Perturbation collocation method

To further demonstrate the simplicity and the straightforward nature of the orthogonal collocation method, we consider the adsorption problem dealt with in Section 12.2 where the singular perturbation approach was used. The nondimensional mass balance equations are... 

Alternate but similar criteria have been derived using perturbation techniques about the surface concentration by Hudgins [1968] and using collocation methods by Stewart and Villadsen [1969]. Their results are essentially given by ... 

In the present communication, a brief description of the pore plugging model is presented and its differences with that of Ramachandran and Smith (7 ) are examined. An analytical calculation of the time required to plug the pore is presented. In addition, a perturbation solution for small times is used to motivate the formulation of a semianalytical version of the collocation method for two point boundary value problems with steep concentration profiles. 

Historically, the basis of gas permeation module design was first proposed by Weller and Steiner in 1950. Nowadays, modem computation techniques provide numerical solutions to the problems thanks to dedicated routines. Orthogonal collocation methods or perturbation methods " are reported to be particularly attractive when a minimum resolution time and computational efforts are required. Several of these routines have been implemented in commercial process simulation software, where advantage can be taken of thermodynamics or unit operation design packages in order to simulate hybrid or multi-stage operations with gas separation membranes. Nevertheless, much effort has been devoted to... 

However, integrabihty imposes a criterion for obtaining DBs analytically. DBs are obtained analytically for integrable systems, while for non-integrable systems it is obtained by various numerical methods viz. spectral collocation method, finite-difference method, finite element method, Floquet analysis, etc. As evident from many numerical experiments, DBs mobility is achieved by an appropriate perturbation [42]. From the practical application perspective, dissipative DBs are more relevant than their Hamiltonian counterparts. The latter with the character of an attractor for different initial conditions in the corresponding basin of attraction may appear whenever power balance, instead of energy conservation, governs the nonlinear lattice dynamics. The attractor character for dissipative DBs allows for the existence of quasi-periodic and even chaotic DBs [54, 55]. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

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