fluid r)yrmnia, 401 31, N0.

2. 1996

FORMULAS FOR THE HEAT AND DIFFUSION FLUXES TO A CATALYTIC SURFACE IN A CHEMICALLY NONEQUILIBRIUM MULTICOMPONENT BOUNDARY LAYER V L. Kovalev and O . N . Suslov (deceased)

UDC 533 .6.011 .5 :532.526:54112

On the basis of an asymptotic expansion of the solution of the equations of a multicomponent chemically nonequilibrium boundary layer for large Schmidt numbers, formulas are obtained for the heat flux and the diffusion fluxes of the reaction products and chemical elements on a surface with arbitrary catalytic activity . The results are compared with well-known analytic and numerical solutions . The comparison reveals the high accuracy of the formulas proposed . The results of calculating the diffusional separation of the mixture due to the selectivity of the catalytic properties of the surface with respect to recombination of oxygen and nitrogen atoms are presented . Values of the reduction of the convective heat fluxes due to the catalytic properties of the surface are obtained over a wide range of conditions in the free stream .

Approximate formulas are widely used in analyzing heat exchange with catalytic surfaces [1, 2] . For example, in [3] in investigating the catalytic properties of modern heat shield coating the Goulard formula [2] generalized to include the case of a mixture of several dissociated gases was used . However, the use of these formulas is limited by the fact that the FayRiddell formula [1] was obtained for an ideally catalytic surface and the Goulard formula for a frozen flow in the boundary layer. In the present paper, on the basis of an asymptotic solution of the boundary layer equations for large Schmidt numbers [4], formulas are obtained for the heat flux and the diffusion fluxes of the reaction products and chemical elements on a surface with arbitrary catalytic activity and any degree of nonequilibrium in the boundary layer . A comparison of the results with well-known analytic and numerical solutions [5 - 8] reveals the high accuracy of the formulas proposed. Earlier, the authors used this approach for a parametric investigation of the diffusional separation of chemical elements on a catalytic surface as a function of the Damlcoehler number of the homogeneous and heterogeneous reactions and the atom concentration at the outer edge of the boundary layer [9] . 1 . We will consider the case of supersonic gas flow over a blunt body when in the neighborhood of its catalytic surface a multicomponent partially-ionized, chemicallynonequilibrium boundary layer is formed . The system of equations and the boundary conditions describing the flow in the neighborhood of the stagnation point have the form : =- ffY + W,

=-

I

a1 1=0 :

Y=RS,

'1 - ~:

(1 .1)

IIY

a1

1 Z=Z

If we restrict out attention to the zeroth term of the series expansion of the solution in (1/Sc)' 13 and the fast term of the series in calculating the integrals of Laplace type, then the asymptotic expansion of the solution of the problem (U), (L2), constructed on the assumption I/Sc 0, W=O((1/Sc)'13) [4], makes it possible to obtain formulas for the concentration drops of the reaction products and chemical elements and for the enthalpy in the form : Z - Z, _ ° (l/Sc) -°MJ + N h), J=(Jt,

, Jn

J„-n, .p Jn

W = (w1 , .. ., w~ _ ~~, 0, r = (rt ,

1'=f ° J,

, r,, _ n , 0 0, 0, re)',

~°=(~µwPw(v + 1)) -

Jq) r ,

0, 0, 0)r, t = t P/(µ P),,,

to

W=11 tw

R $ =a°r t = a P(v + 1)

Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaga, No . 2, pp. 171-176, March-April, 1996. Original article submitted February 21, 1995 . 308

0015-4628/96/3102-0308$1250 ® 1996 Plenum Publishing Corporation



Here, the matrices M and N are expressed in terms of the eigenvalues and eigenvectors of the matrix II ; f, Sc, r), and a are the reduced stream function, the characteristic Schmidt number, the Dorodnitsyn variable in Lees form, and the longitudinal velocity component gradient on the outer edge of the boundary layer ; c,, c,', J;, J', J Q, h, p, and µ are the mass concentrations of the reaction products and chemical elements, the components of the diffusion fluxes and heat flux along the normal to the surface, and the enthalpy density and viscosity of the mixture, respectively ; n e and n are the numbers of elements and components, the subscript w relates to the conditions on the surface and the subscript e on unknown functions to the conditions at the outer edge of the boundary layer, while the subscript t denotes the transposition operation. The mass rates of formation of the components in homogeneous and heterogeneous catalytic reactions w, and r, are expressed in terms of the quantities v ;, which represent the deviations from equilibrium of the independent homogeneous chemical reactions only [9] : W=WeB'V,

