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

摘要

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