Solution of the inverse conduction problem

Abstract

Introduction M rocket engines operate in the turbulent boundarylayer regime, and consequently there is considerable interest in calculating convective heat transfer from turbulent boundary layers in nozzles. It is realized that such calculations are essentially empirical because of our limited knowledge of the effect of acceleration on the flow and thermal structure of. turbulent boundary layers and of the simultaneous effects of wall cooling and compressibility on the flow of rocket combustion gases and heated air. In heat-transfer studies, many experimental difficulties may arise in implanting heatflux sensors or thermocouples at the surface for heat-transfer measurements. Furthermore, the presence of a probe at the surface disturbs the condition of the boundary and the flow process adjacent to it and thus actual wall heat flux. It is therefore desirable in these circumstances that the prediction of surface temperature and heat flux be accomplished by inverting the temperature as measured by a probe located interior to the surface of the solid material. Such a problem is termed the inverse problem. Problems of the foregoing kind have been studied by several investigators over the past two decades. Stolz and Beck considered the numerical inversion of the integral solution for semi-infinite and spherical bodies. Other papers using least squares were written by Frank and Burggraf. Carslaw and Jaeger and Shumakov applied different series approaches in which generally the local temperature and local heat flux at an interior location and their higher derivatives are required. Sparrow et al. and Imber and Khan utilized the transform method. Beck used a finite-difference approximation in conjunction with a least-squares fit procedure as well as a nonlinear estimate method for the inverse conduction problem. Most of these analyses assumed a onedimensional model, but in reality the temperature field is distorted to become twoor three-dimensional when a cavity is drilled to accommodate the thermocouple leads. The degree of distortion may be influenced by the dissimilar properties of the thermocouple and surrounding material, and by the diameter and depth of the cavity. The present Note reports an iterative scheme to obtain values of surface temperature and convective heat-transfer Received March 24, 1977. Index categories: Solid and Hybrid Rocket Engines; Heat Conduction. * Engineer, Propulsion Engineering Division (PSN). coefficient from the measured temperature history at the outer surface of the nozzle.