Solution of the Nonlinear Heat Equation for Circular-Foil Heat Flux Gauges

Abstract

A series solution of the steady nonlinear heat equation for a circular-thin-foil heat flux gauge subjected to a constant-radiation heatflux is presented. The dimensionless formof the heat equation involves a dimensionless gauge parameter and a dimensionless irradiation parameter. A linear solution is extracted from the series solution, which yields a simple-yet-accurate formula that is useful for calibration of thin-foil heat flux gauges. The dimensionless temperature at the center of the foil is determined from the linear solution as a function of the preceding two parameters. The dependence of the irradiation parameter on the dimensionless temperature at the center of the circular foil is presented graphically for awide range of the parameters. The values of the dimensionless temperature at the center of the foil obtained with the linear solution are in good agreement with those obtained with a numerical solution of the heat equation usingRunge–Kutta integration. Typical temperature profiles also showgoodagreement with the numerical solution. To compensate for inherent gauge errors and assumptions made in the analysis, individual gauges are calibrated against standards. To curve fit the calibration data, it is suggested that a correction factor be introduced in the argument of the modified Bessel function that appears in the linear formula for the dimensionless irradiation parameter.