On Direct and Inverse Initial Boundary Value Problems for Intensive Quenching Processes

In case of intensive quenching, when calculating initial thermal fluxes on the basis of classic heat conduction equation (that is, parabolic type equation), the results of computations are always overrated and significantly exceed the critical values. There is an opinion that for the right solution of this issue it is necessary to use the corresponding initial-boundary problems for hyperbolic equation of the heat conduction equation. In this work there is a strict mathematical grounding (by the way of constructing a mathematical model “from scratch” in Euler variables) of using the hyperbolic heat conduction equation when describing various aspects of steel intensive quenching process, also when determining the correct initial thermal fluxes, instead of classic (that is, parabolic) equation of heat conduction. As intensive quenching anticipates non-stationary bubble boiling, non-linear boundary conditions appear when modelling. This significantly encumbers finding of analytical solutions of the corresponding initial-boundary value problems. In this work, apart from the above mentioned foundation of use of hyperbolic equation of heat transfer in the processes of intensive quenching, one constructive approach is being elaborated, which allows obtaining analytical solutions of the corresponding non-linear initial-boundary value problems. The essence of this constructive approach is reduction of the initial non-linear model to the problem of solution of non-linear integral Volterra equation, for which the existence of fixed-point is proved and therefore, it is possible to apply Picard method – the method of successive approximations. This work is elaborated within the framework of the ESF project No 2009/0233/1DP/1.1.1.2.0/09/APIA/VIAA/142