THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DISSOCIATION-RECOMBINATION NON-EQUILIBRIUM IN THE LAMINAR HYPERSONIC BOUNDARY LAYER. George R. Inger A dissertation submitted in partial. fulfill.m.ent of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 May, 1960 IP-433

FOREWORD The author would like to express a sincere appreciation to the following faculty members for their assistance during his graduate studies at the University of Michigan: Dr. A, M. Kuethe, Dr. S, We Churchill, Dr, W, W. Willmarth, Dr, R, C, F, Bartels, Dr, R. V4 Churchill and Dr, J, E, Broadwell. A special note of gratitude is due Dr. T. C. Adamson, Jr., whose teaching, guidance and encouragement have all served as a welcome stimulus in the conception and execution of this thesis. Finally, the participation of Dr, J. L. York on the thesis committee is gratefully acknowledged,

TABLE OF CONTENTS Page FOREWARD <.o o o. o o o.. o... O. a O O O..... a a O. 0 4 O O Q ii Oiii LIST OF ILLUSTRATIONS. a,,e..,*.. o 0, O o 0 o, 4 0 a 00 o. o.. a o,. o 00 0 0 0 0 vi LIST OF SYMBOLS.O.,..Q,...o., o.......O. o........o....... o..o.oO viii I, INTRODUCTION.* a. a a o. 0. 0 a 0 a 0.a o 0a a a.. a. 0 a a 0 0 o 1 II, THE BASIC GOVERNING RELATIONS FOR CHEMICALLY REACTING LAMINAR BOUNDARY LAYER FLOW, a. 0...,.0 o o o 0 o a a o. o o O a 8 General Conservation Equations and Boundary Conditionsa,, 8 Specie Conservation,.......... o,, 0.0.,. *., o a O. oo 0 d o 8 Overall Mass Conservation Oa o,,,,, a,,. o. O O o 0 00 10 Momentum Conservation.... o a....,,, o. 0 o 4..O a0 o 0 a o o. a a a. a a 10 Energy Conservation. 0,0 e e * o a 0 o. * 0. av o, o 0 0 o 11 Thermal Equation of State a 0..*. 0 O 0 0.0. 0 o o 0 a 14 Theoretical Representation of a Reacting Gas Mixture.,0.a.. 15 Multicomponent Gas Diffusiona a, o 0o o o. o *o.... o 15 Four Component Mixture Relations,,00000~a00000o000.0 oOo0 19 Volumetric Dissociation-Recombination Rates o,, o o,,,,,, 20 Heterogeneous Reaction Rates 0,..4 o.. a....... o,. a o Q. 26 Chemically Reacting Boundary Layer Equations, ao.,oa0......o. 27 Boundary Layer Equations,,0 0^ o, ^..... o 0 00ao 0. o,, 28 Boundary Conditions,,.*.,a, *a*,4. *.a<0e a0000xo0 50 0 30 Some Special Features of the Equations ~ 0 oo o.0... 0 0 31 Characteristic Non-Equilibrium Parameters and Hypersonic,. Environment 0.,..,............ a0.,,,. o.. *o. a oa.. aa 35 Characteristic Homogeneous Reaction Parameter for the Boundary Layer*, a^*.~, a....I 0a o. o 0 00 o~ a o 38 Surface Reaction Parameter a.0 4 0 O a U0 4 0,.. a 0 * Oa 4 0. o ao e 46 III TRANSFORMATION TO THE SIMILARI.TY PLANE o o. o,. o....,., 50 Application of the Mangler-Stewartson-Blasius Transformation 50 iii

TABLE OF CONTENTS CONT D Page Exact Conditions for Similarity with Chemical NonEquilibrium.,, o0 lao o,o0 o aao o o...0aoooo...o.. ao.ao.ao.ao.a a. 56 Non-Equilibrium Stagnation Point Similarity Solution,.o.o o 60 IV, CHEMICALLY FROZEN PERTURBATION ANALYSIS o o. o o a a oo o o o. o o 63 The Frozen Perturbation Equations,...40....o..a oooo...... 64 Assumptions,, o o 0 0 eo.. 0.00.o B ao eo.o o o ao 4 o 64 Characteristic Non-Equilibrium Parameter Expansion..., 4 o 66 Reduction to Ordinary Differential Equations. ooo,,,,,,,,,o 72 Solution to the Frozen Perturbation Equations o.a.. oo..00 oo 79 Chemically Frozen (Zeroth Order) Solutions a,.o a. o o * 79 First Order Deviations,^Oa~o...000.0ooo 0 0.00 4 0 o 4 4 0a o o 94 Second Order Deviations. o oO..oo.....o, o o o o 126 Further Discussion of Results.. o. o..........ooooo04 o 0....0. 132 Application of First Order Theory to the Calculation of Local Non-Equilibrium Deviations in the Laminar Boundary Layer4o0o <,......a.....o00a000o0~e00ooo o.. 400o0oa o 132 The Error in the Use of Fick s Law,., o a o0o........0, 137 V, DEVIATIONS FROM THERMODYNAMIC EQUILIBRIUM oo00.o......, o...00 140 Some Features of Non-Equilibriijm Couette Flow o 0 o 4 ao o o oa 140 Couette Flow Equations,.00 0,,o 0< 000o<44oao000 oo044 0 140 The Thermodynamic Equilibrium State., o o 0 o o o o a.... 145 Small Deviations from Thermodynamic Equilibriumo 0 o o 0oo 147 Approximate Solution. o 4000. 0 0 0.o o 4 o. OO o 4 0 a O o o o o 150 Properties of the Approximate Solution......o o o. 0a 4. oo 154 Laminar Boundary Layer Flow.. ooO... O.....o o.oo o o o o o, 160 Deviations from Thermodynamic EquilibriumO. o o0 o a o 0o o 161 Approximate Solution........... o o o.....o o aa t o o 167 Further Discussion of Results..oooo0. oo0o0 oo0oo oa., oo. 173 iv

TABLE OF CONTENTS CONT'D Page VI, CONCLUSION...o. 0, 0......... 0 0 0.. o o a o o0 0.0 0. 0. o o0 4 175 Summary, 4...,,oo,4 e~.....44.....0,,oo....Ooo o.... oo...aooa 4O 175 General Conclusions a,,.... oo 0 o oo a a. a q 0. b o 176 Chemically Frozen Deviations.,.,. * a........... o o o4...4 o 179 Thermodynamic Equilibrium Perturbation,, o4 o,o. o o. 181 Limitations. o.4a.......o 4....*.... O 4 o.0.,,,4..o...... 4 o 44 182 APPENDICES A.o MULTICOMPONENT MIXTURE TRANSPORT COEFFICIENTS, o o,,,.., o o o, 184 B DIFFUSION COEFFICIENT RATIO ES MATES,.,,,,.,....186 C CATALYTIC SURFACE REACTIONS.,,..,, 40 a o. o oo o. o 0.. 188 Do REACTION RATE DERIVATIVES AT FROZEN FLOW,..O..,..O. o.,..O.. o 190 E ANALYTICAL COMPLEMENTARY INTEGRAL SOLUTIONS. O a o. o, o o o o... o. 191 F, APPROXIMATE CORRECTION METHOD FOR VARIABLE pp, EFFECTo,,,O,,,O,, 192 G, DERIVATION OF EQUILIBRIUM RELATION BETWEEN MOLECULAR AND ATOMIC SPECIES44,o...4 0 4., * o. 0 o. o. o.0 0 o o 4o 4 a4 195 H, REACTION RATE DEVIATIVES AT THERMODYNAMIC EQUILIBRIUM.., o,. 196 BIBLIOGRAPHY, a o <, 0 4 a a o 0 0 0 0 o4 4, q o a a o 40 a o 00 4 o 0 ~ o 197 V

LIST OF ILLUJSTRATIONS Figure Page 1 Illustration of the Reaction Rate Effect on the Boundary Layer Specie and Temperature Profiles NearFre Florozen Flo w < f........ *........ 34 2 Stagnation Point N'onmEquilibrium Parameter Versus Flight Altitude and Mach 4Nmber,............ o.o.... o......... 41 3 I:nviscid Flow Pressure Distribution for a Hypersonic Hemisphere-Cylinder o............................... 43 4 Varziation of Boundary Layer NonEEquilibrium Parameter Along a Hemisphere Cylind.er-....................... 45 5 Stagnation Point Surface Catalysis Parameter Versus Flight Altitude and Mach Nhmeer...................47 6 Variation of the Boundary Layer Non-Equilibrium Parameter Derivative Factor Q/e Along a HemisphereCylinder~....o o... o.... o.*...*..*.a*<.. o..*...o *......o* 75 7 Atomic Specie Distribution in the Chemically Frozen Laminar Boundary Layer8............................ 81 8 Molecular Nitrogen Distribution in the Chemically Frozen Lamar a L odary Layer^............ ^...... 83 9 Temperature Distribution in the Chemaically Frozen Laminar Boumdary Layerf,,.^.o o,.f 0 *.*... * o. 85 10 First Order Reaction Rate Distribution in a Catalytic Wall Boundary Layero....... 92 11 First Order Reaction Rate Distribution in a Non-Catalytic Wall Boundary Layero e..9......... o - 93 12 First Order Complementary Integral Function Zcl,.x 97 13 First Order Complementary Integral Function z.2...o^ 98 14 First Order Catalytic Wall Atomic Specie Gradient Deviation Versus Q/e and <................. *... 100 15 First Order Nopn-Catalytic Wall Atom Concentration Deviatixon Versus Q/E and................ I..,,, I. *, 101 vi

LIST OF ILLUSTRATIONS CONT D Figure Page 16 Freestream Dissociation Level Effect on the Catalytic Wall Deviations o.0.......O. o.... o.....oO o.r oo. o 102 17 Schmidt NTsmber Effect on the First Order Perturbations, 103 18 The Effect of Activation Energy Parameter on the First Order Perturbations o...........................,.>.. 105 19 JThe Effect of Viscous Dissipatio n on the First Order Pert;ubations *.....,..... O. o.... O o.. a O o.... 0. o * a 107 20 First Order Atomic Specie Deviation Distribution in a Catalytic Wall Boundary Layer**.................. 109 21 First Order Atomic Specie Deviation Distribution in a NonG-Catalytic Wall Boundary Layer....................... 110 22 Total Atomic Specie Deviations Versus the Non-EquilP ibritim Parameter,............... 0 o.............. 112 23 First Order Non-Catalytic Wall Temperature Gradient Deviation Versus Q/e and......c................*... 117 24 Wall Temperature Gradient Deviations Versus the NonEquilibrium Parameter....o....0........o....o......* 119 25 Local First Order Composition and Heat Transfer Deviations from Chemically Fr e FrozenFlow Along a Hypersonic Hemisphere-Cylinder. o * oo.........o oo.. o.. o.......... 0 136 26 Atoic Specie Deviation Profiles in a Couette Flow Near Ther ymc E quil riumfi............ 157 27 Illu.stration of Thermodynamic Equilibrium Deviations in the Highly Cooled Laminar Boundary Layero. o o o, 0.0 171 vii

LIST OF SYMBOLS a. Mass fraction of i-th specie a Total atomic specie mass fraction ( Z ai) ATOMS 63 ITnviscid flow velocity gradient parameter( de/ ) &6:Boundary layer or Couette flow channel thickness ~ Bomndary layer transformation parameter ( —/) Similarity coordinate for transformed boundary layer equations 5 ElBoundary layer non-equilibrium parameter Cc Boundary layer surface catalysis parameter Yi Catalytic efficiency with respect to i-th specie 7 Mean specific heat; ratio for a real gas X Mixture thermal conductivity j. ~ Mixture coefficient of Viscosity (A Recombination rate temperature exponent Boundary layer stream function $ ~Viscous energy dissipation rate per unit volume pi Mass density of i-th specie p Mixture density (p - pi) Transformed X - coordinate of boundary layer TW Shear stress 0 Non-dimensional boundary layer temperature variable (T/Te) AQ Temperature perturbation from thermodynamic equilibrium viii

GA Non-dimensional activation energy parameter (TA/Te) X Non-dimensional atomic Nitrogen. variable (a4/a4e) B Inviscid flow velocity gradient (due/dX) cPi Constant pressure specific heat of i-th specie per unit mass Cp "Frozen" mixture specific heat ( acpi ) C^> Non-dimensional specific heat parameter (~ cp ) C.Chapman-Rubensin parameter (Pi/PRIR) d Diameter D Total or substantial time derivative following a fluid particle Dt Dij Multicomponent diffusion coefficient ~ ~. ~Binary diffusion coefficient T Di Thermal diffusion coefficient DW, 3&J ~ ur E Reaction rate derivatives evaluated at chemically frozen conditions 2J, ^ XX 2 y. Reaction rate derivatives evaluated at thermodynamic equilibrium conditions EA Activation energy f Boundary layer stream function variable Fi Body force per unit mass acting.on i-th specie #2 Functions of the Equilibrium Reaction Rate Derivative g Non-dimensional stagnation enthalpy variable (hs/hse) Non-dimensional reaction rate function hi Enthalpy per unit mass of i-th specie h Enthalpy per unit mass of mixture ix

hI~) Heat of formation of i-th atomic specie at zero degrees absolute hs Stagnation or total enthalpy (h + u2/2) Iz, I% First order frozen perturbation analysis reaction rate integrals yfP~ 4,Second order frozen perturbation analysis reaction rate integrals Boltzmann constant KT Recombination rate constant R K Surface catalysis rate parameter for i-th specie KN Knudsen Number kR Recombination rate parameter kF Dissociation rate parameter K Equilibrium constant r.Net rate of i-th specie production by reaction per unit volume per unit mass L Characteristic length -Le Lewis number (pa122>/) m; Mass of i-th specie MO Flight Mach Number Mi Molecular weight of i-th specie M Molecular weight of mixture.ni Number density of i-th specie n Number of density of mixture Ni Number of i-th specie moles N Number of moles of mixture N Avogadrots number Pi Partial pressure of i-th specie p Hydrostatic pressure x

PR Prandtl number P Viscous stress tensor qWr Heat flux rate accepted at interior of a solid boundary $QW Gaseous heat flux at wall q Non-dimensional dissociation energy parameter for the boundary layer theory Q Non-equilibrium, parameter gradient factor for boundary layer /d./-X ) Q Net heat flux rate in. gas phase ro Local body surface radius in a transverse plane R Local radius of curvature of body in meridian plane Ri I-th atomic specie reaction rate function; first order perturba. tidn CPt II-th atomic specie reaction rate function; second order perturbation R Universal gas constant S Schmidt number (PR/Le) Si Net rate of i-th specie appearance per unit surface area Si. Net rate of catalytic produ.tion t Time T Absolute temperat"-ure TA Characteristic activation temperature (EA/Ro) Boundary layer total enthalpy perturbation UV Velocity components along x,y respectively V Mass average velocity of mixture Diffusion velocity of i-th specie xi

w Non-dimensional molecular Nitrogen variable (a3/%3e) Vl- Molecular Nitrogen perturbation from thermodynamic equilibriu xy Orthogonal space coordinates along a body z Non-dimensional atomic Nitrogen variable (a2/a2e) Z Atomic Oxygen perturbation from thermodynamic equilibrium iI yAtomic Nitrogen perturbation from thermodynamic equilibrium Subscripts A Denotes atom c Complementary integral CAT. Catalytic wall value e Conditions at outer edge of boundary layer EQo Denote thermodynamic equilibrium condition F Denotes chemically frozen condition G Gas i Denotes i-th chemical specie of the mixture I, II, ooH First, second, etc. order perturbations from frozen flow M Denotes molecule NON-CAT. Non-catalytic wall value p Particular integral R Recombination or reference value REF. Reference condition S Stagnation conditions W Denotes wall conditior xii

CHAPTER I INTRODUCTION The origin of real gas chemistry effects in aerodynamic problems, ignored in the chemically inert ideal gas model of air, lies in the possip bility that the kinetic energy of flight, converted into the various internal energy modes, may be great enough to excite such phenomena as molecular vibration, dissociation and ionization. At hyvpersonic speeds, either shock wave compression or viscous deceleration may provide the appropriate conversion mechanism (the former dominates in the blunt-nose region near the stagnation point, while the latter may become important downstream on the longer hypersonic bodies). The actual energy distribution between the various internal energy modes is described by the governing conservation equations and boundary conditions for the flow; real gas effects not only modify existing terms in these equations but generally require the addition of new equations and data as well. The most important new physical features which must be accounted for are: (a) variable molecular weight, as reflected in the thermal equation of state, (b) the presence of a multicomponent gas mixture; this introduces the need for considering the individual behavior of the various component species by accounting for diffusion processes and the proper definition of thermodynamic variables as averages over the component specie properties, (c) chemical reactions between the various species, such as homogeneous dissociation and recombination, and between the gas and flow boundary surfaces (heterogeneous wall catalysis), and (d) molecular transport coefficients; not only must new coefficients be considered for the diffusion processes, but the usual expressions for viscosity and heat

conductivity are modified in the presence of a multicomponent mixture0 The relative importance of mnany of these real gas features to the calculation of the important aerodynamic quantities, such as heat transfer, skin friction, pressure distribution and thermodynamic state profiles in the flow, depends on the comparative rates at which the various physical processes occur, There arises, for this reason, the particular possibility -of steady state chemical non-equilibrium in a dissociated gas flow, due to the "competition" between convection arid. diffusion and the local net volumetric reaction rates in the gas (a similar type of "competition" may exist between gaseous diffusion and heterogeneous surface reaction at a flow boundary). This paper is concerned with some aspects of such chemical non-equilibrium effects in the dissociated laminar hypersonic boundary layer, The potential importance of the influence of real gas effects on certain problems in hypersonic aerodynamics, notably heat transfer rates, has spurred a considerable amount of theoretical and experimental work in recent years, some of which is reviewed in the survey articles by Rosner(l) Griffith(2) and Adams(3) In particular, a number of important theoretical studies have been produced. Lees() in a notable papery defined many of the main theoretical features attending the real gas boundary layer flow problem and explored many useful analytical approximations, such as the use of the "local" similarity boundary layer solutions and the direct neglect of pressure gradient and viscous dissipation effects, which grossly simplify the calculation of heat transfer on a highly cooled hypersonic vehicle surface Further, approximate solutions for the two extremes of

thermodynamic equilibrium and chemically frozen flow heat transfer were given for a completely catalytic surface, assuming a Fick law for the diffusion processes. An experimental verification of the equilibrium heat transfer distribution calculations of Lees, particularly concerning (5) the local similarity arguments, has been given by Kemp, Rose and Detrao The work of Fay and Riddell (6 at Avco constitutes a more thorough treatment of the general case of dissociation-recombination non-equilibrium in the laminar boundary layer for either extreme of wall catalysis, under the conditions of an exact similarity solution possible at a symmetric body stagnation point. Assuming an effective binary mixture of air atoms and air molecules, using variable pp. and specific heat data and a particular (23) form of the recombination rate law due to Davidson( the variation of the heat transfer was found as a function of an appropriate non-equilibrium parameter, where ~ is defined as the ratio of a characteristic boundary layer convection time to a characteristic net reaction time. The results, (4) which approximately confirm those of Lees in the two extremes of thermodynamic equilibrium ( = co) and chemically frozen flow (I = 0), indicate that in the Lewis number range.8 < Le < 1.4, the effect of non-equilibrium on heat transfer is very small for the catalytic wall case) but is quite noticeable for a non-catalytic surface when 10'5 < r < 1. Two reports by Adamson, Nicholls and Sherman(718) also consider some of the features of non-equilibrium in the laminar hypersonic boundary layer. In reference 7, the multicomponent behavior of a three specie mixture, as it affects the calculation of diffusion fluxes and net reaction rates, was discussed, and an approximate account of deviations from the binary

Fick diffusion law given, Further, the destruction of a similarity type of solution due to the non-equilibrium reaction rates, particularly for the flat plate boundary layers was discussed, In reference 8, the significance of the characteristic'non-equilibritm parameter ~ is discussed, and.a method of analyzing the non-equilibrium deviations in the: flat plate laminar boundary layer in terms of a series involving increasing powers of ~ was outlined, Using the recombination rate law proposed by (9) Hirschfelder ( some of the details of the zeroth, first and second order deviation equations were worked out, but no solutions given, A report by Scala(l) consid.ers the solution of the chemically frozen stagnation point boundary layer equations for a binary mixture in the presence of an arbitrary degree of surface catalysis and thermal diffusion in the gasp The results indicate the effect of catalytic efficiency on heat transfer and thermodynamic state profiles, and show the effect of thermal diffusion on heat transfer to be very small for a. cooled wall. A similar but more comprehensive analysis has been given by Goulard(ll), excluding the thermal diffusion. using the highly cooled wall approximations suggested (4) by Lees ) The role of the surface catalysis is very clearly displayed, entering the problem as it does through the specie boundary condition at the ~wall, in terms of an appropriate wall catalysis parameter c analogous to the non-equilibrium parameter ~, The influence of surface material on catalytic efficiency is discussed, and the restriction on similarity type of solutions due to the presence of an arbitrary surface (12,13) reaction rate is considered. Several articles by Rosner( 3) have also dealt with gas flow over a catalytically reacting surface: in particular,

reference 13 stresses the physical significance of the parameter Ad as a ratio of characteristic diffusion and surface reaction times, and presents approximate local solutions for heat transfer in the presence of a general surface catalysis reaction rate. Finally, the subject of real gas boundary layer flow and heat transfer has been notably unified and extended in a (l4) recent and very-comprehensive paper by Lees, which considers both laminar and turbulent reacting boundary layer flow in the presence of mass transfer at the wall, An impressive amount of ground is covered in this paper; concerning the laminar boundary layer problem, the important conclusions can be briefly summarized as follows, First, it was found that if one assumes a Fick law for the diffusion of each specie in a multicomponent gas mixture, Lewis and Prandtl number equal to one, and a known gas composition at the surface, then the total enthalpy distribution, the heat transfer, and the distribution of each of the total chemical element mass fractions are explicitly independent of reaction in the gas, regardless of the viscosity law, The consequent similarity of the total enthalpy and various total chemical element profiles leads to a greatly simplified solution for the heat transfer in the presence of mass transfer at the wall, The Fick law approximation is supported by the results of an investigation into multicomponent mixture diffusion by Knuth () Second, the local similarity approximation for a reacting boundary layer was discussed., it was shown that non-equilibrium in the gas would restrict similarity solutions to the stagnation point whenever any of the conditions (a) Fick diffusion law, (b) PR = Le = 1, or (c) known surface composition are * Each total chemical element mass fraction must necessarily be independent'of chemical reaction, either in the gas or on the surface,

noticeably violated., Third, a survey -of various flat plate and. stagnation point boundary layer solutions, in which detailed account of the variation of transport properties across the layer has been taken, leads to the conclusion that emperical correction of constant property approximate solutions on the basis of these particular results is a satisfactory procedure provided that species having physical properties radically different from those in the main flow are not injected or ablated into the boundary layer, Fourth, the usually neglected pressure diffusion mass flux and Von Karman diffusion stress component terms were subjected to a boundary layer order ~of magnitude computation, and shown to be negligable to the same order of approximation as the boundary layer equations themselves, As a result of these and many other efforts, it appears that many of the major theoretical problems involving real gas effects in boundary layer flows have been solved to good approximation or otherwise shown to be unimportanto However, it is felt that a closer investigation of the dissociation-recombination non-eqyilibrium effects in the laminar hypersonic boundaay layer, particularly in the presence of non-catalytic walls for which these effects appear quite pronounced, is warranted for the purpose of showing more clearly how deviations from equilibrium occur *and how they are influenced by the various aerodynamic and chemical data relevant to the problem, Therefore, this thesis is addressed to a theoretical investigation of deviations from both the chemically frozen and thermodynamic equilibrium extremes, in the absence of surface mass transfer, to show the effect of dissociation-recombination on heat transfer and thermodynamic state profiles in the highly cooled boundary layer, The role of

-7activation energy, recombination rate temperature dependence, and variable'free.Stream conditions causing local non-similarity effects due to the chemical reaction rate terms are considered. The dissociated boundary layer gas is represented as a four component mixture of two atomic and two molecular species, which is a good. approximation to the real gas behavior of air when the presence of Nitric Oxide can be neglected.

CHAPTER II TE BASIC GOVERNING REILATIONS FOR CHEMICALLY REACTING LAMINAR BOUNDARY LAYER FLOW General Conservation Equations and Boundary Conditions The basic governing relations represent the laws of mass, chemical specie, momentum and energy conservation in a flowing multicomponent reacting gas mixture. We will assume a radiation-free cntinuum. flow in the following statements of these laws, Specie Conservation If we denote the difference between the total velocity of the i-th specie component and the mass average velocity of the whole mixture as the diffusion velopity 4., then-the statement of conservation of i.-th specie mass is /0^-^*t ^..^ ~/?.~ 76 @; qf(21) accompanied by the subsidiary relations o<. = 0'(2.2) S,m, =0 (2*3) which express the fact that diffusion and net chemical reaction cannot alter the total mixture mass, Since f-c;<ffl i/y ~ 0/, (2 4) one needs only N1l of such specie conservation equations to completely describe an N component mixture. ~~8M

According to Hirschfelder, et al (1), the i-th specie diffusion mass flux is given by _ < /_ =(20 5) where the three terms under the summation represent ordinary mass, pressure and selective body force diffusion fluxes, respectively, and the last term is the thermal diffusion flux,,If is the multicomponent diffusion coefficient between species i, j in the presence of the other N-i-j species, and. is the thermal diffusion coefficient of the i-th specie. -DI is not equal to the binary diffusion coefficient o6fj for a mixture of three or more components, since the presence of other species will alter the diffusion between any given two. The relationship between AjL, and the binary diffusion coefficients in general is given in Appendix Ao At any boundary between the gas and some other phase (such as a solid body surface), a specie boundary condition in the form of a conservation statement across the interface is required, Equating the net flux of i-th specie mass away from the surface due to diffusion and convection to the net rate of surface specie production per unit area, Si, we have ~./Yf- + ~ ) N =A (2.6) where N is the interface unit normal (positive outward), In the absence of surface mass transfer, Si would equal the net rate of catalytic reaction and 2 Si = 0 (In the presence of a mass source at the interface, however, Z Si would equal the net mass flow rate per unit area). 1

10Overall Mass Conservation (Continuity) A summation of Equation (2.1) over all component species, using Equations (2.2) and (2o3), produces the mass conservation equation for the ehtire mixture: A C7+ 7 /' = 0O. (2.7) The same procedure applied to the boundary condition (2.6) gives j.(/~c^ =B /P - * (2.8) Momentum Conservation If we denote by Fi the i-th specie body force per unit mass and by Pi the symmetric viscous stress tensor acting in the X direction a surface whose normal lies in the I direction, then the momentum conservation equation can be expressed in the form,/O~14 X9- /,) dI ^ d +/O(/ (2.9) The relationship between viscous stress and the rates of strain and dilitation of the flow for a continuum is 104 j _ 6 r __) (2.10) where p is the coefficient of viscosity and r =- 2p/3 assuming the average normal stress on a fluid element is the negative of the local hydrostatic pressure and independent of the rate of change of density in the flow. The viscosity coefficient in a gas mixture is.a certain weighted average of the component viscosities (see Appendix A).

