GATE 2019 BT Q.55: solution and answer

Official answer: 30Source: Official answer key, GATE 2019 BT (IIT Madras)

1. Geometric similarity means every length, including the impeller diameter $D_i$, grows by the same scale factor. The question sets that factor as the cube root of the volume ratio, where $V_1$ and $V_2$ are the lab and large working volumes: $$\frac{D_{i,2}}{D_{i,1}}={\left(\frac{V_2}{V_1}\right)}^{1/3}={\left(\frac{8000\text{L}}{1\text{L}}\right)}^{1/3}=20$$

2. The impeller-to-vessel diameter ratio stays at 0.3 in both vessels, so the impeller diameter also scales by 20. The lab impeller is $0.3\times 20\text{cm}=6\text{cm}$, so the large one is $6\times 20=120\text{cm}$. Only the ratio of the two affects the answer.

3. Impeller tip speed: $u_{tip}=\pi N{D}_i$, where $N$ is the agitator speed (revolutions per unit time) and $D_i$ is the impeller diameter.

4. Constant tip speed: $\pi N_1{D}_{i,1}=\pi N_2{D}_{i,2}$, so $N_2=N_1\frac{D_{i,1}}{D_{i,2}}$.

5. $$N_2=600\text{rpm}\times \frac{1}{20}=30\text{rpm}$$

6. Check: $u_{tip}=\pi \times 600{\text{min}}^{-1}\times 0.06\text{m}\approx 113{\text{m min}}^{-1}$ in the lab vessel and $\pi \times 30{\text{min}}^{-1}\times 1.2\text{m}\approx 113{\text{m min}}^{-1}$ in the large one, so the tip speed is the same in both.