Chapter 8, Problem 34P

Beyond the Colebrook equation, other relationships, such as the Fanning friction factor f, are available to estimate friction in pipes. The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which can all be represented by another dimensionless quantity, the Reynolds number Re. A formula that predicts f given Re is the von Karman equation,

$\frac{1}{\sqrt{f}}=4{\text{ log}}_{10}\left(\text{Re}\sqrt{f}\right)-0.4$

Typical values for the Reynolds number for turbulent flow are 10,000 to 500,000 and for the Fanning friction factor are 0.001 to 0.01. Develop a function that uses bisection to solve for f given a user-supplied value of Re between 2500 and 1,000,000. Design the function so that it ensures that the absolute error in the result is $E_{a.b}<0.000005$.