An integral transform solution for unsteady compressible heat transfer in fluids near their thermodynamic critical point

Abstract

The classical thermodynamic model for near critical heat transfer is an integral- differential equation with constant coefficients. It is similar to the heat equation, except for a source term containing the time derivative of the bulk temperature. Despite its simple form, analytical methods required the use of approximations to generate solutions for it, such as an approximate Fourier transformation or a numerical Laplace inversion. Recently, the generalized integral transform technique has been successfully applied to this problem, providing a highly accurate analytical solution for it and a new expression of its relaxation time. Nevertheless, very small temperature differences, on the orderof mK, have to be imposed so that constant thermal properties can be assumed very close to the critical point. The present paper generalizes this study by relaxing its restriction and accounting for thestrong dependence on temperature and pressure of supercritical fluid properties, demonstrating that (a) the generalized integral transform technique can be applied to realistic non-linear unsteady compressible heat transfer in fluids with diverging thermal properties and (b) temperature and pressure have opposite effects on all properties, but their variation causes no additional thermo-acoustic effect, increasing the validity range of the constant property model.