Concept of the theoretical plate

When the variance is expressed in units of time or volume (a, = F ) then, under the above conditions (a, r ), it holds approximately that 

It follows from equation 31 that the discussion of spreading factors in terms of length variance (see section 1.6.2) can easily be converted to the discussion in terms of H by dividing the corresponding equations by the quantity L. 

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The description of the model up to this point is analogous to a description of a chromatographic model which incorporates the concept of the theoretical plate ( 5). ... 

It should be noted, however, that the concept of the theoretical plate is a hypothetical one. It is used to help in understanding the separation processes occurring within the column environment. Theoretical plates can be used as a physical measure of the degree of separation of any given column under a specific set of parameters. This is often referred to as column efficiency and can be calculated as follows ... 

The concept of a theoretical plate is based on the number of equilibria that may have taken place during the separation process and this number is related to the number of times the effective volume of a column is greater than the peak volume. The variance (a2) for the peak is a measure of the broadening of the injection volume while the square of the retention time (fR2) is a measure of the effective column volume for that compound. Hence the number of theoretical plates (AO may be calculated from the following equation ... 

The concept of the effective plate number was introduced and employed in the late nineteen fifties by Purnell (7), Desty (8) and others. Its introduction arose directly as a result of the development of the capillary column, which, even in 1960, could be made to produce efficiencies of up to a million theoretical plates (9). It was noted, however, that these high efficiencies were were only realized for solutes eluted close to the column dead volume, that is, at very low k values. Furthermore, they in no way reflected the increase in resolving power that would be expected from such high efficiencies on the basis of the performance of packed columns. This poor performance, relative to the high efficiencies produced, can be shown theoretically ( and Indeed will be, later in this book) to result from the high phase ratio of capillary columns made at that time. That is the ratio of the mobile phase to the stationary phase in the column. The high phase ratio was... 

Classical treatment of mass transfer is to consider a unit, of mass transfer as a measure of the interphase equilibrium changes needed to produce a desired degree of diffusion [13], This concept is best applied to the concept of a theoretical plate in distillation [4], Defining Gm as the gas superficial molar velocity (mole/hr/ft2 of tower cross section) and dy as the change of concentration of the diffusing species, then... 

Cooling water in process plants is most commonly and effectively obtained using a cooling tower. The principle of operation is the simultaneous transfer of mass and heat. Colburn (1939) introduced the idea of a unit of mass transfer which is a measure of the number of equilibrium changes to required to effect a given amount of diffusion. It is identical with the concept of a theoretical plate in distillation (Kern, 1950). 

As mentioned earher, the plate theory has played a role in the development of chromatography. The concept of "plate" was originally proposed as a measmement of the performance of distillation processes. It is based upon the assumption that the column is divided into a number of zones called theoretical plates, that are treated as if there exists a perfect equilibrium between the gas and the Hquid phases within each plate. This assumption imphes that the distribution coefficient remains the same fi-om one plate to another plate, and is not affected by other sample components, and that the distribution isotherm is hnear. However, experimental evidences show that this is not true. Plate theory disregards that chromatography is a dynamic process of mass transfer, and it reveals httle about the factors affecting the values of the theoretical plate number. In principle, once a sample has been introduced, it enters the GC column as a narrow-width "band" or "zone" of its composite molecules. On the column, the band is further broadened by interaction of components with the stationary phase which retains some components more than others. Increasing... 

The concept of theoretical plates in chromatography is described elsewhere, and is not repeated here (18,19). Pieri et al. have described the application of the theoretical plate concept to column scale-up and demonstrated its utility in scale-up of pheromone separation over silica gel (15). We have found this same approach to be useful in separations over ion exchange resins used as chromatographic supports. In essence, this approach gives an estimate of column length for a given separation if the particle size is changed. This is based on empirical correlation of the number of theoretical plates, JV, ... 

The efficiency of a real GPC system depends above all on the rate of mass transfer between mobile phase and gel phase, as well as on the extent of secondary processes. Quantitatively, the efficiency can be expressed by the terms like width (w) or deviation (o) of the chromatographic peak, as well as by other terms of the theoretical plate concept. Since the diffusion rate of solute molecules decreases with an increase of their dimensions, one has to expect generally lower efficiency in gel chromatography of macromolecules in comparison with any other mode of liquid chromatographic separation of low molecular substances. 

