Equation (1) can be rearranged into the form where the term on the left is known as the velocity defect. When one assumes that the flow is essentially all in the turbulent core, Equation (1) can be integrated across the pipe to yield the classical von Karman friction factorReynolds number correlation (1 ). Equations ( 1 ) and (3), being applicable to all Newtonian fluids, provide a useful basis for comparing Newtonian and non-Newtonian velocity profiles. In the case of Equation ( 1 ) (and also for any friction factor correlations which may be devised) one must choose an appropriate non-Newtonian viscosity or a non-Newtonian Reynolds number. For convenience in displaying and using friction factor data a Reynolds number which is natural in laminar flow can be extended empirically into the turbulent flow regime ( 4 ) . For mechanistic purposes however it is more helpful to use the apparent viscosity evaluated at the wall shear stress (1, 2, 3 ) , the motivation being that the viscous effects in the wall region are the most important viscous effects which occur in the pipe. This apparent viscosity, calculated by assuming the existence of a Iaminar sublayer sufficiently close to the pipe wall, is obtained by dividing the wall shear stress, as calculated from pressure drop data, by the laminar shear rate, as obtained from a laminar shear diagram: D. C. Bogne is i t the University of-Tennessee, Knoxville, Tennessee.The use of this viscosity has been found to reduce to a considerable extent the dependence of the correlations on arbitrary parameters based on rheological models, although for prediction purposes it is more difficult to use in that a trial-and-error technique is necessary if the viscosity is changing rapidly in the shear stress range of interest. Three previous studies involving the measurements of velocity profiles in non-Newtonian fluids have been carried out (1, 5, 6 ) . The fluids used included pseudoplastic aqueous solutions of carboxypolymethylene (Carbopol) , carboxymethylcellulose (CMC ) , and ammonium alginate (alginate) , and aqueous suspensions of clay (Attagel). The CMC solution is known to exhibit viscoelastic behavior in the sense that it exhibits normal stress and birefringence effects under shear (7, 8).The first measurements of velocity profiles were made by Shaver (5) using CMC and alginate. He presented the data graphically in the form of the velocity defect equation using as a parameter the non-Newtonian flow behavior index ( n ) of the power law rheological model. Shaver's results (Figure 1) showed a progressive change in the turbulent velocity profile toward a less blunt (more laminar) velocity profile with decreasing values of the rheological constant n. In addition to profiles Shaver obtained friction factors over a range of mean velocities. These results have been replotted in Figure 2 against the Reynolds number based upon the wall shear stress viscosity. It is noted that the friction factors lie well below the Newtonian line.Bogue (1) measured the velocity profile and friction fact...

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Oak Ridge Notional Laboratory, Oak Ridge, TennesseeWhen an aqueous flocculated suspension of thoria is pumped through a pipe with a sufficiently high velocity, shear-induced turbulence will develop just as for Newtonian fluids. The interparticle attractive force responsible for flocculation, which produces a shear-dependent viscosity in laminar flow, is overcome sufficiently by large shearinduced forces to permit the random eddying of turbulence to occur.As with Newtonian fluids the turbulent behavior cannot be predicted theoretically, and recourse is made to phenomenological or empirical treatments. The Fanning friction factor-Reynolds number correlation for Newtonian fluids provides a useful framework for prediction of the turbulent Aow pressure drops to be expected in pumping flocculated suspensions. The purpose of this paper is to extend the utility of this approach so that high temperature friction factors can be predicted from low temperature suspension properties. For this purpose a Reynolds number is defined containing a viscosity which can be readily extrapolated to elevated temperatures and which is phenomenologically meaningful.

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