Figure 9.15 Idealized break-through curve.
Correspondingly we call the time required for the ion exchange zone to establish itself and move out the bed, Te, which can then be calculated from:
If we call the height of the entire ion exchange bed, Z(m), and, Tf, the time required for formation of the ion exchange zone, we get:
The quantity of solid removed from the water in the ion exchange zone from the break-point to exhaustion is U kg solid / m2. This area is areas 1 and 2 in Fig. 9.15.
If all the ion exchanger in the zone was saturated with solute, it would contain Yo * Wa kg solute / m2.
Consequently at the break-point, the zone is still within the column. The fractional ability, f, of the adsorbent in the zone still to adsorb is:
If f = 0 it means that the ion exchanger in the zone is saturated, and the time of formation of the zone at the top of the bed, Tf, should be the same as the time required for the zone to travel a distance equal to its zone height, Ta. On the other hand, if f = 1.0 so that the solid in the zone has essentially not taken up anything of the component considered, the zone formation should be very short.
These limiting conditions are described by:
The ion exchange column is Z m tall of unit cross sectional area, and contains Z * Q kg adsorbent, where Q is the apparent packed density of the solid in the bed. If the column was in complete equilibrium and saturated at an ion concentration of Xt kg / kg solid, the weight of the component taken up would be Z * Q • Xt kg. At the break-point the adsorption zone of height, Za, is at the bottom of the column, but the rest of the column, Z - Za (m), is substantially saturated. At the break-point therefore, the removed amount of the considered component is:
The fractional saturation of the column at the break-point is:
In the fixed bed of ion exchange, the active zone moves through the solid in the flow direction as we have seen.
The operating line of the entire tower is:
Since the operating line passes through (0,0) of Fig. 9.16 at any level in the column, the concentration of solute in the water, Y, and the removed component on the solid, X, are then related by the equation:
Over the differential height, dZ, the rate of ion exchange is:
where Kt = the overall transfer coefficient, a = the outside surface area of the solid particles and Y+ = the equilibrium concentration.
For the entire ion exchange zone:
YE dY Za Za
YB Y-Y+ Ht S/Kt * a where Nt = the overall number of transfer units in the ion exchange zone.
The success of this analysis hinges upon the constancy of Ktor Ht for the concentration within the adsorption zone. This will of course depends upon the relative constancy of the resistance to mass transfer in the fluid and within the pores of the solid. An alternative method to determine Ht will be described below; see page 295.
The ion exchange rate can be limited by external diffusion, internal diffusion or by the actual ion exchange process. The external diffusion controls the transfer of solute from the water to the boundary layer of fluid immediately adjacent to the external surface of the ion exchanger. The external diffusion is governed by molecular diffusion and in a turbulent flow by eddy diffusion.
The process can be described by the following equation:
where Va = the rate of ion exchange; Y = the concentration of the ion in the fluid and Y+ = the concentration of the ion in the fluid in equilibrium with the existing concentration in the ion exchanger, keis the external mass transfer coefficient.
Internal diffusion processes control the transfer of solid from the exterior of the ion exchanger to the internal surface. This condition is represented by the following equation:
where I = the interparticle void ratio; Xx = the concentration of the ion in the solid phase that is assumed to be in equilibrium with the coexisting liquid phase at concentration, Y; X = the actual concentration of ion in the solid phase.
If the internal and the external diffusions occur at comparable rates the respective mass transfer coefficients, measured individually, may be added (King, 1965):
K ke ki
The diffusion coefficient as used in the design of a practical column must be found in the literature or by determined experimentation. The internal diffusion can be found by equilibrium experiments by use of equation (9.34).
Ht may be found alternatively by a series of experiments, where the capacity (expressed as volume of water, which can be treated with a required efficiency, i.e., YB is given) is found for different flow rates. The values found are expressed as a percentage of the theoretical capacity, which gives the percentage of the total column "not used," which is equal to Z-nu = (1-f)*Za in Fig. 9.6. Equation (9.33) is used to find Nt by graphic integration, f is furthermore found by graphic Integration of Y / Yo versus (W -\NB), corresponding to equation (9.24). Hence Za can be found, since (1-f*Za) is known. Finally is Ht determined (see also the example in Appendix C2) as a function of the flow rate from equation (9.33): Ht = Za / Nt and can be used for design of full-scale columns. The method is published in Jorgensen et al. (1978) for the exchange ammonium-sodium on clinoptilolite. Ht as function of the flow rate may furthermore be used for an economic optimization of the ion exchange column. Higher flow rate means that the required column volume is reduced but that the utilization of the column is also reduced, which in turn means, that a more frequent regeneration is required to obtain the same effluent quality. A lower flow rate, on the other hand, means that more ion exchange volume is needed, but the frequency of elution is decreased.
The design of an ion exchange column will be exemplified as mentioned above in Appendix C2. The steps to be followed may be summarized in 6 points:
1. Z-nu is determined in laboratory or pilot scale tests as function of the flow,
1.e., for various S-values.
3. Nt is determined by graphic integration of equation (9.33).
4. f is determined by graphic integration of equation (9.24)
5. Za is determined from a combination of the results in points 2 and 4.
6. Ht as a function of the flow rate is determined by use of the expression
Ht = Nt / Za. Ht as function of the flow rate can now be used for any design.
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