R=W5eS 1 V

The components of the matrix W8 , the column V, and the diagonal matrix a are given in [9] . In [10], within the framework of the Langmuir theory of an ideal adsorbed layer, it was shown that the elements of the diagonal matrix e and the matrix W depend on the temperature, pressure, and concentrations, the temperature dependence being due to the dependence of the elements of these matrices on the reverse-reaction rate constants of all the elementary stages on the surface and the equilibrium constants in the adsorption-desorption and independent homogeneous reactions . Substituting the fluxes from the boundary condition on the body (L2) in relation (1 .3), we obtain a system of algebraic equations for determining the reaction products, the diffusional separation of the chemical elements, and the enthalpy on the surface : Z - Z, = BLAB -1(AdWsCs + BL -1t3B -1A w(g W)V ~ =e-1t3eg1,

(5 =

_2I3

,

e=1/Sc

where the columns of the matrix B are eigenvectors of the matrix II, and the elements of the diagonal matrix L are eigenvalues of the matrix II . The matrices Ad and A,, are functions of B, L, and f'(0) [9] . For a known composition and enthalpy on the surface, all the fluxes can be found from the formula K=!_(±) (±) [A d + BL-1t3BaAw(W 9(R'S{~-1I-1BL- B-1

J=K(Z - Z),

(1 .5)

We note that K is the heat transfer coefficient matrix a multicomponent partially-ionized nonequilibrium boundary layer on a surface with arbitrary catalytic activity . 2 . In the case of five-component dissociated air flow over the forward stagnation point of a blunt body when the following dissociation-recombination reactions : OZ + M=20 + M, N2 + M=2N + M, NO + M=N + O + M ; M=0 2, N2 , NO, O, and N take place between the components, with respect to diffusion the mixture can be quite accurately divided into two groups : O and N atoms (a) and 02, N2, and NO molecules (m) [6] . In this case the diffusion is not binary, since it is characterized by three Schmidt numbers S am , S,,,,,,, and S„ calculated from the corresponding binary diffusion coefficients. In this case the eigenvalues of the matrix II and nonzero elements of the matrix B required for calculation purpose are A 1 =SiwISa&

A2 =SJSg ,

b11 =b 12 =bat =-l ; 1=1,2,3 ;

A3 =Le - a/Sc,

b22 =b31 =b41 =b5S =1 ;

baa =(c 1 + c3/2)/c,, ;

bS2 =Q2 - Q1 + h5 - h4,

A 4 =1 b,,=c/c am

b51 =Q3 - Qt + (h5 - h4)n b33 =h4 - h5,

bS1 = -

A

ac

A = -ca(co h4 + cNh5) + c4ha + c5h 5 + ca(c 1 Q1 + c2 Q2 + c3Q3)

