A fermentor is filled with medium at a rate of . A leak develops at the bottom of the fermentor when the medium in the fermentor reaches . The rate of medium leakage is , where '' is the time at which the leak begins.
The volume of medium in the fermentor after 10 min of leakage is ________ L (Answer in integer).
ANSWER
The fermentor holds 110 L after 10 min of leakage.
Official answer: 110Source: Official answer key, GATE 2024 BT (IISc Bengaluru)
Measure in minutes from the moment the leak starts, which is what the question means by " is the time at which the leak begins". At the fermentor holds . Nothing in the question says filling stops, so the feed keeps running at while the leak runs.
Write the unsteady-state volume balance (accumulation = in − out), where is the liquid volume and is the leak rate:
Integrate from , where :
At :
Check: over the 10 minutes the leak removes and the feed adds , a net loss of , so . peaks at (at ) and falls to at . It stays positive throughout, so the fermentor never empties and the balance holds for the whole interval.
KEY IDEA
This is an unsteady-state volume balance, dV/dt = F_in − F_out, with an outflow that changes over time. Measure t from the moment the leak starts. The feed of 1 L/min keeps running, so V(t) = 200 + t − t², and V(10) = 110 L.
WATCH OUT FOR
The usual mistake is to assume filling stops when the leak starts. That gives 200 − ∫₀¹⁰ 2t dt = 100 L instead of 110 L, but the question only says a leak develops. Another mistake is multiplying the rate by the time (2 × 10 × 10 = 200 L lost) when the leak rate grows with time and has to be integrated. A third is measuring t from the start of filling: the leak would then already be about 400 L/min, which empties the tank almost at once and makes no physical sense.
From the GATE 2024 BT paper. Every question of this paper · Download the full paper · All 24 Bioreactor operation questions, with notes
