Friction factor Metzner-Reed Reynolds number

There is also a friction factor chart drawn by Dodge and Metzner plotting the Fanning friction factor versus the Metzner-Reed Reynolds number (although this has been subsequently redrawn for convenience with (j), see ref. 4). This chart shows that the friction factor decreases with n in turbulent flow—that is, it decreases as shear-thinning character increases. If you are studying this chart, be aware that the only experimentally verified data for turbulent flow is that given for m = 1.0 to n = 0.4. 

The next question is, how do you define a Reynolds number for a fluid that has a variable viscosity Metzner and Reed [16] proposed that the known relation between the friction factor and the Reynolds number for the laminar flow of Newtonian fluids may be applied to the laminar flow of non-Newtonians as well ... 

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

Figure 3.40. Metzner and Reed correlation of friction factor and Reynolds number...

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

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 ... 

Dodge and Metzner (16) presented an extensive theoretical and experimental study on the turbulent flow of non-Newtonian fluids in smooth pipes. They extended von Karman s (17) work on turbulent flow friction factors to include the power law non-Newtonian fluids. The following implicit expression for the friction factor was derived in terms of the Metzner-Reed modified Reynolds number and the power law index ... 

Metzner and Reed [26] introduced a generalized Reynolds number Re such that the Fanning friction factor for fully developed laminar pipe flow is given by... 

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