Solving the Model Equations

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

Newton s method for solving systems of equations is summarized in Algorithm C.2 and has been used elsewhere in this book for solving mass and energy transport problems. For solving the modified MESH equations we may identify the vector of variables (x) as 

The vector of discrepancy functions for the column as a whole is given by 

Algorithm 13.1 Procedure for Solving Equilibrium Model Equations with Multicomponent Tray 

Mole fractions of vapor and liquid phases Set efficiency of all components to 0.75. 

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Wynn (1992, 1993) have presented sliort-eut studies of solids proeess flowslieet-ing. Custom-written software to solve the model equations has been used by a number of authors. It has also been proposed that symbolie manipulation paekages, like Mathematiea, ean be used to solve the population balanee equations as a stand-alone modelling environment or as a set of modules to be linked to a generie proeess simulation paekage sueh as SPEEDUP (Hounslow, 1989 Sheikh and Jones, 1996). 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

Russell and Denn solved the model equations analytically, assuming constant liquid density, and showed that... 

As shown by Dang et al. (1977), solving the model equations by Laplace transformation gives... 

Pseudo-experimental data can be generated by solving the model. Equations 1-4, for a chosen set of parameters and initial conditions, and then adding random noise to the model solution. For a given choice of measurement variables, the simulated data is then used in the parameter estimation problem. This procedure provides a means by which to evaluate the measurements that are required and the amount of measurement noise that is tolerable for parameter identification. 

The equations of the developed model need to be solved for certain inputs, certain design objectives, and given physico-chemical parameters in order to predict the output and, for design purposes, the variation of the state variables within the boundaries of the system. In order to solve the model equations we have two tasks ... 

Below we present and discuss our simulation results obtained by numerically solving the model equations (7.29) to (7.47) as described in the ten step adaptive procedure above. The results are based on manipulating equation (7.61), so that the heat removal line becomes independent of the feed temperature and its slope independent of the reactor temperature and the other variables. Once this has been achieved, we can assume that the slope of the heat removal line is constant over time. 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

The pneumatic drying model was solved numerically for the drying processes of sand particles. The numerical procedure includes discretization of the calculation domain into torus-shaped final volumes, and solving the model equations by implementation of the semi-implicit method for pressure-linked equations (SIMPLE) algorithm [16]. The numerical procedure also implemented the Interphase Slip Algorithm (IPSA) of [17] in order to account the various coupling between the phases. The simulation stopped when the moisture content of a particle falls to a predefined value or when the flow reaches the exit of the pneumatic dryer. 

Powers MF, Vickery DJ, Arehole A, Taylor R. A nonequilibrium-stage model of multicomponent separation processes—V. Computational methods for solving the model equations. Computers Chem Eng 1988 12 1229-1241. 

Adomian s Decomposition Method is used to solve the model equations that are in the form of nonlinear differential equation(s) with boundary conditions.2,3 Approximate analytical solutions of the models are obtained. The approximate solutions are in the forms of algebraic expressions of infinite power series. In terms of the nonlinearities of the models, the first three to seven terms of the series are generally sufficient to meet the accuracy required in engineering applications. 

Comparison of the Computation Results. As indicated above, Gear s method was used to solve the model equations only for a fraction of the total residence time in the reactor which took 8.59 minutes of machine computation time. The same set of equations was solved by the approximate iterative technique for the same time interval in 5.8 seconds of computer time. As a comparison of the accuracy overall devolatilization Vj = Z Z v j as predicted by the two techniques are plotted on a dimensionless scale in Figure 1. The definitions for the dimensionless quantities used are ... 

In equilibrium-stage models, the compositions of the leaving streams are related through the assumption that they are in equilibrium (or by use of an efficiency equation). It is important to recognize that efficiencies are not used in a nonequilibrium model they may, however, be calculated from the results obtained by solving the model equations. 

In order to solve the model equation, we must complete it with the univocity conditions. In some cases, relations (3.100)-(3.107) can be used as solutions for the model particularized for the process. The equivalence between both expressions is that c(x,t)/C(j appears here as P(x,t). Extending the equivalence, we can establish that P(x, t) is in fact the density of probability associated with the repartition function of the residence time of the liquid element that evolves inside a uniform porous structure. 

Specification of the numerical methods to be used for solving the model equations... 

In this chapter we have considered models of multicomponent condensation. In particular, we have considered various approaches to calculating the rates of mass and energy transfers in the vapor and condensate, respectively. Methods of solving the model equations have also been discussed. 

Powers, M. F., Vickery, D. J., Arehole, A., and Taylor, R., A Nonequilibrium Stage Model of Multicomponent Separation Processes—V. Computational Methods for Solving the Model Equations, Comput. Chem. Eng., 12, 1229-1241 (1988). 

Luo [73] and Luo and Svendsen [74] extended the work of Coulaloglou and Tavlarides [16], Lee et al [66] and Prince and Blanch [92] formulating the population balance directly on the macroscopic scales where the closure laws for the source terms were integrated parts of the discrete numerical scheme used solving the model equations. 

Several algorithms have been proposed to solve the model equations with chemical reactions (Holland, 1981 Saito et al., 1971 Tierney et al., 1982). Venkataraman et al. (1990) applied the two-tier method for this purpose. 

In all cases, it is essential to solve the model equations efficiently and accurately. Some techniques are discussed in this book and in the appendices, for the solution of the highly non-linear algebraic, differential and integral equations arising in the modelling of fixed bed catalytic reactors. The most difficult equations to solve are usually the equations for diffusion and reaction in the porous catalyst pellets, especially when diffusional limitations are severe. The orthogonal collocation technique has proved to be very efficient in the solution of this problem in most cases. In cases of extremely steep concentration and temperature profiles inside the pellet, the effective reaction zone method and its more advanced generalization, the spline collocation technique, prove to be very efficient. 

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