Methods of Computational Fluid Dynamics

The three most widespread methods for solving the transport equations for momentum, matter and heat are the finite-difference [76], the finite-element [77, 

78] and the finite-volume [79] approaches. In computational fluid dynamics, typically equations of convection-diffusion type  

Within the finite-difference method (FDM), the derivative terms appearing in Eq. (32) are approximated by finite-difference expressions at each grid node. As an 

The finite-element method (FEM) is based on shape functions which are defined in each grid cell. The imknown fimction O is locally expanded in a basis of shape fimctions, which are usually polynomials. The expansion coefficients are determined by a Ritz-Galerkin variational principle [80], which means that the solution corresponds to the minimization of a functional form depending on the degrees of freedom of the system. Hence the FEM has certain optimality properties, but is not necessarily a conservative method. The FEM is ideally suited for complex grid geometries, and the approximation order can easily be increased, for example by extending the set of shape fimctions. 

In computational fluid dynamics only the last two methods have been extensively implemented into commercial flow solvers. Especially for CFD problems the FVM has proven robust and stable, and as a conservative discretization scheme it has some built-in mechanism of error avoidance. For this reason, many of the leading commercially available CFD tools, such as CFX4/5, Fluent and Star-CD, are based on the FVM. The oufline on CFD given in this book wiU be based on this method however, certain parts of the discussion also apply to the other two methods. 

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The key reactive separation topics to be addressed in the near future are a proper hydrodynamic modeling for catalytic internals, including residence time distribution account and scale-up methodology. Further studies on the hydrodynamics of catalytic internals are essential for a better understanding of RSP behavior and the availability of optimally designed catalytic column internals for them. In this regard, the methods of computational fluid dynamics appear very helpful. 

The majority of models for treating fireballs is based on correlations for its diameter and duration [2, 40]. More fundamental models are discussed in [2] and the application of methods of computational fluid dynamics ( CFD ) to fireballs is treated, for example, in [41]. 

This method has been devised as an effective numerical teclmique of computational fluid dynamics. The basic variables are the time-dependent probability distributions f x, f) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a detenninistic local rule. A carefiil choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the fomiation of lamellar phases in amphiphilic systems [92, 93]. 

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

Numerical methods are a working tool which we encounter almost daily in the fields of engineering and science. The complexity of numerical methods ranges from simple spreadsheets to the solution of complex, non-linear differential equation systems that occur in flow dynamics. The aim of computational fluid dynamics, or CFD, is to obtain a deeper understanding of the flow processes that take place within the extruder and to combine the findings with experiments to produce reliable and economic extruder designs. 

Xu, B.H. and Yu, A.B. (1997), Numerical simulation of gas solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chem. Eng. Sci., 52, 2785. 

Software In view of the rigor involved in numerical solution, many commercial software packages have been developed that serve the purpose of computational fluid dynamics (CFD). Appendix 2. A gives a listing of sources for various commercial as well as free CFD codes. These CFD codes may be broadly categorized into either finite volume method based or finite element method based. For a detailed account of computational methods, see the books by Patankar (1980), Ferziger and Peric (2002), Ranade (2002), Chen (2005), Reddy (2005), and so forth. 

Measurements have been made of turbulence structure by a number of workers using laser-Doppler methods and using hot-film anemometry Application of computational fluid dynamics to turbulent flow in stirred tanks is developing rapidly and involves using assumptions inherent in Kolmogoroff s theory and turbulence measurements to supply boundary conditions. 

Jafari A, Shirani E, Ashgriz N (2006) An improved three-dimensional model for pressure calculation in volume tracking methods. International Jotmial of Computational Fluid Dynamics 21(2) 87-97... 

The immersed boundary method is a numerical method in computational fluid dynamics where the flow boundary, e.g., the surface of a solid body in contact with the fluid or the interface between two immiscible fluids, is immersed in the mesh that does not conform with the boundary. In the immersed boundary method, special treatment has to be taken at the boundary to incorporate the boundary conditions. 

Increased power of present computers and progress in numerical methods and programming enables application of more sophisticated computer codes to some industrial problems. Before such aplication is made, the computer codes must be validated, especially when solving the problems of nuclear safety. Also the NRI therefore started validation and application of Computational Fluid Dynamics (CFD) codes to some selected problems encountered in NPP safety analyses. The commercial code FLUENT 5 was the first code undergoing such validation. 

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