Effective medium approach viscosity

Note that err = y (crr)a3/k Tand recall that in a concentrated dispersion the Peclet number is Pe = 67ry (crr)a3/k T. The use of the suspension viscosity implies that the particle diffusion can be estimated from an effective medium approach. Both Krieger and Cross gave the power law indices (n and m) as 1 for monodisperse spherical particles. In this formulation, the subscript c indicates the characteristic value of the reduced stress or Peclet number at the mid-point of the viscosity curve. The expected value of Pec is 1, as this is the point at which diffusional and convective timescales are equal. This will give a value of ac 5 x 10 2. Figure 3.15 shows a plot of Equation (3.57a) with this value and n = 1... 

Moreover, v is the fluid velocity, pf and p, are the density and the viscosity of the fluid, respectively (f) and K are porosity and permeability of the core c/.,s are specific heat of the fluid and of the solid respectively Cfast>siow are the sound propagation speed of the fast and slow waves I fast,slow are the intensities of the fast and slow waves, while a.fast,siow are their damping coefficients. We use an effective medium approach for the liquid, describing the effect of the acoustic waves as source terms. There are two source terms. First there... 

Approach (iii) listed above refers to the use of effective medium theory (Kirkpatrick, 1973 Koplik, 1982 Levine and Cuthiell, 1986) for calculating certain average flow properties in idealised porous media models—usually simple networks. Cannella et al (1988) have recently applied this approach to the flow of power law fluids through networks of capillaries. They use this method to derive an expression for the apparent viscosity of the polymer in the porous medium which has the same overall form as the capillary bundle expression (e.g. Equation 6.18). They then adjusted the parameters in the effective medium formula in order to match their particular form of the capillary bundle formula with C = 6 (Equation 6.18). The values of the effective medium parameters are physically interpretable, and Cannella et al (1988) deduced from these that the effective radius for the flow of a power law fluid is larger than that for the flow of a Newtonian fluid. They also... 

Considerable red-edge effects exhibiting a dependence on the viscosity of the medium are observed for model solutions of indole and tryptophan(33) (Figure 2.11a), which permits this approach to be applied to studies of the dynamics of the environment of tryptophan residues in proteins. In discussion... 

Coagulation, the result of approach, contact and coalescence of the particles of the suspensoid, is evidently hindered by any factor which may retard one of these three actions. The approach of one particle to another is brought about by the thermal or Brownian movement of the particles within the medium and factors such as low temperature, high viscosity of the medium or large particle size will evidently diminish the rate of approach. When the particles are in close proximity to one another, actual contact will be prevented if the particles possess electric charges similar in sign, due to the action of electrostatic repulsion. The particles will possess no net charge, i.e. their surface will be covered with the same number of cations and anions and will not repel one another at the isoelectric point when the capillary attraction can operate effectively (Hardy, Proc. Roy. Soo. LXVI. 110,1900). 

As in other oxidation processes, increased temperatures have an adverse effect which aiso ascribed to the resuiting increased vapour pressure ieading to easier cavitation, but iess vioient coiiapse, as a consequence of the decreased viscosity and surface tension. As the temperature approaches the soivent boiiing point, a iarge number of cavitation bubbies are formed concurrentiy that act as a barrier to sound transmission and dampen the effective US energy from the source to enter the iiquid medium. A temperature dose to room ievei is easy to maintain and ensures proper deveiopment of the process. 

Any increase in temperature will raise the vapor pressure of a medium and so lead to easier cavitation but less violent collapse (see above). This effect will be accompanied by a decrease in viscosity and surface tension. However, at temperatures approaching the solvent boiling point, a large number of cavitation bubbles are generated concurrently. These will act as a barrier to sound transmission and dampen the effective ultrasonic energy from the source which enters the liquid medium. The combination of all these effects shows a shape of maximum and the optimum temperature depends on the experimental conditions used and reaction studied. 

In Equation 3.54, r] is the viscosity of the polymerizing medium, t]o the viscosity of the pure monomer, and k rj represents the rate coefficient of translational diffusion at zero conversion fit to experimental data [10]. Equation 3.55 has been used to fit, for example, styrene polymerization rate data to high conversion over a range of temperatures [16]. In the free-volume (vf) approach [48, 49], parameters are fitted to experimental data, with the effect of polymer MW on system viscosity being captured by expressing Ci as a function of Mw Further details and variations of these modeling approaches can be found in the literature [10, 50-53]. 

The fluorescence quantum yields of certain compounds exhibit a strong dependence on the viscosity of the medium. As early as 1913, Stark (5) noticed that some dyes which do not fluoresce in ordinary solvents will, however, fluoresce strongly in highly viscous media such as glycerol at low temperatures. A number of studies on the viscosity-dependence of the fluorescence quantum yields of various compounds have appeared (6-12). Particularly noteworthy among these are the work of Oster and Nishijima (6), Forster and Hoffman (7), and Sharafy and Muszkat (8). However, Loutfy (13,14) was the first to exploit this effect for monitoring polymerization reactions. He applied the viscosity-dependent fluorescence probe approach for study of polymerization of vinyl monomers and discussed the potential of such probes for the study of polymeric systems... 

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