The condition for power transmission by flow through a pipeline to be maximum is that the loss of head of the flow due to friction throughout the pipeline length is:

Net available head (h) at the outlet of the pipe is given by:

h = H - hf

Where,

H is the total head at the inlet and hf be the head loss in the transmission.

\(\rm{h = H - \frac{{4fl{V^2}}}{{2gd}}}\)

Power available at the pipe outlet is given by:

\(\rm{P = w\times Q\times h = w\times\frac{\pi }{4}\times{d^2}\times V\left[ {H - \frac{{4fl{V^2}}}{{2gd}}} \right]}\)

For maximum power

\(\rm{\frac{dP}{{dV}} = 0\; \\\to \;H = 3{h_f}}\)

∴ hf = H/3.