Transactions on Modelling and Simulation vol 16, 1997 WIT Press, ISSN X

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1 An approximate method to evaluate the surface catalycity in high enthalpy flows G. Zuppardi & G. Verde Dipartimento di Scienza edlngegneria dello Spazio "Luigi G. Napolitano" - University of Naples "Federico II" P.le V. Tecchio 80, Naples - Italy Abstract An integrated numerical/experimental method is proposed to evaluate the surface catalycity. The method is approximated and relies in matching the measured heat flux with the computed one. It is iterative; the iteration parameter is the atom mass fraction at the surface. This is a boundary condition needed to integrate, numerically, the system of boundary layer differential equations. The atom mass fraction at the surface, fulfilling the match, is used to evaluate the surface catalycity. This is computed as the ratio of the atom mass fraction at the surface to the atom mass fraction at the boundary layer edge. The method has been applied to the evaluation of the catalycity of a hemisphere calorimeter in nitrogen flow, in the enthalpy range [MJ/kg]. The computation of the heat flux relies on an improved version of the Fay-Riddel procedure. The results have been compared, by means of the Goulard formula, with the experimental ones. 1 Introduction During the re-entry path of a space vehicle, the high stagnation temperatures, produced by strong shock waves, generate endothermic chemical processes of dissociation and ionization. Dissociation is followed by an exothermic process of recombination when the gas expands or flows along a cool surface. Consequently the heat flux is made up also of a diffusive term due to the atom recombination at the surface. The surface recombination efficiency is named catalycity. This is, by definition, the ratio of the number of atoms recombining at the surface to the number of atoms striking the surface. The higher is the surface catalycity, the higher is the heat flux. The knowledge of the surface catalycity is important for the design and the choice of the material of a space vehicle Thermal Protection System (TPS). In fact, an heat flux reduction

2 590 Computer Methods and Experimental Measurements by means of low catalycity surfaces may be an attractive possibility for a safe reentry or for long range hypersonic vehicles. The evaluation of the surface catalycity of a model or a calorimeter is also important both from an experimental and a numerical point of view. In fact, catalycity changes the gas composition close to the surface and influences a correct interpretation of the measured heat flux. The determination of the flow enthalpy, scaled from the heat flux measurement, is misleading whether the recombination process at the calorimeter surface is assumed to be fully catalytic or the gas composition at the calorimeter surface is assumed to be frozen (no recombination process). Knowledge of surface catalycity makes also possible to fix correctly the boundary conditions in solving numerically hypersonic flow fields. Barbato et al. [1] investigated the effects of different catalytic boundary conditions on the numerical simulation of hypersonic flows over a blunt body. Catalycity depends on a number of factors: gas composition, flow conditions, temperature, material and roughness of the surface. As the last factors can not be easily modelled, the theoretical evaluation of catalycity is still today an open problem. Barbato referenced some theoretical models trying the evaluation of the surface catalycity. A current approach is to use fits or correlations coming from experimental data. The purpose of the present paper is to propose a method to evaluate the catalycity of the surface of a test model (i.e. calorimeter). The method relies on an integrated numerical/experimental procedure. The philosophy of the method is to provide for the difficulties in modelling the effects of the surface material and roughness with experimental data. The method consists in matching the measured and computed heat fluxes. It is iterative; the iteration parameter is the atom mass fraction at the surface. This is a boundary condition needed to integrate, numerically, the system of the boundary layer differential equations. The atom mass fraction at the surface, fulfilling the match, is used to evaluate the surface catalycity. This is computed as the ratio of the atom mass fraction at the surface to the atom massfractionat the boundary layer edge. As the method has been applied to the evaluation of the catalycity of a hemisphere calorimeter, the computation of the heat flux relies on an improved version of the Fay-Riddel procedure [2]. By the present method the arc-heater wind tunnel experimenter is provided with an easily available tool to get a preliminary insight into the catalycity of the surface of hemisphere calorimeters. The computer code, developed on the present procedure is not time consuming and does not require a large core storage. It can easily operate, in fact, also on personal computers. The method has been used to evaluate the catalycity of a calorimeter in nitrogen flow in the total enthalpy range [MJ/kg]. The results have been compared, by means of the Goulard formula [3], with experimental data 2 Improved Fay-Riddel computing procedure The pioneering work by Fay and Riddel [4] is a milestone in the computation of the heat flux at the stagnation point of a blunt body, in

