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GATE 2024 CE (CE2) – Question 42

Water and Waste Water Quality and Treatment · Unit processes and operations · 2 marks · Multiple choice

A hypothetical multimedia filter, consisting of anthracite particles (specific gravity: 1.50), silica sand (specific gravity: 2.60), and ilmenite sand (specific gravity: 4.20), is to be designed for treating water/wastewater. After backwashing, the particles should settle forming three layers: coarse anthracite particles at the top of the bed, silica sand in the middle, and small ilmenite sand particles at the bottom of the bed. Assume
(i) Slow discrete settling (Stoke's law is applicable)
(ii) All particles are spherical
(iii) Diameter of silica sand particles is 0.20 mm

The CORRECT option fulfilling the diameter requirements for this filter media is

  1. diameter of anthracite particles is slightly less than 0.35 mm and diameter of ilmenite particles is slightly greater than 0.141 mm.
  2. diameter of anthracite particles is slightly greater than 0.35 mm and diameter of ilmenite particles is slightly less than 0.141 mm.
  3. diameter of anthracite particles is slightly less than 0.64 mm and diameter of ilmenite particles is slightly less than 0.10 mm.
  4. diameter of anthracite particles is slightly greater than 0.64 mm and diameter of ilmenite particles is slightly less than 0.10 mm.

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Show answer and explanation

Correct answer: (A) diameter of anthracite particles is slightly less than 0.35 mm and diameter of ilmenite particles is slightly greater than 0.141 mm.

Explanation

By Stokes' law the settling velocity is proportional to $(G_s - 1)d^2$. For the layers to form, the anthracite must settle slowest and the ilmenite fastest. For silica: $(2.6 - 1) \times 0.2^2 = 0.064$. Anthracite needs $0.5d_a^2 < 0.064$, that is $d_a < \sqrt{0.128} = 0.358$ mm (the option says slightly less than 0.35 mm). Ilmenite needs $3.2d_i^2 > 0.064$, that is $d_i > \sqrt{0.02} = 0.141$ mm (slightly more than 0.141 mm).