Solutions of the Model Equations

As noted above, Newton s method can be used to solve the entire set of nonlinear equations simultaneously. To use Newton s method, we repeatedly solve Eq. 14.2.10 linearized about a current guess (x ) of the vector (x). 

The method may be assumed to have converged when either of the following two criteria are satisfied 

Newton s method requires the evaluation of the partial derivatives of all equations with respect to all variables. The partial derivatives of thermodynamic properties with respect to temperature, pressure, and composition are most awkward to obtain (and the ones that have the most influence on the rate of convergence). Since pressure is an unknown variable in this model, the derivatives of K values and enthalpies with respect to pressure must be evaluated. Neglect of these derivatives (even though they are often small) can lead to convergence difficulties. 

In the above formulas m and /Wg are the steady-state values of wa and wb in (19.42) and (19.43). These steady states must be nonnegative and therefore physically significant only if 5 1. Note that for of = 0 the above equations give us the solution of the deposition problem if one neglects the equilibration processes between the two phases. Such quantities are noted using an asterisk( ). 

The system of ordinary differential equations (19.57) to (19.62) with initial conditions given by (19.63) is solved numerically using standard Ordinary Differential Equation (ODE)-soIving routines. 

Neglecting the gas-aerosol phase equilibration processes is equivalent to setting (T = 0 in (19.57) to (19.62). The corresponding solutions of this simplified system are denoted by using an asterisk( ) and are 

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Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

Therefore, efficient computation schemes of the state and sensitivity equations are of paramount importance. One such scheme can be developed based on the sequential integration of the sensitivity coefficients. The idea of decoupling the direct calculation of the sensitivity coefficients from the solution of the model equations was first introduced by Dunker (1984) for stiff chemical mechanisms... 

The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

D. SOLUTION OF THE MODEL EQUATIONS. We will concem ouTselves in detail with this aspect of the model in Part 11. However, the available solution techniques and tools must be kept in mind as a mathematical model is developed. An equation without any way to solve it is not worth much. 

Despite the assumptions and simplifications we have made in arriving at a model we feel that the physical basis we have adopted is sufficiently realistic to give good predictions, certainly as far as our present experimental results eneible us to make tests. The numerical solution of the model equations we have used presented no difficulties using a fast computer ( v 5 secs per solution). ... 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

Solution of optimisation problems using rigorous mathematical methods have received considerable attention in the past (Chapter 5). It is worth mentioning here that these techniques require the repetitive solution of the model equations (to evaluate the objective function and the constraints and their gradients with respect to the optimisation variables) and therefore computationally can be very expensive. 

Approximate methods. In these a great many simplifying assumptions are made to obtain solutions of the model equations. These methods were the subject of the immediately preceding subsection. 

Our judgment is that feasible path methods in which the solution of the model equations over time is carried out by conventional integration software, which has been extensively developed and refined, are at present more reliable than infeasible path methods. Feasible path optimization methods are also easier to implement as the size of the optimization problem is much smaller. For these reasons, we have pursued feasible path methods despite evidence that infeasible path methods are more efficient on some problems. 

In this section we present a number of examples designed to illustrate the use of a nonequilibrium model as a design tool. In view of the large number of equations that must be solved it is impossible to present illustrative examples of the application of the nonequilibrium model that are as detailed as the examples in prior chapters. In the examples that follow we confine ourselves to a brief summary of the problem specifications and the results obtained from a computer solution of the model equations. In most cases several different column configurations were simulated before the results presented below were obtained. 

Pales and Stroeve [31] investigated the effect of the continuous phase mass transfer resistance on solute extraction with double emulsion in a batch reactor. They presented an extension of the perturbation analysis technique to give a solution of the model equations of Ho et al. [29] taking external phase mass transfer resistance into account. Kim et al. [5] also developed an unsteady-state advancing reaction front model considering an additional thin outer liquid membrane layer and neglecting the continuous phase resistance. 

Algorithm for the numerical solution of the model equations Modelling for the different configurations of the ammonia converters... 

Development of the necessary algorithms for the solution of the model equations,... 

With todays computers and the state of the art regarding numerical techniques, it does not seem that the numerical solution of the model equations presents any serious problems. With the fast development of computer hardware and software, this problem will become almost trivial in the near future. 

For the most simple column models under certain simplifying conditions there are analytical solutions of the model equations available. Related to the equilibria this holds for problems vhere all components of interest are characterized by linear isotherm equations, in vhich the Henry constants are not affected by the presence of other component. Then all kinetic effects causing band broadening can be described by a single lumped parameter, for example, the number of theoretical plates. Consequently, the usage of two or more kinetic parameters is not justified. 

Approximate Solution. Approximate solutions of the model equations (1) - (3) were obtained under pseudo steady state conditions and it can be shown that the... 

Modelling is useful only if it can provide information which is not readily available, or difficult to obtain. This information can usually be obtained from the solution of the modelling equations or from the specific points which characterize these equations. It is worth remembering that the solutions of algebraic equations are numbers, whereas the solutions of differential and integral equations are functions. These specific points can be ... 

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