Example Problems: algebraic equations

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

Transformation replaces a differential equation with independent variable, t, to an algebraic equation in variable, s. The latter relation can be solved algebraically for the transform f s). Then f(t) is found by inversion with Table 1.4. Problems PI.04.01 ff are examples. Cases of multiple reactions are treated this way in problems P2.02.08 and P2.02.10. 

The model involves four variables and three independent nonlinear algebraic equations, hence one degree of freedom exists. The equality constraints can be manipulated using direct substitution to eliminate all variables except one, say the diameter, which would then represent the independent variables. The other three variables would be dependent. Of course, we could select the velocity as the single independent variable of any of the four variables. See Example 13.1 for use of this model in an optimization problem. 

The simplest case of this parameter estimation problem results if all state variables jfj(t) and their derivatives xs(t) are measured directly. Then the estimation problem involves only r algebraic equations. On the other hand, if the derivatives are not available by direct measurement, we need to use the integrated forms, which again yield a system of algebraic equations. In a study of a chemical reaction, for example, y might be the conversion and the independent variables might be the time of reaction, temperature, and pressure. In addition to quantitative variables we could also include qualitative variables as the type of catalyst. 

We can easily integrate this equation numerically to find the sblutions for the PFTR. We can also solve the algebraic equation for the CSTR. We will solve this problem numerically using a spreadsheet in the next example, but in Figure 5-5 we just plot the solutions X(r) and T(t). 

Models can have the characteristic of different types and sizes of equation sets relative to a general set of algebraic equations. Some common example situations include physical property models and models containing differential equations. In posing the mathematical problem to be solved, a completely simultaneous solution approach can be used or a "mixed mode" that combines specialized solution techniques within the overall EO approach. 

For implicit schemes, we will obtain a system of linear algebraic equations that must be solved. As mentioned in Example 8.1, one-dimensional diffusion problems generate tri-diagonal matrices, that can be solved for using the Thomas algorithm or other fast matrix routines. Equation (8.83) can be written as... 

As a rule, more than two dimensionless numbers are necessary to describe a phys-ico-technological problem they cannot be produced as shown in the first three examples. The classical method to approach this problem involved a solution of a system of linear algebraic equations. They were formed separately for each of the base dimensions by exponents with which they appeared in the physical quantities. J. Pawlowski [5] replaced this relatively awkward and involved method by a simple and transparent matrix transformation ( equivalence transformation ) which will be presented in detail in the next example. 

Fundamental to almost all applications of quantum mechanics to molecules is the use of a finite basis set. Such an approach leads to computational problems which are well suited to vectoris-ation. For example, by using a basis set the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients - a set of matrix equations. The absolute accuracy of molecular electronic structure calculations is ultimately determined by the quality of the basis set employed. No amount of configuration interaction will compensate for a poor choice of basis set. 

The next step in the problem solving procedure is to outline a solution procedure for the Equations listed in Table 3.2.2. There are algorithms available for determining in what order to solve a set of algebraic equations, which is called the precedence order. See, for example, Rudd and Watson [17] and Myers and Seider [18] for a discussion of some of these algorithms. Sometimes, we can develop a procedure by inspection of an equation set, as in the procedure given in Table 3.2.3. 

In the foregoing example we have found it necessary to solve a quadratic algebraic equation. Many of the simple problems of chemical equilibrium lead to quadratic equations, which can of course be solved easily. Sometimes we meet more complicated algebraic equations, which can be solved only by trial or by a method of successi e approximations. 

Unfortunately, chemical processes are seldom relatively simple. For example, most problems you have encountered in this text could eventually be reduced to the solution of linear algebraic equations. 

This appendix explains how to use DDAPLUS to solve nonlinear initial-value problems containing ordinary differential equations with or without algebraic equations, or to solve purely algebraic nonlinear equation systems by a damped Newton method. Three detailed examples are given. 

TliCf riiiite difl erence fomiulatioii of steady heal conduction problems usu ally results in a system of iV algebraic equations in /V unknown nodal temperatures that need to be solved siiiiullaneously. When Af is small (such as 2 or 3), we can use the elementary elimination method to eliminate ail unknowns except one and then solve for that unknown (sec Example 5-1). The other unknowns are then determined by back substitution. When W is large, which is usually Uie case, the elimination luelliod is not practical and we need to use a more systematic approach that can be adapted to computers. 

Variables in the algebraic model can be identical to those in the approximated system of differential equations [228]. Another possibility is to approximate only a subset of the original variables, including the important variables and those which have the greatest influence on the important variables of the model. For example, a repro-model can be constructed which describes only the evolution of the slow variables [229]. This means an indirect utilization of the very different time-scales characteristic of combustion systems. In most applications of time-scale separation the basic problem is to derive expressions which describe the time derivative of slow variables as a function of the same slow variables. The algebraic equations, resulting from the repro-modelling, contain these dependencies and therefore constitute an implicit application of slow manifolds. 

When you are given a problem to solve, it often can be written as an algebraic equation. You can use letters to represent measurements or unspecified numbers in the problem. The laws of chemistry are often written in the form of algebraic equations. For example, the ideal gas law relates pressure, volume, amount, and temperature of the gases the pilot in Figure 10 breathes. The ideal gas law is written... 

Solution of a multiple-equilibrium problem requires us to develop as many independent equations as there are participants in the system being studied. For example, if we wish to compute the solubility of barium sulfate in a solution of acid, we need to be able to calculate the concentration of all the species present in the solution. There are five species [Ba ], [SOj"l, [HSOj], [H30" ], and [OH ]. To calculate the solubility of barium sulfate in this solution rigorously, it is then necessary to develop five independent algebraic equations that can be solved simultaneously to give the five concentrations. 

For unsteady problems the discretized algebraic equation, for the upwind scheme example, is generally written ... 

The examples are made with the Chemical Engineering addition to FEMLAB, version 3.1. Appendix F describes the finite element method in one dimension and two dimensions so you have some concept of the approximation going from a single differential equation to a set of algebraic equations. This appendix presents an overview of many of the choices provided by FEMLAB. Illustrations of how FEMLAB is used to solve problems are given in Chapters 9-11. Thus, you may wish to skim this appendix on a first reading, and then come back to it as you use the program to solve the examples. A more comprehensive account of FEMLAB is available in Zimmerman (2004). 

---