3 Computer Methods and Experimental Measurements 591 dissociating air. They found the solution of the laminar boundary layer by the theory of self-similar solutions, and provided also the related computing procedure. Because of the limitation of the knowledge about the kinetic parameters and the transport coefficients, and the lack of computer resources at that time (1958), the original Fay-Riddel procedure suffers from inaccuracies that reduce the validity of the results. Zuppardi and Verde [2] developed a renewed Fay-Riddel procedure, getting over the above mentioned limitations. The improvements are related to: i) the overcoming of some operating limitations (like the evaluation of the equilibrium mass fraction of each chemical specie), ii) the using of updated and more reliable thermodynamic and kinetic parameters and iii) the computing of the Prandtl number and the Lewis number in the boundary layer. The Zuppardi- Verde computing procedure, originally written to operate with simulated air (i.e. a mixture of oxygen and nitrogen), for the purpose of this paper has been specialized to operate with pure nitrogen. The independent, curvilinear space variables x and y are transformed by the Lees-Dorodnitsyn transformation in and r variables. These read: x (1) 0 0 u, p u, and r are the velocity, the density, the viscosity and the cylindrical body radius, respectively. Subscripts e, and w are for boundary layer edge and wall, respectively. Fay-Riddel procedure considers air just like one component gas; the gas is a binary mixture of "air" atoms and "air" molecule. This assumption simplifies the problem because it reduces the number of species continuity equations to just one. The non-dimensional, dependent variables are f, 8 and C* f is defined in such a way that f ' = u/u@ (superscript ' is for partial derivation with respect tot ), 8 is a non-dimensional temperature (8 = T/T^), C is the massfractionand subscript A is for atom. The system of ordinary differential equations is: Momentum: (xf) +ff"+0.5k-f = 0 (2) Energy: ((ol/pr)&') +of»' +8(LeX/Pr)» C^ +XiX2V= 0 (3) Atom continuity: ((AJLe/Pr)c]J + fc^ -%%v = 0 (4) Pr is the Prandtl number, Le is the Lewis number (Pr = Cpji/k, Le = pdcp/k, Cp is the constant pressure specific heat, k is the thermal conductivity, D is the diffusion coefficient),, X, 5, a, v, xi and %2 are defined as: (5)

4 592 Computer Methods and Experimental Measurements where: (6) *) (7) 8 =CA-CMCPW =3/7-2/7e--' (8) e-^^ (l-ca) (9) (10) Xi=KpT;Rdue/dx (ll) subscripts M, E and s are for molecule, equilibrium and inviscid stagnation point, respectively. Subscript v is for vibrationai and superscript - is for average with respect to the gas composition. RO is the universal gas constant and K the recombination rate constant. The velocity gradient at the stagnation point is computed, by the Newtonian theory, as: R is the body nose radius. -pj/ps (12) ;) (13) hf3 is the average atom dissociation enthalpy in external flow: Ah is the heat of formation of i* chemical species. The boundary conditions are: (15) Fay-Riddel provided the following equation to compute the heat transfer at the wall:

5 Computer Methods and Experimental Measurements 593 I i J ri=0 /dx)j(hs - h J/Pr] (16) s is the ratio of the current mass fraction and the mass fraction at the boundary layer edge. 3 Evaluation of the surface catalycity The system of the three ordinary differential equations (2), (3) and (4) is integrated by a 4& order Runge-Kutta algorithm in the range of the independent variable 0<r <3. Beside C^w other boundary conditions, to be fixed for the numerical integration, are the derivatives (d^f/drf)^, (d&/dr )^ and (dc^/d7i)w It is necessary to use the numerical, iterative procedure, named shooting technique, to state the values both of C^w and of the unknown derivatives. In the present method the shooting technique relies on four nested iteration cycles. The inner cycles are on the above mentioned derivatives. These start from guessed values, and stop when df/dr], 8 and C*, at rj=3, are equal, with a convergence criterion of 0.001, to 1, 1 and C^, respectively. The boundary layer solution is achieved at the convergence of these cycles. The outer iteration cycle is on the atom mass fraction at the wall. This cycle starts from C*w = 0 (fully catalytic surface), and stops when the wall heat flux, computed by Eq.16, matches, with a convergence criterion of ±0.05, the measured one. The value of C^w fulfilling the match is used to compute the surface catalycity. As already said, the catalycity (y) is the ratio of the number of atoms recombining at the wall (N,.), per unit area and time, to the total number of atoms striking the wall (N), per unit area and time:,.* As CAW is representative of N,. and C^ is representative of N, y can be approximately evaluated as: y=^aw (18) CAS Often the recombination efficiency is defined in terms of the catalytic reaction rate (k): where m is the atom weight.

