R-FUNCTIONS APPLICATION FOR MATHEMATICAL MODELLING OF HEAT TREATMENT PROCESSES
Journal Title: Прогресивні техніка та технології харчових виробництв ресторанного господарства і торгівлі - Year 2015, Vol 2, Issue
Abstract
The investigations on the establishment of optimal mode of heat treatment process of semi-finished products are carried out in this paper. A mathematical model of heat treatment process of semi-finished products is considered in the form of non-stationary heat equation with the initial conditions and main types of boundary conditions. The presence of time component in the initial-boundary problem and the desire to take into account real geometric shapes of semis, complicate the mathematical model of the problem. There are three types of boundary conditions in the problem statement, which are corresponding to different types of external conditions. In the first type of boundary condition the temperature of external environment is given, in the second heat flux is given, and in the third – heat exchange with external environment of the given temperature. The basis of the calculation algorithm is projective Bubnov-Galyorkin method in conjunction with R-function method, which will enable to consider arbitrary geometric forms of semi-finished products and to take into account wide range of tools for efficient supply of heat treatment process. The structures of problem solution are constructed by means of R-function method. These structures exactly satisfy the boundary conditions. To determine unknown coefficients of approximate solution of Bubnov-Galyorkin projective method is proposed. Reduction of the initial-boundary problem to Cauchy problem to a system of differential equations allows carrying out numerical experiments with the use of PC in order to select optimal modes of heat treatment of semis. The solution of Cauchy problem for the differential equations system can be obtained by means of Runge-Kutta method.
Authors and Affiliations
Валерій Михайлов, Микола Синєкоп, Лариса Пархоменко, Світлана Михайлова, Марина Софронова
Keywords
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