GATE 2022 BT Q.34: solution and answer

Official answer: 20000Source: Official answer key, GATE 2022 BT (IIT Kharagpur)

1. Impeller Reynolds number: $R{e}_i=\frac{\rho N{D}_i^2}{\mu }$. Here $\rho$ is the broth density, $N$ is the impeller speed in revolutions per second, $D_i$ is the impeller diameter and $\mu$ is the broth viscosity.

2. Data in SI units: $\rho =1000{\text{kg m}}^{-3}$, $N=2{\text{s}}^{-1}$, $D_i=0.1\text{m}$ (so $D_i^2=0.01{\text{m}}^2$) and $\mu =10^{-3}{\text{kg m}}^{-1}{\text{ s}}^{-1}$.

3. The characteristic length is the impeller diameter, not a dimension of the tank. The $100\text{L}$ vessel volume therefore does not enter the calculation.

4. $$R{e}_i=\frac{(1000)(2)(0.1{)}^2}{10^{-3}}=\frac{20}{10^{-3}}=2\times 10^4$$

5. Units check: $\frac{{\text{kg m}}^{-3}\cdot {\text{s}}^{-1}\cdot {\text{m}}^2}{{\text{kg m}}^{-1}{\text{ s}}^{-1}}=\frac{{\text{kg m}}^{-1}{\text{ s}}^{-1}}{{\text{kg m}}^{-1}{\text{ s}}^{-1}}$, which is dimensionless, as a Reynolds number must be.

6. $R{e}_i=20000$. This is above about $10^4$, the value usually taken as the start of fully turbulent flow in a stirred vessel. In a baffled vessel the power number is constant in that regime.