Numerical solutions to equations

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

The computing time required to evaluate Equation 4.19 in a Newton-Raphson iteration increases with the cube of the number of equations considered (Dongarra et al., 1979). The numerical solution to Equations 4.3 1.6, therefore, can be found most rapidly by reserving from the iteration any of these equations that can be solved linearly. There are four cases in which equations can be reserved ... 

Numerical solutions to Equations 22 and 23 have been obtained for various values of R,/ , i, and a2. The changes in ellipticity (R /R) developed by various draw ratios D = ai/a2 are shown in Figure 8 for several initial ellipse orientations p and an initial ellipticity R = 0.2. The curve corresponding to p = 0 is given by R /R = 1/D as can be readily obtained from Equations 14, 21, and 22. For all other orientations, that... 

Tafel lines, A vs. In I(y), give constant slopes at small polarizations, but the theoretical values of the relative exchange current densities (from the ordinate) are dependent on y. For large polarizations, we need numerical solutions to Equations 16.59 or 16.62 with fixed values of r2, rx, j0, Db C°, and x- Figure 16.9 shows some of the results. 

For the cases where the inverse of X X does not exist or if X X is ill-conditioned (that is, X X is nearly singular), there is always a numerical solution to Equations (3.27) and (3.29). However, this does not mean that this solution is always desirable from a statistical or practical point of view. Specifically, the estimated regression vector b tends to be uncertain because the solution is mostly governed by the noise part of the data. This can lead to high variances of predicted y values for new samples or objects. [Belsley et at. 1980],... 

The numerical solutions to Equations 2.18 through 2.20, along with a set of realistic deposition conditions (T,, aii=600K, Ts b=300K, Re =10), are plotted... 

Numerical solutions to equation (11.2.9) have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for... 

If filtration takes place at constant pressure then the equations presented for the tube press in Section 6.2.1.3 are also valid for the multi-element candle filter noting that = n,ndh where n, is the total number of candles in the filter. When filtration takes place at variable pressure then it is necessary to impose the pump curve characteristics to relate Ap and q and a numerical solution to equation (6.17) is generally required. For the special case of variable pressure/constant rate filtration with a positive displacement pump... 

Equation 4.127 well approximates the exact numerical solution to Equation 4.122 in... 

FIGURE 5.17 (a) Points the exact numerical solution to Equation 5.125. Line the fitting equation (Equation 5.126). (b) Real part of the CCL impedance at the point corresponding to parameter phf = ft//s (the junction of the semicircle with the straight HE line, see Eigure 5.16b). 

In this section, we obtain numerical solutions to Equation 7-14 for several different flow problems. For simplicity, we consider constant meshes, with lengths Ax and Ay in the x and y directions. The resolution achieved with such meshes systems near wells is limited. The programs are given for illustrative purposes, but the question of resolution will be addressed when we deal with curvilinear meshes. Now, the second derivative in Equation 7-3 applies to a function F(x) at x = x i, but P(x,y) depends on an additional y, say indexed by j. At any point (i,j), use of Equation 7-3 in both x and y directions with Equation 7-14 leads to the simple model (P -2P,+P. 

Chapter 8 Use of the Energy Balance in Reactor Sizing and Analysis APPENDIX 8-A NUMERICAL SOLUTION TO EQUATION (8-26)... 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

N concentrations. Years ago, this fact was useful since numerical solutions to ODEs required substantial computer time. They can now be solved in literally the blink of an eye, and there is little incentive to reduce dimensionality in sets of ODEs. However, the theory used to reduce dimensionality also gives global stoichiometric equations that can be useful. We will therefore present it briefly. 

Fortunately, it is possible to develop a general-purpose technique for the numerical solution of Equation (3.9), even when the density varies down the tube. It is first necessary to convert the component reaction rates from their normal dependence on concentration to a dependence on the molar fluxes. This is done simply by replacing a by and so on for the various... 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

The first step in developing the numerical method is to And a formal solution to Equation (8.63). Observe that Equation (8.63) is variable-separable ... 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz. ... 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

One of the most important issues is concerned with a smaller number of the iterations performed in the numerical. solution of equations with variable coefficients. It was shown in Section 7 that the number of the iterations required during the course of ATM is proportional to where Cj and are the smallest and the greatest values of coefficients, respectively. The operator R in question can be put in correspondence with the operator A with variable coefficients such that... 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

Although Hq. (152) can in principle be solved by the development of y(x) in a power series, the periodicity of the argument of cosine, namely, 2jc = Na complicates the problem. The most important application of Mathieu s equation to internal rotation in molecules is in the analysis of the microwave spectra of gases and vapors. The needed solutions to equations such as Eq. (152) are usually obtained numerically. 

For a general introduction to some of the techniques employed to solve difference equations of the types encountered in chemical engineering, consult the text by Lapidus (116). Detailed numerical examples of one method of numerical solution to the two-dimensional reactor problem are contained in the texts of Smith (117) and Jensen and Jeffreys (118). 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

We consider two cases, one with a higher Peclet number than the other. Disper-sivity tt[, in the first case is set to 0.03 m in the second, it is 3 m. In both cases, the diffusion coefficient D is 10-6 cm2 s-1. Since Pe L/oti., the two cases on the scale of the aquifer correspond to Peclet numbers of 33 000 and 330. We could evaluate the model numerically, but Javandel el al. (1984) provide a closed form solution to Equation 20.25 that lets us calculate the solute distribution in the aquifer... 

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