In power requirement calculations for mixing in stirred tanks, which dimensionless group(s) are fundamentally important for correlating power number and operating regime?

Introduction / Context:
Power correlations in mixing use dimensionless groups to capture viscous, inertial, and gravitational effects that influence flow patterns and power draw.

Given Data / Assumptions:

• Re = ρ N D^2 / μ (viscous vs inertial).
• Fr = N^2 D / g (inertial vs gravity), important for vortex formation/free-surface effects.

Concept / Approach:
Reynolds number classifies laminar, transitional, and turbulent mixing, influencing Po. Froude number becomes important when free-surface vortices or gas–liquid interfaces are present, altering power and flow. Thus both groups are key in many mixing problems.

Step-by-Step Solution:
Identify roles of Re (viscosity) and Fr (free surface/rotation).Note their appearance in empirical power charts.Conclude both are important.

Verification / Alternative check:
Power number charts often plot Po vs Re with Fr (or vortex depth) as a parameter in free-surface systems.

Why Other Options Are Wrong:
Single-parameter views miss regimes where gravity matters.

Common Pitfalls:
Ignoring Fr in open tanks, leading to unexpected vortexing and power draw changes.

Final Answer:
Both (a) and (b)