Finite series Volume

For a finite sample volume V, the sum includes a sequence of discrete values of k which can be selected as usually, by adopting periodic boundary conditions. When writing a Fourier series in terms of exponential functions, the amplitudes are complex numbers. We represent them as... 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

Numerical simulations of the thermal performance of the module were performed using finite element analysis. In the present model, the fluid path is represented by a series of interconnected nodes. Convection processes are modeled as transfer processes between these nodes (or volumes) and surfaces of the geometrical mesh. In this case, a series of analyses based on knowledge of the fluid properties, flow rates, and the relative sizes of the fluid passages and solid phase interconnections led to the value of 3.88 W/cm -K for the effective heat-transfer coefficient. Convective heat transfer using this coefficient was used on all of the internal free surfaces of the module. 

The dipole moment induced in a molecule, or in a group of molecules, is a finite range function of the intermolecular separations, R, which falls off faster than R-3 for R —> oo. Van Kranendonk has argued that, therefore, it is possible to expand the above equation in a series of cluster functions [400, 402]. If ft( 1 n) designates the dipole moment induced in the cluster of molecules 1 n when these are present alone in the given volume V, cluster functions 1/(1 n) can be defined according to... 

This set of equations can be solved by a variety of approaches. Historically they were solved analytically by a separation-of-variables method, which is tedious, time-consuming, and, for most, an error-prone task. The results presented here were computed using a finite-volume discretization of the momentum equation on a 10 by 10 mesh, which was solved iteratively in a spreadsheet. The programming time was a couple of hours, and the solution is found in about a minute on a typical personal computer. The results are accurate to within one percent of the exact series solutions. The details of the spreadsheet programming for this problem are included in an appendix. 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

The sum in Eq. (5) was taken over all coordination spheres. A model of the atomic distributions may be constructed, based on Eq. (5), in terms of a series of peaks convoluted with the peak-shape fiinction P(r) to include the effects of data termination at Kmax- In order to compensate for the edge effects in a finite-sized model the correction term t(r), which corresponds to probability of finding an atom at a distance r fi om another atom lying inside the volume of the considered model, is introduced as multiplier of the RRDF computed for the infinite model. According to [13], for the model in the form of a rectangular prism with the edged a, b and c, s(r) is given by ... 

This process was first recognized by Ostwald and is known as Ostwald ripening. The mathematical details were worked out independently by Lifshitz and Slyozov and by Wagner ° and is known as the LSW theory. However, this theory is based on a mean field approximation and is restricted to low volume fraction systems. Voorhees and coworkers extended the LSW theory to finite volume fraction systems and conducted a series of flight experiments designed to test this and similar theories. ... 

The principles of IGC as well as experimental details have been reviewed elsewhere [8]. The attention of the reader is simply drawn to two possible ways to operate IGC either under infinite dilution, or finite concentration conditions. For the determination of 7, exploiting IGC under infinite dilution conditions, minute amounts of a homologous series of n alkanes are injected in the GC and their retention volumes (Vn) are detected (volume of carrier gas necessary to push the solute through the column containing the solid to be examined), -yf is then simply obtained by application of the following relationships ... 

Effects of the gas - solid potential corrugation on the behaviour of monolayers formed on the (100) face of an fee crystal at finite temperatures have been recently studied by Patrykiejew et al. [163] with the help of Monte Carlo method. They have considered three-dimensional systems of constant volume and containing fixed number of particles interacting via the Lennard-Jones potential (1). The gas - solid interaction potential has been assumed to be represented by the two-fold Fourier series [88]... 

There are two important features associated with the generation of power series representations of functions. First, a value of jc lying in the domain of the function must be chosen for the expansion point, a second, the function must be infinitely differentiable at the chosen point in its domain. In other words, differentiation of the function must never yield a constant function because subsequent derivatives will be zero, and the series will be truncated to a polynomial of finite degree. The question as to whether the power series representation of a function has the same domain as the function itself is discussed in a later subsection. The next subsection is concerned with determining the coefficients, c for the two kinds of power series used to represent some of the functions introduced in Chapter 2 of Volume 1. 

If initial molecule concentrations are specified together with the concentrations at the compartment boundaries, RDEs can be solved numerically using finite-difference or finite-element discretizations of the compartment volume. The solution consists of molecular concentrations across the discretized compartment volume in a series of time points. RDEs and variants thereof have been used in a range of applications, including studies of pattern formation [Meinhardt and Gierer 2000 Myasnikova et al. 2001] and calcium transport [Smith, Pearson, and Keizer 2002]. 

It is clear that, for deep-shell excitation, there is not much difference between the destruction of Rydberg states due to the finite volume available around the atom and the termination of series due to Auger broadening. Furthermore, since the ionisation threshold no longer concides with the series limit even for the free atom, one has to be careful to distinguish... 

These relations support our earlier assertion that for the same overall conversion the total volume of a cascade of CSTRs should approach the plug flow volume as the number of identical CSTRs in the cascade is increased. For a finite (low) number of CSTRs in series, equation (8.3.52) can be rewritten as... 

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