6 594 Computer Methods and Experimental Measurements By the simplifying hypotheses: T^ «Tg, X = 1, Pr = const., Le = const, allowing the uncoupling of the boundary layer equations (at the stagnation point of blunt bodies), Goulard [3] found, analytically, a formula linking the convective heat flux with partially catalytic surface (q^) to the one with fully catalytic surface (4wfc)" 4wfc 1 h0g is the flow total enthalpy, <p is the catalytic factor. This, as reported by Pope [5], reads: (21) Ps Ps Sc is the Schmidt number (Sc = uvpd), Z is the compressibility factor: 4 Analysis of the results In order to work in physically congruent conditions, in terms of temperature, density, gas composition and so on, the computer code, developed on the present procedure, has been interfaced with a computer code [6] modelling the jet of an arc-heater wind tunnel. The original version of the code is able to operate with simulated air (i.e. a mixture of nitrogen and oxygen). For the purpose of this paper the code, that works like a pre-processor, been specialized to operate with pure nitrogen. The simulated wind tunnel is supposed to be made of: 1) a heater, 2) a chamber to mix hot nitrogen (coming from the heater) and cold gas (e.g. oxygen) if any, and 3) a supersonic nozzle (area ratio 1:4). It solves an one-dimensional, steady, inviscid flow field. The thermochemical model considers the dissociation and the vibration of the N2 molecule; the gas is a mixture of two species (% and N) in chemical non-equilibrium. The thermo-fluid-dynamic parameters and the gas composition at the nozzle exit are considered to be the asymptotic ones. The boundary layer edge parameters, at the stagnation point, are the ones downstream a normal shock wave, due to the calorimeter. The gas composition is supposed to be frozen both through the shock and along the stand-off distance up to the boundary layer edge (no relevant chemical relaxation process was found). The flow total enthalpy at the stagnation point is the same as the asymptotic one: hgg =ho^. The total flow enthalpy will be labelled as hg. The heat flux measurements have been performed by Rose and Stankewics [7] (also reported by Zoby [8]) in an arc-driven shock-tube by a

7 Computer Methods and Experimental Measurements calorimetric gage (thick resistance thermometer) of platinum, mounted at the stagnation point of an hemisphere model (R=0.635 [cm]). Unfortunately it has to be expected that the test conditions and the numerically simulated test conditions (computed by the above described pre-processor) are different. Only the total flow enthalpy is reproduced accurately. The temperature of the calorimeter is not provided. For the present runs a typical value of T^=350 [K] was assumed. Moreover, because of the shortness of the test time (this was on the order of 10 [us]), the measured heat fluxes could be uncertain. The method has been applied in the range of the flow total enthalpy: 48<ho<59 [MJ/kg]. This enthalpy range has been chosen because for flow enthalpy lower then 48 [MJ/kg] the catalycity process could be not relevant (the nitrogen dissociation energy is 34 [MJ/kg]). For flow enthalpy higher than 59 [MJ/kg] a gas ionization process occurs and the code is not able to consider this process (the ionization energy of nitrogen molecule and nitrogen atom is 54 and 100 [MJ/kg], respectively). More specifically the data shown here are related to the following five values of the flow total enthalpy: 48, 53, 55, 58, 59 [MJ/kg]. Table I shows some asymptotic parameters as a function of ho- Table I - Asymptotic thermo-fluid-dynamic parameters ho [MJ/kg] V= [m/s] ToofKl Poo [kg/m^] 5.99x10^ 5.77x10^ 5.68x10-^ 5.54x10^ 5.47x10^ %* CNOC The measured heat fluxes and the computed ones, with fully catalytic surface, as a function of the flow total enthalpy, are shown in Fig 1. Table II shows the atom concentration at the wall (C^wX providing the match of the measured and the computed heatfluxes,the atom concentration at the boundary edge (CNS), the surface catalycity (y), the catalytic reaction rate (k^), the measured (q^m) and the computed (q^c) heatfluxes.as shown, the match of the computed and the measured heat flux is good enough, the percentage difference ranges between -1.8% and 2.6%. Figs.2 and 3 show, as a typical example of the boundary layer solution, the non-dimensional temperature (8), and the mass fraction profiles of nitrogen atom (CN) and molecule (C^) as a function of rj, respectively. The comparison of the catalytic reaction rate by the present method with the experimental one is performed indirectly in terms of the ratio q^/awfc (Fig.4). The ratio qw/(wc, labelled as measured, has been computed as the ratio of the measured heat flux to the one computed by the present code for a fully catalytic surface. The ratio, labelled as present method + Eq.20, has been