S =x0S

+ xj,~_,

.S'a =XjS + Xa,Sa,

Sc =Sa,

Xa =x + X5

309



/ max

4 ~4 H, km

60 0,6 50 0,55 0,7

F 0, 2 ,

10

10

40 2

K p,, cm/s

4

5

Fig . 1

Fig . 2

Fig. 3

Fig. 4

X a =X 1 + XZ + X3,

C g =C4 + CS ,

9

V,,, km/s

C, =C 1 + CZ + C3

where Le and a are the Lewis and Prandtl numbers, numbers from 1 to 5 are assigned to the components 0 2, N 2, NO, O, and N, respectively; and c;, x, c 0 , c N ', Q ;, and h are the mass and molar concentrations of the components, the concentrations of the chemical elements, the formation heats of the reaction products, and the enthalpies of the basic components . We note that is a double eigenvalue . In Fig. 1 we have plotted graphs of the reduction of the convective heat fluxes Jq /4~ due to the catalytic properties of the surface in nonequilibrium (curve 1) and frozen boundary layers obtained using the asymptotic formulas (L4), (1.5) in the neighborhood of the forward stagnation point of a sphere of radius 0 .5 m with a surface temperature T =700°K. Here, q and 1Q are the convective heat fluxes to a surface with finite catalytic activity and an ideally catalytic surface, respectively. It was assumed that at the outer edge of the boundary layer the temperature T =6400° K and the pressure p~=0 .216 atm . It was also assumed that the heterogeneous recombination of oxygen and nitrogen atoms is a first-order reaction: r1 =-pK~c4 ,

r2 =-rK,lr c s ,

r3 =0,

Here, the dimensionality of the coefficients Kand Kis cms' .

3 10

K, N =K, o =Kw,

cm/s

K k ~ , em/s

Fig . 5 The difference of the order of 3-10% from the results of the numerical calculations [8] (broken curves in Fig . 1) can be attributed to the use of different data for the rate constants of the homogeneous reactions and the transport coefficients . A comparison with the results of papers [5-7] of the diffusional separation of a mixture showed that the co obtained in the present paper differ by no more than 2-3% . The use of the asymptotic formulas proposed makes it possible to investigate the heat and mass transfer between a chemically nonequilibrium multicomponent boundary layer and a catalytic surface over a wide range of the key parameters of the problem . For various models describing the catalytic properties of the surface the heat flux reduction isolines were obtained in the (flight velocity V~ flight height H) plane . In Fig . 2 we have plotted the isolines of the heat flux ratios 9/qm" for a surface covered with a reactively treated glass-material having a high silica content [3] . The numbers on the curves correspond to the values of the ratio J9 /Jqm" . The surface of the body was assumed to be an equilibrium radiating surface with emissivity e =0 .8 . The model used to describe the catalytic properties of the surface was obtained on the basis of a detailed consideration of the catalytic reaction mechanism [10] . In this case the effective catalytic activity coefficients Kand Kare functions of the temperature, pressure, and composition . Figure 3 illustrates the behavior of K(continuous curves) and K,~ (broken curves) for H=40 km as functions of the flight velocity. Curves 1 correspond to the results obtained using the temperature dependences [3] for the recombination coefficients . It can be seen that in this case the catalytic activity coefficients rapidly increase with the velocity. Curves 2 were obtained using the model for describing the catalytic properties of the surface proposed in [10] . We note that here the velocity dependence is nonmonotonic and the quantities Kand Kare in agreement with the laboratory experimental data [11] (K= 1 .2 m In Fig . 4 we have plotted the ratios J9/9~ as functions of the flight velocity in the neighborhood of the stagnation point for a body with a bluntness radius R o =1 m flying at a height H=60 km for certain model descriptions of the catalytic properties of the surface . In Fig. 4 curve 1 shows that neglecting the pressure and composition dependence of the catalytic activity coefficients, as in [3], leads to a substantial overestimation of the heat flux for real flight velocities. In this case the specific heat fluxes are 80-85% of the corresponding values obtained for an ideally catalytic surface, which is in good agreement with the results of the numerical calculations carried out in [12] (7388%) . However, the results obtained using the model description of the catalytic properties of the surface outlined in [10] (curve 2) differ from those obtained for a noncatalytic surface (curve 3) by no more than 20% . Curve 4 was obtained for K=10 .4 ms' and K=114 ms' [13] . We note that the results presented in Figs . 2 and 4, like the calculations carried out in 311