In a continuum flow, the mass average flow velocity and pressure in the gas adjacent to a boundary are equal to the corresponding values on the surface of the interface. Energy Conservation This statement of the first law of thermodynamics will be first given in terms of the mixture enthalpy h per unit mass, defined as the following mass average over the component enthalpies: T /h = i j<, -E A]n, (2.11) (o) where h is the heat of formation of the i-th specie at zero degrees (o) absolute(h, will be taken positive for atomic species and zero for the molecular species). The energy conservation statement is then~ oft+ A f Hi. 7pi S C mi- Fe (2.12) where f Jtd x d e i (2.13) is the viscous energy dissipation rate per unit volume, and Q is the net heat flux per unit area through the gas. The effect of chemical reaction in the gas is implicit in Equation (2.12) by virtue of the above definition of h. The heat flux vector is related. to the state,of the gas, (16) according to Hirschfelder et al, as follows o = ->7T tP. ^ - - ). (2.14)

^12^ The three terms on the right side are the heat conduction, mass diffusion heat flux, and thermal diffusion heat flux, respectively. The coefficient of thermal conductivity X is an appropriately defined.average of individual specie values in a multicomponent mixture; however, it is convenient to replace X by the Prandtl Number.-* - / /A' =>' ~ / A. (2.15) It has been shown (8) that the Prandtl Number is, for practical purposes, fairly constant over a wide range-of temperatures for air; thus as long as we exclude mass injection of gases with radically different physical properties, we can express X in terms of: and c with known PR. p At an interface between the gas and (say) a solid boundary, one must prescribe an energy balance including any possible heat of phase change (if this statement is in terms of the enthalpy, surface reaction effects will be implicit in that variable). Several recent articles have considered the formulation of this heat balance( 4'9); in general it reads %2 _ c - (/P 7W 4 -9 4 ) (2.16) where X is the net rate of heat input per unit area into the boundary material depths, QW is the value of Equation (2.14) at the interface, Sj is the mass rate of disappearance of the j-th specie boundary phase constituent of enthalpy hj, and hG is the gas mixture enthalpy, The supposition of a continuum inipli-es equality of gas and boundary temperatures at the interface. In conjunction with the use of Equation (2.16), one may cp is the so-called "frozen" mixture specific heat, P

-135 pursue two courses: either prescribe the interface temperature and subsequently calculate qW from the solutions to the governing equations, or prescribe qu and calculate the surface temperature that the resulting flow solution assumes. The latter corresponds to the recovery temperature or thermometer problem when qW is set equal to zero. Several alternative forms of the energy Equation (2ol2) prove useful in certain applications, namely the total enthalpy form and the "temperature" form in which the net gas phase reaction rates appear explicitly. Regarding the latter, the definition of h permits one to write S -h= Xg A' + As afar (2.17) Further, obgt may be replaced by the diffusion and chemical reaction terms from the specie conservation equation to produce S Ad~3 =^ 4h6L-^r v'to) (2.18) which, when inserted into Equation (2o17) and the result then used in the energy equation, eliminates the enthalpy in favor of the temperature and exhibits the reaction rates explicitly. Similarly, the use of the relation vh = E he ov a /c+ vr (2.19) permits the gas heat flux to be rewritten in terms of the enthalpy as Q - =/ ~-LL j/c L 52 - ) (2.20) ^^ ^-^- Lr

Thermal Equation of State If we assume each component specie. to behave as a perfect gas" A L A/n r ^ r * (2.21) so application of Daltons Law gives / -= 7 = /- 7,/'. T (2.22) Since,.4 = /'- = Az< A then'/ /3 / ( 2(2.23) =,t4, t _= /W M < /V where 44M = > /' /~, (2.24) is the molecular weight of the mixture, defined as the appropriate molar average of the component molecular weights, Consequently. we have the thermal equation of state (2.22) as,b= /~(@ )T -. (2.25) The volumetric fraction is related to the mass fraction of a given specie as follows: (2.26) The inverse relation of the mass fraction to the mole fraction is ~ = (c/Al)(,/~t), (2.27) * ni/N. R-/k - Ng' Avogadros Number

Theoretical Representation of a Reacting Gas Mixture Multicomponent Gas Diffusion The mathematical description of multicomponent diffusion is gen, erally quite complicatedj accordingly, one must employ suitable approximations which adequately simplify the analysis without omitting the significant real gas effects. Several schemes have been suggested to meet this need: (1) The use of the binary Fick diffusion law for each specie in a multicomponent mixture of three or more species. This approximation possesses the two-fold advantage of the inherent simplicity of the binary form and an applicibility to any number of components for all temperatures. (2) The "quasi-two component" approximation scheme described by Adamson(7), which attempts to account for deviations fromFick's law in formulating the diffusion fluxesg originally developed for a three component mixture, this approach may readily be extended to four or more Sp/6cies, Before taking up the four component case, consider first the multicomponent diffusion properties for a three component mixture of molecular oxygen and nitrogen and atomic oxygen (species 1, 3 and 2 respectively), for which we make the reasonable approximations ml = m3 2m and M1 = M3. =2M2. Using the relations between volume and mass fractions for this case, /r^ = 2ie/ 7) the general three component diffusion coefficient expression yields the following values

J^~/ = /, (2- )",,- (2'30) DY /- (2 2.3/^2 - 06'-e- (2.31)!92 2./z / The corresponding diffusion fluxes for the atomic and second molecular species ignoring pressure, body force and thermal diffusion) are then obAnined u from Equation (2.5) as follows: (/.2 8)= -(2.29). an2d t3. -(V43)2 (2i32) _ ~= -7)2, =] 2 rd3rX+~I:~= -:2p(/,f- v~,13@/ 2- 93)-S 2 0~j(2,33 Now regardless of specithe diffusion coefficients /a2 are gdl should be equal when mi a m3 and the collision cross-sections are the same, and this is verified by the diffusion coefficient estimates given in Appendix B. An immediate consequence of the single assumption oC%2 =C from Equations (2.28), (2.29) and (2-32), is that -2 =r S/ 2 -2D and so that the atomic specie diffusion obeys Fick's law regardless of the

-.17value for 0/oea Further, the assump-tion 4/'2 =?3 simplifies Equations (2~30) and. (2 31) to 2,...- ^ /,+' t / 4: /2X2fi ~8 -/) Z)3//Ao8 = ~J^~yJ/2-3 _(2.35) 2211/7L Pi t2 jl+^ /2 2/ which in turn permits the third, specie diffusion flux to be written as follows: /~~N^~~~ t- -/~^ -^ y^2^2S~5- -(2q36) Equation (2,.36) shows that the third specie diffusion will also obey Fick's law when either a13 =-oC or when a3/(lH2) is a constanto However, the former condition, while giving a Fick diffusion flux, does not yield -32? or 3 equal to o z (that is, the assumption D,o, does not imply Fick diffusion for all the species present). On the other hand, the assumption O/z =- 1/2, which appears to be nearer the actual value (Appendix B), gives Z3j2 = Cz and / —~ 3 but does not give Fick diffusion for a3 Regarding the second condition ao3/(12a) constant, it can be shown that this relation is exact in the -case of a completely noncatalytic wall in frozen flow,, The combination of this fact with the fact that 1/2-` P/~/ 2 1 leads to the conclusion that the use of Ficks law for 03, which implies neglect of the second term in brackets in Equation (2D36), may not introduce any significant error in an analysis of near frozen non-equilibrium behavior, particularly if a3 itself has a weak effect on the calculations. Further discussio n on this point is given later.

-18 The quasi-two component scheme() for a three component mixture also employs the assumption 4a. -cd3 and hence also yields Fick diffusion for a2 regardless of /3Y// However, this scheme differs from the foregoing in that it assumes e /=0, which consequently gives // - 0, D)32o^r = ^/c- ~2)J and r~G t = /)VC/2, (2237) Now when the Fick law diffusion for a3 is assumed (P/- X=-/< Fr9 ), Equation (2.36) shows that the resulting a3 diffusion flux error is tC3 - Pi33 ~-) )Eoe - T4 2 1 -/>J -7/3 S (2.38) whereas the use of the quasi-two component diffusion approximation above gives an error of PU f -/v tsaus)-rvo = ^ // }k v< 3- (2 3M) Hence the quasi-two component approximation is exact (and reduces to the Fick result) when C3/(1-a2) = constant0 Equations (2,38) and (2,39) show, assuming /3/O/= -f for the sake of comparison, that the a3 diffusion flux error for the quasi^two component approximation is always opposite in sign to and greater than (by a factor 1 + a2/l<.a) the Fick law error, The Fick law therefore always has less absplute errorj, and the simpler mathematical consequences of vsimng the Fick law would there., fore definitely favor its use, The use of the quasi-two component scheme would be of interest, however, in that it represents the upper limit on the a3 diffusion flux and so could be compared with the Fick

results (representing the lower limit), in order to "bracket" the actual value Regarding a mixture of four,components, one would suspect on the basis of the foregoing discussion that each atomic specie diffusion flux obeys Fick's law, regardless of the individual molecular species behavior, as long as all the atomic specie diffusion coefficients are the same. Four component mixture calculations based on the general diffusion coefficient in Appendix A confirm this to be true.. If all the specie diffusion coefficients are assumed equal, the Fick law is valid for every specie, as shown for any number of components by Knuth (15) The Fick law approximation in conjunction with the four component mixture representation described in the next paragraph will be used in all the subsequent boundary layer theory given in this paper. Four Component Mixture Relations Consider a four component air mixture (neglecting NO)J with al for 2, 2 for O, 23 for N2 andI G for N, The relation between the volume and mass fractions is obtained from Equation (2 26) as follows: Vfig_ / / =( 3/9(o4/)A^j 1 (2.40o) 2/MA -ig-(aX2Ji/w=tS;/ZJ~/q/ iW J where 4 ll....-...... (2.41)

.20O The reasonable approximations ml m3 2 2m =.2mn are seen to greatly simplify these relations, since the entire bracketed term in Equation (2o41) vanishes. The mixture molecular weight, assuming Mi/Mj mi/mj, becomes -/ M//M, = (/+i2+L e). (2042) The assumption of Fick's law} /. = - /o4 (7c -'=/) gives the diffusion heat flux term as ~'-,-)-_ )AI (2,43) and the total heat flux becomes -= - /y^^4+z-/A2-^r<^-^^^^ih~v~gg (2 44) where Le pS;^/X is the Lewis number. As shown by Equation (2o45), the case Le 1 allows the heat flux to be expressed entirely by-the enthalpy gradient%'The additional approximations c1 cp3 and cp= cp4, with (o) (o) (O) (O. h43bi and h hfa > 0, give h3 hl (but h4 h2) and therefore drop out the explicit contribution of a3 in Equations (2o43) to (2.45). Volumetric Dissociation4Recombination Rates The general real gas flow conservation equations require a specification of the net chemical production rates of the various species per unit volume of gas; for the air mixture under consideration, this

-21introduces the dissociation-recombination chemistryo Consider the dissociation-recombination reaction / /x Z + / (2,k46) KK in the presence of an inert specie Nx, NM and NA being the molecular and atomic species, respectively; this reaction is of the form encountered for the present four specie air mixture. The net reaction rates are(20) M4/1 = K f xK - 4 <% 1 -= 3-N,[^ N~ 7 \ (2.47) d ^/=I-i ( A/2) J where the mass rate of formation is found from. the molar rate by twix Mx e @ (2.48) The equilibrium prasameter k., kf/kR:for this reaction is of the form (9)0 Npc -. -^-rJ (2,49) where p is in atmospheres,. is a constant, and TA EA/Ro is a characteristic activation energy temperature. Equatims'(2,4.7) show that while the inert third specie Nx does not affect the equilibrium composition of the,NA d N gas (dt d.t d 0), it does play a role in any chemical non-equilibrium process. The value of the recombination rate parameter kR is established by detailed. theoretical analyses of the ternary collision processes (supported by extensive experimental study)o It appears from current work with Oxygen and Nitrogen that kR has the general form XR = K T eT (2.50) with kR and CD of the ord.er 1015 and -1, respectively, although widely differing values'have been proposed by various investigators~ ( )

.22a Table I presents typical oxygen recombination rate data to illustrate the present day uncertainty. It is therefore very desirable to assess the sensitivity of the results of real gas analyses involving the reaction rates to the parameters kR and ao TABLE I RECOMBINATION RATE DATA FOR OXYGEN Source (cm6 mole2-sec,) Value at 40000K Davidson(23) 5 x 1014 (T/3000K)^1 ~5 1.05 x 1013 Logan(21) 3 x 1015 (T/3000K)5 1010 x 1016 Hirschfelder() 1.2 x 1016 (T/300~K)25 780 x 1018 The recombination-dissociation rates above, if applied to oxygen and nitrogen in air, would imply the assumption of no direct chemical interreaction between the two and-constitutes a neglect of Nitric Oxide formation, It appears that the nitric oxide chemistry is not completely understood at present; an examination of current literature on the subject,, however, indicates the following situationo (a) Several investigations (21,22,23) into NO reactions 2 ) apparently produce favorable agreement with experiment when the chemical interreaction between the 02 4 20 and N2 42N processes is neglected, (b) Neglect of the coupling reactions due to NO, however, does not necessarily mean that NO alone, acting as a fifth specie, may not be as important as the oxygen and nitrogen dissociationehrecombinat tion rates in determining the non-equilibrium behavior. Some recent estimates by Logan (21) imply three main facts that suggest that NO may

play a very minor role in the present investigation0 First, the concentration of NO in a high temperature air mixture at the edge'of the boundary layer for low pressures (high altitudes) is very small and is noticeable only in a temperature range of approximately 3000-60000~K for example, the maximum fractional concentration is about 4% at 5000~K for p - 100 lb/ft2o As long as the recombination constant and temperature dependence for NO are not very much greater than those for 0 and N, this would imply that the recombination rate of NO (which is proportional to the square of the concentration) is much less significant to a non-equilibrium analysis of the boundary layer than the 0 and N recombination rates. Second, the corresponding maximum amount of total internal energy possessed by the NO is 5%,, which indicates that the neglect of NO would have a very small effect on the non-equilibrium heat transfero Third, it was found that the low pressure production rate of nitric oxide is several orders of magnitude slower than the oxygen dissociation rate; this also suggests the NO chemistry plays a minor role in non-equilibrium air behavior in the boundary layers The foregoing facts seem to indicate that it is reasonable to neglect the -presence of nitric oxide in the present analysis, assuming the individual oxygen and nitrogen dissociation-recombination chemistry to be the important mechanisms. We therefore take the four specie dissociated air mixture chemistry to be governed by the following two independent reactions~ N. (2.51) /y A/l NZ. /

where the inert specie N (total number of particles) participates in both reactions indirectly as a third body collision partner. Since these two reactions are independents ^Zj X, f z= O ) which satisfies the:S miE = 0 requirement. Since Equations (2,51) are of the form (2Q46), the two net atomic specie reaction rates can be written, using Equations (2.47) to (2,49), as follows: A ~ _ ____z) -- 2/r) Mam (2532) Using the thermal equation of state, the volume-mass fraction relations and the molecular weight expression previously developed for this four specie mixture, these rate expressions can be written in terms of pressure, density, temperature and mass fractions as follows: <2zz -2rt24)1 +9f 46XP,, (2-54) Af+=-21/O41<c// 7$) L(4/ )_ (2055) where 7 tt= ^4 * It is consistent with the assumed equality of physical properties between species 1 and 3, and species 2, 4 (except the heats of formation)

that we also assume a2. a a and kR -l kR T T. The last equality is not necessary but simplifies the non-equilibrium analysis ( = o4 a and'! ki = ln 4 should be a good approximation, since the N and 0 recombination processes are independent of EA and. EA4)O The reaction rates given by Equations (2054) and ~o55) are especially convenient forms, since they show the net rate is essentially the product of a state function (proportional to density cubed) of the recombination chemistry and another function which vanishes in. the classical thermodynamic equilibriumo The thermodynamic equilibrium composition is governed by the equations (mini = 0) Y —— aL,L - ) (2/56) / —~ - (- /f F-q) XP 4 (2~57) (The specie a 9 w:hich is also requ.ired for the calculation of the equilibrium composition, is determined from. an additional statement of the conservation of both total oxygen and nitrogen molecules in a closed systemo See Appendix G). The net production of total atomic specie mass by homogeneous chemical reaction, which is of interest when comparing the present formulation with'that of other reacting boundary. layer theories, is obtained by adding Equations (2 54) and. (2,55): -(c~*~~ 2~ ~&2 (2.58) - - -g$p 0 ( ^( } - e 7~4J74\ 1 2 /+~C cnr(-~~~~~~~~~~~~-/-cxrp-j: ~~~~~~~~~~~F-0

.26where a - a + a4 is the total atomic specie The first bracketed term in Equation (2<58) is a binary atomic reaction rate form based on aO, such (6)s as used by Fay and Riddell o The second term is an increment due to the importance of the individual second atomic and molecular specie behavior, not accounted for by a alone This second bracket is composed of two parts: the first accounts for the reduction of the recombination rate from that based on a alone, while the second shows the effect of the molecular nitrogen due to the difference in activation energy between nitrogen and oxygen. When there is a significant atomic nitrogen popular tion in the gas mixt-ure, Equation (2o58) shows that the total atomic reaction rate can be seriously overestimated by the use of only the binary rate form based on the total atomic specie concentration a. HeternE.eneoaus Reaction Rates The effect of possible chemical reaction between the flowing gas and the material of a boundary surface enters through the specie equation boundary conditions. Of particular interest is surfacecatalyzed recombination of atomic oxygen and nitrogen in the gas, since the amount of diffusion heat flwc into the surface depends on the degree of such catalytic activity. In an "open" system, in which diffusion and convection exist to "compete" with chemical reactions, surface catalysi caan notice.ably alter the composition of the gas near the surface (something it could not do in the steady state in a closed system), The catalytic production rate of any particular specie j in the gas can be written in the form. — ) (o 9 )

27where kcj is a characteristic surface reaction parameter, a function of surface temperature and material, and ajEQ is the equilibrium specie concentration at the surface temperatureo (A more detailed discussion of surface catalysis is given in Appendix C. ) Regardless of the detailed behavior, overall mass conservation requires the following relation between the individual catalysis rates: z S. = o o0 For the previously chosen four specie model of a dissociated air mixture, the development in Appendix C shows that; Jc^ =-Apw (<4 \s[#eJ v (2.60) sac =/ e,2 (16 ^- ~ Q.) J which are generally required for the three specie conservation equation boundary conditions. It has been shown(lO) that for cooled walls with temperatures below about 2000cK, the catalytic dissociation affect is negligable, i.e., that a2EQw and OI E can be set equal to zero and only catalytic recombination considered for the atomic species Chemically Reacting Boundary Layer Equations The usual boundary layer order of magnitude analysis is assumed to be applicable to a reacting multicomponent gas mixture flow. In addition to the usual assumptions involved, the presence of diffusion and reaction rates in a real gas requires two other restrictions: first, the Schmidt number Se must be of order one so that the composition boundary layer thickness is of the same order as the velocity and thermal layers; second, the streamwise variation of the reaction rates must be small

.28compared to variations in a direction normal to the body surface. Furthermore, boundary layer~ inviscid flow interreaction phenomena such as induced pressure gradients, vorticity at the edge of the boundary layer, and any coupling between inviscid flow and boundary layer chemical reactions, will be neglected. The flow at the boundary layer edge is assamned a known inviscid solution for the given body' Boundary Layer Equations The boundary layer approximations applied to the general flow equations yield the following relations governing axi-symmetric or two dimensional real gas flow (body forces, radiation, thermal and pressure diffusion, and overall mass transfer at the body surface are omitted)' (1) Continuity* jx (/4 ),I4o(y)g -~o. (2.61) (2) First Atomic Specie dq12 d9/22 d4 2 2' where Z 2/+,2_ /4. y(2. 63) Bt' /J - ExP ( e- 7/r) _ o;2_ 0 E Pet / ) (2.64) (3) Second Atomic Specie flow X o y-2T A (265) * K O for a two dimensional flow and K: 1 for an axially=symmetric flowi

2.9(4) Second. Molecular Specie 0f 4d' +t ^1n]- (13 V t'a (1. (2.66) arS X f / AJ =dy d..e RI O (5) Momentum V/r dX = d - / (2s67) (6) Energy Equations In terms of the static enthalpy (with cpl cp3 and. cp2 *- cp4 ) given by = h, ~i&41)- < -434 -, (2.68) we have - f' V'J (2-69) The stagnation enthalpy form of this equation, on the other hand, reads /^P'. ek ^)N-( - 2-^2-70) where hs - u22. + ho Equation (2o69) can alternatively be written in a form involving the temperature in which the reaction rates appear by expressing the enthalpy derivatives in terms of temperature and concentration derivatives (see Equations (2o17) to (2o19)): dr(drvA2+oG )f 3-2-nJJ/XH y (2-71) where cp = cp +a(cp2 cp1) (2. 72) p Pl P2 P l2 Pi (7) Thermal Equation of State A = /4(,) /($S/?i)7T (2.75)

Boundary Conditions At the outer edge of the boundary layer (y - oo), the solution must match the known inviscid flow: u(X,oo) = Ue(X), T(X,oo) = Te(X), i. (Xoo) = Oie(X) and hs(X,oo) = hs = constant. Further, when the viscous transport terms and interreaction phenomena are equated to zero at the edge of the layer, the foregoing equations give ~ ahL _ A | eX - 2X,e 4 < = d /Ga ge (6z/2A j (2d74) The two specie relations show that the assumption of an equilibrium inviscid free-streamn flow (g2e g4e = 0) permits the neglect of the daie /dX terms in the boundary layer equationsO At the surface of the body, on the other hand, we have u(X,0) = V(XO) 0, T(X,0). T, the specie diffusion-surface reaction rate balance (neglecting catalytic surface dissociation) ^;~a ^X-) - -^ [ (^' O,

531and finally the heat flux of the gas into the surface, -W 4 ){;C d7-*/Z -2.) d^,) -}7 (2.76) - 8dY1 70//e-%z< * f*- iyj /=_O~ f^ ^-^-^ ^ M. ~~~~~(2~77) Some Special Features of the Equations Several special cases of the foregoing boundary layer equations warrant attention before taking up any detailed, solutions, (a) Inspection of the stagnation enthalpy form of the energy Equation (2o70) shows that when the surface enthalpy is known (known surface temperature and gas composition), and when Le = PR = 1, the use of the Fick law diffusion approximation causes hs to be governed. by a reaction-independent differential equation of the same form as for low speed, perfect gas flow~ and the surface heat flux is given entirely in terms of the enthalpy gradient, Therefore, when Le = 1 and the gas composition at the surface is known, the sum of the conduction and diffusion heat fluxes (although each is individually sensitive to reaction) remains invariant to chemical reaction of any form in the gas. This special case has been thoroughly exploited by Lees,(4 4) Futhermore, if there is a negligable pressure gradient,

the hs profile is completely similar to the velocity profile regardless of the viscosity law (as seen by a comparison of the enthalpy and the momentum equations) when hW = constants hs 1 + C2 a u, All of these conclusions are for arbitrary gas phase reaction rates. However, when any one or all of the assumed conditions above are substantially untrue, then the hs variable will not be a suitable one for the energy equation and heat transfer calculationo (b.) An interesting qualitative similarity between the effects of the reaction rates in the gas and the pressure gradient effect in the momentum equation may be shown. In the gas adjacent to the wall, the momentum equation gives (2.78) whereas from the atomic specie equations, A s2=o (/5aa)(Caged (2.79) Equation (2.78) indicates the well known fact that an unfavorable (dp,/dX > 0) pressure gradient causes an inflection point at some Y > 0 in the velocity profile0 Equation (2.79) illustrates a similar fact, namely that when a2 are positive in the neighborhood of the wall (recombination dominating dissociation in the gas near the wall), a point of inflection will exist in each of the atomic specie profiles. The initial increase in slope near the surface will be most prominent in regions of high pressure and in cases for which the recombination

533rate temperature dependence co is the largest positive nutibero This reaction rate effect for near frozen flow is illustrated in Figure la da2 p for a highly cooled catalytic (cz = 0) and non-catalytic (dY = wallo A similar conclusion can be drawn regarding the temperature profile in the gas near the surface when the viscous dissipation is considered negligable; from Equation (2.71) (assuming cp2 = cp), one has - / 0 2d12 = )- (/1woT2 (2.80) Here, the predominance of recombination acts in the opposite sense to that in the specie profiles (ice., like a favorable pressure gradient), and an inflection point in the T(Y) profile will occur only when dissociation is the dominant reaction in the gas near the wall This behavior is illustrated in Figure lb for a cooled wallo (e) Several combinations of the specie mass fractions are governed by equations that are independent of the detailed gas reaction rates, and therefore show a general similarity to the enthalpy and velocity profile under certain conditions. Addition of Equations (2,65) and (2.66) gives This equation is free of reaction rate terms and therefore yields a general similarity between a3 + a4 and hs when Sc. Le PR = 1 and when the gas composition and enthalpy at the wall are constant, Further. more, the ao + Q4 profile (and the al. a = 1 - a o4 profile as well)

-34a2 ac2 DISSOCIATION DISSOCIATION - DOMINANT \OMIAN.DOMINANT DOMINANT 20 \ i< we a =__ /'// CHEMICALLY CHEMICALLY / // FROZEN (NO REACTION) FROZEN // / -RECOMBINATION RECOMBINATION DOMINANT DOMINANT 0 C o N CA o Y O Y CATALYTIC WALL NON -CATALYTIC WALL a) ATOMIC SPECIE PROFILE T RECOMBINATION \DOMINANT Te ////f - CHEMICALLY FROZEN // / TEMPERATURE PROFILE // / - DISSOCIATION // / DOMINANT Tw O. 0 Y b) TEMPERATURE PROFILE Figure 1. Illustration of the Reaction Rate Effect on the Boundary Layer Specie and Temperature Profiles Near Frozen Flow.

