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Rectangular notch

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

  • 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.

\therefore Area of the strip

We know know that the theoretical velocity of water through the strip,

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

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

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

Example - Discharge over a rectangular notch
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.
  • b = 0.5 m
  • H = 400 mm = 0.4 m
  • Cd = 0.62

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