GATE 2023 CH – Question 52

Fluid Mechanics and Mechanical Operations · Equations of continuity, motion and mechanical energy, Euler and Bernoulli equations · 2 marks · Numerical answer

A cylindrical tank with a diameter of 500 mm contains water (density = 1 g cm$^{-3}$) upto a height $h$. A 5 mm diameter round nozzle, whose center is 1 cm above the base of the tank, has its exit open to the atmosphere as shown in the schematic below. The pressure above the water level in the tank is maintained at 2 bar (absolute). Neglect all frictional and entry/exit losses. Use acceleration due to gravity as 10 m s$^{-2}$ and atmospheric pressure as 1 bar. The absolute value of initial $\frac{dh}{dt}$ (in mm s$^{-1}$) when $h = 51$ cm is equal to ________ (rounded off to two decimal places).

a closed tank of diameter 500 mm, filled with water to the height $h$, with 2 bar (abs) above the water. A nozzle of diameter 5 mm leaves the side wall 1 cm above the base.

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Correct answer: 1.44 to 1.46

Explanation

The head above the nozzle centre is $0.51 - 0.01 = 0.50$ m. The Bernoulli equation from the water surface (velocity negligible) to the jet gives $v^2 = 2\left[\frac{P_1 - P_{atm}}{\rho} + g\Delta z\right] = 2\left[\frac{10^5}{1000} + 10 \times 0.5\right] = 210$, so $v = 14.49$ m/s. The continuity equation gives $\frac{dh}{dt} = v\left(\frac{d_n}{D}\right)^2 = 14.49 \times \left(\frac{5}{500}\right)^2 = 1.449 \times 10^{-3}$ m/s $= 1.45$ mm/s.