The so-called dimensionless Moody pipe friction factor f, plotted in Fig. 6.13, is calculated in experiments from the following formula involving pipe diameter D, pressure drop Δp, density ρ, volume flow rate Q, and pipe length L:
f = \frac{π^2}{8} \frac{D^5 Δp}{ρQ^2 L}Measurement uncertainties are given for a certain experiment: D = 0.5 percent, Δp = 2.0 percent, ρ = 1.0 percent, Q = 3.5 percent, and L = 0.4 percent. Estimate the overall uncertainty of the friction factor f.
The coefficient π^2/8 is assumed to be a pure theoretical number, with no uncertainty. The other variables may be collected using Eqs. (1.41) and (1.42):
\delta P=\left[\left(\frac{\partial P}{\partial x_1} \delta x_1\right)^2 + \left(\frac{\partial P}{\partial x_2} \delta x_2\right)^2 + … + \left(\frac{\partial P}{\partial x_N} \delta x_N\right)^2\right]^{1/2} (1.41)
\frac{\delta P}{P} = \left[\left(n_1 \frac{\delta x_1}{x_1}\right)^2 + \left(n_2 \frac{\delta x_2}{x_2}\right)^2 + \left(n_3 \frac{\delta x_3}{x_3}\right)^2 + …\right]^{1/2} (1.42)
U=\frac{\delta f}{f}=\left[\left(5 \frac{\delta D}{D}\right)^2 + \left(1 \frac{\delta Δp}{Δp}\right)^2 + \left(1 \frac{\delta \rho}{\rho}\right)^2 + \left(2 \frac{\delta Q}{Q}\right)^2+\left(1 \frac{\delta L}{L}\right)^2\right]^{1/2} = [\{5(0.5 \% )\}^2+(2.0 \%)^2+(1.0\%)^2+\{2(3.5\%)\}^2+(0.4\%)^2]^{1/2} \approx 7.8 \%By far the dominant effect in this particular calculation is the 3.5 percent error in Q, which is amplified by doubling, due to the power of 2 on flow rate. The diameter uncertainty, which is quintupled, would have contributed more had \deltaD been larger than 0.5 percent.