Analytical solution for mixed convection in a vertical circular duct with partially insulated wall

Mixed convection flow in a vertical circular duct is investigated for fully developed regime, under the assumption of laminar parallel flow. The duct wall is assumed to be half subjected to a uniform heat flux and half adiabatic. An analytical method based on Fourier series expansions of temperature and velocity fields is adopted to determine the velocity and the temperature distributions as well as the friction factor and the average Nusselt number. Plots of both the dimensionless temperature and the dimensionless velocity as functions of the angular coordinate ϑ and of the dimensionless radial coordinate r are presented for some values of the ratio between the Grashof number Gr and the Reynolds number Re. The positive and negative threshold values of Gr/Re for the onset of the flow reversal phenomenon are determined and a comparison between the average Nusselt number for the considered non−axisymmetric case and that for the case of a duct subjected to a uniform wall heat flux is performed.