Scaling criteria for wall thermal boundary conditions of air- and water-cooled free convective thin plates with volumetric heat generation

This study proposes scaling criteria for the wall thermal boundary conditions of thin plates with volumetric heat generation based on free convective/conductive conjugate heat transfer analysis. The choice of boundary conditions - uniform wall temperature (UWT) or uniform heat flux (UHF) - plays a crucial role in estimating the peak temperature of a free convective plate with volumetric heat generation. The parameters for the scaling criteria for the wall thermal boundary conditions were derived analytically using the perturbation method to examine the effects of conjugate heat transfer. To quantify these parameters, the governing equations were numerically solved using the Runge-Kutta method for free convective flow and the finite volume method for solid conduction. The results showed that, when the modified conjugate parameter, defined as the product of the longitudinal conduction-to-convection ratio with half of the plate thickness divided by the plate length, was >0.5, the solution converged to the UWT solution. Whereas, when the ratio was <0.01, the solution aligned more closely with the UHF solution. The scaling criteria were validated against the existing experimental data for both air- and water-cooled free-convective plates with volumetric heat generation. This study not only offers guidelines for scaling analysis of wall boundaries to evaluate the peak temperature of a free-convective plate with volumetric heat generation but also analytically proves that such a plate has the highest and lowest peak temperatures under UHF and UWT conditions, respectively.