Dimensionless groups in mixing: The power number (Po) is best interpreted as the ratio of which effects in an agitated vessel?

Introduction / Context:
The power number Po is a key dimensionless parameter for correlating agitator power in geometrically similar vessels across different scales and operating conditions. It connects measured power to fluid properties and impeller dimensions.

Given Data / Assumptions:

• Standard definition: Po = P / (ρ * N^3 * D^5).
• P is shaft power, ρ is density, N is rotational speed, D is impeller diameter.
• Fully baffled tank where Po is relatively independent of Reynolds number at turbulent conditions for a given impeller type.

Concept / Approach:
Po is analogous to a drag coefficient for mixing: it scales the imposed hydrodynamic resistance (torque/power) against characteristic inertial scaling (ρ * N^3 * D^5). Thus, it reflects the relative magnitude of impeller-imposed forces compared with inertial forces in the flow.

Step-by-Step Solution:
Start from Po = P / (ρ * N^3 * D^5).Recognize ρ * N^3 * D^5 as inertial scaling for power in rotating flows.Interpret Po as a dimensionless power coefficient (imposed resistance vs. inertia).Conclude the best phrasing: imposed (drag/torque) forces to inertial forces.

Verification / Alternative check:
In turbulent regimes, Po is approximately constant for a given impeller geometry (e.g., ~5–6 for a Rushton turbine), consistent with a drag-like coefficient relative to inertial scaling.

Why Other Options Are Wrong:
Buoyant to inertial or gravitational to inertial: those relate to Froude or Archimedes numbers, not Po.

Imposed to gravitational: misses the inertial basis central to Po’s definition.

Common Pitfalls:

• Confusing Po with Reynolds or Froude numbers; each captures different physics.
• Using Po correlations outside their valid geometry range (blade count, pitch, baffling).

Final Answer:
Imposed (drag/torque) forces to inertial forces