GATE 2021 ME (ME2) – Question 56

Fluid Mechanics · Bernoulli's equation and fluid acceleration · 2 marks · Numerical answer

Water flows out from a large tank of cross-sectional area $A_t=1$ m² through a small rounded orifice of cross-sectional area $A_o=1$ cm², located at $y=0$. Initially the water level, measured from $y=0$, is $H=1$ m. The acceleration due to gravity is 9.8 m/s². Neglecting any losses, the time taken by water in the tank to reach a level of $y=H/4$ is _____ seconds (round off to one decimal place).

Tank of area At with initial level H and a small rounded outlet of area Ao at y=0. See the attached source image.

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Correct answer: 2247.51 to 2270.09

Explanation

Torricelli gives exit speed √(2 gy). Conservation gives $A_tdy/dt=-A_o\sqrt{2gy}$. Integrating yields $t=2A_t(\sqrt H-\sqrt{H/4})/(A_o\sqrt{2g})$.

With Aₒ=10⁻⁴ m², t=2258.76976 s, rounded to **2258.8 s**. The changing head must be integrated rather than held constant.