Correct Answer - Option 3 : 2/3

**Explanation:**

The velocity of water in suction or delivery pipe is given by equation:

\(v=\frac{A}{a}\omega r\sin \omega t=\frac{A}{a}\omega r\sin θ\) ....(i)

Loss of head due to friction in pipes is given by:

\(h_f=\frac{fLV^2}{2gD}\) ....(ii)

where f = co-efficient of friction, I = length of pipe, d = diameter of pipe, and V = Velocity of water in pipe.

Substituting equation (i) into equation (ii), we get

\(h_f=\frac{fL}{2gD}\left[\frac{A}{a}\omega r\sin θ\right]^2\)

The variation of h_{f }with θ is parabolic.

At θ = 0° and 180°, h_{f} = 0 and at θ = 90°, h_{f} is maximum.

\((h_f)_{max}=\frac{fL}{2gD}\left[\frac{A}{a}\omega r\right]^2\)

\((h_f)_{avg}=\frac{2}{3}\frac{fL}{2gD}\left[\frac{A}{a}\omega r\right]^2\)

Thus the ratio of the average frictional head to the maximum frictional head in the delivery pipe is 2/3.