Cylindrical core model

Using a cylindrical core model open at both ends, for the adsorption isotherm the mutually perpendicular radii are given by rj =, t2 =  

To obtain the core size distribution either the adsorption or desorption branch of the isotherm can be used but, as before, it is preferable to use the desorption branch. 

As the pressure falls from to /V-1 condensed volume Vf. is desorbed where  

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The 30-mm sediment slices of the segmented cylindrical cores obtained from box coring at the seven stations were dried, pulverized, and thoroughly mixed to yield a uniform sample for analysis. Sediment from each of these slices was analyzed by two independent methods. The first method used a Perkin-Elmer model 5000 atomic absorption spectrophotometer (AA) for the elements Fe, Mn, Ti, Pb, Zn, Cu, Cr, Ni, Co, Hg, and Cd (9). The second method utilized a Philips PW 1410 X-ray fluorescence spectrometer for the analysis of elements Fe, Mn, Ti, Ca, K, P, Si, Al, Mg, Na, Pb, Zn, Cu, Cr, V, and Ba (10). The AA analysis was chosen because of the known accuracy and sensitivity to a wide spectrum of elements. The XRF analysis was chosen for its accuracy and similar nondestructive mode of analysis equivalent to the shipboard XRF analysis. Good agreement between the AA and the XRF values was felt to be imperative because the Philips XRF equipment was to be used in the land-based multielement analysis of the CS -collected sediment samples. 

The unreacted core model, suitably modified for cylindrical geometry, was used to describe the behavior... 

The simplest model for fission energy generation corresponds to a bare, homogeneous core. The geometry of most practical importance is the cylindrical core, for which the distribution of (radial and axial) energy generation is given by... 

It would be expected that the results of a calculation for a cylindrical core with a diameter-to-height DjH) ratio of 1 would be similar to that of a spherical core having the same core volume. Calculations were performed using the spherical and cylindrical versions of the modified Bethe-Tait method, and the results show good agreement between the two models for this case. One example of such a comparison is shown in... 

Suau et al. studied the conformation of chromatin as a function of ionic strength and Braddock et al. fitted model calculations for the nucleosome core particle in solution to these experimental scattering curves. The best fit to the data was found for a model in which there were 1.7 0.1 turns of DNA wrapped around a hydrophobic core. Models in which this core was cylindrical or wedge-shaped were compatible with the measured scattering curves. However, spherical or ellipsoidal core models were incompatible and had to be rejected. 

If the transport of the gaseous reactant through the product layer of the cylindrical particle is the rate-determining step (fast chemical reaction, no mass transfer resistance by external diffusion), the concentration at the surface of the shrinking core almost reaches zero. Thus the reaction is confined to a front. In contrast to the shrinking core model with the influence of reaction (combined model. Section 4.6.3.3), the reactant concentration is zero at the reaction front, and no reaction occurs within the core. Equation (4.6.56) simplifies as we can assume a negligibly small value of the term and we obtain ... 

For simplicity we ignore the molecular structure of the rodlike block and model it as a cylinder of length Nb and diameter d such that I,od d. The minimal surface area per rod is attained when the micellar core assumes a cylindrical shape. The B blocks are close packed with their axes aligned and their tips towing a single basal plane. The cylindrical core thus formed carries two... 

This reactor model is within the range of concepts studied by NRPCT (Project Prometheus Reactor Module Final Report, 2006). Figure 2 contains the schematic of the RELAP5-3D input model for a representative gas reactor. Some of the features of this reactor concept are surmnarized in Table 1. The reactor core model is composed of 354 fuel pins of diameter 1.777 cm. Each fuel pin passes through a cylindrical core block with an aimular flow chaimel between the pin and the core block. Each fuel pin has a gas plenum to acconunodate fission products. The reactor vessel has both the inlet and outlet nozzles at one end of the... 

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

In order to allow for the thinning of the multilayer, it is necessary to assume a pore model so as to be able to apply a correction to Uj, etc., in turn for re-insertion into Equation (3.52). Unfortunately, with the cylindrical model the correction becomes increasingly complicated as desorption proceeds, since the wall area of each group of cores changes progressively as the multilayer thins down. With the slit model, on the other hand, <5/l for a... 

To convert the core area into the pore area ( = specific surface, if the external area is negligible) necessitates the use of a conversion factor R which is a function not only of the pore model but also of both r and t (cf. p. 148). Thus, successive increments of the area under the curve have to be corrected, each with its appropriate value of R. For the commonly used cylindrical model,... 

Figure 3.S Schematic diagram of packing side chains In the hydrophobic core of colled-coll structures according to the "knobs In holes" model. The positions of the side chains along the surface of the cylindrical a helix Is pro-jected onto a plane parallel with the heUcal axis for both a helices of the coiled-coil. (a) Projected positions of side chains in helix 1. (b) Projected positions of side chains in helix 2. (c) Superposition of (a) and (b) using the relative orientation of the helices In the coiled-coil structure. The side-chain positions of the first helix, the "knobs," superimpose between the side-chain positions In the second helix, the "holes." The green shading outlines a d-resldue (leucine) from helix 1 surrounded by four side chains from helix 2, and the brown shading outlines an a-resldue (usually hydrophobic) from helix 1 surrounded by four side chains from helix 2.

F l G U RE 6.11 (See color insert following page 390.) Schematic of the application of x-ray CT analysis to provide density profile images of different sections of a hydrate core contained in a cylindrical high pressure aluminum cell. (From Gupta, A., Methane Hydrate Dissociation Measurements and Modeling The Role of Heat Transfer and Reaction Kinetics, Ph.D. Thesis, Colorado School of Mines, Golden, CO (2007). With permission.)... 

Some attention should be also paid to the fact that some copolymers with special sequence distribution do not assume cylindrical shape within the HA model. For example, this is the case for protein-like sequences. Protein-like sequences correspond to a copolymer which forms globules with a hydrophobic core and a hydrophilic shell showing no tendency to aggregation. Proteinlike copolymers have been previously studied within the HP model [32-34], Application of the more realistic HA model showed that the globules formed by protein-like copolymers under worsening solvent quality assume conventional spherical shape and show no tendency to aggregate [23]. The stability for HA model protein-like copolymers is much higher than for those within the HP model. 

Based on the above general principles, quite a number of models have been developed to estimate pore size distributions.29,30,31-32,33 They are based on different pore models (cylindrical, ink bottle, packed sphere,. ..). Even the so-called modelless calculation methods do need a pore model in the end to convert the results into an actual pore size distribution. Very often, the exact pore shape is not known, or the pores are very irregular, which makes the choice of the model rather arbitrary. The model of Barett, Joyner and Halenda34 (BJH model) is based on calculation methods for cylindrical pores. The method uses the desorption branch of the isotherm. The desorbed amount of gas is due either to the evaporation of the liquid core, or to the desorption of a multilayer. Both phenomena are related to the relative pressure, by means of the Kelvin and the Halsey equation. The exact computer algorithms35 are not discussed here. The calculations are rather tedious, but straightforward. 

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