Mass Conservation for the Fluid Phase

If we include the porosity n, the discussions of Sect. 3.6 can be directly used. Assume that within the REV of volume AV all variables are homogeneous. If the mass of the ath component of the fluid phase is denoted as na, the volume fraction (occasionally referred to as the volume molar concentration) coa and the component mass density Pa are defined by 

Let Ya be the mass supply of the ath component due to, for example, a chemical reaction, and let be the mass that is absorbed per unit area on the surface of the porous fabric that composes the solid phase. Then the mass conservation law of the ath component can be written as 

If we apply Reynolds transport theorem to the Lh.s. of (5.16) under (5.7), the local form of the mass conservation law can be obtained as 

Here we have assumed that a chemical species exchange can take place between the fluid phase and the solid phase only through adsorption. Therefore the total mass is balanced, giving 

The material time derivative of a function at a space-time (at, t) with reference to the mean velocity v is given by 

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