This paper describes a numerical method developed to estimate the heat flux at the surface of a solid, using temperature measurements inside it. This inverse problem is solved while taking advantage of the available analytical solution of the direct problem. Moreover, the efficiency of the proposed algorithm appears when analysing several measurements because most of the calculations are made only once for all the records. INTRODUCTION During a current interruption process in a circuit-breaker, the energy dissipated in the electric arc is transmitted to the surrounding media (quenching gas, insulating nozzle, metallic contacts, ...). Specifically this paper deals with the problem of estimating the heat flux flowing from the arc column to the electrodes : this information is indeed very important to characterise the arc-electrode interaction. Unfortunately a direct measurement of that heat flux is practically impossible due to the high temperatures involved by the electric arc. Nevertheless that information can be obtained using temperature measurements inside the electrode. This paper doesn't deal with the measurement problems; but it describes a numerical method to determine the unknown heat flux when the measurements are available. In the practical studied situation, one can assume that: • the heat flux is different from zero only during a given period of time (corresponding to the arcing time); • the heat flux is uniformly distributed over the electrode surface; • the material properties are supposed to be constant (no phase changes). Transactions on Engineering Sciences vol 5, © 1994 WIT Press, www.witpress.com, ISSN 1743-3533 376 Heat Transfer MATHEMATICAL FORMULATION A rather general approach can be derived using the concept of transfer function of a system. Here the relation between the heat flux (excitation) and the temperature T(t) (response) can be noted: Q(t) H T(t) (1) For a given geometry, the operator H depends on the material properties and on the probe position. First, one can choose a set of N linearly independent excitation functions E(t) that are different from zero only in an interval [0,t] (outside which the heat flux is a priori equal to zero). The response functions R(t) are then defined by : E(t) H R(t) (2) Since the problem is linear, these response functions are also linearly independent. They can be linearly combined to define a set of functions S(t), orthonormal over a chosen time interval [ti,t2J : S(t) = A-R(t) (3) This set of orthonormal functions is the response of the system to a set of excitation functions defined as follows : F(t) = A-E(t) and F(t) ** ) S(t) (4) The elements of the matrix A have to be determined to satisfy : Si (t) Sj (t) dt = 8jj (5ij is the Kronecker symbol) (5)