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In principle, fluidized ion exchange beds are similar to stirred tank chemical reactors. The general equations of kinetics and mass transfer can be applied to the individual fluidized units in an identical manner to those for chemical reactors. The primary difference lies in accounting for the behavior of suspended particles in the turbulent fluid.'371

The operation of these fluidized ion exchange beds is identical to that of the fixed beds, with the exception that the resin of each stage is confined by perforated plates and maintained in a fluidized suspension using liquid flow or impellers.

The critical design parameter for fluidized beds is the loss or leakage of the solute through a given stage. The design equation for a single stage bed has been described by Marchello and Davis.1381

2.3 Chromatographic Theory

Mathematical theories for ion exchange chromatography were developed in the 1940's by Wilson,^ DeVault[40] and Glueckauf.[41][42l These theoretical developments were based on adsorption considerations and are useful in calculating adsorption isotherms from column elution data. Of more interest for understanding preparative chromatography is the theory of column processes originally proposed by Martin and Synge[43] and augmented by Mayer and Thompkins,'44' which was developed analogous to fractional distillation so that plate theory could be applied.

One of the equations developed merely expressed mathematically that the least adsorbed solute would be eluted first and that if data on the resin and the column dimensions were known, the solvent volume required to elute the peak solute concentration could be calculated. Simpson and Wheaton'451 expressed this equation as:

Eq.(7) VUAX = KdVrl+Vl where FMAX is the volume of liquid that has passed through the column when the concentration of the solute is maximum (the midpoint of the elution of the solute). Kd, defined in Eq. 1, is the distribution coefficient of the solute in a plate of the column; Vrl is the volume of liquid solution inside the resin and V; is the volume of interstitial liquid.

The mathematical derivation of Eq. 7 assumes that complete equilibrium has been achieved and that no forward mixing occurs. Glueckauf*461 pointed out that equilibrium is practically obtained only with very small diameter resin beads and low flow rates. Such restricting conditions may be acceptable for analytical applications, but would severely limit preparative and industrial chromatography. However, column processing conditions and solute purity requirements are often such that any deviations from these assumptions are slight enough that the equation still serves as an adequate first approximation for scaled-up chromatography applications.

Theoretical Plate Height. A second important equation for chromatography processes is that used for the calculation of the number of theoretical plates, i.e., the length of column required for equilibration between the solute in the resin liquid and the solute in the interstitial liquid. If the elution curve approximates a Gaussian distribution curve, the equation may be written as:

where P is the number of theoretical plates; c (= KdVrlIVi) is the equilibrium constant; W is the half-width of the elution curve at an ordinate value of 1/eofthe maximum solute concentration. For a Gaussian distribution, W = 4a, where a is the standard deviation of the Gaussian distribution. The equilibrium constant is sometimes called the partition ratio. An alternate form of this equation is:

Here if is measured in the same units as Vmax • This form of the equation is probably the easiest to calculate from experimental data. Once the number of theoretical plates has been calculated, the height equivalent to one theoretical plate (H.E.T.P.) can be obtained by dividing the resin bed height by the value of P.

The column height required for a specific separation of two solutes can be approximated by:[47J

where H is the height of the column, P is the number of plates per unit of resin bed height and c is the equilibrium constant defined above. Note that the number of plates in a column will be different for each solute. While this equation may be used to calculate the column height needed to separate 99.9% of solute 1 from 99.9% of solute 2, industrial and preparative chromatography applications typically make more efficient use of the separation resin by selectively removing a narrow portion of the eluted solutes, as illustrated in Fig. 7.t4*J

A .6 .8 1.0 1.2 1.1 1.6 1.3 2.0 2.2 2A Vbmc of Eluate Collected Ofe) / Mxue of FtesiN Bed (Vr)

Figure 7. Distribution of eluate into fractions for product, recycle, and waste forNaCl and glycol separation.[48]

Table 7 shows how the theoretical plate number for a chromatographic system may be calculated from various combinations of experimental data. The band variance, af2, is calculated from the experimental data and combined with the retention time, tR, for a given solute. Figure 8 shows the different experimental values which may be used to calculate a,.

