Rectangular

These weirs have large capacity, but have less sensitivity for flow measurement. The rectangular weir is also known as a notch weir. The weir coefficient is based on ratio between the head on the weir crest and the height above the bottom of the pond. Based on the standard formula, in cases where the length is small and the depth is great, the standard formula can become negative. When the adjusted length is less than 0.2 ft, the program will assume that the weir is operating as an orifice with the opening area defined by the current water surface elevation and the weir length. The orifice formula is applied to the centroid of the weir (stg-crest) divided by 2.

Figure 8: Rectangular Weir

Practical design limits for the rectangular weir are:

• H >= 0.03 meters (about 1.2 inches)
• H/p1 <= 2 and p1 >= 0.1 meters  (about 4 inches)
• L >= 0.15 meters  (about 6 inches)
• The tail water level should be at least 0.05 meters below the weir crest.  (about 2 inches)

Where:

• H is the depth of flow above the weir.
• p1 is the height from the weir crest to the bottom of the approach channel
• L is the length of the weir

Discharges are computed based on the following equation:

Where

• Q is the discharge rate in cfs; L is the crest length (ft);
• H is the stage above the crest (ft);
• and c is given by the equation:

Where p is the height of the weir above the bottom.

Note that L is an adjusted length shortened by 0.2H! This means that the length can be adjusted to zero and the Q can be reduced to zero! When the adjusted length is less than 0.2ft, the program computes flow through the weir using the orifice equation. The “head” on the imaginary orifice is the average water surface elevation. The orifice coefficient is assumed to be 0.61.  (Practical limits are taken from: T.J. Casey, "Water and Wastewater Engineering Hydraulics", Oxford Science Publications, Oxford University Press, 1992).

Wright/Baldwin Equation

In May 2001, Steven Wright and Braden Baldwin published Report UMCEE 01-07 titled Notch Weirs for Use in Stormwater Detention Basin Control Structures  It is a report out of:

Department of Civil and Environmental Engineering
The University of Michigan
Ann Arbor, Michigan

In the report, they presented an equation that approximated measured data to within three percent.  We simply refer to the equation as the Wright/Baldwin Equation.

Since the report is not well known, the Results section of the report is duplicated here.

RESULTS

Q = (3.27 + 0.4H/P)(L - 0.2H)H^3/2  . . . . . . (1)

For reasons of convention, results are presented in terms of a weir coefficient C defined by:

C = Q/L*Pow(H,3/2) . . . . . . (2)

Note that C is not dimensionless in this representation and values therefore depend on the system of units employed.  The values presented in this report utilize the U.S. Customary system of units because of the convention in engineering practice in the U.S.  In order to convert to S.I. or other units, one only need recognize that C has units of the square root of gravity.  To convert to S.I. units, for example, the C values reported herein must be multiplied by 0.3048^1/2 where 0.3048 represents the conversion from feet to meters.

With the notch weir in the side of a circular pipe, there is a choice on how to represent L.  It could be given by the circumferential distance from one side of the weir to the other or it could equally well be given by the straight-line distance.  Logically, the first definition may be more valid at very low weir heads in which the flow passing over the pipe circumference determines the nature of the flow.  At high weir heads, the straight-line distance is perhaps more appropriate as it defines the area that the approach flow "sees" as it approaches the weir.  The King and Kitsap County Surface Water Design Manuals suggest that the circumferential length is the appropriate one to be used in the discharge equation.  In the data analysis, both definitions of L were employed and the second definition proved to be more capable of collapsing the data in a non-dimensional sense; i.e. to provide more similar values of C for each weir width.  The data for the 12 inch pipe and the various weir widths with each definiton of L are presented in Figures 8 and 9.  When the weir width is very short such as 1.5 inches, the differences between the two definitions of L are minimal but become more significant when the weir width is a larger proportion of the pipe circumference.  From these two figures, it appears that the use of the straight line distance provided a more consistent presentation of experimental results. Therefore, this definition of L is utilized in all further results presented herein.

Weirs Without Baffles

The majority of the experiments were performed without the baffle installed since it is intended that the baffle not affect the head-discharge relation for the weir.  A typical result for a weir width of 1.5 inches is given in Figure 10.  Also included on this figure is data from Carter at a similar weir width (1.42 inches).  The two data sets are reasonably consistent in terms of the general magnitude of the weir coefficient.  All sets of experiments performed in the current study clearly demonstrated the trend that the weir coefficient decreases with increasing head.  This may be due to the issue discussed above where the crest length at low heads may be more appropriately correlated with the circumferential length in the sense that a longer length in Equation (2) would reduce the C value; hence using the straight line distance for L yields too large a weir coefficient.  This effect would not occur for a weir in a flat plate and is consistent with the observations for the one experiment performed with a flat plate (and discussed below).  Carter's Georgia Tech data were inconsistent in the trends of C vs. H for the various weir configurations, in some cases decreasing and in other, increasing with H.  This may be due to other effects not investigated in the current study such as small values of P.

