Required number of plates

For conditions in which the relative volatility is constant, Ienske 29-1 derived an equation for calculating the required number of plates for a desired separation. Since no product is withdrawn from the still, the equations of the two operating lines become ... 

The reduction in the required number of plates as R is increased beyond Rm will tend to reduce the cost of the column. For a column separating a benzene-toluene mixture, for example, where Xf = 0.79, xd = 0.99 and xw = 0.01, the numbers of theoretical plates as given by the McCabe-Thiele method for various values of R are given as follows. The minimum reflux ratio for this case is 0.81. 

REQUIRED PLATE NUMBER. Knowing the capacity factor, k1, and the separation factor, a, one is able to calculate the required number of plates (n ) for the separation of two components (the k value refers to the more readily sorbed component). 

Equations 2.109 and 2.111 illustrate that the required number of plates will depend on the partition characteristics of the column and the relative volatility of the two components, that is on K and 3. Table 2.9 gives the values of the last term of Equation 2.109 for various values of k. These data point up a few interesting conclusions If k <5 the plate numbers are controlled mainly by column parameters if k >5 the plate numbers are controlled by relative volatility of components. The data also illustrate that k values greater than 20 cause theoretical number of plates, n, and effective number of plates, N, to be of the same order of magnitude, that is,... 

Another factor that contributes to Rs is the plate count N. However, we have seen in section 1.5 that optimization through an increase in N is expensive, not only in terms of equipment and columns, but also in terms of analysis time. Therefore, as long as the shape of the peaks and the plate height (length of the column divided by N) are satisfactory, we should not rely on the number of plates for optimization, unless as a last resort. Methods which may be used to optimize the chromatographic system with respect to the required number of plates will be described in chapter 7. 

Once we have realized optimum capacity factors and optimized the selectivity, we can use the resolution equation to calculate the number of plates that is required to achieve baseline resolution (11,= 1.5). The required number of plates will to a large extent determine the kind of column and instrumentation needed to perform the separation. This will be briefly discussed in chapter 7. 

In the case, where the dimensions of the column are to be optimized after completion of the selectivity optimization process, another time factor may be even more appropriate. The first step to be taken after the completion of the optimization process is to establish the required number of plates (JVne). If the lowest resolution value encountered in the chromatogram is Rs min, and if the required resolution is Rs ne, then... 

Since all continuous criteria (see table 4.7) require knowledge of the Rs or S values, eqn.(4.34) or (4.35) can be used immediately to calculate the required number of plates. In many cases the required number of plates will only be used to estimate the length of the final analytical column, while all other parameters are being kept constant. 

The above simple proportionality between tm and Nne (eqn.4.36) is not always obeyed. For instance, at constant flow rate and particle size, the number of plates that can be achieved is limited by the maximum allowable column pressure. In that case, we are forced to vary the flow rate, the particle size, or both. If we do so, the analysis time (tj) will no longer be proportional to the required number of plates (JVne). 

In chapter 7 (section 7.2.3) we will see that in the case where the pressure drop over a packed column is kept constant and both the flow rate and the particle size are allowed to vary in order to realize optimum operating conditions, the retention time will be proportional to the square of the required number of plates, i.e. 

In the reality of LC practice, eqn.(4.38), which is based on a different optimum particle size for a different required number of plates, will not usually be realistic. For LC the truth may be somewhere in between the two extremes described by eqn.(4.36) (constant flow rate and particle size) and eqn.(4.38) (constant pressure drop). Relatively long columns with 10 pm particles may be used for difficult separations, requiring relatively large numbers of plates. For more convenient separations, somewhat shorter 5 pm particle columns may be used, and for relatively simple separations requiring modest numbers of plates 3 pm particles packed in very short columns may provide fast analysis. 

Figure 4.11 Calculated characteristics for optimum chromatograms (r = 1) containing 10 equally resolved peaks as a function of the separation factor S. Plotted on a logarithmic scale are the capacity factor of last peak (1 +k eqn.4.46), the required number of plates (Afne eqn.4.47), the required analysis time under conditions of constant flow rate and particle diameter (rne f>(ji eqn.4.48), and required analysis time under conditions of constant pressure drop fne p eqn.4.49). For explanation see text.

