Numerical solutions fluid model equations

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

Another detailed method of determining pressures is computational fluid dynamics (CFD), which uses a numerical solution of simplified equations of motion over a dense grid of points around the building. Murakami et al. and Zhoy and Stathopoulos found less agreement with computational fluid dynamics methods using the k-e turbulence model typically used in current commercial codes. More advanced turbulence models such as large eddy simulation were more successful but much more costly. ... 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

Specification of necessary information/data related to flow process under consideration. Once a suitable grid is generated, the user has to specify the necessary information concerning the physicochemical properties of fluids such as molecular viscosity, density, conductivity etc. for the solution of model equations. If the process under consideration involves chemical reactions, all the other necessary data about reaction kinetics (and stoichiometry, heat of reaction etc.) need to be supplied. In addition to system-specific data, specification of boundary conditions on the edges/external surfaces of the solution domain is a further crucial aspect of the solution process. It is also necessary to provide all the information related to the numerical method selected... 

Lathouwers D, van den Akker HEA (1996) A numerical method for the solution of two-fluid model equations. In Numerical methods for multiphase flows. FED-Vol 236, ASME 1996 Fluids Engineering Division Conference, Volume 1, ASME 1996, pp 121-126... 

A property of the finite volume method is that numerous schemes and procedures can be designed in order to solve the two-fluid model equations. In addition, the coupling terms can be approximated and manipulated in different ways. Besides, it is very difficult to predict the convergence and stability properties of novel solution methods. These aspects collectively increase the possibility of devising a numerical procedure which, when put to the test, does not converge. 

To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

We present and discuss results for MD modeling of fluid systems. We restrict our discussion to systems which are in a macroscopically steady state, thus eliminating the added complexity of any temporal behavior. We start with a simple fluid system where the hydrodynamic equations are exactly solvable. We conclude with fluid systems for which the hydrodynamic equations are nonlinear. Solutions for these equations can be obtained only through numerical methods. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

The use of the compressibility term can be described as follows. The greater the stiffness a system model has, the more quickly the flow reacts to a change in pressure, and vice versa. For instance, if all fluids in the system are incompressible, and quasi-steady assumptions are used, then a step change to a valve should result in an instantaneous equilibrium of flows and pressures throughout the entire system. This makes for a stiff numerical solution, and is thus computationally intense. This pressure-flow solution technique allows for some compressibility to relax the problem. The equilibrium time of a quasi-steady model can be modified by changing this parameter, for instance this term could be set such that equilibrium occurs after 2 to 3 seconds for the entire model. However, quantitative results less than this timescale would then potentially not be captured accurately. As a final note, this technique can also incorporate flow elements that use the momentum equation (non-quasi-steady), but its strength is more suited by quasi-steady flow assumptions. 

Figure 3.31 As (due to orientational response of aqueous solvent) versus e, calculated for ET in a large binuclear transition metal complex (D (Ru2+/3+) and A (Co2+/3+) sites bridged by a tetraproline moiety) molecular-level results obtained from a nonlocal polarization response theory (NRFT, solid lines) continuum results are given by dashed lines, referring to numerical solution of the Poisson equation with vdW (cont./vdW) and SAS (cont./SAS) cavities, or as the limit of the NRFT results when the full k-dependent structure factor (5(k)) is replaced by 5(0) 5(k) for bulk water was obtained from a fluid model based on polarizable dipolar spheres (s = 1.8 refers to ambient water (square)). For an alternative model based on TIP3 water (where, nominally, 6 = ), ambient water corresponds to the diamond. (Reprinted from A. A. Milishuk and D. V. Matyushov, Chem Phys., 324, 172. Copyright (2006), with permission from Elsevier).

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