Parabolic Profile in Heat-Conduction Problems. 4. Semi-Bounded Space with a Convective Boundary Condition
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New approximate solutions of the heat-conduction problem for a semi-bounded space with a convective boundary condition have been obtained with the use of the heat balance integral method, the refined integral method, and the combined integral method as well as with new hybrid schemes involving a new generalized integral relation, the temperature momentum integral, and the heat-flow momentum integral. The solutions obtained were represented as a parabola with a time-dependent exponent n(t), and they are entirely explicit in form. It is shown that the optimization of the exponent n(t) on the basis of minimization of the Langford norm EL provides, as a rule, the obtaining of solutions with a low approximation accuracy. The new hybrid schemes proposed are more efficient, because they make it possible to obtain solutions in the explicit form with a fairly high approximation accuracy. It is proposed to define parabolic solutions of the heat-conduction problem with a convective boundary condition with the use of new generalized variables and without recourse to the Biot number as an independent variable. A triple hybrid scheme based on three different-kind integral relations was used for the first time in solving the heat-conduction problem with a convective boundary condition. All the solutions obtained with the hybrid schemes proposed are characterized by good approximation properties. The new hybrid schemes can be used in solving different practical problems.
Keywordsheat-conduction problem semi-bounded space convective boundary condition parabolic solution generalized integral relation triple hybrid scheme
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- 4.H. S. Carslow and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn., Oxford University Press, Oxford (1992).Google Scholar