Numerical Integration Results

Equations (9.11, 9.13,9.14) have been numerically integrated for a spherical bead of gel using the above boundary conditions. Here, we only present a short summary. More details may be found in Refs. [29-31]. 

The values of the following parameters were kept fixed  

We have checked that, with this model and in the range of A values used in the computations that follow, these conditions give rise to oscillations of u and t in a sphere of nonresponsive gel. 

The experimental determination of such profiles presents numerous challenges already found in the study of reaction-diffusion patterns in inert gels. In active gels, the problem is even more acute as shown in the experimental Section 9.5. 

Another set of simulations was carried out, at fixed /3, while varying the concentrations of the substrates. For instance, it is found that u(0, t) increases monotonously with A. This induces a corresponding enlargement of the maximum and mean radii of the sphere. On the other hand, it is well known, for a BZ system, that the dependence of the period of oscillations, T, on the concentration of the substrates (bromate ion, malonic acid, and free protons) provides useful information [1]. In our simulations, we have obtained that T increases with the decrease in concentrations of the substrates following power laws with negative exponents. Similar behaviors are obtained for a sphere of passive gel, where 

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It is also possible to integrate Eq. (1-29) directly by numerical means and to subtract the result from 1.0 to obtain the proportion of particles with speeds in excess of v /v p. In this project we shall use numerical integration of G v)dv over various intervals to obtain /(v) as a function of v /vmp. Because v p = [Eq. (1-28)], Jq G(v)dv can be written... 

The most popular of the SCRF methods is the polarized continuum method (PCM) developed by Tomasi and coworkers. This technique uses a numerical integration over the solute charge density. There are several variations, each of which uses a nonspherical cavity. The generally good results and ability to describe the arbitrary solute make this a widely used method. Flowever, it is sensitive to the choice of a basis set. Some software implementations of this method may fail for more complex molecules. 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

To measure a residence-time distribution, a pulse of tagged feed is inserted into a continuous mill and the effluent is sampled on a schedule. If it is a dry miU, a soluble tracer such as salt or dye may be used and the samples analyzed conductimetricaUy or colorimetricaUy. If it is a wet mill, the tracer must be a solid of similar density to the ore. Materials hke copper concentrate, chrome brick, or barites have been used as tracers and analyzed by X-ray fluorescence. To plot results in log-normal coordinates, the concentration data must first be normalized from the form of Fig. 20-15 to the form of cumulative percent discharged, as in Fig. 20-16. For this, one must either know the total amount of pulse fed or determine it by a simple numerical integration... 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-(ej-48. pp. 76-81 some numerical calculations. Farrow and Edelson and Porter... 

These calculations employ a grid of points in space in order to perform the numerical integration. Grids are specified as a number of radial shells around each atom, each of which contains a set number of integration points. For example, in the (75,302) grid, 75 radial shells each contain 302 points, resulting in a total of 22,650 integration points. 

The semiempirical methods represent a real alternative for this research. Aside from the limitation to the treatment of only special groups of electrons (e.g. n- or valence electrons), the neglect of numerous integrals above all leads to a drastic reduction of computer time in comparison with ab initio calculations. In an attempt to compensate for the inaccuracies by the neglects, parametrization of the methods is used. Meaning that values of special integrals are estimated or calibrated semiempirically with the help of experimental results. The usefulness of a set of parameters can be estimated by the theoretical reproduction of special properties of reference molecules obtained experimentally. Each of the numerous semiempirical methods has its own set of parameters because there is not an universial set to calculate all properties of molecules with exact precision. The parametrization of a method is always conformed to a special problem. This explains the multiplicity of semiempirical methods. 

A numerical integration scheme has produced the following results ... 

These can be solved by classical methods (i.e., eliminate Sout to obtain a second-order ODE in Cout), by Laplace transformation techniques, or by numerical integration. The initial conditions for the washout experiment are that the entire system is full of tracer at unit concentration, Cout = Sout = L Figure 15.7 shows the result of a numerical simulation. The difference between the model curve and that for a normal CSTR is subtle, and would not normally be detected by a washout experiment. The semilog plot in Figure 15.8 clearly shows the two time constants for the system, but the second one emerges at such low values of W t) that it would be missed using experiments of ordinary accuracy. 

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... 

This expression was numerically integrated and regressed to the experimental data using a nonlinear least-squares fitting procedure. The resulting integrated equation is... 

When all else fails, recourse to numerical methods is indicated. Some of the classic methods of numerical integration are described in Chapter 13. However, it should be emphasized that numerical methods are to be used as a last resort Not only are they subject to errors (often not easily evaluated), but they do not yield analytical results that can be employed in fbrther derivations (see p. 43). 

Equation (3-124) has been integrated numerically, and results can be presented. Figure 3.41 is a plot of liquid holdup EL versus W+ for various values of the parameter R+. 

Using the values of /CA0 determined from this relation, it is possible to integrate equation F numerically. The result is... 

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