The second field of application involves the occasional analysis of wide range samples. In this category we find samples which only occur in the laboratory occasionally and in small numbers, so that only a small number of chromatographic analyses have to be performed. For samples of this kind it is usually sufficient to realize a separation and it is not rewarding to try and optimize the selectivity, not even if the analysis time is rather long and the required number of plates high. 

The required number of plates (Nne) is the most relevant factor for the selection of the type of column and the column dimensions. However, there are various other factors which we need to consider in the selection of the most suitable column for a given analysis ... 

The optimization of the efficiency of the chromatographic system involves the selection of a column with a sufficient but not excessive number of plates. If a column is used with twice the required number of plates, then the observed resolution would exceed the required value by 40%, but both the analysis time and the pressure drop over the column would be double the required value. Moreover, the sensitivity of the detection would be decreased by 40% (see section 7.3). It is clear that we should aim to use a column with the optimum (i.e. the required) number of plates. 

If the required number of plates is moderate (say several thousands), then short capillary columns may be used to provide fast analysis of the sample. The required column length and retention time can easily be calculated from eqns.(7.3) and (7.6). For example, if we operate a capillary column with a diameter of 200 pm at a reduced velocity of 5 with a reduced plate height of 1.5, then 2000 theoretical plates require a column with... 

In LC, packed columns are used for practical separations. Eqn.(7.6) shows the speed of analysis to be proportional to the required number of plates and to the square of the (particle) diameter. We used this equation in chapter 4 to derive an expression for the required retention time under conditions of constant flow rate and particle size (tne fiC i eqn.4.48). However, eqn.(7.6) suggests that the speed of analysis may always be increased by decreasing the particle size. In other words, it suggests that the smallest available particles should always be used in packed columns. This interpretation is too simple. 

An important conclusion from eqn.(7.10) is that the analysis time is proportional to the square of the required number of plates. Eqn.(7.10) also reveals that the analysis time may be decreased by three factors ... 

Figure 7.2 Example of the relationship between the required number of plates and the required column length (h=4) for four different particle sizes in HPLC. Thin lines indicate the maximum column lengths for pressure drops of 200 and 400 bar, with v= 10, Dm = 10 9 m2/s and r/= 1 mPa.s. Heavy dots indicate possible columns for separations requiring up to 10,000 plates. Open circle represents a column with 20,000 plates.

At this level, the flow rates required by an ideal TMB (which mainly means that kinetic and hydrodynamic dispersive effects are assumed to be negligible) to get 100% pure products for a feed of a given concentration are known. The final flow rates will be extremely close. This procedure is sensible because it has been proven that a TMB or SMB performance is only slightly sensitive to the number of plates.29 In most cases, the required number of plates can easily be achieved and optimum flow rates are then available. 

The basis for determining the required number of plates or column efficiency at infinite dilution is to solve the simplified Knox equation in terms of the optimum d /L (Eq. (7.19)). 

Assuming a difficult separation of two components with a = 1.05 and a reasonable goal of obtaining a resolution R, of 1.0, the number of theoretical plates required at various capacity factor values can be calculated. For a capacity factor of 1.0, a minimum of 28,200 plates is required. For a capacity factor of 2.0, the required number of plates drops to 15,880. Because many liquid chromatographic systems usually do not exhibit efficiencies greater than 25,000 theoretical plates, it is evident from these calculations that developing methods for difficult separations should be performed when the capacity factors of the components to be separated are adjusted so they are all greater than 2. 

These models assume that the chromatographic column can be divided into a series of a finite number of identical plates. Each plate contains volumes and of the mobile and stationary phases, respectively. The sample is introduced as a solution of known concentration in the mobile phase used to fill the required number of plates. Plate models are essentially empirical, and cannot be related to first principles. Depending upon whether one assumes continuous or batch operation. two plate models can be considered The Craig model and the Martin and Synge model. 

By such rematching procedure, it is possible to obtain exact agreement at the feed plate for all the light components and all the heavy components because they are arbitrarily chosen in one portion of the tower. However, it may be impossible to obtain an exact match of the key components because an even number of theoretical plates will not be consistent with the design chosen. In this case, it is possible to bracket the required number of plates within a difference of one, plate. While this rematching operation gives a more consistent set of... 

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