This paper introduces an improved hybrid (numerical-analytical) model for simulating microwave (MW) heating of laminar duct flow. The proposed procedure links numerical results to analytical calculations, providing a tool for accurate prediction of the bulk temperature distribution in a relatively reduced computation time, enhancing the design of MW heating of continuous flow water systems. The hybrid solution was obtained by first numerically solving the Maxwell equations in correspondence of an average dielectric permittivity; discrete values of the cross-section averaged heat generation arising from the numerical solution were first corrected by a suitable weighting function and then interpolated by a function resulting from the discrete Fourier series. The momentum and the energy equations fed by the above calculated heat generation distribution were uncoupled from Maxwell equations. The problem being linear, the analytical thermal 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 second one, driven by the heat generation, was solved in closed form by the variation of parameters method. On the other hand, the same problem was solved by a complete numerical approach in order to have a reference solution; thus, the Maxwell equations and the energy balance for the flowing fluid were simultaneously solved considering temperature dependent dielectric permittivity. Fully developed velocity, thermally developing conditions and no phase transition during the heating process were assumed for both the hybrid and the numerical solution. The availability of the reference solution allowed to prove the substantial enhancement of the hybrid solution in describing the bulk temperature distribution along the pipe when compared to the one related to the classical constant properties approach. Results, presented and discussed for different inlet velocities, show that increasing velocities provide a better agreement due to the smoothing effect realized by higher frequencies fluctuations in heat generation distribution felt by the flowing fluid.
Numerical solution of Williamson fluid flow past a stretching cylinder and heat transfer with variable thermal conductivity and heat generation/absorption