Natural convection flow due to a heat source in a vertical channel

An analysis is presented of the laminar natural convection flow due to a localized heat source on the centerline of a long vertical channel or pipe whose walls are kept at a constant temperature. Stationary solutions are obtained for infinitely long and finite length channels, the asymptotic limit of infinite Rayleigh numbers is discussed, and an optimal height of the channel is found leading to maximum mass flux and minimum temperature for a given heat release rate.