Application of Finite Volume Method

For physically realistic and bounded results, it is necessary to ensure that all the coefficients of the discretization equation are positive. This requirement imposes restrictions on the time step that can be used with different values of 0. It can be seen that a fully implicit method with 0 equal to unity is unconditionally stable. Detailed stability analysis is rather complex when both convection and diffusion are present. In general, simplified criteria may be used when an explicit method is used in practical simulations  

Implementation of the basic steps of the finite volume method discussed above to solve the governing equations of a flow model requires development of a computer 

FIGURE 6.11 SIMPLE family of algorithms for unsteady flows. 

FIGURE 6.12 Outline of vessel (isometric and two-dimensional approximation). 

The next issue is the formulation of appropriate boundary conditions. The availability of suitable boundary conditions may also affect the decision concerning the extent of the solution domain. Obviously in practice, the inlet and outlet of any vessel will be connected to the associated pipe work. It is essential to decide the extent of the solution domain in such a way that it does not affect the simulated results. Generally for high velocity inlets, conditions in the process vessel do not affect the flow characteristics of the inlet pipe, and therefore it is acceptable to set the inlet boundary conditions right at the vessel boundary. More often than not, some piping at the outlet section may have to be considered if the outlet boundary condition is to be used. Alternatively, one may use constant pressure boundary conditions. Possible boundary conditions and solution domain are shown in Fig. 6.13. Before examining the influence of the solution domain on the simulated results, it is necessary to identify an adequate number of grids to resolve all the major features of the flow. 

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R.D. Lonsdale, R. Webster, "The application of finite volume methods for modelling three-dimensional incompressible flow on an unstructured mesh", in Proceedings of 6th International Conference on Numerical Methods in Laminar and Turbulent Flow, Swansea, United Kingdom, July 1989. 

The finite volume method, which returns to the balance equation form of the equations, where one level of spatial derivatives are removed is the method of choice always for the pressure equation and nearly always for the saturation equation. Commercial reservoir simulators are, with the exception of streamline simulators, entirely based on the finite volume method. See [11] for some background on the finite volume method, and [26] for an introduction to the streamline method. The robustness of the finite volume method, as used in oil reservoir simulation, is partly due to the diffusive nature of the numerical error, known as numerical diffusion, that arises from upwind difference methods. An interesting research problem would be to analyse the essential role that numerical diffusion might play in the actual physical modelling process particularly in situations with unstable flow. In the natural formulation, where the character of the problem is not clear, and special methods applicable to hyperbolic, or near hyperbolic problems are not applicable, the finite volume method, in the opinion of the author, is the most trustworthy approach. 

Also a simulation of the flow field in the methanol-reforming reactor of Figure 2.21 by means of the finite-volume method shows that recirculation zones are formed in the flow distribution chamber (see Figure 2.22). One of the goals of the work focused on the development of a micro reformer was to design the flow manifold in such a way that the volume flows in the different reaction channels are approximately the same [113]. In spite of the recirculation zones found, for the chosen design a flow variation of about 2% between different channels was predicted from the CFD simulations. In the application under study a washcoat cata-... 

Raithby, G.D. Discussion of the finite-volume method for radiation, and its application using 3D unstructured meshes. Numerical Heat Transfer, Part B, 1999. 35, 389-405. 

Examples of CFD applications involving non-Newtonian flow can be found, for example, in papers by Keunings and Crochet (1984), Van Kemenade and Deville (1994), and Mompean and Deville (1996). Van Kemenade and Deville used a spectral FEM and experienced severe numerical problems at high values of the Weissenberg number. In a later study Mompean and Deville (1996) could surmount these numerical difficulties by using a semi-implicit finite volume method. 

The discussion in the previous section assumed that the velocity field required to calculate the necessary coefficients of the discretized equations was somehow known. However, generally, the velocity field needs to be calculated as part of the overall solution procedure by solving momentum conservation equations. The governing equations are discussed in Chapters 2 to 5. The basic momentum transport equations governing laminar flow are considered here to illustrate the application of the finite volume method to calculation of the flow field. The governing equations can be written ... 

The finite volume method ensures integral conservation of mass, momentum and energy and is, therefore, attractive for reactor engineering applications. The steps in applying the finite volume method to solve transport equations are listed below. 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

When the governing model is given by the convection-diffusion equation (no electrical migration effects are considered), well-established numerical methods can be used directly in electrochemical cell design. When using commercial software, it should be remembered that the code has probably been benchmarked for applications different from those found in metallization, where spatial distributions of flux at high Schmidt numbers may be of more interest than the spatial average flux. Freitas has recently provided a comparison of several commercial CFD codes. Many of these codes are based on a finite-volume method (FVM) or a finite-element method. West jj yg discussed the application of... 

In addition to the topics reviewed above, which form the vast majority of the articles published to date in the field of electrochemical simulation, there are a number of other alternative methods that have been exploited by workers. These include, statistical techniques such as the Monte Carlo method [174-179], which has been exploited to examine the fractal nature of electrode surfaces and electrodeposited polymer film growth. The finite volume method, which has found significant application in the engineering literature [180, 181], remains poorly exploited in the electrochemical field [182, 183] as does the multidimensional upwinding method, which has been applied by Van Den Boss-che and coworkers [184, 185] to multi-ion systems at the rotating disc electrode. For recent advances, readers are referred to the review of Speiser [19]. 

The finite volume method FVM should probably be studied more than it is for electrochemical applications, but it has been applied [ 161,162]. It is possibly related to the box method. A good text on FVM is that of Patankar [11],... 

This method is popnlar for use with structural analysis codes and some CHD codes. In the early days of CFD, when structured orthogonal grids were used for most applications of the finite volume method, the finite element method offered the luxury of unstructured meshes with nonorthogonal elements of various shapes. Now that the use of unstructured meshes is common among finite volume solvers, the finite element method has been used primarily for certain focused CFD application areas. In particular, it is popular for flows that are neither compressible nor highly turbulent, and for laminar flows involving Newtonian and non-Newtonian fluids, especially those with elastic properties. 

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