Gibbs-Duhem equation used to describe uncompensated and apparent heat transfer applied to spray cooling

Entropic analysis makes it possible to determine the equilibrium of a system with its environment, but heat exchange at boundaries is a difficulty in this exercise. In this article, we establish the energy balance of a water liquid and air mixture: the Gibbs-Duhem equation is used to describe the balance. Diffusion terms are expressed while accounting for volume changes and their entropic contributions, leading to an expression for fugacity. After mixing, the mixture spontaneously evolves combining mechanical, chemical and thermal changes towards equilibrium. This latest is solved assuming either an isenthalpic or isentropic transformation in the balance, to highlight the energy exchange between the system and its environment. Some irreversibilities are included in transformations and primarily affect the chemical path. The comparison reveals heat transfer terms corresponding to uncompensated heat. This analysis illustrates the concept of uncompensated heat and introduces its complement, the apparent heat, which corresponds to enthalpy. The models explain the limiting phenomena involved in spray cooling, whose efficiency varies depending on whether the process follows an adiabatic or an isenthalpic path. The isenthalpic model was applied to literature data to estimate experimental cooling efficiencies over a wide range of operating conditions. However, further modelling work is needed to express the scaling factor for fugacity as a function of the droplet size distribution. The mechanical pathway also requires more attention and dedicated measure-ments are needed to include pressure drops in the analysis.