Derivation of Conduction Heat Transfer in Thin Shell Parabolas

This paper presents the derivation of the equations for circumferential, longitudinal and radial heat transfer conductance for a right circular thin shell parabola or a segment of the parabola. A thin shell parabola is one in which the radius to thickness ratio is greater than 10. The equations for the surface area of a parabola or of a parabolic segment will also be derived along with the equation to determine the location of the Centroid. The surface area is needed to determine the radial conductance in the parabola or parabolic segment and the Centroid is needed to determine the heat transfer center of the parabola or parabolic segment for circumferential and longitudinal conductance. These equations can be used to obtain more accurate results for conductive heat transfer in parabola which is a curved spacecraft components.

A comparison will be made (1) using the equations derived in this paper which takes into account the curvature of the parabola and (2) using flat plates to simulate the parabola. The method of finite difference will be used for both cases which places the nodes at the centroid of the element. The results show that a significant error can be introduced when using flat plates unless the mesh size is small.