Concentration profiles, constant pattern

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

The input signal is composed of 2n concentration profiles of the partaking substrates. In this study it was considered that each of these concentration profiles either follows the pattern described in Figure 4.2 or is constant. The... 

At sufficiently large values of X the saturation curves approach a constant pattern form, and thereafter the concentration front progress through the columii at a steady velocity, governed by the capacity of the adsorbent and the feed concentration, with no further change in the shape of the curve. Such behavior is characteristic of systems with a favorable equilibrium isotherm (12). The constant pattern limit is reached when the dimensionless concentration profile in fluid phase and adsorbed phase become practically coincident, and the asymptotic form of the break-... 

Constant Pattern Behavior In a real system the finite resistance to mass transfer and axial mixing in the column lead to departures from the idealized response predicted by equilibrium theory. In the case of a favorable isotherm the shock wave solution is replaced by a constant pattern solution. The concentration profile spreads in the initial region until a stable situation is reached in which the mass transferrate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-phase profiles become coincident. This represents a stable situation and the profile propagates without further change in shape—hence the term constant pattern. 

Length of Unused Bed. The constant pattern approximation provides the basis for a very useful and widely used design method based on the concept of the length of unused bed (LUB). In the design of a typical adsorption process the basic problem is to estimate the size of the absorber bed needed to remove a certain quantity of the adsorbable species from the feed stream, subject to a specified limit ((/) on the effluent concentration. The length of unused bed, which measures the capacity of the adsoibei which is lost as a result of the spread of the concentration profile, is defined by... 

Proportionate Pattern Behavior. If the isotherm is unfavorable (as in Fig. 1,111), the stable dynamic situation leading to constant pattern behavior can never be achieved. The equilibrium adsorbed-phase concentration then lies above rather than below the actual adsorbed-phase profile. As the mass transfer zone progresses through the column it broadens, but the limiting situation, which is approached in a long column, is simply local equilibrium at all points (c = c ) and the profile therefore continues to... 

Finally, nonlinear wave can also be used for nonlinear model reduction for applications in advanced, nonlinear model-based control. Successful applications were reported for nonreactive distillation processes with moderately nonideal mixtures [21]. For this class of mixtures the column dynamics is entirely governed by constant pattern waves, as explained above. The approach is based on a wave function which can be used for the approximation of the concentration profiles inside the column. The wave function can be derived from analytical solutions of the corresponding wave equations for some simple limiting cases. It is given by... 

In this study, the crystallization temperature and the heating rate were varied using the milled precursor, and ZSM-5 crystals could be synthesized. For example, the temperature was elevated from 160 to 210°C with a constant heating rate of 0.2°C/min (Method 2). The crystals prepared by Methods 1 4 had about same BET-surface area of 385 11 ma /g and the XRD patterns of ZSM-5. The average size of crystals reduced from 8 urn for Method 1 to 1 ym for Method 4. The concentration profiles of Si and A1 from outside to inside the crystals became uniform with reducing size. The activity of methanol conversion, the yield of gasoline fraction, and the content of aromatics in the gasoline clearly increased for the product of Method 4 (Fig. 6). 

Assuming that the temperature and concentration profiles do not change with time and further that there are no heat losses (i.e constant-pattern propagation), we can set dT/dt and drj/dt equal to zero. Then Eq. (11) takes the following form ... 

The same approaches that were successful in linear chromatography—the use of either one of several possible liunped kinetic models or of the general rate model — have been applied to the study of nonlinear chromatography. The basic difference results from the replacement of a linear isotherm by a nonlinear one and from the coupling this isotiienn provides between the mass balance equations of the different components of the mixture. This complicates considerably the mathematical problem. Analytical solutions are possible only in a few simple cases. These cases are limited to the band profile of a pure component in frontal analysis and elution, in the case of the reaction-kinetic model (Section 14.2), and to the frontal analysis of a pure component or a binary mixture, if one can assume constant pattern. In all other cases, the use of numerical solutions is necessary. Furthermore, in most studies found in the literature, the diffusion coefficient and the rate constant or coefficient of mass transfer are assumed to be constant and independent of the concentration. Actually, these parameters are often concentration dependent and coupled, which makes the solution of the problem as well as the discussion of experimental results still more complicated. 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

There are simple algebraic solutions for the linear ideal model of chromatography for the two main coimter-current continuous separation processes. Simulated Moving Bed (SMB) and True Moving Bed (TMB) chromatography. Exphcit algebraic expressions are obtained for the concentration profiles of the raffinate and the extract in the columns and for their concentration histories in the two system effluents. The transition of the SMB process toward steady state can be studied in detail with these equations. A constant concentration pattern can be reached very early for both components in colimm III. In contrast, a periodic steady state can be reached only in an asymptotic sense in colunms II and IV and in the effluents. The algebraic solution allows the exact calculation of these limits. This result can be used to estimate a measure of the distance from steady state rmder nonideal conditions. 

Constant pattern The asymptotic solution in frontal analysis or displacement chromatography. Each point of the concentration profile moves at the same velocity, so the profile migrates but its shape remains constant. 

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