Numerical Solution of the Governing Equations

Equation (5.23) is solved numerically either by linearizing the nonlinear term and discretizing the resulting equation or by discretizing Equation (5.24) as an extension of the mixing problem. In the former case, the resulting equation is 

As in the case of decretizing fluid flow equations. Equation (5.27) requires the appropriate upwinding and as a result Equation (5.28) represents the upstream donor cell difference whereby 7=1 gives a full upstream effect. For 7 = 0 the equation becomes numerically unstable (Anderson et al., 1984). 

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Table III shows that the experimental and predicted evaporation rates are in good agreement at all beam intensities. There is some inconsistency at the highest power levels. It was difficult to maintain the droplet in the center of the laser beam at the highest power level, and the measured evaporation rate is somewhat low as a result of that problem. Additional computations demonstrate that the predicted evaporation rate is quite sensitive to the choice of the imaginary component of N, so the results suggest that this evaporation method is suitable for the determination of the complex refractive index of weakly absorbing liquids. For strong absorbers, the linearizations of the Clausius-Clapeyron equation and of the radiation energy loss term in the interfacial boundary condition may not be valid. In this event, a numerical solution of the governing equations is required. The structure of the source function, however, makes this a rather tedious task.

Approximate solutions for the two limiting cases discussed above can be obtained (see below). However, most real flows are not well described by either of these two limiting solutions. For this reason, a numerical solution of the governing equations must usually be obtained. To illustrate how such solutions can be obtained, a simple forward-marching, explicit finite-difference solution will be discussed here. 

Figures 21 and 22 show the normalized pressured drop estimated by equation 128 for a packed D = 5.588 mm, ds = 3.040 mm, and e = 0.5916. The experimental data are taken from Fand and Thinakaran (92). We can observe that the approximate solution, equation 128, predicts fairly well the experimental results and is very good in representing the exact numerical solution of the governing equations. For clarity, Figure 21 is an expanded region for the small Rem values. Even with this scale, we observe that the approximate solution is very close to the exact numerical solution.

The numerical solution of the governing equations is achieved by a combination of the finite... 

The computational approach is based on a colocated, finite-volume, energy-con-serving numerical scheme on unstructured grids [10] and solves the low-Mach number, variable density gas-phase flow equations. Numerical solution of the governing equations of continuum phase and droplet phase are staggered in time to maintain time-centered, second-order advection of the fluid equations. Denoting the time level by a superscript index, the velocities are located at time level f and... 

Any solution of equation (6.3.36) or (6.3.37) requires detailed information about the flow field around the filter bed coUector/fiber. The flow field may be available via the three velocities Vx, Vy, (or v, Vg, Vz, etc.) or via the stream function ip, if it can be assumed that the particle motion does not affect the flow field. The solution of such a problem generally requires a numerical solution of the governing equations (e.g. equation (6.2.6b)) for the chosen velocity field around the fiber in the depth filter. 

The flow becomes highly complex in a spiral-wound module containing a feed-side spacer screen. Numerical solutions of the governing equations incorporating most of these complexities have been/are being implemented (Wiley and Fletcher, 2003) using computational fluid dynamics models (see Schwinge et al. (2003) for the complex flow patterns in a spacer-filled channel). 

IDA and MSA procedures require the numerical solution of the governing equations of motions of the structural model together with constitutive equations in each time step for each response history analysis. As a consequence, it is time-consuming and computationally expensive. Thus, there is a need for simple to apply, but yet sufficient accurate methods to predict the global collapse of MDOF stmctures under seismic excitations (Lignos and Krawinkler 2012). 

The flow field of the impacting droplet and its surrounding gas is simulated using a finite-volume solution of the governing equations in a 3-D Cartesian coordinate system. The level-set method is employed to simulate the movement and deformation of the free surface of the droplet during impact. The details of the hydrodynamic model and the numerical scheme are described in Sections... 

Equations (50) and (51) are in good agreement with the empirical correlations suggested by Churchill and Ozoe [17] to represent their numerical solution of the governing transport equations. Tables I and II compare the results. 

For nonlinear systems the solution of the governing equations must generally be obtained numerically, but such solutions can be obtained without undue difficulty for any desired rate expression with or without axial dispersion. The case of a Langmuir system with linear driving force rate expression and negligible axial dispersion is a special case that is amenable to analytical solution by an elegant nonlinear transformation. 

There are a number of model codes that have been developed that allow users to conveniently apply analytical and numerical solutions of the governing transport equations to various contamination scenarios. The following discussion will look at the capabilities and limitations of some of these codes. We will restrict our discussion to the more popular numerical codes that can simulate 2- or 3-D contaminant transport, as analytical models and 1-D solutions are usually too simplified for application to real contamination scenarios. 

Theoretical criteria normally contain an explicit expression of the intrinsic chemical rate, and optionally also a measured value of the observed reaction rate. Thus, these criteria are useful only when the intrinsic kinetics are available, and one is, for example, interested in whether or not transport effects are likely to influence the performance of the catalyst as the operating conditions are changed. If it is not possible to generate a numerical solution of the governing differential equations, either due to a lack of time or to other reasons, then the use of theoretical criteria will not only save experimental effort, but also provide a more reliable estimation of the net transport influence on the observable reaction rate than simple experimental criteria can give, which do not contain any explicit... 

The obvious question is whether we can say anything about the behavior of the air hockey system when Re is not small. A direct solution of the governing equations, (5-134)—(5—137), would require a numerical approach because of the nonlinearity of (5-135), if no additional approximation were made. However, as indicated earlier, we may expect that for large-enough values of the pressure difference, p R— p a, it may be possible to neglect the radial variation in the blowing velocity under the disk. Indeed, referring back to the analysis of the present subsection, we see that... 

Schematic concentration profiles for the general case are shown in Figure 14.4.6. The limiting current with all of the processes contributing can be obtained only by numerical solution of the differential equations governing the system. However, in most experimental systems only one or two of the processes will be important. Which limiting case or subcase applies (i.e., which factors are rate-determining) is determined by the relative magnitudes of the characteristic currents, or more explicitly, by the ratios / // and ip/i, ...

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 ... 

Quantitative optimization or prediction of the performance of photoelectrochemical cell configurations requires solution of the macroscopic transport equations for the bulk phases coupled with the equations associated with the microscopic models of the interfacial regions. Coupled phenomena govern the system, and the equations describing their interaction cannot, in general, be solved analytically. Two approaches have been taken in developing a mathematical model of the liquid-junction photovoltaic cell approximate analytic solution of the governing equations and numerical solution. 

This transfer function can now be studied in the frequency domain. It should be noted that these are linear partial differential equations and that the process of frequency domain analysis is appropriate. The range of values of e = 0.01 to 0.2, M = 5 to 20, and R = 0.75 have been established [Grant and Cotton, 1991] in a numerical finite difference solution of the governing equations. Having established these values the frequency response can be completed. 

A short chapter is dedicated to the solution of the governing equation. Since a detailed description of numerical methods dealing with the solution of the governing equations can be found in many textbooks only the underlying principles will be explained. 

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