For power law fluids

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Figure 3.39. Fully-developed laminar velocity profiles for power-law fluids in a pipe (from equation 3.134)...

Hartnett, S. P. and Kostic, M. /nr. Comm. Heat Mass Transfer 17 (1990) 59. Turbulent friction factor correlations for power law fluids in circular and non-circular channels. 

Irvine, T. F. Chern. Eng. Comm. 65 (1988) 39. A generalized Blasius equation for power law fluids,... 

ReMR Metzner and Reed Reynolds number for power-law fluid (see Chapter 3) ... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

For power law fluids, the basic dimensionless variables are the Reynolds number, the friction factor, and the flow index (n). If the Reynolds number is expressed in terms of the mass flow rate, then... 

Figure 11-5 Plot of 1 jCy vs. n for power law fluid. Line is Eq. (11 -29). Data points are from Chhabra (1995) and Tripathi et al. (1994).

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... 

Velocity profiles for power law fluids showing the effect of the power law index, n... 

The corresponding equation for power law fluids is [Dodge and Metzner (1959)]... 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n ... 

There have been very few studies of the effects of non-Newtonian properties on flow patterns in hydrocyclones, although Dyakowski et al.,AU have carried out numerical simulations for power-law fluids, and these have been validated by experimental measurements in which velocity profiles were obtained by laser-doppler anemometry. 

Extensive comparisons of predictions and experimental results for drag on spheres suggest that the influence of non-Newtonian characteristics progressively diminishes as the value of the Reynolds number increases, with inertial effects then becoming dominant, and the standard curve for Newtonian fluids may be used with little error. Experimentally determined values of the drag coefficient for power-law fluids (1 < Re n < 1000 0.4 < n < 1) are within 30 per cent of those given by the standard drag curve 37 38. ... 

Rei)n Reynolds number for power-law fluid in a granular bed (equation 4.28) — —... 

Heat transfer involving non-Newtonian fluids has not been studied in rotating devices. Models have been developed for gravity-driven heat transfer for power-law fluids (46). These models may be useful as a starting point to evaluate performance in higher-gravity fields. 

Equations for a number of non-Newtonian fluid types are available in the literature [213,359]. They tend to be somewhat unwieldy and require a knowledge of the fluid rheology. For power-law fluids in smooth pipes, the friction factor can be estimated by using a modified Reynolds number in Eq. (6.57). The Metzner-Reed modified Reynolds number, Re, is given by ... 

Figure 12.7 gives the velocity gradients of a Newtonian and a Power Law fluid with fi = nr, isothermal flow is assumed. It is clear that, for Power Law fluids, viscous heating may be intense near the capillary wall, whereas the central portion of the fluid is relatively free of this effect. 

Inserting Eqs. 12.3-7 and 12.3-10 into the general design equation 12.3-3, we obtain the specific design equation for Power Law fluids ... 

Of interest is a recent theoretical relation for eg in gas-liquid bubble columns based on liquid circulation and claimed to be valid both in the homogeneous bubble flow regime and in the chum-turbulent regime, also for non-Newtonian fluids. For power law fluids with... 

For power-law fluids, the above equations can be used if Re is defined as... 

Conventional stirred-tank polymeric reactors normally use turbine, propeller, blade, or anchor stirrers. Power consumption for a power-law fluid in such reactors can be expressed in a dimensionless form, Ne = Reynolds number based on the consistency coefficient for the power-law fluid. Various forms for the function f(m) in terms of the power-law index have been proposed. Unlike that for Newtonian fluid, the shear rate in the case of power-law fluid depends on the ratio dT/dt and the stirrer speed N. For anchor stirrers, the functionality g developed by Beckner and Smith (1962) is recommended. For aerated non-Newtonian fluids, the study of Hocker and Langer (1962) for turbine stirrers is recommended. For viscoelastic fluids, the works of Reher (1969) and Schummer (1970) should be useful. The mixing time for power-law fluids can also be correlated by the dimensionless relation NO = /(Reeff = Ndfpjpti ) (Tebel et aL 1986). 

Figure 8-3 The Friction Factor-Generalized Reynolds Number (GRe) Relationship for Power Law Fluids Under Laminar Flow Conditions. It can also be used for Newtonian fluids in laminar flow.

Figure 8-3 illustrates the friction frictor versus GRe relationship for power law fluids under laminar flow conditions. It can also be used for Newtonian fluids in laminar flow with the Reynolds number being used in place of GRe. In fact, the Newtonian/ versus Re relationship was established much earlier than extension to non-Newtonian fluids. Once the magnitude of the friction factor is known, the pressure drop in a pipe can be estimated from Equation 12. 

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