Partially and Fully Contracted Rectangular Weirs
Kindsvater and Carter (1959) developed an improved method for calibration rating of rectangular thin-plate weirs. The method applies to both fully and partially contracted rectangular weirs. The method also rates the equivalent of a suppressed weir. The capability of rating partially contracted weirs provides design versatility, especially in selection of low crest heights to reduce head drop and side contraction needed to measure flow. Thus, these weirs can reduce head loss and conserve delivery head. These weirs have coefficients that vary with measuring head as well as geometry. The resulting calibrations are at least as accurate as the equations and tables for "standard" fully contracted weirs. Weir use and dimension limits are defined by the curves for determining the calibration ratings.
The basic equation for the Kindsvater-Carter method is:
(7-1)
where:
Q = discharge, cubic feet per second (ft3/s)
e = a sub denoting "effective"
Ce = effective coefficient of discharge, ft1/2/s
Le = L + kb
h1e = h1 + kh
In these relationships:
kb = a correction factor to obtain effective weir length
L = measured length of weir crest
B = average width of approach channel, ft
h1 = head measured above the weir crest, ft
kh = a correction factor with a value of 0.003 ft
The factor kb changes with different ratios of crest length, L, to average width of approach channel, B. Values of kb for ratios of L/B from 0 to 1 are given on figure 7-4. The factor kh is a constant value equal to 0.003 ft.
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Figure 7-4 -- Value of width-adjustment factor from Georgia Institute of Technology tests (courtesy of American Civil Society of Engineers).
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The effective coefficient of discharge, Ce, includes effects of relative depth and relative width of the approach channel. Thus, Ce is a function of h1/p and L/B, and values of Ce may be obtained from the family of curves presented on figure 7-5. p is the vertical distance from the weir crest to the approach pool invert.
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Figure 7-5 -- Effective coefficient of discharge, Ce, as a function of L/B and h1/p, from Georgia Institute of Technology tests (courtesy of American Civil Society of Engineers)..
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The straight lines on figure 7-5 have the equation form:
(7-2)
where:
Ce = effective coefficient of discharge
C1 = equation coefficient
h1 = head on the weir (ft)
p = height of crest above approach invert (ft)
C2 = equation constant
For convenience, the coefficients and constants for straight lines of each L/B on figure 7-5 are given in the following tabulation for interpolation:
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Table 7-1. Coefficient and constants used in determining the effective coefficient of discharge for the Kindsvater-Carter method |
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L/B |
C1 |
C2 |
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0.2
0.4
0.5
0.6
0.7
0.8
0.9
1.0 |
-0.0087
0.0317
0.0612
0.0995
0.1602
0.2376
0.3447
0.4000 |
3.152
3.164
3.173
3.178
3.182
3.189
3.205
3.220 |
The straightforward, comprehensive, and accurate Kindsvater-Carter method of determining discharges for rectangular weirs is well suited for discharge rating use. It is particularly useful for installations where full crest contractions or full end contractions are difficult to achieve.
Traditional rectangular weirs that do not meet crest height limits or that are using the older methods of correcting for velocity of approach should be recalibrated using the Kindsvater-Carter method. Weirs that fall out of the limits of the Kindsvater-Carter rating curves need replacement or field calibration by thorough current metering.
Limits on usage and dimensions are:
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The calibration relationships were developed with rectangular approach flow and head measurement sections for these weirs. For applications with other flow section shapes, the average width of the flow section for each h1 is used as B to calculate discharges.
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The crest length, L, should be at least 6 in.
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The crest height, p, should be at least 4 in.
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Like all weirs used for head measurement, h1 should be at least 0.2 ft .
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Values of h1/p should be less than 2.4.
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All the requirements in section 5 apply.
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The downstream water surface elevation should be at least 2 in below the crest.
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All the approach flow conditions in chapter 2 apply .
Fully Contracted Standard 90-Degree V-Notch Weir
The triangular or V-notch, thin-plate weir is an accurate flow measuring device particularly suited for small flows.
(a) Traditional Equation for Standard 90-Degree Contracted V-Notch Weirs
The Cone equation is commonly used for 90degree V-notch weirs. This equation is reliable for small, fully contracted weirs generally encountered in measuring water for irrigation.
The Cone equation is:
Q=2.49h12.48 (7-6)
where:
(b) Discharge of 90-Degree Contracted V-Notch Weirs
Table A7-4 contains discharges in cubic feet per second for the standard 90-degree, fully contracted V-notch weir (figure 7-1) from the Cone equation for a range of heads ordinarily used in measuring small flows. To be fully contracted, all the overflow plate edges and the point of the notch must be located at least a distance of 2h1max from the approach flow boundaries.
