Analysis of heat transfer in a cylinder being treated in a gas nitriding furnace using a new inverse problem solving method

This paper presents a new method for solving the inverse heat conduction problem (IHCP) for the Helmholtz equation. The purpose of undertaken research was developing a method of solving the nonstationary IHCP that could be applied to online determine boundary conditions for complex geometries without the necessity of determining the regularization parameter. The temperature distribution is sought in a form of a linear combination of basis functions satisfying the Helmholtz equation. The unknown boundary condition is determined from the minimum of the quadratic function, which is a measure of the distance between the measured and calculated temperatures at the measurement points. A stable solution of the IHCP was obtained, and distributions of the relative error of heat flux on the boundary did not exceed 15 % for a significant part of heating time. The application of the aforementioned method made it possible to determine boundary conditions, on the surface of a cylinder heat-treated in a gas nitriding furnace. The differences between the heat flux on the boundary obtained for disturbed measurement data and for data without disturbances decreased significantly in the first units of time to reach values below 6 W/m2 after 200 s of the heating process. The discussed for determining boundary conditions in nonstationary heat transfer problems is numerically stable and is characterized by a short calculation time. Hence, it can be used in online mode to control thermochemical treatment processes as well as to determine temperature in components of machines and energy devices.