A sensitivity analysis was undertaken prior to the experimental study; for the purpose of demonstrating the impact of instrumentation error on the outputs of the key equations employed within the current study, namely the C-W equation. This would allow the intended instrumentation to be vetted and the operational range to be refined in accordance with the established instrumentation limitations. The principal variable and equation under review were ks and the C-W equation, respectively. Based on the traditional C-W equation ks is a function of pipe diameter, average freestream velocity, headloss, pipe length, kinematic viscosity, and the gravity acceleration constant:
𝑘𝑠 = 𝑓(𝐷, 𝑈̅, 𝐻𝑓, 𝜐, 𝑔); 𝑣 = 𝑓(𝑇) Equation 3.18
The impacts of each of the parameters in Equation 3.18, with the exception of g on ks were reviewed discretely and a total uncertainty was established. The total uncertainty was determined using the root-square-sum (RRS) method (Abernethy et al. 1985). Total ks
uncertainties were established for three different scenarios, including; i) a varying flow scenario (i.e. ReD); ii) a varying surface roughness scenario (i.e. ks) and iii) a varying temperature scenario. It should be noted that a total ks uncertainty was estimated for each variable increment for each of the aforementioned scenarios. Table 3.2 provides a summary of the predefined and fixed uncertainties by which each parameter was varied within each of the three scenarios.
Table 3.2 Parameter Uncertainties and Values Parameter Symbol Uncertainty Average Freestream Velocity 𝑈̅ ±1.5%
Hydraulic Headloss Hf ±0.57mmH2O Diameter (=102.08mm) D ±0.67mm
Pipe Length (= 5.00m) L ±0.5mm
Fluid Temperature T ±0.22ºC
These uncertainties were derived from the manufacturer’s specifications and worst case measurements. The upper and lower boundaries for each scenario were selected based upon estimated conditions during testing. For instance, the upper and lower limits for the the varying flow scenario were ReD = 3.00x104 (𝑈̅ ≈ 0.3 m/s) and ReD = 1.50x105 (𝑈 ̅ ≈ 1.5 m/s), respectively (see Section 3.2.2). In the case of the varying roughness scenario the upper and lower limits were selected based upon the expected ks values for a non-fouled and fouled pipe, respectively. Principally, for the upper limit ks value was taken as 0.600 mm (Wallingford and Barr 1994), and for the lower limit a ks value of 0.012 mm (Grann-Meyer 2010). The upper and lower limits of the varying temperature scenario were 17 and 22ºC, respectively and were based upon estimated temperature range within the pipeline during testing.
Figure 3.24 to Figure 3.26 summarise the results of the sensitivity analysis for each of the three scenarios. In particular, Figure 3.24 illustrate the results of the varying flow conditions scenario, for fixed values of ks equal to 0.012 and 0.600 mm. Figure 3.25 present the results of the varying surface roughness scenario, for ReD = 3.00x104 and ReD = 1.50x105. Fluid temperature within the varying flow and surface roughness scenarios was fixed at 20ºC.
Figure 3.26 show the results varying temperature scenario, for ReD = 3.00x104 and ReD = 1.50x105, respectively. The value of ks within the varying temperature scenario was equal to 0.012 mm. In addition to showing the total ks uncertainties, Figure 3.24 to Figure 3.26 present the proportional impacts of each of the discrete parameters on the totaluncertainty, as a percentage. It is evident that the contribution from the uncertainty in L was insignificant as a proportion of total uncertainty. This was a direct result of the uncertainty in L being several orders of magnitude less than its absolute value (i.e. 5 m). Similarly, the impacts of v and D on the total uncertainty were typically insignificant, though to a lesser extent than L.
Generally, it was found that uncertainties in 𝑈̅ and Hf had the greatest contribution to the overall uncertainty in ks.
Figure 3.24 Sensitivity analysis on ks: varying flow scenario, for fixed values of a) ks = 0.012 mm and b) ks = 0.600 mm (T = 20ºC). Highlighting the proportional impacts of U, Hf, D and v on the
total ks uncertainty.
Figure 3.25 Sensitivity analysis on ks: varying surface roughness scenario for fixed values of a) ReD
= 3.00x104 and b) ReD = 1.50x105(T = 20ºC). Highlighting the proportional impacts of U, Hf, D and v on the total ks uncertainty.
0%
Figure 3.26 Sensitivity analysis on ks: varying temperature scenario for fixed values of a) ReD = 3.00x104 and b) ReD = 1.50x105(ks = 0.012mm). Highlighting the proportional impacts of U, Hf, D
and v on the total ks uncertainty.
Figure 3.27 Impact of the error in Hf and 𝑈̅ on total uncertainty in ks for a) varying ReD and ks =
The analysis has indicated that the nature of the uncertainties played a significant role in their overall uncertainty in ks. For instance, the uncertainty in 𝑈̅ was specified by the manufacturer as a fixed percentage of the absolute reading (±1.5%). Whereas, the uncertainty in Hf was specified as a fixed value and for pressure transducers used within the current study, which equalled ±0.57 mmH2O. This meant that the uncertainty associated with the pressure transducer would dominate at low values of ks and/or ReD, where absolute headloss is inherently small, as shown by Figure 3.27.
It is evident that the total uncertainty in ks was proportionately large at the lower ends of the spectrum either in terms of ks or ReD. For instance, Figure 3.24 illustrates that total ks
uncertainty at ReD = 3.0x104 can be an order of magnitude greater than the absolute ks value (i.e. 0.012 mm). This highlights the difficulty determining accurate values of ks for very smooth pipes, particularly when operating at ReD. Consequently, all low ReD investigations (i.e. ReD < 5.0x104)in the current study were repeated several times and were mostly viewed with caution. This was particularly relevant of the non-fouled phase of the current study.
The sensitivity analysis has indicated that surface roughness and ReD have the greatest impact on total ks uncertainty. In particular, it was found that as surface roughness or ReD increases the total uncertainty in ks decreases, as shown 3.7, which present the contributed impact of surface roughness and ReD on total ks uncertainty. The impact of temperature was found to be minor in comparison, particularly for the expected range (i.e. 17ºC < T < 22ºC). The sensitivity analysis has indicated that the intended equipment was suitable for measuring ks
although, caution should be taken when investigating low ReD and smooth surfaces.
Figure 3.28 Sensitivity analysis on ks: impact of ks and ReD on total ks uncertainty.
0.00 0.10 0.20 0.30 0.40 0.50 0.60
3.00E+04 5.00E+04 7.00E+04 9.00E+04 1.10E+05 1.30E+05 1.50E+05 ks (mm)
ReD
10-15%
25-50%
> 100%
50-100%
15-25%
Total % uncertainty in ks