The Lumped Capacitance Model for Unsteady Heat Conduction in Regular Solid Bodies With Natural Convection to Nearby Fluids Engages the Nonlinear Bernoulli Equation

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to nearby fluids, the two feasible models are (1) the differential or distributed model and (2) the lumped capacitance model. In the latter model, the suited lumped heat equation is linear, separable, and solvable in exact, analytic form. The linear lumped heat equation is constrained by the lumped Biot number criterion $Bil=h¯(V/S)/ks$ < 0.1, where the mean convective coefficient $h¯$ is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation turns nonlinear because the mean convective coefficient $h¯$ depends on the instantaneous mean temperature in the solid body. Undoubtedly, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the classical Runge–Kutta method. Also, due to the variable mean convective coefficient $h¯ (T)$⁠, the lumped Biot number criterion $Bil=h¯(V/S)/ks$ < 0.1 needs to be adjusted to $Bil,max=h¯max(V/S)/ks$ < 0.1. Here, $h¯max$ in natural convection cooling stands for the maximum mean convective coefficient at the initial temperature T_in and the initial time t = 0. Fortunately, by way of a temperature transformation, the nonlinear lumped heat equation can be homogenized and later channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. This simple route paves the way to an exact, analytic mean temperature distribution T(t) applicable to a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder, and sphere; all solid bodies constricted by the modified lumped Biot number criterion $Bil,max<0.1$⁠.