Constant pattern wave

For a constant pattern wave, all concentrations within the wave have the same velocity. 

For nonconvex equilibrium functions, combined wave solutions are possible which consist in parts of spreading waves and in others of constant pattern waves. In the equilibrium diagram combined wave solutions represent the convex hull of the equilibrium line, as illustrated in Fig. 5.5(c). 

Fig. 5.5. Construction of wave solutions in the scalar case, (a) Constant pattern wave (b) spreading wave (c) combined wave solution.

Let us first focus on a nonreactive system with constant separation factors. Typical examples are distillation processes with constant relative volatilities or adsorption processes described by competitive Langmuir isotherms. For nonreactive systems with constant separation factors, the constant pattern waves and spreading waves are... 

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... 

Each term under the sum represents a single constant pattern wave, as illustrated in Fig. 5.15. Therein, s( s) represents the position, p k) the slope and xjk>, xjk+li the... 

Wave models were successfully used for the design of a supervisory control system for automatic start-up of the coupled column system shown in Fig. 5.15 [19] and for model-based measurement and online optimization of distillation columns using nonlinear model predictive control [15], The approach was also extended to reactive distillation processes by using transformed concentration variables [22], However, in reactive - as in nonreactive - distillation, the approach applies only to processes with constant pattern waves, which must be checked first. 

Another type of analysis is based on the pattern of the concentration front in the bed, essentially the reverse image of the breakthrough curve, particularly when this front passes through the bed with a eonstant velocity. Some of this we mentioned in Chapter 4. Such constant pattern waves are not limited to ion exchange or adsorption, for they are often encountered in the poisoning of fixed-bed reactors, in either isothermal [A. Wheeler and A.J. Robell, J. CataL, 13, 299 (1969) H.W. Haynes, Jr., Chem. Eng. ScL, 25, 1615 (1970)] or nonisothermal operation [H.S. Weng, G. Eigenberger and J.B. Butt, Chem. Eng. Sci., 30, 1341 (1975) T.H. Price and J.B. Butt, Chem. Eng. Sci., 32, 393 (1977)]. 

The analysis can be extended to the multi-component case [43, 44, 51, 65, 66]. The number of the fronts is directly related to the number of components in the mixture. For ideal and moderately non-ideal mixtures the concentration and temperature profiles consist of n,. - 1 fronts connecting two pinch points. Again, constant pattern waves occur for ideal and moderately non-ideal mixtures. Addi-... 

Under steady state conditions, at most one of these wave fronts can be located in the middle of a column section, whereas the others are located at either one of the boundaries, where they can overlap and interact. A multicomponent example [51] is shown in Fig. 10.19. Initially, two distinct constant pattern waves can be identified in the concentration profiles of both components traveling at the same velocity to the top of the rectifying section. Close to the upper boundary, the waves start to interact and form some combined steady state pattern. 

Despite the analogy between reactive and non-RD, the type of wave pattern may not necessarily comply for similar types of mixtures. In particular, combined waves as described above for non-ideal non-reactive mixtures might occur for ideal, reactive mixtures. Conversely, constant pattern waves may arise for highly non-ideal reactive mixtures. 

An example of wave propagation in an RD column is shown in Figs. 10.21 and 10.22. Constant pattern waves can be observed in case of ideal phase equilibrium, kinetically controlled mass transfer and a single reversible reaction close to chemical equilibrium (compare Figs. 10.21). In contrast, at low Da in the kinetically controlled regime of the chemical reaction a different, more complex type of dynamic behavior can be observed (compare Fig. 10.22). The behavior in the kinetic regime is not sufficiently understood today and needs further research. 

Since most adsorption and ion exchange separations of commercial significance operate in the nonlinear region of the isotherm, the previous analysis needs to be expanded to nonlinear systems. Nonlinear behavior is distinctly different than linear behavior since one usually observes shock or constant pattern waves during the feed step and diffuse or proportional pattern waves during regeneration. Experimental... 

For an isotherm with a Langmuir shape, if the column is initially loaded at some low concentration, c q, (ciow = 0 if the column is clean) and is fed with a fluid of a higher concentration, C] (see Figure 18-IZA), the result will be a shock wave. The feed step in adsorption processes usually results in shock waves. Experiments show that when a shock wave is predicted the zone spreading is constant regardless of the column length (a constant pattern wave). With the assunptions of the solute movement theory (infinitely fast rates of mass transfer and no axial dispersion), the wave becomes infinitely sharp (a shock) and the derivative dq/dc does not exist. Thus, the Aq/Ac term in the denominator of Eq. f 18-141... 

In experiments tFigure 18-15A1 the oudet concentration profiles are not sharp as shown in Figures 18-17B and 18-18B. Instead the finite mass transfer rates and finite amounts of axial dispersion spread the wave while the isotherm effect (illustrated in Exanple 18-71 counteracts this spreading. The final result is a dynamic equilibrium where the wave spreads a certain amount and then stops spreading. Once formed, this constant pattern wave has a constant width regardless of the column length. 

In general, we cannot obtain analytical solutions of the complete mass and energy balances for nonlinear systems. One exception to this is for isothermal systems when a constant pattern wave occurs. Constant pattern waves are concentration waves that do not change shape as they move down the column. They occur when the solute movement analysis predicts a shock wave. 

Experimental results (Figure 18-IS) and the shockwave analysis showed that the wave shape for constant pattern waves is independent of the distance traveled. This allows us to decouple the analysis into two parts. First, the center of the wave can be determined by analyzing the shock wave with solute movement theory. Second, the partial differential equations for the column mass balance can be sirtplified to an ordinary differential equation by using a variable = t - z/u jj that defines the deviation from the center of the wave. This approach is detailed in more advanced sources (e.g., Ruthven. 1984 Sherwood et al.. 1975 Wankat. 19901. 

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