Flux Explained:

So what is the origin of this expression for R_B? To answer that question, we must examine the flux, W, of species B through the membrane. Consider the following diagram:

We see that the flux of species B is the same for each region in the diagram; i.e., for bulk side tranport, for transport through the membrane itself, and for shell side transport. The general expression for flux is:

The driving force usually takes the form of a concentration difference, while the transport coefficient is (usually) a constant that represents the ease or difficulty of transport through a given material.

The flux of species B from the bulk fluid to the interior membrane interface is given by:

Similary, the flux of species B from the exterior membrane interface to the the shell side fluid is given by:

As we shall soon see, the most important flux expression is the one for the flux of species B through the membrane itself, from the interior membrane interface to the exterior membrane interface:

We want to combine our three flux expressions, but first let's rearrange them:

Now let's combine them and see what drops out:

Rearranging some more we get:

If we look at the transport coefficients as being analogous to electrical resistances, then we have three "resistances" in the transport of species B through the membrane. We can safely assume that the greatest resistance to transport will occur in the membrane itself. In other words:

Or:

Applying this to our flux expression, we get:

If we make the additional assumption that , and we note that , then we get:

Of course, R_B is another way of expressing the flux, W, of species B.