Finite-Volume Methods

In the finite volume method, discretized equations are obtained by integrating the governing transport equations over a finite control volume (CV). In this section, general aspects of the method are briefly discussed using a generic conservation equation for quantity, 0. Patankar (1980), Versteeg and Malalasekara (1995) and Ferziger and Peric (1995) may be referred to for a more detailed description. 

Let us consider the task of solution of a generic transport equation of the following form in a two-dimensional solution domain  

FIGURE 6.1 Finite volume grid, nodes centered in control volumes (left) and feces centered between 

FIGURE 6.2 Typical finite volume CV and the notation used for a Cartesian grid. 

The governing transport equations are integrated over each computational cell (employing a divergence theorem) and over the considered time interval At  

The FVM is similar to FEM in the sense that it is based on the integral formulation of the PDEs describing the system. The solution steps involved in FVM can be summarized as follows  

The solution strategy is somewhat varied by the last step since the approach used to linearize and solve the discretized equations varies with the solver type. The two commonly employed solvers in the FVM 2se pressure-based and density-based solvers [ 12,16]. In both methods the velocity field is obtained from the momentum equations. In the density-based approach, the continuity equation is used to obtain the density field, while the pressure field is determined from an equation of state. On the other hand, the pressure-based solver extracts the pressure field by solving the pressure or pressure correction equation, which is obtained by manipulation of the momentum and continuity equations [16]. Implementation of the pressure-based solver via the so-called Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [12] is explained later. Details of the density-based solver are extensively covered elsewhere [16] and will not be discussed here. 

The pressure-based solver and the SIMPLE algorithm, as well as the density-based solvers, are implemented into ANSYS Fluent, another CFD package used in many applications [16]. While it is successful in simulating many continuum problems, some additional steps needs to carried out for simulation of the catalytic reactors in ANSYS Fluent. If the catalytic phenomena are described by mechanistic rate laws, these expressions have to 

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Finishing Finish removers Finite-volume method Fin quel Fintrol... 

The commercial CFD codes use the finite volume method, which was originally developed as a special finite difference formulation. The numerical algorithm consists of the following steps ... 

Versteeg, H. K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics—The Finite Volume Method, Addison Wesley Longman Ltd., 1995. 

The finite volume method, a very eommon method for solving fluid flow problems (Versteeg and Malalasekera, 1995). The balanee equations are solved for eaeh grid eell using an iterative solution approaeh, as the underlying physieal phenomena are eomplex. 

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-... 

The, J. L, Raithby, G. D., Stubley, G. D., Surface-adaptive finite-volume method for solvingfree-sutface flows, Numer. Heat Transfer B26 (1994) 367-380. 

G. D., Prediction of incompressible free Surface flows with an element-based finite volume method, Comput. Fluid Dyn. J. 4 (1995) 353-371. 

Finite Volume Methods Finite volume methods are utilized extensively in computational fluid dynamics. In this method, a mass balance is made over a cell, accounting for the change in what is in the cell, and the flow in and out. Figure 3-52 illustrates the geometry of the ith cell. A mass balance made on this cell (with area A perpendicular to the paper) is... 

The equations for both laminar and turbulent flows, and the finite volume methods used to solve them, have been presented extensively in the literature (Patankar, 1980 Mathur and Murthy, 1997 Ranade, 2002 Fluent, 2003). The following summary focuses on aspects of particular concern for simulation of packed tubes and also those options chosen for our own work. 

The discretized equations of the finite volume method are solved through an iterative process. This can sometimes have difficulty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and s are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced ... 

Hsu, A. T., M. S. Anand, andM. K. Razdan (1990). An assessment of PDF versus finite-volume methods for turbulent reacting flow calculations. In Paper 96-0523,... 

While Heisig et al. solved the diffusion equation numerically using a finite volume method and thus from a macroscopic point of view, Frasch took a mesoscopic approach the diffusion of single molecules was simulated using a random walk [69], A limited number of molecules were allowed moving in a two-dimensional biphasic representation of the stratum corneum. The positions of the molecules were changed with each time step by adding a random number to each of the molecule s coordinates. The displacement was related... 

Mathematically, the PPDF method is based on the Finite Volume Method of solving full Favre averaged Navier-Stokes equations with the k-e model as a closure for the Reynolds stresses and a presumed PDF closure for the mean reaction rate. 

R. Li et al.. Generalized Difference Methods for Differential Equations Numerical Analysis of Finite Volume Methods (2000)... 

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

Eymard R., Gallouet T., Herbin R., 2000. Finite volume methods. InHandbook of Numerical Analysis, Vol. VII, P.G. Ciarlet, J.L. Lions (Eds.). North Holland, Amsterdam, pp. 713-1020. 

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

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