Finite volume term

Ternary moments consist of an intermolecular force or finite volume term (the singly primed terms), and an interference or cancellation term (the doubly primed terms) [400, 401, 402]. As was mentioned above,... 

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... 

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. 

Vincent et al.(3) used a simplified configurational entropy term Ass = -k ln(4>f /4>.). For a dilute dispersion, the In 4>d term is probably correct, but for the floe phase, with of the order of 0.5, a term In 4>f certainly can overestimate the entropy in the floe, because hard spheres with finite volume have at high concentration much less translational freedom than (volumeless) point... 

The discretized equations of the finite volume method are solved through an iterative process. This can sometimes have difficulty converging, especially when the nonlinear terms play a strong role or when turbulence-related quantities such as k and s are changing rapidly, such as near a solid surface. To assist in convergence a relaxation factor can be introduced ... 

The actual volume of the gas is less than the ideal gas. This is because gas molecules do have a finite volume and the more moles of gas present, the smaller the real volume. The volume of the gas can be corrected by the V -nb term, where n is the number of moles of gas and b is a different constant for each gas. The larger the gas particle, the more volume it takes up and the larger the b value. 

From this equation, we can see that the total nonideality correction (in braces) contains a negative contribution (first bracketed term) that is indeed proportional to the attractions constant a, while the positive contribution (second bracketed term) is proportional to the finite-volume repulsions constant b, as was supposed in the interpretation of experimental Z behavior in Fig. 2.2. One can also see that the attractions term is linearly proportional to density n/V, whereas the repulsions term is proportional to squared density (,njV)2, so that the former must always prevail at low density (low P) and the latter at high density (high P), as was shown in Fig. 2.2. Furthermore, one can recognize from the 1 /RT prefactor that the entire nonideality correction must diminish with increasing P, as was noted in Fig. 2.3. Thus, regardless of the particular values chosen for a and b, the Van der Waals equation is expected to exhibit both pressure and temperature dependences that are qualitatively consistent with the observed Z(P, T) behavior. 

Any departure introduced into an assumed steady state of a system. The magnitude of the departure is often assumed to be small so that product terms in the dependent variables may be neglected the term perturbation is therefore sometimes used as synonymous with small perturbation. The perturbation may be concentrated at a point or in a finite volume of space or it may be a wave (sine or cosine function) or, in the case of a rotating system, it may be symmetric about the axis of rotation. 

Such intermolecular forces also account for the deviations of real gases from the ideal behaviour required by the equation PV — RT. Deviation arises from two causes appreciable intermolecular attraction and the finite volume occupied by the molecules themselves, which is another way of saying that repulsive forces come into play when two molecules approach one another closely. In van der Waals equation, these effects are respectively covered by the additional terms in (P+a/V2)(V—b)=RT Because of their relationship to the a/V2 term, secondary attractive forces are often referred to collectively as van der Waals forces. 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

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... 

When integrated over a finite volume, the divergence term becomes a surface integral of the Poynting vector [318]. Expressed in covariant notation, 3V ° = — F0vjv, showing the explicit modification of energy-momentum conservation due to the final dissipative term here. 

Many detection principles require a finite volume of eluent. For example, a UV absorption detector yields a signal that is directly proportional to the optical pathlength (Beer s law, see eqn.5.21). The volume of the detector flow cell is usually well-defined and its contribution to aejc, and hence its effects on the observed dispersion ctg, can be discussed in quantitative terms (see section 7.4.2). 

Correction Due to Volume of Gas Molecules The ideal gas equation PV = nRT is derived on the assumption that the gas molecules are mass points, i.e., they do not have finite volume. Van der Waals abandoned this assumption and suggested that a correction term nb should be subtracted from the total volume V in order to get the ideal volume which is compressible. In order to understand the meaning of the correction term nb, let us consider two gas molecules as unpenetrable and incompressible spheres, each of which has a diameter d, as shown in the following figure. 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

Figure 3.26. Modelled main cell-potential loss terms. Ohmic bulk losses in electrolyte Act. an activation losses at negative electrode Act. cat activation losses at positive electrode. (Reprinted from S. Campanari and P. lora (2004). Definition and sensitivity analysis of a finite volume SOFC model for a tubular cell geometry. /. Power Sources 132,113-126. Used by permission from Elsevier.)...

After finalizing the model equations and boundary conditions, the next task is to choose a suitable method to approximate the differential equations by a system of algebraic equations in terms of the variables at some discrete locations in space and time (called a discretization method). There are many such methods the most important are finite difference (FD), finite volume (FV) and finite element (FE) methods. Other methods, such as spectral methods, boundary element methods or cellular automata are used, but these are generally restricted to special classes of problems. All methods yield the same solution if the grid (number of discrete locations used to... 

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