for the vbratonal CR model. Fgure 3 shows the rotatonal-translatonal temperature and vbratonal temperature as a functon of tme for the mult-temperature model and the vbratonal CR model. The defnton of the vbratonal temperature for the CR calculaton gven n Eq. (2.6 s justfed consderng that the thermodynamc state of the gas s mostly characterzed by the

7 Temperature [ K ] Exctaton and dssocaton of molecular ntrogen x1-2 4x1-2 6x1-2 8x1-2 1x1-2 Dstance from the shock [ m ] Fgure 1. Post-shock translatonal temperature (unbroken lnes and vbratonal temperature (dashed lnes for a flud partcle as a functon of the Lagrangan tme startng at t= s wth the shock, based on the free stream LTE condtons gven n Table 1. Mult-temperature model usng the Park (1993 rate coeffcents for a 5-speces mxture (N, N 2, N +, N + 2 and e wth standard mechansm (thck lnes and for a 2-speces mxture (N and N 2 wth smplfed mechansm (thn lnes. Lagrangan tme [s] Dstance from the shock [m] Table 2. Lagrangan tme for a flud partcle and correspondng dstance from the shock for the vbratonal CR model. populaton of the lowest energy levels, hghly populated. Although descrbed by dfferent dynamcs, the vbratonal relaxaton tmes for the two models, defned as the length of tme requred for thermalzaton among the vbratonal and rotatonal-translatonal energy modes, are very smlar. Thermalzaton s descrbed dfferently by the two models: whle the two-temperature model of Park predcts an overshoot for the vbratonal temperature and a slow relaxaton to equlbrum, the CR model allows for a chemstryvbraton couplng leadng to a monotonc relaxaton to equlbrum. Fgure 4 shows a slower dssocaton rate for the Park model than for the CR model at the early stages of dssocaton, whereas ths trend s nverted at later stages. Ths behavor can be explaned by a strong underpredcton of the dssocaton rate for large dfferences among the vbratonal and translatonal temperatures.

8 66 T. E. Magn et al Pressure [ Pa ] Velocty [ m s -1 ] x1-5 5.x x1-5 1x1-5 Tme [ s ] x1-5 5.x x1-5 1x1-5 Tme [ s ] Fgure 2. Post-shock pressure (left and velocty feld (rght for a flud partcle as a functon of the dstance from the shock, based on the free stream LTE condtons gven n Table 1. Vbratonal CR model (thck lne and mult-temperature model (thn lne for the 2-speces mxture wth smplfed mechansm. Temperature [ K ] x1-5 5.x x1-5 1x1-5 Tme [ s ] Temperature [ K ] Tme [ s ] Fgure 3. Post-shock translatonal temperature (unbroken lnes and vbratonal temperature (dashed lnes for a flud partcle as a functon of the Lagrangan tme startng at t= s wth the shock, based on the free stream LTE condtons gven n Table 1. Vbratonal CR model (thck lnes and mult-temperature model (thn lnes for the 2-speces mxture wth smplfed mechansm. Left: lnear tme scale; rght: logarthmc tme scale. Fgure 5 shows the post-shock populaton for the vbratonal energy levels of the ntrogen molecule n functon of ther vbratonal energy, computed by means of the vbratonal CR model. Ths fgure clearly ponts out devatons from a Boltzmann dstrbuton. As expected, the populatons of the hghest vbratonal levels ncrease frst, and, due to multquantum transtons, we notce the formaton of a plateau for hgh vbratonal states. The populaton densty of ths plateau ncreases wth tme up to t = 1 6 s whle the populatons of the frst vbratonal levels reman unchanged. For t > 1 6 s, a sgnfcant dssocaton occurs and the populaton of all vbratonal states decreases. Then, almost all the vbratonal energy levels, except those very close to the dssocaton energy, follow a Boltzmann dstrbuton at a vbratonal temperature that slowly converges toward 89 K.

9 Mole fractons [ - ] Exctaton and dssocaton of molecular ntrogen x1-5 5.x x1-5 1x1-5 Tme [ s ] Mole fractons [ - ] Tme [ s ] Fgure 4. Post-shock ntrogen atom mole fracton (unbroken lnes and ntrogen molecule mole fracton (dashed lnes for a flud partcle as a functon of the Lagrangan tme startng at t = s wth the shock, based on the free stream LTE condtons gven n Table 1. Vbratonal CR model (thck lnes and mult-temperature model (thn lnes for the 2-speces mxture wth smplfed mechansm. Left: lnear tme scale; rght: logarthmc tme scale. Populaton number densty n v [ m -3 ] x1-18.8x x x1-18 Vbratonal energy E v [ J ] Fgure 5. Post-shock populaton of the vbratonal energy levels of the ntrogen molecule as a functon of ther vbratonal energy for the Lagrangan tmes of a flud partcle, t =, 1 14, 1 13,..., 1 3, 1 2 s, obtaned by means of the vbratonal CR model based on the free stream LTE condtons gven n Table Future plans In ths work, we have developed a 1D vbratonal state-to-state model to descrbe the nternal energy relaxaton and dssocaton processes behnd a strong shockwave n a n-