Zone Spreading. The net forward progress of each solute is an average value with a normal dispersion about the mean value. The increased band or zone width which results from a series of molecular diffusion and non-equilibrium factors is known as zone spreading.

The plate height as a function of the mobile phase velocity may be written as a linear combination of contributions from eddy diffusion, mass transfer and a coupling term:

Ion Exchange 403 Table 7. Calculation of Plate Number from Chromatogram

Measurements Coversion to Variance Plate number tR and ax ------------A' = (tR/crt)2

tR and baseline width Wb ert = Wb / 4 N = 16(tR / Wb )2

tR and width at half height Woi °t = ^0.5 1 V8 ta 2 N = 5-54('R I wo.5)2

tR and width at inflection points aX = W| ! 2 N = 4(tR / W, )2 (0.607 h)Wi tR and band area A and height h at = A/h^T>c N = 2«(tRh/A)2

Figure 8. Identification of chromatographic peak segments for the calculation of column performance.

A plot of Eq. 11 for any type of linear elution chromatography describes a hyperbola, as shown in Fig. 9.1491 There is an optimum velocity of the mobile phase for carrying out a separation at which the plate height is a minimum, and thus, the chromatographic separation is most efficient:

where DM is the diffusion coefficient of the solute molecule in the mobile phase, Ds is the diffusion coefficient in the stationary phase, dp is the diameter of the resin bead and Rt = L/vt, where L is the distance the zone has migrated in time t.

Corrier velocity, cm/sec Figure 9. Relationship between late height and velocity of the mobile phase.1491

Resolution. A variation on calculating the required column height is to calculate the resolution or degree of separation of two components. Resolution is the ratio of peak separation to average peak width:

The numerator of Eq. 13 is the separation of the two solutes' peak concentrations and the denominator is the average band width of the two peaks. This form of the equation is evaluating the resolution when the peaks are separated by four standard deviations, a. If R = 1 and the two solutes have the same peak concentration, this means that the adjacent tail of each peak beyond 2a from the ^max would overlap with the other solute peak. In this instance there would be 2% contamination of each solute in the other.

Resolution can also be represented'501 by:

Resolution can be seen to depend on the number of plates for solute 2, the separation factor for the two solutes and the equilibrium constant for solute 2.

In general, the larger the number of plates, the better the resolution. There are practical limits to the column lengths that are economically feasible in industrial and preparative chromatography. It is possible to change P also by altering the flow rate, the mean resin bead size or the bead size distribution since P is determined by the rate processes occurring during separation. As the separation factor increases, resolution becomes greater since the peak-to-peak separation is becoming larger. Increases in the equilibrium constant will usually improve the resolution since the ratio c2/( 1 + c2) will increase. It should be noted that this is actually only true when c2 is small since the ratio approaches unity asymptotically as c2 gets larger. The separation factor and the equilibrium factor can be adjusted for temperature changes or other changes which would alter the equilibrium properties of the column operations.

Equation 14 is only applicable when the two solutes are of equal concentration. When that is not the case, a correction factor must be used where A1 and A2 are the areas under the elution curve for solutes 1 and 2, respectively. Figure 10 shows the relationship between product purity 0i), the separation ratio and the number of theoretical plates. This graph can be used to estimate the number oftheoretical plates required to attain the desired purity of the products.

For example, when the product purity must be 98.0%, then rj = Am/m = 0.01, when the amount of the two solutes is equal. If the retention ratio, a, is equal to 1.2, then the number of theoretical plates from Fig. 10 is about 650. With a plate height of 0.1 cm, the minimum bed height would be 65 cm. In practice, a longer column is used to account for any deviation from equilibrium conditions.

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