Particle in power-law fluid

Graham, D. I. and Jones, T. E. R. J. Non-Newt. Fluid Mech, 54 (1994) 465. Setfling and transport of spherical particles in power-law fluids at finite Reynolds number. 

Determination of the Free Settling Parametres of Spherical Particles in Power Law Fluids, Chem. Eng. Process. 24 (1988) 4, p. 183-188... 

Motion and Mass Exchange of Particles in Power-Law Fluids... 

For mobile-surface particles in power-law fluids. Hirose... 

For spheres of equal terminal falling velocities, obtain the relationship between diameter and density difference between particle and fluid for creeping flow in power law fluids. 

Prior to 1993 the results of theoretical and experimental smdies on the flow of non-Newtonian fluids past a sphere have been reviewed by Chabra". Since then a number of research smdies have been published, most notably by Machac and co-workers at the University of Pardubice. They investigated experimentally the drag coefficients and settling velocities of spherical particles in power law and Herschel-BuUdey model fluids, in Carreau model fluids (spherical in ref. 13 and non-spherical in ref. 14) and also the effect of the wall in a rectangular ceU, for power law fluids. ... 

At present, insufficient data are available to put forward predictive expressions for wall factor for different shaped particles in purely viscous fluids. Suffice to say that, in general, the wall effects are less severe in power-law fluids than that in Newtonian fluids under otherwise identical conditions. This finding is consistent with the behavior observed for spherical... 

Analytical and experimental results on drag on thin rods in power-law fluids in creeping flow region Experimental correlations for drag coeflicient in terms of equal volume sphere diameter. Re < 100 No difference between drag on particles of different shape but same volume, up to Re< 10... 

An empirical correlation has been proposed by Shah and Lord [10] for predicting deposition and resuspension velocities for slurries of relatively coarse particles ( d = 0.63 and 1 mm) in power law fluids of the type sometimes employed as drilling fluids. In these experiments, a minimum pressure gradient criterion was used to determine the deposition or resusjjension velocity. The pipe diameters used in the experiments ranged between 38 amd 70 mm and because most of the fluids were viscous, many of the flows were laminar. The correlation does not distinguish between turbulent and laminar flows and is not applicable to other fluids. 

In a series of papers, Chhabra (1995), Tripathi et al. (1994), and Tripathi and Chhabra (1995) presented the results of numerical calculations for the drag on spheroidal particles in a power law fluid in terms of CD = fn(tVRe, ). Darby (1996) analyzed these results and showed that this function can be expressed in a form equivalent to the Dallavalle equation, which applies over the entire range of n and tVRe as given by Chhabra. This equation is... 

Table 11-1 Procedure for Determining Unknown Velocity or Unknown Diameter for Particles Settling in a Power Law Fluid...

The wall effect for particles settling in non-Newtonian fluids appears to be significantly smaller than for Newtonian fluids. For power law fluids, the wall correction factor in creeping flow, as well as for very high Reynolds... 

Only a very limited amount of data is available on the motion of particles in non-Newtonian fluids and the following discussion is restricted to their behaviour in shear-thinning power-law fluids and in fluids exhibiting a yield-stress, both of which are discussed in Volume 1, Chapter 3. 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

From Table 3.9 it is seen that, depending on the value of n, the drag on a sphere in a power-law fluid may be up to 46 per cent higher than that in a Newtonian fluid at the same particle Reynolds number. Practical measurements lie in the range 1 < Y < 1.8, with considerable divergences between the results of the various workers. 

Thus, in equation 3.65 only Re n includes the terminal falling velocity which may then be calculated for a spherical particle in a power-law fluid. 

In addition to temperature, the viscosity of these mixtures can change dramatically over time, or even with applied shear. Liquids or solutions whose viscosity changes with time or shear rate are said to be non-Newtonian, that is, viscosity can no longer be considered a proportionality constant between the shear stress and the shear rate. In solutions containing large molecules and suspensions contain nonattracting aniso-metric particles, flow can orient the molecules or particles. This orientation reduces the resistance to shear, and the stress required to increase the shear rate diminishes with increasing shear rate. This behavior is often described by an empirical power law equation that is simply a variation of Eq. (4.3), and the fluid is said to be a power law fluid ... 

Power-law fluids. Here we briefly discuss the motion of spherical bubbles, drops, and particles at a constant velocity U in a power-law non-Newtonian. 

The expressions (6.9.2)—(6.9.5) allow one to calculate the drag coefficients for particles, drops, and bubbles in a Stokes flow of a power-law fluid. 

In the case of mass exchange between a particle and a translational flow of a quasi-Newtonian power-law fluid, the mean Sherwood number can be estimated using the relation... 

This approach was also found to correlate power law fluid flow through granular beds of column to particle diameter ratio 5.8 < D/ds < 20 in the creeping (Darcy) flow regime (99). 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

At the point of incipient fluidisation, the bed voidage, s f, depends on the shape and size range of the particles, but is approximately equal to 0.4 for isometric particles. The minimmn fluidising velocity V f for a power-law fluid in streamline flow is then obtained by substituting s = Smf in equation (5.67). Although this equation applies oifly at low values of the bed Reynolds numbers (<1), this is not usually a limitation at the high apparent viscosities of most non-Newtonian materials. 

Estimate the hindered settling velocity of a 30% (by volume) defloccu- (a) lated suspension of 50 /rm (equivalent spherical diameter) china clay particles in a polymer solution following the power-law fluid model... 

KushaUcar KB, Pangarkar VG. (1995) Particle-hquid mass transfer in mechanically agitated three phase reactors power law fluids. Ind. Eng. Chem. Res., 34 2485-2492. 

Particle Re)aiolds number for the terminal velocity of the particle in the fluid Reynolds nvmiber for power-law fluids Underflow-to-throughput ratio Resistance of filter medium (1/L)... 

The power law fluid yield stress is zero, and the fluid is deformed as long as the effect of a small force on the fluid. Particle density is greater than that of the fluid. In addition there is a vertical downward force formed by particle gravity and buoyancy force of the particle fluid. Therefore, particles settle. When the particle diameter is small to a certain extent, it will not overcome the yield stress and get a suspension in the fluid. Then sedimentation does not occur, which is known as natural suspended state. When the fluid stops circulating, it can make the solid phase suspension in the annulus to prevent the deposition of the solid phase at the bottom of the borehole. In this case, accidents can be avoided. Conditions for particles sedimentation is shown as follows ... 

As for settling of single particles in Newtonian liquids, the fundamental hydrodynamic characteristic for particle motion in non-Newtonian fluids is again the drag coefficient. Its prediction allows calculations of terminal settling velocities. Note that equation 18.10, which applies to low particle concentrations (below 0.5% by volume) in Newtonian liquids at low Reynolds numbers, can, in principle, also be used non-Newtonian fluids where viscosity // then becomes the apparent viscosity but, depending on the type of the non-Newtonian behaviour (= model), its determination may require an iterative procedure. Each model redefines the particle Reynolds number so that, for example, for a power law fluid characterized by constants n and K... 

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. 

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