Finite differences staged equations

Chapter 5 Staged-Process Models The Calculus of Finite Differences homogeneous equation are... 

Equations for Large Stage Separation Factors. The preceding results have been obtaiaed with the use of equation 2 and by replacing the finite difference, by the differential, chc/ dn both of which are vaUd only when the quantity (a — 1) is very small compared with unity. However,... 

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

At this stage, we need to discuss the actual task of calculating the Jacobian matrix J. It is always possible to approximate J numerically by the method of finite differences. In the limit as Akz approaches zero, the derivative of R with respect to k, is given by Equation 7.16. For sufficiently small Ak the approximation can be very good. 

At this stage, two approaches are possible. The first one calculates solutions of the mass balance equation (Eq. 10.60) and uses finite-difference schemes that give a numerical error of the second order. The second approach calculates solutions of the mass balance equation of the ideal model (Eq. 10.72) and uses finite-differences schemes that give an error of the first order. The parameters of the numerical integration are then selected in such a way that the numerical error introduced by the calculation is equivalent to the dispersion term, so the approximate numerical solution of the approximate equation and the exact solution of the correct equation are equal to the first order. 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

The above equations are limited to cases of constant flowrates and linear equilibrium relationships. For situations where there are small deviations from linear phase equilibrium and/or changes in flow from stage to stage, the above equations can be applied over sections of the cascade in series. For situations where this approach is not reasonable, finite difference mathematical analysis can also be applied to equilibrium-stage calculations. 

The final stage in the adiabatic reduction is the solution of Eq. (4.24). Given the adiabatic potential of Eq. (4.26) this cannot be done analytically, but the resulting ordinary differential equation may be solved numerically using the finite difference method. As an example, we show in Fig. 20 a comparison between the even-parity adiabatic eigenvalues and the exact ones, obtained by solving the full coupled channels expansion, using the artificial channel method.69... 

A finite difference scheme for discretization in time is used at this stage. In order to reduce the set of ordinary differential equations to algebraic equations, a time weighting coefficient is introduced, that allows to use several schemes explicit, implicit or the Crank-Nicolson scheme. 

The model equations were numerically solved using a finite difference technique. The predicted movement of the interfaces, R and S, with respect to time is shown in Fig. 27. The corresponding concentration profile is shown in Fig. 28. These results show the existence of three distinct stages in the dissolution process. In the first stage, there is solvent-penetration inducing swelling... 

A nonlinear equation, which arises in both continuous and staged (i.e., finite difference) processes, is Riccati s equation... 

The last equation may be so solved as to eliminate the concentrations of the rafiinates of adjacent stages by the calculus of finite differences (19). If ntB/A 5 1,... 

It has been proposed to represent the TF with 10 CSTR staged units 0.3 ft in length. Develop solutions to this problem using a finite difference method of solving an ordinary differential equation and a lumped parameter model employing a method of solution of simultaneous linear algebraic equations. 

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