ANS:C - N²

In a stirred tank reactor, when the power number becomes constant at high Reynolds numbers, the variation of power input with impeller rotational speed NNN is typically proportional to N3N^3N3.

Explanation:

• Power Number (NpN_pNp​): The power number is a dimensionless parameter that relates the power consumption in a stirred tank to the fluid dynamic conditions. It is defined as: Np=Pρ⋅N3⋅D5N_p = \frac{P}{\rho \cdot N^3 \cdot D^5}Np​=ρ⋅N3⋅D5P​ Where:
  • PPP is the power input,
  • ρ\rhoρ is the fluid density,
  • NNN is the impeller rotational speed,
  • DDD is a characteristic length (often the impeller diameter).
• Behavior at High Reynolds Numbers: At high Reynolds numbers (Re), the flow in the stirred tank becomes turbulent, and the power number NpN_pNp​ tends to become constant. This indicates that the power input is primarily dependent on the impeller speed and less on other parameters.
• Variation with Impeller Speed (NNN): In this regime, the power input PPP is proportional to N3N^3N3. This means that doubling the impeller speed NNN would increase the power input by a factor of 23=82^3 = 823=8.
• Other Options:
  • N0N^0N0 (N°): Implies no dependence on impeller speed, which is not the case because power input increases with impeller speed.
  • N1N^1N1 (N1): Implies a linear relationship, which typically does not hold at high Reynolds numbers where turbulent effects dominate.
  • N2N^2N2 (N2): Implies a quadratic relationship, which may be observed in some regimes but not typically at high Reynolds numbers where N3N^3N3 dependence is more common.

Therefore, at high Reynolds numbers in a stirred tank reactor, the variation of power input with impeller rotational speed NNN is proportional to N3N^3N3. This reflects the dominance of turbulent flow effects and the resulting power consumption in the system.