## Power Law Fluids

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) modelsâ€”the power law and Bingham plasticâ€”can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results.

where and

Both laminar and turbulent pipe flow of highly loaded slurries of fine particles, for example, can often be adequately represented by either of these two models over an appreciable shear rate range, as shown by Darby et al. (1992).

A. Laminar Flow

Because the shear stress and shear rate are negative in pipe flow, the appropriate form of the power law model for laminar pipe flow is

By equating the shear stress from Eqs. (6-42) and (6-4), solving for the velocity gradient, and introducing the result into Eq. (6-7) (as was done for the Newtonian fluid), the flow rate is found to be

Q = JiwY/n [\2+i/n dr = JV3n+1)/n (6-43) \mR) J0 \mr) \3n + 1/

This is the power law equivalent of the Hagen-Poiseuille equation. It can be written in dimensionless form by expressing the wall stress in terms of the friction factor using Eq. (6-5), solving for f, and equating the result to 16/ NRe (i.e., the form of the Newtonian result). The result is an expression that is identical to the dimensionless Hagen-Poiseuille equation:

fNRe.pl = 16 (6-44) if the Reynolds number for the power law fluid is defined as

8D"V2-"p

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 p 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.

B. Turbulent Flow

Dodge and Metzner (1959) modified the von Karman equation to apply to power law fluids, with the following result:

Like the von Karman equation, this equation is implicit inf. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for f for a given flow rate and pipe diameter, in turbulent flow.

Note that there is no effect of pipe wall roughness in Eq. (6-46), in contrast to the case for Newtonian fluids. There are insufficient data in the literature to provide a reliable estimate of the effect of roughness on friction loss for non-Newtonian fluids in turbulent flow. However, the evidence that does exist suggests that the roughness is not as significant for non-Newtonian fluids as for Newtonian fluids. This is partly due to the fact that the majority of non-Newtonian turbulent flows lie in the low Reynolds number range and partly due to the fact that the laminar boundary layer tends to be thicker for non-Newtonian fluids than for Newtonian fluids (i.e., the flows are generally in the "hydraulically smooth'' range for common pipe materials).

### C. All Flow Regimes

An expression that represents the friction factor for the power law fluid over the entire range of Reynolds numbers (laminar through turbulent) and encompasses Eqs. (6-44) and (6-46) has been given by Darby et al. (1992):

where

/Tr = 1.79 x 10-4 exp[-5.24n]NRe41P}+0'757" (6-50)

The parameter a is given by 1

and NRe plc is the critical power law Reynolds number at which laminar flow ceases:

Equation (6-48) applies for NRe pl < NRe plc, Eq. (6-49) applies for 4000 < NRepl < 105, Eq. (6-50) applies for NRe>plc < NRe>pl < 4000, and all are encompassed by Eq. (6-47) for all NRe pl.

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