Theoretical plate model

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... 

Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

The main difference between the chromatographic process carried out in the linear and the nonlinear range of the adsorption isotherm is the fact that in the latter case, due to the skewed shapes of the concentration profiles of the analytes involved, separation performance of a chromatographic system considerably drops, i.e., the number of theoretical plates (N) of a chromatographic system indisputably lowers. In these circumstances, all quantitative models, along with semiquantitative and nonquantitative rules, successfully applied to optimization of the linear adsorption TLC show a considerably worse applicability. 

Golay equation 21, 611 gradient (LC) 490 height equivalent to a theoretical plate 11 longitudinal diffusion 16 mass transfer resistance 17 nonlinear chromatography SOS plate model 14 rate theory IS reduced parameters 78, 361, 611... 

The process is as described in Section 3.3.3.2 and consists of a distillation column containing seven theoretical plates, reboiler and reflux drum. Distillation is carried out initially at total reflux in order to first establish the column concentration profile. Distillate removal then commences at the required distillate composition under proportional control of reflux ratio. This model is based on that of Luyben (1973, 1990). 

The efficiency of a well packed column should be about 95 000 m 1 5% for a 5 pm octadecyl-bonded silica gel column. The equations for the calculation of the number of theoretical plates (AO and peak asymmetry (.. s) are as follows. A model chromatogram is given in Figure 3.5. 

The proposed model can be readily related to the plate theory. The number of theoretical plates can be deduced from the moment expressions and when Pe and the K are large then Equation 33 follows from Equation 23. 

The ideal model and the equilibrium-dispersive model are the two important subclasses of the equilibrium model. The ideal model completely ignores the contribution of kinetics and mobile phase processes to the band broadening. It assumes that thermodynamics is the only factor that influences the evolution of the peak shape. We obtain the mass balance equation of the ideal model if we write > =0 in Equation 10.8, i.e., we assume that the number of theoretical plates is infinity. The ideal model has the advantage of supplying the thermodynamical limit of minimum band broadening under overloaded conditions. 

A number of kinetic models of various degree of complexity have been used in chromatography. In linear chromatography, all these models have an analytical solution in the Laplace domain. The Laplace-domain solution makes rather simple the calculation of the moments of chromatographic peaks thus, the retention time, the peak width, its number of theoretical plates, the peak asymmetry, and other chromatographic parameters of interest can be calculated using algebraic expressions. The direct, analytical inverse Laplace transform of the solution of these models usually can only be calculated after substantial simplifications. Numerically, however, the peak profile can simply be calculated from the analytical solution in the Laplace domain. 

Single-component adsorption isotherms can also be evalnated from the diffnse rear bonnd-ary recorded in FA or from the rear part of overloaded band profiles. Techniqnes snch as FA by characteristic point (FACP) and elntion by characteristic point (ECP) ntilize this approach. Their major advantages over FA and PP are the smaller amonnt of material needed and the faster determination of an isotherm. However, both FACP and ECP are only snitable for singlecomponent systems and can be nsed only when the efficiency of the chromatographic system used is at least several thousand theoretical plates. This is becanse both methods rely on the nse of an eqnation of the ideal model, and hence include a model error that becomes important at lower efficiencies. 

To develop an HETP equation it is necessary to first identify the dispersion processes that occur in a column and then determine the variance that will result from each process per unit length of column. The sum of all these variances will be (H), the Height of the Theoretical Plate or the total variance per unit column length. There are a number of methods used to arrive at an expression for the variance resulting from each dispersion process and these can be obtained from the various references provided. However, as an example, the Random-Walk Model introduced by Giddings (5) will be employed here to illustrate the procedure.The theory of the Random-Walk processes itself can be found in any appropriate textbook on probability (6) and will not be given here but the consequential equation will be used. 

The model is directly related to the widely used theoretical plate concept, which, in principle, is only valid for Gaussian peaks. 

