The transient natural-convection process is analyzed using an integral method of analysis. Differential equations are derived which relate average surface temperature and time for either heating or cooling for vertical elements having arbitrary thermal capacity. The equations are applicable to laminar flow for all fluids. The coefficients are Prandtl number dependent and are estimated for Prandtl numbers in the range 0.01 to 1000. A solution of the equations is presented for the extreme case of a vertical plate of negligible thermal capacity subjected to a step in flux at its surface. Fluids having Prandtl numbers of 0.01, 0.1, 0.72, 1.0, 5, 10, 100, and 1000 are considered. The results, in terms of generalized variables, are practically independent of Prandtl number. Simple one-dimensional transient behavior is followed for approximately 20 per cent of the transient with a subsequent quick approach to the asymptotic value. The results show no substantial overshoot of the average surface temperature. It is doubted that significant temperature overshoot actually occurs for vertical surfaces even for a step in flux.
