On the definition and derivatives of macroscale energy for the description of multiphase systems

Modeling of flow and transport in environmental systems often involves formulation of conservation equations at spatial scales involving tens to hundreds of pore diameters in porous media or the depth of flow in a channel. Quantities such as density, temperature, internal energy, and velocity may not be uniform over these macroscopic length scales. The external gravitational potential causes gradients in density, pressure, and chemical potential even at equilibrium. Despite these complications, it is important to formulate the thermodynamic analysis of environmental systems at the macroscopic scale. Heretofore, this has been accomplished primarily using the approach of rational thermodynamics whereby the thermodynamic dependence of macroscale internal energy on macroscale variables is hypothesized directly without development of any systematic method for transforming microscale energy dependence from the microscale to the macroscale. However when thermodynamic variables are inhomogeneous at the microscale, the functional dependence of macroscale internal energy on macroscale variables is not a simple extension of the microscale case. In the present work, the relation between the definitions of microscale and macroscale intensive thermodynamic variables is established. Expressions for the material derivatives of macroscale internal energy of phases, interfaces, and common lines are derived from and consistent with their microscopic counterparts by integrating to the macroscale. The forms obtained and the consistency required will be important for use in analyses of systems at scales where microscopic heterogeneities cannot be neglected.