GATE 2025 CH – Question 49
A leaf filter is operated at 1 atm (gauge). The volume of filtrate collected ($V$ in m$^3$) is related with the volumetric flow rate of the filtrate ($q$ in m$^3$/s) as: $\frac{1}{q} = \frac{1}{dV/dt} = 50V + 100$
The volumetric flow rate of the filtrate at 1 hour is ____ $\times 10^{-3}$ m$^3$/s (rounded off to 2 decimal places).
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Correct answer: 1.63 to 1.65
Explanation
From $\frac{dt}{dV} = 50V + 100$, integrating gives $t = 25V^2 + 100V$. At $t = 3600$ s: $25V^2 + 100V - 3600 = 0$, that is $V^2 + 4V - 144 = 0$, so $V = \frac{-4 + \sqrt{16 + 576}}{2} = 10.166$ m$^3$. Then $q = \frac{1}{50 \times 10.166 + 100} = \frac{1}{608.3} = 1.64 \times 10^{-3}$ m$^3$/s.