Jiv Y igK7S

Figure 5-30. Comparison of experimental friction factors with those predicted. (From Dodge and Metzner.^ Courtesy of A.I.Ch.E.)

The use of the Newtonian /-A'«, relationship when interpreting data from pipe flow experiments led to various misconceptions about the viscosity of non-Newtonian fluids in turbulent flow - such as that it equalled the plastic viscosity, or some multiple thereof, or that it equalled the viscosity of the liquid phase, or that it was a function of the concentration of solids. The values thus obtained were generally too low, and did not appear to change with flow velocity. The latter observation led some authorities to recommcnd the use of a constant turbulent viscosity of three centipoises in hydraulic calculations. Actually, according to Dodge and Metzner, their experimental data showed that turbulent flow viscosity may vary threefold over a livefold range of Reynolds numbers. Fortunately, such large variations in viscosity do not lead to equally large errors in the value of /, because / is relatively insensitive to the value of N,]f. as shown in Figure 5 31.

1,000 10,000 100,000 REYNOLDS NUMBER, N' =

Figure 5-31. Relationship between Fanning friction factor and the generalized Reynolds number. Note that, for a given Reynolds number, f is strongly dependent on the value of n'. (From Dodge and Metzner.25 Courtesy of A.I.Ch.E.)

1,000 10,000 100,000 REYNOLDS NUMBER, N' =

Figure 5-31. Relationship between Fanning friction factor and the generalized Reynolds number. Note that, for a given Reynolds number, f is strongly dependent on the value of n'. (From Dodge and Metzner.25 Courtesy of A.I.Ch.E.)

The Onset Of Turbulence

Figure 5 -31 shows that turbulence in a non-Newtonian fluid with an n value of 0.4 would not start until the Reynolds number reached a value of 2900, compared to a value of 2100 for a Newtonian fluid. The difference is significant because it means that the flow velocity would have to be 3K",, higher, all other factors being equal. These figures emphasize the importance of using the generalized Reynolds number when determining whether or not a non-Newtonian fluid is in turbulent flow. The generalized Reynolds number for the system may be determined from Equations 5 54 or 5-55, the critical value may be determined from Figure 5-31, and the value of« for the fluid.

An alternative criterion, the Z stability parameter, which determines the point at which turbulence is initiated, has been introduced by Ryan and Johnson. Its chief advantage is that the critical value is always the same, regardless of the value of n. Ryan and Johnson theorized that turbulent flow will start at a point on the radius of the pipe, r/R, at which the 7 parameter is maximum, and that the local Reynolds number at that point will be 2100, The maximum value of Z is 808 for all fluids. With Newtonian fluids r/R at Z maximum is 1 j yj 3; with non-Newtonian fluids r/R increases with decrease in the value of n. Thus the critical Z number of 808 is independent of the value of n, but the mean flow velocity required for turbulence increases with a decrease in the value of n, which is in accordance with the findings of Dodge and Metzner, Ryan and Johnson substantiated their theory with experimental data.

Unfortunately, determination of the Z parameter involves complex calculations, but Walker37 has published an approximate method for flow in the annul us, which takes the rotation of the drill pipe into account. His equation is

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