8 Computer Methods and Experimental Measurements i Computed:ftillycatilytic suffice ; J Figure 1: Heat fluxes vs. flow total enthalpy Table II - Recombination efficiency and catalytic reaction rate h<)[mj/kg] CNW CNS T = CNW^NS 4.70x x x10"! 7.54x x10-2 kw [cm/s] q^[kw/m2] q^[kw/m2] i y/^ -! ] / ^~ - - o.oo f Figure 2: Non-dimensional temperature profiles in the boundary layer

9 Computer Methods and Experimental Measurements J97 Figure 3: Nitrogen mass fraction profiles in the boundary layer Present method + Eq.20 Measured (Ref.7) ho[mj/kg] Figure 4: Heat fluxes vs. flow total enthalpy computed by Eq.20 by using the catalytic reaction rates (shown in table II) by the present method. As suggested by Goulard, a constant Lewis number (Le=1.4) and a constant Schmidt number (Sc=0.485) have been used. Typically the computed, asymptotic Lewis number ranges from 0.61 to 0.58, the Schmidt number ranges from 0.89 to At the wall the Lewis number ranges from 0.93 to 0.96, the Schmidt number ranges from 1.28 to Both sets of data show the same profile. The mismatch of the values (the percentage discrepancy range between 19% to 30%) had to be expected. In fact, besides the intrinsic approximations of both the present method and the Goulard formula (Eq.20), there is uncertainty in the measured heat flux and mismatch of the computed and the experimental asymptotic flow conditions. Because of these difficulties a reliable check of the method can be performed, by using the results of thermo-chemical, non-equilibrium Navier-Stockes codes (e.g. TINA, VIRGIN1).

10 598 Computer Methods and Experimental Measurements 5 Conclusions An integrated numerical/experimental method is proposed for an approximate evaluation of the surface catalycity. The method relies in matching the measured heat flux with the computed one and in evaluating the surface catalycity as the ratio of the atom mass fraction at the wall (fulfilling the match) to the atom mass fraction at the boundary edge. The method has been used to evaluate the surface catalycity of a calorimeter, mounted on a hemisphere cylinder. Numerical tests have been performed by considering heat fluxes, measured in nitrogen flow in the total enthalpy range [MJ/kg]. The computation of the heat flux is performed by an improved version of the Fay-Riddel procedure. As the computer code requires limited resources both in terms of core storage and processing time, it could be easily used by an arcwind tunnel experimenter to get a preliminary insight into the surface catalycity of hemisphere calorimeters. The method seems to be sound, but the reliability of the results is closely connected to the accuracy of the measured heat flux and to the capability in reproducing numerically the test conditions. 6 References 1. Barbato, M., Giordano, D, Muylaert, J & Bruno, C Comparison of Catalytic Wall Conditions for Hypersonic Flow, Journal of Spacecraft and Rockets, Vol. 33, Sept.-Oct., Zuppardi, G & Verde, G. An Improved Fay-Riddel Procedure to Compute the Stagnation Point Heat Flux (Submitted for publication on Journal of Spacecraft and Rockets) 3. Goulard, R On Catalytic Recombination Rates in Hypersonic Stagnation Heat Transfer, Jet Propulsion, Vol. 28, No 11, Nov. 1958, pp Fay, J A & Riddel, FR Theory of Stagnation Point Heat Transfer in Dissociated Air, Journal of Aeronautical Sciences, Vol. 25, No 2, Feb. 1958, pp Pope, B R Stagnation-Point Heat Transfer in Frozen Boundary Layer, AIAA J., Vol.6, N. 4, Apr pp Zuppardi, G, Verde, G & Esposito A., Numerical Modelling of an Arc Jet Wind Tunnel, DISIS Report J-96-1, Jan (in Italian) 1. Rose, P.H. & Stankewics JO Stagnation Point Heat Transfer Measurements in Partially Ionized Air, AIAA J., Vol.1, No 12, Dec 1963, pp Zoby, E V Empirical Stagnation Point Heat Transfer Relation in Several Gas Mixtures at High Enthalpy Levels, NASA TND-4799, Oct. 1968