Detailed Explanation:

Question:
Scale-up of a fermenter is done based on constant impeller tip speed. If the diameter of the impeller is increased by 10 fold, the agitator speed will:

• (1) Decrease by 10 fold

• (2) Decrease by 100 fold

• (3) Increase by 10 fold

• (4) Increase by 100 fold

Correct Answer:

(1) Decrease by 10 fold

Explanation:

In a fermenter scale-up process, one of the key considerations is maintaining constant impeller tip speed. The impeller tip speed plays a crucial role in ensuring adequate mixing and mass transfer within the reactor, affecting the overall performance of the fermentation process.

When scaling up a fermenter, the relationship between the impeller diameter (D) and the agitator speed (N) is governed by the principle of maintaining the same tip speed. The impeller tip speed (Vt) is defined by the equation:

$$Vt=\pi \times D\times N$$

Where:

• $Vt$ is the impeller tip speed,

• $D$ is the diameter of the impeller,

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

Scale-Up Principle:

• Constant Impeller Tip Speed:
  In a scale-up process, if the impeller diameter (D) is increased, the agitator speed (N) must be adjusted in such a way that the tip speed remains constant.

• Relationship between Diameter and Speed: Given that the impeller tip speed ($Vt$) is proportional to both the impeller diameter ($D$) and the impeller speed ($N$), if the diameter is increased by a factor of 10, the speed of the agitator will need to decrease by a factor of 10 to keep the tip speed constant.

Thus, if the diameter of the impeller is increased by 10 fold, the agitator speed will decrease by 10 fold in order to maintain the same tip speed.

Why the Other Options are Incorrect:

• Option (2) Decrease by 100 fold:
  This would suggest an overcorrection in the relationship between the diameter and speed. A 100-fold decrease is not necessary based on the scale-up principle outlined above.

• Option (3) Increase by 10 fold:
  This would imply that as the diameter increases, the speed increases as well, which contradicts the concept of maintaining constant tip speed during scale-up.

• Option (4) Increase by 100 fold:
  Again, this would suggest an overcompensation in speed, which is not in line with the principle of scale-up based on constant impeller tip speed.

Conclusion:

When scaling up a fermenter, maintaining constant impeller tip speed requires the adjustment of the agitator speed. If the diameter of the impeller increases by 10 fold, the agitator speed will decrease by 10 fold to preserve the same tip speed. This relationship is crucial for ensuring effective mixing and mass transfer during the fermentation process.

For more information on optimizing fermenter design and scale-up, feel free to dive deeper into bioprocess engineering principles!