A number of theoretical treatments of gel chromatography have been published, including applications of the theoretical plate concept similar to those outlined in Chapter 2. It is appropriate here to mention only a few simple parameters of most value in practical work. 

It was not, however, until the concepts of the theoretical stage and the transfer unit had been introduced that it became possible to express in numerical terms the efficiency of a column and the difficulty involved in a separation. A theoretical (or ideal) plate (or stage) is defined to be such a section of a column that the vapour leaving it upwards (towards the next higher theoretical plate) and the liquid leaving it downwards (towards the next lower theoretical plate) are in thermodynamic-equilibrium (Fig. 24). 

The correct answer is to minimize those contributions to peak broadening These factors are interpreted in terms of contributions to the height equivalent to a theoretical plate (HETP). This line of reasoning leads to the need to define a HETP that, in turn, requires that we introduce the concept of a theoretical plate in chromatography. So let us get started ... 

Continuous distillations usually take place in towers containing packing or plates to promote vapor-liquid contact. With vapor and liquid in countercurrent flow through the tower, analysis usually is in terms of the number of theoretical plates required to effect the desired separation. Figure 10.5.12 illustrates the concept of a theoretical plate. The vapor and liquid leaving the plate are in physical equilibrium. Distillation calculations then depend on accurate knowledge of the equilibrium relationship. When activity... 

The concept of a theoretical plate is introduced to characterize separation of compounds in chromatographic columns. A theoretical plate is a virtual discrete section of the chromatographic column in which equilibrium between the fraction of the analyte in the mobile phase and the stationary phase can occur at the provided conditions. The number of theoretical plates (V) is a measure of column efficiency. It depends on the length of the column (L) and the height equivalent to a theoretical plate (HETP) ... 

The use of the concept of the reduced plate height, h, which is the quantity H divided by dp and the reduced velocity, V, given by V = udp/Dm is of practical advantage in the comparison of the efficiency of columns packed with particles of different diameters and in theoretical treatments. However, the discussion of the reduced plate height is out of the scope of this book and one who is interested in this problem should consult other related books [e.g., ref. 9]. 

The concept of a theoretical plate as used in distillation theory is completely different from the one employed here in distillation the number of theoretical plates defines the ease of separation of two components and represents the number of equilibriiifn states on the equilibrium curve of the... 

Since a large number of theoretical plates are desired, h should be as small as possible. Naturally, there are no real plates in a column. The concept of a theoretical plate is a variable, the value of which depends on the particle size, the flow velocity, the mobile phase (viscosity) and, especially, on the quality of the packing, h can be calculated using the following equation ... 

Because the transfer unit is defined on a differential countercurrent basis, it is usually assumed to be more correct for the design of packed towers than the stepwise countercurrent procedure of the theoretical plate. This is probably true, but there is a serious question whether the transfer unit is on a sound theoretical basis. Most distillation operations are of a degree that requires a number of theoretical plates, and for su h cases it is doubtful whether at the present stage of development the transfer-unit concept has any advantage over the theoretical-plate basis for the design of packed towers. The latter is easier to employ. 

The idea of the effective plate number was introduced and employed by Purnell [4], Desty [5] and others in the late 1950s. Its conception was evoked as a direct result of the introduction of the capillary column or open tubular column. Even in 1960, the open tubular column could be constructed to produce efficiencies of up to a million theoretical plates [6]. However, it became immediately apparent that these high efficiencies were only obtained for solutes eluted at very low (k ) values and, consequently, very close to the column dead volume. More importantly, on the basis of the performance realized from packed columns, the high efficiencies did not... 

Time-dominated processes inherently govern chromatography. The horizontal axis of a chromatogram is time (and not energy as in spectroscopy). To describe the quality of a chromatographic system the concepts of the height equivalent to a theoretical plate, HETP or H, and the number of theoretical plates N are used (Equation 4.1) ... 

The continuous annular chromatograph can be described mathematically by a theoretical plate approach similar to the one developed by Martin and Synge [40] and exemplified by Said [41] for stationary columns [5]. The mathematical description results in algebraic expressions for the elution position of each solute relative to the feed point and for the bandwidth of the eluting zone as a function of the elution position or other system parameters. However, a series of simplifications have to be made in order to describe the CAC with the theoretical plate concept ... 

GLUECKAUF, E. Trans. Faraday Soc. 51 (1955) 34. Theory of chromatography, Part 9 the theoretical plate concept in column separation. 

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