A fast and accurate hybrid model for simulating continuous pipe flow microwave heating of liquids

Use of microwave technology in several industrial heating processes is a relatively new approach. Considering that scaling-up is hard to recover, theoretical simulations can be of help in order to study and optimize the process at hand while reducing the mass of experimental work. This paper aims to speed up prediction of bulk temperatures for an incompressible liquid as it flows continuously in a circular duct that is subjected to microwave heating. Usually, temperature increases are desired which require temperature dependent dielectric permittivity; thus, studying the problem at hand involves the simultaneous solution of the electromagnetic, fluid flow and heat transfer problems. In contrast, a hybrid model is introduced which links numerical results to analytical calculations, providing a tool for accurate prediction of the bulk temperature distribution while noticeably reducing the required computation time. The hybrid solution was obtained by first numerically solving Maxwell equations in correspondence of a fixed average dielectric permittivity; discrete values of the cross-section averaged heat generation arising from such solution were first corrected by a suitable weighting function and then interpolated by a function resulting from the discrete Fourier series. Then the momentum and the energy equations fed by the above calculated heat generation distribution turned out in a linear problem; the related analytical solution was sought as the sum of two partial solutions, each one affected by a single non-homogeneity. The former solution turned out to be the classical Graetz problem, while the latter, driven by the heat generation, was solved in closed form by the variation of parameters method. Fully developed velocity, thermally developing conditions and no phase transition during the heating process were assumed for both the hybrid and the numerical solution. Simulations are intended to validate the hybrid solution when compared to the corresponding numerical one. Results, presented and discussed for different inlet velocities, ensured the accuracy of the proposed model meanwhile showing that computational times are reduced at least by one- tenth.