-35is similar to the velocity profile in the absence of a pressure gradient, regardless of the viscosity law, if the wall gas composition is constant. Characteristic Non-Equilibrium. Parameters and Hypersonic Environment Previous investigations ('25) have indicated the utility of dealing with reactions in flow problems in terms of the following characteristic parameter representing the ratio of a characteristic flow time to a characteristic reaction time::S~~ t-~~ - (2.o8l) tR where tF and tR are defined, according to the particular problem at hando This parameter can be used to indicate qualitatively the reaction rate effect on composition changes relative to diffusion and convection -effects, Two extremes of possible behavior are indicated by the definition of 5: when ~ approaches zero, flow induced composition changes are much faster than the reaction rate effect, whereas when becomes very large, the reaction rate effect on the local flow composition completely predominates over diffusion and, or convectiono Such extremes may occur for either gas or surface reaction rates and may be illustrated by means of the governing equations for both types of reaction.s Consider a typical i-th atomic specie conservation equation involving a net reaction rate term in steady flow: __ = ji = - _ _V + _ - (2.82) in which mig. for dissociation-recombination can be expressed in the form 1) g/;r "~~~~~~";~~~~~p~

.36. where gi is a dimensionless function of temperature and concentration that identically vanishes in classical thermodynamic equilibrium. Assuming that one can define an appropriate characteristic flow time tF for convection-diffusion, and a reaction time hRZ~ = 4-Z )Z based on suitably chosen reference conditions, then Equation (2.82) may be written as (2Zm~ o//* ) /i,)) / ~-',' ~ (2 83) When -> O with gi j 0, the net reaction rate term becomes negligable and the flow composition is determined by the action of convection and diffusion alone; this is called a chemically frozen flow, On the other hand, when -e ooi the function gi must everywhere approach zero (li - aiEQ ) to maintain finite convection and diffusion terms on the left side of Equation (2,8 the flow approaches a thermodynamic equilibrium solution. The net reaction rate term, however, does not necessarily vanish at this extreme, since the product of g and gi as - -oo and a.i a -i -0 must equal a finite term on the EQ left given by [fd 17)^ t t.l-B%+47tt1 t..(2,84) The role of g is therefore an important one in understanding the qualitative features of the gas phase reaction rates; as g departs from zero, the reaction rates enter in proportionately to modify diffusion and convection effects, whereas when g is not large enough to be considered infinite, the

thermodynamic equilibriunm composition is no longer a solution and the entire chemical specie conservation equation must be solved to accurately calculate the gas composition distribution. Catalytic surface reactions on a flow boundary may also compete with diffusion and convection rates in determining the gas composition at the surface, and an analogous characteristic parameter to the above gas phase reaction case can be defined based on a characteristic surface reaction timeo This has been discussed for the boundary layer by Rosner3) The role of the parameter can be shown quite generally, however, by considering an i-th atomic specie boundary condition at a surface (see Equation (2o6)) in the absence of overall mass transfer: /{ e fx v4a) =~:, (2.85) where the net catalytic reaction rate Si per unit area is of the form c s C = 4 it i. (~2 86) If we define a characteristic dif~fusion time td based on an appropriate flow length L normal to the surface, and define a characteristic surface catalysis time then Equation (2,85) can be used to express the amount of i-th specie diffused into the flow layer of depth L as ^ i^=X^-^. (2.87)

~58~ When c -tD/tR approaches zero, the net catalysis rate becomes c D c very small and the composition at the surface is determined solely by diffusion (in this case, without any mass injection, the diffusion velocity vanishes at the surface). This extreme is appropriately called a noncatalytic wall. When C becomes very large, however, the maintenance of: c a finite diffusion on the left side of Equation (2.87) requires ciw to iw approach the equilibrium composition aiEQ corresponding to the wall EQW temperature TW; the surface reaction rate completely predominates over'the diffusion (which, however, does not vanish in general) and the wall is said to be completely catalytic. Characteristic Homogeneous Reaction Parameter for the Boundary Layer The average flow and reaction times must be chosen such that the characteristic parameter G indicates the gross non-equilibrium effect of the boundary layer deceleration of the external flow. Consequently, should be defined as the smallest value possible throughout the boundary layer to insure that a very large value of ~ would indeed indicate thermodynamic equilibrium everywhere in the boundary layer, and hence "truly apply to the boundary layer as a whole, Now for a boundary layer, the characteristic diffusion time tD is of the order l/ but since 6 Sa' /- and c. 2j,, we can write this equally well as the characteristic convection time z 3 ft /; )_ _ (2o88) Then if % is to be a minimum, the characteristic reaction time tR should be defined in terms of reference conditions which yield a maximum value.

.39The reaction rates encountered in the dissociated boundary layer theory are of the form 2110/ KR T cJ where gnever exceeds unity so only the remaining factor contributes to order'of magnitude changes in the reaction rate effecto Consequently, take 74~/ 2 70f A;ow, JI~ AC(2.89) c~2 with the reference temperature T chosen so as to minimize Ta1 (clearly REF PEF. making the choice of T dependent on whether cs is greater or less than 2)~ For co < 2, the maximum boundary layer temperature should be used, which for a highly cooled boundary layer is the outer edge temperature Teo This gives exactly the parameter employed by Fay and Riddell, who used the Davidson recombination rate model.- 3/2, In the case c > 2, on the other hand, the minimum boundary layer temperature should be used (thus TR. = TW for the highly cooled. situation). For the important case of a cooled boundary layer, the proper boundary layer non-equilibrium parameter is 4=ZxR~~~Ce30 (2.90) where T T,T < 2 RE. e:= TW, @-c > 2 o This parameter will appear naturally in the course of actually solving the boundary layer equations in the presence of homogeneous reactions. Equation (2.90), evaluated at an axially-symmetric stagnation point for

,=i40:0 various flight Mach Number- altitude conditions, is plotted in Figure 2 for several different values of co It is apparent from this graph that significant departures from equilibrium flow can occur at altitudes above 100,000 fto, depending on co, and that the altitude at which non-equilibrium effects appear (. < 102 for purposes of illustration) decreases noticeably as Xu decreases, Clearly, the conclusions regarding boundary layer equilibrium depend on the recombination rate model employed. Also shown in Figure 2 is a plot:of the Knudsen number of the gas behind the shock, based on nose radius,* defined as mean free path R When KN approaches unity, we are in doubt concerning the assumption of continuum flow, so the presence of this curve in Figure 2 will indicate roughly whethether or not the boundary layer non-equilibriun occurs in the continuum regime of flights Because of the delay of non-continuum effects at higher altitudes due to hypersonic bow shock compression, it is seen from this figure that the boundary layer is nearly completely chemically frozen for cD < 2 before any possible non-continuum effects occur; for c > 2, a noticeable degree of freezing has still occurred before KN reaches the order of one. It appears that the analysis of chemical nonequilibrium can be confidently carried out with the usual continuum flow theorys' Reference 27 shows that R (rather than boundary layer thickness) is a more satisfactory dimension to use in defining the regimes of high altitude hypersonic gas flows at the stagnation pointa

I0\ w ~~~~~~R=IFT 104 CK./2 —- (Sc<R) 3 10 102 w 1/2 I0 z s. I I \ \X u' I sX______V X.NI _____Kn- -3/2 (I ) 10 -2~~~~~~~~~~c rO~~~~~~~~~/ I0 I 0 10: — ^ ^ 50 100 150 200 250 300 350 ALTITUDE, ls FT Figure 2. Stagnation Point Non-Equilibrium Parameter Versus Flight Altitute and Mach Number.

The parameter ~ will in general be a function of the distance along the body, because of the dependence on the external inviscid flow shown in Equation (2.90), and thus must be treated as a variable with respect to X. Therefore the degree of gas phase chemical non-equilibrium in the boundary layer is expected to depend on X also. Only in the particular case of the stagnation point flow around a symmetrical body will 5 be independent of X; in this case ue/X Bs and all the thermos dynamic quantities are constants, so 3aR= p <C-Z().(2.91) On the other hand, ~ is not a constant for a uniform inviscid flow (constant Pe, Te' ue),such as occurs on a flat plate or cylinder, but rather is proportional to X; =PEATE, z*d. &d (txA) (2q,92) The inviscid flow over a hypersonic body such as a hemisphere-cylinder is of such a nature, however, that 5 may not be intermediate in value between those given by Equations (2.91) and (2,92)> For example, Figure 3 shows the approximate inviscid flow pressure as a function of distance along a hemisphere-cylinder; the initial expansion follows the very rapid pressure d.rop predicted by the Newtonian theory, and this fairs into the pressure field due to the blunt nose effect (given by blast wave theory) about two body diameters downstream(26) The subsequent die-off of the pressure show is based upon sneglect of the boundary layer-induced pressure gradient5 The significant drop in pressure clearly will reduce the local

1.0.8.6.4 [ j.2 0.-.08 1 *.06 - - -.004 20 —----- --- — _.01 -_ _ ---- --.00 - 0.1 1.0 0 C — 100.004 -....001 x/d Figure 5. Inviscid Flow Pressure Distribution for a Hypersonic Hemisphere - Cylinder.

G by one or more orders of magnitude; to indicate the extent of this effect in this example, consider the ratio s/s given by (2093) Assuming an isentropic equilibrium expansion around the body in terns of some effective specific heat ratio 7 such that r/7- (>e/, ) Equation (2.93) becomes 2 f( 3as (-:^~~ / sA ) Me-~,(2,94) For the pressure distribution shown in Figure 3, this ratiQ is plotted as a function of X for several values of cC (assuming 7 = 1,2) in Figure 4 to illustrate the typical variation which could be expected on a practical hypersonic body. The results clearly show a very significant reduction in ~ over the frontal part of the body, indicating -the local boundary layer to be relatively further from equilibrium than at the stagnation points Decreasing c, however, tends to make less sensitive to pressure variations, The possibility that a short hypersonic body may be covered by an entirely non-equilibrium boundary layer is suggested by this example. When the inviscid flow becomes uniform,:/5s subsequently begins to rise proportionately to X. The chemical state of the boundary layer then tends back toward the stagnation point condition, For example,

-451.0 -.8.61 ~ I I I I --.2_I 04.01 oe_.008.006 — ____:- -.004 - _ OD =20.002.0008 -0008 -_ ____ _ —_.0006 _ __ __ _ _.0004 -_____ 0. I 1.0 10 100 x/d Figure 4. Variation of Boundary Layer Non-Equilibrium Parameter Along a Hemisphere-Cylinder Parameter Along a Hemisphere-Cylinder.

..46. if the inviscid flow becomes uniform when the local flow experiences only very small hypersonic disturbances, ue u u and pe I p, Equation (2094) estimates the X/d for which / Ls - 1 to be: -d / = U a (u (295 ) which is a very large number for hypersonic flowo Surface Reaction Parameter The proper reaction rate parameter Cc involves the behavior of the gas adjacent to the boundary surface. The appropriate flow time is that of the diffusive mass flux into the surface, given by 2, ~s/ 5 x (2.96) in terms of the boundary layer thickness 5, The surface catalysis rates previously discussed are of'the form /i Wc a) where gC never exceeds unityo Since the significant flow length in this case is 5, the characteristic reaction time is 2 -97) Consequently, the surface catalysis parameter for the boundary layer is 6-C -14sz C g (2.98) in agreement with Reference 13. The axially-symmetric stagnation point value of c is plotted in Figure 5 as a function of altitude (the dependence on flight Mach Number is very small for Mn > 1), assuming ptJ = constant and TW/Te = 05, for several different surface materials * A * Based on surface recombination data for atomic oxygen and nitrogen given in Reference 11.

4 10 2' Io' ~,,.,.._ Tw =300K 102 Al ox0 CIR NROGEN 10 Figur - g i 4 C EACH CURVES HOLDS Flight AltituO adE.vA OVER A MACH NUMBER JJ-) I" -XYG-.., SPREAD 10 I Mo<:20 EN OR 10 10 O SURF4CE -4 IO tS ((=1/2) -5 I0 50 100 150 200 250 300 350 ALTITUDE, 10 FT Figure 5. Stagnation Point Surface Catalysis Parameter Versus Flight Altitude and Mach Number.

significant non-catalytic effect is indicated at the higher altitudes with the use of non-metallic surfaces, such as the frequently employed plastic or glassy-like thermal protection coatings on missleso Figure 5 permits a rough assessment of the flight regimes in which significant catalytic inhibition can be expected with a given surface material or poisoning treatment. For the sake of comparison the w = 1/2 case of the gas phase parameter; is also shown in the figure; examination of the curves suggests that non-catalytic surface effects for non.metallic walls are important when the boundary layer is out of equilibrium in the chemically frozen regime. The dependence of c-on the local inviscid flow indicated by Equation (2498) causes it to vary along the body, changing the catalytic effect with X; e is a constant only in the stagnation point, for which ECc/ - d/ Hi l(2.99) Otherwise, Cc will vary in the following approximate manner with X (using PW -P (R-, with RW constant, and assuming negligable wall temperature variations and an isentropic inviscid flow): ~~~v _~ tC CA * *, ) (2oo100) The ratio K (X)/Kc will be unity except for some x-variation (such as a discontinuity) in the surface material. Equation (2o100) indicates for the hemisphere-cylinder example that the exapnsion over the frontal portion of the body produces a local surface reaction effect that is

-49more non-catalytic relative to the stagnation point for the same material, Far downstream. when the inviscid flow becomes uniform, / Cs will grow as /X and may therefore reach and exceed unity for very long hypersonic bodies.

CHAPTER III TRANSFORMATION TO THE SIMILARITY PLANE Application of the Mangler-Stewartson-Blasius Transformation There are several means of solving the boundary layer equations, such as the KormLanPohlhausen integral method. and the Blasius series expansion methods Another fruitful approach is to deal with the equations in a new coordinate plane in which the three dimensional body effects and thlEe majority of the compressibility effects have been eliminated. by means of the combined Mangler-Stewartson transformation. Further, a functional form for the stream function is introduced. this form being that suited to an investigation of the possibilities for "similarity" (reduction of the governing equations in the new variables to a set of ordinary differential equations in a single coordinate)o We will transform the boundary layer problem to a new ~, q "similarity plane," and subsequently attack the chemically reacting boundary layer problem in this new set of variables. First, a continuity-satisfying stream function f is defined as follows: ^ — /^. _hX/, SA-_.S (5.31) Next we introduce the transformation? AX- J; fa e ko (p32) 7 T / / (3 3) and write the stream. function in the form:5O0 = -r / J @. (34)..5:0~

As a consequence of these transformations, we get the following relations: -~/ YJ -?'/;e O, 1YA, 4w and' u!/) J/ IX (3.6) Now introduce the variables tie C = f?/ /^ I^te in^8e l/f ~d /^ ^y/ /,. /) - S and. the following relations expressing equilibrivmm in the freestreamn (with consequent neglect of the freestream composition dcerivative terms in the

"52* resulting equations ) e/t+ c ) t= 4kJ (,) 4ie/g/+ i - < ~e ~X/;4 ) Aes where ae a2e +4e a e e The application of these transformations to the bouindary layer equations yield the following set of relations: (1) First Atomic Specie /<;MJ@( 6i) k wj r J, d J22 where (2) Seond tom /-ic Sp e (2) Second Atomic Specie where ffe)'__ _, —-'/i;-~. (35 ll)

~535 (3) Second Molecular Specie 4/ d2 / dS)__ )- 4 f 2f' / J) (3 12) d/ dt(:c A/ H ^e 6 le {S(/+ Jr/e ) JT I (4) Momentum IfC +Lk^}C - ) /(y2 /tS e )! E^^ ^ ^-^ dfd (3513) (5) Energy Equations In terms of the variable g. h /h e we have the stagnation enthalpy form // d d_ (a) P (tre/4 (+)e Z Oh (3o414) The constancy of h provides the following relationship (when the freestream composition dervivaties are neglected.) _i_ _ _ _ _ _ _ (3.15) 7/x /?

The temperature form can be written 6j^//4 // ^ ^ 7 _ - /ele; r Ce 7 /+/e evef ~4fe e)X It is im~plied f:.'Ther that f/: -':/ —. ~/5i.=./~:',.X/^-. w/~: 0 as o and that. the variations of 2 e and are negligable in the2 36) 2 <, 3,#e-2 ceel re F or (6) Thermal EqaEaion of Stab= 1T1he ter bo~mday eonditions at o - are -g -Q z' X - W > af/ 1 It is implied furthher tLah atf/a, ag/a > a g/an = at/a? E ax/a% s aw/an.- o as r _4 oo and tbe.- the variatsions o C ~sei a~4e and ax3e negligible in the boundary layer eq xationse At the inner boaundary s.urface, the following conditions apply in the similarity plane (neglecting catalytic surface dissociation): Q^r)-~= ^j %faiF-^~ j (3~1D8)

C55gqJ F _ -r: V^ 1 g e FJ' / R/ lr) j^J^r); 1(3f19) - ^'^r)= f'> 4()]J 2'Fr) The heat flux at the wall is -^ A ^ 4^l4l^w _4 I +(4Eg)/Oe(/~L44J 4V X (3.20) e-r s4 dii^ej (3,21) This expression clearly illustrates that the recovery temperature on a surface in a dissociated gas flow, ie, that on an adiabati, e wall, is not in general defined by zero temperalaxre gradient as in the ideal gasJ indeed from Equation (3521)' we have __ )L e 8o Oe. 22) The shear stress is 2~ i~)(52)

m561 — Exact Conditions for Similarity with Chemical Non-Equilibrium The exact conditions which permit the neglect of all a/c of the dependent variables, reducing the governing equations to a set of ordinary differential equations in the single independent variable rqare twofold: first,, the coefficients in the equations themselves must be independent of ~ (ioe,, constants or functions of q only)- second. the boundary conditions at both inner and outer boundaries must likewise be independent of 6o If these conditions are met, it would be plausible to neglect the | dependence of all the dependent variables, The essential accomplishment of the Stewartson-Mangler-Blasius transformation in this case has been to completely absorb the direct effects of compressibility, axi-symmetry (three dimensionality) and inviscid flow variation along the body into coordinate stretchingo The resulting equations in the stretched., B plane describe the boundary layer effect relative to the freestream variations by a set of equations simpler than the original partial differential equations0 These exact conditions are met by a limited number of physical situations, since only certain particular combinations of inviscid flowwall temperature variations will make the coefficients in the flow equations and boundary conditions constants or functions of q only0 A number of these cases have received attention by various theoretical investigators~ It is the purpose here to evaluate the exact similarity possibilities for a chemically reacting four component gas mixture boundary layer flow in the absence of overall surface mass transfer.

.57Similarity of the three species Equations (3o8), (310) and (3,12) requires: f, Q, z, X and W all functions of, C and S. constants or functions of, (c, C 2e 4e and C3e constant (324) ~ constant constant The neglect of the inviscid flow composition gradients (daie/dX O) is exact for either a symmetric body stagnation point or for a locally frozen flow at the edge of the boundary layer. Otherwise, it implies that the local composition is slowly-varying and dCaie/dX terms may be neglectedo Such an assumption is consistent with the boundary layer approximation that streamwise reaction rate variations are small in comparison to variations across the boundary layer The condition = cons-bar t implies = constant X)' and therefore the integral equation X = COAvSr4T 4f 1e A J (3.25) which is satisfied, for example, by PORIjR. ue and ro all having the form canstant~XNo The condition = constant, which necessarily enters when a general real gas is considered, will be satisfied when (a) 0, corresponding to the case of a chemically frozen flow, or (b) t g corresponding to the thermodynamic equilibrium extreme, or (c) ue constant X pPe and Te constants this is met at the stagnation point, of a symmetric body,

f58Examination will show that there are no other realistic pe(X), ue(X), and Te(X) variation combinations for which 5 will be a constants The condition for similarity of the momentum Equation (31s3) now becomes = constant, i.e, ue constant * Xf (3526) -which is the well-known condition for inviscid subsonic flow on a wedge of semi-angle ~p/(1+p )o The temperature form of the energy equation, usually the necessary form when dealing with a reacting gas mixture, adds the following requirements: P/&, /Cp, (Bp constants or functions of, (3527) U/ 7 y constant, (3.28) 0 I 7J (ose) constants or funetions of t* (3,29) The condition on the specific heats is clearly satisfied when cp and cp are taken as constants, or when the freestream and wall temperatures are.con stant.o Condition (5,28) enters when the viscous dissipation of energy into heat is important; it is clearly satisfied when ue = (stagnation point), or when the external flow is independent of X (flat plate, cylinder or cone flow). Conditions (3529) are satisfied either at a stagnation point or for a uniform inviscid flow (the latter, however, will not give similarity of'the species or temperature equations except when = 0 or 5 ->Oo The similarity of the total enthalpy form of the energy equation, which is of interest in some particular real gas flow problems, requires g a function of T, (3o30) ue2/2hse constant, (3531) Le constant or a function of'. (3352)

The case PR 1 eliminates the necessity of condition (3o31), whereas the case Le 1 eliminates condition (3~32) as well as the similarity restrictions of the species equations The ultimate usefulness of the total enthalpy equation depends on whether or not g(O, |) is known (and this in general depends on the chemistry)0 The inner surface boundary conditions (3018) and (53,19) will give similarity if $W- Z^G Xy 1W and WE(o) are constant, (3o33) C = constant. (334) The condition on the catalytic parameter is obeyed when (a) c O 0 a completely non-catalytic wall, or (b) c 00, the opposite extreme of a completely catalytic wall, or (c) Ue ^ constant X and Kc TW and p are constant~ This occurs only at the stagnation point with a given surfaCe material, Otherwise, one is left only with the case in which the surface material is distributed along X such that for a constant surface temperature, Co /SA/r4I -w- (5 35) A nvimber of interesting real gas similarity cases can be formed by combinations of the foregoing similarity conditions involving the extremes of gas reaction effect. Both chemically frozen and thermodynamic equilibrium flows for the two extreme conditions of surface catalysis have similarity solutions for stagnation point and runiform inviscid flow (and foz^ arny

-60o wedge flow if PR = 1). H-owever, the only similarity solution which includes arbitrary chemical reaction is obtained for the stagnation point flow of a symmetrical body, as pointed out by Fay and Riddell, ) The formulation of this problem for the present four-speeie air mixture reprpesentation is given below, Non-Equilibrium Stagnation Point Similarity Solution In the stagnation point flow on a symmetrical body (X = 0), we have the following values: u' B X ue sB X, 2, two-dimensional ~ 2(1+K), J (556):4 axially-symmetrical -= 1, Ip Ri constant R R All other freestream quantities are independent of X, the viscous dissipation (of order X2) drops out, and the non-equilibrium and catalysis parameters ~ and 5'assume constant values s and c c, respectively. One c s CS therefore obtains the following exact similarity solution equations (denoting derivatives with respect to q by primes): (1) Atomic Specie f/; + (SC./?^) S;; A^/f wze (35D7) where 2 and..are given by Equations (3*9) and (311), respectively. (2) Second Atomic Specie 2/v^@_/C idttv = ^6 Je,8 t(s 53,8)

(3) Second Molecular Specie LQz fhf7jW ae= 244e~ t - (3f39) (4) Momentum //f'" 2Z/'z= 2 ^/- ~e~~ ~ (ejY- (3,40) (5) Energy S' + jrC 4 ^+^^ ce2 = -,-' —~,'"2 f (^-^)- ( ) =-^; s-/5 F7ef / ^2 (3.41) _'- C Cel) /. —, /2 where 7 /c/ 0/+ f. In the above equations, C(t) - C(9,;z, X, W), Se(), s (~, z, X, W), PR(R) G P(g z, X, W) 2, 4 l1 >l a,4(g) are assumed known ~unctions. The corresponding boundary conditions are (cool wall):

-62'f() = ^- = f ) ( -, = J) = / j/hj = f /()- 0 fr=)= 6(3,42) A -= -L/4(o),^)]7/ @('~ Fay and Riddell solved these equations only for the two extremes, = 0 (z (o) = 0, non-catalytic wall) and c- o (z(o) 0, a completely catalytic wall surs face). Finally, the surface heat transfer is given by _ 4 W^/<p7 -r* _A 4 - i (3.45) o./~, -2 t,' 7 f r,' and the skin friction is C>)2"46ss/2' = C / (3.44) These equations govern the only possible exact similarity solution which includ.es arbitrary gas phase and surface reaction rates, They constitute a generalization of the problem formulated by Fay and Riddell in three respects: (a) the details of a four specie air mixture in the diffusion and reaction rate terms are more clearly brought out, (b) an arbitrary recombination rate temperature dependence is included, and (c) a general rate of catalytic wall recombination is allowed in the specie boundary conditions.

