4. Uncertainty Quantification: In a fully rough turbulent pipe flow, frictional head loss $h_f$ (that determines the pressure drop in the pipe and hence the pumping power needed) is estimated using Darcy-Weisbach equation
$$h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g},$$
where the Darcy friction factor $f$ is estimated using the Colebrook equation for highly turbulent flows as:
$$f = \left[-2 \log_{10} \left(\frac{\phi_r}{3.7}\right)\right]^{-2}$$
where $f$ is the Darcy friction factor (dimensionless), $\phi_r$ is relative roughness (dimensionless), $L = 100[\text{m}]$ is pipe length, $D = 0.1 [\text{m}]$ is pipe diameter, $V = 2.0 \pm 0.1 [\text{m/s}]$ (5% uncertainty) is fluid velocity, $g = 9.81 [\text{m/s}^2]$ is gravitational acceleration, $\phi_r = 0.02 \pm 0.01$ (50% uncertainty) is the relative roughness.
(a) Compute the nominal value of friction factor $f$ and head loss $h_f$.
(b) Using first-order Taylor series uncertainty propagation, estimate the uncertainty $\Delta h_f$ due to uncertainties in $\phi_r$ and $V$. Show all steps.
(c) Compute the relative uncertainty in percent and identify the dominant source of uncertainty.
