GATE 2024 CH – Question 59

Fluid Mechanics and Mechanical Operations · Flow through pipes, friction factors, flow meters and pumps · 2 marks · Numerical answer

Water of density 1000 kg m$^{-3}$ is pumped at a volumetric flow rate of $3.14 \times 10^{-2}$ m$^3$ s$^{-1}$, through a pipe of inner diameter 10 cm and length 100 m, from a large Reservoir 1 to another large Reservoir 2 at a height 50 m above Reservoir 1, as shown in the figure. The flow in the pipe is in the turbulent regime with a Darcy friction factor $f = 0.06$, and a kinetic energy correction factor $\alpha = 1$. Take $g = 9.8$ m s$^{-2}$. If all minor losses are negligible, and the pump efficiency is 100%, the pump power, in kW, rounded off to 2 decimal places, is ______

a pump at Reservoir 1 pushes water up through a 100 m long inclined pipe to Reservoir 2, whose water surface is 50 m above that of Reservoir 1.

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Correct answer: 30.31 to 30.61

Explanation

The velocity in the pipe is $v = \frac{Q}{A} = \frac{0.0314}{\frac{\pi}{4} \times 0.1^2} = 4.0$ m/s. The friction head is $h_f = f\frac{L}{D}\frac{v^2}{2g} = 0.06 \times \frac{100}{0.1} \times \frac{16}{19.6} = 48.98$ m. The energy balance between the two free surfaces (both at rest) gives the pump head $H = 50 + 48.98 = 98.98$ m. The power is $\rho gQH = 1000 \times 9.8 \times 0.0314 \times 98.98 = 30458$ W $= 30.46$ kW.