Volume fraction density

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

The mean particle length calculated from this volume-fraction density function is... 

Urick (71) has shown that the sonic velocity through a mixture of solid particles in a fluid is dependent upon the volume fraction, / , density, q2 and compressibility, K2 of the particles and upon the density, et, and compressibility, Kt, of the fluid. His equation is... 

The interaction parameter of poly(methyl methacrylate) (PMMA) in toluene at 300 K is reported to be 0.450. Calculate the free energy resulting from mixing a PMMA sample with molecular weight 50,000 g moT in toluene at 10% volume fraction (density of PMMA = 1.13 g cmr, density of toluene = 0.862 g cm , and total volume of solution = 1 cm ). 

Dynamic Mobility - Variation with Volume Fraction Density l.I g/cnu 1 Micron Dia. 

Volume fraction Density difference between continuous and dispersed phases. Container height... 

Where a, p, pif, u- are the volume fraction, density, transfer coefficient (viscosity), and velocity vector of the phase respectively (indices 1 - continuous phase, and 2 - dispersed phase). 

The volume fraction, density and viscosity of T phase respectively... 

The denominator above represents the total volume fraction of all particles. Similarly, mass fractions can also be readily defined. For the case of scalar internal state using only particle size (volume) denoting the number density by /i(v, r, t), the volume fraction density of particles of volume v becomes... 

Sample Fiber Volume fracture/% ZrC volume fraction /% Density g/cin Open porosity/% Bending stress /MPa Elastic modulus /GPa... 

Fig. XI-7. Volume fraction profile of 280,000-molecular-weight poly(ethylene oxide) adsorbed onto deuterated polystyrene latex at a surface density of 1.21 mg/m and suspended in D2O, from Ref. 70.

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

Samples can be concentrated beyond tire glass transition. If tliis is done quickly enough to prevent crystallization, tliis ultimately leads to a random close-packed stmcture, witli a volume fraction (j) 0.64. Close-packed stmctures, such as fee, have a maximum packing density of (]) p = 0.74. The crystallization kinetics are strongly concentration dependent. The nucleation rate is fastest near tire melting concentration. On increasing concentration, tire nucleation process is arrested. This has been found to occur at tire glass transition [82]. 

The units of [77] reveal the concentration units in this experiment to be grams of protein per cubic centimer of solution. Dividing this concentration unit by the density of the unsolvated protein converts these concentration units to volume fractions ... 

Physical Properties. Physical properties of importance include particle size, density, volume fraction of intraparticle and extraparticle voids when packed into adsorbent beds, strength, attrition resistance, and dustiness. These properties can be varied intentionally to tailor adsorbents to specific apphcations (See Adsorption liquid separation Aluminum compounds, aluminum oxide (alumna) Carbon, activated carbon Ion exchange Molecular sieves and Silicon compounds, synthetic inorganic silicates). 

As a good first approximation (187), the heat conduction of low density foams through the soHd and gas phases can be expressed as the product of the thermal conductivity of each phase times its volume fraction. Most rigid polymers have thermal conductivities of 0.07-0.28 W/(m-K) and the corresponding conduction through the soHd phase of a 32 kg/m (2 lbs/fT) foam (3 vol %) ranges 0.003-0.009 W/(m-K). In most cellular polymers this value is deterrnined primarily by the density of the foam and the polymer-phase composition. Smaller variations can result from changes in cell stmcture. 

Foams have a wide variety of appHcations that exploit their different physical properties. The low density, or high volume fraction of gas, enable foams to float on top of other fluids and to fiU large volumes with relatively Httle fluid material. These features are of particular importance in their use for fire fighting. The very high internal surface area of foams makes them useful in many separation processes. The unique rheology of foams also results in a wide variety of uses, as a foam can behave as a soHd, while stiH being able to flow once its yield stress is exceeded. 

Eig. 10. The variation of the density of carbon-fiber reinforced epoxy resin with the fiber volume fraction, based on the rule of mixtures. 

Cake Dryness In dewatering, usually the cake needs to be as diy as possible. Cake diyness is commonly measured by the solids fraction by weight W or by volume ,. The moisture content is measured by the complement of W or ,. The volume fraction of the pores and void in the wet cake is measured by the cake porosity (= 1 — ,) whereas the volume fraction of the liquid in the pores of the cake is measured by the saturation S. For welf-defined solids in the cake with sohd density (bone diy) p, and hquid density Pl, and given that the cake volume... 

FIG. 20-71 Classification of agglomeration processes by agitation intensity and compaction pressure. Relative density is with respect to primary particle density and equals (1 — e) where e is the solid volume fraction. Reprinted from Granulation and Coating Technologies for High-Value-Added Industties, Ennis and Litster (1996) with permission of E G Associates. All rights reserved. 

The properties of a foam are determined by the properties of the polymer, and by the relative density, p/p - the density of the foam (p) divided by that of the solid (p ) of which it is made. This plays the role of the volume fraction Vf of fibres in a composite, and all the equations for foam properties contain p/p. It can vary widely, from 0.5 for a dense foam to 0.005 for a particularly light one. 

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