Finite volume elements simulation

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

Within a finite volume element given by the Eulerian simulation of the extracellular liquid phase, the velocities of the liquid phase in each spatial coordinate can be linearly approximated, which allows one to obtain an analytical solution of Eq. (3.21). The random movement caused by turbulent dispersion is superimposed on the convective flow represented by the velocity field v. In the Lagrangian... 

The advantages of this type of system are obvious the pore space is of sufficient complexity to represent any natural or technical pore network. As the model objects are based on computer generated clusters, the pore spaces are well defined so that point-by-point data sets describing the pore space are available. Because these data sets are known, they can be fed directly into finite element or finite volume computational fluid dynamics (CFD) programs in order to simulate transport properties [7]. The percolation model objects are taken as a transport paradigm for any pore network of major complexity. 

The focus of the remainder of this chapter is on interstitial flow simulation by finite volume or finite element methods. These allow simulations at higher flow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method. 

The inclusion of chemical reaction into CFD packed-tube simulations is a relatively new development. Thus far, it has been reported only by groups using LBM approaches however, there is no reason not to expect similar advances from groups using finite volume or finite element CFD methods. The study by Zeiser et al. (2001) also included a simplified geometry for reaction. They simulated the reaction A + B - C on the outer surface of a single square particle on the axis of a 2D channel (Fig. 16). 

At the other boundary, bulk concentration of A must be maintained at some finite distance from the electrode, while the concentration of B will be zero at the same point. This distance may be regarded as the diffusion layer thickness. In terms of the simulation, the establishment of the semi-infinite boundary condition requires the determination of the number of volume elements making up the diffusion layer. This will be a function of the number of time iterations that have taken place up to that point in the simulation. At any time in the physical experiment, the diffusion layer thickness is given by 6(Dt)1/2. This rule of thumb may be combined with Equation 20.7 to calculate Jd, the number of volume elements in the diffusion layer ... 

In this example, one periodic element (a cross-over) of the laboratory scale version of Katapak -S was selected for the detailed CFD simulation with CFX-5. This solver uses the finite volume discretization method in combination with hybrid unstructured grids. Around 1,100 spherical particles of 1 mm diameter were included in the computational domain. As the liquid flows through the catalyst-filled channels at operating conditions below the load point (cf. Moritz and Hasse, 1999), permeability of the channel walls made of the wire mesh is not taken into account by this particular model. The catalyst-filled channels are considered fully wetted by the liquid creeping down, whereas the empty channels are completely occupied by the counter-current gas. It means that the bypass flow... 

The method normally used in simulation is the finite difference method. The solution is divided into small volume elements within which concentrations are assumed to be uniform. Time is also incremented. 

Sucessful simulations have been performed with computerized fluid dynamics programs (CFD), based on the fundamental Navier-Stokes equations, with appropriate volume element grids and/or a finite elemet approach, and in some cases were backed up by isothermal physical modeling (hydraulic modeling) experiments [455]. Examples for the CFD software used are FLUENT (Fluent Inc.) [450] and CFDS-FLOW3D [458] and others, usually modified by the contractor or licensor [448] to adapt them to the conditions in a secondary reformer. Discussion of simulation with CFD can be found in [444], [448], [449]. 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

Macroscopic phenomena are described by systems of integro-partial differential algebraic equations (IPDAEs) that are simulated by continuum methods such as finite difference, finite volume and finite element methods ([65] and references dted therein [66, 67]). The commonality of these methods is their use of a mesh or grid over the spatial dimensions [68-71]. Such methods form the basis of many common software packages such as Fluent for simulating fluid dynamics and ABAQUS for simulating solid mechanics problems. 

Elements of the Finite Volume Method for Flow Simulations... 

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