4.3 Momentum Balance for the Gas as a Whole

The momentum balance for component B is obtained from Equation 14 by replacing the subscript A with B to obtain Equation 15.

\displaystyle -\frac{dp_{B}}{dl}=\frac{RTJ_{B}}{D}+\frac{RTN_{B}^{D}}{D_{B}^{K}} (15)

The sum of Equations 14 and 15 for the individual components is the momentum balance for the gas as a whole, and is given by Equation 16.

\displaystyle -\frac{dp}{dl}=RT\left \{ \frac{N_{A}^{D}}{D_{A}^{K}}+\frac{N_{B}^{D}}{D_{B}^{K}} \right \} (16)

The first term on the right side of Equation 14 and Equation 15 sum to zero as required by Equation 7, and is an expression of conservation of momentum for intermolecular collisions within the gas, as a whole as noted previously. Thus, Equation 16 expresses the momentum balance for diffusion-initiated, molecule-particle collisions for the gas as a whole. No momentum loss due to viscous shear is accounted for in Equation 16. Under isobaric conditions Equation 16 becomes Equation 17.

\displaystyle \frac{N_{B}^{D}}{N_{A}^{D}}=-\frac{D_{B}^{K}}{D_{A}^{K}} (17)


Flux Equations for Gas Diffusion in Porous Media Copyright © 2021 by David B. McWhorter. All Rights Reserved.