Further Discussion of General Turbulent Pipe Flow Scale-Up Correlation for Rheologically Complex Fluids

This paper may be divided into two main parts: (1) analysis and data to show that Eq. 6 adequately correlates the authors' experimental data, and (2) the assertion that Eq. 20 "should permit scale-up with data taken from one pipe diameter for Newtonian, inelastic non-Newtonian, and viscoelastic fluids, such as dilute polymer fracturing fluids, which produce considerable pressure reductions." The authors might well recommend that this broad grouping of fluid types could be correlated by using Eq. 6 (requiring data from at least two pipe diameters). On the other hand, Eq. 20 is only shown to be adequate for conditions within the range of the present data set. Specifically, predictions by Eq. 20 for diameter scale-up to casing sizes would give larger deviations than implied in Table 2, and scale-up for some other fluids that give large drag reduction could be grossly in error. The description of a class of fluids "which produce a diameter family of straight parallel lines on a log (d?p/4L) vs log (8 v/d) plot. . ." defines a necessary condition for using Eq. 6; i.e., that s shall be constant for each fluid. The proposal that Eq. 20 is applicable to turbulent flow of such fluids is equivalent to saying that the spacing of these parallel lines, with A being constant, will be found to correspond to m = -1.20 for all such fluids. Table 1 shows the values of m derived from data on 16 non-Newtonian fluids. The key statement of this paper is that the diameter exponents "do not vary significantly from . . . -1.20." In statistical terms, this says that the m value for each fluid, which was obtained from all the measurements made on that fluid. has not been shown to be different from the arithmetic mean of the m values for the 16 fluids tested. This implies that the group mean, -1.20, is a more reliable value to use for an individual fluid than is the m value derived from the data on that fluid alone. No statistical test is described to support the key statement that the diameter exponents do not vary significantly from -1.20. Such a test would have to take into account that each individual m value is shown to correlate the pressure gradients for its fluid, with only about 3 percent standard deviation over the diameter range covered. That is, the m value is not the result of a single independent measurement. Thus, it would not be sufficient, or valid, to compare the deviation of one of the m values with the variance of the 16 values of m about their mean. An alternative interpretation of the key statement is that the diameter exponents do not vary appreciably from -1.20; i.e., that using -1.20 does not cause serious inaccuracy. Table 2 seems to be offered in support of such an assertion. However, such a statistical treatment (1) is unnecessary and (2) cannot lead to any valid scale-up generalization. The amount of error introduced by using a 1"wrong" value of diameter exponent — i.e., -1.20 instead of m — is determined by Eq. 6. The ratio of the pressures predicted by Eqs. 6 and 20 for scale-up from ds in small pipe to d, in casing is The pressure for 30-percent calcium carbonate slurry (m = -1.112) in 10-in. ID pipe would be predicted 30 percent higher by Eq. 6 than by Eq. 20, if based on tests in 0.5-in. ID pipe. The data set tested in Table 2 represents only a limited sample of pipe diameters (ratio of about 4:l) out of the population of diameters that would