In a well stirred contacting vessel, the mass exchange analysis is rather simple.

For mathematical simplicity we will restrict our attention to
dilute systems where we can ignore changes in *overall*
material flows, as the concentrations change (i.e., the
concentration changes are small enough that the resultant changes
in total flow are negligible). This restriction is easily removed,
but is an unnecessary complication.

In a well-mixed tank (sometimes called a CSTR: continuously
stirred tank reactor), the concentration inside the tank is assumed
to be uniform and equal to the concentration in the **exit**
stream.

Performing a balance on species A in the liquid phase of the continuous process shown above, one can start with a "word equation":

rate of change of mass of species A | = | net rate of inflow of species A |
+ | net rate of reaction of species A |

Assuming no reaction and steady state conditions, this reduces to

0 | = | net rate of inflow of species A |

The inflow of A to liquid 1, can occur from two sources: (1) the
actual inflow of the liquid phase and (2) transfer of species A
between phases (**NOTE**: in both cases here we are talking
about a *net* inflow, so each item could easily be negative
and therefore technically be an outflow)

$\displaystyle{0 = L_1C_{A_o}-L_1C_A+N_AA_i}$

Here, the area, A_{i}, is the interfacial area which is
the area perpendicular to the mass flux. This quantity is difficult
(impossible) to measure so we instead use the product A_{i}
= aV, where a is the interfacial area per unit volume, a quantity
that can be experimentally determined and used directly for
scale-up.

Using aV for the interfacial area and plugging in for the
expression for the mass flux, N_{A}, we get:

$\displaystyle{0=L_1C_{A_o}-L_1C_A+K_LaV(C_A^*-C_A)}$

Recall that we use C_{A} in our expression for mass flux
since that is the concentration everywhere within the contacting
apparatus.

The product, K_{L}a, is called the **capacity
coefficient** and is often used as an experimentally determined
parameter for scale-up of mass transfer equipment.

Rearranging this expression, we get an equation for the outlet concentration for this type of contactor:

$\displaystyle{C_A = \frac{L_1C_{A_o}+K_LaVC_A^*}{L_1+K_LaV}}$

Performing a balance on species A in the liquid phase of the semi-batch process shown above, one can start with a slightly different "word equation":

rate of change of mass of species A | = | net rate of inflow of species A |
+ | net rate of reaction of species A |

Assuming no reaction, but **not** steady state we can
write:

$\displaystyle{\frac{d}{dt}(C_AV) = K_LaV(C_A^*-C_A)}$

since this semi-batch process has no inflow of liquid
(L_{1}=0). If the total volume in the tank remains constant
in time (which is typically true), we get:

$\displaystyle{\frac{dC_A}{dt} = K_La(C_A^*-C_A)}$

Solving this differential equation subject to the condition that
the initial concentration is given as C_{Ao},
yields:

$\displaystyle{\ln \frac{C_A^*-C_{A_o}}{C_A^*-C_A} = K_Lat}$

Calculate the mass exchanged in a well-mixed contactor or the time necessary to achieve a particular exchange