Local energy balance, specific heats and the Oberbeck-Boussinesq approximation

TL;DR

This paper discusses the correct form of the local energy balance within the Oberbeck-Boussinesq approximation, emphasizing the appropriate thermodynamic property for different fluids and extending the analysis to porous media.

Contribution

It clarifies the thermodynamic properties needed for energy balance equations in the Oberbeck-Boussinesq approximation for gases, liquids, and porous media.

Findings

For perfect gases, use specific heat at constant volume.

For liquids, use specific heat at constant pressure.

No explicit pressure work term is needed in the energy balance.

Abstract

A thermodynamic argument is proposed in order to discuss the most appropriate form of the local energy balance equation within the Oberbeck-Boussinesq approximation. The study is devoted to establish the correct thermodynamic property to be used in order to express the relationship between the change of internal energy and the temperature change. It is noted that, if the fluid is a perfect gas, this property must be identified with the specific heat at constant volume. If the fluid is a liquid, a definitely reliable approximation identifies this thermodynamic property with the specific heat at constant pressure. No explicit pressure work term must be present in the energy balance. The reasoning is extended to the case of fluid saturated porous media.