Determination of thermophysical characteristics in a nonlinear inverse heat transfer problem

This paper presents a detailed and developed method for finding soil nonlinear thermo-physical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junc-tion of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measure-ments were carried out in the time interval ( 0 , 4tmax). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coeffi-cients of thermal conductivity, heat capacity, heat transfer and material density is inves-tigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and non-linear. The linearized difference scheme is implemented numerically by the scalar Thomas' method, and the nonlinear difference problem is solved by the Newton's method. Assum-ing the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0 , tmax), the quadratic convergence of Newton's method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton's method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (tmax, 2tmax), the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2tmax, 3tmax), the density of the soil is determined. Also, based on the measured initial data on the segment (3tmax, 4tmax), the heat transfer coefficient is determined.Based on the experimentally measured data, at each time interval, the corresponding func-tional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gra-dient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compila-tion and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed.The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 -3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the en-vironment and the soil for the heat transfer coefficient was shown.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper presents a detailed and developed method for finding soil nonlinear thermo-physical characteristics using two-layer container complexes and numerical simulations. The method aims to accurately solve the boundary value problem and provide initial data for inverse coefficient problems. The paper also discusses the differences between the solutions of linearized and nonlinear difference schemes. Rating: 9/10