General Numerical Solutions Multicomponent Systems

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

This model has recently been used to interpet the experimental adiabatic breakthrough curves obtained by Wright and Basmadjian for COj-CjH -Nj on 5A sieves. N2 is treated as a nonadsorbing carrier. The binary equilibrium data were derived from the isotherms given by Glcssner and Myers and the temperature dependence of the overall rate coefficients was assumed to be given by an Arrhenius-type expression with activation energy 

Helfferich and G. Klein, Mul komp men Chromalography. Marcel Dekker, New York, 1970. 

T- Vermeulen, in Chemical Engineers Handbook, 5th ed., R. H. Perry and C. H. Chilton (eds.). McGraw-Hill. New York, 1973, section 16. 

in Percolation Processes, NATO ASI No. 33, Espinho, Portugal 1978, A. E, Rodrigues and D. Tondeur (eds.). Sijthoff and Noordhoff, Rockville, Md., 1981. 

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The design and optimization of adsorptive processes typically require simultaneous numerical solutions of coupled partial differential equations describing the mass, heat, and momentum balances for the process steps. Multicomponent adsorption equilibria, kinetics, and heat for the system of interest form the key fundamental input variables for the design. " Bench- and pilot-scale process performance data are generally needed to confirm design calculations. 

A general conclusion that can be drawn from this short survey on the many attempts to develop analytical theories to describe the phase behavior of polymer melts, polymer solutions, and polymer blends is that this is a formidable problem, which is far from a fully satisfactory solution. To gauge the accuracy of any such approaches in a particular case one needs a comparison with computer simulations that can be based on exactly the same coarse-grained model on which the analytical theory is based. In fact, none of the approaches described above can fully take into account all details of chemical bonding and local chemical structure of such multicomponent polymer systems and, hence, when the theory based on a simplified model is directly compared to experiment, agreement between theory and experiment may be fortuitous (cancellation of errors made by use of both an inadequate model and an inaccurate theory). Similarly, if disagreement between theory and experiment occurs, one does not know whether this should be attributed to the inadequacy of the model, the lack of accuracy of the theoretical treatment of the model, or both. Only the simulation can yield numerically exact results (apart from statistical errors, which can be controlled, at least in principle) on exactly the same model, which forms the basis of the analytical theory. It is precisely this reason that has made computer simulation methods so popular in recent decades [58-64]. 

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