# Mathematical aspects of transient solutions to the heat diffusion equation

### Aquino, V. M. de; Iwamoto, H.;

Problems involving the dynamics of heating or cooling of physical systems submitted to certain boundary conditions are of great importance in a wide range of situations. Since simple situations involving cooling of food where the temperature of each part of the system needs to be controlled and, in general, does not involve the presence of heat sources, even more complex situations like nuclear reactors, where there is a heat source which is dependent on position and the dependence of the reactor reactivity on the temperature requires a good knowledge of the dynamics of this quantity, both for safety and efficiency. Problems of the steady-state heat diffusion equation for a series of physical systems are easy to solve and can be found in a vast literature on the subject. To find the time evolution of temperature T (x, t) of a system submitted to a heat source, however, is not a simple task and this type of problem is usually treated in an approximate way. In this work, for one-dimensional and homogeneous system, equilibrium T(x) and transient T (x, t) solutions are presented for cases of the existence of sources permeating throughout the medium. Each system considered is supposed to have constant temperature in the edges. Calculations are presented for two cases: one is the case of a source non-dependent on position and time and the other is the case of a source with sinusoidal spatial dependence, a situation close to what would occur in a one-dimensional nuclear reactor. The extension to a three-dimensional case, homogeneous sphere with constant temperature on its surface will also be presented. The case involving the contact of two subsystems of different materials will also be analyzed.

Palavras-chave: heat diffusion equation, heating a sphere, transient temperature,

Palavras-chave:

DOI: 10.5151/meceng-wccm2012-20293

##### Referências bibliográficas
• [1] G. Kessler, “Steady state and transient profiles in a multishell spherical system heated internally by reactor-grade plutonium”. Nuclear Engineering and design 239, 2430-2443, 2009.
• [2] Ibrahim Dincer, “Simplified solution for temperature distributions of spherical and cylindrical products during rapid air cooling”. Energy Convers. Mgmt Vol. 36, N0 12, pp 1175-1184, 1995.
• [3] P.M. Morse, H. Feschbach, Methods of theoretical physics, McGraw-Hill, New York, 195
• [4] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series, and products. Academic Press, New York, 1980.
##### Como citar:

Aquino, V. M. de; Iwamoto, H.; "Mathematical aspects of transient solutions to the heat diffusion equation", p. 5052-5058 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-20293

últimos 30 dias | último ano | desde a publicação