“Derivation of Conduction Heat Transfer in Thin Shell Toroids” 2000-01-2487
This paper presents the derivation of the equations for circumferential, longitudinal and radial heat transfer conductance for a thin shell toroid or a segment of the toroid. A thin shell toroid is one in which the radius to thickness ratio is greater than 10. The equations for the surface area of a toroid or of a toroidal 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 toroid or toroidal segment and the centroid is needed to determine the heat transfer center of the toroid or toroidal segment for circumferential and longitudinal conductance. These equations can be used to obtain more accurate results for conductive heat transfer in toroid 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 toroid (true geometry) and (2) using flat plates to simulate the toroid. 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.
Because of the complexity of the equations, they will be solved using numerical methods. The segments will be divided into N number of increments and the trapezoidal rule will be used to determine the various parameters using radians as the angular measurement.
The derivations in this paper will use angular measurement in radians.