Accuracy of numerical solutions

The gain in accuracy provided by refining the step h is limited by requirements of economy. Such an approach is equivalent to minimizing the execution time necessary in this connection in obtaining the solution. But if the solution of the original problem u and / both are smooth functions of X, the accuracy of numerical solution can be increased by performing calculations for the same problem (12) on a sequence of grids, , < h ... 

Besides examining these properties of numerical methods, specific efforts need to be made to assess the accuracy of numerical solutions of flow processes. Various types of errors and possible ways of estimating and controlling these errors are discussed... 

There exists only a few analytical solutions of the PB which have a considerable significance for testing the accuracy of numerical solutions. One of the solutions was derived for a single plate (more precisely, half-space) immersed in an symmetric electrolyte. The electric potential distribution in this case is given by the formula derived by Vervey and Overbeek [24] ... 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

The above analytic solution has two applications (i) to investigate the concentration evolution under the special conditions given above, and (ii) to check the accuracy of numerical programs. One application is given in Example 2-1. 

In order to determine the accuracy of the solution proposed in Eq. (3.101) for the case of a microdisc electrode, in Fig. 3.13 numerical results are compared with this equation and also with the Oldham Eq. (3.95). Fully reversible, c[Jisc ss = 1000 /4, quasi-reversible, cj[lsc ss = nj4, and fully irreversible, cj[lsc ss = 0.001 /4, heterogeneous kinetics were considered under steady-state behavior. It is seen that, for fully reversible kinetics, both equations give almost identical results which are in good agreement with the simulated values. As the kinetics becomes less reversible, however, the results given by the two equations diverge from each other, with the simulated result lying between them. The maximum error in the Oldham equation is 0.5 %, and for Eq. (3.101), the maximum error is 3.6 %. 

Some graphic examples justifying these statements are presented in Figures 4.12 and 4.13, where the components of two nonlinear complex susceptibilities are plotted as the functions of the parameter ct. For a given sample, a in a natural way serves as a dimensionless inverse temperature. In those figures, the solid lines correspond to the above-proposed asymptotic formulas where we retain the terms, including a 3. The circles show the results of numerical solutions obtained by the method described in Ref. 67. Note that even at ct 5 the accuracy is still rather high. 

A30/Ai = 0, k /k2 = 1. This problem was first analyzed numerically by Brian and Beaverstock (12). Approximate analytical solution to this problem is also available (Ramachandran (13)). Figure 3 has been mainly presented to test the method for multi-step reaction scheme and hence detailed comparison of the accuracy of the solution and the influence of the number of collocation points was not done for this case. 

However, numerical integration of the ordinary differential equations also possesses advantages. Foremost among these are the inherently greater accuracies of the solutions, when they can be obtained one-dimensional... 

Discussion We used a relatively crude numerical model to solve this problem to keep the complexities at a manageable level. The accuracy of Ihe solution obtained can be improved by using a finer mesh and thus a greater number of nodes. Also, v/hen radiation is involved, it is more accurate (but more laborious) to determine the heat fosses for each node and add them up instead of using the average temperature. 

This chapter starts with an introduction to modeling of chromatographic separation processes, including discussion of different models for the column and plant peripherals. After a short explanation of numerical solution methods, the next main part is devoted to the consistent determination of the parameters for a suitable model, especially those for the isotherms. These are key issues towards achieving accurate simulation results. Methods of different complexity and experimental effort are presented that allow a variation of the desired accuracy on the one hand and the time needed on the other hand. Appropriate models are shown to simulate experimental data within the accuracy of measurement, which permits its use for further process design (Chapter 7). Finally, it is shown how this approach can be used to successfully simulate even complex chromatographic operation modes. 

In addition, an exact algebraic solution for the incubation period of bottom-up filling has been described for the situation when catalyst is pre-adsorbed and negligible consumption occurs during subsequent metal deposition [344], As with the string model, this analytical solution captures the essential aspects of the shape transitions that accompany the trench superfilling CEAC dynamic. The analytical solution also provides a metric for evaluating the accuracy of numerical models and associated computer codes. 

Chan, B.K.C. Chan, D.Y.C. Electrical doublelayer interaction between spherical colloidal particles an exact solution. J. Colloid Interface Sci. 1983, 92 (1), 281-283 Palkar, S.A. Lenhoff, A.M. Energetic and entropic contributions to the interaction of unequal spherical double layers. J. Colloid Interface Sci. 1994, 165 (1), 177-194 Qian, Y. Bowen, W.R. Accuracy assessment of numerical solutions of the nonlinear Poisson-Boltzmann equation for charged colloidal particles. J. Colloid Interface Sci. 1998, 201 (1), 7-12 Carnie, S.L. Chan, D.Y.C. Stankovich, J. Computation of forces between spherical colloidal particles Nonlinear Poisson-Boltzmann theory. J. Colloid Interface Sci. 1994, 165 (1), 116-128 Stankovich, J. Carnie, S.L. Electrical double layer interaction between dissimilar spherical colloidal particles and between a sphere and a plate nonlinear Poisson-Boltzmann theory. Langmuir 1996,12 (6), 1453-61. 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

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