A case of combined radial and axial heat flow in composite cylinders

Abstract

Introduction. Although several problems of heat flow in composite cylinders have been studied, all the cases considered treat the heat flow in the radial direction only [1,2,3]. The case of combined radial and axial heat flow in composite cylinders presents an interesting boundary value problem which has also considerable significance in the theory of vibrations and propagation of electromagnetic waves [4,5,6]. In this paper, we consider a case of combined radial and axial heat flow in the unsteady state in finite cylinders composed of two coaxial parts of different materials. The temperature distribution in the cylinder at any instant under the assumed boundary and initial conditions has been obtained by the use of the Laplace transformation.The procedure is illustrated by a numerical calculation in a particular case.The Problem. Composite cylinder made of two different materials, the inner cylinder 0 < r < a and the outer cylinder a < r <b having thermal conductivity and diffusivity coefficients ex and fc, and e2 and k2 respectively.f Boundary conditions: The flat ends of the cylinder x = 0 and x = I kept at zero temperature with the outer surface insulated and perfect thermal contact at r = a between the two coaxial parts. Initially the cylinder is assumed to be heated to constant unit temperature. Required the temperature distribution in the cylinder for any time t > 0 (see Fig. 1).

The transient temperature field in a composite semi-space resulting from an incident heat flux

“…To effect a dual inversion of the two-dimensional part of the temperature solution in the form given by (12) and (13) is extremely difficult. The form of these equations does, however, suggest some simplification providing we make the temporary restriction (ultimately to be relaxed): 2 The case of arbitrary time-dependent heat inputs is accessible through the additional use of…”

“…Solutions to problems involving the conduction of heat in two distinct materials for heat flow described by a single spatial dimension have been extensively treated for both planar and cylindrical geometries, [1], Thiruvenkatachar and Ramakushna [2] appear to be the first to consider the transient temperature field in two space dimensions in the case of a cylindrical composite. Kumar and Thiruvenkatachar [3], in their attempt to improve the response of thermocouples and hot-wire anemometers, investigated the response of a finite composite cylinder to both radial and axial heat flow in the presence of harmonic variation of surface temperature.…”

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“…To effect a dual inversion of the two-dimensional part of the temperature solution in the form given by (12) and (13) is extremely difficult. The form of these equations does, however, suggest some simplification providing we make the temporary restriction (ultimately to be relaxed): 2 The case of arbitrary time-dependent heat inputs is accessible through the additional use of Duhamel's theorem [1].…”

“…Solutions to problems involving the conduction of heat in two distinct materials for heat flow described by a single spatial dimension have been extensively treated for both planar and cylindrical geometries, [1], Thiruvenkatachar and Ramakushna [2] appear to be the first to consider the transient temperature field in two space dimensions in the case of a cylindrical composite. Kumar and Thiruvenkatachar [3], in their attempt to improve the response of thermocouples and hot-wire anemometers, investigated the response of a finite composite cylinder to both radial and axial heat flow in the presence of harmonic variation of surface temperature.…”

A general class of unsteady heat flow problems in a finite composite hollow circular cylinder

“…We note that the technique of solution employed in [1], [2], [3] and [7] is the method of separation of variables, whereas the procedure followed in [4], [5], [6] and [8] employs the method of Laplace transform. It is well known that in the event of general and/or complex problems of heat flow, the application of the Laplace transform technique generally results in such complicated expressions for the inverse transform as to render its use prohibitive.…”