We have seen that columns in series are operated past the breakthrough point to the exhaustion point. If the graph is steep, there will be little error in neglecting the volume that passes after breakthrough. There is a method to account for this volume that will be presented here. Note that a rate constant is needed for the main equation. It is more trouble to measure this rate than to do experiments with a laboratory column for the scale up already mentioned. Nevertheless, following the logic of th e mathematical method improves understanding of countercurrent adsorption.
The shape of the breakthrough curve relates to the adsorption
isotherm. The time for the adsorption zone to become defined and to
move to the end of the column is:
t = Ve / Qt
where t = time as exhaustion zone exits
Ve = volume applied to column, volume liquid/area
Qt = flow rate to column, volume/time-area
The time for an established adsorption zone to move a distance corresponding to its own thickness is:
t = ( Ve - Vb ) / Qt
where t = transit time
Vb = volume, starts at conc. for breakthrough
The amount adsorbed in the adsorption zone is the integral from Vb to Ve of ( Co - C ) dV
We next need to construct the shape of the adsorption zone from the adsorption isotherm. This means that equilibrium is assumed.
As is customary in the analysis of staged separations, we consider a differential element in the column. The liquid and the solid adsorbent are assumed to have velocities entering the element even though the solid is actually motionless. A material balance for the entire column is:
Qt ( Co - 0 ) = Bt ( q - 0 )
A material balance for the section of the column containing the adsorption zone just as it is exiting is:
Qt ( C - 0 ) = Bt ( q - 0 )
These material balances define an operating line shown in the figure.
Each point on the operating line represents compositions of liquid and solid in contact at some point in the column. For the differential element:
Qt dC = K ( C - C* ) dZ
where K = a rate coefficient
Z = distance
C* = equilibrium solute concentration
The equation for the column is:
where C =
concentration deemed the breakthrough
C = concentration at the other end of the element
The quantity ( C - C* ) is the horizontal distance between the operating line and the isotherm line. It can be integrated graphically by taking the area under the function as in the following figure :
There are two integrations, one from the top of the column to the end and one for the adsorption zone.