Suspensions conditional probability density

Equations (8.10)—(8.12), tensorial ranks and boundary conditions (8.14)-(8.15) notwithstanding, embody a structure similar in format and symbolism to their counterparts for the transport of passive scalars, e.g., the material transport of the scalar probability density P (Brenner, 1980b Brenner and Adler, 1982), at least in the absence of convective transport. As such, by analogy to the case of nonconvective material transport, the effective kinematic viscosity viJkl of the suspension may be obtained by matching the total spatial moments of the probability density Pu to those of an equivalent coarse-grained dyadic probability density P j, valid on the suspension scale, using a scheme (Brenner and Adler, 1982) identical in conception to that used to determine the effective diffusivity for material transport at the Darcy scale from the analogous scalar material probability density P. In particular, the second-order total moment M(2) (sM, ) of the probability density P, defined as... 

Figure 11. Distance required to remove 99% of particles from suspension (L ) as a function of suspended particle radius for two chemical conditions (attachment probabilities). Flow rate = O.lm/day, media radius = 0.025 cm, temperature = 25 °C, particle density = 1.05 g/cm3, and aquifer porosity = 0.4. (Reproduced with permission from reference 29. Copyright 1987.)...

As an example, let us consider suspension polymerization. In suspension polymerization, the monomers from the monomer droplets transfer into the polymer particles as long as monomer droplets exist. Thus, regarding the change of monomer concentration in the space of polymerization, that is, polymer particles, this reaction can be considered a semibatch reactor from the viewpoint of the change of monomer concentration. Figure 10 depicts the calculated results of crosslink density distribution of suspension polymerization and homogeneous phase polymerization under Flory s simplified conditions [48, 49]. In Fig. 10, the abscissa indicates the reactivity (0) at ttie time of each primary polymer formation and the ordinate shows the probability of crosslink density of each primary polymer. In the figure, for example, the line with l/ = 0.6... 

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