Equations Arising in Chemical Problems

Numerical Methods for the Solution of 1D, 2D and 3D Differential Equations Arising in Chemical Problems 

Many mathematical models of chemical applications are expressed via onedimensional, two-dimensional or three-dimensional differential equations.A well known example is the Schrddinger type differential equations. 

For example we have mathematical models of chemical applications which are expressed via the one-dimensional Schrddinger type differential equations 

Finally one can find mathematical models of ehemical applieations whieh are expressed via the three-dimensional Sehrddinger type differential equations which 

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T. E. Simos, Numerical methods for ID, 2D and 3D differential equations arising in chemical problems. Specialist Periodical Reports—Chemical Modelling Applications and Theory, The Royal Society of Chemistry, 2002, 170—269. 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

The value of symmetry arguments in chemical problems arises mainly because the conceptual or computational difficulties involved in applying quantum mechanical methods to complicated molecular systems can often be partially resolved by relatively simple symmetry techniques. However, before these can be applied it is necessary to master the formal methods of handling symmetry transformations. Before discussing the mathematical tools required for this, it is useful to consider one fairly lengthy example of the power of symmetry in simplifying a complex problem. Consider the Schrodinger equation for a system of particles. 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

Another type of stability problem arises in reactors containing reactive solid or catalyst particles. During chemical reaction the particles themselves pass through various states of thermal equilibrium, and regions of instability will exist along the reactor bed. Consider, for example, a first-order catalytic reaction in an adiabatic tubular reactor and further suppose that the reactor operates in a region where there is no diffusion limitation within the particles. The steady state condition for reaction in the particle may then be expressed by equating the rate of chemical reaction to the rate of mass transfer. The rate of chemical reaction per unit reactor volume will be (1 - e)kCAi since the effectiveness factor rj is considered to be unity. From equation 3.66 the rate of mass transfer per unit volume is (1 - e) (Sx/Vp)hD(CAG CAl) so the steady state condition is ... 

Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares). 

This type of problem arises in a chemical context quite frequently. For example, the activation energy of a chemical reaction can be determined by measuring the rate constant for a particular reaction at two different temperatures. The relationship between rate constant and temperature is given by the Arrhenius equation ... 

It is possible to show that the criterion for chemical equilibrium developed here is also applicable to systems subject to constraints other than constant temperature and pressure (Problem 8.4). In fact, Eq. 8.8-1, like the phase equilibrium criterion of Eq. 8.7-9, is of general applicability. Of course, the difficulty that arises in using either of these equations is translating their simple form into a useful prescription for equilibrium calculations by relating the partial molar Gibbs energies to quantities of more direct interest, such as temperature, pressure, and mole fractions. This problem will be the focus of much of the rest of this book. 

As we have seen, the Eulerian description of turbulent diffusion leads to the so-called closure problem, as illustrated in (18.4) by the new dependent variables (njcj),/ = 1, 2, 3, as well as any that might arise in (/ , ) if nonlinear chemical reactions are occurring. Let us first consider only the case of chemically inert species, that is, R, = 0. The problem is to deal with the variables ( c ) if we wish not to introduce additional differential equations. 

Multicomponent diffusion The mathematical description of diffusion of more than one component is a complex problem and not germane to our subject. However, with the present trend toward increasing use of solid catalysts in chemical synthesis, many situations do arise in which multicomponent diffusion is involved. We left the prediction of the multicomponent diffusivity outside the scope of this book, which can be found elsewhere (Doraiswamy, 2001). Once the multicomponent diffusivity is determined, the effective diffusivity can then be found from Equation 6.7. 

Two important challenges exist for multiscale systems. The first is multiple time scales, a problem that is familiar in chemical engineering where it is called stiffness, and we have good solutions to it. In the stochastic world there doesn t seem to be much knowledge of this phenomenon, but I believe that we recently have found a solution to this problem. The second challenge—one that is even more difficult—arises when an exceedingly large number of molecules must be accounted for in stochastic simulation. I think the solution will be multiscale simulation. We will need to treat some reactions at a deterministic scale, maybe even with differential equations, and treat other reactions by a discrete stochastic method. This is not an easy task in a simulation. 

This chapter returns to the subject of diffusion per se and examines what happens when the rate of diffusion varies with both time and distance (Section 4.1) and when diffusion occurs simultaneously with a chemical reaction (Section 4.2). These are more advanced topics, which in the case of Section 4.1 lead to partial differential equations, notably Pick s equation given in Chapter 2 (Equation 2.18c). We do not attempt to solve it here, which would merely distract us from the main task, and confine ourselves instead to a presentation of the more important results in either analytical or graphical form. These are then used to solve a range of practical problems, a task that is far from trivial in spite of the appearance it gives of applying a set of convenient "recipes." Section 4.2 is confined to steady-state processes in which the state variable varies only with distance. Hence no partial differential equations arise here. We do, however, have to deal with ordinary differential equations, which sometimes require going beyond the elementary separation-of-variables technique seen in previous chapters by using the so-called D-operator method. This procedure is outlined in the Appendix at the end of the text. 

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The very short time constant in the second equation requires a very small h to follow both y and z in time. Problems of instability can easily arise. To solve stiff problems, the most typical approach is to use an implicit method, which is known to exhibit excellent stability properties (recall the backward Euler method). Special software packages are available for solving stiff systems. Fortunately, many of the straightforward chemical engineering problems enconntered in practice do not yield stiff systems, but when difficulties arise, stiffness might well be the culprit. 

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