The power output of a water turbine is given by P =ρgQh, where P is power, ρ is density of water (kg /m3), g is acceleration due to gravity (m/s2), Q is water flow rate (m3/s), and h is the available water head (m). What is the appropriate unit for P ?
The power output of given turbine is given by
$$ P=\rho g Q h $$
Where, \(P=\) Power
\(\rho=\) Density of water \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\)
\(g=\) Acceleration due to gravity \(\left(\mathrm{m} / \mathrm{s}^{2}\right)\)
\(Q=\) Water flow rate \((\mathrm{kg} / \mathrm{s})\)
\(h=\) Available water head \((\mathrm{m})\)
Substituting all values in the above equation:
$$ \begin{aligned} &P=\rho g Q h \\ &P=\left(\frac{\mathrm{kg}}{\mathrm{m}^{3}}\right)\left(\frac{\mathrm{m}}{\mathrm{s}^{2}}\right)\left(\frac{\mathrm{m}^{3}}{\mathrm{~s}}\right)(\mathrm{m}) \\ &=\frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^{2}} \frac{\mathrm{m}}{\mathrm{s}} \quad\left(\because 1 \mathrm{~N}=1 \frac{\mathrm{kg} \cdot \mathrm{m}}{\mathrm{s}^{2}}\right) \\ &=\frac{\mathrm{N} \cdot \mathrm{m}}{\mathrm{s}} \end{aligned} $$
Thus unit of power is \(P=\frac{\mathrm{N} \cdot \mathrm{m}}{\mathrm{s}}\)