Core radii

Note tliat here /c, = 2jt/4 and a is tire core radius. The parameter Vdetennines tire number of modes supported by... 

Fig. 3.8 Relation between r of the Kelvin equation (Equation (3.20)) and the core radius r for a cylindrical pore with a hemispherical meniscus 6 is the angle of contact.

Thus, as pointed out by Cohan who first suggested this model, condensation and evaporation occur at difi erent relative pressures and there is hysteresis. The value of r calculated by the standard Kelvin equation (3.20) for a given uptake, will be equal to the core radius r,. if the desorption branch of the hysteresis loop is used, but equal to twice the core radius if the adsorption branch is used. The two values of should, of course, be the same in practice this is rarely found to be so. 

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. 

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 ... 

Psueudopotentials should satisfy several basic requirements. For example, the pseudo and real

wave functions must be identical outside the core radius (>rc), not only in their spatial dependence but also in their absolute magnitudes such that two wave functions generate identical charge densities. The equality of the two types of wave functions outside the core radius in this context is guaranteed by imposing the following constraint ... 

Although the pseudopotential is, from its definition, a nonlocal operator, it is often represented approximately as a multiplicative potential. Parameters in some chosen functional form for this potential are chosen so that calculations of some physical properties, using this potential, give results agreeing with experiment. It is often the case that many properties can be calculated correctly with the same potential.43 One of the simplest forms for an atomic model effective potential is that of Ashcroft44 r l0(r — Rc), where the parameter is the core radius Rc and 6 is a step-function. 

Horio s scaling law derivation was based on the requirement that two similar circulating fluidized beds have equal values of voidage distribution, dimensionless core radius, gas splitting to core and annulus, solid splitting to core and annulus, and cluster voidage. The CAFM equations were then examined to determine how these requirements could be met. 

The earlier evaluation of the core radius was in terms of Bohr s impulse condition (see Sect. 2.3.3) at (relatively) high energies. This gives the core radius as 30 A at a particle energy of 10 MeV/amu. For much lower energies, this relation is unrealistic, since electrons ejected in glancing collisions penetrate... 

The hard-core radius of the solvent molecules is absent in FFK. 

If quark matter is in the ferromagnetic phase, it may produce the dipolar magnetic field by their magnetic moment. Since the total magnetic dipole moment Mq should be simply given as Mq = fjq (47t/3 rq)nq for the quark sphere with the quark core radius rq and the quark number density nq. Then the dipolar magnetic field at the star surface R takes the maximal strength at the poles,... 

Solution of the entire pseudo-steady state problem (commonly referred to as the shrinking core model) is achieved by analytical integration of eqn. (53) and substitution of the result into eqn. (55), subsequently eliminating the unknown Ca by the use of eqn. (54). Substitution into eqn. (56) then gives the overall reaction rate in terms of CAg, and r. This result is not particuleirly useful, however, until the shrinking core radius, r, is related to time. Recalling the chemical stoichiometric relationship [eqn. (50)] the rate of consumption of A in terms of the core radius is... 

For a cylinder with an original radius Rq reacting with a core radius... 

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