CHAPTER IV CHEMICALLY FROZEN PERTURBATION ANALYSIS The investigation of the deviations from chemical equilibrium in the laminar boundary layer is impeded, by the lack of a similarity solution to the equations away from the stagnation point and by the need for computer solutions to study the influence of the various basic parameters even if one does have a set of similarity equations, Accordingly, an analysis of the problem is offered which overcomes these difficulties to some extent; the role of the dissociationsrecombination rate in causing deviations from either extreme of chemical equilibrium is investigated by a chemical perturbation analysis, Many features of the non=equilibrium behavior are revealed in a fairly simple and straightforward manner by this method. In this chapter, the case of chemically frozen deviations are considered in detail for a highly cooled wall, The chemical perturbation approach includes three particularly important features. First, it shows the influence of the basic chemical parameters involved in the gas phase reactions: activation energy, recombination rate constant, and recombination rate temperature dependence. Second., the theory accounts for the detailed behavior of the various species in dissociated air, as reflected in the previously given diffusion.and reaction rate terms for a four component mixture, Third. the perturb bation method provides differential equations for the dependent variable deviations that. admit a wider range of exact and approximate conditions for which simailarity type of solutions can be mad e, Thus local non.equilibrium ra6j>

in the boundary layer away from the stagnation point can be treated. The results ofe the analysis will be primarily aimed toward calculation of the heat transfer deviationss since this boundary layer parameter is potentially the most greatly affected by non-equilibrium for a highly cooled wall, Thermodynamic state profile deviations are also obtained as a result of the solutions. The Frozen Perturbation Equations Assumptions (7,25) (28 Previous investigations by Adamson(7 ) and Broadwell(2) have indicated that deviations from the chemically frozen flow extreme may be treated in terms of a power series of the characteristic parameter For example, the first atomic specie behavior is of the form a2(X Y) m. OF(XY) +' i(X,)+ 1((XXY) +. (X. where aC is the chemically frozen atomic specie profile, and o2I. O2II' etc., are the first, second, etc., order deviations due to non-equilibrium, Similar expansions would apply to the other dependent variables, Substitution of such series into the governing boundary layer equations and subsequent equating of the net coefficient of each power of: equal to zero will1 produce a succession of equation sets for the zeroth, first, second. etc. order deviation functions, Each of these equation sets are then solved, subject to appropriate boundary conditions, for the desired quantities such as the wall gradients of temperature and species. In order to retain, as simply as possible, only those features which appear to be of major importance to the influence of the gas phase

-65reaction rates on non-equilibrium, the following assumptions will be introduced at the start, (a) Assume C pt/PRItR = 1, which is exact when p c (1 + a)T. While the finer details of the effect of the viscosity law are certainly of interest, particularly in the presence o:f surface injection of gas species of noticeably different physical properties, inclusion of the exact pp variation swould demand digital computation to solve the resulting governing equations, and would add little to the understanding of the chemical non-equilibrium analysis, The assumption C l1 eliminates a common coupling effect between the viscous transport terms in the momentum. specie and energy equations. (b) Assume Le and PR are constants While these quantites do vary somewhat, it is consistent with the above assumption and the main objectives of the thesis to ignore the detailed dependence of these parameters on temperature and composition0 (c) The specific heat cp and cp are assumed constant and the c Pc contribution to the hb2-hl and h4-hl coefficients of the reaction rate terms in the energy equation compared to that of the heats of formations will be neglected. This seems a reasonable approximation when studying the main effects of the chemical non. equilibrium due to such reaction terms, Further, we shall neglect the remaining cp-2cl terms ((- 1) and consequently eliminate coupling between the energy and specie equations, Approximate calculations have shown that the specific heat perturbations contribute negligably to the temperature gradient deviation in comparison to the direct effect of the reaction rate; the exclusion of these terms (which otherwise greatly complicate the perturbation equations) does not alter the basic dissociation-recombination

-66chemistry effects that are the main object of inquiry. (d) We will neglect velocity profile perturbations due to non -equilibrinum in comparison to those experienced by the composition and temperature profiles. Lees (414) has pointed out that the velocity profile solution has been found very insensitive to the pressure gradient ternm in the similarity plane momentum equation, and the specie and temperature profiles and their surface gradients are even less sensitive to the pressure gradient effect on the velocity pro^file for'a highly cooled favorable pressure gradient boundary layer flow, Hence the pressure gradient term in the momentum equiation is neglected by Lees, Similarly, it would seem that a satisfactory approximation for the purposes of studying:non-3equilibrium cooled boundary layers would be to ignore the pressure gradient term in the momentum equation and neglect all velocity perturbations due to -chemical reaction, taking the velocity functions f(,I) and f' () to be the known Blasius solutions. An important consequence of this is the linearization of the remaining specie and energy -equations, since the velocity terms in these equations become known variable coefficients, (e) Finally, the effects of surface reaction shall be considered only for the two extremes of catalytic actions ignoring the intermediate surface reaction rate details but'still accounting for the ma.ximumn possible differences due to surface recombination. Characteristic NTon=Equilibriu Parameter Expansion The aforementioned perturbation of the boundary layer equations is carried out with the similarity plane (5, B) forms given in Chapter III, Together with the foregoing assumptions, then, we introduce into these equations the following expansions:

z = ZF(, T).+ t' Z(i i).+ 7- o Zz(S, T) + ~~.~.'F+ r;) z+ G;D z( )II( + zo(,) |. X - XF + 5 e Xi + 2 * XII +.O ( 2p~~~~~~' ^ (4o1) w WF + t o wI +. WII +.... o= F + 5 o I + I2 9 II+ I o where the subscript."F" denotes the chemically frozen solution and where it is required that all the various deviation functions vanish at the outer "edge" (q - oo) of the boundary layer, XI(oo) = II(oo) a.o0 e:0 OI(~~) Wii(o) =.... 0 o gI( ) QI(m):0 ~~ O Corresponding to these perturbations, the net reaction rate functions ^ and may be written as the following expansions: 2^ - e [~wl 0 (0 =( 2,)F I$//24 ((4~:) ^4^!(^ - L Ti^ ^^~rB~ 4~

The various derivatives in Equation ()4,3) are given in Appendix D. Substitution of expansions (41l) and (435) into the governing boundary layer equations, collecting terms and setting the net coefficient of each power of % equal to zero, will produce sets of equations governing the frozen, first order, second order. etc,, deviation functions Intro.dtucing the symbols / (o). Q - (Cd x)/ ~[."), J" _ ) =,o,, o ) then the resulting perturbation equations for zeroth, fi-rst, and. second. order deviations, successively, are as follows: (1) Zeroth Order (Chemieally Frozen Flow) $~ X-F d + d =an = 2xf id (4a5) */^/'^^/"^=^ Ct.~4,8)

4/, 6+z~f'~j L5. t/P~- (4,9) (2) First Order Frozen Deviations sc / d-i, +,z < 4l' (4'10) (e) d2e = 2ize F t 74;2S+= t., ^^/y / cfM i.w^^s0-k 4 ( 12) - 1 ^\2^ ^j/.t^^/'41 "~~(^/^^7^ de) ^-^^ ^ ^^^ -^//^~~~~~~~~~~~~~~I "<^^^^-^^6 6xfl5 41I.J.-ZF AT

(3) Second Order Frozen Deviations L= 6aP i'^.)^j f % 4 f + SS C f Xt erV f 2 f s 22az X f'Rz (4,15) - )s$ft~r^zi^ tvS + AS +i v 3 + &) 4 +_$,wz+ 41; _.-J~ feS rl4 - -rc ( J+eT6)'+ i^-'e 1/) —~ 7/T =' +: Zi ft"'^/'fle / -/&1k^ -)7 (4.17) + e /34) /. 2 ) =~~~-;t/Ff~*/^^4^^4

-71The boundary conditions are as followso At the outer edge of the boundary layer: fz(^) =zF(oo) = XW(o) o WF(O~) =g(O) 0 1) ZI(<x) xi X(o) W=I() = =I(00) = o F (4.18) IIT(KV X~)3 ) II(~~) ='QI(0) = ~ J At the body surface r = O, on the other hand, we have the conditions, fo)= 9-6 a \ i (4. 19) ^(o)= &w / (O)- 4ri() = 4h e= ( e)= bro o)= o) a= e, 7. (420) o e o) = l ^, j, z-/)= dei)= cOLC AZtions// C J ( Io)= O = Hc' 7) = O) = k% r ) (\ 2 (4 - ~^Y/ o))- F W = t.-Cv4zr4Z7c IAkL The -surface skin friction, hich remains unperturbed to the approximations used, is ~/y 2 e~~~~ /ti A ^ // / 7 ~ (4e22) The surface heat transfer, however, shows the following non=-equilibrium behavior in the neighborhood of the chemically frozen value, to second order deviations:

-72*Z.,...... z__, (4__3) Reduction to Ordinary Differential Equations Each.of the foregoing chemiical perturbation eqguation sets canI be exanined by sixilarit. y arg-mients of the kind previously discnssed. in Chapter III, Because of th.e foregoing simplifying;assumptions and the fact that the. characteristic non-equilibriumn parameter. does not appear explicitly in these equationsj the exact and approximate simrSilarity possibilities (reduction to sets of ordinary differential eqpations) are nFv samewhat greater, In general, an inspection indicates that neglect of g derivatives and the use of local constants for the c y dependent coefficients is juxstified. when the follo wing parameters are either constant or slowly-varying f-unctions of Xo t e gas composition at the edge of the boand ary layer, the parsmeters c, P and ue,/ep Te, the freestream and -wall temperatures, and the non.equ ilibrium parameter daerivative factor Q- The boundary conditions (4.18) through (4.21) ('with the possible exception of the frozen temperature bonmdary cond.ition ~oW) are all independent of t The similarity conditions are exactly satisfied for constant s.urface temperattres regardless of the degree of surface.cooling in two cases. First, the symmetric stagnation point flow, for which p = 1,

.735 ue. — O,e = 2(K+l), and Q = ~/peTe = 0. Second,, a uniform inviscid flow (such as a flat plate or cylinder) for which ue and u / Te are constants, 3 =0, and Q=s 1, Approximate similarity can be obtained., however, over a wider range of local conditions, Regarding the specie equations, the freestream composition has already been assumed slowly-varying to be consistant with the chemically reacting boundary layer flow approximations0 The frozen flow specie equations can, be taken locally similar in view of the boundary conditions. The first and second order specie perturbation equations can likewise be approximated by ordinary differential equations if the reaction rate functions ( 4 )F and their derivatives 2 2, J ~ r 4 and c2, ean be assumed functions of Tr only and if the parameter Q/e is constant or a slowly-varying function of X. The former condition is.met if the frozen specie and temperature profiles are approximately similar and A2,4 is constant or a slowlly-varying function of X (since QA - TA/Te, this in turn requires the freestream temperature to be constant or slowly-varying), These temperature conditions are taken up in the examination of the energy equations, Concerning the latter condition, Qf/ will be exactly constant when the parameters' and e vary with X as XxW If such a form locally describes the variation of these quantities at various portions of the body, then as long as N itself varies slowly between these regions, Q/e can be taken as a local constant at each X0 The local value of the parameter Q can be obtained by differentiation of Equation (2.94): Q = / -/ f/24-9)],/4/Y) (4024)

"747 As an example of the typical variation of Q/e on a hypersonic vehicle, the pressure distribution shown in Figure 3 has been used. to calculate c and Q; the resulting:Q/f distribution is plotted in Figure 6 as a function of distance along the hemisphere-cylindero The results indicate that Q/E is exactly constant at X = O (stagnation point, Q = O), at X/d - oo (uniform inviscid flow, Q/ - + 1) and at X/d C < 7 (in the neighborhood of the sonic point) with a minimum value around -U2, depending on the value of c., Otherwise, Q/c appears to be everywhere a slowly-varying function of X with the possible exception of a region of approximately one body diameter in width on either side of the minimum point. In addition to the foregoing conditions, the similarity of the energy equations imposes further requirements* The inner boundary condi" tions on the temperature deviations are independent of ~, whereas the frozen temperature boundary condition OF(o) = ~W will give similarity when OW is constant or slowly-varying (further, GW itself has a very small effect on the frozen temperature profile, since we assume the wall to be highly cooled, W << l)o Concerning the coefficients in the equations, the approximate similarity of the temperature variable will depend on the factors Te and uec pTe being constant or slowly-varying functions of X, (The parameters q2 and q4 are local constants if Te can be considered so), Now considering the inviscid flow as isentropic with some average y, 7.1 Te - (Pe) 7 e which indicates that even in the presence of significant pressure gradients (such as shown in Figure 3) the corresponding inviscid flow temperature

1.0 1'0 "~|~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x/d = c MOD 10 ~ = 10^.2 IN ^ I 2 I -10 _____ -__ __\\__ __ _ ^^\ __ _ _ ______ _ __ ________!~ \^ 7/v w s 2= -2.0 -3.0O _ _ _ _ _ _ _ ___ ___i K_ _K___0_ _00.I 1.0 10 100 x/d Figure 6. Variation of the Boundary Layer Non-Equilibrium Parameter Derivative Factor Q/e along a Hemisphere-Cylinder.

,76. variations should be much more gradual and can be taken as slowly-varying for the bondary layer theory0 The kinetic energy/internal energy ratio ue2 /p Tea using the fact that 1/2 e2 + cplT(l + q + aq) " hse constant, (4.25) can be written as P~e e ~' ( 2 -2(?/ se)(4.26) This relation indicates two situations under which the ue2 /pTe term gives similarity of the energy equationso The first is the case in which Ue2/hs e e is very small(because of a low speed, high temperature inviscid flow)and can be neglected, regardless of the X-dependenceo Such a situation would occur in the frontal region of a blunt-nosed hypersonic body where the flow originates from the strong parts of the bow shock and is at a very high temperatureo The second case occurs when the magnitude of ue/h is important but the X-dependence of the termn is comparatively weak because ue(X) becomes a slowly-varying function, For many hypersonic flows of. interest, it appears that in the regions where u e2/p Te varies significantly with X it is small enough to neglect, whereas when its size builds up to a significant effect, the X-variation has noticeably weakened (P - 0)o We shall therefore make the approximation that the u e2/ Te term can be taken as a local constant in the energy equation while neglecting the O9/6- terms at each X. When including this local term in the energy -equation for a cooled boundary layer flow, we shall further neglect the P/C(lz)Q:f' tand B/c(1-X)QF' terms in the frozen energy equation (com. pared to f" ) and the perturbations of these terms in the first and second

.-77order energy equations, since they enter in proportionally to p/s (Ue.pe Te) and are therefore negligable'when uep eT is of significant size (accepting the above approximation that this.latter occurs only when P becomes very small ) Accepting the approximations (and limitations) in the foregoing similarity arguments for a cooled non-equilibrium laminar boundary layer, the zeroth, first and second order specie and temperature profile deviations are governed by the following sets of ordinary linear differential equations (denoting d/d, by a prime): (1) Zeroth Order SC /4 + " = (47) s/ r', =?" (4,,28) $f P <', 4 =o (4I29) ^/^'^ 3L t$/+4'^ y/= ke^2 (4,30) where f, f' and f" are the known Blasius functionss ie,, solutions to the equation f f" + f' 1 o with f'(oo) = 1 and f() () f(o) 0, (2) First Order Deviations,/^ 2_ - ^_/@r e (4,31) 5/ X-' X Or -e eel )2

.-78^ _/ _ __~_ // =/ -2 e9 As (4033) vv2 a,, +/ de)od4F 72 32 X14)34) La + ~- ( e/ (3) Second Order Deviations AmpJ4r^I -'t: (4,36) -= Ich < (>r)&"-2) + 4? +,$/6C,'^ A"- M -v S IA (4.37) 5/6 6- 68/ d (Ip8) go /^ ^ We L</9 ^^~ 4 L (48) _Y2 4;- 42 7 9)} gtt c? g)^ ^^)*iX

,790 The boundary conditions (4al8) through (4,21) still a.pplyJ the shear -stress is given by Equation (4-,22). and the heat transfer is given by Equation (4.23) when the partial derivatives are made total derivatives with respect to r. Solution to the Frozen Perturbation Equations Chemically Frozen (Zeroth Ord0er).Solutions This group of equations and boundary conditions can be solved easily in terms of known integrals of the Blasius velocity functions, We shall consider each of the equations in turn The atomic specie.Equtation (4.27), viewed as a first order equation in zF, can be integrated by means of the integration factor EXP(Sc j fd-r) and subsequently integrated again to obtain the general solut ion 4 w' - ^ + 4 L-I ( ( 409) he integral hs been g orGo29) The integral has been given, ~for example, by Goldstein(29 for a slightly different scale factor in qI and f wvhich can easily be adjusted to the present case; in parti. ular, reference 30 gives t he approximation jWF-rM Xj;~)h (.S 17 s7 YC3) (44o0) Applying the boundaxy conditions ZF(oo). 1 and either zF(o) 0 O (catalytic wall) or ZF(o) 0 (non-catalytic wall) to Equation (4039) gives the following two solutions: [t774 ) A7 r(^ ^/^ ^ |f d-P (4 1) L 4^ ^

..80o 1/3 with zF (o)] 47 S', and UzF (n)) 1. (4.42).F NONTCAT.(44) The frozen second, atomic specie Equation (4.28), when compared to Equation (4o27), shows that the variables XF anad ZF are linearly related. In view of the boundary conditions on Z and. XF being the same for either wall catalysis condition, then XF zpF (4~43) In the catalytic case, the atomic specie concentrations are zero on the surface, whereas in the non.catalytic case, the build-up of atoms in the absence of surface reaction gives a uniform steady state concentration distribution, The total atomic specie profile becomes _ <e K =4e =, (4,^4) Using the tabulated values of the integral in Equation (4,41), this solution is plotted versus, for various Schmidt nimabers in Figure 7, Also shown for comparison is the total atomic specie profile obtained by Fay and Riddell(6) for Le. l4. (Se =. 5) The discrepancy in the catalytic wall profiles, including a 44% greater wall slope than found by Fay and Riddell, is the result of the pt/pRpR:. C 1 assumption in the present theory, (Application of an approximate method, for correcting C = 1 bodundary layer calculations to account for variable ppt affectsf described in Appendix F, brings the present theory value of zj(o) AT down to within a few percent of the Fay and. Riddell value), It is interesting to note cfrom Figure 7 It can be shown t his result is also obtained if the quasi-two component diffusion flux termn is used instead of Fick s law0

-811.0 NON- CATALYTIC WALL 0. 9 - Sc'l - 1(6) / o0s8 e,,/ / CONST. // / / 0.7 0.6 U. 05 ___ _/ CATALYTIC WALL /// ^- zp(o)..373 0.2 -- Z;(0).209 0. 1 ----- 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 r) Figure 7. Atomic Specie Distribution in the Chemically Frozen Laminar Boundary Layer.

-82P that the C = 1 assumption approximately doubles the boundary layer thickness. The second molecular specie Equation (4p29) integrates in the same way; applying the boundary conditions WF(oo) 1 and either WF(o) - WEQ(o) or WF(o) = 0, we have = Aj', (~+/- J, (445) which can be used for either catalysis extreme since zF = 1 (non-catalytic solution) gives the correct answer when inserted into the equation. This result shows that when WEQ(o) - 1 (ie o, l3EQ(o) - 3), the catalytic wall solution coincides with that for the non-catalytic case and there is no steady state diffusion of molecular nitrogen~ Such a conclusion is, of course, based on the use of the Fick law diffusiono* Some typical WF('r) profiles are shown in Figure 8; when WEQ(o) > 1, the catalytic wall provides an excess of molecular nitrogen at the wall over that in the freestreamo The energy Equation (4~30) is a second order non-homogeneous differential equation that can be treated as a first order non-homogeneous equation for Gp(l) and therefore integrated to yield the following result: VI.) d$d^ f $WSXS-)^o/i -^ 0&^f|. (4.46) The integral Fp constitutes the non-homogeneous particular integral of the energy equation contributed by the dissipation term and is a tabulated function(29) Applying the boundary conditions F(oo) = 1 and F(o) -=, If the quasi-two component diffusion flux term were to have been used instead in Equation (4.37), it would be found that the solution (4,.45) is again obtained provided one takes WEQ(o) = (1 cge)l

-831.8. l —. 1.7 1.6 _ Sc- PR=.70 1.5 (0) 2.0 EQ. 1.4 ] 1.3 8 M- -l.50 -t- - - - -- - - 1.2 1. ___________ w E (0) = 1.0 OR NON-CATALYTIC 1.0 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.5 77 Figure 8. Molecular Nitrogen Distribution in the Chemically Frozen Laminar Boundary Layer.

the following solution is obtained: Q4= ^= ^^ h7fl.2i/-,4)X,^! )^'( _ _ _~ _ o. _ _ _ _ _ _ _ _(4.47) _ _ _ __ J with Fp ) -~J ='^[74@fi- iv +W -(448) where This solution is plotted in Figure 9 for PR 7 70, OW 05, and several values of the ue2/e Te parameter0 The viscous dissipation causes a bulge in the temperature profile (and consequent increase in the wall gradient) which would be increasingly prominent for larger OW values. However, for highly cooled walls and ue2/SpeTe < 2, the viscous dissipation heating does not appear to drastically distort the profile, and will therefore be foanmd to have a very minor effect on the dissociation rate in the non-eqluilibrium perturTbation analysis., Also shown in Figure 9 is the t stagnation point catalytic wall temperature profile computed by Fay and Riddell using variable pp. and specific heat data6 A comparison indicates the error in the present theory due to the combined assimptions of C = 1 and equal constant Zmolecular and atomic specie specific heatsj the present theory gives a 24% higher wall temperature gradient in this example due to these

-851.0 FAY AND RIDDELL — /. 0.9 - (6) ---- / pL 9 CONST. / 0.8 cP2 s=.500 U 0 SC:=.50 c-'TP o CATALYTIC / WALL 0.7'_eF: CpeTe / I. 70 0.4 -- ~ / / — -------- - -=P05 - -0.1 - --- ---- -- 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2.6 4.0 4.4 4.8 77 Figure 9. Temperature Distribution in the Chemically Frozen Laminar Boundary Layer.

-86* two ass'mpti.ons However the general temperature profile shape for a cooled wall is not notably different and proves quite accurate for -se in the perturbation theory, The chemically frozen temperature solution (4 47) is independent of the gas composition, since an energy equation coupling terr. involving the atomic species and proportional to the difference between the atomic:and molecular specific heats has been neglected. However, if we return to the general energy Equation (5,16) and again apply the previously dis. cussed similarity approximations without the assumption cp2 -C the following chemically frozen flow energy equation is obtainedt: /7i/ +<7e/)/. 7L 71 ^=O 4) /+, -/ 3, (4.50~) and where P2 and c are assumed constant, Since the frozen atomic specie profile zF is known, Equation (4, 9) can be solved to show the effect of the specific heat inequality on the frozen temperature profile (particilarly the wall gradient)s For the non:catalytic surface (ZF - c. w1), we see nrimediately that cp2^ cP1 has absolutely no effect on the equations solution (4-47) is exact in this case, In the opposite:extreme of a completely catalytic surface, however, the effect is presents Viewed as a first order equation for:F and subject to the boundary conditions GF(oo) s1 and QaF(o) = GW, Equation (4,49) in this case can be integrated

twice to obtain the solution, In the ecase of negligable dissipation, for example, we have _,<-^^,^^7 > with p/2 PiYe),rya) = (4=5(b) axd wlhere Np/$(o) 1 + (cp./pl'1 < l)xaezF for a eatalyti. TallD Since (cp2/cpL l)aezF varies from zero at %he li11a to a value:of o 2 at the edge of the boundary layer (for ae',50, ep/epl 1/w), we an make the approximation -(?ZJ to simplify the calculations, In particular,. ETqation (4.52 ) simplifies tc FA /. 7 (4)53)/-_ /-.^^^/^(%^-/^^-^/^^ -.... which clearly shows that the eff:.eCt -of - >'> is to inerease the catalytic P2 P1 Vwall temperature gradient an.d raise the entlre en etemperature profile above the non-catsJltic wall sol.tiono Direat evtalation of the integral in Equation (4 535) shows that,.5 0P' 70e =/ 50..a P For the example Sc o.50 PR: 70i a.*50 and cpg/cpl/ 1.4, we have f. 4&i;~N0 /~

.88which is a 31%a increase over the Cp2= cP (or noncatalytic) value, a significant alteration in the temperatkure gradient. The frozen heat transfer function (423) acomputed from the fore" going specie and temperature (C2 -ep) solutions, is as follows: Q = $ + ) 6 )ie K 2 Sf)> t). (4,55) For'the catalytic wall case, we therefore have l7 = / 4' L4Le j )/ t56) which has, for example, been given by Lees In the case of a cp C on the other hand, the original heat flux expression (3521) would give [Q^176<4$~~~~~wi - Xbf gO) $ 9 Ze (4,57) Using soliations (4.)53) and (4,54), "this becomes (for negligable dissipation)!/&,/ =., p ijf1,('< /- _ _:-)T L — (4*58)LrJ^,' f~x~tp/~~~,/Hr~jd~f^'6^^ ^ (4=58) Heece -the c p effet.; on heat transfer will be less pronoeunced than that on the ~timperatre grad.iert',(O) alone, becauose'of the factqr- (0), For the previously cited example (ole =:50. S'..51 P- 7. ep7, /./ 1c4)p we have,tJ~ct =c. 40)1/^^ E = (jtoAnbL 4P/7;/ no fd, 7

The unequal specific heat effect in this example increases the temperature gradient by 51%, whereas the increase in heat transfer is 9% o1r only one third as much. The c Cp approximation for computing heat transfer p2 pl may be acceptable for many practical purposes. On a non-catalytic wall (z o 0), there is complete inh.ibition of strface atom recombination, and only the heat condaction mode remains s E6A/a-/C,; = 97" *^ ^ i- A P Z 7 (4.59) which is exact whether cp and c axe eqal or not,'sing the foregoing functions Q,. the local heat transfer var.iation for a particular frozen flow problem *@an be calc alated by the ~use of the X-aependent term on the left side -of Equ.ation (4, 23), as demonstrated in detail by Lees(4) for example. The:foregoing frozen boundary layer solutiones. can now be used to rewrite the reaction rate funactions appearing in the first ordcer deviation eq.gationso, The fainetions ( ) givern by Equations (359) and (3511), for example, can be written in, the following formtns _ X 7H-'-D^C } (4_61) where e-,^-e'- ef-'F -J (4,62)

eJ/-^ SL~~.W/ L:r\ (4,63) / I l t63 ) I = ^i^- (i-e] (4u 64) Now for a highly cooled wall, the recombination rate (z2 term) dominates these reaction rate ters completely over the inner half of the boundary layer (q < 2). the exponential dissociation terr contributes significantly to the function only in the -outer (T. > 2) regiono This situation allows some convenient simplifications to be made in Equations (460o) and (4,6l). First, the factor (1 a.z)l/(l a2) will be taken as unityr it is exactly so for a non-catalytic wall, whereas for the catalytic wall the major error resulting from this.assiM-ption occurs when the exponential ter which the factor rmultiplies is extremely small (7when the dissociation exponential becames of significant size fo > 2. the subject fawtor approaches unity). Second. the specie factors C1 and C3 will be neglected. Again, this is exact,for a non-catalytic -wallo For the catalytic wall case, these factors become compaxable to unity only in the inner part of the boundary layrer where the exponential term they multiply becomes vanishingly small. The error incurred by the neglect of these factors in the pertubation analysis has been found to be extremely small. In passing^. it is interesting to note that the ise of tehe guasi-two component diffusion scheme for ca (WEQ(<) (le)^ l)causes C1 and C5 to be exactly zero every. where. Third, the factor C2O which is a correction to the dissociation rate for the viscous dissipation heating distortion of the frozen temperature

991l profile, will be assumed, unity for a highly cooled wallo The main distortion (largest value of C2> 1) occurs in the inner part of the boundary layers calculations have verified that the effect of C2 > 1 on the d.issociation exponential is extremely small in the perturbation analysis, The recombination rate temperature dependence function Q' will also be affected by the dissipation3 using QF ='QF + +2 e j w e have - - X (4'65) The effect of dissipation on this recombination rate temperature dependence term is by far the most prominent way in which dissipation can influence the highly cooled wall non-eequilibrium perturbations (tahe d.egree of the effect depending on the values of u, ue/cIeT ad fW)d., The complete expression (4o65) will be retained in the subsequent theory in order to assess the significance of the dissipation in the non-eqguilibriumn behavior where Wo Z 2. The foregoing approximations simplify the reaction rate functions used in the perturbation analysis to the following. " --,x - eii j^ W e1d-] IPt~ 4~ tt ~i/74$^il4/(4.u66) where gFc is the Ue/ Tp = 0 temperature profile. The function is plotted versus q for various values of the activation energy parameter Q in Figure 10 (catalytic wall) and Figure 11 (non-catalytic wall). The parameter QAis seen to noticeably influence these funtions only in the outer part of a cooled boundary layer, this effect being more prominent in the atalytic wall case,

-920.8 ------ PR=SC = 70 0.7 -- w- =.05 ae =.50 0.6 cg 20 0.5 -,... 0.4 N 0.3 0.2 0.1 0 L —-- 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 Figure 10. First Order Reaction Rate Distribution in a Catalytic Wall Boundary Layer..

-93Pp = S =.70 0.9 -- ------— O --- - w =.05 0.8 - \A -20 0.7 I o - 0.6 0.5 " 0.4 0.3 0.2 0.1 0 — *~ 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 77 Figure 11. First Order Reaction Rate Distribution in a Non-Catalytic Wall Boundary Layer.

