GATE 2021 BT – Question 55
Fermentation is scaled from 1 L to 1000 L at constant power/volume. Small-vessel agitation is 300 rpm and linear scale factor is 10. Large-vessel speed (nearest integer rpm) is _______.
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Correct answer: 65
Explanation
**Scale-up rule.** At constant power per unit volume ($P/V$) in turbulent stirred tanks, the power is $P\propto N^3D^5$ and the volume is $V\propto D^3$, so
$$\frac PV\propto N^3D^2=\text{constant} .$$
**Step 1: speeds.** With a linear scale factor of 10, $D_2=10D_1$:
$$N_2^3D_2^2=N_1^3D_1^2\;\Rightarrow\;N_2=N_1\left(\frac{D_1}{D_2}\right)^{2/3}=300\times10^{-2/3} .$$
**Step 2: number.**
$$10^{2/3}=4.642,\qquad N_2=\frac{300}{4.642}=64.6\approx\mathbf{65\ rpm}.$$