The heat-balance integral: 1. How to calibrate the parabolic profile?

The Heat-Balance Integral Method (HBIM) of Goodman under classic prescribed temperature boundary conditions has been studied towards it optimization. Because the parabolic profile satisfies both the boundary conditions and the heat-balance integral at any value of the exponent the calibration is of a primary importance in generation of the approximate solution. The simple 1-D heat conduction problem, enabling one to demonstrate the HBIM performance with the entropy generation minimization (EGM) concept in calibration of a parabolic temperature profile with unspecified exponents, has been developed. The EGM concept provides constraints that impose addition boundary conditions at the approximate parabolic profile. Additionally, entire domain optimizations based on the mean-squared error concept has been performed in two versions – the method Myers and through a similarity transformed diffusion equation.