We will derive a relationship between the time rate-of-change of total amount of substance inside the control volume (Concentration times Volume) and the difference in transport across the sidewalls:
Since the volume does not change:
First, let us just consider the contribution to the total rate of concentration change due to the x-component of flux shown in the figure above. Remember that transport is flux times area. For the x-component of flux (), the relevant face has area :
Now divide by the volume (the cancels out of the numerator and denominator on the right side):
Now change the sign on the right so that the numerator represents a in the direction of increasing x. Then allow the control volume to become infinitesimally small so that the ratio of discrete changes in the x-direction can be represented by the continuous first partial derivative.
There are similar contributions due to variations of fluxes in the other two coordinate directions:
Add these all together for how the total time rate of change of concentration relates to the flux divergence:
- 1D Flux Divergence © Susan Hautala is licensed under a CC BY-NC-SA (Attribution NonCommercial ShareAlike) license