# 64 Changing Concentration related to Flux Divergence

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: