Collocation method Column

The collocation method is based on the assumption that the solution of the PDE system can be approximated by polynomials of order n. In this method the whole space domain (the column), including the boundary conditions, is approximated by one polynomial using n + 2 collocation points. Polynomial coefficients are determined by the condition that the differential equation must be satisfied at the collocation points. This approach allows the spatial derivatives to be described by the known derivatives of the polynomials and transforms the PDE into an ODE system. 

Raghavan, N.S., and Ruthven, D.M., Numerical simulation of a fixed-bed adsorption column by orthogonal collocation method, AIChE J 29(6), 922-925 (1983). 

While most of the batch unit operation models involve ordinary differential equations some unit operations like batch adsorption column encounters partial differential equations, orthogonal collocation method can be used to reduce set of partial differential equations to ordinary differential equations. 

Figure 10.12 Comparison of the numerical solutions of the equations of the equilibrium-dispersive model calculated with the forward-backward algorithm (dashed line) and the method of orthogonal collocation on finite elements (dotted line). 250 iL injection of a 15 g/L solution of (+)-Tr6ger s base. Column length, 25 cm column efficiency, Np = 146 plates F = 0.515 u = 0.076 cm/s. Isotherms in Figure 3.25...

For the solution of the PDE models of the columns, a Galerkin method on finite elements is used for the liquid phase and orthogonal collocation for the solid phase. The switching of the node equations is considered explicitly, that is, a full hybrid plant model is used. The objective function F is the sum of the costs incurred for each cycle (e.g., the desorbent consumption) and a regularizing term that is added in order to smooth the input sequence in order to avoid high fluctuations of the inputs from cycle to cyde. The first equality constraint represents the plant model... 

A dynamic model of reactive absorption column is developed in non-dimensional form. Three discretization methods OC, FD, OCFE are used to solve the model equations. For steady state process synthesis and optimisation, orthogonal collocation — based methods are found accurate and robust. 

Thus the inth element of the matrix, L)in, is the approximate value of the differential operator, L, of the nth basis function (where n is the column index) evaluated at the ith collocation point (where i is the row index). If L is a nonlinear operator, on the other hand, then the algebraic Eqs. (11) are nonlinear, too. In such a case, they should be solved by iteration methods such as the Newton-Raphson method. 

The balance equations for column reactors that operate in a concurrent mode as well as for semibatch reactors are mathematically described by ordinary differential equations. Basically, it is an initial value problem, which can be solved by, for example, Runge-Kutta, Adams-Moulton, or BD methods (Appendix 2). Countercurrent column reactor models result in boundary value problems, and they can be solved, for example, by orthogonal collocation [3]. The backmixed model consists of an algebraic equation system that is solved by the Newton-Raphson method (Appendix 1). 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

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