Concentrated suspensions viscosity

PROBSTEIN, R.F., SENGUN, M.Z. c TSENG, T-C. 1994. Bimodal model of concentrated suspension viscosity for distributed particle sizes. J. Rheol. 38, No. 4 (1994). 

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) ... 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

Fig. 13. Pattern of variation of concentration dependence of suspension viscosity when the ratio between the length and diameter of aniso-diametricity of filler s particles increases. The arrow indicates the direction of growth of 1/d of filler s particles. The slop of the initial part of line A (for spherical particles) is 2.5...

The polymer chain concentration and the polymer chain radius were employed to assess the rotational mobility of a molecule within a swollen gel [14]. To this purpose the gel is considered a thick, viscous polymer suspension. Its viscosity can be evaluated with the following equation, proposed by Nicodemo and Nicolais for concentrated suspensions of polymeric fibers [147] ... 

Viscosities of concentrated suspensions of carbon black in a white mineral oil (Fisher "paraffin" oil of 125/135 Saybolt viscosity) were measured with a Brookfield viscometer as a function of OLOA-1200 content. Figure 13 shows the viscosities of dispersions with 30 w%, 35 w% and 70 w% carbon black. In all cases the viscosity fell rapidly as the 0L0A-1200 content increased from 0 to 1%, then fell more gradually and levelled off as the 0L0A-1200 content approached 2%. In many respects the reduction in viscosity with increasing OLOA-1200 content parallels the conductivity measurements both phenomena are sensing the buildup of the steric barrier, and this steric barrier weakens, softens, and lubricates the interparticle contacts. As evidenced in foregoing sections, the particles are still flocculated but can be easily stirred and separated mechanically. The onset of electrostatic repulsion at OLOA-1200 contents in excess of 2.5% did not affect viscosities. 

Adsorption of nonionic and anionic polyacrylamides on kaolinite clay is studied together with various flocculation properties (settling rate, sediment volume, supernatant clarity and suspension viscosity) under controlled conditions of pH, ionic strength and agitation. Adsorption and flocculation data obtained simultaneously for selected systems were correlated to obtain information on the dependence of flocculation on the surface coverage. Interestingly, optimum polymer concentration and type vary depending upon the flocculation response that is monitored. This is discussed in terms of the different properties of the floes and the floe network that control different flocculation responses. Flocculation itself is examined as the cumulative result of many subprocesses that can depend differently on system properties. 

With some concentrated suspensions of solid particles, particularly those in which the liquid has a relatively low viscosity, the suspension appears to slip at the pipe wall or at the solid surfaces of a viscometer. Slip occurs because the suspension is depleted of particles in the vicinity of the solid surface. In the case of concentrated suspensions, the main reason is probably that of physical exclusion if the suspension at the solid surface were to have the same spatial distribution of particles as that in the bulk, some particles would have to overlap the wall. As a result of the lower concentration of particles in the immediate vicinity of the wall, the effective viscosity of the suspension near the wall may be significantly lower than that of the bulk and consequently this wall layer may have an extremely high shear rate. If this happens, the bulk material appears to slip on this lubricating layer of low viscosity material. 

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. 

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... 

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 ... 

A dimensionless relationship has been proposed to describe the bulk viscosity T (( ), y) of concentrated suspensions taking into account particle size and shape effects [29] ... 

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). 

The parameter c was found to be a linear function of Reynolds number with regression coefficients between 0.98 and 1.00. The shear rate constant, k, was within 10% of the values found by Donnelly (15) and Rieth (16) for a double-helical ribbon impeller. Furthermore, the Power Law could be used to describe corn stover suspension viscosity with correlation coefficients above 0.99 for all four concentrations tested. Finally, the yield stress predicted by direct data extrapolation and by the Herschel-Bulkley model was similar for each concentration of com stover. 

The colloidal chemistry of very diluted clay suspensions is usually described by the double layer theory (3). In more concentrated suspensions or thick pastes, where the average distance between sheets is shorter, the Van der Waals forces have to be considered. The combination of all these effects influences the distribution of the individual sheets and, thus, the viscosity of the slurry. From the viewpoint of this review, these spatial distributions of the individual sheets affect the availability of the surface area with the important consequence that the surface area is no longer an extensive property of the material. Disordered... 

Now the technique provides the basis for simulating concentrated suspensions at conditions extending from the diffusion-dominated equilibrium state to highly nonequilibrium states produced by shear or external forces. The results to date, e.g., for structure and viscosity, are promising but limited to a relatively small number of particles in two dimensions by the demands of the hydrodynamic calculation. Nonetheless, at least one simplified analytical approximation has emerged [44], As supercomputers increase in power and availability, many important problems—addressing non-Newtonian rheology, consolidation via sedimentation and filtration, phase transitions, and flocculation—should yield to the approach. 

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

Analysing the rate of energy dissipation in concentrated suspensions of solid spheres, Frankel and Aerivos [69] also arrived at the conclusion that the relative viscosity of suspensions is a function of the relation a/h ... 

In concentrated suspensions, the particles touch each other. If there is also an attraction between the particles, the suspension may not flow when the shear stress is small it is a solid (Figure C4-14). The stress at which the liquid starts moving is known as the yield stress. Once the liquid yields, it often behaves like a Newtonian liquid with a constant differential viscosity. The behaviour of such Bingham fluids is similar to that of shear thinning fluids ... 

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