Application of finite volume

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. 

Electroanalytical application of hemispherical [35,36], cylindrical [37,38] and ring microelectrodes [39] has been described. A hemispherical iridium-based mercury ultramicroelectrode was formed by coulometric deposition at -0.2 V vs. SSCE in solution containing 8 x 10 M Hg(II) and 0.1M HCIO4 [35]. The radius of the iridium wire was 6.5 pm. The electrode was used for anodic stripping SWV determination of cadmium, lead and copper in unmodified drinking water, without any added electrolyte, deoxygenation, or forced convection. The effects of finite volume and sphericity of mercury drop elecPode in square-wave voltammetiy have been also studied [36]. 

Solution. As will be demonstrated in the next chapter, an infinite (oo) number of finite volume CSTRs connected in series approaches the behavior of a tubular flow (TF) reactor. Furthermore, an infinite number of CSTRs of differential (approaching zero) volume also approaches the behavior of a TF reactor. The volume of a TF reactor for this application as shown in Illustrative Example 10.16 is ... 

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-... 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

For molecules containing light atoms, we accordingly neglect this effect of finite nuclear volume or field shift, but other effects prevent exact application of isotopic ratios that one might expect on the basis of a proportionality with in formula 13 instead of total F. For this reason we supplement term coefficients in formula 8 for a particular isotopic species i with auxiliary coefficients [54],... 

It is seen from equation (1) that under the conditions considered, where the charge is place on the first plate, (Xn) can never equal zero and pure mobile phase will never elute from the column. However, the sample is rarely injected solely on the first plate. As it occupies a finite volume of mobile phase when it is injected onto the column, it will also occupy a finite number of theoretical plates. If (p) plate volumes of pure mobile phase are injected onto a column that has been equilibriated with mobile phase containing a concentration (X0) of a solute, then, on the application of a charge of pure mobile phase,... 

An experimental study was performed to determine the applicability of the theory. Oil-in-water (o/w) emulsions, stabilised with anionic surfactants, were prepared, with known quantities of added electrolyte, and were creamed by either gravitation or centrifugation. The results can be summarised as follows at low electrolyte concentrations, where h would have a finite value, <(> was less than 0.74. Over a range of concentrations, where it was assumed that both 0 and h were negligible, = 0.74 ( 0.02). The emulsions were found to be polydis-perse, so this did not appear to affect the volume fraction to a great extent. In addition, < > was found to be independent of the method of cream formation. 

Diazinon has a finite vapor pressure (see Chapter 3) and thus will be present in the air. A method for diazinon in air has been reported that is based on the use of polyurethane foam (PUF) to adsorb the pesticide from the air as the air is pulled through the PUF (Hsu et al. 1988). The PUF is then Soxhlet-extracted and the extract volume reduced prior to capillary GC/MS analysis. An LOD of 55 ng/m3 (5.5 m3 sample) and recovery of 73% were reported. Another study was described in which the diazinon levels in indoor air were monitored following periodic application of the pesticide for insect control (Williams et al. 1987). In this method, air is pulled through a commercially available adsorbent tube to concentrate diazinon. The tube is then extracted with acetone prior to GC/NPD analysis. No data were provided for the LOD, but recoveries in excess of 90% were reported at the 0.1 and 1 pg/m3 levels. This paper also indicated that diazinon can be converted to diazoxon by ozone and NOx in the air during the sampling process. 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

For many applications, interpolations of functions of two or three variables defined in two-and three-dimensional domains must be considered. For example, global interpolations in two- and three-dimensional systems are analogous to polynomial interpolation in onedimensional systems however, global interpolants do not exist in 2- and 3D. This is a big drawback in numerical analysis because a basic tool available for one variable is not available for multivariable approximation [21], The best developed aspect of this theory is that of piecewise polynomial approximation, associated with finite element and finite volume approximations for partial differential equations, which will be examined in detail in Chapters 9 and 10. 

Proceeding from an Ogden-type material formulation, which is extended towards an inelastic porous media application, volumetric extension terms are developed which describe the finite volume change including the concept of a volumetric compaction point. Thus, the equilibrium part of the mechanical... 

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. 

The modelling of super critical water oxidation (SCWO), up to now not been used in large scale industrial applications, is important for design of pilot plants and, later, industrial plants. The applied programme to model the continuous flow in a reactor is called CAST (Computer Aided Simulation of Turbulent Flows [8]) and is based on the method of the finite volume. That means that the balance equations were integrated over the surfaces of each control volume. 

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. 

Particle in Cell Methods. An alternative to the direct finite-diflFerence solution of (7) is the so-called particle in cell (PIC) technique. The distinguishing feature of the PIC technique is that the continuous concentration field is treated as a collection of mass points, each representing a given amount of pollutant and each located at the center of mass of the volume of material it represents. The mass points, or particles, are moved by advection and diffusion. It is convenient but not necessary, to have each of the particles of a given contaminant represent the same mass of material. The application of the PIC technique in hydrodynamic calculations is discussed by Harlow (32). Here we consider the use of the PIC technique in the numerical solution of (7). 

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