Problem Description

From the table given in the theory section, the surface roughness of wrought iron pipe (ε) is 0.00015 ft.

The average velocity of the crude oil can be determined from the volume flow rate as:

     V = 4Q/πD²

Substitute the above expression for the velocity into the Darcy-Weisbach equation to yield:

     h_L = f(L/D)(V²/2g)

     h_L = f(L/D)(4Q/πD)²/2g

Rearrange terms and the diameter of the pipe is given by

     D⁵ = 8LQ²f / h_Lgπ²

          = 8(9,500)(10)²f / (80)(32.2)π²

          = 298.9f

(a) It is found that the required diameter depends on the friction factor, which can be determined from the Moody chart. The friction factor, however, depends on the Reynolds number and relative roughness, both functions of the diameter. Hence, an iterative method is needed to solve this problem. First start with an assumed value of f, and calculate the diameter from the above equation. Based on this diameter, calculate the Reynolds number and relative roughness, then obtain the f value from the Moody chart. Compare the assumed value of f with the one from the Moody chart. Repeat this process until both values fall within an assumed error tolerance. The iterative process is summarized in the following table; the diameter of the pipe is determined to be 1.43 ft.

(b) An alternate way to determine the required pipe diameter is to use the Colebrook correlation:

Substitute the expression for f (=D⁵/298.8) into the above equation to yield:

Again, substitute the expression for Reynolds number and plug in the known quantities (e.g., ε, ρ and μ), the equation can be rewritten implicitly in terms of the diameter as follows:    

A trial and error process (or a root finding algorithm) is then executed via a spreadsheet (as shown in the table below), and the diameter of the pipe is calculated to be 1.42 ft, which is within 1% of the results obtained in part (a).