# MHE: Analyzing Mass Transfer Equipment (well-stirred)

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.

##### NOTE:

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.

### A Continuous Contactor

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)

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

##### NOTE:

Here, the area, Ai, 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 Ai = 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, NA, we get:

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

Recall that we use CA in our expression for mass flux since that is the concentration everywhere within the contacting apparatus.

##### DEFINITION:

The product, KLa, 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:

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

### A (semi-)Batch Contactor

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 (L1=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 CAo, yields:

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

##### OUTCOME:

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