CHAPTER 2 - BASIC CONCEPTS RELATED TO FLOWING WATER AND MEASUREMENT

13. Discharge Equation for Broad-Crested Rectangular Weirs

The discharge equation for the rectangular broad-crested weir will now
be derived similar to Bos (1989). The width, *L _{b}*, of a
rectangular flow section is the same as

(2-32)

or:

(2-33)

Conversely:

(2-34)

Multiplying both sides of equation 2-27 by the area, *A _{c}*

(2-35)

To get unit discharge, *q*, this equation is divided by the width
of flow, *L _{b}*, resulting in:

(2-36)

Solving for *h _{c}*:

(2-37)

Using equation 2-34 to replace *h _{c}* with

(2-38)

Discharges in equations 2-35 through 2-38 are usually considered actual, assuming uniform velocity throughout the critical depth cross section and assuming that no correction of velocity distribution is needed.

Because specific energy is constant in a fairly short measuring structure
with insignificant friction losses, specific energy, *H _{c}*,
at the critical location can be replaced with specific energy,

(2-39)

For measurement convenience, the total head, *H*_{1}, is
replaced with the depth, *h*_{1}. To correct for neglecting
the velocity head at the measuring station, a velocity coefficient, *C*_{v},
must be added, resulting in:

(2-40)

This equation applies to both long-throated flumes or broad-crested weirs and can be modified for any shape by analyses using the energy balance with equation 2-31.

These equations differ only in numerical constants that are derived
from assumptions and selection of basic relationships used in their derivation.
However, experimental determination of the coefficient values for *C*
and *C _{v}* would compensate, making each equation produce
the same discharge for the same measuring head. Either equation could be
used.

The examples given above show that traditional discharge equations are often a mixture of rational analysis and experimental coefficient evaluation. However, recent development of computer modeling of long-throated flumes (Clemmens et al. [1991]) precludes the need for experimental determination of coefficients. These long-throated flumes are covered in chapter 8.