**Units: **ft^{3}=cubic foot, gal=US gallons, m^{3}=cubic meter, mm=millimeter, s=second

**Introduction**

Weirs are typically installed in open channels such as streams to determine discharge
(flowrate). The basic principle is that discharge is directly related to the water
depth above the crotch (bottom) of the V; this distance is called head (h). The
V-notch design causes small changes in discharge to have a large change in depth allowing
more accurate head measurement than with a rectangular weir.

**V-Notch Weir (Triangular Weir) Equations**

V-notch weir equations have become somewhat standardized. ISO (1980), ASTM (1993),
and USBR (1997) all suggest using the Kindsvater-Shen equation, which is presented below
from USBR (1997) for Q in cfs and heights in ft units. All of the references show
similar curves for C and k vs. angle, but none of them provide equations for the curves. To produce automated calculations, LMNO Engineering used a curve fitting program to
obtain the equations which best fit the C and k curves. Our equations are shown
below. The graph shown is from our fits. If you compare it to the graphs shown
in the references, it looks nearly identical which implies that our fits are very good.

C = 0.607165052 - 0.000874466963 θ + 6.10393334x10^{-6} θ^{2}

k (ft.) = 0.0144902648 - 0.00033955535 θ + 3.29819003x10^{-6} θ^{2}
- 1.06215442x10^{-8} θ^{3}

where θ is the notch angle in degrees

**Installation Guidelines and Equation Applicability**

USBR (1997) suggests using the V-notch weir equations for the following conditions:

Head (h) should be measured at a distance of at least 4h upstream of the weir.

It doesn't matter how thick the weir is except where water flows over the weir through
the "V." The weir should be between 0.03 and 0.08 inches (0.8 to 2 mm)
thick in the V. If the bulk of the weir is thicker than 0.08 inch, the downstream
edge of the V can be chamfered at an angle greater than 45^{o} (60^{o} is
recommended) to achieve the desired thickness of the edges. You want to avoid having
water cling to the downstream face of the weir.

Water surface downstream of the weir should be at least 0.2 ft. (6 cm) below the bottom of the V to allow a free flowing waterfall.

Measured head (h) should be greater than 0.2 ft. (6 cm) due to potential measurement error at such small heads and the fact that the nappe (waterfall) may cling to the weir.

The equations have been developed for h<1.25 ft. (38 cm) and h/P<2.4.

The equations have been developed for fully contracted V-notch weirs which means h/B should be ≤ 0.2.

The average width of the approach channel (B) should be > 3 ft. (91 cm).

The bottom of the "V" should be at least 1.5 ft. (45 cm) above the bottom of the upstream channel.

If your weir does not achieve some of the above criteria, you may have a
"partially contracted V-notch weir" where h/B needs only to be ≤ 0.4, the
bottom of the "V" only needs to be 4 inch (10 cm) above the bottom of the
upstream channel, the approach channel only needs to be 2 ft. (61 cm) wide, and h can be
up to 2 ft. (61 cm) instead of 1.25 ft. (38 cm). Partially contracted weirs use a
different graph for C which is a function of h/P and P/B and is only valid for a notch
angle of 90^{o}. In the graph (not shown - see USBR, 1997), C varies from
0.576 to 0.6; whereas, for a fully contracted 90^{o} notch, C is 0.578 from our
graph shown above. Our calculation does not account for partially contracted weirs,
but for most practical purposes the difference in C is inconsequential.

**Error Messages**

*"All inputs must be positive".* This is an initial check
of user input.

*"Angle out of range".* The notch angle must be between 20^{o}
and 100^{o} (0.35 and 1.75 radians) for the equations to be valid.

*"Infeasible input".* Input results in a negative head due to the
compiler's machine precision. Occurs if head is being computed and a very low Q is
entered (e.g. 1.0e-20).

**References**

ASTM. (1993). American Society for Testing and Materials. ASTM D5242.
Standard method for open-channel flow measurement of water with thin-plate weirs.
1993.

ISO. (1980). International Organization of Standards. ISO 1438/1-1980(E). Water flow measurement in open channels using weirs and venturi flumes - Part 1: Thin plate weirs. 1980.

USBR. (1997). U.S. Department of the Interior, Bureau of Reclamation. Water Measurement Manual. 3ed. Available from http://www.usbr.gov/tsc/techreferences/mands/wmm/index.htm .

**© 1999-2024 LMNO Engineering, Research, and Software, Ltd. All rights reserved.**

LMNO Engineering, Research, and Software, Ltd.

7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 707-2614

LMNO@LMNOeng.com
https://www.LMNOeng.com