First Order Deviations The governing linear non-homogeneous differential equations can be solved by standard methods once a homogeneous solution (complementary integral) is foundo The form of the variable coefficients in these equations2 however. generally prevents a closed form solution and requires that a numerical integration procedure be used. (i) First Atomic Specie (Oxygen) The complementary integral of Equation (4031) a solution to the associated homogeneous differential equation -s / f c i @ - 24f 4/ a = 0, (4.68) is of the form- zI =A ~ zc1(B) + B o Zc2(q ) where A and B are arbitrary constants and zc,, Zc2 are twq linearly independent functions each of which is determined by a numerical integration of Equation (4.68) by Milnes Method (30) for arbitrary non-zero values of Q/e.* This method requires four values of zcl and Zc2 near the wall with which to start the numerical procedure; these are provided by a series solution for zI in the neighborhood of the wall (r. << 1), Since in this region the c Blasius velocity profile behaves as f = alr/2 - BP5 + ~.~0 (a = o472 p = o00018)2 direct substitution into Equation (4.68) produces the following series solution for ZIc (i << 1). fc2 = ////' 3- - - f zl j /,/, (4.69) * In the two particular cases Q 0 and Q/ = -1/2, analytical solutions to equation 4.68 can be obtained; see Appendix EO

-95This solution provides the desired starting values of zl and zc2 for the numerical integration, the typical esult;s of which are shown in. Figures 12 (zcl) and 13 (Zc2) for various values of Q/e and SC - 70. With these complementary integrals known, a standard method yields the following.complete solution to the full non-homogeneous differential Equation (4o31.) (see, for exampl.e Page 3, Chapter 1 of Reference 31)~ 2=++fi/4.)hoi-ew~gg + s r.^ (47~0) where ng fJ ~Xp(S cyJI ) (4071) lo p - " J — / J 4zc, ~'KYS ^fA/& (4,72) with the reaction rate function R2 given by Equation (4~66). Application of the boundary conditions z-(oo) = 0 and either zi(o) = 0 or zI(o) = 0 requires that B. 0 in either case in order t.hat solultion (470o) be free of arbitrary constants,, As a result, the c.atalytic and non-catalytic wall solutions are~ r&<)c ~r""/+l/e- --.3C2 *Jc^-r. (4-73) with ) (4074) with g-= 4 el -,) - (' It), - ____= (4..75) fc<.^-^/% 4O-C47, ^

-96where 1z and T are the integrals I z I 2 respectively, in which z is z!'Z 2 Zi Z2 Ic z2 replaced by z c The imposed boundary conditions therefore require the use of only the zl part of the complementary integral, zc having the properties z.(o) = 1 z' (a) = Oo The corresponding behavior of the integrals at the wall is seen to be (o) = I (o) () = and (o) Using the Z solutions (Figure 12), evaluation of the integrals 1 and I for desired combinations of the parameters Q/,E, Cu and 6A completes the solution, Since 2F is positive throughout the boundary layer (recombination dominating dissociation),as shown in Figures 10 and 11, the integral I is positive (and Iz will also be positive) as long as Z1 is positive (which it always is in the neighborhood of the wall)o Therefore, zI and z. are negative for small rq and recombination-dominated non-equilibrium deviation tends to reduce both the atom concentration in the vicinity of anon-catalytic wall and the atomic diffusion flux at a completely catalytic wall, agreeing with previous qualitative predictions given in Chapter II (See Figure l), Equations (4.73) and (4o75),when multiplied by the parameter, give the non-dimensional first order atomic oxygen perturbations~ Typical deviation profiles throughout the boundary layer will be shown later for the total atomic specie, Regarding the maximum deviation quantities at the wall, the first order specie gradient perturbation at a catalytic surface is fti C27/ 2^ie~ (/jf (4077) On the other hand, the first order specie concentration perturbation at a non-catalytic wall surface is 4C rI (A o )I^. 1 t2 rk2 r2" J( (4k78) ir54^J^.i: 2 OPT^J4 /-2-^ -^Cr:

-972.6 Q/e =2 2.4 2.2 1/2 2.0 1.8 1.6 __ S C.70 1.4 ___ Sc' 1.2 Q/e =0 1.0 0.4 \~\.\' Q/e 1/2 0.6 - 0.2 _-\_ -0.4 -0.8 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 r) Figure 12. First Order Complementary Integral Function z 1.

-982.6'i - - - 2.4 Q/E= 2 2.2 2.0 Q/e zO Sc=.70 1.8 1.6 1.4 - 0.8 0.6 0.4 2 0.2 _ -0.2 -0.4 -0.6 I/. \ -0.7 ---- 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 77 Figure 15. First Order Complementary Integral Function zc.

.'99The surface values of the perturbations involve only the total values of I z1 and Iz across the entire boundary layer, I l(o) and I (oo) The calculated values of the deviations zI(o)]CAT and I(o) ]NON-CAT referred to the value for a stagnation point flow with a Davidson recombination rate (Q = 0, c =- 3/2), are shown in Figure 14 (catalytic wall) and Figure 15 (non-catalytic wall) for various values of Q/e and the recombination parameter ct It is seen from this data that the first order solutions are very sensitive to negative values of the non-equilibrium parameter derivative factor Q/es, belig several orders of magnitude larger than the stagnation point value when Q/e.<'- 2, and about 50% of the stagnation value when Qi~ = + 1 with the same recombination rate law, The recombination parameter C also has a noticeable effect on these results, more so for a non-catalytic wall than for a catalytic one; an increase in.-@ significantly reduces the non-equilibrium perturbation for a given Q/co This occurs because c( enters the reaction rate integral as 02 and for a cooled wall (6 ~<< ), will decrease if Xc is increased, particularly at small values of 0F in the neighborhood of the wall. A comparison of the two Figures, in. view of the difference in the values of Iz2 (o0; Q = 0, = 3/2) shows that the noncatalytic wall atom concentration perturbation is over 100 times greater than the catalytic wall atom concentration gradient deviation This is due to the absence of any surface reaction "restraint" on the gas composition in the neighborhood of the wall in the latter caseo For a given value of,9 the non-catalytic wall boundary layer will experience a much larger deviation from chemically frozen flow due to gas phase chemical reaction, The calculated effect of the freestream dissociation level (Ce) on the catalytic wall atomic specie gradient deviation is shown in Figure 16;

-100Z (O w,Q/ ) Zi (o;-3/2,0 ) CAT 100 A 51.6 L Izi (co) J I0 --- --- -- _- -1 t Figure 1 First Order Catalytic Wall Atomic Speie Graient50 Deviation verss Qe ad.70 eA2 = 8 I~1-S ~ ~ ~ ~ ~ e =.05 w-^ ~^- =-3/2 0.001 -2 -1 0 I 2 Q/~ Figure 14. First Order Catalytic Wall Atomic Specie Gradient Deviation versus Q/e and o.

-101ZI (0o,-3/2,o0) ZI (O;w,Q/E) 1 NON-CAT. 100 [iz, (0 -3/2,0)] = 6606 NON-CAT 10 Sc =.70 _____ ss^-2 - ____ ___ 0.1 -I 0 I 2 0.0010.0001-2 -l 0 I 2 Q Q/E Figure 15. First Order Non-Catalytic Wall Atom Concentration Deviation versus Q/E and to.

-1021.15 1_ 1.10 1.05 o 1.00 ae 0.95 N N.90. 85 _ —--.80.75 0.25.50.75 1.00 aZe Figure 16. Freestream Dissociation Level Effect on the Catalytic Wall Deviations.

1.15 -0130.1 4S.Q I.- Z 0F 1.005 cmt W- \C w D IF —z,~ 0. U: <095 I- S i~n ________ _________________ ^s^^ _______ NON-CAT. TEMP. GRADIENT 0.85 1.00 0.905 0. 6 —-------------- Sc Figure 17. Schmidt Number Effect on the First Order Perturbations.

this curve applies equally well to the catalytic wall temperature gradient solution. The effect of Schmidt number on the first order deviations is shown in Figure 17- Although this effect is a minor one in the perturbation analysis, it is interesting to note that itts direction is reversed when the wall surface is changed from catalytic to non-catalytic because of the change 2 in the recombination rate term zF in the reaction rate functionThe integral I (oo) has also been evaluated for all the previously 2 indicated combinations of 9/c and XD as a function of 0A over a range from 8 to 20 (a range that includes the'A values as well) for the highly cooled A4 wall example OW = ~05; the effect on the perturbation solutions is shown in Figure 1.8 It is clear from this Figure that the integral is not very sensitive to A2 for 9A2 > 5; doubling the value from 8 to 16, for examples oA2 A2 increases I2 (o) by less than five percent. This slight increase is to be expected from the exponential manner in which ~A enters the integrand for a p 2 cooled wall (See Equation (4~66))o The dissociation rate which contains this term exhibits a very rapid rise in the outer portion of the boundary layer primarily because of the influence of the cooled wall temperature function.Fo This behavior is not changed very muchs as far as it contributes to the overall reaction function R2, when:0 is changed by a factor of two or less, The sensitivity of z (co) to this activation energy parameter will be much 2 greater when the wall is not highly cooled; the calculations have indicated that Figure 18 is applicable for x < o20o The assumption of a highly cooled wall. is progressively worse for higher OW values and 3A will exert An increasingly more prominent.effect on the perturbation solutions0 m-2' The recombination rate temperature dependence term eF in the integrand of I~. causes the latter to be very sensitive to the value of B.

-1051.05 1.04 I.I t " 0.05 FZ 1.03 E F C 0.99 0.98 0.98................ 0 5 10 15 20 Figure 18. The Effect of Activation Energy Parameter on the First Order Perturbations.

,0lo6when it is small compared. to one and when cw departs significantly from a value of two. The calculated effect of the QW variable for a highly cooled wall was fomnd to be representable by the following formula: Wall Perturbation (lo, 1 ) -(4 79) Wall Perturbati6n _'(c, - - QW2 for both catalytic and non-catalytic surfaces, For a highly cooled wall with XC < 2, this equation predicts a significant correction due to small changes in OW. For example, a 20% correction from QW =.05 to QW1 = o04 would increase the first order atomic specie and temperature deviations by a factor of 2.20 or 120%. The viscous dissipation also enters the recombination temperature dependence term (Equation 4465) and has been evaluated as a function of the parameter ue2/p Te. The results, which clearly depend on the recombination rate exponent co and to a far lesser extent on the wall temperature ratio Q6, are shown in Figure 194 The effect of increasing ue2/ep Te is to reduce the size of the first.order perturbations when cr < 2, this reduction being slightly greater for smaller values of QW. When ue2 /s Te> 1, the dissipation effect on the recombination term becomes significant to the accurate calculation of local non-equilibrium specie and temperature deviations in the highly cooled boundary layer, The catalytic wall deviations are approximately one third as sensitive to the dissipation correction shown in Figure 19 for the non-catalytic wall case, due to the smaller recombination rate specie term in the neighborhood. of a catalytic wall (compare Figures 10.and 11), (2) Second Atomic Specie (Nitrogen) Equation (4.32) is similar to the first.atomic specie Equation (4,531); since the boundary conditions on XT are exactly the same as for Zy,

-107O1.0 -- 2 - 0.9 NON -CATALYTIC WALL 0.8 e 0.7 I-. —' V^,-^.20 20 w m 0.5 0 | 0.4 0.2 0.I 0 I 2 3 4 5 2 Ue Figure 19. The Effect of Viscous Dissipation on the First Order Perturbations.

.108the two solutions will be analogous. The complementary integrals are exactly equal (Xc = ZIc); and the non-homogeneous reaction rate term for XI is A i instead of 2F o Consequently, the XI solution is given by Equations (4.73-4076) by merely replacing 2e by O4e and Iz2 by I4, where Iz4 is the integral given by Equation (4o72) with eA2 replaced by 0A49 io.e, I4 = Iz2(0A4) The value of eA4 ranges from 16 to 18 (roughly twice A29) The insensitivity of the Iz2 integral to large changes in this activation energy parameter for highly cooled walls indicates that the approximation Iz4, Iz2, based on a common average value of eA, would be acceptable, This gives a very simple dolution for Xi, namely a direct proportionality between the two atomic specie deviations: KI' -. ~2 o(4.80) (3) Total Atomic Specie The total.atomic specie deviation is (^ q~;P~~ =$ ^^+^4~ ^", ^ (4.81) with zI given by Equations (4o735-476)o Some representative profiles, (a/%e)I versus r, are shown in Figures 20 (catalytic wall) and 21 (non-4 catalytic wall) for a value of: = 10 o These curves illustrate the typical distribution of the non-equilibrium atomic specie deviation throughout the boundary layer and clearly exhibit the previously discussed sensitivity to the parameters C and Q/E, Also shown in Figure 21 is the corresponding profile obtained in Reference 6, which is seen to be in qualitative agreement with the present theory. The difference between the two results, as will be

-109-.00028.00026 N 60!Lae.00024 / I/ \ w =) -3/2 Sc =.50.00022 / \ Q/E=- P =-10 // \ ae =.50.00020 --- - 2 / \.00018 T /.00014 ~~fU.00012 i.00010 /a\ PRESENT THEORY.00008 --- — I0 (wt =-3/2,Q 0).00008 I /w - SO.00006 -.00004 \ 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.8 4.0 -.4 4.8 770 0.4 0.8 1.2 1.6 Figure 20. First Order Atomic Specie Deviation Distribution in a Catalytic Wall Boundary Layer.

-110-.13.12 Sc =.50 ___ PR.70 = 05 ~ I U.10 \ =-3/2 Q/E = - 2.09 4.08.07 - - FAY AND RIDDELL w- - 3/2.06Q -- ~= o yla".05.03 20[ ] - -- Oe \a PRESENT THEORY.02 Q - 3/2 Q — 0.01 -.02 -.02 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 77 Figure 21. First Order Atomic Specie Deviation Distribution in a Non-Catalytic Wall Boundary Layer.

subsequently discussed, is primarily due to the use of a larger value for the total atomic specie reaction rate in Reference 60 A comparison of Figures 21 -4 and 22 shows that the catalytic wall specie profiles for 4 = 10 would barely be discernable.on the scale of Figure 22; the non-catalytic wall perturbations are much larger for the same value of Co The first order total atomic specie gradient and concentration deviations at a catalytic or non-catalytic wall, respectively, are ~I(~(zeE d = t9 Z- S sST7 @ ).(4q82) and /NL. - -I2e/ L /i > ir/-C4r 6- // i (eI/z (40 85) These are plotted versus i in Figure 22 for several example combinations of the parameters X and Q/c (for the first order deviations, the curves are necessarily straight lines, the curvature being introduced by the second and higher order perturbation terms)o The slopes of the lines are drastically increased when either W or Q/c is increased negativelyo Also shown for comparison is the Q = OX = -3/2 solution of Fay and Riddell, corrected from eW =.04 (which they used) to. = o05 by means of Equation (4o79)a Their result varies linearly with ~ for very small. and therefore independently confirms the assumed non-equilibrium perturbation expansion in the present analysis, Furthermore, the Fay and Riddell curve develops a negative curvature when; > 2 x 10"5; with further increase in L, the wall deviations are progressaively less than predicted by the linear first order theoryo This indicates that the second order perturbations are opposite in sign to those given by the first order theory and constitute increasingly larger subtractive

-112-.15 - Sc =.50.14 =.=70... 9w =.05 Ue /-p Te 0.13 Ce 536 2e= 21 - ae=.536 4e.326 AY.12 B.. 8 RIDDE.0J 0 0,,<,/.'(" > a =- 3/2 ~11,, 8 ew-.05 c3 ^.oh^I a o/ Q/o =- -- -2 ---- I,.,.i ts. 09 w__3/2 _ ___ ______ O.07' _ THEORY aI ~ Q,'w - -3/2,-.06 __-.0 5 ^ ^_~.05 _____ ---- __9____ ___ I/'~/ /a.04 l^ ^ ^-^'0il = -3/2 0 - -5 -- ----— x10 5_ --- 0 I0' 5 5x 10'5 IO(4 Figure 22. Total Atomic Specie Deviations Versus the NonEquilibrium Parameter I.

-113corrections as t increases. On the scale of Figure 22, the catalytic wall total atomic specie gradient deviations are extremely small when t < 10 4 again illustrating the negligible size of catalytic wall perturbations in comparison to those on a non-catalytic wall at the same values of,,Q/e and w. Of course, the noncatalytic wall deviation for Q/E > O and c > -3/2 is itself a small absolute number; for the stagnation point example (Q = O, (c = -3/2), the effect of -5 non-equilibrium at: = 5 x 10 decreases the chemically frozen wall atom concentration by 4% according to the present theory. When Q/e is negative, however, this local perturbation would be much larger (for the same value of:) because of the greater initial slope of the deviation curve shown in Figure 22, The small size of these deviations is an expected consequence (and limitation) of a perturbation type of solution; nevertheless, analysis of these small increments reveals the important basic mechanisms (and the significant physical parameters therein) which govern the non-equilibrium behavior in the chemically frozen regime. A comparison of the value of [a/Oe(o) I with that found by Fay and Riddell in the stagnation point example (Figure 22) shows the latter to be -5 72.7% greater at, = 10. Analysis has shown that this discrepancy is almost entirely due to the difference between the reaction rates used in the two respective theories. The total atomic specie reaction rate used by Fay and Riddell, as previously pointed out in Chapter II, was a binary form based on the total atomic specie concentration variable a (the first term in Equation (258)), thereby neglecting a subtractive correction due to therey ele a individual atomic specie behavior which is important in the presence of a significant amount of atomic nitrogen at the edge of the boundary layero In

the present comparative example, where aC4 = o326 and a2e =,210, calculations indicate that the neglect of this second reaction rate term by Fay and Riddell leads to too large, a total atomic specie reaction rate (and non-catalytic 2 wall concentration perturbation) by approximately the factor 2Q2ea4e/Ce -5 o47o When the Fay and Riddell value at 5 = 10 is reduced by 47%, a much better agreement with the present theory occurs, The corrected Fay and Riddell curve in this case drops slightly below the present theory, However, some difference between the two theories would be expected because the former theory includes a variable p. expression in the digital computations (whereas pp has been assumed constant in the present theory)o However, provided we take PRJLR = PW= W (as Fay and Riddell did), the effect of variable pp. on the perturbations is small in comparison with the effect of the reaction rate difference between the present theory and Fay and Riddellp work, This particular example clearly brings out the importance of accounting for individual atomic specie behavior in formulating the reaction rates for a non-equilibrium theory of a multicomponent gas mixture, In the foregoing example, for instance, the application of a more accurate reaction rate expression predicts roughly one-half the initial rate of non-equilibrium deviation from frozen flow found in Reference 6 for the same flow conditions0 Consequently, it appears that the chemically frozen regime at the stagnation point persists to larger values of 5 than shown in Reference 6,.(4) Second Molecular Specie A direct comparison of Equations (4,33) and (-452), since WI and XI share the same boundary conditions, gives the following solution: vt2= _e- a (4084)

.115This agrees with the previous statement (Chapter II) that a3 and a4 nmust be linearly related regardless of the gas phase reaction rates; The negative sign in Equation (4,84) means that non-equilibri"nm perturbation from frozen flow increases the molecular nitrogen whenever the atomic nitrogen is decreased in the boundary layer. (5) Energy Equation Equation (4.34) is quite similar to the species equations, The associated homogeneous equation for ~j is the same as that for zI if Se is replaced by PRp hence the complementary integral GQ of Equation (4,34) can be written rc = / PR). Furthermore, the inner boundary condition on Qr -1(o) = 0, is analogous to the catalytic wall specie solutioni so that the full solution to Equation (4,34) is similar in form to Equations (4,73) and (4,74): $w - < L()i it h g -e Z e -. r1g Is(X) - C }7 (4.85) with /r / _ as/2, /J|;2 Z e z i| (4 86)

and where I (P)I and The integrals IQ24 and IZ 4 (also Ig and. 1 are actually equal when S, PR(Le =.1). Recombination - dminated non-equilibrium perturbation, since the IQ integrals are positive, increases the wall temperature gradient near frozen flowo The typical temperature perturbation profiles (GI versus i]) have the same general shape as the catalytic wall atomic specie profiles shown in Figure 20, The first order temperature gradient deviations at the wall are given by the product of: and Equation (4~86) and involve only the total values of the temperature integrals across the entire boundary layer, If, (oo), These integrals exhibit the same insensitivity to the activation energy parameter for highly cooled walls as found for the corresponding specie integrals (Figure 18), so that the assumption I- I4 is a good,2 0 approximation; hence yn tt 4oJ ~) z-eie Aid f he A l ( 4e88) The calculated values of [e(o)]ON CAT referred to the value for Q -0 and.= -.3/2, are shown in Figure 23 (the values of [ ())]CAT, are nearly the same as the values of [z (o)]CAT shown in Figure 14 except for a small Schmidt number correction given in Figure 17) T The results in Figure 23 show, like the previous atomic specie solutions, the following features,

-117E i (0o, Q/e) 81 ( o,-3/2,0) 0L _ _ ( 2 NON - CAT. e2(^ i -3/2.0) 1= 2757 SC - PR =.70 A2 = 8 ew =.05 w ~ -3/2 0.01 -2_ -______ ____ o 0.00 2___ _ I0.0001__ -2 -I 0 I 2 Q/~ Figure 23. First Order Non-Catalytic Wall Temperature Gradient Deviation versus Q/e and c.

-118(a) A profound sensitivity to the parameters Q/e and c of nearly the same extent as the atomic specie solutions. The non-catalytic temperature gradient perturbation is slightly less sensitive to Q/e in comparison to [z (o)] CA for Q/e < - 1. (b) The non-catalytic wall temperature gradient perturbation is over 50 times greater than that on a catalytic wall for the same ~ and flow conditionso While this is one-half of the difference between non-catalytic and catalytic wall effects exhibited by the atomic specie deviations, one still can conclude that the catalytic 4 wall non-equilibrium effects are vanishingly small when. < 1 0 in comparison to the corresponding non-catalytic behavior. (c) Figure 19 can also be used to correct the temperature gradient perturbation solltions for the dissipation effect on the recombinationrate temperature term, Equation (4079) is likewise applicable for corrections to different highly cooled wall temperature values. Equation (4~88) is plotted versus ~ in Figure 24 for several combinations of the parameters Q/e and w; the slopes of the lines, as in the corresponding atomic specie case (Figure 22), increase notably when either Q/: or c is decreased. Also shown in this figure are the results of Fay and Riddell,(6) corrected to =.05. Their temperature data also exhibits the expected linearity with G for small I and indicates that the second order effect, as in the atomic specie deviations, begins at about = -2 x 10'5 and subsequently constitutes a subtractive correction to the first order theory~ The temperature gradient deviations are approximately the same size as (but opposite in sign to) the corresponding wall atomic specie deviations; at g 5 x 10"5, the non-equilibrium effect

-119-.15 Sc =.50.14 P =.70 =e.05 2 T Ue/cp Te= 0.13 e 536 /aze =.21 ae =536 - ~ e 536 (ae 326 X FAY.12 —I ^ -- I I/- ----- a RIDDELL 5 Le(0)NON-CAT VALUE (6)!, 1 - 3/2~ Q/Eci2 0e-.05.os09! "~/,,, / // PRESENT.08 THEORY /0I.07 C: =-3/2.06 I.05 /' /.04,,//.45?,;,OCA Q= 0.02 C = -3/2 0//z-. _ __ __N- ____ __, 0 li 5 x106' Id4 Figure 24. Wall Temperature Gradient Deviations Versus the NonEquilibrium Parameter [.

.120. increases the wall temperature gradient over the chemically frozen value by 4%o The disagreement with Fay and Riddell's results in the stagnation flow example shown in Figure 24 is again primarily due to the larger total.atomic specie reaction rate used in Reference 6, Furthermore, when this rate term is applied to the energy equation, Fay and Riddell do not correctly average the atomic specie reaction rate terms over the corre, sponding heats of formation (see Equation (A-10), page 40, Reference 6)~ These authors assume that Aro^v /o XAT6a Ar~t / = e + _ _ _ _ (4 K89) or he fo compone mix sed in his thesis in his cas, te for the four component mixture used in this thesiso Now in this case, the first bracket alone is the correct representation of the summation on the left side of Equation (4,89). Thus, Fay and Riddell err in this energy equation termn by using too small a value (i.e,, e < 1). The second term is negligable in comparison to the first and therefore does not contribute significantly to the error, This is the reason that al times the Fay and e Riddell result is shown in Figure 24; the difference between this and the present theory is now reconcilableon thee basis of the difference in reaction rates themselves. Reduction of the Fay and Riddell results in Figure 24 by approximately 47v% (the preaviously estimated error in

-1212 their rate for the comparative flow example) produces very good agreement with the present theory, considering the difference in the ppi laws used. It is worth noting that the two errors in Reference 6 mentioned above (the average heat of formation coefficient of the energy equation reaction term is too low by a factor ae and the reaction rate itself is about 47% too high) tend to cancel to a great extent; this means that the temperature non-equilibrium behavior is actually more accurate than the corresponding specie deviation solutions (which contain only the reaction rate overestimate). Again, however, we must stress the necessity to carefully and accurately formulate both specie and energy reaction terms when dealing with a multicomponent mixture if the physical effects of dissociation-recombination in the boundary layer are to be clearly understood, The specific heat effect, particularly the inequality of atomic and molecular specie specific heats, is very small in the non-equilibrium temperature gradient perturbations, as indicated by the excellent agrees ment between the present theory (which assumes cpl c- P1 constant) and the Fay and Riddell results (in which variable and unequal specific heat data are included in the digital computations), This might be expected for a non-catalytic wall on the basis of the complete absence of the specific heat inequality effect found in the chemically frozen case, Furthermore, it is confirmed by purely theoretical reasonings The retention of the cp2 - cl terms in the original energy equation can be shown to add the following additional term on the left side of the first perturobation energy Equation (4i.4) when c-p and cp- are constant:

.122" ( _f/_i t __ ~, — t-iF(,. (_o9o) where _Gp _ =. R2/zbW ^ /^ ~ d~e ~l)^ + c/- ^g J(4 92)'When a_ ofcours/, thisterm When oe(c p/cpl 1) is very small compared to one, of course, this term is obviously negligable. However, for the flow examples considered in Figure. 22 and 241 this parameter is about.209 which is not very small compared to imityj hence, a further examination of Equation (4,90.) is necessary. For a non-catalytic wall (zF' ( 0,. z:.'F. 1), Eqaation (4.90) reduces to _(2 ^ (a-)( -/^lT W (4., 93) / ~<2e e//*/ — E /) which is extremely small in comparison to the non-catalytic value of the reaction rate term on the right side of the energy Eqguation (40-34), particularily in the neighborhood of the wall where zI: O, Hence, the cp cp effect on the temperature gradient perturbation should be very'P2 l small. On the other hand, in the catalytic wall case, the entire term (4,.9 ) remains and is comparable in size to the corresponding catalytic value of the reaction rate ter. on the right side of Equation (4i534)

.-123 However, because the catalytic wall temperature deviations are in themselves so extremely small (approximately 1/50 of the corresponding noncatalytic wall deviations), the specific heat effect would hardly be noticeable even if it changed the QI(o) solution by 100%, Based on the effect of Cp2 C cp on the frozen flow catalytic wall temperature gradient, a 30% effect would seem nearer the actual value. It appears that inclusion of the specific heat effect is a refinement which adds negligably to the accuracy of the present perturbation analysis and constitutes an improvement, mainly in the catalytic wall case, which is expediently handled only by digital computation, (6) Heat Transfer The first order wall heat flux perturbation, defined in Equation (4.253) (for equal specific heats) as p, Z= re{'d + Lel[% -J + Yt r)] }, can now be written by means of the above atomic specie and temperature gradient deviation solutions in the following forms for highly cooled catalytic and non-catalytic walls, respectively: 4= - 4 - ^20 fatomicQ~~j scie and' 4epra7. e g i (na) 2= 2 ~8eie4f('O \ Jo The catalytic wall heat transfer perturbation, being the difference between atomic specie and temperature gradient deviations which are nearly equal in

