Temperature field of an elliptical cylinder with an internal stationary heat source under mixed boundary conditions

Аuthors

Kanareykin A. I.

Sergo Ordzhonikidze Russian State University for Geological Prospecting, Moscow, 117997, Russia

e-mail: kanareykins@mail.ru

Abstract

The processes of heat exchange and related mass transfer play an exceptional role in nature and technology. To date, an impressive amount of work has been devoted to the study of heat exchange processes. Of particular scientific interest are works describing modern heat exchange elements of heat exchange equipment with a detailed description of their manufacturing methods, as well as control of a heat exchanger with a variable heat exchange surface area. Today, a new direction is being formed, which implies the inclusion of material features and properties in the mathematical models being deve-loped. Thus, functional dependencies are increasingly being included in the modeling of heat and mass transfer processes. At the moment, it is known that for the best cooling of the elements, a large surface for heat transfer is required. As you know, for the best cooling of the elements, a large surface for heat transfer is required. The surface can be increased either by finning, or by replacing rods of circular cross-section, which have minimal surface area, with other rods with an increased cross-section, for example, with an oval or elliptical cross-section.

Elliptical tube heat exchangers occupy a special place. Elliptical tube heat exchangers are increasingly being used in various industrial products. Their peculiarity is that by manipulating the length change of the semi-axes of the ellipse, it is possible to obtain accurate analyses of stationary thermal conductivity problems for a very wide range of shape changes: from a cylinder (the semi-axes of the ellipse are equal) to a thin plate (one of the semi-axes is significantly larger than the other).

However, the temperature distribution in a body of elliptical cross-section under given boundary conditions has not been sufficiently studied. The article calculates the temperature field of an elliptical cylinder with internal heat sources. The boundary conditions are set by the law of heat exchange between the cylinder surface and the environment, and an unevenly incident plane-parallel heat flow on one side. At the same time, the task itself is stationary. The solution is found analytically when moving to the system of elliptical coordinates. Based on the obtained solution, the temperature distribution over the surface of the elliptical wall was obtained. The obtained result can be useful in engineering calculations of heat exchange devices, solar collectors and fuel rods of nuclear reactors.

Keywords:

heat transfer, temperature field, elliptical cross section, boundary conditions of the third kind, heat flow, elliptical integral

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