Modelling of the cylindrical geometry cooling process based on the solution of the inverse problem

Processes of thermo-chemical treatment, such as nitriding, are used to create a surface layer of high mechanical values. When the nitriding process, often consisting of a multi-stage heating and soaking, is ended, elements being under treatment are cooled. The cooling rate depends on the massiveness and geometry of the given element. Too fast cooling can result in the formation of high temperature gradients, which leads to the element damage. This paper presents numerical analysis of a cylinder cooling. The non-linear, unsteady inverse problem for the heat equation was solved. Test examples were chosen based on experimental research conducted in the furnace for thermo-chemical treatment. * Corresponding author: magda.joachimiak@put.poznan.pl E3S Web of Conferences 321, 02017 (2021) ICCHMT 2021 https://doi.org/10.1051/e3sconf/202132102017 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).


Introduction
Machine elements are submitted to thermo-chemical treatment, such as nitriding, to create a surface layer of high mechanical properties. After the process phase consisting in a series of heating and soaking is ended, the element is cooled. It is necessary to adjust the speed of the cooling process to the massiveness and complexity of the heated element surface in such a way that the permissive thermal stresses are not exceeded. So far, research on heating the cylinder in the furnace for thermo-chemical treatment was conducted. To determine thermodynamic parameters, the inverse problem for the heat conduction equation was solved [1,2]. Research on the treatment of metal elements with laser are presented in papers [3,4]. To analyze the heat flow, the solution to the inverse problem was applied. Inverse problems were also used to investigate the heat flow in the aluminum solid block during the process of water cooling [5]. Computing thermal stresses can be done based on temperature distribution in the element being cooled. They can be determined by solving the direct heat conduction problem if the temperature, the heat flux or the combined radiation convection heat transfer coefficient (CHTC) on the boundary is known. Temperature measurement on the boundary of the element being treated is impossible during the processes of thermochemical treatment or is subject to a great error. However, it is possible to measure temperature inside the cooled element and to determine the boundary condition by solving the inverse problem [6]. Inverse problems are ill-conditioned numerically [7]. Many studies on methods for solving inverse problems and on their stability were conducted [8][9][10][11], with particular focus on cylindrical geometries, what was discussed in papers [12][13][14]. Determination of the heat transfer coefficient for the cooling process by solving the inverse problem with the use of the Trefftz method was presented in the paper [15]. Today, inverse problems are widely used to determine the temperature, the heat flux and the heat transfer coefficient in elements of machines and thermal devices during heating and cooling processes [16][17][18][19][20][21][22]. In this paper, numerical research on the solution to the inverse problem for cooling the cylinder is presented.

Calculation model
The process of heat flow in the cylinder is described by the heat conduction equation with the initial condition Temperature on the boundary of the cylinder during the time of cooling was sought , To that end, the boundary inverse problem was solved. Temperature measurement inside the cylinder was assumed. The distance between the calculated and measured values at points of temperature measurement is described by the functional of the following form T T (6) Details of the calculation model for the heating process were given in papers [1,13].

Numerical tests
For calculations, the assumed temperature of the boundary was of the following form  Values of the error in temperature measurement in the function of the measured temperature are shown in figure 4. To verify the program operation, the following numerical tests were carried out: x Test 1 (INVERSE PROBLEM) no disturbance to measured values x Test 2 (INVERSE PROBLEM PLUS) Results in the first time units differ from the assumed distribution of the heat flux. These differences decrease in next time units (Fig. 6). We obtained values of the CHTC similar to the assumed distribution for test 3 and significantly different values for test 2 (Fig. 7). For the time interval [t 2 , t 3 ], values of the CHTC (Fig.  8) and of the heat flux ( Fig. 9) were investigated with the maximum overstating (test 2) and the maximum understating (test 3, Fig. 10, 11) of the measured values resulting from the maximum error in the measurement and from the thermocouples displacement. Despite the assumed disturbances, a stable solution to the inverse problem was obtained. Differences between the solution to the direct problem and to the inverse problem were described by the maximum absolute norm (8) and by the relative norm

Conclusion
This paper presented numerical tests for a non-linear, unsteady inverse problem. Research investigated the process of cooling the cylinder. Tests were performed with respect to cooling massive elements. A stable solution to the inverse problem for the temperature with regard to maximum possible disturbances to measurement was obtained. For the case of undisturbed input data, the CHTC and the heat flux on the cylinder boundary were achieved with a great accuracy. For initial time units the solution to the inverse problem is more sensitive. Values of the CHTC and the heat flux gave satisfying results for the second time interval (from 1 000 to 10 000 s).
Nomenclature f -function FC -function of temperature of the cooling rate setpoint ET -error in temperature measurement, [ᵒC] IP -inverse problem Subscript: 0 -initial (for t = 0) b -boundary c -calculated value DP -direct problem g -gas IP -inverse problem m -measured value