GATE 2024 BT Q.14: solution and answer

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

1. The impeller Reynolds number for a stirred tank is $$N_{Re}=\frac{\rho N{D}_i^2}{\mu }$$ where $\rho$ is the broth density, $N$ the impeller speed (revolutions per unit time), $D_i$ the impeller diameter and $\mu$ the broth viscosity.

2. (A) $N_{Re}\propto 1/\mu$, so it depends directly on the viscosity of the medium. A is false.

3. (C) $N_{Re}\propto N$, so it is directly proportional to the impeller speed, not inversely proportional. C is false.

4. The mixing time $t_m$ is the time needed after a tracer is added for the tank contents to reach a stated degree of uniformity. It is correlated as the dimensionless mixing time $N{t}_m$ (the number of impeller revolutions needed to mix) against $N_{Re}$. In the laminar and transitional ranges, $N{t}_m$ decreases as $N_{Re}$ rises. A higher $N_{Re}$ means faster stirring or a thinner broth, so blending is quicker and $t_m$ gets shorter. Nothing in this range makes $t_m$ rise with $N_{Re}$. B claims the opposite, so B is false.

5. In the fully turbulent range ($N_{Re}$ above roughly $10^4$ for a standard turbine), the curve flattens: $N{t}_m$ is a constant set only by the tank and impeller geometry, whatever the value of $N_{Re}$. So if $N_{Re}$ changes because the viscosity or density changes, $t_m$ stays the same. If the impeller speed is raised, $t_m$ does fall, but only as $1/N$, because each revolution takes less time. The number of revolutions needed to mix does not change. This is the standard sense in which mixing time is independent of $N_{Re}$ in turbulent flow.

6. D is the only correct statement.