Lumped mass transfer problems are quite similar to those of heat transfer.

Consider the case where a solid with some concentration of
species A, C_{A}^{S}, is losing material to a bulk
fluid.

If we focus on the interface between the phases, we could make the same argument that we did for the case of heat transfer to explain why the diffusive flux to the surface must be equal to the convective flux off of the surface:

$\displaystyle{-D\frac{dC_A^S}{dr} = k_c(C_A^f - C_A^{f\infty})}$

where we need to note that there exists a "jump" in concentration at the interface (because at steady state the concentration in the solid and fluid phases would not be equal, but instead would be related via $\displaystyle{mC_A^S=C_A^f}$).

Using this relation and defining a fictitious solid
concentration far away from the solid surface (i.e., the solid
concentration that *would* be in equilibrium with the bulk
fluid concentration), we get

$\displaystyle{-D\frac{dC_A^S}{dr} = mk_c(C_A^S - C_A^{S\infty})}$

We could define a dimensionless concentration and spatial coordinate as

$\displaystyle{\phi = \frac{C_A^S}{C_A^{S\infty}}\;\;\xi = \frac{r}{R}}$

which can be plugged into our equation to give its dimensionless form

$\displaystyle{\frac{d\phi}{d\xi} = -mBi_m(\phi-1)}$

where the Biot number is now defined (for mass transfer problems) as

$\displaystyle{Bi_m = \frac{k_cR}{D}}$

For consistency, we will continue to define the characteristic length in the Bi as V/A, so for a sphere it should have been R/3, above.

As with heat transfer, if Bi<<1, we get a "lumped" problem, whereby we can write the total rate of change of the mass/moles from our "point mass" as being equal to the external rate of mass flow:

$\displaystyle{\frac{d(C_A^SV)}{dt} = V\frac{dC_A^S}{dt} = -mk_cA(C_A^S-C_A^{S\infty})}$

If we use both the bulk and initial concentrations to make C
dimensionless (as we did in heat transfer), and notice that
V/(mk_{c}A) has units of time so that our dimensionless
variables are:

$\displaystyle{\theta\frac{C_A^S-C_A^{S\infty}}{C_{A_o}^S-C_A^{S\infty}}\; \; \tau = \frac{mk_cAt}{V}}$

we get the same lumped equation that we did for heat transfer

$\displaystyle{\frac{d\theta}{d\tau} = -\theta}$

which, of course, yields the same solution

$\displaystyle{\theta = e^{-\tau}}$

Use the "lumped" equation to solve "1D" transient mass transfer problems