## Equilibrium curve

Figure 9.10. Operating line and equilibrium curve for a single-stage operation.

If sufficient time of contact is allowed, so that equilibrium is almost reached, the final liquid and solid concentration will correspond to a point (see Q, Fig. 9.10), which is quite close to the equilibrium curve.

The mass balance assumes that the amount of liquid mechanically retained with the solid after filtration or settling is negligible. This is usually the case.

If Freundlich's isotherm can be used, we can, at the final equilibrium condition, set up the following equation:

Since the adsorbent (ion exchanger) normally used contains no initial adsÃ³rbate, that is Xo = 0, then the two equations yield:

As can be seen, this permits analytical calculation of the adsorbent solution ratio for a given change in solution concentration, provided that the constants in the equation system are known.

However, removal of a given amount of solutes may be accomplished by less adsorbent, if the solution is treated with separate small batches of ion exchanger rather than a single large batch. This method is the multi-stage cocurrent operation. The savings are greater the larger the number of batches, but the expense of equipment and even handling costs will increase with the number of stages. It is therefore rarely economical to use more than two or three stages. A schematic flowchart and operating diagram for two ideal stages of cocurrent adsorption are shown in Fig. 9.11. As seen, the same quantity is treated in each stage, but by two different amounts of adsorbent Ai and A2. The mathematical balances are given by the following equations:

These two equations provide the operation lines as shown in Fig. 9.12.

When Freundlich's expression is used as a description of the adsorption isotherm and fresh adsorbent is used in each stage, Xo = 0, the two-stage system can be computed directly:

A2 Xo

 S Y1 1 f A2 Stage 2 X2 Y2

Figure 9.11. Flowchart for a two stages cocurrent operation.

A1+A2 Y0-Y1 Y1-Y2

0 0