Turbulence Micro-Scales — How does the Kolmogorov eddy size vary with impeller speed in a stirred tank?

Introduction:
The Kolmogorov micro-scale characterizes the size of the smallest dissipative eddies in turbulent flow. In mixing, its magnitude influences potential shear damage to delicate materials such as cells or polymer structures. Understanding how it responds to speed helps set safe operating windows.

Given Data / Assumptions:

• Turbulent regime in a baffled stirred tank.
• Increasing impeller speed increases specific energy dissipation.
• Fluid properties (density, viscosity) are constant.

Concept / Approach:

The Kolmogorov length scale eta scales with (nu^3 / epsilon)^(1/4), where nu is kinematic viscosity and epsilon is energy dissipation rate per mass. As impeller speed rises, epsilon increases, so eta decreases. Thus, higher speeds generate smaller dissipative structures and potentially stronger local velocity gradients.

Step-by-Step Solution:

Verification / Alternative check:

Power correlations (P = Np * rho * N^3 * D^5) imply higher N increases epsilon in the tank, leading to reduced eta. Measurements in mixing studies confirm smaller micro-scales at higher speeds.

Why Other Options Are Wrong:

A: Not constant; depends on epsilon. B: Opposite of correct trend. D: Never zero; finite even at low speeds. E: There is a clear dependence via energy dissipation.

Common Pitfalls:

Assuming micro-scale equals bubble size or impeller diameter; it is a turbulence property tied to dissipation, not a geometric feature.

Final Answer:

It decreases as stirrer speed increases (smaller dissipative eddies at higher energy dissipation).