Core volume

The steps may be so chosen as to correspond to consecutive points on the experimental isotherm. In practice it is more convenient to divide the desorption process into a number of standard steps, either of relative pressure, or of pore radius, which is of course a function of relative pressure. The amount given up during each step i must be converted into a liquid volume i , (by use of the normal liquid density) in some procedures the conversion is deferred to a late stage in the calculation, but conceptually it is preferable to undertake the conversion at the outset. As indicated earlier, the task then becomes (i) to calculate the contribution dv due to thinning of the adsorbed film, and thus obtain the core volume associated with the mean core radius r by the subtraction = t ... 

For each group of pores, the pore volume 6v is related to the core volume by means of a model, either the cylinder or the parallel-sided slit as the case may be. Allowance is made for the succession of film thicknesses corresponding to the progressive thinning of the multilayer in each pore, as desorption proceeds. Thus for group i, with radius rf when the film thickness is tj j > i) and the core volume is the pore volume 6vf will be given by... 

Equation (3.52) is applied in succession to all steps from step 1 onwards, commencing from the uptake n, where all pores are deemed full (often at p/p° = 0-95 cf. p. 132), to obtain the values of 5 4, 6A2 etc. If no correction is made for the thinning of the multilayer as the emptying process continues, the core volumes will be given by Svf = ( — and the uncorrected... 

Table 1 contains technical data for the newer plants of the Magnox and AGR type. These are operated in the United Kingdom by Nuclear Electric pic. The electrical power output of the AGR is almost three times that of the Magnox, whereas its core volume is less than half as large. 

In the second decrement the liquid volume desorbed iVuq)2 i st be corrected for the decrease in the adsorbed film depth remaining on the walls of previously emptied pores. By assuming the pores are cylindrical, the core volume V )2 can be calculated from the decrease in statistical thickness t, as... 

Fig. 3—Glassy core volume decrease of tablets KET-R vs. time , pH 1.2 pH 6.8.

Solution Equation (4.41) gives the Einstein relationship between [r/] and , the volume fraction occupied by the dispersed spheres. The volume fraction that should be used in this relationship is the value that describes the particles as they actually exist in the dispersion. In this case this includes the volume of the adsorbed layer. For spherical particles of radius R covered by a layer of thickness 8R, the total volume of the particles is (4/3) + 4ttR2 8R. Factoring out the volume of the dry particle gives Vdfy(1 + 38RJRS), which shows by the second term how the volume is increased above the core volume by the adsorbed layer. Since it is the dry volume fraction that is used to describe the concentration of the dispersion and hence to evaluate [77], the Einstein coefficient is increased above 2.5 by the factor (1 + 36/Vfts) by the adsorbed layer. The thickness of adsorbed layers can be extracted from experimental [77] values by this formula. ... 

As a first approximation we postulate the existence of an equally sized hard core volume located in the center of each cylinder that defines the excluded volume per segment, according to the degree of interpenetration of neighboring cylinders. This simple construction includes complicated local intersegmental configurations, but it imposes a unique local orientation correlation of segments within the domains of a PE melt The shape of those impenetrable correlation cylinders is assumed also to be cylindrical. 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

Fig. 4.24 Phase diagram for PS PI diblocks in decane in terms of PS core volume fraction (4>c) as a function of the ratio of coronal layer thickness to core size (RAIRa). Unspecified structures are labelled ( - ) (McConnell et al. 1993).

Fig. 4.45 Structure factors versus wavevector for rfPS-PI diblocks in core-contrast matched decane solutions (Gast 1996 McConnell et al. 1994) (a) dPS,W3PI >o( at cote volume fractions of 0.012 (A), 0.02(+), 0.03( ), 0.04 (A) and 0.05 (o) (b) f/PS Pl at core volume fractions of 0.006 (o), 0.013 ( ) and 0.019 (A). The lines are theoretical fits from the self-consistent field interaction potentials and the Rogers-Young closure to the Ornstein-Zernike equation.

Molecules that self-assemble into reverse micelles with low surfactant properties are generally efficient extractants (such as HDEHP, TBP, malonamides, etc.). Their adsorptions at the interface permit the complexation of the aqueous solute and their low surfactant properties permits the avoidance of the formation of very stable emulsion. Hence, ions are extracted, but typically there is less than one water molecule per ion extracted. Exact determination of coextracted water is still important, however, for interpreting the conductivity values and for evaluating the polar core volumes. Typical values are found for the Hamaker constant, because polar cores are supersaturated salt solution. 

Here is the effective surface fraction of molecules and the tilde ( ) denotes reduced variables. All quantities, except v in the EOS, are reduced by the parameters in Equation 4. The specific volume v, is reduced by v, the molecular hard-core volume,... 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

Mixtures of hydrocarbons are assumed to be athermal by UNIFAC, meaning there is no residual contribution to the activity coefficient. The free volume contribution is considered significant only for mixtures containing polymers and is equal to zero for liquid mixtures. The combinatorial activity coefficient contribution is calculated from the volume and surface area fractions of the molecule or polymer segment. The molecule structural parameters needed to do this are the van der Waals or hard core volumes and surface areas of the molecule relative to those of a standardized polyethylene methylene CH2 segment. UNIFAC for polymers (UNIFAC-FV) calculates in terms of activity (a,-) instead of the activity coefficient and uses weight fractions... 

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