Elemental volume

Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius e about a point in the media, where e is sufficiently large that Darcy s law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, e is the minimum radius that satisfies such a condition. If e is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume. 

Heterogeneity, nonuniformity and anisotropy are based on the probability density distribution of permeability of random macroscopic elemental volumes selected from the medium, where the permeability is expressed by the one-dimensional form of Darcy s law. 

Consider the elemental volume S6l at length 1 of the plug flow. Applying the general energy balanee in differential form gives... 

A material balanee on an elemental volume AV of the reaetor between ages X and A. -i- dA. follows. 

Macromolecules Containing Metal and Metal-Like Elements, Volume 8 Boron-Containing Polymers, edited by Alaa S. Abd-El-Aziz, Charles E. Carraher Jr., Charles U. Pittman Jr., and Martel Zeldin. Copyright 2007 John Wiley Sons, Inc. 

Motaium E., Badawy S.H. Effect of irrigation using sewage water on the distribution of some heavy metals in bulk and rhizopshere soils and different plant species Cabbage plants (Brassica Oleracea L.) and organge trees (Citrus sinensis L). Proceeding of the 5th International Conference on the Biogeochemistry of Trace Elements, Volume 1,1999, Vienna, Austria. 

C. Sneden, I. I. Ivans, J. P. Fulbright Globular Clusters and Halo Field Stars . In Origin and Evolution of the Elements Volume 4, Carnegie Observatories Astrophysics Series, ed. by A. McWilliam, M. Rauch (Cambridge, 2004)... 

Figure 3-1 Relation between concentration increase in an element volume and fluxes into and out of the volume. The flux along x-axis poinfs fo fhe righf (x-axis also points to the right). The flux af x is and fhaf af x + dx is /x+dx. The net flux info fhe small volume is (/x —/x+dx), which causes the mass and density in the volume to vary.

The following is a list of those elements volumes have been determined. 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

Dimensionless current representations also exist. If the electrode is placed at the center of the first volume element in the model, this current will be proportional to the material flux into the first element from the second. For electroactive species A, this flux is given by DMA [fA(2) - fA(l)], where fA(J) is the fractional concentration of A in the Jth element. This fractional flux may be converted to moles of flux through multiplication by C (bulk concentration) and A Ax (volume-element volume, assuming a planar electrode of area A). Appropriate electrochemical conversion and the recognition that this material flux occurs during the interval At yield the current expression... 

In this model, instead of the uniform and interpenetrating continuous phases of the gas and the solids, a distinct heterogeneous structure is assumed. The elemental volume in the flow field, which has displayed observable heterogeneity, is partitioned into fractions occupied by the gas-rich, dilute phase (denoted by subscript "f") and the particle-rich, dense phase (denoted by subscript "c"), respectively. Within each "phase," uniformity is assumed, and the dense "phase" is assumed to occur as spherical clusters. That is, the dense phase is discrete, surrounded by the continuous dilute phase. In this way, eight variables... 

The direct tomographic technique can be further subdivided according to the detector area into which a certain elemental volume in the object can radiate. For the case of a primary beam confined to a 1-D pencil beam, these include the point-to-point, point-to-line, line-to-line and point-to-plane variants [35],... 

FIGURE 10. A schematic representation of the dolichol (Dol) pathway for the glycosylation of proteins. Reproduced from K. L. Kirk, Biochemistry of the Elements, Volume 9B Biochemistry of Halogenated Compounds, by permission of Plenum Publishing Corp. 

To derive an expression for a measure of the magnitude of the buoyancy force, consider an elemental volume of the fluid as shown in Fig. 1.15. First, consider the forces acting on this control volume when the fluid is unheated and at rest. 

Fig. 1-2 Elemental volume for one-dimensional heat-conduction analysis.

Fig. 1-3 Elemental volume for three-dimensional heat-conduction analysis (a) cartesian coordinates (to) cylindrical coordinates (c) spherical coordinates.

Table 3-2 Summary of Nodal Formulas for Finite-Difference Calculations (Dashed Lines Indicate Element Volume.)...

The resistance formulation is also useful for numerical solution of complicated three-dimensional shapes. The volume elements for the three common coordinate systems are shown in Fig. 3-11, and internal nodal resistances for each system are given in Table 3-3. The nomenclature for the (m, rt, k) subscripts is given at the top of the table, and the plus or minus sign on the resistance subscripts designates the resistance in a positive or negative direction from the central node (m, n, k). The elemental volume AV is also indicated for each... 

Table 4-2 Explicit Nodal Equations (Dashed lines indicate element volume)... 

Fig. 5-6 Elemental volume for energy analyse of laminar boundary layer.

In nonequilibrium systems, the intensive properties of temperature, pressure, and chemical potential are not uniform. However, they all are defined locally in an elemental volume with a sufficient number of molecules for the principles of thermodynamics to be applicable. For example, in a region A , we can define the densities of thermodynamic properties such as energy and entropy at local temperature. The energy density, the entropy density, and the amount of matter are expressed by uk(T, Nk), s T, Nk), and Nk, respectively. The total energy U, the total entropy S, and the total number of moles N of the system are determined by the following volume integrals ... 

C.H. Evans, Biochemistry of the Elements, Volume 8 Biochemistry of the Lanthanides, Plenum, New York, 1990. 

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