## Operating line 1 2 operating line 2 3 equilibrium curve

Figure 9.12 Operating diagram for two stages adsorption.

The minimum total adsorbent is found by setting d(Ai + A2)

This reduces to:

cocurrent ion exchange or

Equation (9.16) can be solved for Yi, and the adsorbed quantity can be found by equations (9.13) and (9.14).

Even greater economy in the use of adsorbent / ion exchanger can be achieved by a countercurrent operation. Figure 9.13 shows a diagram of this operation and Fig. 9.14 shows the operation line and equilibrium curve for this case. The operating line can be set up as follows:

and if Freundlich's adsorption isotherm can be used and Xo = 0, then a combination of this equation and (9.17), provides the following expression:

An equation for calculating Yi can be found by eliminating S/A:

Yo Yi Yi

Y2 Y2 Y2

It is then possible to calculate S/A directly from (9.17).

If Freundlich's adsorption isotherm cannot be used, it is of course possible to use the diagram for the necessary calculation as shown in Fig. 9.14.

In the continuous operation the water and the adsorbent / ion exchanger are in contact throughout the entire process without a periodic separation of the two phases. The operation can either be carried out in strictly continuous steady-state fashion by movement of the solid as well as the fluid or in a semi-continuous fashion characterized by moving fluid but stationary solid, the so-called fixed bed adsorption / ion exchange, which is widely used in waste water treatment, including by the removal of ammonium and proteins from waste waters. It is generally found more economical to use a stationary bed for waste water treatments due to the relatively high cost of continuously transporting solid particles. Only this case will therefore be treated mathematically.

Stage 1 "1-i

 1 f A, X2 Stage 2 S, Y2 i A, Xo

Figure 9.13. Flowsheet for a two stages countercurrent adsorption.

The design of a fixed bed ion exchanger and the prediction of the length of the cycle requires knowledge of the percentage approach to saturation at the break point. Figure 9.15 shows an idealized break-through curve.

Let us consider a case where the flow of water through an ion exchange bed Is S kg/h m2 - entering with an initial solute concentration of Yo kg solute / kg solvent. The total, solute free, effluent after a given time is W kg/m2 (see Fig. 9.15). The break-through curve should be steep and the solute concentration in the effluent rises rapidly from close to zero to that of the incoming water. Some low value Yb is arbitrarily chosen as the break-point concentration and the column is considered exhausted when the effluent concentration has risen to some other arbitrarily chosen concentration of value Ye, close to Yo. The critical values are the quantity of effluent Wb and We (see Fig. 9.15).  