Multicomponent problem

The basic bacl round and understanding of binary distillation applies to a large measure in multicomponent problems. Reference should be made to Figure 8-1 for the symbols. 

The method proposed by Lewis and Matheson (1932) is essentially the application of the Lewis-Sorel method (Section 11.5.1) to the solution of multicomponent problems. Constant molar overflow is assumed and the material balance and equilibrium relationship equations are solved stage by stage starting at the top or bottom of the column, in the manner illustrated in Example 11.9. To define a problem for the Lewis-Matheson method the following variables must be specified, or determined from other specified variables ... 

To quantify the diffusion profiles is a difficult multicomponent problem. The activity-based effective binary diffusion approach (i.e. modified effective binary approach) has been adopted to roughly treat the problem. In this approach. 

Czok and Guiochon [49] called these three schemes Methods I, II, and HI, respectively. They discussed their physical sense and their consequences for the calculation of numerical solutions of the single and multicomponent problems. 

Chapter 10, which provides satisfactory accuracy and is the simplest and fastest calculation procedure. This method consists of neglecting the second-order term (RHS of Eq. 11.7) and calculating numerical solutions of the ideal model, using the numerical dispersion (which is equivalent to the introduction in Eq. 11.7 of a first-order error term) to replace the neglected axial dispersion term. Since we know that any finite difference method will result in truncation errors, the most effective procedure is to control them and to use them to simplify the calculation. The results obtained are excellent, as demonstrated by the agreement between experimental band profiles recorded with single-component samples and profiles calculated [2-7]. Thus, it appears reasonable to use the same method in the calculation of solutions of multicomponent problems. However, in the multicomponent case a new source of errors appears, besides the errors discussed in detail in Chapter 10 (Section 10.3.5). 

Thus, in the calculation of the individual band profiles in the case of multicomponent mixtures, there is a third source of errors, besides the two classical error sources observed with finite difference methods, which we have discussed in the study of the single-component problem. Obviously, these two sources are also found in the calculations of solutions of the multicomponent problems. As can be seen from Eqs. 11.17 to 11.19, this new error increases with the difference between the retention factors of the two components, and it decreases with decreasing Courant number. The error would disappear with the second and third schemes (Eqs. 11.18 and 11.19) and the numerical dispersion for the two solutes would become equal and correspond to the proper value of H if a was dose enough to 0. This observation is important because, for these two schemes (Eqs. 10.87 and 10.88), we can always select low values of the Courant raunber if needed by combining a large space increment h and a suitably small time increment t (Eq. 11.10). 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

Our task here is to derive an expression that describes how the composition of a multicomponent mixture changes with time in a Loschmidt diffusion apparatus of the kind described in Section 5.5. The composition profile for a binary system is given by Eqs. 5.5.5 and 5.5.6) the solution to the binarylike multicomponent problem is given by the same expressions on replacing the binary diffusivity in those equations by the effective diffusivity. The average composition in the bottom tube after time Z, for example, is given by... 

Estimate convective mass-transfer coefficients for multicomponent problems based on binary correlations. 

Abstract. In the present paper the problem of reuse water networks (RWN) have been modeled and optimized by the application of a modified Particle Swarm Optimization (PSO) algorithm. A proposed modified PSO method lead with both discrete and continuous variables in Mixed Integer Non-Linear Programming (MINLP) formulation that represent the water allocation problems. Pinch Analysis concepts are used jointly with the improved PSO method. Two literature problems considering mono and multicomponent problems were solved with the developed systematic and results has shown excellent performance in the optimality of reuse water network synthesis based on the criterion of minimization of annual total cost. 

In this work, the PSO algorithm was properly modified in order to satisfy the requirements of leading with discrete type variables and other strategies were also included to solve MINLP-based models. In addition, as criteria for obtaining the synthesis of the reuse water network, the minimization of the total cost was applied. At first, the WAP problem definitions and its mathematical formulation are presented. Then, the proposed modified PSO is shown, and finally applied in two literature case studies, mono and multicomponent problems, considering the minimization of annual total cost as optimization criteria. 

We have discussed in the last two chapters about the various transport mechanisms (diffusive and viscous flows) within a porous particle (Chapter 7) and the systematic approach of Stefan-Maxwell in solving multicomponent problems (Chapter 8). The role of diffusion in adsorption processes is important in the sense that in almost every adsorption process diffusion is the rate limiting step owing to the fact that the intrinsic adsorption rate is usually much faster than the diffusion rate. This rate controlling step has been recognized by McBain almost a century ago (McBain, 1919). This has prompted much research in adsorption to study the diffusion process and how this diffusional resistance can be minimized as the smaller is the time scale of adsorption the better is the performance of a process. 

For a binary distillation system of components A and B, Ny == Ny,g and Noyjt- For multi-component systems, it is possible for each component to have a different value of the transfn unit. For a discussion of the multicomponent problem, see Krishnamurthy and Taylor. The usual practice is to deal with the multicomponent mixture as if a staged column were to be used and then convett from theoretical stages to transfer units by the relationships... 

Based on the principles of a simultaneous optimisation model developed by Yee and Grossmann (1990) for heat exchanger network synthesis, a robust optimisation model for mass exchange network synthesis has been developed. Superstucture, essential modelling equations, and example problems with their solutions are presented. The new model is fairly linear and is applicable to systems of both packed and staged vessels, and multicomponent problems as well. 

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