Calibration and Uncertainties of Pipe Roughness Height

Transcription

1 9 th IWA/IAHR Conerence on Urban Drainage Modelling Calibration and Uncertainties o Pipe Roughness Height Kailin Yang, Yongxin Guo, Xinlei Guo,Hui Fu and Tao Wang China Institute o Water Resources and Hydropower Research Beijing,China

2 Outline Introduction Estimation o roughness height and uncertainty 3 Application 4 Conclusion

3 Introduction The average roughness height k o pipe is a key parameter in hydraulic calculation o pipe systems. For any type o pipes, as soon as k is known, the capacity o water conveyance o urban water-supply and drainage systems can be predicted in the design. Thereore, or a new type o pipes, the irst thing rom the hydraulics view o point is to calibrate this value in hydraulic labs precisely. I lows are in ully rough region, k can be obtained by the Nikuradse equation (933). Otherwise, it can be estimated by the Colebrook equation (939), which covers not only the transition region but also the ully developed smooth and rough pipes. 3

4 Introduction Recently, several authors carried out the urther investigation or estimation o the roughness coeicient. Author Shockling, Allen and Smits (006) Yang and Joseph (009) Contents the roughness coeicient or honed suraces ollows the Nikuradse (933) orm with dips and bellies rather than the monotonic relations seen in the Moody diagram based on the Colebrook equation derived an accurate composite riction actor versus Reynolds number correlation ormula or laminar, transition and turbulent low in smooth and rough pipes. We did the experiment to calibrate the values o k or the three types o ductile cast iron pipes lined with cement mortar, epoxy resin and polyethylene. The results shown that the values o k ound by the Colebrook equation varied signiicantly with the change in Re(Reynolds number). 4

5 Photos o three types o pipes Cement mortar lining pipes Epoxy resin lining pipes Polyethylene lining pipes 5

6 Introduction Author Yang Kailin et al. Contents The values o k were between 0.04mm and 0.040mm or the pipes lined with cement mortar, between 0.05mm and 0.078mm or the pipes lined with epoxy resin,and between mm and 0.006mm or the pipes lined with polyethylene. As well known, however, its value should be constant or the given pipe within a short time. Why does the contradiction occur? One o the reasons is that there are the uncertainties in the measured data. Figure. Found by the Colebrook equation 6

7 Introduction Our Work It is mainly to demonstrate how to determine the value o k reasonably by the systematical analysis o the uncertainties or the measured parameters, such as pipe diameter, length, low rate and headloss as well as width, height and head above crest level o weir or low rate.

8 Estimation o roughness height and uncertainty. Estimation o roughness coeicient The head loss may be estimated using the Darcy-Weisbach equation LQ h = gda h depends on the piezometer heads at the inlet and the outlet o a pipe The dimensionless riction number depends on both the pipe roughness and the Reynolds number k = lg 3.7D +.5 Re k is roughness height Thus = gda LQ h k = Re

9 Estimation o roughness height and uncertainty. Uncertainty o The dispersions o the value can be estimated rom the above equation, on the basis o the theory o uncertainty propagation or independent variables (EA-4/0, 999), as = h h + Q + D + L Q D L It may be written as h = h Q Q D + 5 D gda = h LQ gda h = L L Q L L = h =... L h Q D L In uncertainty analysis, the values h Q D L are reerred to as dimensionless standard uncertainties Q u D u ( L) and have the notation u ( ) u ( ) u ( ) ( ) Since the uncertainties are independent o each other, its summation can be obtained as u ( ) ( ) ( ) 5 4 ( ) = u h + u Q + u D + u ( L) h

10 .3 Uncertainty o k Similarly, the dispersions o the values k can also be expressed as It may be written as Thus, the dimensionless standard uncertainty o k should be D D k Q Q k k k + + = + =.5.5 Re ln0 3.7 D k Re Q D Q k = L L a D D a Q Q a h h a k k + = 4 3 = Re D k ( ) ( ) ( ) ( ) ( ) 4 3 L u a D u a Q u a h u a k u = Estimation o roughness height and uncertainty

11 Estimation o roughness height and uncertainty.4 Uncertainty o Q or suppressed sharpcrested weirs The Rehbock equation (Rehbock, 99; BS ISO 438, 008) is recommended or the low rate or the weir orm. It may be written as ollows H Q = B H + P ( 0.00). 5 The dispersion o the value Q can be written as Q Q = b H H B + B b P P b P = + H H P H b = 0.4 H P P the dimensionless standard uncertainty o Q should be u ( Q) = b u ( H ) + u ( B) + b u ( ) P H

12 Estimation o roughness height and uncertainty Uncertainties roughness coeicient roughness height discharge u u u Expression ( ) ( ) ( ) 5 4 ( ) = u h + u Q + u D + u ( L) ( k) = a u ( h ) + a u ( Q) + a u ( D) + a u ( ) 3 4 L ( Q) = b u ( H ) + u ( B) + b u ( ) P In summary, the uncertainties o and k come rom the uncertainties o the measurement or D, L, H, B and P as well as H and H. I electrometric measurements, such as low meter and pressure transducers, are used or Q, H and H, u*(q), u*(h ) and u*(h ) equal their accuracy reading. The piezometer tubes are usually used or the measurement o H and H, the point gauges or H, and measuring scales or D, L, B and P. The uncertainties o them are basically caused by eyeballing random errors.

