Numerical Solution of Equations

It will be assumed in discussing the solution of the differential equations relating conditions within a tubular reactor that the calculations are to be carried out with some sort of automatic digital computer. The problem is just manageable with a desk computing machine in the one-dimensional approximation when there is only one independent reaction, but even in this simplest case, this is an expensive way to work, particularly if a fairly large number of solutions is required. 

If the one-dimensional approximation is adequate, the problem is reduced to a routine integration of a set of ordinary differential equations. Procedures for integrating such sets of equations are given in standard works on numerical analysis (see, for example, H5, M2, and M3). The working equations for simple forward-difference equations are given in the next paragraph. 

It must be remembered that heat capacity, heat of reaction, density, and the rates are, in general, variables in these equations, being known functions of the dependent variables. 

When the cooling medium is a boiling liquid, the dependence of its temperature on the length of the tube must be calculated from the conditions in the pool of liquid. When it is a circulating liquid, flowing concurrently with the reacting fluid, its temperature can be calculated from a heat balance. This heat balance is expressed by the equation 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem, 

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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... 

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... 

Note that in the numerical solution of equation 22.2-19, /B must be less than 1 to avoid ln(0) in the use of E-Z Solve, an initial guess of 0.5 is entered, and the upper and lower limits may be set to 0 and 0.9999, respectively. Since upper and lower limits must be specified, a user-defined function is not applicable to this case. 

Ve have obtained numerical solutions of Equations 1-5, with boundary values which satisfy Equations 7-9 for a wide range of state points. At each state the pressure tensor was computed from Equation 10 and used to evaluate Equations 12 and 13 and Equations 14-17. 

The band profile obtained as a numerical solution of Equation 10.10 gives the concentration distribution as a function of the reduced time at the column end, i.e., at location x= 1, regardless of the column length. The band profile depends only on the column efficiency, the boundary conditions, the phase ratio, and the sample size (which is part of the boundary conditions). The mobile phase velocity has been eliminated from the mass balance equation and the apparent axial dispersion coefficient has been replaced by the plate number. 

Some examples of equation (2.2) numerical solutions are shown in Fig. 2.1. It should be emphasized here that numerical solutions of equation (2.2) exhibit a greater variety of shapes, peak positions, and magnitudes than shown. 

Figure 2.1 Numerical solutions of equation (2.2) for the case F, = 0.4eV and = 10 and 10 s (curves 1-4 and l"-4", respectively). Heating rate is 0.1, 0.3, 0.5, and 1.0K/s (cnrves 1-4, respectively). Also shown are the dielectric relaxation currents (maximum M2) for the same heating rates (curves l"-4", respectively AE = l.SeV, C = lO crn ).

Extraction calculations involving more than three components cannot be done graphically but must be done by numerical solution of equations representing the phase equilibria and material balances over all the stages. Since extraction processes usually are adiabatic and nearly isothermal, enthalpy balances need not be made. The solution of the resulting set of equations and of the prior determination of the parameters of activity coefficient correlations requires computer implementation. Once such programs have been developed, they also may be advantageous for ternary extractions,... 

Numerical solution of equations (13)—(19) for polypropylene extrusion was made in 29,34> using approximation of the flow function (flow curve) by a piecewise power function. To find the root of b(f, M) of Eq. (13), the authors used a formal search algorithm compiled as a standard program for computer M-20 (USSR). Figure 2 gives dependency of b/f upon M (M is the specific moment of a core s rotation, i.e., the moment related to the length of the channel). It can be seen in Fig. 2 that (b/f) is a strictly decreasing function. 

When a larger fraction of HOC was associated with the particulate phase, numerical solutions of Equation (6.132) and Equation (6.133) were required to predict PCB transformation rates. The instantaneous concentration of dissolved PCB was estimated by incorporating the terms S and OH into the rate constant expressions for Equation (6.133) and Equation (6.134). 

