Pipe Friction Between Two Reservoirs (Moody Chart): 04-BS-7, May 2013, Q8

Question 8: Pipe Friction Between Two Reservoirs (Moody Chart) (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $D=2.0\ \text{m}$, $L=1560\ \text{m}$, roughness $e=1.5\ \text{mm}$, $Q=8.0\ \text{m}^3/\text{s}$; water $\nu=\mu/\rho=1.0\times10^{-6}\ \text{m}^2/\text{s}$.

Find. Required elevation difference between the two reservoir surfaces to drive this flow (major friction loss; entrance/exit minor losses neglected as no loss coefficients are supplied).

Approach. Compute velocity and Reynolds number, obtain the Darcy friction factor from the Moody relationship (solved here via the Colebrook–White equation the chart itself plots), then apply the Darcy–Weisbach head-loss formula.

Check: the friction factor is obtained by solving the Colebrook–White equation the supplied Moody diagram graphically represents, to the same relative roughness and Reynolds number a chart reading would use.

Velocity and Reynolds number. $A=\pi D^2/4=\pi(2)^2/4=3.1416\ \text{m}^2$; $V=Q/A=8.0/3.1416=2.546\ \text{m/s}$. $$Re = \frac{VD}{\nu} = \frac{2.546\times2.0}{1.0\times10^{-6}} = 5.09\times10^6$$
Relative roughness. $e/D = 1.5/2000 = 7.5\times10^{-4}$.
Friction factor (Colebrook–White). $\dfrac{1}{\sqrt f}=-2\log_{10}\left(\dfrac{e/D}{3.7}+\dfrac{2.51}{Re\sqrt f}\right)$ solved iteratively gives $f=0.0184$ (fully-turbulent region of the Moody chart, consistent with $Re>5\times10^6$).
Head loss (Darcy–Weisbach). $$h_L = f\frac{L}{D}\frac{V^2}{2g} = 0.0184\times\frac{1560}{2.0}\times\frac{2.546^2}{2\times9.81} = \boxed{4.75\ \text{m}}$$

Quantity	Value
Velocity	2.55 m/s
Reynolds number	5.09×10⁶
Friction factor f	0.0184
Required reservoir elevation difference	4.75 m

Topic: Darcy–Weisbach pipe friction and the Moody chart.

Key relations: $h_L=f(L/D)(V^2/2g)$; Colebrook–White equation.

Why this works: the Moody chart is simply a pre-solved graphical lookup of the (implicit) Colebrook–White equation across Reynolds number and relative roughness; solving that equation directly gives the identical friction factor a careful chart reading would, without the parallax error of reading a log–log plot by eye. At $Re\approx5\times10^6$ with $e/D\approx7.5\times10^{-4}$, the flow sits in the upper "complete turbulence, fully rough" region where f is nearly independent of further Re increases.

Common pitfall: using the pipe's nominal or radius value instead of diameter in $e/D$ and $Re$, or forgetting that $h_L$ (not some other head) is exactly the required difference in reservoir surface elevations when both reservoirs are large (negligible approach/exit velocity).

Source: Crowe, Elger & Roberson, Engineering Fluid Mechanics, Ch. 8 (pipe flow, Moody chart).

Question 8 of 13: Pipe Friction Between Two Reservoirs (Moody Chart)

Question 8: Pipe Friction Between Two Reservoirs (Moody Chart) (5 marks)