(c) Limits of 90-Degree Contracted V-Notch Weirs
The crest of the weir consists of a thin plate beveled 45 degrees or greater from the vertical to produce an edge no thicker than 0.08 in. If heads will be frequently near the 0.2-ft lower limit, then the bevel-ing should be 60 degrees. This weir operates as a fully contracted weir, and all conditions for accuracy stated for the standard contracted rectangular weir apply. To be fully contracted, all the overflow plate edges and the point of the notch must be located at least a distance of two measuring heads from the approach flow boundaries. The head measuring station is located a distance of at least four measuring heads upstream from the weir crest. This 90degree V-notch weir should only be used for discharges between 0.05 and 4.25 ft3/s and should not be used consistently near the high end of this range because a 2-ft fully contracted rectangular weir will deliver the same flow at 40 percent less head for the same approach channel width. All the requirements of section 5 apply. All the approach flow conditions in chapter 2 apply.
The use of the Kindsvater-Shen method for rating V-notched weirs can considerably extend the limitations described above
-Notch Weirs of Any Angle
The Kindsvater-Shen relationship can be used for fully contracted notches of any angle between 25 degrees and 100 degrees (Kulin and Compton, 1975). The equation which includes the angle 
as a variable is written as:
(7-3)
where:
Q = discharge over weir in ft3/s
Ce = effective discharge coefficient
h1 = head on the weir in ft
h1e = h1 + kh
= angle of V-notch
The head correction factor, kh, is a function of (figure 7-6a). However, for fully contracted traditional 90-degree V-notch weirs, equation 7-6 and the rating table discussed later produce comparable accuracy.
Figure 7-6a -- Head correction factor,
kh, for V-notches of any angle (courtesy of National Bureau of Standards, Kulin et al. [1975])..
Figure 7-6b -- Effective coefficient,
Ce, for fully contracted V-notches of any angle (courtesy of National Bureau of Standards, Kulin et al. [1975])..
For fully contracted V-notch weirs, the value of kh is related to as given on figure 7-6a, and values of Ce are read from figure 7-6b. Partially contracted 90-degree V-notches only can be rated using figure 7-7 to obtain Ce values. The calibration relationships were developed with rectangular approach flow and head measurement sections for these weirs. For applications with other flow section shapes, the average width of the flow section for each h1 is used as B to determine coefficients.
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Figure 7-7 -- Effective coefficient, Ce, for partially contracted 90-degree V-notches (courtesy of National Bureau of Standards, Kulin et al. [1975])..
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Bos (1989) and International Organization for Standardization (1983) explain and define limits basic to the use of these figures. Precautions and restrictions concerning the use of V-notch weirs are as follows:
(a) V-notch weirs should not be designed beyond the range of the parameters plotted on figures 7-6 and 7-7. Only the 90-degree V-notch weir can be made partially contracted through the use of figure 7-7.
(b) The water surface downstream from the weir should always remain at least 0.2 ft below the notch. Lower discharge readings should be rejected if the contraction is not springing underneath for the entire nappe length.
(c) The measuring head should be greater than 0.2 ft because precision of head measurement error is large relative to smaller head depths, and the nappe may cling to the weir plate.
(d) For the fully contracted V-notch, the maximum measuring head should be less than 1.25 ft.
e) For the partially contracted V-notch, the maximum head should be less than 2 ft
f) For fully contracted V-notches, the h1/B ratio should be equal to or less than 0.2.
(g) For the partially contracted 90-degree notch, h1/B should be equal to or less than 0.4
h) The average width of the approach channel, B, should always be greater than 3 ft for the fully contracted Vnotch.
(i) For the partially contracted 90-degree V-notch, the approach channel width should be greater than 2 ft.
(j) The V-notch of the weir should always be located at least 1.5 ft above the invert of the weir pool for fully contracted weirs.
(k) Only the 90-degree V-notch can be partially contracted, but the point of the notch must be located at least 4 in from the channel invert.
(l) All the requirements in section 5 apply.
(m)All the approach flow conditions in chapter 2 apply.
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Cipoletti Weir
A standard Cipoletti weir is trapezoidal in shape. The crest and sides of the weir plate are placed far enough from the bottom and sides of the approach channel to produce full contraction. The sides incline outwardly at a slope of 1 horizontal to 4 vertical. A Cipoletti weir is shown on figures 7-1 and 7-9
Figure 7-9 -- Cipoletti weir with a well-type head-measuring station.