The numerical solution of Equation (2.67) determines the power density in combination with Equation (2.52) as well. 

Unfortunately, the Debye-Hiickel approximation (ztp < c. 25 mV) is often not a good one in the treatment of colloid and surface phenomena. Unapproximated, numerical solutions of equation (7.11) have been computed.88... 

Figure 3. Kinetics of primary particle appearance calculated from full numerical solution of Equations 13-15 (Curve C), and from approximative Equation 16 (Curve A ) irreversible capture.

We use the simplest method (Eyler method, with the step of integration h) to write the algorithm for numerical solution of equations (3.52)... 

This equation reduces to equation (31) when r1 = t2 because then a = 0. It reduces also naturally to equation (37) when < t2. For the case t2 < it has to be compared with the numerical solution of equation (38). 

The results for a < 0 correspond to t2 < t,. The full lines present the numerical solution of equation (38), and the dotted lines present the calculations using equation (40). These curves clearly show, for low values of 8, an increase in tile rate of deposition. This increase is expected because of the higher rate of deposition on the covered surface and is more pronounced as the value of a approaches — 1, which means that i2 becomes much smaller than Tj. Three values of fi were used in the numerical solutions of equation (38) 2-25 x I0 2, 4-5 x 10-2 (which corresponds to the experiments of Weiss Harlos, 1972) and 9 x 10 2. All the curves corresponding to these... 

This may be written F(J) = B J(J+l)- DVJ2(J+1)2. If rotational levels are required to a greater degree of accuracy, higher terms may be included, but this is rarely justified by the experimental data. Thus, in principle, the energy levels for nuclear motion may be calculated exactly from a potential-energy curve, either by numerical solution of equation (1) for different values of J, or by numerical solution of the simplified rotationless equation,... 

If we replace p - pi by its uncertainty u in equation (1.11) then w2 is the minimum value of the mass fraction of the impurity that could be reliably detected. The uncertainty in p - p, includes the uncertainty in the measurement of p but also the uncertainty in pi, the density of the pure sample. The value of pi must be determined for a sample whose purity is known, by some independent means, with an accuracy considerably greater than the one being tested. An example using equation (1.11) and Table 3 is the determination of a small impurity of 1,2-diethylbenzene in 1,4-diethylbenzene. Table 3 contains numerical solutions of equation (1.11) at various densities and uncertainties. 

The signal Scoh represents a convolution integral of the intensity of the probing pulse oc EP(t — to) 2 with the molecular response the latter is governed by the autocorrelation functions 0Vib and 0r- Numerical solutions of Equations (2)-(5) are readily computed and will be discussed in the context of experimental results. 

Numerical solution of equations was performed for a differential bed model using the IMSL integration subroutine DIVPAG (method of Gear). 

Numerical solutions of equations allowing for the Coriolis coupling, give, instead of the factor 4.2, in equation (88), considerably lower values of 2.25-2.28 [84-8],... 

Numerical solutions of equation (10) subject to the boundary conditions (11) and (13) are available (see, for example ref. 4) and hence the self-consistent field V(r) in heavy positive ions is established. 

A description of pair substitution by a numerical solution of Equation 9.3, after appropriate modification, is always feasible. However, that frequently employed procedure has one major drawback it depends on many parameters, some of which are often not known very precisely. The usual remedy is to determine them by a multiparameter nonlinear fit, but the uniqueness of a many-component solution vector obtained in that way is questionable if the curves do not have very characteristic shapes, as in Fig. 9.4. As an alternative approach, one can exploit the fact that the same parameters are contained in the polarization intensities in the limits of no pair substitution and of infinitely fast pair substitution. For the system of Fig. 9.3, recasting the equations in terms of these experimental quantities leads to a closed-form expression that contains most parameters implicitly and only has a single adjustable parameter, namely, the rate constant of pair substitution divided by the intersystem... 

There are two points of interest associated with the numerical solution of equations (2) - (6) (i) the x-domain is infi-... 

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