[14, 15] using the model [10], show that the heat flux values are in good agreement with the data of flight experiments [16] . In Fig. 5 we have plotted the variation of the concentration of the chemical element oxygen c 0 ' on the surface of a sphere with R0 =1 m and T,=3000° K as a function of the catalytic properties of the surface for a free-stream velocity V_= 10 km s' and H=60 km . Under these conditions, at the outer edge of the boundary layer the air is almost completely dissociated, inside the boundary layer the chemical reactions are frozen, and on an ideally catalytic surface there is no diffusional separation of the mixture [5-7] . However, from Fig. 5 it can be seen that where there is considerable difference between the catalytic properties of the surface with respect to recombination of oxygen and nitrogen atoms diffusional separation occurs . For an ideally catalytic surface the diffusional separation is associated with the conditions at the outer edge of the boundary layer and is due to the higher mobility of atoms as compared with molecules . For example, the maximum accumulation of oxygen on the body takes place when at the outer edge of the boundary layer the oxygen is completely dissociated, while the nitrogen has not yet begun to dissociate_ In these circumstances the diffusional separation of the mixture is associated with the conditions on the body. For example, if K,~ is large and K,~ =0, the surface promotes the recombination of oxygen atoms and opposes the recombination of nitrogen atoms, which results in an accumulation of the chemical element oxygen on the body . REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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J. A. Fay and F F Riddell, Theory of stagnation point transfer in dissociated air" J . Aeronaut., Sc 25, No. 2, 73 (1958) . K Goulard, "On catalytic recombination rates in hypersonic stagnation heat transfer" Jet Propulsion, 1B, 737 (1958) . C. D . Scott, "Catalytic recombination of nitrogen and oxygen on high-temperature reusable surface insulation," AIAA Paper, No. 1477 (1980) . O. N . Suslov Asymptotic integration of the equations of a multicomponent, chemically nonequilibrium boundary layer" in : Aerodynamics of Hypersonic Flows with Blowing [in Russian], Izd. MGU, Moscow (1979), p . 6 . N . A. Anfmov, "Some effects accociated with the multicomponent nature of gas mixtures," Izv Akad Nauk SSSR, Meth Mashinosm ., No. 5, 117 (1963) . G. A . Tirskii, "Determination of the effective diffusion coefficients in a laminar multicomponent boundary layer," Dokl. Akad. Nauk SSSR, 155, 1278 (1964) . V G . Gmmo "Chemically nonequilibrium laminar boundary layer in dissociated air" Izv. Akad Nauk SSSR, Mekh. Zhidk Gaza, No . 2, 3 (1966) . V G . Voronkin and L. K Geraskina, "Nonequilibrium laminar dissociating-air boundary layer on axisymmetric bodies," Izv. Akad. Nauk SSSR, Mekk Zhidk Gaza, No . 3, 144 (1969) . V L. Kovalev and O . N. Suslov, "Diffusional separation of chemical elements on a catalytic surface," Izv. Akad. Nauk SSSR, Mekh. Zhidk Gaza, No . 4, 115 (1988) . V L . Kovalev and O . N . Suslov, "Model of the interaction between partially ionized air and a catalytic surface," in : Investigation of Hypersonic Aerodynamics and Heat Transfer with Allowance for Nvnequilibrium Chemical Reactions [in Russian], Izd . MGU, Moscow (1987), p .58. P. N. Baronets, A. N . Gordeev, A. F Kolesnikov et al . "Developing heat shield materials for the "Buran" orbital vehicle on inductive plasmatrons," in: Gagarin Scientific Studies in Aviation and Cosmonaudcs, 1990, 1991 [in Russian], Nauka, Moscow (1991), p. 41. C . D . Scott, "Space Shuttle laminar heating with finite rate catalytic recombination," AIAA Paper, No. 1144, (1981) . H . Tong, H. L. Morse, and D. M . Curry Application of a nonequilibrium viscous-layer computational procedure to the evaluation of space shuttle TPS requirements and material performance," AIAA Paper, No . 757 (1974) . V L. Kovalev and A_ A . Krupnw, "Multicomponent chemically-reacting turbulent viscous shock layer in the neighborhood of a catalytic surface," Izv. Akad. Nauk SSSR, Mekk Zhidk Gaza, No. 2, 144 (1989) . V L. Kwelev, "Modeling of the diffusion processes in describing chemically nonequilibrium flows in the neighborhood of catalytic surfaces," Vestn . MGU, Ser. 1, Matenwtika, Mekhanika, No . 1,86 (1995) . J . V Rakich, D. A . Stewart, and M . J. Lanfranco, "Results of a flight experiment on the catalytic efficiency of the "Space Shuttle" heat shield," AMA Paper, No . 0944 (1982) .