BAE 3313 Natural Resource Engineering
Fall 2013
Laboratory Number 6 - H-Flume Rating Curve
Due: November 5, 2013 by 5:00 pm
Total Points 50
Objectives
1. Learn how to read a point gage.
2. Introduce the concept of measuring flow using flumes and orifices.
3. Develop a rating curve using regression analysis.
Procedure
Each student must take one set of point gage readings and set the flow rate using the
manometer.
1. Install an orifice plate; use two orifice plates (1.5 inch or 2.5 inch). Take 14 sets of readings
(different flow rates) per orifice plate. The equations for the orifice plates at the USDA ARS
Hydraulics Laboratory are:
1.5" Diameter Orifice: 𝑄 = 0.016917√∆ℎ
2.5" Diameter Orifice: 𝑄 = 0.04721√∆ℎ
Q = flow rate, cfs
Δh = difference in manometer readings, inches
2. Take readings in the flume to determine B0, B1, and D; record. Pour enough water in flume to
initiate flow. Record point gage when flow is at or near zero; this is the H-Flume Zero Reading.
3. Set the flow rate using the manometer. Record flow rate by reading the manometer and
adjusting it for temperature. Assume the flow rate going through the orifice plate is known.
4. Measure the depth of flow in the H-flume using the point gage. Repeat for a total of five
readings per manometer setting.
Assignment
1. Using ALL data points from BOTH orifice plates, develop ONE rating curve for the H-flume
using MS Excel, i.e. Flow Rate = f (stage height). Adjust measured depths using the H-Flume
Zero Reading.
a) Develop regression equations using the following forms: logarithmic, power, linear,
second-order polynomial
b) Develop a Standard H-flume equation neglecting the velocity of approach. See details
under the H-FLUME STANDARDIZED DISCHARGE EQUATION section.
b) For each equation, provide the following:
i) Graph containing observed data points (symbol) and equation (as line)
ii) Regression equation, standard error (Se), Coefficient of Determination (R2)
∑𝑛 (𝑂𝑖 − 𝑃𝑖 )2
𝑆𝑒 = √ 𝑖=1
𝑛−𝑘
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where
O = observed or measured discharge
P = predicted discharge using an equation
n = number of data points
k = number of estimated parameters. For second-order
polynomial, k = 3; for all other regressions in this exercise, k = 2.
iii) Residual plot, i.e. Qpredicted - Qobserved (Y axis) vs Qpredicted (X axis).
2. Present a summary table with your regression statistics and Standardized Flume equation.
3. Select the most appropriate regression model. Explain and justify your selection. Is your
selected regression equation acceptable?
4. Compare your selected regression equation with the Standardized Flume equation. Discuss
and explain any differences.
5. Suppose you required to establish a network of discharge gaging stations consisting of 50 Hflume structures. Would you perform a calibration for all 50 H-flumes? Things to consider may
include: 1) time and resources to develop rating curves, 2) differences between laboratory and
field conditions, and 3) quality control of flume fabrication, 4) etc. Discuss and explain.
Example H-Flume Rating Curve Using a Linear Equation
Figure 1. Discharge as function of flow depth. Red square symbols are observed data and the
solid black line is the regression equation.
Figure 2. Plot of residuals for a linear equation.
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Background
Vernier Scales
Verniers are a mechanical means of increasing the level of precision on a reading. For more
detailed information on how to read a Vernier scale see Cline (2000), Field (2011) or Lindberg
(2003).
Figure 1. Vernier scale of point gage with steps for reading to the nearest 0.001 ft.
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H Flumes
H flumes, developed by the Natural Resources Conservation
Service, are made of simple trapezoidal, flat surfaces. These
surfaces are placed to form vertical converging sidewalls. The
downstream edges of the trapezoidal sides slope face upward
toward the upstream approach, forming a notch that gets
progressively wider with distance from the bottom. These flumes
should not be submerged more than 30 percent. This group of
flumes, including H flumes, HS flumes, and HL flumes, has been
used mostly on small agricultural watersheds. The primary
advantage of an H flume is its ability to measure flow over a wide
range with reasonable accuracy. Construction is relatively simple
and installation is easy.
Figure 2. Commercially
fabricated H flume in
laboratory setting.
Figure 3. Dimensions of standard H flumes. From Grant (1989).
