GATE 2024 CH – Question 58
A metallic spherical particle of density 7001 kg m$^{-3}$ and diameter 1 mm is settling steadily due to gravity in a stagnant gas of density 1 kg m$^{-3}$ and viscosity $10^{-5}$ kg m$^{-1}$ s$^{-1}$. Take $g = 9.8$ m s$^{-2}$. Assume that the settling occurs in the regime where the drag coefficient $C_D$ is independent of the Reynolds number, and equals 0.44. The terminal settling velocity of the particle, in m s$^{-1}$, rounded off to 2 decimal places, is ____
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Correct answer: 14.35 to 14.49
Explanation
At the terminal velocity the drag balances the weight less the buoyancy: $\frac{1}{2}C_D\rho\frac{\pi d^2}{4}v^2 = (\rho_p - \rho)g\frac{\pi d^3}{6}$. So $v = \sqrt{\frac{4(\rho_p - \rho)gd}{3C_D\rho}} = \sqrt{\frac{4 \times 7000 \times 9.8 \times 0.001}{3 \times 0.44 \times 1}} = \sqrt{207.9} = 14.42$ m/s.