Column dimensionless parameters

Let us now assemble the complete set of dimensionless parameters for the problem. These are set out in Table 11.1, where the last column indicates the nature of their dependence on the external pressure p, the mean pore diameter and the pellet radius a. Symbols ft and 0... 

Figure 33. Dimensionless spoutdiametersasafunctionof dimensionless height for small columns. Case A test case Case B all dimensionless parameters matched, bed diameter halved Case C particle Reynolds number mismatched Case D Froude number mismatched Case E density ratio, Reynolds number mismatched Case F bed Reynolds number mismatched Case G internal friction angle, loose packed voidage mismatched. (From He et al., 1995.)...

Eqns (16), (20) and (22) were integrated numerically to obtain the separation performance of the one- and two-column processes The GEAR package (13) was used for the integration after determining that it was faster than, say, Runge-Kutta methods For all calculations N 50 and e - 0.40 Dimensionless parameters varied were 8, ph pL Yf and, for the two-column process, H. Combinations of the parameters of 6 and Pr/Pl were chosen to correspond to the me thane-helium system on BPL carbon Adsorption isotherm data for methane at 25°C (14) were represented by... 

The introduction of parameters Da, yN0 and kN0 in the gas-phase balance equations might seem unnecessarily complicated. However, it allows inclusion of the column volume Vin only one dimensionless parameter (Da), and leads to simple form for the bulk-liquid model. 

Figure 12.4 shows the use of a linear equilibrium approach (equation 12.8) for several Kd values (ranging from 0.5 to 3 cm3/g) in order to describe the BTC from the Bs-I column (C0 = 0.005 M). It is obvious that a linear approach failed to describe the shape of the BTC results. We therefore attempted to use the Freundlich approach of equation 12.1 in the transport equation where the dimensionless parameter n was allowed to vary from 0.5 to 4. As shown in Figure 12.5, the use of n < 1, which is... 

To illustrate more clearly the effect of these variables on analysis time, reduced parameters can be used for the plate height and velocity. Reduced parameters effectively normalize the plate height and velocity for the particle diameter and the diffusion coefficient to produce dimensionless parameters that allow comparison of different columns and separation conditions. The reduced plate height and reduced velocity are expressed, respectively, as... 

Comparisons of columns of different lengths that are also packed with different sized particles may be made by use of the reduced plate height Qi) which is a dimensionless parameter defined by equation (2.29). 

Dimensionless formulation of model equations for a chromatographic column can be found in Chapter 6. For clarity, the dimensionless parameters and the phenomena described by them will also be mentioned here ... 

The main idea behind the design and optimization strategy for batch chromatographic columns is the equality of concentration profiles if their dimensionless parameters (i.e. number of stages and loading factor) are identical. As mentioned... 

According to Wade et ah [46], the dimensionless band profile depends on three dimensionless parameters, fcg, 7 and bC fi. This last parameter characterizes the degree of column overloading. Equation 14.60 is easily rewritten as ... 

Loading factor Ratio of the sample size to the column saturation capacity. Dimensionless parameter characterizing column overloading (Eq. 7.26). 

The analytical or graphical solutions for concentration are usually obtained in a dimensionless form, which provides the greatest generality. This requires that the column and solution variables be assembled into the following dimensionless parameters ... 

The relative steepness or sharpness improves as the volume of the resin bed increases and also as the exchange rate increases. In order to use generalized mathematical results, these two factors must be combined into a dimensionless column capacity parameter 2, which is entirely analogous to the number of transfer units, NTU or N, as defined by Chilton and Colburn for differentially continuous separations (C3). 

Figure 13. Normalized stable isotope composition of a rock column infiltrated by a reactive fluid, (a) The solution assumes local equilibrium, with a Peclet number (Npe) of 100 and infinite Damkohler 1 number, (b) solution calculated for the case of a Peclet number of 100 and a Damkohler 1 number (No) of 1. The dimensionless parameters are given by normalized concentration c = (5, - >j )/(8j -5y ) distance z=xlL dimensionless time T =i( I 3jL) (after Bowman and Willet 1991—note that captions for their Figures 1 and 3 are switched).

Third, the conditions for scale-up from lab to process plant are constant figures for the dimensionless parameters. But in practice it is not certain that the packing of the columns is always identical. Slight variations of the void fraction and HETP may occur. Additionally, differences in the fluid dynamics, especially at the column inlet and outlet, have to be taken into account. The theoretical scale-up strategy ignores these deviations. But in order to make sure that real numbers of plates of both plants are really the same, it is recommended to determine the Van Deemter plot, void fraction, and friction number for the new packing and to correct the interstitial velocity, the flow rate, and the injection volume. 

The concepts of reduced velocity v and reduced plate height h are powerful ideas that allow us to compare columns to each other under a broad range of mobile-phase conditions and over a range of particle sizes. We use the principle of corresponding states to form dimensionless parameters from the HETP and the linear velocity. The HETP has the dimension of length. To make it dimensionless, we simply divide it by the particle diameten... 

For the reliable prediction of RD process performance it is crucial to identify the dominant local mass and energy transport resistances between the phases in a packed catalytic column section. Dimensionless parameter groups are an efficient tool for this purpose, which allow estimation of the relevance of certain transport resistances from experimental data. Here only the most important qualitative results are given [33]. 

The comparison of SEC columns that differ in length and diameter is simplified by converting the relevant volumes to porosities, dimensionless parameters defined in Equations (5) through (8) ... 

The values of Vj and Vo are usually determined by the respective standard markers. For example, in the Sephadex G-lOO/aqueous 0.1 M NaOH system, blue dextran (BD) and p-nitrophenol (pNP) are often used as standard markers for Vo and Vj, respectively. In the standardized elution profiles depicted in Fig. 2, the elution volume has been converted to a slightly different dimensionless parameter, termed relative retention volume (Vr), which is operationally defined as Vr = Vj x scaling factor, where Vr = (V — Vo)/Vo. The Vb is taken as the starting, rather than the apex position, of the BD peak. " The scaling factor normalizes Vr, which may vary slightly from ran to ran due to experimental background variations, by setting the Vr value of pNP to 2 (since the Vr value of pNP eluted from a 2.5 x 100 cm column packed with Sephadex G-lOO with aqueous 0.1 M NaOH is close to 2). Thus, an equivalent expression is Vr = 2 x (V — Vq)/ (K,np - Vo), where VpNp is the elution volume of... 

X = 0.5 and = 0.6 these are dimensionless parameters that have the indicated values for a well-packed column (no gaps) ... 

A related dimensionless parameter that is also very useful in evaluating and comparing columns is the separation impedance E (Bristow 1977) ... 

We next illustrate three column experiments using Ottawa sand where little or no mobilization occurred because of the relatively high interfacial tension. Mobilization actually depends on the trapping number [41], which is a dimensionless parameter defined as the magnitude of the vector sum of the buoyancy and viscous forces acting on the trapped... 

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