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(a) Equation for Cipoletti Weirs
The Cipoletti weir is a contracted weir. However, its discharge calibration resembles that of a suppressed weir because the effects of side contractions are intentionally compensated for by sloping the sides of the weir plate outward. Thus, discharge calibrations are nearly equivalent to suppressed weirs of the same crest lengths.
The Cipoletti equation, neglecting velocity of approach, is:
Q = 3.367 L h13/2 (7-7)
where:
The accuracy of measurements obtained by use of Cipoletti weirs and the above equation is considerably less than that obtainable with suppressed rectangular or V-notch weirs (Shen, 1959). The accuracy of the discharge coefficient is +5 percent.
(b) Discharge of Cipoletti Weirs
Table A7-5 contains discharges in cubic feet per second for standard Cipoletti weirs neglecting velocity of approach, for heads and lengths of weirs generally used in measuring small quantities of irrigation water. For the 0.5-ft, 1ft, 2-ft, and 3-ft weirs, and heads greater than one-third the crest length, the discharges have been taken from experiments performed at the Boise Project. All other discharges were computed from the Cipoletti equation. The data in the table may be considered accurate to +5 percent for weirs of the above listed lengths. The same accuracy applies to weirs of other lengths which are listed on the table with heads not over one-third the crest length.
(c) Limits of Cipoletti Weirs
All conditions for accuracy stated for the standard contracted rectangular weir apply to the Cipoletti weir. The height of the weir crest above the bottom of the approach channel should be at least twice the maximum head over the crest, and the distances from the sides of the notch to the sides of the channel should also be at least twice the maximum head. This weir should not be used for heads less than about 0.2 ft or for heads greater than one-third the crest length unless calibrations exist beyond this range for specific size weirs. The head is measured at least a distance of four measuring heads upstream from the crest.
All the requirements in section 5 apply. All the approach flow conditions in chapter 2 apply
Standard Suppressed Rectangular Weir
A standard suppressed rectangular weir has a horizontal crest that crosses the full channel width. The elevation of the crest is
high enough to assure full bottom crest contraction of the nappe. The vertical sidewalls of the approach channel continue downstream past the weir plate, preventing side contraction or lateral expansion of the overflow jet.
Special care must be taken with suppressed weirs to secure proper aeration beneath the overflowing sheet at the crest. Aeration is usually accomplished by placing vents on both sides of the weir box under the nappe (figure 7-8).
Figure 7-8 -- Section through suppressed weir with air vent in wall
Other conditions for accuracy of measurement for this type of weir are identical to those of the contracted rectangular weir, except those relating to side contraction and the crest height. The crest height should be equal to at least 3h1max. A suppressed weir in a flume drop is illustrated on figure 7-3.
(a) Equation for Standard Suppressed Rectangular Weirs With Full Bottom Contraction
The Francis equation for the standard suppressed rectangular weir (figure 7-1) is:
(7-5)
The variables in this equation have the same significance as in the equations for contracted rectangular weirs discussed in section 9. Francis obtained the coefficient of discharge from the same general set of experiments as those stated for the contracted rectangular weir. No extensive tests have been made to determine the applicability of these equations to weirs less than 4 ft in length. Similar to the contracted rectangular weir, heads less than 0.2 ft do not give accurate flow readings because the nappe of water going over the crest may not spring free of the crest. Also, at smaller head depths, heads that are large, relative to precision of head measurement, cannot be measured. The equation should not be used to compute discharges for heads less than 0.2 ft or greater than one-third the crest length.
(b) Discharge of Standard Suppressed Rectangular Weirs
Table A7-3 contains discharges in cubic feet per second for full bottom contracted suppressed rectangular weirs. These discharges were computed from the Francis equation for lengths and heads commonly used in measuring small quantities of irrigation water.
(c) Limits of Standard Suppressed Rectangular Weirs
Equation 7-5 must not be used beyond the maximum discharges shown in table A7-3 or for measuring heads greater than one-third the crest lengths. All the requirements in section 5 apply. All the approach flow conditions in chapter 2 apply. The crest height, p, should be at least equal to 3h1max (three maximum heads). Head is measured at an upstream distance of at least 4h1max from the weir. The sidewalls must extend at least a distance of 0.3h1max down-stream from the crest, and the overflow jet must be adequately ventilated to the atmosphere.
However, the Kindsvater-Carter method discussed in section 6 is ideally suited for use with suppressed rectangular weirs. This method provides the capability of using partially bottom contracted suppressed weirs and automatically corrects for velocity of approach. This method is recommended for general use and is discussed in section 6. This method provides the opportunity to conserve delivery head by using crest heights less than 3h1max, within limits.
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