Transient laminar natural convection along rectangular channels

An analytical approach is proposed to investigate the transient behaviour of a Newtonian, single-phase fluid in natural laminar convection, in a rectangular open-ended duct. The continuity, momentum and energy equations, with the classical Boussinesq approximation, are solved using a twofold sine Fourier transform and the Laplace transform. The unsteady state is due to a step variation of the temperature in the four walls of the duct, which can assume four different, uniform arbitrary values. Considering hydrodynamically developed flow and uniform wall temperatures (UWTs), the velocity and temperature of the fluid are given as series containing two spatial co-ordinates and the time. Some plots show the transient evolution of the velocity and temperature distribution. Then the induced volumetric flow rate, the exchanged power, the mixing cup temperature, and the average Nusselt number are evaluated, as a function of time, emphasising the influence of the duct aspect ratio and the irrelevance of the channel height.