Single-sphere model

A number of authors [46 to 48] employ the single sphere model in which the packed bed is considered as a set of equal spheres that are under the same state of extraction, and the fluid flowing around them is solute-free. That is, equation (3.4-90) would be valid, but without the generation term [46], The transport at the solid-fluid interface obeys the boundary condition (Eqn. 3.4-94) with C = 0 (fluid-flows at a large velocity). Under these assumptions, there is an analytical solution to the above problem (without axial dispersion) in terms of the Biot number (Bi = k, R/De), included in the following equation ... 

Extraction of Thyme and Rosemary with C02 at 25 and 40°C and pressures up to 250 bar is described and the results modelled, Single Sphere Models being best. Good data fits were obtained for both herbs when effective diffusivities De were fitted at each temperature and pressure but variations of De with temperature and pressure were physically more convincing for Thyme. 

Three models were used to simulate the observed extraction rates for the oily components. Each of employed the effective diffusivity De as an adjustable parameter. They were the Characteristic Time Model (3) two other published models here designated Single Sphere Models I (4,5) and II (6). In these models it is assumed that the solute is extracted from a particulate bed composed of porous... 

Single Sphere Model I The problem of diffusion of matter from a sphere initially at a uniform concentration when the surface concentration is maintained constant has been solved by Crank (7) and his equation (6.20) is (on substituting De for D) the same as the expression for the mass extracted as a function of time given by single sphere model I. 

Single Sphere Model II (Equations 4, 5, 8, 9 and 10 in reference 6) In this model allowance is made for the resistance to mass transfer offered by the surface film surrounding the herb particles. The mass transfer coefficient kf was obtained from correlations proposed by Catchpole et al (8, 9) for mass transfer and diffusion into near-critical fluids. An average of the binary diffusivities of the major essential oil components present was used in calculating kf (these diffusivities were all rather similar because of their similar structures). 

The Characteristic Time Model was always found to give the worst fit and the Single Sphere Model II (which is not shown and which allows for film resistance) was not found to be appreciably better than the computationally simpler Model I for these systems. The values of De obtained were about two orders of magnitude lower than the estimated binary diffusivities. 

The single-sphere model in which there is a fusion between the two spheres eorresponding to each reactant to generate one sphere eartying the sum of charges and of radius... 

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

The parameters R and Rj in equation (5.32) are the radii of the equivalent hard spheres representing biopolymers i and y, respectively (where i = j for interactions between the same macroions). The equivalent hard sphere corresponds to the space occupied in the aqueous medium by a single biopolymer molecule (or particle) which is completely inaccessible to other biopolymers. In practice, the hard sphere model is a highly satisfactory description for many globular proteins. 

The model looks like a geodesic dome since all the nodes lie on a single sphere whose radius is, interestingly, equal to Rx. Therefore the curved bars which separate two... 

The figure is an illustration to help you visualize the solvation process. However, solvation is not a crystalline and orderly process. There are not a fixed number of solvating molecules, nor is it a highly organized process, nor is there a single sphere of solvation molecules. In short, solvation is a very messy process, and whoever comes up with a workable model will certainly win the Nobel Prize in chemistry. 

While powerful contemporary techniques permit the use of molecular cavities of complex shape [3], it is instructive to note a few cases based on idealized representations of solute cavity and charge density. Cavities are typically constructed in terms of spherical components. Marcus popularized two-sphere models, [5,38] which can be used to model CS, CR, or CSh processes (see Section 3.5.2), where the two spheres are associated with the D and A sites, and initial and final charge densities are represented by point charges (qD and qA) at the sphere origins. If a single electron is transferred, Ap corresponds to A = 1 in units of electronic charge (e), and Aif is given by [5,38]... 

Fig. 3. A sphere model (a) illustrates the tip paths encountered during the lateral manipulation on a fcc(lll) metal surface, (b) Single atom manipulation signals taken on a Ag(lll) surface show a sudden transition from various pulling modes to a sliding mode at = 30° [5],...

G. Duration of a Collision. The hard sphere model is very useful because it permits us to describe molecular collisions in terms of a single, simple, molecular parameter, the collision diameter. It is, however, insufficient to permit a detailed description of a chemical reaction, which is an event that transpires during a collision between two molecules, because the duration in time of a hard sphere collision is precisely zero. 

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