GATE 2024 CE (CE1) – Question 48
A 2 m × 2 m tank of 3 m height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. At t = 0, an inflow of a salt solution of concentration 5 g/m$^3$ at the rate of 2 litre/s and an outflow of the well stirred mixture at the rate of 1 litre/s are initiated. This process can be modelled using the following differential equation:
$\frac{dm}{dt} + \frac{m}{6000 + t} = 0.01$
where m is the mass (grams) of the salt at time t (seconds). The mass of the salt (in grams) in the tank at 75% of its capacity is _____________ (rounded off to 2 decimal places).
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Correct answer: 24.88 to 25.12
Explanation
The tank has a capacity of $2 \times 2 \times 3 = 12$ m³, and 75% of it is 9 m³. It starts with 6 m³ and gains 1 litre per second, so it takes 3000 litres, that is $t = 3000$ s, to reach 9 m³. Solve the equation: the integrating factor is $6000 + t$, so $\frac{d}{dt}[(6000 + t)m] = 0.01(6000 + t)$ and $(6000 + t)m = 0.01\left(6000t + \frac{t^2}{2}\right) + C$. With $m(0) = 0$, $C = 0$. At $t = 3000$: $9000m = 0.01(18 \times 10^6 + 4.5 \times 10^6) = 225000$, so $m = 25$ g.