Viscosity of a suspension

When discrete particles are present in a fluid, they cannot take part in any deformation the fluid may undergo, and the result is an increased resistance to shear. Thus, a suspension exhibits a greater resistance to shear than a pure fluid. This effect is expressed as an equivalent viscosity of a suspension. As the concentration of solids increases, so the viscosity increases. Einstein [21] deduced the equation  

The equation is found to hold for very dilute suspensions but requires modification for c greater than 1%. In the very dilute suspensions used in sedimentation analysis, the effect is smaller than errors inherent in the determination of rj by conventional methods. 

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A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

Viscosity. Because a clump of particles contains occluded Hquid, the effective volume fraction of a suspension of clumps is larger than the volume fraction of the individual particles that is, there is less free Hquid available to faciHtate the flow than if the clumps were deagglomerated. The viscosity of a suspension containing clumps decreases as the system becomes deagglomerated. This method is not very sensitive in the final stages of deagglomeration when there are only a few small clumps left. 

For suspension of rapidly setthng particles, the impeller turbine diameter should be Df/3 to Dfl2. A clearance of less than one-seventh of the fluid depth in the vessel should be used between the lower edge of the turbine blade tips and the vessel bottom. As the viscosity of a suspension increases, the impeller diameter should be increased. This diameter may be increased to 0.6 Df and a second impeller added to avoid stagnant regions in pseudoplastic slurries. Moving the baffles halfway between the impeller periphery and the vessel wall will also help avoid stagnant fluid near the baffles. 

Rheological methods of measuring the interphase thickness have become very popular in science [50, 62-71]. Usually they use the viscosity versus concentration relationships in the form proposed by Einstein for the purpose [62-66], The factor K0 in Einstein s equation typical of particles of a given shape is evaluated from measurements of dispersion of the filler in question in a low-molecular liquid [61, 62], e.g., in transformer oil [61], Then the viscosity of a suspension of the same filler in a polymer melt or solution is determined, the value of Keff is obtained, and the adsorbed layer thickness is calculated by this formula [61,63,64] ... 

The effective viscosity of a suspension of particles in a fluid medium is greater than that of the pure fluid, owing to the energy dissipation within the electrical double layers. 

This is a well known result The intrinsic viscosity of a suspension of elastic dumb-bells is equal to that of another suspension containing rigid dumb-bells of uniform length equal to the root mean square end-to-end... 

You can change the viscosity of a suspension by adding a suitable electrolyte. The nature and concentration of this electrolyte determine the surface charge of the particles and consequently the particle size and viscosity. When you plot the viscosity in a graph as a function... 

In order to overcome this difficulty, Rudin and Strathdee (1974) developed a semi empirical method for predicting the viscosity of dilute polymer solutions. The method is based on an empirical equation proposed by Ford (1960) for the viscosity of a suspension of solid spheres ... 

Ford equation for the viscosity of a suspension, 602 Form birefringence, 300 Formula of Seitz and Balazs, 450 Fourier number, 59 transform, 361... 

In the suspension process, the polymer is suspended in the monomer propylene. This process offers the advantages of being able to operate at higher solids owing to the lower viscosity of a suspension compared with a solution at comparable solids. Other advantages are simple heat removal by the evaporative cooling of the propylene, more uniform... 

The viscosity in the low shear regime depends mainly on the effective volume fraction of the particles in the suspension. There are many expressions given in the literature which relate the low shear viscosity of a suspension rjo to the viscosity of the suspending fluid ris. Two formulas which are independent of parameters specific... 

The effective viscosity of a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor [22] proposed a theory of the electroviscous effect in a suspension of uncharged liquid drops. This theory has been extended to the case of charged liquid drops by Ohshima [17]. Natraj and Chen [23] developed a theory for charged porous spheres, and Allison et al. [24] and Allison and Xin [25] discussed the case of polyelectrolyte-coated particles. 

Approximate results calculated via Eq. (27.57) are also shown as dotted lines in Fig. 27.2. It is seen that Ka > 100, the agreement with the exact result is excellent. The presence of a minimum of L Ka, la, alb) as a function of Ka can be explained qualitatively with the help of Eq. (27.57) as follows. That is, L Ka, la, alb) is proportional to 1/k at small Ka and to k at large Ka, leading to the presence of a minimum of L Ka, la, alb). As is seen in Fig. 27.3, for the case of a suspension of hard particles, the function L ko) decreases as Ka increases, exhibiting no minimum. This is the most remarkable difference between the effective viscosity of a suspension of soft particles and that for hard particles. It is to be noted that although L Ka, la, alb) increases with Ka at large Ka, the primary electroviscous coefficient p itself decreases with increasing electrolyte concentration. The reason is that the... 

Correction for the finite extent of the fluid is negligible in most cases and errors due to discontinuities in the fluid are only of importance for gas systems. Similarly the increased viscosity of a suspension over that of a pure liquid has negligible effects at low concentrations. Reproducible data are possible at high Reynolds numbers, high concentrations and with submicron particles. These data may be highly inaccurate and in general precise values of erroneous sizes and percentage undersize are of limited worth. 

However, they may not indicate the true bulk viscosity of a suspension that forms a thin layer of the continuous phase (e.g., serum of tomato juice) around the immersed probe or when the probe is covered by a higher viscosity gel due to fouling. Vibrational viscometers are suitable for measuring viscosities of Newtonian fluids, but not the shear-dependent rheological behavior of a non-Newtonian fluid (e.g., to calculate values of the power law parameters). 

The viscosity (or apparent viscosity) of a suspension can be related to the viscosity of the continuous medium in terms of the relative viscosity, rjr. ... 

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

Problem 6.1 (Worked Example) Estimate the zero-shear viscosity of a suspension of hard spheres 100 nm in diameter at a volume fraction of [Pg.318]

Problem 6.8 (Worked Example) Estimate the steady-state uniaxial viscosity of a suspension of 0.1% by volume of rod-like particles L = 6 yum long and d = 10 nm in diameter in a Newtonian oil of viscosity 100 P at an extension rate of 1 sec . ... 

Formulation of pharmaceutical suspensions to minimise caking can be achieved by the production of flocculated systems. A flocculate, or floe, is a cluster of particles held together in a loose open stmemre a suspension consisting of particles in this state is termed flocculated (Fig. 7.27). There are various states of flocculation and deflocculation. Unfortunately flocculated systems clear rapidly and the preparation often appears unsightly, so a partially deflocculated formulation is the ideal pharmaceutical. The viscosity of a suspension is obviously affected by flocculation. 

In deriving an equation for the viscosity of a suspension of spherical particles, Einstein considered particles which were far enough apart to be treated independently. The particle volume fraction (p is defined by... 

The viscosity of a suspension of nonsettling spherical particles larger than the mean free path of a gas increases linearly with the volumeiric concentration, 0, expressed as volume fraction of the particles according to a relation first given by Einstein ... 

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