Particle stress, suspensions

Consider a suspension of identical, spherical (radii a), force- and couple-free particles. Upon neglect of both inertia and Brownian movement, the proper rheological starting point is Eq. (2.33) for the particle stress. As in Eq. (4.1), the contribution arising when particle-particle interactions are absent is known. Its explicit inclusion in yields... 

The previous analysis may be extended to spatially periodic suspensions whose basic unit cell contains not one, but many particles. Such models would parallel those employed in liquid-state theories, which are widely used in computer simulations of molecular behavior (Hansen and McDonald, 1976). This subsection briefly addresses this extension, showing how the trajectories of each of the particles (modulo the unit cell) can be calculated and time-average particle stresses derived subsequently therefrom. This provides a natural entree into recent dynamic simulations of suspensions, which are reviewed later in Section VIII. 

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 ... 

Many industrial processes are affected by the influence of particulate materials on the flow properties of material. Flow properties of materials can be adjusted by fillers to meet the requirements. Flow properties can also be adversely affected by numerous phenomena related to the presence of filler in formulations.One common example is related to the flow of industrial slurries which contain concentrated suspensions of small particles. Such suspensions are usually non-Newtonian fluids with a yield stress which is formed through strong interactions between particles. During flow, these interactions are continuously broken and rebuilt. A solid deposit formed on the slopes and walls is an adverse effect of this property. 

We have seen that it can be difficult to reach the critical concentration required to observe an isotropic-anisotropic transition because concentrated suspensions of colloids are not always stable. However, orientation of flexible polymers as well as of anisotropic particles in suspension can be induced by flow, a phenomenon that has long been observed, reported, and studied. This phenomenon is especially strong when a pretransitional effect exists, which can be easily observed by the naked eye on a sample that is shaken between crossed polarizers (see for example the section on clays). In these systems, birefringence is induced via mechanical forces, like the shear stresses in a laminar flow ( Maxwell-dy-namo-optic effect ). 

The elastic modulus (G ) of a system is a measure of how it reacts to an applied stress that is not severe enough to break any structure. This makes it a valuable method to quantify the degree of dispersion of particles in suspension. A lower G implies less interparticle structure or flocculation and therefore a greater degree of dispersion. Figure 23 shows the behavior of G as a function of pH for mature fine tailings from a commercial oil sands extraction process. The minimum corresponds to a maximum in the electrostatic repulsion of the particles as determined by electronic amplification (34). 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

When there are elements causing increased roughness, the flow separates, and only a part of the shear stress determined from the energy slope is effective in moving the particles in suspension. The work of gravity is then used partly to overcome friction at the bed. 

Goddard [173] derived a formula to describe the stress field for dilute suspensions of oriented slender fibers in a non-Newtonian fluid. The treatment was quite rudimentary but brought out an important result that particle-stress effect was considerably smaller in a shear-thinning non-Newtonian fluid compared to the Newtonian case, possibly due to tensile stiffening in the fluid itself. Qualitative agreement with the experimental data of Charrier and Rieger [176] was observed. A more sophisticated analysis of the same problem was presented by Goddard... 

In comparison to suspensions of rigid spheres, the overwhelming additional effect with axisymmetric particles is orientation. Obviously the orientation of a nonspherical particle with respect to the flow will greatly affect the velocity field around it and thus the particle stress, Tp in eq. 10.2.8. For example, if the particle is a rod with its long axis aligned in the flow direction, the alteration of the... 

The same basic approach used to calculate the constitutive equation for dilute suspensions of spheres can be applied to spheroids. The difficulty lies in calculating the particle stress tp in eq. 10.2.8. Not only is the velocity field more complex, but Xp depends on the orientation. Thus, to get the bulk value of the stress contribution of the particles, we need to integrate over all orientations, weighting by the distribution function... 

Prasad, D. and Kytomaa, H. K. 1995. Particle stress and viscous compaction during shear of dense suspensions. Int. J. Multiph. Flow, 21, 775. 

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) ... 

The two steps in the removal of a particle from the Hquid phase by the filter medium are the transport of the suspended particle to the surface of the medium and interaction with the surface to form a bond strong enough to withstand the hydraulic stresses imposed on it by the passage of water over the surface. The transport step is influenced by such physical factors as concentration of the suspension, medium particle size, medium particle-size distribution, temperature, flow rate, and flow time. These parameters have been considered in various empirical relationships that help predict filter performance based on physical factors only (8,9). Attention has also been placed on the interaction between the particles and the filter surface. The mechanisms postulated are based on adsorption (qv) or specific chemical interactions (10). 

Each stage of particle formation is controlled variously by the type of reactor, i.e. gas-liquid contacting apparatus. Gas-liquid mass transfer phenomena determine the level of solute supersaturation and its spatial distribution in the liquid phase the counterpart role in liquid-liquid reaction systems may be played by micromixing phenomena. The agglomeration and subsequent ageing processes are likely to be affected by the flow dynamics such as motion of the suspension of solids and the fluid shear stress distribution. Thus, the choice of reactor is of substantial importance for the tailoring of product quality as well as for production efficiency. 

How does yield stress depend on the size of particles We have mentioned above that increasing the specific surface, i.e. decreasing an average size of particles of one type, causes an increase in yield stress. This fact was observed in many works (for example [14-16]). Clear model experiments the purpose of which was to reveal the role of a particle s size were carried out in work [8], By an example of suspensions of spherical particles in polystyrene melt it was shown that yield stress of equiconcentrated dispersions may change by a hundred of times according to the diameter d of non-... 

The hydrodynamic forces acting on the suspended colloids determine the rate of cake buildup and therefore the fluid loss rate. A simple model has been proposed in literature [907] that predicts a power law relationship between the filtration rate and the shear stress at the cake surface. The model shows that the cake formed will be inhomogeneous with smaller and smaller particles being deposited as the filtration proceeds. An equilibrium cake thickness is achieved when no particles small enough to be deposited are available in the suspension. The cake thickness as a function of time can be computed from the model. 

Adsorption on Kaolinite. For kaolinite, the polymer adsorption density is strongly dependent on the solid/liquid ratio, S/L, of the clay suspension. As S/L increases, adsorption decreases. This S/L dependence cannot be due totally to autocoagulation of the clay particles since this dependence is observed even in the absence of Ca2+ at pH 7 and at low ionic strength where auto-coagulation as measured by the Bingham yield stress is relatively weak (21). Furthermore, complete dispersion of the particles in solvent by ultra-sonication before addition of... 

Example 15-1 Determine the pressure gradient (in psi/ft) required to transport a slurry at 300 gpm through a 4 in. sch 40 pipeline. The slurry contains 50% (by weight) solids (SG = 2.5) in water. The slurry contains a bimodal particle size distribution, with half the particles below 100 pm and the other half about 2000 gm. The suspension of fines is stable and constitutes a pseudohomogeneous non-Newtonian vehicle in which the larger particles are suspended. The vehicle can be described as a Bingham plastic with a limiting viscosity of 30 cP and a yield stress of 55 dyn/cm2. 

In dense systems such as encountered in solids suspension, particle-particle interaction may be important as well. Then, the closure of solid-phase stresses is an important issue for which kinetic theory models and solids phase viscosity may be instrumental (see, e.g., Curtis and Van Wachem, 2004). 

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