124the Lewis number range.8 < Le < 1i4, is immediately seen to be a very small increment. In fact, when Le 1 (Ig2 Iz and Iz2 3 I since Sc, PR in this case), the perturbation vanishes, This, of course, agrees with the known fact that the increase in heat conduction is just balanced by the decrease in the diffusive heat flux when all the atoms are recombined on the' wall and the Lewis number is unity, When SC < o70 (Le > 1 for PR:'70), which is the normal case for air, a very small decrease in heat transfer due to non-equilibrium perturbation from frozen flow exists since [~(o)]A is slightly less than [zi(o)]CAT because of the slightly different complimentary integral function zc present in the integrands of the reaction rate integrals (see Figure 17), This decrement in heat transfer is extremely small in comparison to either the catalytic heat conduction or heat diffusion perturbation (which are in themselves very small) The noncatalytic wall heat transfer perturbation aonsists entirely of heat conduction in the absence of any surface recombination; therefore, Figure 24 also represents the heat transfer perturbation for this case when equal specif ic heats is assemed. The various T"correction" curves (Figure 16P19) and wall temperature correction formula (4179) can be applied directly to Equation (4o95) for various local flow situations, Figure 25 can be used to assess the effects of the local non-e'qcuilibrium parameter derivative factor Q/E and possible changes in recombination rate law on the nonYequilibriui heat transfer perturbation0 In addition to a very small indirect effect on the temperature gradient perturbation, the unequal specific heat term also can alter the conduction heat transfer as a result of a small perturbation in

-125the specific heat coefficient of the temperature gradients If one includes this specific heat term in the wall heat flux expression (assuming cp and cp are Unequal constants), the heat conduction part of the perturbation ~QI assumes the following form: (cjc ri U = f/ r(o)ik( )+ f $' ~] (4.96) with (gD,~ /O) = C ^ /D2/^ - J OJIje h~The specific heat perturbation therefore adds a separate increment to the heat conduction, (o).. (.o), in addition to modifying QI(o) itself, This increment vanishes for a catalytic wall (C-I' 0) but constitutes a subtractive correction to the equal specific heat non-catalytic wall heat transfer that is proportional to the chemically frozen wall temperat-ure gradient, the difference in specific heats, and the wall total atomic specie concentration deviation, For the Q = =0, C - -3/2 stagnation point flow shown in Figures 22 and 24, for example, Equation (4.96) predicts the following ratio for a non-catalytic wall: for Cp/cI = 1.4, This is a 153% reduction in wall heat transfer pertur. - bation due to unequal specific heats, Since the specific heat effect on;j;(o) is negligable, a direct and simple account of the specific heat inequality in the non-o.atalytic heat transfer can be taken by merely subtracting from [ Q(O)] the first term in Equation (4,96), which is proportional to known values of the frozen and perturbation solutions4

Second Order Deviations The second order frozen deviations are analyzed in the same way as the first order solutions. However, the presence of much more complicated non-homogeneous reaction terms in the second order specie and energy equations makes the numerical task of evaluating the particular integrals which arise very laboriousalthough still possible to do on a desk calculator, The form of the second order deviation solutions will be given to indicate the nature of this increased complexityg and to show that the secoxnd order perturbations are inherently negative with respect to the foregoing first order deviations~ (1) First Atomic $pecie The second order atomic specie Equation (4I35) is of the same form as the first order specie Equation (4o31), the only difference between them being that twice the value of Q/ appears in the second. order equation0 Therefore, the second order complementary integral is obtained directly from the first order case by using 2 o Q/c instead of Q/c in the latter z (Ri Ss Q/1)6 - Iz (TJq Sn 2~Q/c) Since the bou'ndary conditions on zI and zi. are the same, the complete solution to Equation (4355) is formally identical to that for zI when Zl (Q/C) is replaced by Zc (2 Q/c) and when the first order reaction rate function on the right side of Equation (4351) is replaced by the second order reaction rate function on the right side of Equation (4o35)o The following solution results~ i~h< 0,=~7 -A -4^- (4297)

4*127T" with f / ~ ~-cL ~. - -r) Z ~'/~ ~j' -'- L-r (4.99) with T^CR -- e fr4 (j/, J, (4o100) where and 4P4= 1 —' F'P -4$) -2...^. (4-10l) The values of the reaction rate derivative functions 2 J,, and )^~ are given in Appendix D, When these relations are used in conjunction with the first order soluttions (4.80) and (4,84), the second order reaction function ~z appearing in the integrand of c; can be expressed as follows: (/~_ _r).( /L )Sj)Z (v,e ) (/ ) (4,104) 7 fr |' Z r @2 )/-<^ / d* (4F 0 F

.128. It is apparent from this expression that the:computation of the second order solutions involves a great deal more labor than the preceeding first order solutions, Since Ze is always positive near the wall, the second order specie deviation will be negative when 412 (o) is positive (ioe., when the integrand of t2, involving the second order reaction rate functions, is.predominaintly positive throughout the boundary layer)o The function. deterines the sign of V4 iand. hene of the second order specie deviation function zII- Now the coefficient of zI in.., for o < 2 and a highly cooled wal, was found to always be predominantly positive tlhroughout the boundary layery whereas the coefficient of'o is a negative valaue, Since the previous first order results have shown ZI to be negative and QI positive, the second order reaction function Vz (and the second order integral ) is therefore a negative function throlughua t the boundary layero Thu s, the second order specie deviation function is opposite in sign to the first order deviation and acts as a su]btractive correction to the overestimate of the echemically frozen deviations predicted by the first order theory0 On physical grounds, this would be expected from the fact that the second order adeviations are proportional to the products of frozen reaction rate function derivatives and the first order deviations; since these derivatives have a sign Iwhich is opposite to that Qf the first order deviation fumctions in the highly cooled boundary layer, the second order deviations tend to "drive" the solution back toward frozen equilibriumo (2) Second Atomic Specie The solution to Eqation (4Q36)^ in which XII is subject to the sanme boundary conditions as Inposed on the first atomic specie function

-129zII' is xowpletely analogous to the foregoing solution.for zII the Complementary integrals are the same and XII is given by Equations (4,97) through (4,103) when ae is replaced by a4e'and the integral.2 is replaced. by 4j, where W^^ is obtained from 42 by replacing ~A2 by ~O4 and the reaqtion function V by }= {0 + f/ ~,+ f (^-2J% +S]} (40tO5) The fInction canvf be rewrlttfen, with the aid of the values of 4. )4.~'and. given in Appendix B and the first order solutions (4, 79) and (4.83), in the following frm: /i:= 2 / / 0), + -x /-de / 4 ^ A The pred.mianant sign of, like, is forad. to be negative for IC < 2 and. a highly dooled. wallj acaordinglyS XiJ is positive in the neighborhood of the wall (whereas XI is negative), (3) Total Atomic Specie The second. order total atomic specie deviation is given by ( eo2 e-in L /

As a single illustrative example of the typical second. order.effect, the integrals b (2)j and (o) have been evaluated for the non-catalytic wall (which exhibits the greatest non-equilibrim d.eviation ) with Q = O and C. 35/2 (stagnation point flow with the Davidson recombination rate law)3 The increased complexity of the second order reaction rate tems makes the pursuit of as comprehensive a parametric study as the first order ease'nfeasible in this thesis (although it would be expeatedg based on an inspection of the above second order solution integrals, that the second perturbation exhibits at least as much if not more sensitivity to the parameters'Q/s anad c as the first order perturbations and about the same sensitivity to the parameters S., w~Aw' and UeC Te) e. Pe e This specific calc:ulation has confirmed the insensitivity of the results to the 0A parameter and further has shown that it is an accurate approximation to take - for a highly cooled wallo The second pert'urbation of the total atomic specie concentration at the wall, given by fSed -));6&es) /Zj t j YU (4-to107) is shown for this example in Figure 235. The difference between this result and the second order:.effect shown by the Fay and Riddell calculation reflects the reaction rate.discrepaey previously discussed for the first order theory. At 5 = 10, the second order effect in this.example constitutes a 220% subtractive correation to the value predicted by the first order linear variation,,

(4) Second Molecular Specie The WIi solution, as far as heat transfer is concerned, is actually needed only if the third perturbation were to be computed (to evaluate the non-homogeneous reaction rate terms), However, it is readily obtained in terms of the XII solution by inspection of Equations (4,3-6) and (4537) and the respective boundary conditions: 4 ~=3 ( a' (4,1o8) (5) Energy E quation Just like. the.first order case, the complementary integral of Equation (43,8) is found from that of the specie Equation (4,35) by replacing Sc by PR in the latter: gII c zI-c(PR). (4o109) The full solution to Equation (4o38), in view of the GII(o) = 0 inner boundary condition, is similar in form to the catalytic wall second order atomic specie deviation finction solution: 70-, (4 -lio) (4,110) t i(sMt go - r h jag with ^CjL f^/^/..... ll

and where 4 ()- and 2 -/- I^^^ x^^^y^. (Le 022 Using the same arguments employed for the specie integrals, b and f are found to be predominantly negative in the highly cooled boundary layer, Conseq.uemtly, GIi and G. (o) are negative and therefore opposite in sign to the first order temperature perturbations. The integrals (J ) and ~g, (c) have been evaluated for the Q = O., c = -3/2 non-catalytic wall case (Qw= 05) and found to be very nearly equal, again verifying the insensitivity of the non-equilibrium deviations to the activation energy parameter for a highly cooled wall, The equal specific heat non-catalytic wall second order temperature gradient perturbation (2 times Equation (4Ql11)) for this special case is shown in Figure 25. The present second order deviations are smaller than those predicted by Fay and Riddell in accord with the discrepancies between the two theories noted in the first order case. Further Discussion of Results Application of First Order Theory to the Calculation of Local NZonEquilibrium Deviations in the Laminar Boundary Layer The foregoing analysis has shown the first order nondequilibrium deviations from chemically frozen flow to be proportional to the product of the characteristic nonmequilibrium parameter G and an integral of the first order reaction rate function across the boundary layer, These integrals were found to be extremely sensitive to the recombination rate law exponent Cu and the non-equilibrium parameter derivative function Q s- >. / However,

also depends on c and the local inviscid flow and the sensitivity of: times the deviation integrals to o.and the local inviscid flow is different than that of the integrals alone. This net behavior of the atomic specie and temperature deviations must be taken into account when applying the theory to a calculation of local atomic specie concentration and heat transfer deviations in a highly cooled hypersonic boundary layer. Since the theory has shown that a non-catalytic wall exhibits a far greater non-equilibrium behavior in the chemically frozen regime than a catalytic wall, the present discussion will be confined to the former case, Consider the ratio of the first order deviations for any two values of the recombination rate parameter, w1 and u2 (non-catalytic wall): If -trX2@, o)A 3 ) 4i ffi y c~o) (4)113) for the atom concentration and /~ &z'4A 34v1 2fe (4.114) for the temperature gradient. The dependence of _Ap and Z2 on C is given in Figures 15 and 23, respectively, and the definition of ~ (Equation (2.90)) provides? - 2(4, 5 all other conditions remaining the same, As an example of the effect of a change in the recombination rate temperature dependence, the ratios (4.113) and (4.114) have been evaluated as a function of cl for o = -3/2 and

-134several freestream temperatures at the stagnation point (Q = 0). The results, which are approximately the same for both Equation (4.133) and (4.114), are given in Table II. These data show that the net deviations still remain sensitive to the parameter co but the trend of the effect is opposite to that on the deviation integrals alone because the factor exerts the dominant influence and causes the deviations to increase with increasing,. The heat transfer add surface composition deviations are equally affected by w for the non-catalytic case. Both of these deviations are greater when the freestream temperature is increased, as indicated by EHquation (4.115). These results suggest that local non-equilibrium deviations calculated on the basis of a recombination rate temperature dependence exponent significantly greater than -3/2 could be largpr than those calculated on the basis of w = -3/2. TABLE II EFFECT OF RECOMBINATION RATE LAW ON TIE FIRST ORDER HEAT TRANSFER DEVIATIONS FROM FROZEN FLOW (eW =.05, ue2/CpeTe = ) [ i (o) ] / [5i (o~) a -3/2 CO I02(oo0))/Ie2(o -3/2) Te = 3000K' Te = 60000K -3/2 1 1 1 0 1.85 x 10-2.585 1.624 2 6.51 x 10-4 2,06 23.3 The non-catalytic wall heat transfer (equal specific heat value), for example, would increase by a factor of 23 for Te = 6000~K if w were to be changed from -3/2 to 2.

-13 -5 The parameter Q/c also has a pronounced effect on the deviation integrals and it is a sensitive function of the local inviscid flow (Figure 6). The products o zI(o) and g ~I(o) exhibit a different variation with the local inviscid flow than predicted by the effect of Q/~ on ZI(o) and QI(o), however, since ~ also may vary along a hypersonic body (such as shown in Figure 4). As an illustration of this, consider the following ratios which represent the local wall atom concentration and temperature gradient deviations expressed as a fraction of the corresponding stagnation point values: " 4<'(o) r i3/4) /7! r4i(O~jQ-o and r d These are plotted versus the distance along a hypersonic hemisphere-cylinder at Moo 10 in Figure 25 for D =.-3/2, based. on the inviscid flow data shown in Figures 4 and 6 and the Q/e dependence of the deviation functions given by Figures 15 and 23, This example is intended to illustrate two facts: (a) the local non-equilibrium frozen flow deviations for a given recombination rate law can change significantly over the length of a blunt-nosed hypersonic body, and (b) the non-equilibrium state of the stagnation point is a good indicator of the local non-equilibrium behavior on such a body only in the immediate neighborhood of the stagnation point (X/d <,25) and at a point approximately one body diameter downstream of the stagnation points Figure 25 shows that in the nose region X/d >,25, particularly in the neighborhood of the sonic point (X/d s <60 j o70), the boundary layer may noticeably "unfreeze" relative to the stagnation point, Accordinfgly, the heat transfer perturbation becomes roughly three times greater than the stagnation value, Further downstream, the flow subsequently tends

-136I0 __ L _ X1 1 1 1 1 1 11 I | I E l INCE I = 1.2 w = -3/2 / r> \ MZCOOM 10 - w.05 2 - Ue __ _ = 0 0 1.0 1.0 Z - - -/ ATOMIC SPECIE CONCENTRATION z z ~~~~0 CDc~Cldr ^ __\ - TEMPERATURE GRADIENT ~_o -- - \-' " " ""H —-7 (HEAT TRANSFER) - - W *0bI - - -I -. ~~~Q~~~~~~~x d \ Figure 25. Local First Order Composition and Heat Transfer Deviations from Chemically Frozen Flow Along a Hypersonic HemisphereCylinder.

to rapidly "freeze" relative to the stagnation pointr dropping the local heat transfer perturbation to as much as 1/100 of the stagnation value in the region X/d. 10, Finally, at very large X/d, the local boundary layer deviations begin to gradually increase, tending to approach and exceed the stagnation pont point value. If the body were extremely long, the boundary layer would continue to unfreeze and.eventually approach a state of thermo. dynamic equilibrium regardless of the stagnation point situation, This trend toward thermodynamic equilibrium at X/d -> oo is further accelerated by the increased pressures associated. with the boundary layer ~inviscid flow interreaction that undoubtedly occur in this region~ The Error in the Use of Fick' Law The perturbation analysis is based on governing equations in which the mass diffusion terms for each of the four component species have been assumed to obey Fiek's law., While it has been found that the assumption of equal physical properties (except atomic specie heats of formation) between the two molecular species and between the two atomic species leads exactly to Fick's law for each atomic specie diffusion flux, the molecular diffusion fluxes do not obey this law under these conditions; The error in assuming Fick law diffusion for the a3 specie consequently feeds into the non-equilibrium an, alysis in two ways. First, it causes a direct error in both the frozen and the non-,equilibrium perturbation molecular nitrogen solutions. The largest error occurs for the catalytic wall case, Second, it has an indirect effect on the atomic specie and temperature perturbation, since a3 enters the reaction rate as part of a term multiplying the dissociation rate exponentipl, We can show, however, that this second effect is negligible in the perturbation4

-156 r analysis (this is done below) Then the error in a3 can be considered separately from the atomic specie and temperature solution if a particular interest in a more accurate molecular specie profile arises, The exact multicomponent mixture diffusion expression (Appendix A), when applied to the present.approximate four specie mixture, gives the following expression for the a23 diffusion flux term. (/3 # /Z )~ lea 0 =' 4G 116 This in turn yields -the following boundary layer specie conservation (^ Vj /)d A 7, 2SW e w g cd-Z/1;.;7@-2CiJ~g (4:117) This equation reduces to Equation (2,66) only when F i/ck - 1 or when a and a are co.nstant Comparison with Equation (2,65) shows that a3 and o4 are linearly related. independenti of chemical reactiona only when either,of these two special conditions hold.0 The boundary conditions are fixed independently of the diffusion law for either a catalytic or non-catalytic wall, Consequently, the error due to the use of Fick law diffusion for ao distorts the slope y(o) at,a catalytic wall but distorts only the nonequilibrim wall perturbations of a3 at a non-catalytic wall, since the chemieally frozen solution 5lF. a3e prevails in the latter case for any form:of the diffusion law,, The non-Fick diffusion effect would therefore appear to bee of most interest in the former case0 Equation (4117) can be transformed to the similarity plane: form (as in Chapter 3) and subjected to the similarity approximations and perturbation analysis to evaluate the

-139non-Fick effect on the chemically frozen catalytic wall a3 profile and the first, second, etc., order non-equilibrium deviation profiles. Since the prime interest of the present work lies in the atomic specie and temperature variables, a detailed investigation of such a solution is not considered hereo The effect of the error in a3, which is greatest in the vicinity of the wall, can be neglected as far as the atomic specie and temperature non-equilibrium perturbations are concerned because of the following considerations- (a) In general, az3 enters these perturbation solutions as a terra inthe coefficient of the exponential dissociation part of the reaction rate integrals. As we have previously seen, errors in this coefficient in the inner part of the boundary layer have an extremely small effect on the perturbation solutions for a highly cooled wall (the dissociation term coxntri'butes significantly to the integral only for' > 2, where the error in.3 becomes small), (b) Furthermore, in the first perturbation case, the a3 term enjters as the chemically frozen value- since the non-catalytic wall solution for 3F is exact, there is no error in the first perturbations in this case. A slight effect on the dissociation term coefficient for a catalytic wall would be present because the non^Fick difftusion altxers the slope of the O3(rB) profileo However, it would be a negligable effect because of'the extremely small size of the dissociation exponential. The error in the molecular specie profile, as it affects the other perturbations, might be mr-e prominent when the wall is not highly cooled.

CHAPTER V DEVIATIONS FROM TERMOD'YNAMIC EQUILIBRIUM Some Features of Non-Equilibrium Couette Flow The relative simplicity of the governing equations and the qualitative similarity to a boundary layer flow makes the Couette flow problerj q valuable theoretical research tool, and as such it has proven partiqcilaqly use-ful in obtaining a better qualitative understanding pf various hypersonic real gas effects (15 282 5 3 We wish to consider deviations from dissociation-ecombi'nation equilibrium in such a CQuette flow with the sole object of indicating the general features of the non-equilibrium behavior which may serve as a guide in a subsequent appraisal of the same problem in the laminar boundary layero Couette Flow Equations The Couette flow constitutes the motion of a gas between two parallel infinite planes spaced a distance 6 apart2 the upper plane moving at a velocity U with respect to the stationary lower plane. The equations describi ng this problem are obtained from the previously given boundary layer equations by setting the derivatives in the X direction and the normal flow velocity equal to zero2 so that only gradients in the normal direction Y remaino If we assume a Fick law diffusion in a four component air mixture, then, we obtain the following governing equations~ dy( ( 7d ) = - Z ( ) d- A.KLr ~~~14s ~ ~ ~ ~ ~ (.

d/m/ d3)= (5.3) where /r B -L - I, ~) z - r -;~Y'r)] 7 bust. = -2/or' d22J -I - r J (5.5) -2 W - (/~~ ^ W =(5,6) 6 = cosr r_, = -e, / )7 (5, 7) ly (< d + dI The boundary conditions to be imposed are analogous to.those for a boundary layer problemo, At the moving plane Y -= 6 u = U and T = Tee and we shall prescribe the gas composition to be known, and equal to the equilibrium composition for the given values of p and Te where </or) -c^ -^ (.,9)

3c^ -^~ v7^~?-~ ~, ~(5.11) At the stationary inner boundary, u = 0, T(o) = TW and ~i(.) ZE(O) g j ^d)= oo4,) Ok. 3J = <~ 0() for a catalytic wall or io ) _ = _ = - for a non-catalytic wall. The gas heat flux at the stationary wall is given by -Q)V ___5 ~ =-S z:- (5,12) /w - 27' + - +((% It is convenient to simplify the problem, retaining only those physical features of major importance to the non-equilibrium chemistry, by the introduction of the following assumptions: -//,2 - cdn.,, & = coAr ovr. Fr J i2-nt r, o e -/Z c sn r- ara. Further, introduce the following new variables:

1l4317 - *u- -fir - 2,* ~ -2 - - ___ J - SC (;/ - Finally, introduce the following coordinate transformation. _/./_L /.(513) Applying these transformations to the foregoing Couette flow equations, using relations (5.9) and (5.10) to simplify the reaction rate terms, we get the f9llowing simplified forms of the governing equations (denoting derivatives with respect to q by primes): _/ Z- _. (5.14) -'= ~ (5,15) -: "3c- / 5e - (5016) where ~ ( J/+~)?^a,.(l 2.~,^, 7 (5.17) ode t //l 2e C s Ye_ _ //>' S< ~ 1 ( /- -7e (5,8 _ _ _ ____ - __ L ^4/-_ (5A18)

UL-t~ Off'~ [~ <(5.19) Aff^e ( J^/f no -^|^ -f@(2(5.20) The corresponding boundary conditions now read: A_ =X+JgJ C.- = =" - 4 = M = f7o; = te o) 4 ) CA/f4z'r/C pALdL; iAo)' () — {j. = K, A d- ( TA.lk'7T/E C f ZL. Regardless of chemical reaction, the momentum Equation (5,19) and the energy Equation (5.20) (using the boundary conditions) can be integrated twice to give the following results: 4% = /. +_ (5,21) uwr/-y X7' /-A ^ (- (5.22) The latter provides a relation between temperature and the mixture composition that holds for an arbitrary degree of non-equilibrium. The heat transfer can therefore be written entirely in terms of the composition

tl45 variables as followss ___ / - _ __/- _ ^ ^ }^AM]s[IWJ} (5.23) This equation indicates that the Couette Flow heat flux to a cool catalytic inner surface (z(o) and X(o) zero) is invariant to chemical reaction in the gas, The non-catalytic case is not, however, since z(o) and X(o) must in general be obtained from a solution of the reaction rate-influenced atomic specie equationso Non-equilibrium effects on heat transfer would therefore be expected to be of most interest in conjunction with the non-catalytic wall case, The Thermodynamic Equilibrium State This extreme of dissociation-recojibination behavior arises, as previously pointed out, when G becomes very large and the corresponding factors in the reaction rate terms in the species equations vanish0 Referring to Equations (5,19) and (5~20) for example, we see that as -oo the net reaction functions on and y must approach zero to maintain finite diffusion on the left sides~ $ = provide the classical thermodynamic equilibrium relations from which the specie concentrations are calculated as functions of pressure and temperature, as followso* * This^d.i su ss. -o_ equ b m atosbwe the s e (5,2) * This discussion of equilibrium relations between the species and temperature applies equally well to a boundary layer flow problem,

____ ____ __-jrlf.- A \ (5425) /+ 4 o. e 06. 8 / e 4 The thermodynamic equilibrium composition solution is completed by the following relationship between c3, cn and ch (i e, between WER, ZEQ and XEQ) in the assumed absence of NO formation. Q N79 -.7 or - (5426) The derivation of this relation is given in Appendix:G, This result is based on the absence of diffusion; any deviation from exact thermodynamic equilibrium must therefore introduce deviations from Equation (5-26) due to the actual diffusion processes which occur in the gas, Equations (5o24), (5~25) and (5,26) provide three relations with which to calculate the three unknown equilibrium specie concentration functions as a function of the temperature distribution, Consequently, the thermodynamic equilibrium energy equation can be expressed entirely in terms of the temperature; the Couette flow temperature profile (5,39) for equilibrium can be found from Equation (5,22) when Z(%EQ) and X(eEQ) are substituted, Once the equilibrium temperature is known, one could calculate from the specie equations the actual value of the reaction rate functions for a large but non-infinite,o For example, from Equation (514), _=o the_ > =.~eq ilibriJ. (5027) Also, the equilibrium temperature profile permits one to return to

-147Equations (5.24-5Q26) and calculate the chemical specie concentration profiles in the flow, This equilibrium composition is determined solely by the temperature distribution and consequently there is no provision for actually applying separate specie boundary conditions such as would be required in the presence of arbitrary surface catalysis~ The exact equilibrium concentration profiles therefore cannot be influenced by the wasll chemistry (this is consistent with the above neglect of diffusion currents in calculating the specie-temperature relationships, since the wall-gas specie boundary conditions -in general are concerned with a matching of the gas diffusion rate with the surface reaction rate), Only in the extreme case of a completely catalytic wall (where ai(o) = a. (Tw)) will i-L) = C~iEQo W the equilibrium solution in the gas be strictly compatible with the surface chemistry0 The equilibrium diffusion current Q -d EQx r k in the dEEQ, E gas approaching a non-catalytic wall will not agree with the zero diffusion requirement right at the surface unless the wall is so highly cooled that dZEQO (o) -~O0 Consequently, an infinitesimally thin gas layer may exist, dOEQ adjacent to tthe wall, across which the specie concentration gradient adjusts from z Q (o) to zero, and outside of which the flow is everywhere, in exact thermodynamic equilibrium at -> cn. This behavior leads one to expect that this sublayer will spread outward and assume a finite thickness when the flow departs from thermodynamic equilibrium0 Small Deviations from Thermodynamic Equilibrium Any deviation from the thermodynamic equilibrium solution must involve just those features that have been neglected in the equilibrium calculations A study of the prominent physical features -of near-equilibrium Couette flow will be made by considering small deviatlions from the

equilibrium solution as follows: 4 - ~.~ f+.Z~ j = Xy +Z., (5s28) =-. —, - - -,..=,-'"" accompanied by corresponding expansions in the reaction rate functions (noting that )~ at equilibrium), 9,<* ~~ TV+t OaZ )v'41 (5,29) where.=e_,... -,' =/. (5,30) Substitution of these expansions into the Couette flow equations and retention only of first order deviation terms produces the following differential equations~ V8"_ 7viwZ7L fi7>i;t+)-/$a: (5)31) ~~d~ 3"K= A r s (s, ir~~s,)- (532) I3'' 4,e~'WSv ^ ^ tz'~%)B'BrarJ) l-7Kk (5-33) ~B ~ ^ [(~d- _ ^^/)-z ^ - H)-z. +(54)

