GATE 2021 BT – Question 57
A 100 cm sedimentation tank separates spherical sludge of diameter 0.5 mm. Liquid viscosity is 1 cP and density difference 0.1 g/cm³. Assuming instantaneous terminal speed, settling time (two decimal places min) is _______.
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Correct answer: 1.20 to 1.30
Explanation
Assume the sludge particle reaches its terminal (Stokes) velocity at once.
**Stokes' law** for a small sphere in laminar flow:
$$v=\frac{\Delta\rho\,g\,d^2}{18\,\mu} .$$
**Convert to SI units.**
- $\Delta\rho=0.1\text{ g/cm}^3=100\text{ kg/m}^3$
- $d=0.5\text{ mm}=5\times10^{-4}$ m
- $\mu=1\text{ cP}=10^{-3}$ Pa s
$$v=\frac{100\times9.81\times(5\times10^{-4})^2}{18\times10^{-3}}=\frac{2.4525\times10^{-4}}{0.018}=0.013625\text{ m/s}.$$
**Settling time** over the 100 cm (1 m) depth:
$$t=\frac{1}{0.013625}=73.4\text{ s}=\mathbf{1.22\ min}.$$
Official GATE 2021 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2021/bt_2021.pdf