138. For a Rushton turbine impeller (Reynold’s number greater than 10,000) when RPM is doubled, the power
absorption increases by:
(1) 2 fold
(2) 4 fold
(3) 8 fold
(4) 32 fold

Detailed Explanation:

Question:
For a Rushton turbine impeller (Reynold’s number greater than 10,000), when RPM is doubled, the power absorption increases by:

• (1) 2 fold

• (2) 4 fold

• (3) 8 fold

• (4) 32 fold

Correct Answer:

(3) 8 fold

Explanation:

A Rushton turbine is commonly used in bioreactors and mixing applications where high shear and efficient mixing are required. The power absorption ($P$) of an impeller is a key factor that influences the mixing efficiency in industrial reactors. It is governed by a number of variables, including the Reynolds number (Re), which determines the flow regime, and the rotational speed (RPM) of the impeller.

Understanding the Relationship Between RPM and Power Absorption:

The power absorption of a Rushton turbine impeller is related to the RPM and can be described by the following empirical relationship:

P∝N3P \propto N^3P∝N3

Where:

• $P$ is the power absorbed by the impeller.

• $N$ is the rotational speed (RPM) of the impeller.

This means that when the RPM is doubled, the power absorption increases by a factor of 23=82^3 = 823=8. Hence, the power absorbed by the impeller increases 8 fold when the RPM is doubled.

Why Option (3) is Correct:

• Doubling the RPM results in an 8-fold increase in power absorption, which aligns with the formula P∝N3P \propto N^3P∝N3.

• This increase in power absorption is significant, particularly in high-shear applications where efficient mixing and energy input are crucial.

Why the Other Options are Incorrect:

• Option (1) 2 fold:
  This would be true if the relationship between power absorption and RPM was linear, but in fact, the relationship is cubic, as shown by the P∝N3P \propto N^3P∝N3 equation.

• Option (2) 4 fold:
  A 4-fold increase would be expected if the power absorption increased with the square of the RPM, but this is not the case for Rushton turbines.

• Option (4) 32 fold:
  A 32-fold increase would imply that the power absorption increases with the fourth power of the RPM, which is not consistent with the behavior of Rushton turbine impellers.

Conclusion:

For a Rushton turbine impeller with a Reynold’s number greater than 10,000, doubling the RPM results in an 8-fold increase in power absorption. This relationship highlights the significant impact that rotational speed has on energy consumption in industrial mixing processes. Understanding this relationship is crucial for optimizing the efficiency of reactors in biotechnology, chemical engineering, and other industries requiring effective mixing and power management.