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GATE CE 2013 Official Paper

Option 2 : 0.02 mm

CT 1: Ratio and Proportion

2672

10 Questions
16 Marks
30 Mins

__Concepts:__

**Surface Overflow Rate or Surface Loading:**

- Surface overflow rate (SOR) is parameter that describes the settling characteristics of solids in a specific water or wastewater which is given by the volume of water flow per unit of time divided by the surface area (or plan area) of the settling basin.
- Alternatively, if settling velocity of a particle equals or exceeds SOR, then the particle is completely removed from the settling tank.

**Stoke’s law of settling velocity for spherical particles: **Stoke had given a law for settling velocity for particles having spherical in shape whose diameter is less than 0.1 mm and Reynold’s number (Re) of the flow is less than 0.5. The final form of the law is,

\({{\bf{V}}_{\bf{s}}} = \frac{{\left( {{\bf{G}} - 1} \right){{\bf{\gamma }}_{\bf{w}}}{{\bf{d}}^2}}}{{18{\bf{\mu }}}}\)

Where V_{s}is the settling velocity in m/sec, G is the specific gravity of the settling particles, γ_{w} is the unit weight of water, d is the diameter of the particles and µ is the dynamic viscosity of water in N-s/m^{2}

__Calculation:__

Given, SOR = \(30\frac{{{{\rm{m}}^3}}}{{{\rm{day\;}}.{\rm{\;}}{{\rm{m}}^2}}} = 30\frac{{\rm{m}}}{{{\rm{day}}}} = \frac{{30}}{{86400}}{\rm{m}}/{\rm{sec}}\) ; specific gravity of sediment particles, G = 2.65; dynamic viscosity of water, μ = 0.001 N-s/m^{2 }and unit weight of water, γ_{w} = 9.81 kN/m^{3}

For complete removal of particles,

\({\rm{SOR}} \le \frac{{\left( {{\rm{G}} - 1} \right){{\rm{\gamma }}_{\rm{w}}}{{\rm{d}}^2}}}{{18{\rm{\mu }}}} \Rightarrow \frac{{30}}{{86400}} \le \frac{{\left( {2.65 - 1} \right) \times 9810 \times {{\rm{d}}^2}}}{{18 \times 0.001{\rm{\;}}}} \Rightarrow {{\rm{d}}^2} \ge 3.861 \times {10^{ - 10}} \Rightarrow {\rm{d}} \ge 1.965 \times {10^{ - 5}}{\rm{\;m}}\)

∴ d ≥ 0.01965 mm ≈ 0.02 mm ∴ d ≥ 0.02 mm

__Note:__