Simultaneous determination of two unknown thermal coefficients through an inverse one-phase Lamé-Clapeyron (Stefan) problem with an overspecified condition on the fixed face

Abstract

Formulas are obtained for the simultaneous determination of two of the four coefficients, k (thermal conductivity), l (latent heat of fusion), c (specific heat), ρ (mass density), of a material occupying a semi-infinite medium. This determination is obtained through an inverse one-phase Lamé-Clapcyron (Stefan) problem with an overspecified condition on the fixed face of the phase change material. To solve this problem, we assume that the coefficients h 0, σ, 0 0 > 0 are known from experiments (where h 0 characterizes the heat flux through the fixed face, σ characterizes the moving boundary and 0 0 is the temperature on the fixed face). Denoting the temperature by 0, the results we obtain concerning the associated moving boundary problem are the following: (i) When one of the triples { 0,k,l}, { 0,k,p} is to be found, the corresponding moving boundary problem always has a solution of the Lamé-Clapeyron-Neumann type. (ii) If one of the triples { 0, k, c}, { 0,l, c}, { 0, l, p}, and { 0, c, ρ} has to be determined, the above property is satisfied if and only if a complementary condition for the data is verified. Formulas are also obtained for the simultaneous determination of other physical coefficients and the inequality σ 2 < Ste/2( Ste:Stefannumber)for the coefiicient ζ of the free boundary s(t) = 2 aζ t½ of the Lamé-Clapeyron solution of the one-phase Stefan problem without unknown coefficients.