Finite volumes

One considers systems for which the energy shell is a closed (or at least finite) hypersurface S. Then tire energy shell has a finite volume ... 

In the above, the sum over N has the upper limit of mfinity. This is elearly eorreet in the thennodynamie limit. However, for a system with finite volume, V, depending on the hard eore size of its eonstituents, there will be a maximum number of partieles, M(V), that ean be paeked in volume V. Then, for all N siieh that N > M(V), the value of (-P ) beeomes infinity and all tenns in the N sum with N> M(V) vanish. Thus, provided the inter-partiele interaetions eontain a strongly repulsive part, the N sum in the above diseussion ean be extended to infinity. 

Brown F R and Yegulalp A 1991 Microcanonical simulation of Ist-order phase transitions in finite volumes Phys. Lett. A 155 252-6... 

Actual polymer repeat units occupy finite volumes and therefore exclude other segments from occupying the same space. 

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

Reduced Properties. One of the first attempts at achieving an accurate analytical model to describe fluid behavior was the van der Waals equation, in which corrections to the ideal gas law take the form of constants a and b to account for molecular interactions and the finite volume of gas molecules, respectively. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

Two general cases are considered (1) adsorption under conditions of constant or nearly constant external solution concentration (equivalent to infinite fluid volume) and (2) adsorption in a batch with finite volume. In the latter case, the fluid concentration varies from cf to when equihbrium is eventually attained. = (cf — =... 

The assumptions involved in dris equation clearly do not accurately describe real gases in which the atoms or molecules interact widr one another, and occupy a finite volume in space the size of which is determined by the complexity and mass of the particles. The first successful attempt to improve on the ideal equation was that of van der Waals... 

The conecting term in the pressure reflects the diminution in tire impact velocity of atoms at the containing walls of tire gas due to the attraction of the internal mass of gas, and the volume term reflects the finite volume of the molecules. Data for these two constants are shown in Table 3.4. 

This assumption is called the continuity condition, and assures that no region of the body with positive finite volume is deformed into one of zero or negative volume. It also excludes discontinuities such as material interfaces and shock waves which require special treatment. 

The injection point is that point when the sample is placed on the column. If the sample has a finite volume, then the injection point corresponds the start of the sampling process. 

Figure 15. The Injection of a Finite Volume of Charge on to an LC Column...

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

Equation (24) shows that when the charge is placed on the first plate, (Xn) can never equal zero and pure mobile phase free of solute will never elute from the column. However, in practice, it is almost impossible to place the sample exclusively on the first plate, and there will be a finite volume of mobile phase that will occupy a finite number of theoretical plates when it is injected onto the column. 

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. 

Numerical Method Finite volume Finite volume Finite volume Finite volume Finite Element... 

Travis. J. R, K L. Lam, and T. L. Wilson, 1994, GASFLOW A Three-Dimensional Finite-Volume Fluid Dynamics Code for Calculating the Transport, Mixing, and Combustion of Flammable Gases in Geometrically Complex Domains, LA-UR-94-2270 Vol. 1,2, and 3 LANL, July. 

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. 

Suppose we have a physical system with small rigid particles immersed in an atomic solvent. We assume that the densities of the solvent and the colloid material are roughly equal. Then the particles will not settle to the bottom of their container due to gravity. As theorists, we have to model the interactions present in the system. The obvious interaction is the excluded-volume effect caused by the finite volume of the particles. Experimental realizations are suspensions of sterically stabilized PMMA particles, (Fig. 4). Formally, the interaction potential can be written as... 

Aslanov, S. K., and O. S. Golinskii. 1989. Energy of an asymptotically equivalent point detonation for the detonation of a charge of finite volume in an ideal gas. Combustion, Explosion, and Shock Waves, pp. 801-808. 

Ideally, the sample should be injected onto the column as an infinitely thin disc, which covers the total cross section of the column. Because this is impossible, PSS has injected finite volumes onto the columns. In theory, these injection volumes should be as low as possible. In order to be able to detect the sample with significance, a certain (high) concentration of the sample has to be injected. This concept works well for low molar mass compounds, which do not generate much sample viscosity. However, when working with samples... 

Finiteness is the basic assumption a finite total volume of space-time and a finite amount of information in a finite volume of space-time. We require universality, of course, since we know that without it nothing much of interest can happen. We can also take a strong cue from our own universe, which allows us to build universal computers. If the underlying micro-physics was not universal we would not be able to do this. Reversibility is desirable because it ensures a strict conservation of information and can be used to create systems that conserve various quantities such as energy and angular momentum despite underlying anisotropies. 

Deviation of methane gas from ideal gas behavior. Below about 350 atm, attractive forces between methane (CH4) molecules cause the observed molar volume at 25°C to be less than that calculated from the ideal gas law. At 350 atm, the effect of the attractive forces is just balanced by that of the finite volume of CH4 molecules, and the gas appears to behave ideally. Above 350 atm, the effect of finite molecular volume predominates and V, > 1C... 

An increase in molar volume above that predicted by the ideal gas law is related to the finite volume of gas particles. These particles contribute to the observed volume, making Vm greater than V . Ordinarily, this effect becomes evident only at high pressures, where the particles are quite close to one another. 

As already stated any sample placed on a column will have a finite volume, and the variance of the injected sample will contribute directly to the final peak variance. It follows that the maximum volume of sample that can be placed on the column must be limited, or the column efficiency will be seriously reduced. Consider a volume Vi, injected onto a column. Normal LC injections will start initially as a rectangular distribution and the variance of the eluted peak will be the sum of the variances of the injected sample plus the normal variance of the eluted peak. 

Combustion is generally initiated by the introduction of a finite amount of energy to raise a finite volume of the material to its ignition temperature. Potential ignition sources for vapour—air mixtures are listed in Table 5.3, and temperatures in Table 5.4. They include ... 

Every gas changes into a liquid if the pressure is high enough and the temperature is low enough. The atoms or molecules of a liquid or solid stick together in a finite volume rather than expanding, as a gas does, to fill all available space. This cohesiveness comes from electrical forces of attraction between the negative electron cloud of each atom and the positive nuclei of other atoms. We describe intermolecular forces in Chapter 11. 

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