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GATE 2026 CH – Question 49

Fluid Mechanics and Mechanical Operations · Settling, centrifuges, cyclones, thickening, filtration, agitation and mixing · 2 marks · Multiple choice

Laboratory filtration is conducted at constant pressure drop of 200 kPa on a slurry of CaCO$_3$ in water at room temperature. The time taken to collect filtrate is shown in the table.

Filtrate volume collected (m$^3$)Time (s)
$1 \times 10^{-3}$40
$2 \times 10^{-3}$100

The filter area is 0.05 m$^2$ and viscosity of filtrate is $10^{-3}$ Pa s. Which one of the following is the filter medium resistance (in m$^{-1}$)?

  1. $3 \times 10^{-11}$
  2. $3 \times 10^{-8}$
  3. $3 \times 10^{8}$
  4. $3 \times 10^{11}$

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Correct answer: (D) $3 \times 10^{11}$

Explanation

For constant pressure filtration, $\frac{t}{V} = \frac{\mu \alpha c}{2A^2\Delta P}V + \frac{\mu R_m}{A\Delta P}$. The values of $t/V$ are $40000$ s/m$^3$ at $V = 10^{-3}$ and $50000$ s/m$^3$ at $V = 2 \times 10^{-3}$. The slope is $\frac{10000}{10^{-3}} = 10^7$ and the intercept is $40000 - 10^7 \times 10^{-3} = 30000$ s/m$^3$. Then $R_m = \frac{30000 \times A\Delta P}{\mu} = \frac{30000 \times 0.05 \times 2 \times 10^5}{10^{-3}} = 3 \times 10^{11}$ m$^{-1}$.