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GATE 2023 MT – Question 39

Mineral Processing and Extractive Metallurgy · Mineral beneficiation · 2 marks · Multiple choice

An elutriator separates particles by size in flowing air, as shown. Assuming spherical particles and turbulent airflow, the suspended-particle diameter $D_{50}$ with 50% chance of reporting to overflow is expressed as follows. Symbols: air–solid friction factor $f$, air velocity $v$, air viscosity $\mu$, particle and air densities $\rho_s,\rho_a$, and gravity $g$.

Diagram for GATE 2023 MT question 39
  1. $D_{50}=\dfrac{3fv^2\rho_a}{4g(\rho_s-\rho_a)}$
  2. $D_{50}=\left[\dfrac{18\mu v}{gf(\rho_s-\rho_a)}\right]^{0.5}$
  3. $D_{50}=\left[\dfrac{9\mu v}{2g(\rho_s-\rho_a)}\right]^2$
  4. $D_{50}=\dfrac{3fv\rho_a}{8g(\rho_s-\rho_a)}$

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Correct answer: (A) $D_{50}=\dfrac{3fv^2\rho_a}{4g(\rho_s-\rho_a)}$

Explanation

Balance effective weight $(\pi D^3/6)(\rho_s-\rho_a)g$ against drag $f(\rho_av^2/2)(\pi D^2/4)$. Cancelling common terms gives $D_{50}=3fv^2\rho_a/[4g(\rho_s-\rho_a)]$, **A**. Turbulent drag scales with velocity squared.