Effective viscosity of a suspension

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. 

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... 

We have shown in the preceding section that the rheological properties of particulate-filled molten thermoplastics and elastomers depend on many factors (1) particle size (t/p), (2) particle shape (a), (3) volume fraction of filler (f)), and (4) applied shear rate (y) or shear stress a). The situation becomes more complicated when interactions exist between the particulates and polymer matrix. There is a long history for the development of a theory to predict the rheological properties of dilute suspensions, concentrated suspensions, and particulate-filled viscoelastic polymeric fluids. As early as 1906, before viscoelastic polymeric fluids were known to the scientific community, Einstein (1906,1911) developed a theory predicting the viscosity of a dilute suspension of rigid spheres and obtained the following expression for the bulk (effective) viscosity of a suspension ... 

Viscosity can depend strongly on fillers added to provide a range of mechanical, transport, electrical, magnetic, or other physical properties. The well-known Einstein relationship, for example, provides the viscosity of a dilute suspension of rigid spheres, obtaining the bulk or effective viscosity of a suspension as (Han 2007) ... 

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. 

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. 

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... 

EFFECTIVE VISCOSITY OF A CONCENTRATED SUSPENSION 527 surface potential J/o, which becomes for large Ka (Eq. (4.29))... 

EFFECTIVE VISCOSITY OF A CONCENTRATED SUSPENSION OF UNCHARGED POROUS SPHERES... 

We have shown that the effective viscosity of concentrated suspension of uncharged porous spheres of radius a and volume fraction in a liquid is given by Eq. (27.68) (as combined with Eq. (27.74)). The coefficient Q(2a, (f)) expresses how the presence of porous spheres affects the viscosity tjs of the original liquid. 

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. 

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 velocity field given by (7-185) will be used later to estimate heat transfer rates for spherical particles in a straining flow. Here, we focus on a different application of (7 185), namely, its use in predicting the effective viscosity of a dilute suspension of solid spheres. To carry out the calculation, it is first necessary to briefly discuss the properties of a suspension in a more general framework. 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

The viscosity of a suspension is also affected by the presence of the dispersed phase. For suspensions of free-flowing solid particles, the effective viscosity may be estimated from the relation ... 

It is important then to recognize that the flow in crystallizers is of a suspension and not a single-phase fluid. There are obvious differences. The effective viscosity of the suspension is larger than that of the solution by itself. The flow velocities everywhere must be large enough so that the particles do not settle appreciably. Then, there are more subtle differences. The presence of particles blunts velocity profiles and affects turbulence. The particles are not in general uniformly suspended, but are distributed in unexpected ways. The particle size distribution is also not uniform throughout the vessel. The net result is that transport projjerties and the variables that affect crystallization most, such as supersaturation, are affected. 

According to this conservation equation, the viscosity of a suspension of N spheres dispersed in pure solvent, treated as a continuous fluid, is equal to the viscosity of a suspension of N — K spheres dispersed in an effective medium consisting of the solvent and the K remaining spheres, to each of which is assigned an appropriately modified diameter. 

In the simplest experimental conditions, the viscosity of a suspension depends on the concentration, shape, and even potential (the "electroviscous." effects) of the particles (63-65). Figure 12 (63) is an example of experimental data reported on different systems the behavior observed is Newtonian except for very high-volume fractions, and in such concentrated suspensions the frequency of particle-particle collisions is so high that the stability of the whole system will be largely affected. It is hence difficult, if not impossible, to control the flow properties of pharmaceutical suspensions without the use of special additives, almost systematically included in practical formulations. 

The list could be made longer, taking the idea of electrokinetics in a wide sense (response of the colloidal system to an external field that affects differently to particles and liquid). Thus, we could include electroviscous effects (the presence of the EDL alters the viscosity of a suspension in the Newtonian range) suspension conductivity (the effect of the solid-liquid interface on the direct current (DC) conductivity of the suspension) particle electroorientation (the torque exerted by an external field on anisotropic particles will provoke their orientation this affects the refractive index of the suspension, and its variation, if it is alternating, is related to the double-layer characteristics). 

Evidently, liquid-phase viscosity has a marginal effect on Fr, . This is expected since both Equations 9.20 and 9.21 from which Fr is derived do not contain liquid-phase viscosity. The small effect observed nonetheless is likely to be due to finite nonidealities and viscosity effects such as the blade slip factor introduced by RieUy et al. (1992). A major application of gas-inducing systems is in solid-catalyzed gas-liquid reactions. Aldrich and Deventer (1994) studied the effect of presence of solid particles on It was found that was unaffected up to solid loading of 15wt%. Above this loading, there was an increase in presumably due to increase in the effective viscosity of the suspension. This study, however, used a relatively very small-size system (T=0.2m and D=0.065m), and therefore, it is difficult to derive quantitative conclusions. 

Bull 1932, Levine et al. 1975), the hydrodynamic drag on single particles (Booth 1954) or the viscosity of a suspension (Booth 1950 Ruiz-Reina et al. 2003). Note that the primary electroviscous effect is related to individual particles and surfaces any double layer interaction is excluded. It is essentially a second-order effect which —to first approximation—obeys the following equation ... 

Viscosity measurements provide another fairly simple method for determining a layer thickness. For non-agglomerated particles, a fairly simple equation (Equation 1.5) exists relating relative viscosity of a suspension to the effective volume fraction of the solid phase ... 

---