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H-FLUME STANDARDIZED DISCHARGE EQUATION
Discharge Equation (Gwinn and Parsons, 1976)
𝒗𝟐
𝒗𝟐
𝑸 = [(𝑬𝒐 + 𝑬𝟏 𝑫)𝑩𝒐 + (𝑭𝒐 + 𝑭𝟏 𝑫)𝑩𝟏 (𝑯 + )] √𝟐𝒈 (𝑯 + )
𝟐𝒈
𝟐𝒈
𝟑/𝟐
Bo = one-half bottom of flume at outlet, feet
B1 = vertical projected slope of control edge (front view), ft/ft
D = flume maximum depth, feet
Table 1. Discharge Coefficients (Gwinn and Parsons, 1976)
Hydraulics Lab Flume Dimensions for Flow Equation
W
The measured height, D, is 9 1/16 inches or 0.755 ft, the
measured bottom is 0.90 inches, and the top width, W is 20
inches.
1
2
1 𝑓𝑡
)
12 𝑖𝑛
𝐵0 = (0.90 𝑖𝑛) (
𝐵1 =
10−0.9 1
1
2
9
= 0.075 𝑓𝑡
= 0.50
16
Since the H-flume is has standard dimensions, the
Standardized equation is valid. Note, however, that entrance
section of the H-flume is larger than the standard flume,
which will cause some error.
Figure 4. Front view of H Flume.
You should not expect the H flume equation (Gwinn and Parsons, 1976) to fit better than
models developed by regression for a particular flume. Very few structures in practice are
calibrated; they rely on standardized equations developed from exercises like the one you
conducted. Note that for large structures, rating curves are developed using velocity profiling,
similar to the one you performed in a previous laboratory exercise.
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Velocity of Approach
The velocity of approach, v, is neglected because the non-standard proportions and poor
entrance conditions likely create velocity profiles unlike those in flumes used to development the
equation. Additionally, it requires an iterative computation, which though not complex, distracts
from the focus of this laboratory exercise. Furthermore, for the velocities encountered in this
exercise, the term is negligible:
The equation can now be simplified as:
𝑄 = [(𝐸0 + 𝐸1 𝐷)𝐵0 + (𝐹0 + 𝐹1 𝐷)𝐵1 𝐻]√2𝑔 𝐻 3⁄2
Use this equation for your laboratory assignment.
Coefficient of Determination, R2
For the regressions, you will find the residuals after developing the regression equation. For the
Standardized H-flume equation, the equation itself is your “trend line”, and thus residuals are
computed directly from observed and predicted flow rates.
For the known (orifice plate) flow rates and predicted H flume flow rates, Coefficient of
Determination, R2, must be computed rather than merely captured from a spreadsheet trend line
development.
𝑅2 = 1 −
∑𝑛𝑖(𝑂𝑖 − 𝑃𝑖 )2
𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒𝑠 𝑜𝑓 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠
=1− 𝑛
𝑡𝑜𝑡𝑎𝑙 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒𝑠
∑𝑖 (𝑂𝑖 − 𝑂̅ )2
References
Cline, Christopher A. 2000. Vernier Caliper Applet.
http://people.westminstercollege.edu/faculty/ccline/courses/resources/wp/vernier/vernier.html
Grant, Douglas M. 1989. Open Channel Flow Measurement Handbook. Instrument Specialties
Co. Lincoln, Nebraska.
Gwinn, Wendell R., and Parson, Donald A. 1976. Discharge Equations for HS, H, and HL
Flumes. Journal of the Hydraulics Division. ASCE.
Lindberg, Vern. 2003. Verniers and the Caliper.
http://people.rit.edu/vwlsps/VernierCaliper/caliper.html
Page 6 of 7
H flume calibration data
Date:
Water temperature:
H flume calibration
H flume parameters:
pt gage at zero flow:
B0:
B1:
D:
°
Manometer
left
right
Orifice
time,
head, in. head, in.
diame
hh:mm
H2O
H2O
ter, in.
i
ft
ft
Date:
Water temperature:
ft
H Flume
Dh, in.
H2O
Orifice
discharge,
cfs
Pt gage, ft
1
2
3
i
4
5
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
Orific
time,
e
hh:m
diam
m
eter,
in.
Page 7 of 7
h
in