Solution of the Governing Equations

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 assodated with the microscopic models of the inter-fadal 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. 

The semiconductor electrode is typically divided into three regions. Surface-charge and electron and hole-flux boundary conditions model the semiconductor-electrolyte interface. The region adjacent to the interface is assumed to be a depletion layer, in which electron and hole concentrations are negligible. The potential 

Integration of Poisson s equation in the depletion layer, for example, results in a depletion layer thickness W in terms of the voltage drop V across the layer  

This can also be written in terms of the charge q held within the space-charge region 

Analytic models of photoelectrochemical devices closely resemble models of solid-state solar cells (see, e.g.. Refs. 133-145). Several analytic current-voltage relationships have been derived which use the general approach described above and differ in their treatment of surface reactions and recombination within the depletion and neutral layers. The model of Gartner, developed for a p-n junction device, is commonly used in the analysis of photoelectrochemical devices. Recombination and thermal generation 

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The standard Galerkin technique provides a flexible and powerful method for the solution of problems in areas such as solid mechanics and heat conduction where the model equations arc of elliptic or parabolic type. It can also be used to develop robust schemes for the solution of the governing equations of... 

It is obvious that a strong coupling between the solution of the governing equations and the generation of the adaptive mesh is of prime importance for a case like this one, since a priori knowledge on how the solution will evolve is not available. 

Solution of the equations in (2.8-2.18) proceeds with an adaptive nonlinear boundary value method. The solution procedure has been discussed in detail elsewhere (10) and we outline only the essential features here. Our goal is to obtain a discrete solution of the governing equations on the mesh Af... 

With the continuous differential operators replaced by difference expressions, we convert the problem of finding an analytic solution of the governing equations to one of finding an approximation to this solution at each point of the mesh M. We seek the solution U of the nonlinear system of difference equations... 

Optimization strategies and a number of generalized limitations to the design of gas-phase chemiluminescence detectors have been described based on exact solutions of the governing equations for both exponential dilution and plug-flow models of the reaction chamber by Mehrabzadeh et al. [12, 13]. However, application of this approach requires a knowledge of the reaction mechanism and rate coefficients for the rate-determining steps of the chemiluminescent reaction considered. 

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... 

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.

The aim now is twofold finding a spatially homogeneous solution of the governing equations (for a given shear rate) and investigating the stability of this solution. In this section we will describe the general procedure and give the results in Sect. 3. 

Note that /ep in Eq. (5.238) is replaced with /Ep for Eq. (5.240), where /Ep is the heat generated by thermal radiation per unit volume and Qap is the heat transferred through the interface between gas and particles. Thus, once the gas velocity field is solved, the particle velocity, particle trajectory, particle concentration, and particle temperature can all be obtained directly by integrating Eqs. (5.235), (5.237), (5.231), and (5.240), respectively. Since the equations for the gas phase are coupled with those for the solid phase, final solutions of the governing equations may have to be obtained through iterations between those for the gas and solid phases. 

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. 

Solution of the Governing Equation for the First-Order Chemical Reaction A first-order chemical reaction can be represented as follows... 

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. 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

The major problem of Markov chains continuous in time and space is that the availability of analytical solutions of the governing equations, which depend also on the boundary conditions, is limited to simplified situations and for more complicated cases, niunerical solutions are called for. 

An example of the type of results that can be achieved with this approach is shown in fig. 12.20, which cleverly depicts four distinct features associated with the solution of a dendrite structure with fourfold symmetry. The upper left hand panel of the figure shows the nonuniform finite element mesh used in the solution of the governing equations. Contours of constant temperature are shown in the... 

Another important characteristic of the gas bubble is its response to a periodic oscillation of the ambient pressure / ,. For large-amplitude oscillations of the pressure, or for an initial condition that is not near a stable equilibrium state for the bubble, the response can be very complicated, including the possibility of chaotic variations in the bubble radius.22 However, such features are outside the realm of simple, analytical solutions of the governing equations, and we focus our attention here on the bubble response to asymptotically small oscillations of the ambient pressure, namely,... 

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... 

Clearly, the result (7 14) is a consequence of linearity alone and does not require a solution of the governing equations and boundary conditions. Of course, if we want to determine the actual value of the drag, we will have to solve the problem in either configuration (a) or (b). All that (7-14) tells us is that the drag in the two configurations is equal. 

To obtain a model of practical utility which still emphasizes the important characteristics of the RCFE 1n the high Pe limit, it is desirable to substitute a dispersion equation Involving the transverse average concentration for Equation 1. The y-dependent velocities are replaced by their transverse averages and the convective dispersion of solute associated with the crescent phenomenon is lumped into a lateral dispersion coefficient, K, which simplifies the analysis considerably and allows analytical solution of the governing equation,... 

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 ... 

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.

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