The following boundary conditions, taking into account those which prevail on the equilibrium solution, must be imposed on the deviation functions: Z04) - (i = W^) = ^W/ = ~ (5,35)' = 0; Z p)^ 0) ^)= (=KCA4r4/c W4L); 1 (5 36) Zap= - )E470J Z'(O)=-^ ) 87j, W^0= J ( c-RTAd/CwiL, S4e U4e Since WEQ = - - XEQ and WEQ - XEQ from Equation (5,26), a comparison - 3e a3e of Equations (5o32)9 (5533) in view of the boundary conditions shows that TW - - 0 Z'. (5.57) The molecular nitrogen deviation is hence proportional to the negative of the atomic nitrogen specie deviation' This set of second order, linear, non-homogeneous differential equations, constitutes a formidable tmathematical problem because of the variable reaction rate derivative coefficients and the coupling between the variablesZ and o The general procedure for solving these equations would be to obtain two uncoupled fourth order equations forZ andZ, respectively, by solving Equations (5o31-5o34) simultaneously; each of these equations would then be attacked by some suitable method, if possible, However, it is found that this general problem becomes so complicated that the original purpose of considering the Couette flow would be defeated, Some further simplifications of the equations, with respect to both the coefficients and coupling between the variables, must be made if any

-150physical insight into the near-equilibrium behavior is to be gained, Therefore, in the spirit of pointing out some of the prime qualitative features, some simplifying approximations shall be used. Approximate Solution The similarity of the boundary conditions and the proportionality between the atomic species previously found in the chemically frozen regime suggest the assumption~ 1 This seems to be a reasonable simplifying approximation for the purpose of evaluating the main qualitative features of the near-equilibrium behavioro Therefore, we shall assume ^ c dNArnr.Z = /Z, (5.38) where the value of the constant A is'chosen so as to be consistent with the boundary conditions. Now Equation (5.38) is compatible with the outer boundary conditions Z(Wj^ LW and the catalytic inner wall boundary conditions Z(O)-= -O for any value of A. However, the inner noncatalytic wall condition necessarily determines A to be (5.39) i, = he'(o).5 39) Assumption (5.38) completely eliminates the coupling between the two atomic specie Equations (5031) and (5.32) in view of Equations (5533) and (5.34) Consequently, we must solve the following single second order non-homogeneous differential equation forZ(4) subject to boundary conditions (5.35-5.36): Jz_- (r Cf=7_ 4 ~ I)34 -. (540) lease see co f

-151l where ^)~ = S^-W^A^ Z ~^~> -6)4; / (5.41) }4&)= 4?(g7+^A)i4j.' (5042) Unfortunately, and vary quite noticeably across the channel; the exact solution to Equation (5o40) would therefore require lengthy and cumbersome numerical integration. To avoid such an unfeasible numerical complexity and bring out the essential physical features of the problem, we shall now introduce two additional simplifying assu nptions, (a) Assume that certain average constant values of the coefficients 6 and (, 4 and r2 respectively, can be used in Equation (5.40o) While such an assumption. will certainly lead to sizable errors in the magnitude of the solutions, it seems a reasonable one when the.main objective is a purely qualitatieve appraisal o'f the near-equilibrium behavior. (b) Assume that the equ.ilibiri=um specie profile zEQ (1) is a cubic.in Q (0 (T) ) 0 This is a good description of the ZEQ (r.) behavior near the wall and can satisfy the ZEQ (Td) = 1 condition and simplify the particular non-homogeneous integral of Equation (5~40) b Taking the derivatives zEQ (o) and. zEQ (o) to be known, the following cubic profile satisfies the conditions ZEQ (j.d).1 and zQn (o) ) O M4 rJ (5o 143) with __= C____r = (5.44) The introduction of these two assumptions into.Equation (5,40) yields the following differential equation with constant coefficients for -, 5

~'152' l /_ r-a d As - ^}~ - <ZrF (5.45) where At The exact values of the reaction rate derivatives 2,, etco, are given in Appendix H.as functions of the equilibrium specie and temperature variables. Inspection of these formulae shows that for a highly cooled wall, the two derivatives ~~z and zL determine the predominate sign of the functions / and f above. Since >, 0O and Ka < 0, the signs of ~ and a across the channel will be predominately positive and negative, respectively; hence g > 0 and < Equation (5.45) can be directly solved in closed form. The complementary integral of the associated homogeneous equation is ( > 0) IV A particular integral of the non-homogeneous equation, when zEQ(r) = 0, is seen to be The compLete solution Z = Zc + Z involves two unknown arbitrary constants C1 and C2 which are determined from the boundary conditions on Z, The following solutions are obtained,, = t ) 4 -t- Pb /."/]/ jC (5046) 7L Vf

,f155where AX foc X fWS d/ r' / (5.47) = /.:- @ y />> and (:~48') where kh~i __a/o0;a @+_t___ 6 ), 2 r _, / ~, h ] 4' c/ Y, /.~f^ ___ &m v K (7 (5049) The remaining specie and temperature deviations may now be computed with Z(r) knobn0

154^ Properties of the Approximate Solution (a) The specie deviations [Z'(o)CAT~ and [Z(o)]NON-CAT vanish for -e ~co behaving as 1/~% for >> ~. The non-catalytic wall specie gradient does not necessarily vas- howver, sinzce -z (o) in the gas may be zero (b) Solutions (5046-5,49) show that the near-equilibrium behavior depends directly on the various derivatives of the equilibrium atomic specie profile at the wallo The catalytic wall specie deviation is proportional to the second derivative ZEQ (o), whereas the non-catalytic wall deviation is proportional to zQ (o) for large.o For a given value of B, these deviations become vanishingly small whenever the wall. is highly cooled enough to cause the equilibrium specie derivatives (as well as the concentration itself) to approach zero. Furthermore, the sign of the deviations depends on the equilibrium specie-temperature relations at the wallo We have /~- = ^ A (5o50) where dzEQ /dEQ is always positive (the equilibrium dissociation level EQ2 20 always increases with a temperature rise) and d ZEQ /dO EQ is positive up to approximately 50% dissociation and hence positive at a cool wall, Therefore, ZEQ (o) and zEQ (o) will have the same sign on a cool wall (aEQ (o) > 0) only when O'Q (o) is positive or a very small negative number, From Equation (5.22), the equilibrium temperature derivatives are found, to be

~155^4 ^ I 7 /'74 (^2 7 )-^(5 52) /Ir ^ - ^, ^ /^ PV77 Z 2 id/^ 7L e 6~' (2 /Jz Substitu.tion of these values into Equation (5.51) gives/, 4 _U%24;; t2) ^ 2val /6 Le U (o)W(l ~ + s (5.54) / iLe ZFh ( J,4 f) ^ If a linear equilibrium specie-temperature relationship is assumed* (dzEQ /deEQ - constant, d2zEQ /doEQ 0), Equation (5.54) shows that ZQ (O) is entirely due to a negative viscous dissipation term. The catalytic wall atomic specie gradient perturbation from equilibrium is therefore negative and opposite in sign to the non-catalytic wall perturbation (the latter., being proportional to z-'Q (o), is always positive),, However, this result conflicts with the catalytic wall diffusion increase found in. Reference 6. Alternatively, when a significant non-linearity in the ZEQ (0EQ ) relation is allowed (which is a much more realistic assumption), the positive second term. in Equation (5.54) can * Such an tiss-umption is made in Reference 28., for example.

~156predominate for highly cooled walls when is small. The catalytic wall for the laminar boundary layer. The present analysis therefore furnishes the following important conclusion: the near-thermodynamic equilibrium behavior in a highly cooled Couette flow will be a relidble qualitative analogue of such behavior in the laminar boundary layer only when a non-linear ZEQ (OEQ ) relationship is accounted for in the former problem, Furthermore, it is the catalytic wall case which is sensitive to this non-linear effect; the noncatalytic wall deviations from equilibrium are proportional to ZEQ (O) ~o dzEQ /dOEQ (o) and therefore not affected by -the assumption z"Q = 0 EQ0oE EQA (c) The typical distribution of the atomic specie deviation Z across the channel as a function of I, computed from Equations (5~46) and (5~48), is shown in Figure 26, The magnitude of the perturbations shown is, of course, not accurate because of the approximations used in the solutions~ However, an interesting qualitative feature of the non-catalytic solution is shown, namely that the adjustment between the gas and reaction-free surface occurs within a kind of sublayer in the channel whose thickness decreases as G increases. Since the slope [Z (O)]NON CAT = - ZEQ (O) is fixed while the specie deviation Z(o) decreases as 1/NT, the sublayer thickness Z)L /3/ ultimately approaches ze,rO when - oo. A similar result has been found by BroadWell (28) and Hirschfelder (9).

-157-.30 _ X2 Zt _, -2 _ 2 d ZEQ.20.10 0.2 0.4 0.6 0.8 1.0 400,ZCAT(= 9) CATALYTIC WALL 0 0.2 0.4 0.6 0.8 1.0 17d Figure 26. Atomic Specie Deviation Profiles in a Couette Flow Near Thermodynamic Equilibrium.

-158(d) The temperature perturbation in the gas is given entirely in term of the atomic specie solution~ ^ = fZef%/jji- 2,)Z' -< (5.55) f A~)/4p = - / ( ^2;/ q 71 2^ (5.56) The catalytic wall temperature deviation is always opposite in sign to the atomic specie deviation, The corresponding temperature gradient deviation for >> 19 substituting relation (5047) into Equation (5.56), becomes ) = e -W,0) (5.57) which vanishes for o -., The catalytic wall heat conduction will decrease due to thermodynamic equilibrium deviation whenever zEQ (o) is positive, which in turn is true whenever the zEQ (0EQ ) relation is highly non-linear in the presence of a cooled wall. On the other hand, the non-catalytic wall temperature gradient perturbation, from Equation (5-,49),, beconmS 2i&2I= - / B^4-/g (5o58) for >> 1., This equation predicts that the heat conduction perturbation in the gas does not vanish as o - o because of the inner boundary condition Z:i(o) =- zEQ (o) required by the absence of surface reactior.n However, the region, adjacent to the wall in which the non.-vanishing temperature perturbation exists will shrink to zero as G -> oo, leaving a temperature gradient jump right at the wallo When the wall is highly cooled enough to give ai (o) - O as well as zEQ (o) 0 O the heat conduction deviation becomes *.5. ao 7+O s el aszE.

"159vanishingly small even when ~ does not approach infinity, Otherwise, Equation (5.58) shows that the non-catalytic wall heat conduction initially increases due to highly cooled wall nOn-rquilibriumn deviation at large values of As decreases, the sign of this deviation will ultimately change and the heat conduction will then decrease with further departure fpom equilibrium, (e) The heat transfer deviation on a catalytic wall is exactly zero; that on a non-catalytic wall is given by Equation (5.23): and is always opposite in sign to the atomic specie deviation at the wall. For large, the use of solution (5I49) gives Li - _ z e 4(Ps4j v) 46'~~)- (5 59) The sum. of the wall heat conduction and diffusion perturbations in tha gas Vanishes as' -- co Furthermore, we see that the heat transfer calculation is not affected by the non-linearity of the ZEQ (eEQ ) relation as far as sign is concerned; Equations (5.50) and (5~52) show that zr (o) is always positive on a cooled wall and the heat transfer therefore decreases as a result of departure from thermodynamic equilibrium in the Couette flow, The above general features should be useful as a guide in analyzing the highly cooled laminar boundary layer near equilibrium and in any future deta.iled numerical solutions (accounting for the variable coefficients) which a-2 may be of interest, The recombination rate temperature dependence term 0EQ

.l6o0would have a powerful effect On the results. Here we have found it to enter ooi-2 -1/2 as (.EO.AV; one would expect a decrease in X. to reduce the size of the deviations on a highly cooled wall, for a given value Of b Laminar Boundary Layer Flow A small perturbation analysis of the boundary layer equations can be carried out near the thermodynamic equilibrium extreme in the same manner as Couette flow problem. The thermodynamic equilibrium solution (I - oo) remains qualitatively the same, with the notable exception. that the energy equation is a second order differential equation in the temperature variable (including effects of convection) instead of an algebraic relationship between the temperature and the concentration variables, The equilibrium specie-temperature relationships'* when inserted into the boundary layer energy equations provide an equation from which thetheermodynamic equilibrium boundary layer temperature distribution is determined. For this purposes the following enthalpy form is used since it is free from explicit reaction rate terms (the temperature distribution can subsequently be calculated from the relation h Z aCihi): tA t98 (tJ j 2T ( A>;e (5.60) where X 4 *t/a/ (5,61) and ^g&/d. = ~</ y) v^./d ), etc. * For equilibrium, these relations can also be v4iewed as giving zi, )i and 4~2 as functions of enthalpy as well as il. since the equilibrium temperature and enthalpy are directly related0

For example, the assumption of similarity (neglect of 5 derivatives) and PR Le = C = 1 permits Equation (5.60) to be integrated to the following solution:'* q.= (a + 47 R I/ SE(t ( d it (5e62) from which the equilibrium temperature distribution can be calculated through Equation (5,61) by using A 4' -fi^ -ri (5 63) The equilibrium heat transfer in general would be found from Equation (3420) using the relations az/ ( - d ) a g/ etc, to express the composition in terms of the stagnation enthalpy variable. Deviations From Thermodynamic Equilibrium The behavior of the boindary layer equations near equilibrium. can be investigated by assuming expansions of the dependent variables as in Equations (5o28-5 30)o While it has been found that deviations from the opposite chemically frozen extreme can be taken in the form of a series of ascending positive powers of 5 for small 5 (Z = zI ~ 4 + ziiT2 +.oo., for example), ananalogous attempt to assume at the start that the thermodynamic equilibrium deviations are a series of negative powers of C for large G fails, (The resulting equations governing the deviation variables turn out to be algebraic and consequently in.capable of satisfying the required surface boundary conditions) Substituting expansions (5.28) and (5,29) into the bound-ary layer eqxuations, uAsing the simplifying assuLmptions that were invoked * A numerical solution to Equation (5o60) for PR, Le > 1 has been given at the stagnation point in reference 6o

.162for the chemically frozen perturbation analysis,* and retaining only first order terms in the deviation functions, we find that the deviations are governed by the following second order, linear, non-haomgeneous differential equations~ G.J: d, ~Z E - S,.. f d (5.64) Wf t dOk= 2~6P 6I104 r- (W++g (5.65) J>' +J * 2 gr td d reo 4 )Oe7 - (s (5,66) +srJ'44 7 C JI ^&^ s-s7ff ^ ^^+^Jr d 0;^ / I 0 / 2/'+ r' L(/ i(F - -0 +' W7-^t _?' + x,-l -;r /' +^ Or+ e d ] L - Oo, + (5d67) 2' - IVt/^^IlM -2IPA/'r^ The heat transfer deviation is given by pc t const. ant, *p| = constant, P~ and S constants. Cp C = constant, and velocity profile perturbations negligible~ 1 P2

The attendinr boundary conditions to be satisfied at the edge of the boundary layer (r - oo) are 2(&= X(Ao) r)_ f = O; those applicable to the inner surface are given by Equations (5o35) and (5,36). An inspection of Equations (5~65) and (5,66), in view of the boundary conditions, leads to the following solution for y: -F- ^e ~ZE, (5.69) the same result found in the chemically frozen deviation regime, Before attempting any detailed solution of the above equations, some general characteristics of the problem are worth noting in advance, Firsts it is seen that the reaction rate derivative coefficients and the o- 2 SEQ term vary throughout the boundary layer, complicating the mathematics in the same way as encountered in the Couette flow problem9 Second, the coupling between, Z. and V demands a simultaneous solution of the above equations to obtain higher order uncoupled differential equations for each variable0 In the boundary layer case, since the energy equation is a second order differential equation, the resulting ordpr of these individual equations for EZ, and will be sixth (instead of fourth order in the. Couette flow problem, where the energy equation was an algebraic relation), In this respect, the boundary layer problem is much more complicated than the Couette flow, This feature of coupling between the variables did not arise in the chemically frozen perturbation analysis because the first order reaction rate terms were non-homogeneous functions,00 evaluated at the frozen flow solution that (contrary to the present thermodynamic equilibrium

-l64,analysis) do not vanish o Consequently, the thermodynamic equilibrium perturbation analysis is far more complicated mathematically than the frozen perturbation analysiso Third, the presence of the convection terms in the boundary layer e-quations adds a further complication in the form of the variable velocitjy funrction coefficients that is absent in the Couette flow problem. The foregoing considerations suggest that the treatment of the general thermodynamic eq.ilibrium deViation problem poses even more formidable a mathemat;ical task than the foregoing Couette flow analysis. A detailed analysis of the pre-sent problem which includes all of the above complicating features would demand digital computation. Accordingly2 some simplifying assumptions must be introduced in order to obtain any physical insight as to the behavior of the laminar boundary layer near thermodynamric equilibrium, Several, of these approximations are similar to those used in the above Cou.ette flow analysis, (a) Assume that the equilibrium solution is locally similar (azEQo /a = 6XE.Q /c -= )EQo /6t. 0)o This is a satisfactory assumption provided t+he freestream temperature is slowly-varying along the body and the wall is highly cooledo (b) Neglect.the B c (UI/cpTe) terms in the temperature deviation equations, according to the.arguments given for the frozen deviation case, (c) AssumLeL = 1, This assumption, which causes the total enthalpy distribution and heat transfer on a catalytic wall to be invariant to gas reaction, permits a simple integration of the energy equation and greatly facilitates the present objective of a quaalitative appraisal of specie and temperature deviations0 Furthermore, this assumption does not significantly alter the individual specie and temperature behavior from that at Le / 1 when

l65P Le is near unity, and has little effect on the non-catalytic wall heat transfer, (d) Assume that the deviation functions depend on g only through the local non-equilibrium parameter (x) for a slowly-varying freestream, ioe,^ Then the t/c terms in the deviation equations assume the form Q (DEVIATION) C a where Q/c (a/a )/(-/ )o Now the Couette flow results have indicated that for ~ >> 1, the near-equilibrium behavior (at least in the vicinity of the wall) depends on ~ as 1.DEVIATION - so that -./Q. (DEVIATION) Q/e Therefore when Q/e - O(l) and >> 1, we can ignore the 5 dependence of the deviations in comparison with the? dependence across the boundary layer, (e) The first and second atomic specie deviations will be assumed to be proportional / ^' 4()J'' (5-70~) This assumption uncouples the two atomic specie deviation Equations (5464) and (5,65). An immediate consequence -of the foregoing assumptions is the integration of the perturbation energy equation, avoiding the need to use Equation (5,67) When Le =1 and. the velocity profile deviation is neglected, the total enthalpy perturbation

P166(5.71) for C = C = constant is governed (from Equation (3.14)) by the following PI P2 equation: /& / * =. (5.72) Equation? (5,72) can be integrated twice with the outer boundary conditions (00) =Z() =Z(oo) = (o(0) = 0 to give;1/g= 4)2[i/4C74 EX}Ghjj (5073) where 7b = Ah~) fWJ = ) - ^fABj2g (5 74) g /='^,J= as,,9!7 //2 abhor ~~(5075) For a catalytic wall, g(o) = 0 and we get the well-known result A+ q2g + q41 = 0 for a Fick diffusion law with Le = 1o On the other hand, a noncatalytic wall leaves ~(o) to be determined as a parameter in the equations and V'(o) is given by-Equation (5~75)~ By means of this solution for, we can express the temperature deviation in. terms of the atomic specie deviation. Sincer Z- Wandti are now expressed in terms of the single unknown Z(O) we have to solve the following linear ordinary non-homogeneous second order differential equation (substituting the aforementioned approximations into Equation (5o64)): ez'+zre" -s _,)z =fJ +A(-'J)- ),, (5.76) where, --.) X (5o77) 6: (H7 pre) e W

-167and f2n - e; ) (5*78) are predominately positive and negative, respectively, throughout a highly cooled laminar boundary layer. Approximate Solution The variable and cumbersome coefficients ~ and ~ would demand a lengthy numerical integration of Equation (5476) if an exact solution for -were desired,. However, we shall sacrifice a certain quantitative error in the solution, as was done in the Couette flow case, by assming average constant values l and AC (< 0) for these funntions. This approximation should still preserve the qualitative features of'the near.equilibrium behavior. Furthermore, it has been found.during the course of a study of the solutions to Equation (5.76) that the convection terms SCfZ' and ScfzEQ exert a very small affect on the behavior of Z(r3) over the inner part of the boundary layer for either wall gatalysis extreme when >> 1o These two terms can be neglected in the present approximate analysis, Finally, the functions ri) and ZEQ(r) will be approximated by the following approximate expansions near a cooled wall: %4r j= y~}-.(/7 a t) (579) These two approximations are adequate for the present objective of describing the nonw-equilibrium behavior only in the vicinity of the wall,. They enable a simple particular solution of Equation (5.76) to be obtained, Utilizing the foregoing approximations, the complementary solution to the associated homogeneous equation SefZ + Zc - Zec = 0 (5.80) is Zc - C1 exp( N4T' ri) + C2 ex(-, ),> (5.81) and a particular nonAhomogeneous integral to Equa.tion (5,76) is

16&=3 UP^ - - 6v'4d-(/_,f74t}+ t V, (5.82) The complete solution Z = Z + Zp involves the two arbitrary constants C1, C2 which are chosen so as to satisfy the boundary conditions Z(oo) = 0 and either Z(o) = (o) O0 (catalytic wall) or Z'(o) ZQ (o) (non-catalytic wall). Since the present theory is really valid only for the inner part of the boundary layer, however, the Z(oo) = 0 cqndition shall be applied at an effective "infinity" r = 4 for which the approximate f(rn) function in Equation (5o79) vanishes: / ^-= c.^^t. The results for the wall deviations* when >> 1 are as follows: [^ a)1 _ Q*"ci (5.83) L^a-cr - B?t J^* r(5.84) These are essentially the same revmlts that were found in the Couette flow case, which is not too surprising when we note that convective effects play a fairly weak role in the above development for >> 1. The deviations (5.83) and (5o84-) exhibit the same general properties that were discussed for the Couette flow; both vanish as 1 v'* when - oo and the catalytic wall case demands a consideration of the non-linearity in zEQ (QEQ.) in order to determine the sign of ZEQ (o). We have * In view of the various approximations introduced, we shall confine further discussion to deviations in the immediate vicinity of the wallo d~suxson -t; deiaion E

-169^'^=~t. (0e + Using the approximate equilibrium energy equation solution (5~62) and Equation (5065) we have the equilibrium temperature derivatives(Cp = C ) P1 as t,'(7 = Xe f 4 7 (5.85) ^ *u k6/ /CI/(7' IS^ "% |(og) e - J (5086) zEQ (o) is always positive; ZEQ (o) on the other hand becomes ll 6 ) d~c~,dtw W ~)+ fp(~iA X ~p;k(5,87) /7L4~ ( { )'( (/ y 42) The viscous dissipation tends to decrease OEQ (o) and the catalytic wall specie gradient deviatinon. whereas a significant non-linearity in ZEQ (GEQ 0 will make 0"Q (o) and [Z (o)CcAT. positive on a highly cooled wall (ZEQ (o) small)a The non-catalytic specie deviation is not sensitive to this non-linear effect; Z(o) approaches zero as -- oo with a constant derivative Z'(o) = - (o) at the wall, This behavior, as previously pointed out,

.,170Q. means that the non-catalytic deviations due to a departure from thermodynamic equilibriun are largest at the wall and decay outward over a "sublayer" thickness Z(o)/zEQ (o) / O^ l ~o The temperature gradient perturbation is related directly to the specie variables byEquation (5075)o ='(o) - )+ ( gJ4Z'/o)= -17 a( n?Z(). Theefore, /f'4<),7r -KY'2*//Y5)2E762J =- C4~ 7o) (5 88) and 0fW =-(BtaAv$. -.. -.. -- (5.89) The catalytic wall gradient will decrease with a departure from thermodynamic equilibrium when ZEQ" () is positive (very highly cooled wall). The noncatalytic wall perturbation, however, does not vanish at -4 co because of the Z(o) = - z (o) )boundary condition0 As the boundary layer departs from EQo equilibrium, the non-catalytic wall temperature gradient will decrease; the temperature deviation correspondingly grows within a sublayer adjacent to the wall whose thickness spreads as 1/~5' * The deviation profile behavior near the wall implied by these simple results is shown qualitatively in Figure 27. This Figure illustrates the sublayer behavior on a non-catalytic wall and the change in the sign of the catalytic wall perturbations when the wall is not highly cooled enough to give ZEQ (o) > 0 (Equation 5e87)o As -> oo, the non-catalytic specie and temperature perturbations vanish at. > 0 leaving Only a discontinuity in Z (o) and G6 (o) at the wall that is

-171z,1 ~+ {; -Io-10 Z ( ZEQ" (0) > ) OR ( ZEQ"(O) < O) \ _ Z (O) C 0 zE:(O) ) @ ) - \5=10 z,'O1 z (o), GE (o) Cc zE(o) 0 O - co 10 50 100 50 L-' 10 Figure 27. Illustration of Thermodynamic Equilibrium Deviations in the Highly Cooled Laminar Boundary Layer.

172proportional to the equilibrium diffusion velocity zEQ(o). For a very highly cooled wall, such a discontinuity is extremely small, The heat transfer perturbation is given by Equation (5~68): Z = ~?~# + 16i Ze/ B9-Ag)Z% The catalytic wall value is exactly zero for Le lo Ulsing the previously found values for Q'(o) and Z'(o) (computed for Le = las estimates, the above equation shows that This predicts that the catalytic wall heat transfer for Le > 1 will increase T, due to non-equilibrium when ZEQ(o) > 0. This agrees with the result of Fay (6) and Riddell The non-c.atalytic heat transfer perturbation, on the other hand, becomes i4, 7 = _- ~f 7 t_ __ f__ _ _IRA_ (5.91) While the non-catalytic temperature and specie gradient perturbations have been shown non-vanishing in general for -> co, the net heat transfer deviation does vanish at the thermodynamic equilibrium extremeo The above heat transfer deviations vary with 5 as 1/ N according to the present highly simplified analysiso This prediction is confirmed by an analysis of the near-equilibrium heat transfer behavior given in Reference 6, which shows a 1/ \1 dependence on when >> 1, This agreement lends some confidence as to the value of a simplified qualitative approach in assessing the main physical features of departures from thermodynamic equilibrium.

