Sphere, diameter volume

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. 

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]

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 effective hard sphere diameter may be used to estimate the excluded volume of the particles, and hence the low shear limiting viscosity by modifying Equation (3.56). The liquid/solid transition of these charged particles will occur at... 

The term pair potential that contains only the attractive potential, because the repulsion effects have been allowed for by the effective volume fraction and hard sphere diameter. The new potential can be defined as... 

A to plug the pore. Assuming the substituent with 6-6.3 A diameter as a sphere, the volume is calculated as being about 78 cm3/mol which is very close to the optimum van der Waals volume estimated from Eq. 59, which is about 76 (= [AVW + VW(H)] x 10). 

Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(pL - pG)dJa, where pL = liquid density pG = gas density, de = bubble diameter, a = surface tension, and the equivalent diameter de is the diameter of a sphere with volume equal to that of... 

Surface Diameter, Volume Diameter, and Sauter s Diameter The surface diameter, ds, volume diameter, d, and Sauter s diameter, d, are defined such that each of them reflects a three-dimensional geometric characteristic of an individual particle. A surface diameter is given as the diameter of a sphere having the same surface area as the particle, which is expressed by... 

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

Mixing by molecular diffusion. This is the ultimate and finally the only process really able to mix the components of a fluid to the molecular scale. The time constant for this process is the diffusion time t = yLy is a shape factor and L is the ratio of the volume to the external surface area of the particle. For instance let us consider various shapes slabs (thickness 2R, case of lamellar structure with striation thickness 6 = 2R), long cylinders (diameter 2R, case of filamentous structure), and spheres (diameter 2R, case of spherical aggregates). 

Compare relative dimensions of a sphere, platelet, and fiber, assuming that the fiber element diameter and platelet thickness are one-tenth the sphere diameter and that the volumes of the sphere, platelet, and fiber are equal. Assume a circular cross-section. 

P. Liquid drops in immiscible liquid, free rise or fall, discontinuous phase coefficient, oscillating drops T L,d,0icdp Nsh- 032(pA (a3g ph010 [E] Used with a log mean mole fraction difference. Based on ends of extraction column. Nfedrop = 411 < < 3114 dp = diameter of sphere with volume of drop. Average absolute deviation from data, 10.5%. Low interfacial tension (3.5-5.8 dyn), [ic < 1.35 centipoise. [141] p. 406 [144] p. 435 [145]... 

A chemical engineer who is designing a drug factory has to solve a problem which concerns the building of a spherical reservoir over a conical support as shown in Fig. 3.72. For this construction, the total volume must not exceed 5 m. At the same time, the relation between the sphere diameter and the cone height is imposed in accordance with the golden section principle (D = I/5). 

Figure 12 shows Chong et al. (28) data for monodisperse and bidisperse (bimodal) suspension systems. In a bidisperse suspension, the volume fraction of small spheres (diameter d) in the mixture is kept constant at 25% of the total solids. The figure shows that the viscosity of a bidisperse suspension is a strong function of the particle size ratio, d/D, where D is the diameter of the large particles. The viscosity decreases substantially by decreasing d/D at a given total solids concentration. The data for the unimodal system fall well above the bimodal suspensions. Also, the effect of particle size distribution decreases at lower values of total solids concentration. 

Fig. 8. Illustration of two monoclonal antibodies bound to a flattened LDL. If a sphere of diameter 208 A was uniformly flattened, holding volume constant, until it had an apparent diameter of 285 A in projection, it might resemble half of an oblate ellipsoid. A truncated oblate ellipsoid with a major axis of 285 A and a semiminor axis of 111 A would have the same volume as a 208-A sphere. Diameters of this size or larger were frequently observed. Monoclonal antibody A has bound to the LDL above the layer of negative stain, and would not be observed. Monoclonal antibody B has bound to the LDL in the plane of the EM grid, and has displaced the layer of negative stain, so that it would be observed in the electron micrograph.

In Fig. 6.18 this distance dependence is plotted for several sphere diameters. The effective ranges of several interaction forces is also indicated. We see that the mean distance becomes of the order of the interaction range at a volume fraction of about 0.2 for particles of 50 nm and about 0.5-0.6 for particles of 1000 nm. Below these volume fractions the suspension can be considered as diluted or semi-diluted and above these volume fractions as concentrated. It appears that the smaller the particle size, the lower the volume fraction for the transition to the concentrated regime. 

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