Krieger-Dougherty equation, viscosity

It turns out that the Krieger-Dougherty equation can also be used for intermediate shear rates with suitable modifications. Experimental data also suggest that [77] and 7 are independent of particle size, although they are stress dependent. Therefore viscosities of monodis-persed colloids of different particle sizes can be represented by a single equation by suitably defining the variables. A discussion of these and other extensions may be found in Barnes et al. (1989). 

For Newtonian flow, the viscosity of a concentrated suspension can be calculated from the Krieger-Dougherty equation ... 

Concentrated Dispersions. For the viscosity of not very dilute systems, the Krieger-Dougherty equation is often useful. It reads... 

By a different choice of e value, one can derive many existing viscosity equations (84). For example, when e - 2, equation 39 can lead to the Mooney equation 32. When e - 1, equation 39 reduces to equation 38 and hence leads to the Krieger-Dougherty equation. 

Kim and Luckham (140) suggested that the relative dynamic viscosity of a bidispersed suspension may be estimated by using the product of the two component relative viscosities each computed from the Krieger-Dougherty equation as if they were alone in the suspension. This treatment has been commonly used since Farris (139). However, it is valid only when concentration of either component is large, that is, eL - 1 or L - 0, and the particle sizes are very different from each other, that is, ds dL- When the two concentrations are similar, that is, eL 0.5, the estimation of the apparent viscosity using this approach gives a much lower value than the experimental values. This behavior is same for dynamic viscosity as the steady shear viscosity. 

Equations 66 and 68 indicate that the droplet behaves like a solid particle only when the viscosity ratio of the dispersed phase to the continuous phase is large. For liquid-in-liquid dispersions, the modified Quemada equation, Krieger-Dougherty equation, and Mooney equation are still applicable provided that the maximum packing limit and the Einstein constant are left as adjustable parameters for a given system. 

For small values, the third and fourth terms in parenibeses are of little importance and Eq. [52] tends to Einstein s equation, which predicts a linear relationship for sufficiently low. As increases, the last term becomes more important, giving an exponential increase of viscosity. This behavior has been verified experimentally in many types of suspensions and emulstons (5.13-16). A. second expression that is often cited i.s the Krieger-Dougherty equation... 

On the other hand, the adlayer thickness has also an influence on the viscosity of the dispersion. It is generally known that the viseosity 7 of a particle dispersion depends on its solids volume fraction O. The particles start to form a network as d> is increased, which leads to an increase in relative viscosity rjr as described in the modified Krieger-Dougherty equation... 

When the particles in suspension are non-spherical, the rotation of the particles due to Brownian motion results in an excluded volume, which is higher than the volume-fraction of the particles. As the degree of anisotropy increases, the effects become more dramatic. Figure 9.6 shows the volume-fraction dependence of the viscosity of silicon nitride, alumina and silicon carbide whisker (SiCw) suspensions. The experimental points were fitted to a modified Krieger-Dougherty equation ... 

Figure 38 The relative viscosities (no/tts) for CCS polymer solutions as a function of the effective volume fraction (peft). The solvent used was diethyl phthalate and all measurements were recorded at 25 0.1 °C. The Krieger-Dougherty equation ( >p = 0.6) is also shown. Reprinted from Goh, T. K. Coventry, K. D. Blencowe, A. Qiao, G. G. Polymer 2008, 49, 5095. ° ...

Good prediction was obtained for the zero-shear viscosity of latexes and plash-sols [56] computed from the Krieger-Dougherty equation, Eq. (2.10), with 0m obtained from Eq. (2.12). 

Assuming that the suspension viscosity can be described by the Krieger-Dougherty equation, determine the relative viscosity for suspensions containing 40 vol% of solid spherical particles when... 

Figure 5.3 Determination of theoretical maximum packing fraction = K" ) in the Krieger-Dougherty equation [eqn. (5.9)] for the experimental data of Figure 5.2. The quantity where q, is the relative viscosity at stress r = 6 Pa, is plotted against the oil volume fraction 0 (Reproduced by permission of Elsevier Science Ltd. from ref 71)...

Polynomial vs. Krieger-Dougherty equations for relative viscosity variations with respect to maximum particle packing fraction. 

In general, these formulas hold well for < 0.10, while empirical models using higher-order polynomials in can be used to fit almost any viscosity data. An equation that is often used for emulsions and concentrated suspensions of rigid spheres is that by Krieger and Dougherty ... 

Instead, a semiempirical approach is used to develop equations that describe the viscosity of concentrated colloidal dispersions. One of the most widely used equations was derived by Dougherty and Krieger and is applicable across the whole volume fraction range (Figure 9) (9, 109). 

The relative viscosity is related to the volume fraction g> by the Dougherty- Krieger equation [40] for hard spheres,... 

Steady-state shear stress-shear rate curves were used to obtain the relative viscosity (//,.)-volume fraction () relationship for the latex and emulsion. The results are shown in Figure 11.19 which also contains the theoretically predicted curve based on the Dougherty-Krieger equation [14],... 

It may be pos sible to correlate the change in the rate of sedimentation with increasing cp with the viscosity of the suspension, as predicted by the Dougherty-Krieger equation [3],... 

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