Size distribution models for

Everett and Powl51 (1976) have developed a pore size distribution model for the slit shaped pores of ultramicroporous carbons. This model has been further elaborated by Horvath and Kawazoe.52... 

The limited available laboratory measurements of microfault capillary pressure supported the selection of this curve (Fig. 5). Permeability was determined using a mixture of minipermeameter, image analysis and flow tests, and capillary pressure by mercury injection, at the University of Leeds for Statoil. The correlation of these limited data with the model prediction is fair for the majority of the results (which are 8 mD and higher) and lends some support to the model results. The correspondence between field and model results is certainly best with the narrow pore size distribution model for the majority of the data. The exponent of the curve selected is closest to the theoretical prediction of 0.5. 

To avoid the singularity present in their daughter size distribution model for d = 0 (i.e., the parent particle does not break), a minimum particle size was defined, dmin- The excess surface area relation reaches a minimum... 

W. G. Krumbein and W. R. James, A lognormal size distribution model for estimating stability of beach fill material. Technical Memorandum No. 16, U.S. Army Coastal Engineering Research Center (1965). 

Hansen JP, McDonald IR (1990) Theory of simple liquids, 2nd edn. Academic Press, London Heesink ABM, Prins W, van Swaaij WPM (1993) A grain size distribution model for non-catalytic gas-soUd reactions. The Chem Eng J 53 25-37... 

Leaist, D., Hao, L. Size distribution model for chemical interdiffusion in water ACT Heptane water-in-oil microemulsions. J. Phys. Chem. 99(34), 12896-12901 (1995)... 

Arena et al. (1983) investigated the coal attrition in a mixture with sand under hot but inert conditions. As they increased the sand particle size while keeping its mass in the bed constant, they observed an increase in the coal attrition rate. They interpreted their results by assuming that the abrasion energy is shared out on the entire material surface. On the same basis Ray et al. (1987a) developed their attrition rate distribution model for abrasion in a fluidized bed. 

The quantum confinement model was extended by Frohnhoff etal to account for the wide distribution of pore diameters of the PS formed on p-Si. Tunneling of holes through silicon crystallites was proposed as a process also involved in the formation of the quantum size porous structure. The tunneling current oscillates with the crystallite size which was considered to be responsible for the uneven pore size distribution and for the stability of very small crystallites in the PS. The quantum confinement model... 

The experimental data of Risso and Fabre [99] also indicate that an equal size daughter distribution is more common for bubble breakage than an unequal one. Contrary, Hesketh et al [33, 34] investigated bubble breakage in turbulent flows in horizontal pipes and concluded that an unequal size daughter distribution is more probable than an equal size one. The daughter bubble size distribution model of Luo and Svendsen [74] rely on the assumption that unequal sized is more probable than equal size breakage in accordance with... 

Table 5 Parameter Estimates for the Particle Size Distribution Models...

The third and final particle size distribution model assumes that growth is linear as in the first but that breakup results in predominantly small particles (thorough breakage) which are too small to measure by the electronic particle counters used to characterize the suspension. Petenate and Glatz (6) have provided analytical solutions for this model. 

Onr discussion of the free energy driving forces will mostly follow the discussions ontlined in Venables et al. [5] and in Markov [3], The extension of the atomistic approach to the organics will come from Verlaak et al. [15]. The elements of the rate equation will follow Venables et al. [5]. Dynamic scaling will then follow as an extension to the rate equations coupled with some assumptions about the island size distributions [23]. For deposition beyond the first layer the comparison of model predictions with experiment will follow Cohen et al. [29]. The height-height correlation description will follow Krim and Palasantzas [30]. 

Compared to the extensive data describing the ocean particulate (10, 11), size distribution data on particulates in fresh water systems and wastewaters are relatively scarce. Particle size distribution data for several low ionic strength solutions are shown in Figure 1 with the water source, particulate counting method, and references as noted. The size frequency distribution of the four heterogeneous suspensions shown can be modeled by a two-parameter power-law distribution function (2) given by the expression... 

