A Numerical Solution to Equation

The sensitivity of the solution to step size must now be explored. Choose a step size that is half of the original, i.e., choose h = 0.05 h, and repeat the calculation. 

The value of a a at r = 1.5 h changed in the fifth significant figure when the step size was changed from 0.10 to 0.050 h. The numerical solution does not depend on step size. 

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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],... 

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... 

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

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 ... 

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. 

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. 

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 ... 

Perhaps even more important is die fact that LEM does not require a numerical solution to die Navier-Stokes equation. Indeed, even a three-dimensional diffusion equation is generally less computationally demanding than the Poisson equation needed to find die pressure field. 

The first group of methods then manipulate a very small subset of vector elements Vi at a time, and a direct method continually updates the affected elements rj. Such methods are collectively known as relaxation methods, and they are primarily used in situations where, for each elements >< to be changed, the set of affected rjt and the matrix elements Ay, are immediately known. This applies in particular to difference approximations and also to the so-called Finite Element Method for obtaining tabular descriptions of the wave function, i.e. a list of values of the wave function at a set of electron positions. (Far some reason, such a description is commonly referred to as a numerical solution to the Schrodinger equation). Relaxation methods have also been applied to the Cl problem in the past, but due to their slow convergence they have been replaced by analytical methods. Even for numerical problems, the relaxation methods are slowly yielding to analytical methods. 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

A number of approximate analytical solutions to Eqs. (10.205) and (10.206) have been obtained. However, those solutions are of limited applicability and it is now usual to obtain a numerical solution to the equations. A very simple finite-difference numerical procedure, basically identical to that used before to solve for the flow in a fluid-filled enclosure, will be discussed here. 

Since the equations for n (t/i) and pn (z) do not readily yield an analytic solution, we obtain a numerical solution. To simplify the numerical computations, we deal with cumulative distributions of the tails,5 that is, with the fractions of polymer having molecular weights above specified levels ... 

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... 

A numerical solution to Eqs. (4-91) through (4-93) is given by Mhaskar.26 The approximate Galerkin and perturbation solutions to these equations for the cocurrent-flow case are recently given by Szeri et al.36 The method of Hlavacek and Hofmann16 outlined for the slow reaction can also be used for this case in a similar way. In many practical cases, Pec is taken to be infinity (i.e the gas phase is assumed to be in plug flow). 

The theoretical prediction of the leakage current requires the solution of Laplace s equation and a closed-form solution for an arbitrary geometry is not straight-forward. The finite-element method (1 0) can be used to obtain a numerical solution to Laplace s... 

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. 

Even in the simple case where only two components are present, many reactions may take place. It is our unhappy experience that taught us that in this realm of fast diffusion-controlled reactions no reaction can a priori be ignored or assumed to be too slow to affect the observed signals. To avoid such negligence, we employ a computer program that produces a numerical solution to the differential rate equations pertinent to the reaction system. This program reconstructs the time-dependent variable and allows to match the measured values with the computed ones. By systematic varying of the various rate constants of the partial reactions, the computed curve can be shaped to superimpose the experimental results to the satisfaction of the experimentalist (or the referee). 

This method, which is also called the Newton-Raphson method, is an iterative procedure for obtaining a numerical solution to an algebraic equation. An iterative procedure is one that is repeated until the desired degree of accuracy is attained. The procedure is illustrated in Fig. 4.9. We assume that we have an equation written in the form... 

Use Mathematica to obtain a numerical solution to the differential equation in the previous problem for the range 0 < x < 10 and for the initial condition y(0) = 1. Evaluate the interpolating function for several values of x and make a plot of the interpolating function for the range 0 < x < 10. 

For the solution of the governing equations, an iterative scheme is followed [69], After determining the intensity at a cell center (see Eq. 7.141), the intensity downstream of the surface element can be determined via extrapolation using Eq. 7.140. The central differencing used in Eq. 7.140 may result in negative intensities, particularly if the change in the radiative field is steep. A numerical solution to this problem was recommended by Truelove [67], where a mixture of central and upward differencing is used ... 

The solution to equation 36 can be found in numerous forms in the literature. Earlogher (44) writes a general solution to equation 36 in terms of the initial pressure, ph for a reservoir of porous rock with a single well with source Q as... 

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