Analytical solutions of 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient – I. Closed-form inverse solution

Closed-form solution of 1-D heat conduction problem for a single straight fin and spine of constant cross-section has been obtained. The local heat transfer coefficient is assumed to vary as a power function of temperature excess. The dependence of the fin parameter N on the dimensionless temperature difference Te at the fin tip for a given exponent n was derived in a form N/N0=Te−μn (where N0 is a well-known N expression for n=0). Coefficient μ was found to be equal to 5/12 according to the exact solution at Te→1 or to 0.4 according to the fitting procedure for the data of the numerical integration. Obtained formula serves as a basis for the derivation of the direct expressions for Te vs N at given n, fin base thermal conductance and augmentation factor presented in the second part of the study.