The Plate Models

The plate models assume that the column is divided into a series of an arbitrary number of identical equilibrium stages, or theoretical plates, and that the mobile and the stationary phases in each of these successive plates are in equihbrimn. The plate models are in essence approximate, empirical models because they depict a continuous column of length I by a discrete number of well-mixed cells. Although any mixing mechanism is dearly absent from the actual physical system, plate models have been used successfully to characterize the column operation physically and mathematically. Therefore, by nature, plate models are empirical ones, which cannot be related to first principles. 

Martin and Synge published their first description of the plate theory in 1941 [1], at the time when Wicke [10], Wilson [11], and Devault [12] were beginning to study the solution of the mass balance equation of chromatography (Section 6.2). 

The Martin and Synge plate model [1] is a continuous plate model. It assumes that the column is equivalent to a series of continuous flow mixers. Mobile phase is transferred from one vessel to the next one as new mobile phase is added to the first vessel. Hence, the mobile phase flows continuously, and in each mixer the volumes of the mobile phase, Vm, and of the stationary phase, Vs, remain constant. The model is also based upon the assumption that, at the beginning of the experiment, only the first plate (rank / = 0) is loaded with the sample and that there are no sample components in the other plates. Said [13] has extended the theory of the continuous plate model to the case in which the solute is initially distributed on several plates, according to a certain distribution fimction. 

The mass balance for the mixer of rank I, when a volume dv of mobile phase is moved through the series of plates, is given by 

Since we assume that complete (linear) equilibrium is reached in each plate, we have 

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In this section we proceed to study the plate model with the crack described in Sections 2.4, 2.5. The corresponding variational inequality is analysed provided that the nonpenetration condition holds. By the principles of Section 1.3, we propose approximate equations in the two-dimensional case and analytical solutions in the one-dimensional case (see Kovtunenko, 1996b, 1997b). 

The theoretical parameters of isocratic chromatography are often described using the plate model. One can imagine the analyte to be distributed... 

The simulation program CHROMPLATE uses the plate model for the same column conditions as the simulation model CHROMDIFF. The results obtained are very similar in the two approaches, but the stagewise model is much faster to calculate. 

Figure 23 Chondrite-normalized abundances of REEs in representative harzburgites from the Oman ophiolite (symbols—whole-rock analyses), compared with numerical experiments of partial melting performed with the Plate Model of Vemieres et al. (1997), after Godard et al. (2000) (reproduced by permission of Elsevier from Earth Planet. Set Lett. 2000, 180, 133-148). Top melting without (a) and with (b) melt infiltration. Model (a) simulates continuous melting (Langmuir et al., 1977 Johnson and Dick, 1992), whereas in model (b) the molten peridotites are percolated by a melt of fixed, N-MORB composition. Model (b) is, therefore, comparable to the open-system melting model of Ozawa and Shimizu (1995). The numbers indicate olivine proportions (in percent) in residual peridotites. Bolder lines indicate the REE patterns of the less refractory peridotites. In model (a), the most refractory peridotite (76% olivine) is produced after 21.1% melt extraction. In model (b), the ratio of infiltrated melt to peridotite increases with melting degree, from 0.02 to 0.19. Bottom modification of the calculated REE patterns residual peridotites due to the presence of equilibrium, trapped melt. Models (c) and (d) show the effect of trapped melt on the most refractory peridotites of models (a) and (b), respectively. Bolder lines indicate the composition of residual peridotites without trapped melt. Numbers indicate the proportion of trapped melt (in percent). Model parameters...

Equations 6.46 and 6.47 supply the concentration profile along the column (f constant, z variable). As shown previously in the case of the plate models, the elution profile can be obtained by letting z — L, which, after some rearrangements, can be written in dimensionless form as... 

The reason for the inconsistency of the plate -model is presumably the circulation pattern that was observed in the film a self-organized motion of regularly spaced, unsteady channels, oriented parallel to the direction of the substrate flow. These channels, about 1 cm in width, extended most of the length of the film and alternated in direction of flow. The circulation velocity was < Ul2. 

We now consider the entire column rather than a small section thereof, as was done in the plate model. Let the column contain a total volume Vs of stationary phase, and a total volume Vm that can be occupied by the mobile phase. Since the column is uniformly coated or packed , the ratio Vm / will be equal to the ratio vml vs we used earlier for a single plate. Therefore the... 

These numerical examples indicate that the plate model predicts both tr = (ttm and ap = tr tr— lm)l Np. This latter result suggests that the number of theoretical plates can be obtained experimentally from... 

The agreement between the theoretical result and a Gaussian curve is fairly poor for Np = 25, especially for p-values close to 1 (where the plate model predicts quite asymmetrical peaks), but it is already quite good for Np = 1000. For columns with many thousands of theoretical plates, it is clearly justified to treat the resulting peaks as Gaussians, with the values of tr and [Pg.246]

The concept of theoretical plate has its origin in the plate model of the chromatographic separation, for which it is assumed that the column can be ideally divided into a number of equally distributed sections, called plates. At each plate the solute partition is sufficiently fast to let equilibrium be established before the solute moves into the next plate. Thus, the theoretical plate is the smallest section of the column in which equilibrium partition is possible. 

Fig. 4.28. Crack tip phase angle ip for the configuration shown in Figure 4.22 versus stiffness ration D for D2 = 0. The dashed line represents an estimate of ip based on the results for the plate model shown in Figure 4.5.

The strength of the plate model is that, for a discrete number of stages, it can predict useful quantities, e.g. which plate will have the highest solute concentration at... 

For a fixed L, the larger the value of N, the more efficient the column, and correspondingly, the smaller the value of the plate height However, in the plate model of a chromatographic column, the number of plates, N, is unknown therefore H is unknown. In fact, H has to he predicted independently via alternative models (Giddings, 1965). 

In the previous finite element parametric study of cracked steel plate with CFRP patching, the material properties of the 1.2-mm-thick CFRP plate (Sika, 2003) were assigned for the CFRP patching. In order to compare results of the plate model and the tube model, the same material properties of the CFRP plate were assigned for... 

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