Question 6.P.2: The rate of discharge of water from a tank is measured by me...

The velocity of fluid discharged as a height h above the bottom of the notch is:

u=\sqrt{(2 g h)}

The velocity therefore varies from zero at the bottom of the notch to a maximum value at the free surface.
For a horizontal element of fluid of width 2w and depth dh at a height h above the bottom of the notch, the discharge rate of fluid is given by:

d Q=\sqrt{ }(2 g h) 2 w d h

If the discharge rate is linearly related to the height of the liquid over the notch, H, w will be a function of h and it may be supposed that:

w=k h^n

where k is a constant.

Substituting for w in the equation for dQ and integrating to give the discharge rate over the notch Q then:

Q=2 \sqrt{(2 g)} k \int_0^H h^n h^{0.5} d h

=2 \sqrt{(2 g)} k \int_0^H h^{n+0.5} d h

=2 \sqrt{(2 g)} k[1 /(n+1.5)] H^{(n+1.5)}

Since it is required that Q \propto H:

n+1.5=1

and:                              n=-0.5
Thus:                           Q=2 \sqrt{(2 g)}  k H

Since Q=0.1 m ^3 / s when H = 0.15 m:

k=(0.1 / 0.15)\left[1 /(2 \sqrt{(2 g)}]=0.0753  m ^{1.5}\right.

Thus, with w and h in m:         w=0.0753 h^{-0.5}
and, with w and h in mm:        w=2374 h^{-0.5}

and using this equation, the profile is plotted as shown in \underline{\underline{\text { Figure 6 a}}}