Exact analytical solutions of a nonlinear nonstationary inverse problem of heat conduction for a body with low heat conduction coefficient

Exact Analytical Solutions for a Nonstationary Linear Inverse Problem of Heat Conduction for Bodies of One-Dimensional Geometry with Boundary Conditions on One Surface, as Well as on Two Surfaces for a Plane Body, a Cold Cylinder and a Hollow Sphere, Obtained in a Closed Recurrent Form

“…Similar solutions for one-dimensional bodies are also given in [5,6] in which solutions for unsteady temperatures are given explicitly, and the heat flux density was determined by differentiating the temperature fields. Subsequently, solutions were obtained to similar problems, partly having not only theoretical but also applied character, including the nonlinear one-dimensional problem of nonstationary heat conduction [7][8][9][10][11][12][13][14][15][16][17][18][19]. As was partly indicated in [2][3][4][5][6], the explicit expression of solutions for a nonstationary linear inverse problem of heat conduction for bodies of one-dimensional geometry is not possible in all cases; therefore, in order to obtain the final solution, additional assumptions have to be applied, for example, as in [2] where the thin-wall assumption is used.…”

“…Solutions for bodies of simple configuration will differ in the values of the radial quasi-polynomials Pn, 1 and Pn, 2. Within the framework of this work, these quasi-polynomials will be solved in recurrent forms, in contrast to [2][3][4][5][6] and [7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…The above relations express the recurrent form of the exact solution to the inverse problem of unsteady heat conductionfor bodies of one-dimensional geometry with non-stationary boundary conditions specified on one side. The recurrent form of writing the solution makes it possible to solve this problem from a unified position in a closed form, since the expression of solutions in explicit form, as, for example, in [7][8][9][10][11][12][13][14][15][16][17][18][19], is possible not in all cases, as indicated in [2,5,6] The issues of correctness of this inverse problem of heat conduction (i.e, the existence of a solution, its uniqueness and its stability) were considered in detail in [5,6], therefore, in this study there is no need to reconsider them. The above solutions of the non-stationary inverse problem of heat conduction for one-dimensional bodies, obtained in this work, were successfully applied in a practical way as an integral part of the conjugate problem in determining the maximum effect of the carbon layer on the surface of the combustion chamber on the non-stationary parameters of the working fluid during radiationconvective heat transfer [20][21][22], and also in the development of the theory of heat transfer in heat-insulating packaging to stabilize the temperature regimes of storage of perishable products [23][24].…”