Question

Water flows through a tunnel from the reservoir of a dam towards the turbine installed in its power plant. The power plant is situated h meter below the reservoir. If the ratio of the cross-sectional areas of the tunnel at the reservoir and power station end is n, find the speed of the water entering into the turbine.

• A.
• B.
• C.
• D. now of these

Answer 1

The correct option is B

Applying Bernoulli's theorem at reservoir and power plant for the flowing water, we obtain,
$P_0+\stackrel{,}{\rho }g{h}_1+\frac{1}{2}\stackrel{,}{\rho }$ $v_1^2=P_0+\stackrel{,}{\rho }g{h}_2+\frac{1}{2}\stackrel{,}{\rho }{v}_2^2$
$\Rightarrow v_2^2-v_1^2=2g(h_1-h_2).$
$putting(h_1-h_2)-h,weobtain{v}_2=\sqrt{v_1^2+2gh}.....\left(1\right)$
$A_1{v}_1=A_2{v}_2....\left(2\right)$
Eliminating$v_1$ from equation (1) and (2), we obtain
$v_2=\sqrt{{\left(\frac{A_2}{A_1}{v}_2\right)}^2+2gh}$
$\Rightarrow v_2=\sqrt{\frac{2gh}{1-{\left(\frac{A_2}{A_1}\right)}^2}}$
$putting\frac{A_1}{A_2}=\eta ,weobtain$
$v_2=\eta =\sqrt{\frac{2gh}{{\eta }^2-1}}$