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# Rectangular notch

Discharge over a rectangular notch

## Theory

Consider a rectangular notch in one side of a tank over which water is flowing as shown in figure.

Let,
• H = Height of water above sill of notch
• b = Width or length of the notch
• Cd = Coefficient of discharge

Let us consider a horizontal strip of water of thickness dh at a depth of h from the water level as shown in figure.

$\inline&space;\therefore$ Area of the strip
$=&space;b.dh$

We know know that the theoretical velocity of water through the strip,
$=&space;\sqrt&space;{2gh}$

Discharge through the strip,
$dq&space;=&space;C_d&space;\times&space;Area\;of\;strip\;\times\;Theoretical\;velocity$
$\Rightarrow&space;dq&space;=&space;C_d.bdh&space;\sqrt&space;{2gh}$

The total discharge over the whole notch, may be found out by integrating the above equation within the limits 0 and H.

$Q&space;=&space;\int_{0}^{H}&space;C_d.b.dh\sqrt&space;{2gh}$
$\Rightarrow&space;Q&space;=&space;C_d.b\sqrt&space;{2g}\int_{0}^{H}&space;h^{\frac{1}{2}}.dh$
$\therefore&space;Q&space;=&space;\frac{2}{3}C_d.b\sqrt&space;{2g}(H)^{\frac{3}{2}}$

Example:
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##### Example - Discharge over a rectangular notch
Problem
A rectangular notch 0.5m wide has constant head of 400 mm. Find the discharge over the notch in liters per second, if the coefficient of discharge for the notch is 0.62.
Workings
Given,
• b = 0.5 m
• H = 400 mm = 0.4 m
• Cd = 0.62

We know that discharge over the rectangular notch,
$Q&space;=&space;\frac{2}{3}C_d.b\sqrt&space;{2g}(H)^{\frac{3}{2}}\;m^3/s$
$\Rightarrow&space;Q&space;=&space;\frac{2}{3}\times&space;0.62\times&space;0.5&space;\sqrt&space;{2\times9.81}(0.4)^{\frac{3}{2}}\;m^3/s$
$\Rightarrow&space;Q&space;=&space;0.915\times&space;0.253&space;=&space;0.231\;m^3/s&space;=&space;231\;liters/s$
Solution
Discharge over the notch = 231 liters/s