10 68 T. E. Magn et al. 2.x1 12 de vb /dt [ J kg -1 s -1 ] 1.5x x1 12.5x x1 12 2x1 6 4x1 6 6x1 6 8x1 6 e vb [ J kg -1 ] Fgure 6. Post-shock vbratonal energy trajectory n the phase space (de vb /dt, e vb for a flud partcle, obtaned by means of the vbratonal CR model (thck lne and mult-temperature model (thn lne based on the free stream LTE condtons gven n Table 1. trogen flow. The 61 vbratonal energy levels for the ntrogen molecule of the NASA Ames database were taken nto account, as well as all the averaged elementary rate coeffcents for exctaton and dssocaton, assumng that, for each vbratonal level, the rotatonal energy level populatons follow a Boltzmann dstrbuton at the translatonal temperature. The results obtaned were compared to the classcal mult-temperature model proposed by Park and often used n the aerospace communty for non-equlbrum flow smulatons. We propose to use the CR model to determne a new macroscopc model of vbratonal energy relaxaton to be used n mult-temperature models, together wth the correspondng chemstry-vbraton couplng term. In a prelmnary step, one can assess the valdty of the Landau-Teller law wdely used n mult-temperature models for non-equlbrum flow smulatons. In Fgure 6, the post-shock vbratonal energy trajectory s shown n the phase space (de vb /dt, e vb for a flud partcle. Ths vbratonal energy s computed by means of Eq. (2.5 for the vbratonal CR model, whle t s computed by means of the vbratonal temperature based on the assumpton of a harmonc oscllator for the multtemperature model. The vbratonal dynamcs descrbed by means of the CR model s dfferent from the dynamcs descrbed by means of the classcal mult-temperature models. We also propose to develop a rovbratonal collsonal radatve model by relaxng the assumpton of the Boltzmann dstrbuton for the rotatonal energy levels. The energy levels wll be lumped nto bns as a functon of ther global nternal energy, ndependently of ther vbratonal and rotatonal contrbutons.

11 Acknowledgments Exctaton and dssocaton of molecular ntrogen 69 The authors have beneftted from helpful dscussons wth Dr. G. Chaban and Dr. W. Huo at NASA Ames Research Center, Dr. A. Brands at Stanford Unversty, and Prof. C. Laux at Ecole Centrale Pars. The authors would lke to thank Dr. N. Mansour and Dr. S. Yoon from NASA Ames Research Center. REFERENCES Appleton, J. P., Stenberg, M. & Lquornk, D. J Shock-tube study of ntrogen dssocaton usng vacuum-ultravolet lght absorpton. Journal of Chemcal Physcs Bourdon, A., Panes, M., Brands, A., Magn, T. E., Chaban, G., Huo, W., Jaffe, R. & Schwenke D. W. 28 Smulaton of flows n shock-tube facltes by means of a detaled chemcal mechansm for ntrogen exctaton and dssocaton, Proceedngs of the Summer Program 28, Center for Turbulence Research, Stanford Unversty, NASA Ames Research Center. Bultel, A., Cheron, B., Bourdon, A., Motapon, O. & Schneder, I. 26 Collsonal radatve model n ar for Earth re-entry problems. Physcs of Plasmas 13(4 11. Chaban, G., Jaffe, R., Schwenke, D. & Huo, W. 28 Dssocaton cross-sectons and rate coeffcents for ntrogen from accurate theoretcal calculatons. AIAA , 46th AIAA Aerospace Scences Meetng and Exhbt, Reno, Nevada. Esposto, F., Armense, I. & Captell, M. 26 N-N 2 state-to-state vbratonalrelaxaton and dssocaton rate coeffcents based on quasclasscal calculatons. Chemcal Physcs, 331(1 1. Jaffe, R., Schwenke, D., Chaban G. & Huo., W. 28 Vbratonal and rotatonal exctaton and relaxaton of ntrogen from accurate theoretcal calculatons. AIAA , 46th AIAA Aerospace Scences Meetng and Exhbt, Reno, Nevada. Jaffe, R., Schwenke, D. & Chaban G. 29 Theoretcal analyss of N 2 collsonal dssocaton and rotaton-vbraton energy transfer. AIAA , 47th AIAA Aerospace Scences Meetng and Exhbt, Orlando, Florda. Laux., C.O. 22 Radaton and non-equlbrum collsonal radatve models. VKI-LS 22-7, Physco-chemcal models for hgh enthalpy and plasma flows, Rhode-Sant- Genèse, Belgum. Macheret, S. O. AND Adamovch, I. V. 2 N-N 2 semclasscal Modelng of State- Specfc Dssocaton Rates n Datomc Gases. Chemcal Physcs, 113( Magn, T. E., Callault, L., Bourdon, A. & Laux, C. O. 26 Non-equlbrum radatve heat-flux modelng for the Huygens entry probe. Journal of Geophyscal Research - Planets, 111 E7S12. Panes, M., Magn, T., Bourdon, A., Bultel, A. & Chazot, O. 29 Analyss of the Fre II Flght experment by means of a collsonal radatve model, Journal of Thermophyscs and Heat Transfer, Park, C. 199 Non-equlbrum hypersonc aerothermodynamcs. Wley, New York. Park, C Revew of chemcal-knetc problems of future NASA msson, I: Earth entres. Journal of Thermophyscs and Heat Transfer, 7( Park, C. 26 Thermochemcal relaxaton n shock-tubes. Journal of Thermophyscs and Heat Transfer, 2(4 689.

12 7 T. E. Magn et al. Schwenke, D. 199 A theoretcal predcton of hydrogen molecule dssocatonrecombnaton rates ncludng an accurate treatment of nternal state nonequlbrum effects. Journal of Chemcal Physcs, Schwenke, D. 28 Dssocaton cross-sectons and rates for ntrogen. VKI LS 28, Non-equlbrum Gas Dynamcs, from Physcal Models to Hypersonc Flghts, Rhode- Sant-Genèse, Belgum.