Concentrated suspensions apparent viscosity

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

Because concentrated flocculated suspensions generally have high apparent viscosities at the shear rates existing in pipelines, they are frequently transported under laminar flow conditions. Pressure drops are then readily calculated from their rheology, as described in Chapter 3. When the flow is turbulent, the pressure drop is difficult to predict accurately and will generally be somewhat less than that calculated assuming Newtonian behaviour. As the Reynolds number becomes greater, the effects of non-Newtonian behaviour become... 

The flow properties of suspensions are complex. The apparent viscosity at a given shear rate increases with increasing solids concentration and rises extremely rapidly when the volume fraction of solids reaches about 50 per cent. The flow properties also depend on the particle size distribution and the particle shape, as well as the flow properties of the suspending liquid. 

Not all suspensions will exhibit wall slip. Concentrated suspensions of finely ground coal in water have been found to exhibit wall slip [Fitzgerald (1990)]. This is to be expected because the coal suspension has a much higher apparent viscosity than the water. In contrast, when the liquid is a very viscous gum, the addition of solids may have a relatively small effect. In this case, the layer at the wall will behave only marginally differently from the material in the bulk. 

Several attempts have been made to predict the apparent settling velocity of a concentrated suspension. In 1926 Robinson(3) suggested a modification of Stokes law and used the density (pc) and viscosity (p.c) of the suspension in place of the properties of the fluid to give ... 

As the shear rate increases, the viscosity of some dispersions actually increases. This is called dilatancy, or shear-thickening. Dilatancy can be due to the dense packing of particles in very concentrated dispersions for which at low shear, the particles can just move past each other but at high shear they become wedged together such that the fluid cannot fill (lubricate) the increased void volume, and the viscosity increases. An example of this effect is the apparent drying of wet beach sand when walked on, the sand in the footprint initially appears very dry and then moistens a few seconds later. Other examples include concentrated suspensions (plastisols) of polyvinyl chloride (PVC) particles in plasticizer liquid and the commercial novelty product Silly Putty (which is a silicone material). 

Because no general theories exist even for concentrated non-food suspensions of well defined spherical particles (Jeffrey and Acrivos, 1976 Metzner, 1985), approaches to studying the influence of the viscosity of the continuous medium (serum) and the pulp content of PF dispersions, just as for non-food suspensions, have been empirical. In PF dispersions, the two media can be separated by centrifugation and their characteristics studied separately (Mizrahi and Berk, 1970). One model that was proposed for relating the apparent viscosity of food suspensions is (Rao, 1987) ... 

Dilatant Fluids. Dilatant fluids or shear-thickening fluids are less commonly encountered than pseudoplastic (shear-thinning) fluids. Rheological dilatancy refers to an increase in the apparent viscosity with increasing shear rate (3). In many cases, viscometric data for a shear-thickening fluid can be fit by using the power law model with n > 1. Examples of fluids that are shear-thickening are concentrated solids suspensions. 

Shear thinning of concentrated suspensions is typical for submicron particles dispersed in a low viscosity Newtonian fluid.At low shear strain rates. Brownian motion leads to a random distribution of the particles in the suspension, and particle collision will result in viscous behavior. At high shear strain rates, however, particles will arrange in layers, which can slide over each other in the direction of flow. This results in a reduced viscosity of the system in agreement with the principles of shear thinning. A pro-noimced apparent yield stress can be found for shear thinning suspensions, if the Brownian motion is suppressed by electrostatic repulsion forces, which result in three-dimensional crystal-like structures of the particles with low mobility. 

In order to model the flow behavior of molten silicate suspensions such as magmas and slags, the rheological behavior must be known as a function of the concentration of suspended crystals, melt composition, and external conditions. We have determined the viscosity and crystallization sequence for a Kilauea Iki basalt between 1250°C and 1149°C at 100 kPa total pressure and f02 corresponding to the quartz-fayalite-magnetite buffer in an iron-saturated Pt30Rh rotating cup. viscometer of the Couette type. The apparent viscosity varies from 9 to 879 Pa.s. The concentration of suspended crystals varies from 18 volume percent at 1250°C to 59 volume percent at 1149 C. The molten silicate suspension shows power-law behavior ... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Viscosity, Fluidity, and Oxygen Supply. When plant cells grow well, they can occupy 40 to 60% of the whole culture volume, and the apparent viscosity becomes very high. Tanaka (1982) examined the relationship between apparent viscosity and concentration of solids in suspension, and concluded that when the cell density exceeds 10 g/1, the slope of the apparent viscosity increases rapidly, and when cell density reaches 30 g/1, the culture medium becomes difficult to agitate and supply with oxygen. 

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. 

Liquid droplets cannot be treated the same as solid particles in their codispersed systems. This behavior has been indicated by equation 66 or 68, in which the Einstein constant increases with increasing viscosity ratio of the dispersed phase to the continuous phase. As is shown by Yan et al. (195, 197, 198), liquid droplet and solid particle effects are additive only when the solid concentration is low, say s < 0.05, and when both solid particles and liquid droplets have comparable sizes. However, when the particle-to-droplet size ratio is large, the particles and the droplets become additive (192) for a wider solid concentration range (Figures 34 and 35). The apparent viscosity of the system may be added in terms of the two distinct model systems pure emulsion characterized by solid-free dispersed phase volume fraction and pure suspension characterized by the volume fraction of the solids. The additive rule for the ternary systems is similar to the rule for bimodal solid particle suspensions due to Farris (139) ... 

Figure la-c shows the dependence of the apparent viscosity of the enzyme hydrolysis suspension on biomass concentration. It clearly demonstrates a dramatic decrease of viscosity at the reloading point (i.e., after the initial 4 h). The experimental determination of the viscosity-shear rate and shear stress-shear rate relationships of the various formulation suspensions with different concentrations was performed with a variable speed rotational viscometer (2 to 200 rpm). 

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