BIE 5300/6300 Assignment #5
Open-Channel Constriction Calibrations
06 Oct 04 (due 12 Oct 04)
Show your calculations in an organized and neat format. Indicate any
assumptions or relevant comments.
You have measured data in a spreadsheet file (attached) for an open-channel
constriction (non-orifice flow). One data set is for free-flow conditions, the other for
submerged-flow.
I.
Develop a free-flow rating for this constriction using the following equation:
Qf = Cf hnuf
(a)
(b)
(c)
II.
Determine Cf and nf.
Make a graph with plotted symbols for (Qf)measured vs. (Qf)calculated. The
ordinate range should be the same as the abscissa range, with a
diagonal line representing (Qf)measured/(Qf)calculated = 1.0.
Comment on the data fit, using correlation or other such indices, as
appropriate.
Develop a submerged-flow rating for this constriction using the following
equation:
Cs (hu − hd )
nf
Qs =
(a)
(b)
(c)
III.
( − log10 S )
ns
Determine Cs and ns, using nf from the free-flow rating.
Make a graph with plotted symbols for (Qs)measured vs. (Qs)calculated. The
ordinate range should be the same as the abscissa range, with a
diagonal line representing (Qs)measured/(Qs)calculated = 1.0.
Comment on the data fit, using correlation or other such indices, as
appropriate.
Solve for transition submergence, St, for the above calibration.
(a)
(b)
(c)
Determine St.
Make a graph of St (from 0.1 to 0.99 on the abscissa) vs. the function
value (Qf - Qs = 0) and indicate where the solution(s) exist, if any.
If you don’t get any solution for St, try adjusting Cs slightly so that you
get a solution. If you do this, show the adjusted Cs value.
IV.
Re-do the submerged-flow rating using the following equation:
Cs (hu − hd )
ns1
Qs =
(a)
(b)
(c)
(d)
( − log10 S )
ns 2
Determine Cs, ns1 and ns2 based only on the submerged-flow data.
Make a graph with plotted symbols for (Qs)measured vs. (Qs)calculated. The
ordinate range should be the same as the abscissa range, with a
diagonal line representing (Qs)measured/(Qs)calculated = 1.0.
Comment on the data fit, using correlation or other such indices, as
appropriate.
Comment on the data fit using this equation, as opposed to using the Qs
equation from (II) above.
Solutions:
I.
Develop a free-flow rating for this constriction using the following equation:
Qf = Cf hnuf
Make two new columns for ln(Qf) and ln(hu) in the spreadsheet. Do a linear
regression using the LINEST spreadsheet function. The regression gives:
Cf= 5.57
nf = 1.61
for Qf in cfs; and hu in ft.
The R2 value is 0.999, indicating a very good fit, and this is also seen in the
comparison graph:
7.0
6.0
(Qf)calculated
5.0
4.0
3.0
2.0
1.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
(Qf)measured
II.
Develop a submerged-flow rating for this constriction using the following
equation:
Cs (hu − hd )
nf
Qs =
( − log10 S )
ns
Make two new columns for ln(Qs/(hu-hd)nf) and ln(-log10S) in the spreadsheet.
Do a linear regression using the LINEST spreadsheet function. The regression
gives:
Cs= 2.62
ns = 1.43
for Qs in cfs; and hu & hd in ft.
The R2 value is 0.999, indicating a very good fit, and this is also seen in the
comparison graph:
7.0
6.5
6.0
(Qs)calculated
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
2.0
3.0
4.0
5.0
6.0
7.0
(Qs)measured
III.
Solve for transition submergence, St, for the above calibration.
Use the equation for f(St) = 0, as shown in the lecture notes. This equation is
derived by setting Qf = Qs. Make a table of St versus f(St), then plot the results.
The only solution is for St = 1.00, which is mathematically correct, but physically
impossible.
Adjust Cs slightly, from 2.62 to 2.639, whereby a solution is found at about St ≈
0.79, as shown in the next graph.
6.0
5.0
f(St)
4.0
3.0
2.0
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
St
0.016
0.014
0.012
f(St)
0.010
0.008
Solution at
St = 0.79
0.006
0.004
0.002
0.000
0.65
0.70
0.75
0.80
0.85
St
0.90
0.95
1.00
Re-do the submerged-flow rating using the following equation:
Cs (hu − hd )
ns1
Qs =
( − log10 S )
ns 2
Make three new columns for ln(Qs), ln(hu-hd), and ln(-log10S) in the spreadsheet.
Do a multiple linear regression using the LINEST spreadsheet function. The
regression gives:
Cs= 2.78
ns1 = 1.54
ns2 = 1.36
for Qs in cfs; and hu & hd in ft.
The R2 value is 0.996, indicating a very good fit, and this is also seen in the
comparison graph, which is very similar to the previous plot:
7.0
6.5
6.0
5.5
(Qs)calculated
IV.
5.0
4.5
4.0
3.5
3.0
2.5
2.0
2.0
3.0
4.0
5.0
6.0
7.0
(Qs)measured
Even though the R2 value is slightly lower than for the previous form of the
submerged-flow equation, the sum of absolute deviations in measured and
calculated discharges is less in this case.