-1753 Further Discussion of Results The foregoing results provide some qualitative understanding of the near-equilibrium behavior in the boundary layer and would no doubt also be valuable as a guide in interpreting more exact.and detailed solutions. No attempt has been made at an exhaustive discussion ofhe results in view of the admitted error in the magnitude of the predictions. However, the simplified analysis does provide three major conclusions of some importance: (a) The non-linearity of the equilibrium atomic specie-temperature relation must be taken into account even in a simple theory if accurate catalytic wall nonequilibrium behavior is to be obtained. When the wall is not highly cooled.and, or viscous dissipation is important, the catalytic wall heat transfer may decrease with a departure from thermodynamic equilibrium in spite of a nonlinear effect, The non-catalytic wall is not sensitive to this non-linearity, howevero (b) The catalytic wall diffusion, heat conduction and heat transfer deviations vanish as 1/ 4 for 5 - oo, The surface reaction requirement z(o) = zEQ(o) is completely compatible with the equilibrium concentration distribution near the wall in this case. When the wall is reaction-free, however, the equilibrium concentration gradient z'Q(o) in the gas (computed solely on the basis of the equilibrium temperature distribution) does not strictly agree with the zero diffusion requirement z (o) = 0 right at the surface. Consequently, there is actually a state of non-equilibrium in the gas within an extremely thin layer adjacent to the wall when: - oo 0 We have found that the non-catalyti- wall specie and temperature gradient perturbations do not vanish at: e oo but remain proportional to the change of z' required across the sublayer (namely - zEQ(o)). The temperature and specie concentration deviations themselves do become vanishingly small within this sublayer,

howeve.r When the boundary deviates from thermodynamic equilibrium because of a decrease in., the sublayer thickness grows as 1/ fvQ and the non-equili. brium perturbations grow in size and spread out laterally from wall into the flow (driven, so to speak, by the specie gradient jump existing at the wall when the remaining portion of the flow is at equilibrium), The total energy flux. across the thin sublayer must be constant and therefore csarnot show any discontinuous behavior across it) the non-equilibrium heat transfer perturbation therefore vanishes as 1/ vNf when 5 -_ oo even though the individual temperature and atomic specie wall gradients do not, (c) The favorable agreement between the approximate Couette flow and laminar boundary layer analysis indicates that the former is,a dependable qualitative model of the dissociationrecombination non equilibrium behavior in the latter problenm

CHAPTER VI CONCLUS I ON Smrmary There are two primary objectives connected with the present study(a) a description of the basic theoretical features which are involved in the fonmulation of -a chelmically reacting gas mixture flow problem, particularly those which are important in the study of dissociation-recombination non. equilibrium, and (b) a detailed study of the non-equilibrium effects in a highly cooled dissociated laminar boundary layer as they affect deviations in composition, temperature and heat transfer from either a chemically frozen or thermodynamic equilibrium flow extreme. Objective (a) has lead to the consideration of a fo.ur component representation of an air mixture with the attending formulation of multicomponent gas mixture diffusion and reaction ratesf It has also required that we consider the effect of possible surfacecatalyzed recombination reactions entering the problem through the composition variable boundary conditions. Finally, objective (a) lead to the definition and interpretation of the basic characteristic reaction time flow time ratios which naturally appear in such flow problems5 These parameters have been used to establish the hypersonic flight regimes in wh.ich gaseous non-equilibrium and non-catalytic surface effects become important (Figures 2 and 6), and also to illustrate the fact that the nonequilibrium state of the boundary -can vary widely over the length of a typical blaint-nosed hypersonic body (Figure 4), Objective (b) has been concerned with the effect of the various significant aerodynamic and 175

-176chemical parameters (such as recombination rate temperature dependence, activation energy of the atomic species, the extremes of the surface -catalysis effect, and the dependence on the local non-equilibrium parameter) on the non,-equilibrium deviations. The governing equations were cast into the well-known "similarity plane" form by means of the StewartsonMangler-Blasius transformation* The similarity limitations for an arbitrary degree of non-equilibrium, plus the desire to study the initial trends from the two chemiical behavior extremes without the use of digital computation, led to the development of a chemical perturbation analysis which is applicable over a fairly wide range of local flow conditions at the edge of the boundary layer. The introduction of suitable approximations for the highly cooled wall case enables the equations governing the deviations from both chemically frozen and thermodynamic equilibrium extremes to be reduced to sets of linear ordinary differential equations which can be solved by standard methods. The effect of the aforementioned significant parameters on the composition, temperature and heat transfer deviations was given in detail for the chemically frozen regime, whereas a qualitative analysis was given of the near-thermodynamic equilibrium behavior because the latter case is so much more cumbersome from a numerical standpoint, However, in both cases, the important physical mechanisms "driving" the non-equilibrium behavior have been clearly displayed* General Conllusions The detailed results of the investigation summarized above can be used to infer a number of rather general conclusions about the important theoretical elements required for any non-equailibrium theory and the actual

177behavior of the non-equilibrium deviations encountered in a highly cooled laminar boundary layer (with surface mass transfer absent). (a) Variable ppi and unequal specific heat data, which can be feasibly includled only when a, digital computation is utilized in solving the governing equations, appear to have a minor effect on the non-equilibrium deviations. These variables do play a role of some significance in the chemically frozen. and thermod.ynamic equilibrium flow extremes for the case of a catalytic wall: the variable ppi tends to reduce the wall composition and temperature gradients by roughly 50%, whereas the unequal specific heat effect acts in the opposite direction in tending to increase the wall temperature gradient proportionazlly to (cpcpl) a e. The heat transfer is affected much less than the temiperature gradient by the specific heat inequality, (b) The results of the present theory, when compared to other solutions for a particular case (6) have already brogoht out the importance of accounting for individual specie behavior in formulating reaction rates in the gas when the detailed non-eqLlibrium behavior is to be studied. The use of an eq.uivalent binary system of lumped. atomic and molecular species cans overestimate the initial rate of deviation from frozen flow by as much as 50% when a significant amount of atomic nitrogen exists in the flow. Care must also be taken when the individual rates are averaged over the respective heats of formation in the reaction rate heat source term which occurs in the temperature form-of the m heenergy equation (c) The use of Fick's diffusion law for each specie in a multicomponent air mixture is a satisfactory approximation, at least as far as the atomic specie and temperature behavior is concerned, for a cooled

-178boundary layer flow when there is no ablation, etc. of species with radically different properties into the main flow, Some error in the detailed molecular specie profiles will result from this assumption, however, (d) It is important to account for the catalytic state of the wall surface when appraising non-equilibrium effects (particularly concerning heat transfer)o The results of the present investigation have shown that the non-equilibrium deviations from the two chemical behavior extremes are much greater for a completely non-catalytic wall than for a completely catalytic one, The state of the wall catalysis is governed by a characteristic reaction time-flow time ratio that involves the surface temperature and material, as well as the diffusion and convection flow properties. For glassy like wall, the non-catalytic condition appears a definite possibility in the hypersonic flight regimes for which significant non-equilibrium effects may occur (Moo > 10, altitudes above 100,000 ft. ) (e) The non-.equilibrium state of the boundary layer can vary notably over the length of a hypersonic blunt-nosed body; the stagnation point situation is generally not an accurate indicator of the non-equilibrium behavior downstream along the bodyo Particularly in the neighborhood of the sonic region, the deviations from frozen flow appear to be as much as three times larger than those at the stagnation point, This variation of effect generally excludes the possibility of a local similarity solution to the full boundary layer equations except in the two chemcial behavior extremes. The use of a perturbation type of analysis, which sacrifices a consideration of non-equilibrium effects over the full range of i, gains however a wider

-179range of local application under varying freestream pressure and temperatu;re (f) The non-equilibrium behavior on a highly cooled wall is sensitive to the value of the recombination rate temperature exponent a, because of the depend.ence of the charac-teristic parameter 5 on i and also because of the effect of the value of:- on the recombination rate in the governing equations. Th eec is ect issmall only when t is very near a value of two (and vanishes when..o 2)4 The results of the frozen pertur. bation analysis will be of use in assessing the effect of any change in the recombination rate on the non-equilibrium behavior (Table II). (g) The non-equilibritum behavior, insofar as can be deduced from the chemically frozen and thermodynamic equilibrium perturbation analyses, is very insensitive to large changes in the activation energy when the wall is highly cooled (OQ <.20).O The difference between atomic oxygen and nitrogen activation energies therefore causes a very small difference between the non-equilibrium behavior of these two species; the deviations can be considered essentially proportionaal to each other. When the wall is not cooled, however, this conclusion is reversed; the activation energy may then play a very prominent role in the non-equilibri=m behavior, Chemical/jr Frozen Deviations Several conclusions specifically related to the frozen perturba~ tion analysis are worth listings (a) The deviations in composition, temperature and heat transfer from the corresponding chemically frozen solutions behave as a series of

ascending powers of I, the initial deviation being linear in and the second order terms being proportional to 52 (these constitute subtractive corrections to the first order results). The coefficients of 2, 2, etc~ are determined from a solution of the boundary layer equations governing each coefficient.fanctiono The analysis of the deviations becomes increasingly complex for the second and higher *order effects. (b) The frozen deviations, in addition to the dependence on described in (a), also depend significantly on the local streamwise derivative of this factor, io.e, the parameter Q/ ec _ When Q/e is negative, the deviations increase rapidly with a decrease in Q/t and can become 100 times the Q 5 0 result (stagnation point) when Q/e -.2 (for a given value of 5), for example. Positive Q/~ reduces the deviations to a much lesser extent (relative to Q = 0)'This parameter Q/e arises from the perturbation analysis of the governing boundary layer equations, (c) The qualitative effects of non-equilibrium on the composition and temperature distribution in the boundary layer were found to be as follows: As ~ is increased, the catalytic wall diffusion current (atomic specie wall gradient) and non-catalytic wall atom concentration both decrease, the heat conduction in both catalysis extremes increases, the non-catalytic wall heat transfer (pure conduction) increases, and the catalytic wall heat transfer decreases very slightly for Le > 1 (the decrease in diffusion of heat to the wall being slightly greater than the increase in heat conduction). While the catalytic wall deviations are always much smaller than those for the non-catalytic case at the same value of; when the Lewis number is of the order of one, both cases were found to exhibit about, the

same insensitivity to the activation energy parameter for a highly cooled wall. Thermodynamic Equilibrium Perturbation The mathematics associated with this opposite extreme of the nonequilibrium effect "spectrum" and the complexity of the variable coefficients in the governing differential equations for the perturbations resulted in a much less detailed analysis of the problem than given in the chemically frozen regime0n However, the main physical features behind the departure of composition and temperature disatributionus fromr exact equilibrium as decreases were displayed and indicated s-veral important conclusions. (a) A series method of solution, in terms of ascending negative powers of g, is not a feasible method of atta.k,. An approximate study of the general small deviations from equilibrium indicated that, at least near the wall, the deviations in heat transfer, catalytic atomic diffusion, catalytic heat conduction (temperature gradient;) and non-catalytic wall atom concentration all vary as 1/f^ when > 1o The non-catalytic specie and temperature gradiexnts do not vanish at c - oo, however; they remain proportional to the equilibrium wall diffusion flux (which is extremely small for a very highly cooled wall). (b) The catalytic wall non-equilibri=m deviations depend significantly on the non-linearity of the equilibrium,spec ie temperature relationship, while the non-catalytic wall results do notp If the wall is not highly cooled, it has been found that this non-linear effect may cause the heat transfer deviation to be opposite in sign to that for the highly cooled situation (in the latter case, the heat transfer increases due to a

1l82deviation from thermodynamic equilibrium for L > 1). This result should be useful as a guide in any future study of non,-equilibrium flow over slightly -cooled or "hot" surfaces. (c) Like the frozen deviations, the thermodynamic equilibrium deviations are affected by -changes in the recombination rate parampeter cWo The results of the qualitative analysis, however, seem to indicate that this effect may be somewhat different in both magnitude and direction from that found in the chemically frozen perturbation analysis, Unlike the frozen case, however, the thermodynamic extreme does not appear to be sensitive to the non-equilibrium derivative factor Q/E when >> 1L Limitations By way of a conclusion to the present chapter, it is desirable to emphasize that the major conclusions listed above depend on several key assumptions made in the present theory. When conditions are not in agreement with these assumptions, many of the present results may be changed" In addition to the neglect of variable fluid properties, mass transfer (ablation, coolant injection, etco ), radiation, thermal difCusion, and electromagnetic body forces, there were three particularly important.assumptions made in the course of formuLating the governing equations for the non-equilibr-ium perturbations, (a) Slowly-varying freestream properties. The freestream temperature and velocity variations with X were assumed gradual enough to permit local similarity approximations to the behavior -of the chemically frozen, thermodynamic equilibriumn. and the non-equilibrium composition and temperature throughout the boundary layer' Such an assumption appears to be fairly reasonable provided there are no

shock waves or flamre fronts Impinging on the boundary layer, nor any abrupt changes in the body shape. (b) Neglect of any Nitric Oxide that may possibly be in the flow, Arguments based on rather sparse data tend to indicate tha4t the small amounts of this constituent present in the air at the edge of th e boundary are not significant to the calculation of none equilibrim effects* However, should future improvements in the knowledge about the NO rates indicate that it may play a more important role, the present perturbation method can be repeated for a five component mixture including the NO recombination=dissociation chemistry. (c) Assumption of a highly cooled wall. This is by far the most critical assumption, since many subsequent simplifications and conclusions are either directly or indirectly based on this condition (such as the neglect of velocity profile perturbations because of the insensitivity of the boundary layer momentum equation to the pressure gradient term, the. predominance of recombination, over dissociation in the reaction rate terms for the inner half of the boundary layer, the minor role generally played by viscous dissipation in the non-equilibrium behavior, and the neglect of surface-catalyzed dissociation reactions), When the wall is not highly cooled, it is expected that the effects of the recombination rate parameter (tt and the activation energy found in the present analysis may be somewhat reversed, the latter parameter playing a very prominant role in the non equilibrium behavior, A nonequilibrium perturbation approach may still be a valuable tool in the "hot wall" problem, but it most certainly would demand a "fresh start" by a reappraisal of the proper theoretical approximations in the governing equations.

APPENDIX A MULTICQMPONENT MIXTURE'TRANSPORT COEFF ICIENTS Viscosity Coefficient Reference 17 gives the following mixture value in terms of the indiviudal component viscosities: ~ /.... lll I L C __ where (A 1) S., _ - A/ 7r M)f Diffusion Coefficient Reference 16 gives the following expression for the N-component mixture diffision coefficient:,) (=.....~ /)'.~' (A.2) where ____ ( ^ { M j T / ^^(As) and where |^':l = r, 2C' *' 2 A 7 F:. ~,,,' ~.,.F 13, *,~18 ~~jlS4~ ~ ~~j;A

-185and / F /2''' -/ i/ ~''' -^,/ ^2''',2 >/-/ VJ'+ /' d -' d / ~ ~ ~-' ^ ^ ^ 5 (A.4) L)-lJ 0~ t'-!, nl = J-I /,../,2''''-LS-/ ^^'' 4 (-4 gR ~ ~Z''*' i-^/ 6;L/ ^''J' a /I ~,,, I and where by definition of binary diffusion, In the particular case of a three-component mixture, the above reduces to the simpler formula /oB / I 29~/' + Mffl,0><2'it~P' - — C~ (A-5) with indices cyclically permutedj clearly this gives,'/ ^ only when /:='0 in general~

APPENDIX B DIFFUSION COEFFICIENT'RATIO ESTIMATES Kinetic theory( ) yields the following expression for the binary diffusion coefficient between two gases: o^ ~ ~CO N Y r ry (B 1) where Qij is a collision integral proportional to the mutal collision crosssection of the gas particles. When i = j 9ii has the physical significance of a "self-dif fusion" coefficientl describing the kinetic motion of particular particles amongst other particular particles, such as isotopes of equal mass but different nuclei. Consider a three component 02, 0, N2 mixture (denoted by species 1, 2 and 3, respectively) with the approximations /I,= f=2; then 0^23 f = 12, /2 3/3 / (B.2) &9AZ -'~22? +A2 -' 2~2 ozp/ - _&,2 ///A1t A'-t _ _ --- -7 2..... ~.~.o _.~. —,B/2 /l/C3 1/-J 3 R 3 *(Bo3) Now take %ij ( (di/2+ di/2) where di is the i-th specie particle diameter. Then, for purposes of a rough estimate, take d d3 2d2 and hence obtain -1/2 2 /,,-,1 S (Bk4) — /22J -^-/^ L

.-l87u# Consequently we have from Equations (B.2) and (B.3) the estimates?/.,2 /2. = / (B.5) 4 ^/3/6no 2 -2, -0 N/ (B.6)

APPENDIX C CATALYTIC SURFACE REACTIONS The catalytic production rate for a j-th gaseous specie has been shown (lo0lll2) to be proportional to the kinetic mass flux p /R T/2xcM. of the specie upon the wall, and is approximately given by the expression -s 0? /7'F 2T/, Q (C.l) where yj is a "catalytic efficiency factor" accounting for the fact that not all collisions with the wall result in a catalytic reaction effect. A general discussion of the definition of yj for'a multicomponent mixture has (12) been given by Rosner 2) It is often convenient to rewrite Equation (Cl1) in terms of a surface reaction parameter Kcj as S-C = / Kr (Io v&- i'. (C.2) The results of vaxious investigations into surface catalysis can be given either in terms of a catalytic efficiency or in terms of the parameter K.? Different surfaces exhibit varying recombination efficiencies and some may have their eatalytie nature altered by suitable coatings or deposits. For (24) exanple, the experimental studies of Marsden and Linnett show that for atomic oxygen recombination in the 300 ^ 700~K wall temperature rage roughly 100 times more heat is released on a platinum surface than on a glassy one, Further data on the catalytic recombination of oxygen and nitrogen atoms may be found in a summary given in a paper by Goulardi 11) Some of the possible surface recombination mechanisms, for example, would be (a) two gas phase atoms colliding in a close enough proximity of the ^188<

-189. the surface toz allow a saurface atom to act as a third. body in transmitting the heat of recombination to the surface material, (b) two separate gas atoms are absorbed in the surface structure and retain sufficient mobility to subsequently meet'and recombine -within this structure, and (c) possible gas phase atom collision with a layer of absorbed but immobile atoms and subsequent recombination The. results of Marsden and Linnett indicate that (b) and (c) are the probable mechanismns for glass and platinum surfaces, respectively. The concept of surface "poisoning" in inhibiting catalytic atom recombination appears in regard to these latter two mechanisms, wherein the poisoner uses up all available absorption sites on the sutrace, Also, the physieal nature of the surface, as well as the material, may decidely influence recombination, Equation (C,2) can be written for each specie of the four specie air mixtre in this paper as follows: s,, — /P z (c<-od<Jv 1 =.- S S4 sine - S In view of the astne (3) S3C =''/ C (<]- ~6^)^ J and S. =.(S2 + S4 + ) since ~ Sj = 0O In view of the assumed eui o p3apr Cer J e equality of physical properties between the atomic species 2 and 4 (and between 1 and 3), we shall take Kc2 d Ke4 and K~1 e K.3 since K refers to an activation energy w independent recombination process, Then as a direct result.of the ~ Sj - 0 requirement,.ne gets K o= -K C3 c2 and. we can write the following approximate net catalytic prodluction rates for the subjeet mixture: SGc - PwKc2( - "aQ)W )IcMF~2p 4Ewc)W2 (c 4) s35 + P Ke2(o3'-'O. W ^^p~a~a^QW

APPENDIX D REACTION RATE DERIVATIVES AT FROZEN FLOW The derivatives of the reaction rate functions Z and 4 (defined by Equation (359) and (311.)) with respect to the specie and temperature variables, evaluated at the chemically frozen flows are found to be the followingo A ((DBA) /fO== F' (g~ / + 7"c~,2_ 9(D,5) > 4/ tZ q/E "'r$ )A (~L~ 7 z)2(D,8) 2 = It (Do ) The frozen flow solution zF = XF has been used in the above relations, 1L90-2

APPENDIX E ANALYTICAL COMPIEMENTARY INTEGRAL SOLUTIONS Equation (4,68) in the text admits analytical solutions for two special values of the non-equilibrium derivative parameter-. Q = 0 This corresponds to stagnation point conditions; Equation (4o68) is in this case sA I4c = o, (EXl) which integrates directly to the following solution: 4/izj = /rco ) *L /C 4^j{ ~KP6-Ksf/M j^T. (Eo2) Since zi = A Zl(n).+ B z2(n), where A = Zi(o) and B zi (o) (Equation 4,69). we see that in this case Q/ l-./ 2 This:corresponds to a slightly favorable pressure gradient flow such as might.ocdur slightly downstream frmn a hypersonic blunt body stagnation point, In this case, Equation (4.68) can be written X4IC + H = ~) ~ (E 53) which integrates to the solution 4grI; EXIP-%k,~) rAjY&- dlc 4jk7 (E.4) Hence, 2 2 = =19t1 2

APPENDIX F APPROXIMATE CORRECTION METHOD FOR VARIABLE pD EFFECT Probstein (34) has outlined an approximate method of accounting for the effect of variable pi on wall temperature, enthalpy and specie gradient solutions calculated on the basis of pkL/pWLW = C = 1i This method, developed for equilibrium flow, should be.also applicable to catalytic wall frozen flow situations as Well, since the pj variation across the boundary layer is about the same in either case, Probstein's approach is.simply to replace the integrated effect of the variable p.i across the boundary layer by some average constant value of C = CAV < 1* such that (WALL GRADIENT)C1 (WALL GRADIENT)C=1, C. Reference,(34) outlines an iterative scheme, based on the original similarity plane form of the energy equation with variable C, by which the appropriate value of CA can be calculated with the use of the best known viscosity versus enthalpy data, The similarity between the specie and enthalpy energy equations for a highly cooled laminar boundary layer flow with negligible dissipation makes this method applicable to the correction of diffusion gradients as well as enthalpy and temperature gradients at the wa]l., ProbsteinYs approximate method, when applied as a correction to enthalpy gradients calculated on the basis of C = 1, gives fairly good agreement with exact variable pAs solutions computed by several investigators, The value of CAV given by.Probstein is not very sensitive to the wall temperature ratio as long as the wall is highly cooled (oW << 1) * For a highly cooled wall, C varies from 1 at the wall to a value of the order 41-a2 at the edge of the boundary layer, when the Variable molecular weight is accounted for4a ~192r

-195> Hence we can use his results to estimate the variable p.i effect on the present frozen flow catalytic specie and temperature gradients with confidence, For W =.032 (Ce = ol8), Reference (34) indicates that CAVo l 550 When this is applied to the present stagnation value [zF(o)]CAT~ = o373, C =1 we get [zF(O)]cAT 205, F CAT.C < 1 as compared to 2= 209 [z(o) ]FAY AND RIDDELL which is a satisfactory agreement considering the slight difference in velocity profiles used in the two theorieso A similar agreement between the catalytic wall temperature gradients is obtained when such a viscosity law correction is applied to the present theory (proper allowance being made for the specific heat inequality effect noted in the text of Chapter IV) In the non-catalytic wall case, the frozen temperature gradient is much less sensitive to the variable pp. effect since there is no molecular weight variation across the boundary layer, The foregoing correction is much too drastic; indeed the C = 1 result of the present theory checks very well with Fay and Riddell (there being no specific heat inequality effect in the non-catalytic case): F() ]NON-CAT, =4 1 C =1 F() ]FAY AND RIDDELL =.415 for the stagnation point~

The above results would seem to indicate that the non-equilibrium deviations would not be significatnly affected by variable p, for the noncatalytic wall case. The effect would be more noticeable for the catalytic wall and would appear (on the basis of the above examples) to further reduce the already extremely small heat conduction and diffusion perturbations that occur in this case, Unfortunately, the mathematical complexity of the non-equilibrium boundary layer perturbation equations makes the application of Probstein.s approach to estimate C. 1 effects in the perturbation solutions quite complicated in. comparison to the frozen solution,

APPENDIX G DERIVATION OF EQUILIBRIUM RELATION BETWEEN MOLECULAR AND ATOMIC SPECIES Consider the four component air mixture (NO neglected) in which m1 = m2 = 2m2 = 2m4, The mole fractions of each specie are then related to the corresponding mass fractions as follows: N1 _ l N2 2c2 N 2-(al+a3) N 2-(c1+a3) (G.l) N3 a3 N4 24. N 2-(a1+a3) N 2-(1+a3) Now in the classical thermodynamic equilibrium (no diffusion) steady state, the number of moles of oxygen and nitrogen atoms is conserved regardless of reaction in the gas, This requirement can be stated ass (NJJ (N2 INITIAL MOLES OF ).+ - ) =2x 21 x ISSOCIATED AIR eN r e \N /N / (G.2) INITIAL MOLES OF N N4 ( ) 2 x.79 x (UNDISSOCIATED AIR N KN) N AIR) Division of these two statements produces the equation 2(N-i/N)EQ + (N2/N)EQo. 21 = - (( 3) 2(N3/N)EQ, + (N4/N)EQ..79 Using the fact INi/N = 1 to eliminate N1/N and substituting relations (Gl), Equation (Go3) yields the following result after some algebraic manipulation: (3)EQ, = *79 ()EQ (Q (G4) This result.is valid when the Nitric Axide is negligible and ml = 2 = -= At" etc, are good approximations, t195

APPENDIX H REACTION RATE DERIVATIVES AT THERMODYNAMIC EQUILIBRIUM The various derivatives of the reaction rate functions > and. (Equations (359) and (3511)) with respect to the specie and temperature variables, evaluated at thermodyaamic equilibrium conditions, are as follows: ~) = 2t/-~+).' ~:4 C(.. e^, /EP,.ti (+. l) /j~4e = p7 ~4 2e Xg4* (H - ~2 e 212 (Ho2) W8 /-eve2ie lEB.'z~ X. fDe) ~B~ EXP/- (H.2) ____ = _ ~. ex9S (H.5) ^....... __ e 1, _ ~9 /f d0e~7Ed7. /*e22$R:,+*XGQ /+ ~et$^Q. + /- -~~~~~~~~(H 7 / + ^^A^e X^J/ (H.e)') = _EXE- i/ Q) (Ho7) relations o -196

BIBLIOGRAPHY 1. Rosner, D. "Recent Advances in Convective Heat Transfer with Dissociation and Atom Recombination," Jet Propulsion, July 1958. 2. Griffith, W. "Recent Advances in Real Gas Effects in Hypersonic Flow," Jet Propulsion, March 1958, 3. Adams.M, "Recent Advances in Ablation," Journal of American Rocket Society, September 1959. 4. Lees, Lo "Laminar Heat Transfer over Blunt-Nosed Bodies at Hypersonic Flight Speeds," Jet Propulsion, 26, 1956. 5. Kemp, N,, Rose, P. and Detra, Ro Laminar Heat Transfer Around. Blunt Bodies in Dissociated Air, AVCO Research Report 15, May 1958, 6. Fay, J, and Riddell, F. Theory of Stagnation Point Heat Transfer in Dissociated Air, AVCO Research Report 1, April 1957. 7. Adamson, To. Nicholls, Jo and Sherman, P. A Study of the Hypersonic Laminar Boundary Layer with Dissociation, University of Michigan Engineering Research Institute Report 1, Project 2606-6-T, May 1957, 8. Adamson, To Nicholls, J. and Sherman, P. A Study of the Hypersonic Laminar Boundary Layer with Dissociation, University of Michigan Engineering Research Institute Final Report, Project 2606-6-F, September 1957. 9, Hirschfelder, Jo "Heat Transfer in Chemically Reacting Mixtures," Journal of Chemical Physics, 26, No. 2, 1957 10. Scala, S. Hypersonic Stagnation Point Heat Transfer to Surfaces Having Finite Catalytic Efficiency, Go E. MOSD Report R-58SD-236, January 1958. 11. GoulardR.R "On Catalytic Recombination Rates in Hypersonic Stagnation Heat Transfer," Jet Propulsion, November 1958. 12. Rosner, D. "Boundary Conditions for the Flow of a Multi-Component Gas," Jet Propulsion, August, 1958. 13. Rosner, D. "Chemically Frozen Boundary Layers with Catalytic Surface Reaction," Journal of the Aero, Scio, August 1959o 14. Lees, L. Convective Heat Transfer with Mass Addition and Chemical Reaction, Third Comb. and Prop, Symp., NATO-AGARD, March 1958. -F197r

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