Sphere number concentration

In particular, variance can be expressed in terms of the chemical potential derivative for a gas composed of hard spheres over the sphere number concentration. 

The frequency of binary encounters during perikinesis is determined by considering the process as that of diffusion of spheres (radius a2 and number concentration n2) towards a central reference sphere of radius a, whence the frequency of collision I is given by 27 ... 

The number of moles of the solid chemical in the sphere is N = (4/3)ttPpslm, and this is diminished by the mass transfer from the falling sphere into the stagnant fluid. At the surface of the sphere the concentration of the solute is 5, the saturation solubility, and far from the sphere it is zero. Thus,... 

The angular scattering approach is the principal aim of this work. The properties of the Lorenz-Mie intensity coefficients are treated in some detail in order to illustrate their utilization for the determination of particle size distribution, refractive index and number concentration. In a related paper the internal structure of polymer latex spheres is considered (8 ). ... 

Over all, the mean ion concentrations of species of valence v in the suspensions differ from that in region m by rv per sphere times sphere number density N ... 

A suspension of long rods with aspect ratio of 50 or more can only be considered dilute if its concentration is very low, less than 1 % by volume. The reason is that diluteness requires that the rods be able to rotate freely without being impeded by neighboring rods (Fig. 6-18a), and the volume that a single long rod can sweep out by rotation about its center of mass must be large, around. Thus, rod-rod interactions should be expected when the number concentration of rods, y, reaches a value proportional to 7. . Experimentally (Mori et al. 1982), found that the transition occurs at more than 30 times this estimate, apparently because a rod can easily dodge several other rods that invade its sphere of rotation. Thus... 

The maximum q-values (first maximum) are plotted in Fig. 53 for both, H+ and Cs+ counter-ions. No clear picture evolves as the slope of the sample with Cs+ counter-ions is 0.44 whereas that for H+ counter-ions is 0.34. The dotted line in Fig. 53 would result according to qmax = - j with < d > the mean distance of the particles if the particles were evenly distributed throughout the solution, i.e. formed a fee lattice of spheres. For rod like particles this is only to be expected if the interparticle distance is much larger than the rod length I, i.e. NPlt-,j, where Ny is the number concentration. This is shown by the arrow in Fig. 53 assuming a cylinder length per mono-... 

The limiting Sherwood number of 2.0 corresponds to an effective film thickness of DJ2 if the mass-transfer area is taken as the external area of the sphere. The concentration gradients actually extend out to infinity in this case, but the mass-transfer area also increases with distance from the surface, so the effective film thickness is much less than might be estimated from the shape of the concentration profile. 

The initial condition assumes that the particles are initially homogeneously distributed in space with a number concentration Nq. The first boundary condition requires that the number concentration of particles infinitely far from the particle absorbing sphere not be influenced by it. Finally, the boundary condition at r = 2 Rp expresses the assumption that the fixed particle is a perfect absorber, that is, that particles adhere at every collision. Although little is known quantitatively about the sticking probability of two colliding aerosol particles, their low kinetic energy makes bounce-off unlikely. We shall therefore assume here a unity sticking probability. 

In order to close these expressions for particulate pressures, we also need equations for the variance of total particle volume concentration in an assemblage of particles belonging to the two different types. For an arbitrary polydisperse particulate pseudo-gas, variances of partial volume concentrations for different particles can be evaluated on the basis of the thermodynamical theory of fluctuations. According to this theory, these variances are expressible in terms of the minors of a matrix that consists of the cross derivatives of the chemical potentials for particles of different species over the partial number concentrations of such particles [39]. For a binary pseudo-gas, these chemical potentials can be expressed as functions of number concentrations using the statistical theory of binary hard sphere mixtures developed in reference [77]. However, such a procedure leads to a very cumbersome and inconvenient final equation for the desired variance. To simplify the matter, it has been suggested in reference [76] to ignore a slight difference between this variance and the similar quantity for a monodisperse system of spherical particles of the same volume concentration. This means that the variance under question may be approximately described by Equation 7.4 even in the case of binary mixtures. 

FIGURE 8.6 The dependence of the structure factor on qR for polystyrene spheres (i =45nm) immersed in deionized water adapted from figures in [83], with the number concentration n/par-ticles-mm =2.53, 5.06, 7.59, and 10.12 for the circles with increasing depth of the shading [10]. Copyright 2000, John Wiley Sons Ltd. 

Here, d is diameter, U is velocity, is particle number concentration (m ) in a control volume AV, ij is coUision efficiency, and At is a time step for a droplet to move from cme positirai to another. The subscripts p and d denote the physical quantities relative to particle and droplet, respectively. The particle-droplet relative velocity and the particle number concentration along the droplet moving path can be obtained based on a Lagrangian tracking. The collision efficiency i/ is defined as the ratio of the number of particles which collide with the spheroid to the number of particles which could collide wifli the spheroid if their trajectories were straight lines. In the case of laminar flow past a sphere, where the particles are uniformly distributed in the incident flow, the collision efficiency can be determined as t] = 2ycildd), where is the distance from the central symmetry axis of the flow, at which the particles only touch the sphere while flowing past it and is the diameter of the sphere (as shown in Fig. 18.30). The particles, whose coordinates in the incident flow are y > jcn will not collide with the sphere. In Schuch and Loffler [33], the collision efficiency rj is correlated with Stokes number (St) as... 

This exercise was reported in reference [72] for the simplest example of a colloidal mixture, namely, a binary mixture of hard-spheres with diameters Oj and 02, and number concentrations and within the PY [79] approximation for 5 p( ). The asymmetry parameter 6 (= 0 /02 < 1), the total volume fraction ( ) = < ), -f < )2 (with ( ) = nria[Pg.24]

Since cloud motion depends on the bulk properties of aerosols, we evaluate first the viscosity of an aerosol cloud relative to that of pure air. Consider a monodisperse aerosol of 1-pm spheres of standard density at a mass concentration of 100 g/m. This is a very high mass concentration and corresponds to the densest smokes that can be produced. The number concentration of such an aerosol is 2 x 10 /cm. The average spacing between particles is 17 pm, or 17 times the diameter of the particles, so foe aerosol is still mostly air, and foe particles are far enough apart to have little effect on each other. The viscosity of such a two-phase system is... 

If P is a point on a given plane and r is a positive number, the circle with center P and radius r is the set of all points of the plane whose distance from P is equal to r. The sphere with center P and radius r is the set of all points in space whose distance from P is equal to r. Two or more circles (or spheres) with the same P, but different values of r are concentric. 

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