Also include in Figure 10 is the prediction of Equation (1) in the limit with H/P=0; this is considered to be the most appropriate description for the experimental configuration studied.  It can clearly be seen that Equation (1) does not capture the trend in the data, supporting the previous conclusion (see Appendix 2) that it is an inappropriate equation for this particular application with relatively large H/L values.

The results for the four different crest widths in the twelve-inch PVC pipe are presented in Figure 8.  The concept implied by Equation (1) is that an "effective crest length" is computed by reducing the length of the crest by a function that depends on the weir head.  The underlying philosophy is cast into doubt by the results in Figure 11 since the weir coefficient should then decrease with crest length since end effects would be greater for smaller crest lengths and the opposite trend is actually indicated.  It should be noted that the Georgia Tech data also supports this conclusion for weirs in flat plates although a bit more ambiguously.  In the Handbook of Hydraulics, 7th edition, (E.R. Brater, H.W. King, J.E. Lindell and C.Y. Wei, McGraw Hill) page 5.11, a correction to the weir coefficient to account for end contractions is discussed based in part on the Georgia Tech Data.  Figure 5.3 on page 5.14 indicates a positive correction to be made to the actual length over nearly the entire range of flow conditions, consistent with the results presented in Figure 8.  Such results call into question the structure of Equation (1), even though it currently is the most commonly used equation for contracted weirs.  The experiments with the 8-inch pipe with crest lengths of two, four and six inches were intended to correspond to geometrically similar conditions for the 12-inch pipe (i.e. the 2-inch crest length corresponding to the 3-inch, the 4-inch to the 6, etc.)  However if the results for the two different pipes are plotted non-dimensionally in terms of H/D where D is the pipe diameter as in Figure 11, there is no correspondence between the results for the two pipes for equal values of H/D.  If, however, the results are plotted simply as a function of head H and crest length L as in Figure 12, then the two sets of results appear to be consistent with each other.  This finding led to the additional experiment with the 6-inch weir width on a flat plate as described earlier.  Figures 13 and 14 show the results for all three six-inch weirs with the two different definitions of L and again support the conclusions based on the data in Figures 8 and 9.  The fact that the data in Figure 13 are reasonably consistent with each other implies that the curvature of the pipe that the weir is installed on is basically irrelevant so long as the crest length is interpreted as the linear distances between the two sides of the weirs.  Additional experiments may be in order to verity this conclusion over a wider range of geometries.

Attempts were made to collapse the data in some sort of dimensionless presentation without success.  Instead, following the development by Kindsvater and Carter as described in Equation 5-35 of the 7th edition of the Handbook of Hydraulics, the only successful means of collapsing the data was by adding a fixed correction to the length as:

Leff = L + Kl . . . . . (4)

Kindsvater and Carter recommends Kl values on the order of 0.008 ft for their studies while a fixed value of 0.045 ft appears to be adequate for collapsing the data for the present study.  The values of Kl can, however, be doubled without a significant change in the standard deviation of the fitted equation (Equation 6 below) and the exact value is subject to some uncertainty.  Figure 15 shows the result of this correction to the crest length and the ability to present experimental results for several crest lengths as a unique function of the weir head.  The differences between the current study and that by Carter (from which the Kindsvater and Carter's correction was derived) appears to be quite substantial in regard to this correction so there is likely to be a difference mechanism associated with the correction in the two studies.  It should be noted that many of Carter's experiments were conducted with fairly small P values (H/P less than one) and this may well have an important and significant difference on the flow downstream of the weir crest.

Kindsvater and Carter also recommends using a weir equation in which the head is corrected in a fashion similare to Equation (4):

Heff = H + Kh . . . . . .(5)

After attempts to nondimensionalize the results were not successful, this approach was ultimately selected for the current study as well.  Again, a much larger value of Kh, on the order of 0.018 ft, was required to collapse the data as indicated in Figure 16 than the value of 0.003 ft recommended by Kindsvater and Carter.  A portion of the difference between the two studies is believed to be due to the curved nature of the notch weir constructed in the side of a pipe, but the limited data with the flat weir do not indicate that this is the case.  It also appears that the conditions downstream from the weir are likely to play a significant role on the pressure distribution at the weir crest and therefore to affect the values of the weir coefficients.  A future study to investigate this issue is planned.  In any case, a flow equation that is adequate to predict the entire range of experimental data to within an accuracy of about three percent (to cover all the experimental data; the standard deviation is about 1.7%) is:

Q = 3.06(L + 0.045)(H + 0.018)^3/2 . . . . . . . . .(6)

This equation is dependent on the system of units employed and should be applied for Q in cfs and L and H in feet.  In the S.I. system of units, a similar equation would be:

Q = 1.69(L + 0.014)(H + 0.0055)^3/2 . . . . . . . . (7)