Numerical integration stiff problems

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

In contrast to our preferred standard mode in this book, we do not develop a Matlab function for the task of numerical integration of the differential equations pertinent to chemical kinetics. While it would be fairly easy to develop basic functions that work reliably and efficiently with most mechanisms, it was decided not to include such functions since Matlab, in its basic edition, supplies a good suite of fully fledged ODE solvers. ODE solvers play a very important role in many applications outside chemistry and thus high level routines are readily available. An important aspect for fast computation is the automatic adjustment of the step-size, depending on the required accuracy. Also, it is important to differentiate between stiff and non-stiff problems. Proper discussion of the difference between the two is clearly outside the scope of this book, however, we indicate the stiffness of problems in a series of examples discussed later. So, instead of developing our own ODE solver in Matlab, we will learn how to use the routines supplied by Matlab. This will be done in a quite extensive series of examples. 

Initial value problems, abbreviated by the acronym IVP, can be solved quite easily, since for these problems all initial conditions are specified at only one interval endpoint for the variable. More precisely, for IVPs the value of the dependent variable(s) are given for one specific value of the independent variable such as the initial condition at one location or at one time. Simple numerical integration techniques generally suffice to solve IVPs. This is so nowadays even for stiff differential equations, since good stiff DE solvers are widely available in software form and in MATLAB. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

The 03, O, and 0(1D) equations are a coupled set of partial differential equations whose time constants differ by many orders of magnitude. Such a set of equations is called a stiff system (see Box 5.1). It cannot be easily solved by simple methods, unless a timestep of the order of the smallest lifetime is used for the numerical integration. In view of this problem, it is very useful to define chemical families, whose lifetimes can be very much longer than those of the constituent members. Adding equations (5.24), (5.25), and (5.26) one obtains an equation describing the behavior of the sum of 0(3P), 0(1D), and 03, which is generally referred to as the odd oxygen family, Ox ... 

Thus, the composition of such a chemical system may be obtained, at time t, by solving the set of non-linear differential equations (76). The solution is generally obtained by numerical integration, which must be highly stable ones, as the problems encountered are locally exponential , giving rise to the phenomenon called. .stiffness . Warner gives a critical review of the different numerical methods... 

In order not to raise wrong expectations in all of our experiments, the direct numerical integration of the unprepared stiff ODE system was much faster than the integration of the split DAE system, since the splitting and associated transformation cost quite a bit. However, as stated already in the beginning, the purpose of the paper is to derive a reliable dimension reduction tool for application in the context of PDEs. The presentation, however, is much simpler in the ODE context. The eventual effect of the herein advocated methods on the actual numerical solution of challenging reaction-diffusion problems will be shown in a forthcoming paper. 

The numerical integration of the resulting set of ordinary differential equations presents a considerable challenge, since the system exhibits a rapid overshoot followed by a slow transient, as will be shown later. Due to the widely different time constants inherent in our problem, we used the GEARIB code (24), which is well suited to handle such stiff problems. The detailed description of the numerical technique can be found elsewhere (25). 

Other popular methods for numerical solutions of DEs are the Runge-Kutta methods. They again come in forms of different order, depending on the number of selected points on each sub-interval for which the function is evaluated and averaged. The development of these methods includes quite sophisticated analyses of errors (deviations from the true solutions) which occur with functions of different properties. A major problem in the numerical integration of rate equations is stiffness. A differential equation is called stiff if, for instance, different st s in the process occur on widely different time scales. It is very in dent to compute with time intervals suitable for the steepest part of the progress curve (see Press et al., 1986, chapter 16 and commercial programs recommended on p. 36). 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

While offering a more inherently realistic method of solution, however, the technique may cause some additional problems in the numerical solution, since high values of Kl can lead to increased stiffness in the differential equations. Thus in using this technique, a compromise between the approach to equilibrium and the speed of numerical solution may have to be adopted. Continuous single-stage extraction is treated in the simulation example EQEX. Reaction with integrated extraction is demonstrated in simulation example REXT. 

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