Upper entropy bounds for transient forced convection

This article develops upper bounds for total entropy associated with convective heat transfer and transient fluid motion in an enclosure. Entropy production includes both friction and thermal irreversibilities due to fluid mixing in the enclosure. An integral formulation of entropy transport is developed in terms of the temperature excess (difference between the point-wise and spatially averaged temperature). The thermal irreversibility of entropy production is written in terms of the squared temperature excess. In this way, an upper entropy bound can be derived with respect to geometrical parameters and initial temperatures. Furthermore, this entropy bound is minimized by re-formulating the minimization problem in terms of a standard form of eigenvalue problem. Several example problems are considered and a spectral method is used to solve the governing energy equation. Theoretical predictions are compared successfully against numerical simulations for cases involving both Neumann and Dirichlet boundary conditions.