In order to estimate resolution among peaks eluted from a chromatography column, those factors that affect N must first be elucidated. By definition, a low value of Hs will result in a large number of theoretical plates for a given column length. As discussed in Chapter 11, Equation 11.20 obtained by the rate model shows the effects of axial mixing of the mobile phase fluid and mass transfer of solutes on Hs. 

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). ... 

Many theories have been suggested to explain the mechanism of migration and separation of analytes in the column. The oldest one, called the theoretical plate model, corresponds to an approach now considered obsolete but which nevertheless leads to relations and definitions that are universal in their use and are still employed today. 

These successive equilibria are the basis for the static model for which the column length L is partitioned into N theoretical plates numbered from 1 to N, all with the same height. For each of these plates, the concentration of analyte in the mobile phase is in equilibrium with the main concentration of analyte in the stationary phase. The height equivalent to a theoretical plate (HETP or II) is thus given by equation (1.5) ... 

Figure 1.5—Theoretical plate model. Computer simulation of the elution of two compounds, A and B, chromatographed on a column with 30 theoretical plates (KA = 0.5 KB = 1.6 MA — 300 pg Mb = 300 pg) showing the composition of the mixture at the outlet of the column after the first 100 equilibria. As is evident from the graph, this model leads to a non-symmetrical peak. However, when the number of equilibria is very large, and because of diffusion, the peak looks more and more like a Gaussian distribution.

By comparison with the model of theoretical plates, this more recent approach also leads to the value for the height equivalent to one theoretical plate, H. Stated in a simpler way, any chromatogram that shows an elution peak with a temporal variance a2, permits the determination of H or, directly, the number of the theoretical plates N — L/H, which is called the theoretical efficiency for the compound under investigation. 

A gold film was also evaluated using pAP as analyte model. The theoretical plate number (AO was 5990 m-1 with a half-peak width (wy2) of 4.1s for pAP. Peak current at gold-film electrode was lower than at SPEs but higher than at gold and platinum wire. 

Seader and Henley (1998) considered the separation of a ternary mixture in a batch distillation column with B0 = 100 moles, xB0 = = <0.33, 0.33, 0.34> molefraction, relative volatility a= <2.0, 1.5, 1.0>, theoretical plates N = 3, reflux ratio R = 10 and vapour boilup ratio V = 110 kmol/hr. The column operation was simulated using the short-cut model of Sundaram and Evans (1993a). The results in terms of reboiler holdup (Bj), reboiler composition profile (xBI), accumulated distillate composition profile (xa), minimum number of plates (Nmin) and minimum... 

Mujtaba (1989) used CMH model to simulate the operations considered by Domenech and Enjalbert (1974). Since the overall stage efficiency in the experimental column was 75%, the number of theoretical plates used by Mujtaba was 3. The column was initialised at its total reflux steady state values. Soave-Redlich-Kwong (SRK) model was used for the VLE property calculations. Vapour phase enthalpies were calculated using ideal gas heat capacity values and the liquid phase enthalpies were calculated by subtracting heat of vaporisation from the... 

Lehtonen et al. (1998) considered polyesterification of maleic acid with propylene glycol in an experimental batch reactive distillation system. There were two side reactions in addition to the main esterification reaction. The equipment consists of a 4000 ml batch reactor with a one theoretical plate distillation column and a condenser. The reactions took place in the liquid phase of the reactor. By removing the water by distillation, the reaction equilibrium was shifted to the production of more esters. The reaction temperatures were 150-190° C and the catalyst concentrations were varied between 0.01 and 0.1 mol%. The kinetic and mass transfer parameters were estimated via the experiments. These were then used to develop a full-scale dynamic process model for the system. 

The most common performance indicator of a column is a dimensionless, theoretical plate count number, N. This number is also referred to as an efficiency value for the column. There is a tendency to equate the column efficiency value with the quality of a column. However, it is important to remember that the column efficiency is only part of the quality of a column. The calculation of theoretical plates is commonly based on a Gaussian model for peak shape because the chromatographic peak is assumed to result from the spreading of a population of sample molecules resulting in a Gaussian distribution of sample concentrations in the mobile and stationary phases. The general formula for calculating column efficiency is... 

---