Nonlinear Thermal Model of Circular Foil Heat Flux Gauges

Abstract

This paper presents the solution to a nonlinear model of a circular foil heat flux gauge that is exposed to a blackbody source in a vacuum environment. This is the scenario typically used to calibrate a circular foil heat flux gauge. The nonlinear model is solved using a Green's function approach. This approach results in an integral equation for the steady-state temperature profile in the gauge, which is solved using the method of successive approximations. A relationship between the incident radiative heat flux and the temperature profile is developed using this model. This relationship is compared to relationships that were derived using linear models. The first and simplest linear model neglects emission from the foil. The second linear model is obtained by linearizing the emissive power of the gauge. It is shown that these linear models only produce accurate results when the gauge design and operating conditions result in a nearly uniform foil temperature. A procedure based on the nonlinear model is proposed for optimizing the design of a circular foil heat flux gauge. A calibration procedure based on the nonlinear model is also proposed. NomenclatureH=T 4 R c = calibration function constants d = calibration function constants f = calibration functions depending on A G = Green's function H = irradiation, W=m 2 h r = radiation heat transfer coefficient, W=m 2 K I o = modified Bessel function k = thermal conductivity of the constantan foil, W=m K N c = conduction-to-radiation parameter, kt=R 2 T 3 R n = calibration function constant R = radius of the constantan foil, m r = radial coordinate, m T = temperature, K t = thickness of the constantan foil, m = relaxation factor = dimensionless temperature difference = Dirac delta function " = emittance = dimensionless temperature, T=T R = dimensionless radial coordinate, r=R = Stefan-Boltzmann constant, 5:67e-8, W=m 2 K 4 Subscripts a = approximate L = linear o = temperature at center of heat flux gauge, K R = temperature of the copper heat sink, K 1-7 = index for calibration functions and constants Superscript i = iteration number

Solving nonlinear heat transfer problems using variation of parameters

“…Circular foil heat flux gauges can be modeled as radiating annular fins. Liechty et al [8] presented a thermal model of a copperconstantan circular foil heat flux gauge exposed to a blackbody source in a vacuum environment. Such a setup is typically used to calibrate a heat flux gauge.…”

“…Fig. 1 shows a comparison of the dimensionless temperature profiles calculated from Equation (14) to those found using the Green's function method used by Liechty et al [8] and to those found from a direct numerical solution using commercial software [28] for various values of the parameters A and B. The accuracy and computational requirements of the method of variation of parameters were compared to those of finite difference methods.…”

“…For the case where T L ¼ 0.5T 0 and k ¼ 1, the nondimensional temperature profiles calculated using Equation (20) are compared to those calculated using a finite difference method [8], the collapsed dimension method implemented by Talukdar and Mishra [31], and the discrete ordinates method [30]. Fig.…”