The participants in this symposium addressed many of these problems. Methods for characterizing size and chemical composition by size were discussed, and several models of metal and virus adsorption were presented. Modeling particulate dynamics in rivers and the ocean provided new insights into the fate of contaminants associated with particulates. Papers on applications of size distribution measurements for selection, process modeling, and control of solid/liquid separation processes demonstrated the analytical value of particle counting compared to cumulative measurements of particulate concentration. 

FIGURE 1.200 Pore size distributions (model of voids between spherical particles) for initial and hydrothermally treated nanosilicas. (Adapted from J. Colloid Interface ScL, 269, Gun ko, V.M., Skubiszewska-Zi ba, J., Leboda, R. et al., Influence of morphology and composition of fumed oxides on changes in their structural and adsorptive characteristics on hydrothermal treatment at different temperatures, 403-424, 2004a. Copyright 2004, with permission from Elsevier.)... 

Figure 9.13 The grain size distribution function for three theoretical distributions and that obtained from a computer simulation employing the Monte Carlo procedure lognormal distribution (solid curve), Hillert s model (dotted curve), Louat s model (dashed curve), and computer simulation (histogram). (From Ref. 22.)...

The isosteric heats of adsorption of methane in BPL activated carbon and ethane in MCM-41 were obtained by Monte Carlo simulation. The simulated absolute isosteric heats were converted into their experimental excess counterparts using a thermodynamie equation, which was derived by the thermodynamic analysis of the Clausius-Clqieyron equation for the isosteric heats. The difference between absolute and excess adsorption is small at low pressure in small pores but becomes bigger as the pressure increases, and is substantial in pores with a pore size bigger than 20 A even at low pressures. Excellent fits were obtained between experimental and simulated isosteric heats of adsorption of methane in BPL activated carbon and ethane in an MCM-41 sample. A pore size distribution model was used to relate simulation results for pores of different sizes to the experimental adsorbent. It is found that the isosteric heat is a more sensitive measure of the structure of activated carbon adsorbents than an adsorption isotherm. 

Nitrogen Capillary Condensation. Application of the Kelvin equation generally assumes circular pores but in reality for non-circular shapes the Kelvin equation evaluates a volume-surface capillary ratio. In view of the uncertainty of the Kelvin equation in terms of the variation observed between the adsorbed and the bulk physical properties of nitrogen, together with the inconsistency of t-curves and BET coefficients excessive refinement of the pore size distribution model has little warranty (13,17) and thus the modelless treatment was chosen (18). 

The computational model used assumes using the adsorption branch, that all pores with a radius larger than a specific value, r, determined from the Kelvin equation and relative pressure of the isotherm, will contain an adsorbed layer of thickness t. Pores smaller than r will be filled with capilllary condensed liquid. The pore size distribution (dN2) for 5A and 820 materials are presented in Figures 1-6. Calculation of pore sizes from the adsorption branch of the isotherm is not usually continued below a pore diameter of 1.0 nm because of the uncertain validity of the Kelvin equation. 

By using the morphology previously established, the authors were able to generate a set of model Au nanoparticles covering the whole range of sizes observed in the experimental distributions shown in Fig. 2.24. They analysed the nanostructural characteristics, i.e. the coordination number, of the Au atoms present at the surface of each of the model particles, and applied the corresponding results to the experimental size distributions determined for Au/CZ... 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Figure Bl.14.13. Derivation of the droplet size distribution in a cream layer of a decane/water emulsion from PGSE data. The inset shows the signal attenuation as a fiinction of the gradient strength for diflfiision weighting recorded at each position (top trace = bottom of cream). A Stokes-based velocity model (solid lines) was fitted to the experimental data (solid circles). The curious horizontal trace in the centre of the plot is due to partial volume filling at the water/cream interface. The droplet size distribution of the emulsion was calculated as a fiinction of height from these NMR data. The most intense narrowest distribution occurs at the base of the cream and the curves proceed logically up tlirough the cream in steps of 0.041 cm. It is concluded from these data that the biggest droplets are found at the top and the smallest at the bottom of tlie cream.

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