In the late 1940s it was discovered that the addition of polymers that form random coiled structures in a fluid would reduce the power required to pump that fluid through a tube or to push an object through the fluid. The phenomenon of drag reduction has become very important and is used in the Alaska pipeline. An excellent review of the subject is given by Virk [27]. Drag reduction depends upon the concentration of polymer in the system, but Virk found that the effect asymptotes and that the minimum friction factor is given by:

\[\frac{1}{\sqrt{f}}=19 \log _{10}\left[\operatorname{Re}_{d} \sqrt{f}\right]-32.4\]

a. Compare predictions for the friction factor at maximum drag reduction with the Prandt1/von Karman universal resistance law. Plot the difference for Reynolds numbers between \(10^{4}\) and \(10^{7}\).

b. Compare the power required to flow oil through the Alaska pipeline at a nominal flow rate of 600,000 barrels per day with and without maximum drag reduction. The pipeline is \(1287 \mathrm{~km}\) long and \(122 \mathrm{~cm}\) in diameter and the temperature of the oil is \(40^{\circ} \mathrm{C}\). The specific gravity of the oil is 0.877 and its viscosity is \(8.77 \times 10^{-3} \mathrm{Ns} / \mathrm{m}^{2}\).