GATE 2024 BT Q.61: solution and answer

GATE 2024 · BTQ.61 · 2 marks · NAT

Consider scale-up of fungal fermentation from a 20 L model-type to 20,000 L prototype stirred tank reactor. The model-type and prototype have the same aspect ratio during scale-up. The impeller speed in the model-type is 500 rpm and the scale-up criterion is constant shear.

The impeller speed in the prototype reactor will be ________ rpm (Answer in integer).

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Answer and solution

ANSWER

The impeller speed in the 20,000 L prototype is 50 rpm.

Official answer: 50Source: Official answer key, GATE 2024 BT (IISc Bengaluru)

Same aspect ratio means the two vessels are geometrically similar, so every length scales by the same factor. Volume then goes as the cube of a length: $V \propto D^{3}$ , where $D$ is the impeller (or tank) diameter.
Diameter ratio: $\frac{D _{2}}{D _{1}} = (\frac{V _{2}}{V _{1}})^{1/3} = (\frac{20000 L}{20 L})^{1/3} = (1000)^{1/3} = 10$ . Subscript 1 is the 20 L model, subscript 2 the 20,000 L prototype.
Constant shear criterion: in a stirred tank the maximum shear on the cells (fungal hyphae are shear-sensitive) is set by the impeller tip speed $v_{t i p} = π N D$ , where $N$ is the impeller speed. Holding the tip speed constant means $N_{1} D_{1} = N_{2} D_{2}$ .
Prototype speed: $N_{2} = N_{1} \frac{D _{1}}{D _{2}} = 500 rpm \times \frac{1}{10} = 50 rpm$

KEY IDEA

With geometric similarity, $D \propto V^{1/3}$ . Constant shear means constant impeller tip speed $π N D$ , so $N \propto 1/ D \propto V^{- 1/3}$ . A 1000-fold volume increase gives a 10-fold larger diameter and one tenth of the speed.

WATCH OUT FOR

Scaling the speed by the volume ratio instead of the diameter ratio (giving 0.5 rpm), or using a different criterion by mistake. Constant power per volume gives $N \propto D^{- 2/3}$ , about 108 rpm. Keeping $N$ itself constant gives 500 rpm. Another slip is taking $D_{2} / D_{1} = 1000$ without the cube root.

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GATE 2024 BT Q.61: solution and answer

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