Number of theoretical plates

To determine the number of theoretical plates of a fractionating still. 

The boiling point at 760 mm Hg of benzene + carbon tetrachloride mixtures and the compositions of the two phases have been determined by Rosanoff and Easley (J. Amer. Chem. Soc, 1909, 31, 971). Their measurements are recorded in table 1 where x denotes mole fraction of carbon tetrachloride in the liquid and y its mole fiaction in the vapour. 

In a fractionating still set up at Reading University by Halberstadt and Whitaker, the mole fraction y of the distillate at the top of the column was found to be 0.56 when the mole fraction x of the residue at the bottom was 0.06. 

We plot y against on a large scale and on the same diagram plot X against x. We then count the number of steps required to connect X = 0.06 with y = 0.56. 

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Used in virtually all organic chemistry analytical laboratories, gas chromatography has a powerful separation capacity. Using distillation as an analogy, the number of theoretical plates would vary from 100 for packed columns to 10 for 100-meter capillary columns as shown in Figure 2.1. 

Otherwise expressed, the number of theoretical plates required for a given separation increases when the reflux ratio is decreased, i.e., when the amount of condensed vapour returned to the colunm is decreased and the amount distilled off becomes greater. 

In their original theoretical model of chromatography, Martin and Synge treated the chromatographic column as though it consists of discrete sections at which partitioning of the solute between the stationary and mobile phases occurs. They called each section a theoretical plate and defined column efficiency in terms of the number of theoretical plates, N, or the height of a theoretical plate, H where... 

A column s efficiency improves with an increase in the number of theoretical plates or a decrease in the height of a theoretical plate. 

The number of theoretical plates in a chromatographic column is obtained by combining equations 12.12 and 12.16. 

It is important to remember that a theoretical plate is an artificial construct and that no such plates exist in a chromatographic column. In fact, the number of theoretical plates depends on both the properties of the column and the solute. As a result, the number of theoretical plates for a column is not fixed and may vary from solute to solute. 

Equations 12.21 and 12.22 contain terms corresponding to column efficiency, column selectivity, and capacity factor. These terms can be varied, more or less independently, to obtain the desired resolution and analysis time for a pair of solutes. The first term, which is a function of the number of theoretical plates or the height of a theoretical plate, accounts for the effect of column efficiency. The second term is a function of a and accounts for the influence of column selectivity. Finally, the third term in both equations is a function of b, and accounts for the effect of solute B s capacity factor. Manipulating these parameters to improve resolution is the subject of the remainder of this section. 

If the capacity factor and a are known, then equation 12.21 can be used to calculate the number of theoretical plates needed to achieve a desired resolution (Table 12.1). For example, given a = 1.05 and kg = 2.0, a resolution of 1.25 requires approximately 24,800 theoretical plates. If the column only provides 12,400 plates, half of what is needed, then the separation is not possible. How can the number of theoretical plates be doubled The easiest way is to double the length of the column however, this also requires a doubling of the analysis time. A more desirable approach is to cut the height of a theoretical plate in half, providing the desired resolution without changing the analysis time. Even better, if H can be decreased by more than... 

Number of Theoretical Plates Needed to Achieve Desired Resolution for Selected Values of ks and a... 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

Efficiency The efficiency of capillary electrophoresis is characterized by the number of theoretical plates, N, just as it is in GC or ITPLC. In capillary electrophoresis, the number of theoretic plates is determined by... 

Sometimes the height equivalent to a theoretical plate (HETP) is employed rather than and to characterize the performance of packed towers. The number of heights equivalent to one theoretical plate required for a specified absorption job is equal to the number of theoretical plates,... 

The required number of actual plates, A/p, is larger than the number of theoretical plates, because it would take an infinite contacting time at each stage to estabhsh equihbrium. The ratio is called the overall column efficiency. This parameter is difficult to predict from theoretical... 

Fig. 25. McCabe-Thiele diagram, (a) Number of theoretical plates, 5 (b) number of actual plates, 8.

For linear equiHbrium and operating lines, an expHcit expression for the number of theoretical plates required for reducing the solute mole fraction... 

This is the one case where the overall column efficiency can be related analytically to the Murphree plate efficiency, so that the actual number of plates is calculable by dividing the number of theoretical plates through equation 86 ... 

The simplest efficiency is the overaH column efficiency which is the number of theoretical plates in a column divided by the number of actual plates ... 

Favorable Vapoi Liquid Equilibria. The suitabiHty of distiUation as a separation method is strongly dependent on favorable vapor—Hquid equiHbria. The absolute value of the key relative volatiHties direcdy determines the ease and economics of a distillation. The energy requirements and the number of plates required for any given separation increase rapidly as the relative volatiHty becomes lower and approaches unity. For example given an ideal binary mixture having a 50 mol % feed and a distillate and bottoms requirement of 99.8% purity each, the minimum reflux and minimum number of theoretical plates for assumed relative volatiHties of 1.1,1.5, and 4, are... 

In the example, the minimum reflux ratio and minimum number of theoretical plates decreased 14- to 33-fold, respectively, when the relative volatiHty increased from 1.1 to 4. Other distillation systems would have different specific reflux ratios and numbers of theoretical plates, but the trend would be the same. As the relative volatiHty approaches unity, distillation separations rapidly become more cosdy in terms of both capital and operating costs. The relative volatiHty can sometimes be improved through the use of an extraneous solvent that modifies the VLE. Binary azeotropic systems are impossible to separate into pure components in a single column, but the azeotrope can often be broken by an extraneous entrainer (see Distillation, A7EOTROPTC AND EXTRACTIVE). 

Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation. 

The design of a plate tower for gas-absorption or gas-stripping operations involves many of the same principles employed in distillation calculations, such as the determination of the number of theoretical plates needed to achieve a specified composition change (see Sec. 13). Distillation differs from gas absorption in that it involves the separation of components based on the distribution of the various substances between a gas phase and a hquid phase when all the components are present in Doth phases. In distillation, the new phase is generated From the original feed mixture by vaporization or condensation of the volatile components, and the separation is achieved by introducing reflux to the top of the tower. 

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Tran.s. Am. Jn.st. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown ... 

Four theoretical plates have been stepped off for the key component (butane) on Fig. 14-9 and are sufficient to give a 75 percent recovery of butane. The operating hues for the other components were drawn in with the same slope and were placed so as to give the same number of theoretical plates insofar as possible. 

The left-hand side of Eq. (14-55) represents the efficiency of absorption of arw one component of the feed-gas mixture. If the solvent oil is denuded of solute so that Xo = 0, the left-hand side is equal to the fractional absorption of the component from the rich feed gas. When the number of theoretical plates N and the hquid and gas rates L i and G, f have been fixed, the uractional absorption of each component may be computed directly and the operating lines need not be placed by trial and error as in the graphic approach described earlier. 

N = number of theoretical plates Gb = allowable vapor velocity in heat exchangers, (lbmol)/(hft ) h = hours of operation... 

Thus, the variance of the peak is inversely proportional to the number of theoretical plates in the column. Consequently, the greater the value of (n), the more narrow the peak, and the more efficiently has the column constrained peak dispersion. As a result, the number of theoretical plates in a column has been given the term Column Efficiency. From the above equations, a fairly simple procedure for measuring the efficiency of any column can be derived. 

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