Temperature field of a cylinder with an internal stationary heat source and an elliptical shell 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

Heat exchangers are widely used in many areas of economic activity, aerospace, energy, chemical, oil refining, food industry, refrigeration and cryogenic equipment, heating systems, hot water supply, air conditioning, various heat engines. Due to the increase in the energy armament of spacecraft, the en-ergy intensity of devices, and the increased requirements for possible modes of regulating systems, the issue of studying possible processes of heat exchange intensification is quite relevant. It is espe-cially important to reduce the weight and size characteristics of heat exchangers due to the intensifica-tion processes.
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 a minimum area, with other rods with an increased cross-section, for example, with an oval or ellipti-cal 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 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 con-ditions has not been sufficiently studied. The article calculates the temperature field of a cylinder with an internal heat source and an elliptical shell surrounding it under boundary conditions of the fourth kind. Convective heat exchange with the environment takes place on the shell surface. In addition, the shell is subjected to unilateral heating by a radiant heat flow. At the same time, the task itself is sta-tionary. The solution is found analytically when moving to the elliptical coordinate system. It is estab-lished that the temperature field along the section of the cylinder and its surrounding shell varies ac-cording to the law of cosine. From the solution, a ratio is obtained for calculating the temperature dis-tribution over the shell surface. The proposed mathematical model and its solution algorithm are in-tended for engineering calculations of heat exchangers, solar collectors and fuel rods of nuclear reac-tors using elliptical channels.

Keywords:

heat transfer, temperature field, elliptical cross section, boundary conditions of the fourth kind, heat flow, heat flow balance equation, elliptic integral

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