Numerical solutions full equations

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.

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... 

As with the two-dimensional workbench problem, the numerical solution of this problem can be found by solving the full turbulent fluid flow equations using the methods described in Chapter 13. 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

The approach has proven to be advantageous in comparison with the conventional" method. Because the reduced equation is a special form of the Reynolds equation, a full numerical solution over the entire computation domain, including both the hydrodynamic and the contact areas, thus obtained through a unified algorithm for solving one equation system. In this way, both hydrod5mamic and con-... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

This problem was first approached in the work of Denisov [59] dealing with the autoxidation of hydrocarbon in the presence of an inhibitor, which was able to break chains in reactions with peroxyl radicals, while the radicals produced failed to contribute to chain propagation (see Chapter 5). The kinetics of inhibitor consumption and hydroperoxide accumulation were elucidated by a computer-aided numerical solution of a set of differential equations. In full agreement with the experiment, the induction period increased with the efficiency of the inhibitor characterized by the ratio of rate constants [59], An initiated inhibited reaction (vi = vi0 = const.) transforms into the autoinitiated chain reaction (vi = vio + k3[ROOH] > vi0) if the following condition is satisfied. 

Full numerical solution of the above equations involves many detailed assumptions and it is not always easy to visualize the effects of these assumptions on the outcome. Therefore it is useful to consider the much simpler approach of the next section. 

Thus, analytic solutions for flow around a spherical particle have little value for Re > 1. For Re somewhat greater than unity, the most accurate representation of the flow field is given by numerical solution of the full Navier-Stokes equation, while empirical forms should be used for C. These results are discussed in Chapter 5. 

At larger Re and for more marked deformation, theoretical approaches have had limited success. There have been no numerical solutions to the full Navier-Stokes equation for steady flow problems in which the shape, as well as the flow, has been an unknown. Savic (S3) suggested a procedure whereby the shape of a drop is determined by a balance of normal stresses at the interface. This approach has been extended by Pruppacher and Pitter (P6) for water drops falling through air and by Wairegi (Wl) for drops and bubbles in liquids. The drop or bubble adopts a shape where surface tension pressure increments, hydrostatic pressures, and hydrodynamic pressures are in balance at every point. Thus... 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

Equation 16 tends to underestimate the number of particles except during the earliest few seconds of reaction, but serves as an extremely useful predictor for assessing the effect of experimental variables on the number of primary particles formed as a function of time. In Figure 3 are shown some calculations for styrene polymerization in which results from this approximative equation (curves A) are compared to those for the full numerical solution (curves C) at two values for jcr (30). It can be seen that when the oligoradical solubility is reduced (jcr = 10- 5), the rate of nucleation and final number of particles are greatly increased. This is, of course, in the absence of change in any other variable. 

The integrated continuity equation is a weaker form of the full continuity equation. This is noticed in numerical solutions of mold filling problems, where continuity is never fully satisfied. However, this violation of continuity is insignificant and will not hinder the solution of practical mold filling problems. The integrated continuity equation reduces to... 

Figure 3.31 As (due to orientational response of aqueous solvent) versus e, calculated for ET in a large binuclear transition metal complex (D (Ru2+/3+) and A (Co2+/3+) sites bridged by a tetraproline moiety) molecular-level results obtained from a nonlocal polarization response theory (NRFT, solid lines) continuum results are given by dashed lines, referring to numerical solution of the Poisson equation with vdW (cont./vdW) and SAS (cont./SAS) cavities, or as the limit of the NRFT results when the full k-dependent structure factor (5(k)) is replaced by 5(0) 5(k) for bulk water was obtained from a fluid model based on polarizable dipolar spheres (s = 1.8 refers to ambient water (square)). For an alternative model based on TIP3 water (where, nominally, 6 = ), ambient water corresponds to the diamond. (Reprinted from A. A. Milishuk and D. V. Matyushov, Chem Phys., 324, 172. Copyright (2006), with permission from Elsevier).

This chapter has mainly been devoted to the solution of the boundary layer form of the governing equations. While these boundary layer equations do adequately describe a number of problems of great practical importance, there are many other problems that can only be adequately modeled by using the full governing equations. In such cases, it is necessary to obtain the solution numerically and also almost always necessary to use a more advanced type of turbulence model [6],[12],[28],[29]. Such numerical solutions are most frequently obtained using the commercially available software based on the finite volume or the finite element method. 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

Regarding the physics of this issue, the task is to formulate the appropriate equations, describing convection, diffusion and conduction. The equations of motion are coupled to electric equations, like Poisson s law. The mathematical task is to solve the ensuing set of differential vector equations, some of which are non-linear, with the appropriate boundary conditions. General analytical solutions do not exist, but there are numerical solutions and good approximate equations for a number of limiting situations. Although the full mathematical anailysis is beyond the confines of the present chapter, we shall present their main elements because these are needed to understand the physics of the phenomena. 

We have remarked earlier that the treatment given above is based on an assumption for the case of that is, they are in an effective parallel combination. This is not strictly correct for a number of conditions, so the logarithmic potential-decay slopes in relation to Tafel slopes must be worked out from the full kinetic equations of Harrington and Conway (104) referred to earlier, based on the relevant mechanism of the electrode reaction. Numerical solution procedures, using computer simulation calculations, are then usually necessary for comparison with observed experimental behavior. 

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