# In a single reciprocating pump without air vessel, the ratio of the average frictional head to the maximum frictional head in the delivery pipe is

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In a single reciprocating pump without air vessel, the ratio of the average frictional head to the maximum frictional head in the delivery pipe is
1. 1/2
2. 1/3
3. 2/3
4. 3/4
5.

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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 hwith θ is parabolic.

At θ = 0° and 180°, hf = 0 and at θ = 90°, hf 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.