Plate, effective plates theory

There are two fundamental chromatography theories that deal with solute retention and solute dispersion and these are the Plate Theory and the Rate Theory, respectively. It is essential to be familiar with both these theories in order to understand the chromatographic process, the function of the column, and column design. The first effective theory to be developed was the plate theory, which revealed those factors that controlled chromatographic retention and allowed the... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

Equation (18) displays the relationship between the column efficiency defined in theoretical plates and the column efficiency given in effective plates. It is clear that the number of effective plates in a column is not aii arbitrary measure of the column performance, but is directly related to the column efficiency as derived from the plate theory. Equation (18) clearly demonstrates that, as the capacity ratio (k ) becomes large, (n) and (Ne) will converge to the same value. 

So far the plate theory has been used to examine first-order effects in chromatography. However, it can also be used in a number of other interesting ways to investigate second-order effects in both the chromatographic system itself and in ancillary apparatus such as the detector. The plate theory will now be used to examine the temperature effects that result from solute distribution between two phases. This theoretical treatment not only provides information on the thermal effects that occur in a column per se, but also gives further examples of the use of the plate theory to examine dynamic distribution systems and the different ways that it can be employed. 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

Primarily the Plate Theory provides the equation for the elution curve of a solute. Such an equation describes the concentration of a solute leaving a column, in terms of the volume of mobile phase that has passed through it. It is from this equation, that the various characteristics of a chromatographic system can be determined using the data that is provided by the chromatogram. The Plate Theory, for example, will provide an equation for the retention volume of a solute, show how the column efficiency can be calculated, determine the maximum volume of charge that can be placed on the column and permit the calculation of the number of theoretical plates required to effect a given separation. 

It is now clear that the plate theory has a wide field of application. Its use, however is not restricted to LC. For example, the plate theory can be used to Investigate temperature changes that take place in a GC column,(3), pressure changes that take place in a GC column, (4), the effect of solute decomposition on band profile and other similar effects that can take place in a chromatographic system. The plate theory has many areas of application that still remain to be explored. 

Now, if the column is to effect a particular separation, where the pair of solutes that are eluted closest together have a separation ratio of (a) and the first of the pair, ( solute A), is eluted at a capacity ratio value of (k A), then the value of (n) is given by the Purnell Equation, which was discussed in the chapter on The Applications of the Plate Theory, viz... 

During the processing of composite materials in a hot press or an autoclave, the laminate is usually kept flat until cure is complete. If the platen surfaces are assumed frictionless, the effect of the constraints is to require that the curvatures K] and k2 be zero throughout cure. To develop the elastic solution under these constrained conditions, the laminated plate theory may be used with conditions of N = 0 and jc, = 0. The resulting midplane strains are given by... 

There are several resources available for designing filament-wound cylinders. In general, filament-wound cylinders are classified as cylindrically orthotropic. Adjacent helical plies, loading conditions can be determined by following the principles of laminated plate theory [7]. When applying laminated plate theory, the plate consists of the cylinder wall. In this case, the effect of cylinder curvature is neglected, and the Q and z axes are considered the planar axes of the plate. Failure criteria applied in laminated plate theory, such as maximum stress or strain, or the quadratic Tsai-Wu failure criteria [7] may also be applied. Several specialized loading cases have been studied. 

The quantitative description of the migration process and resulting soil contamination profile, including the effect of a continuous and variable aerosol input, may be adopted from ion exchange plate theory as presented by Mayer and Tompkins (9). 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

In this paper the stresses in a joint with a functionally graded material (FGM) are analyzed. In the middle of a joint with FGM the stresses have been calculated analytically by using the plate theory. The effect of the thickness of the FGM layer and the effect of the transition function form on the stresses in the joint is discussed. Near the free edges of the interface in a joint with FGM, the stresses are described analytically by using the Mellin transform method. Some examples are presented to show the good agreement of the stresses calculated from FEM and with the analytical description in a joint with graded material. 

Thus the number of equilibrium stages is directly proportional to column length, if the linear flow rate remains unchanged. The role of contact time, which is obscured in the plate theory, now becomes evident. Wherever mass-transfer is independent of flow rate, a diminution of the flow velocity through a bed of constant length will increase the effective number of theoretical plates in inverse proportion. The number of plates in any one column may vary for the different components, just as Nr may, although usually the variation is not great. 

Haijula and Townsend [6] point out that the application of plate theory can be used to quantify column efficiency via the estimation of the number of effective plates (Nr,P)-For a film diffusion controlled exchange,... 

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