Effective volume

Computer Models, The actual residence time for waste destmction can be quite different from the superficial value calculated by dividing the chamber volume by the volumetric flow rate. The large activation energies for chemical reaction, and the sensitivity of reaction rates to oxidant concentration, mean that the presence of cold spots or oxidant deficient zones render such subvolumes ineffective. Poor flow patterns, ie, dead zones and bypassing, can also contribute to loss of effective volume. The tools of computational fluid dynamics (qv) are useful in assessing the extent to which the actual profiles of velocity, temperature, and oxidant concentration deviate from the ideal (40). 

Viscosity. Because a clump of particles contains occluded Hquid, the effective volume fraction of a suspension of clumps is larger than the volume fraction of the individual particles that is, there is less free Hquid available to faciHtate the flow than if the clumps were deagglomerated. The viscosity of a suspension containing clumps decreases as the system becomes deagglomerated. This method is not very sensitive in the final stages of deagglomeration when there are only a few small clumps left. 

A conservative assumption is that O can be set equal to zero. When the stress O equals the characteristic strength O the faUure probabUity is 63.2%. Under conditions other than tensUe loading, the stress distribution in a body is inhomogeneous. To account for this, a loading factor k is used to calculate the effective volume under stress and kVreplaces V. 

The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

Following the idea of van Laar, Chueh expresses the excess Gibbs energy per unit effective volume as a quadratic function of the effective volume fractions. For a binary mixture, using the unsymmetric convention of normalization, the excess Gibbs energy gE is found from6... 

We may use this example of triangular closest packing to derive an expression for the distribution of spheres in successive layers. The form of the expression (number of spheres proportional to the cube of a length, the radius) reflects the assumption of constancy of effective volume per sphere. The expression is... 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

Increasing the bound mbber content increases the effective volume fraction of filler by intimately bonding polymer to the filler. This polymer is no longer available to contribute to viscous flow. As a consequence, the viscosity of the compound increases. 

Tjo is the viscosity of the unfilled polymer ( eff is the effective volume fraction of filler... 

By relating the endpoint of crushed DBF absorption to the void space within and between equivalent spheres of aggregates, and assuming the spheres to be packed at random, Wang et al. obtained the following equation for the effective volume fraction of carbon black ... 

E Taft steric parameter. E reflects the effective volume... 

The frictional coefficient varies with concentration, but at infinite dilution it reduces to the coefficient (/o) for an isolated polymer molecule moving through the surrounding fluid unperturbed by movements of other polymer molecules (see Chap. XIV). At finite concentrations, however, the motion of the solvent in the vicinity of a given polymer molecule is affected by others nearby binary encounters (as well as ones of higher order) between polymer molecules contribute also to the observed frictional effects. The influence of these interactions will persist to very low concentrations owing to the relatively large effective volume of a polymer molecule, to which attention has been directed repeatedly in this chapter. Since the sedimentation con-stant depends inversely on the frictional coefficient, s must also depend bn concentration. 

According to the concept of free volume as the effective volume over which the centers of gravity of the molecules are distributed, the entropy may be taken as that of a perfect gas composed of the same number of molecules confined to a volume equal to the free volume. Since the entropy of a perfect gas consisting of n molecules depends on its volume as nk InF, the increase in entropy owing to the greater free volume available to the solvent molecules in the solution will be... 

According to the interpretation given above, the intrinsic viscosity is considered to be proportional to the ratio of the effective volume of the molecule in solution divided by its molecular weight. In particular (see Eq. 23), this effective volume is represented as being proportional to the cube of a linear dimension of the randomly coiled polymer chain,... 

Influence of reactor size/shape on process performance via heat transfer. Heat transfer between a reaction mixture and its surroundings strongly depends on the size of the reactor. The amount of heat evolved during an exothermic reaction is proportional to the volume of the reaction mixture (Qgcncrated V), ie. the effective volume of the reactor, whereas the amount of heat removed from the mixture is proportional to the heat-transfer surface area (firemovod A,). Morc precisely, the amount of heat transferred is given by ... 

Dead space is undesirable, since it means reduction in effective volume of the machine. 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

A similar effect may exist for hydrophobic interaction between solute and stationary phase, as one solute may adsorb more strongly to the stationary phase than another. It has also been remarked that a flexible polymer confined to a pore should be at a lower entropy than one in bulk solution, leading to exclusion in excess of that expected for a simple geometric solid.23 Even in the absence of interactions, a high concentration of a small component can lead to an excluded volume effect, since the effective volume inside the pore is reduced. 

Similarly, concepts of solvation must be employed in the measurement of equilibrium quantities to explain some anomalies, primarily the salting-out effect. Addition of an electrolyte to an aqueous solution of a non-electrolyte results in transfer of part of the water to the hydration sheath of the ion, decreasing the amount of free solvent, and the solubility of the nonelectrolyte decreases. This effect depends, however, on the electrolyte selected. In addition, the activity coefficient values (obtained, for example, by measuring the freezing point) can indicate the magnitude of hydration numbers. Exchange of the open structure of pure water for the more compact structure of the hydration sheath is the cause of lower compressibility of the electrolyte solution compared to pure water and of lower apparent volumes of the ions in solution in comparison with their effective volumes in the crystals. Again, this method yields the overall hydration number. 

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