Analytical solution to heat transfer in compressible laminar flow in a flat minichannel

Heat transfer in compressible laminar flow in mini-/micro-channels, a classical and general topic in fields of fuel cells, electronics, micro heat exchanger, etc., is revisited. Based on a two-dimensional continuum flow model, analytical solutions of the dimensionless model are achieved in closed-form symbolic algebras of Whittaker eigenfunctions, corresponding to two kinds of boundary conditions with arbitrarily prescribed wall temperature or wall heat flux. As the eigenvalues and eigenfunctions are independent on the dimensionless quantities, which influence the along-the-channel behaviors, the algorithm reveals the common features of compressible laminar thermal flows. The algorithms do not require the assumption of a linear pressure distribution, which is proved to be untenable in some cases (e.g. constant wall heat flux). The algorithms are validated well by the exact (numerical) computations in exemplary cases of both small and moderate Reynolds number, Mach number and Eckert number of air. Although expressed in a series of eigenfunctions, only several terms (sometimes one or two terms) of solutions are required for a practical computation.