Exact numerical solution

Two-dimensional semiclassical studies described in section 4 and applied to some concrete problems in section 6 show that, when no additional assumptions (such as moving along a certain predetermined path) are made, and when the fluctuations around the extremal path are taken into account, the two-dimensional instanton theory is as accurate as the one-dimensional one, and for the tunneling problem in most cases its answer is very close to the exact numerical solution. Once the main difficulty of going from one dimension to two is circumvented, there seems to be no serious difficulty in extending the algorithm to more dimensions that becomes necessary when the usual basis-set methods fail because of the exponentially increasing number of basis functions with the dimension. 

To compare the results of the correlation presented in this article and an exact numerical solution, let us consider the case where air with a wet-bulb temperature of 70°F is used to cool water from 120°F to 80°F. Table 1 summarizes the results for different air-to-water flow rate ratios. 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Butler and Pillingf) calculated an exact numerical solution of the diffusion equation. They showed that the interpolation formula proposed by Gosele et al.e) reproduces the numerical solution with high precision. 

Finally, as described in Box 4.1 of Chapter 4, an exact numerical solution of the diffusion equation (based on Fick s second law with an added sink term that falls off as r-6) was calculated by Butler and Pilling (1979). These authors showed that, even for high values of Ro ( 60 A), large errors are made when using the Forster equation for diffusion coefficients > 10 s cm2 s 1. Equation (9.34) proposed by Gosele et al. provides an excellent approximation. 

The ratio of the rotational flows indicated that the diameter of the extruder is not a factor for the deviation shown in Fig. 7.16. That is, for this channel with a small H/Wthe simplified analysis produces less than a 10% error when compared to the exact numerical solution. Thus, the rotational flow rate can be calculated quite reliably using the simple generalized Newtonian method at these conditions. 

There is no exact numerical solution to Eq. (4.89) when both x and y > 0. In practice, therefore, one variable mnst be kept constant (or zero), while the value of the other changes. This was described for Example 16. Thns, if x is kept constant, the double polynomial is reduced to a simple one... 

Fig. 3.10 External Sh for spheres in Stokes flow (1) Exact numerical solution for rigid and circulating spheres (2) Brenner (B6) rigid sphere, Pe 0, Eq. (3-45) (3) Levich (L3) rigid sphere, Pe 00, Eq. (3-47) (4) Acrivos and Goddard (A 1) rigid sphere, Pe oo, Eq. (3-48) (5) Approximate values fluid spheres.

Fig. 3.11 Local Sherwood number for rigid sphere in Stokes flow (1) Exact numerical solution Pe = 10 (2) High Pe asymptotic solution (L3) Pe = 10 (3) Low Pe asymptotic solution (A2) Pe = 0.1.

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

Figure 4.38. Sabatier volcano-curve The limiting case of the exact numerical solution of the microkinetic Model 1.

This deviates by less than 0.5% from the exact numerical solution at Sc = 100 [23]. Newman [24] proposed another expression... 

The complex phase shift can be obtained from exact numerical solution of the radial Schrodinger equation.2 The following quantities can immediately be given in terms of 8r The differential elastic cross section in the center-of-mass system... 

The overall clearance of hemofiltration is more difficult to calculate than the diffusive clearance or the HF clearance, as it combines diffusive and convective transfers. An approximate equation for this clearance, obtained from an exact numerical solution has been given by Jaffrin et al. [12] as... 

Figure 4.26 presents a selection of cross-sections of the 3D plot of R2 where we compare the exact (numerical) solution with the effective time approximation (4.290). In the low-temperature range, the deviations, although not resolved in the graph, are inevitable. But they are not the main issue of the present study. As to the quadratic SR proper, the agreement again is very good. 

Fig. 12.4 Relation between the equilibrium pressure Peq and the normalized radius r of the afferent arteriole for different values of the muscular activation level [f. The solid curves represent our analytical approximation and the dashed curves represent the exact numerical solution. The area bounded by dotted lines corresponds approximately to the regime of operation for the model.

Obviously, the use of Fig. 7.11 with the generalized Thiele modulus as defined by 7.118 requires a knowledge of the rate equation in order to be able to calculate the integral in the denominator. For partial reaction orders varying from one-half to three the deviations from the exact numerical solution are limited to 15%. These deviations are highest at Thiele moduli around one. 

Exact numerical solutions of the full colloidal electrohydrodynamic problem have appeared in recent years. For computational convenience, the numerical schemes treat as an independent variable the user varies until the predicted and measured values of n agree. The earliest numerical solutions [132] were hampered by convergence difficulties at relatively low values of . O Brien and White [133] resolved these numerical problems, and their solution is widely used today. Some recent publications [134,135] document the evolution of analytical and numerical solutions of the colloidal electrohydrodynamics problem and report numerical solutions for particle mobility, suspension conductivity, and suspension dielectric permittivity for both constant and oscillatory applied electric fields. 

We compare the exact numerical solution to the Poisson-Boltzmann equation (6.6) and the approximate results, Eq. (6.37) for case 1 (low surface charge density case) and Eq. (6.50) for case 2 (high surface charge density case) in Fig. 6.3, in which the scaled surface potential jo = zeij/JkT is plotted as a function of the scaled... 

Nearly exact numerical solutions of the Smoluchowski equation show that for the Maier-Saupe potential, A < 1 when S = S2 > 0.524. For the Onsager potential, A < 1 for all values of the order parameter within the nematic range. Values of A for the Onsager potential are plotted in Fig. 11-18. 

We first discuss the exact numerical solution then we illustrate what we mean by making approximations. In the next section we discuss a method of graphical representation that is expedient in making such calculations. 

Exact Numerical Solution A numerical approach may start out by eliimi-nating [OH ] in equations i and iva. After this substitution we have... 

As in Example 3.3a, the exact numerical solution would lead first to... 

The temperature increase (d i) 1 /dt]+ )t(+=0 is, as Fig. 3.46 also shows, a function

Prandtl number Pr. The numerical results from Ostrach have been reproduced by Le Fevre [3.49] through an interpolation equation of the following form, which deviates from the exact numerical solution by no more than 0.5 % ... 

With the advent of fast computers, more exact numerical solutions have appeared. An original numerical solution of the equilibrium-dispersive model was developed [46], applied first to the prediction of band profiles in gas chromatography and later adapted to liquid chromatography [47,48]. It is used to predict the band profiles of large size samples. It requires prior determination of the column HETP imder linear conditions at the selected flow velocity, the column void volume, the extra-column volume, and the adsorption isotherm [48]. Other similar algorithms are available, and we now give a general presentation. 

Figure 5. The tunnel terms at il 5 meV, kuT = 0.5 meV, Uc = 50 meV. The four curves correspond to V = 0, 25, 50, 75, 100 meV. Thin lines refer to the exact numerical solution, dotted lines to Eq. (32) without logaritmic correction.

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