13 3 Application The ollowing will demonstrate how to determine the value o k reasonably by the systematical analysis o the uncertainties or the measured parameters, by taking the ductile cast iron pipes lined with epoxy resin as an example. The calibrated pipeline consisted o 5 pipes. The length o each pipe is 6m and the internal diameter is 0.30 m. The energy loss was measured by two piezometer tubes and the corresponding pipe length is 6.60 m. The eyeballing random errors are H = ±0.0005m H = ±0.0005m D = ±0.000m L = ±0.00m The low rates were measured by a suppressed sharp-crested weir with P=0.56m and B=.005m. The eyeballing random errors are P = B = ±0.00m H = ±0.000m The upstream gauged head above the crest level and the pipe headloss are measured in each case. Thus, the uncertainties o u*(q), u*(h ), u*() and u*(k) could be calculated and the results are shown in Figures to 8. 3

14 3. Uncertainty o Q 3 Application Figure. Curves o Q and u*(q) versus H As the weir head H increases rom m to 0.9 m, the low rate Q varies rom 0.036m 3 /s to m 3 /s and the dimensionless standard uncertainty u*(q) decreases rom 0.3% to 0.% monotonically. It demonstrates that the precision o the low measurement is quite high. 4

15 3. Uncertainty o h 3 Application Figure 3. Curves o h and u*(h ) versus Re These two igures show the results o calculated head losses and their uncertainties. I the low Q increases rom 0.036m 3 /s to m 3 /s, the corresponding Reynolds number is between 0 5 and As Re rises, h increases rom 0.009m to.035m and u*(h ) reduces rom 7.86% to 0.07% monotonically. It illustrates that the value o u*(h ) is much greater when Q is less. 5

16 3 Application 3.3 Uncertainty o pipe roughness coeicient Figure 4. Curves o and u*( ) versus Re Figure 4 shows the results o calculated Darcy riction actors and their uncertainties. It can be seen that with the increase o Re the pipe roughness coeicient decreases rom to 0.07, and the uncertainty u*() reduces rom 7.88% to 0.3% monotonically. It demonstrates that the calibration o has a greater uncertainty i Re or Q is less. 6

17 3 Application 3.4 Uncertainty o pipe roughness height k Figure 5. Curves o k and u*(k ) versus Re Figure 5 shows that the value o k ound by the Colebrook equation varied signiicantly with the change in Re. When Re increases rom 0 5 to 0 6., k changes between 0.05 mm and mm, and the corresponding dimensionless standard uncertainty u*(k) decreases rom 34% to 3%. When Re is less, k is greater and ruleless. When Re is greater, i.e. Re> 0 5.7, k approximates a ixed number and u*(k) is less than 5%. The ixed number is the calibrated value o k. 7

18 Components o u*(k) 3 Application 3.4 Uncertainty o pipe roughness height k a u*(h ) 50 a u*(q) lg(re) Components o u*(k) a u*(h ) a u*(q) lg(re) Figure 7 Eect o u*(h ), u*(q), u*(d) and u*(l) on u*(k) 7 6 a 3 u*(d) 5 a 4 u*(l) lg(re) Obviously, the eect o u*(l) on u*(k) is negligible. I Re is less, u*(k) mainly depends on u*(h ). I Re is greater, u*(k) is dominated by u*(h ), u*(q) and u*(d). The characteristics o the Colebrook Equation is that when Re is greater, or the low is in ully rough regime, the value o mainly depends on k rather than Re ; but i the low is in smooth rough regime, Re is oten a dominating eect. Components o u*(k) 8

19 3 Application 3.4 Uncertainty o pipe roughness height k Components o u*(k) a u*(h ) 50 a u*(q) lg(re) Components o u*(k) a u*(h ) a u*(q) lg(re) Components o u*(k) 7 6 a 3 u*(d) 5 a 4 u*(l) lg(re) As mentioned above, one comes to a conclusion that or the ductile cast iron pipes lined with epoxy resin k=0.0mm and u*(k)<5%. Similarly, it is calibrated that i u*(k)<5%, k=0.005mm or the ductile cast iron pipes lined with polyethylene and k=0.037mm or the ductile cast iron pipes lined with cement mortar. 9

20 3 Application 3.4 Uncertainty o pipe roughness height k The above Figure shows the curves o versus Re, in which the discrete points are drawn by the measured data and the solid lines are computed by the Colebrook Equation at k =0.005mm, 0.0mm and 0.037mm, respectively. Obviously, except some positions in which there are the greater deviations when Re is less, the deviations between the measured data and computed one is less than % in most o the positions. 0

21 4 Conclusions This paper presents the equations or calculation o the uncertainties o the roughness coeicient, average roughness height and the low rate o suppressed sharp-crested weirs, on the basis o the uncertainties o the measurements o pipe diameter, length and headloss or peizometer head as well as width, height and head above crest level o weir or low rate. It demonstrates how to determine the value o k reasonably by the systematical analysis o the uncertainties, Some important conclusions are obtained as ollows: ) The dimensionless standard uncertainties o headloss, low rate and roughness coeicient all decrease monotonically with the increase o Re; ) When Re>4x0 5, with the increase o Re the roughness height varies slightly and its uncertainty reduces greatly; 3) The tiny uncertainties o measured headloss, low rate, pipe diameter and length can result in a quite great uncertainty o roughness height.

22 Thank you