Sphere, diameter

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

The virial pressure equation for hard spheres has a simple fomr detemiined by the density p, the hard sphere diameter a and the distribution fimction at contact g(c+). The derivative of the hard sphere potential is discontinuous at r = o, and... 

Weak electrolytes in which dimerization (as opposed to ion pairing) is the result of chemical bonding between oppositely charged ions have been studied using a sticky electrolyte model (SEM). In this model, a delta fiinction interaction is introduced in the Mayer/-fiinction for the oppositely charged ions at a distance L = a, where a is the hard sphere diameter. The delta fiinction mimics bonding and tire Mayer /-function... 

This implies, with the indicated choice of hard sphere diameter d, that the compressibilities of tlie reference system and the equivalent of the hard sphere system are the same. 

NflaNsc) < 10,000 Nrs < 1-0. Constant sphere diameter. Low mass-transfer rates. [146] p. 214... 

The results for silver particles show the way in which die average particle size of the spheres modifies the map of the predominating mechanisms which depend on the sphere diameter, a, in differing ways as shown above in the variation in the values of m which can be shown in the form of a general equation... 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

NPei and NRtt are based on the equivalent sphere diameters and on the nominal velocities ug and which in turn are based on the holdup of gas and liquid. The Schmidt number is included in the correlation partly because the range of variables covers part of the laminar-flow region (NRei < 1) and the transition region (1 < NRtl < 100) where molecular diffusion may contribute to axial mixing, and partly because the kinematic viscosity (changes of which were found to have no effect on axial mixing) is thereby eliminated from the correlation. 

Kato (K3) measured so-called critical gas velocities corresponding to the complete suspension of solids, and presents a graphical correlation of the results for glass spheres (diameters from 0.074 to 0.295 mm), magnetite particles (particle size from 0.038 to 0.175 mm), and sand particles (particle size 0.147 to 0.295 mm). 

Figure 6. Pore dlfifuslvlty versus pore width. Theory Is for 6-oo LJ fluid with an effective hard sphere diameter cTgff = 0.972. Units of dlfifuslvlty are (3a/8)...

Before concluding this discussion of the excluded volume, it is desirable to introduce the concept of an equivalent impenetrable sphere having a size chosen to give an excluded volume equal to that of the actual polymer molecule. Two such hard spheres can be brought no closer together than the distance at which their centers are separated by the sphere diameter de. At all greater distances the interaction is considered to be zero. Hence / = for a dey and fa = 0 for a[Pg.529]

GPl7][R21] Thepressuredropinafilamentous-bedreactor(i.d. 7 pm porosity 0.8),whichwasusedforpropanedehydrogenation,wascomparedwitharandomly packed powder bed (sphere diameter from 100 to 160 pm) of similar hydraulic... 

The purpose of this test is the same as the previous one. The critical volume is the sphere diameter of a substance, below which it is impossible to obtain detonation conditions under the influence of a firing blasting charge. These tests can only be conducted on shooting ranges and sometimes with large substance quantities. 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

The GvdW density functional for a binary primitive electrolyte with the hard sphere diameters set to zero will have the form... 

Calculations of the capacitance of the mercury/aqueous electrolyte interface near the point of zero charge were performed103 with all hard-sphere diameters taken as 3 A. The results, for various electrolyte concentrations, agreed well with measured capacitances as shown in Table 3. They are a great improvement over what one gets104 when the metal is represented as ideal, i.e., a perfectly conducting hard wall. The temperature dependence of the compact-layer capacitance was also reproduced by these calculations. 

AB effective hard sphere diameter for i in a CSTR network... 

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