Inl. J .ll eal.l/as' Transfer . Vol. 26. No. 8. PI'. 1151-1157. 1983 Primed in Great Britain 0017-9310;83 S3.00+0.00 © 1983 Pergamon Press ltd . SIMULTANEOUS DETERMINATION OF TWO UNKNOWN THERMAL COEFFICIENTS THROUGH AN INVERSE ONE-PHASE LAME-CLAPEYRON (STEFAN) PROBLEM WITH AN OVERSPECIFIED CONDITION ON THE FIXED FACEt D.A. TARZIA PROMAR (CONICET-UNR), Instituto de Matematica "Beppo Levi", Universidad Nacional de Rosario, Avenida Pellegrini 250. 2000 Rosario. Argentina (Receit'ed 18 June 1982 and in reti sed fo rm 23 November 1982) Abstract- Formulas are obtained for thesimultaneousdetermination oft wo of the four coefficients, k (thermal conductivity),/(latent heat of fusion), c (specific heat) , p (mass density), of a material occupying a semi-infinite medium. This determination is obtained through an inverse one-phase Lame-Clapeyron (Stefan) problem with an overspecified cond ition on the fixed face of the phase ch ange material. To solve this problem, we assume th at the coefficients /'0, CT, 0 0 > 0 are known from experiments (where h o characterizes the heat flux through the fixed face, CT characterizes the moving boundary and 0 0 is the temperature on the fixed face). Denoting the temperature by 0, the results we obtain concerningthe associated moving boundary problem are the follow ing : (i) When one of the triples {O,k,/}, {O,k,p} is to be found, the corresponding moving boundary problem always has a solution of the Lame-Clapeyron-Neumann typ e. (ii) If one of the triples {O,k,e}, {O,/ ,e}, {O,/,p}, and {O,c,p} has to be determined, the above property is satisfied if and only if a complementary condition for the dat a is verified, Formulas are also obtained for the simultaneous determination of other physical coefficients a nd the inequality < Slej2(Ste: Stefan number) for the coefficient of the free bound ary 5(t ) = 1/2 of th e Lame- Clapeyron solution of the one-phase Stefan problem without unkn own coefficients. I\'Ol\IEI\'CLATURE c, specific heat; f, error function; k, thermal conductivity; I, latent heat of fusion; 11 0 , coefficient defined by equ ation (2e); Ste, Stefan number, cOo/I ; s, position of phase change location; t, time variable; x, spatial variable. Greek symbols a, thermal dilfusivity, k]pc (= a 2 ); p, mass density; a, coefficient defined by equation (I); 0, temperature; 00' temperature on the fixed face, _x = 0; dimensionless parameter, a/a. I. 11'1RODUcnON SUPPOSE that two of the four coefficients, k (thermal conductivity).! (latent heat of fusion), c(specific heat), p tTh iswork has been presented a t the Reunion Nacional de Fisica-198I held at San Luis (Argent ina) on 24-27 November 1981 and was written while the author was staying on a fellowship of the Italian C.N.R. (G.N.F.M.) in the Istituto Matematico "Ulisse Dini", Uni v. di Firenze, Viale Morgagni 67A, 50134 Firenze, Italy. (mass density) of a phase (e.g. liquid) of some given materi al ar e known. If , by means of a change of phase experiment (fusion of the material at its meltin g temperature) we are able to measure the quantities 11 0 > 0, a > 0 and 0 0 0, then we will be able to find the formulas for the simultaneous determination of the unknown coefficients. Consider the inverse one-phase Lame-Clapeyron problem (or the inverse one-phase Stefan problem with constant thermal coefficients)[7, 18,35,45,54,60] with an overspecified condition on the fixed face x = 0 [53, 54]. This overspecified condition consists of the specification of the heat flux through the fixed face of the material undergoing the phase change process. Other boundary value problems for the l-dirn. heat equation with an overspecified condit ion on a part of the boundary have been analyzed [5,8,9,11-17,22,23, 29-31 ,42]. (See also the references listed in ref. [54].) In ref. [59], it was shown that the solution of the inverse conduction problem is characterized by a discontinuous dependence on data. Other references dealing with inverse problems are refs.[3, 38,47] ; those on the identification of parameters are refs. [19-21,24, 34, 49], and those on improperly posed problems in partial differentialequationsarerefs.[36,37, 43,50,58]. If we suppose that the melting temperature is zero and the moving boundary is given by s(t) = 2at 1/ 2 , with a> 0, (1) our problem is reduced to finding the temperature 1151