Agitation scale-up fundamentals: a vigorous mixing guideline is about 2–3 hp per gallon for very thin liquids. In dimensionless form, the impeller “Power number” (Po) is defined as which ratio of variables?

Introduction / Context:In mixing, the dimensionless Power number (Po) collapses impeller power draw across scales and fluids. Together with Reynolds number (Re = rho * N * D^2 / mu), it underpins geometric similarity scale-up. Knowing the correct Po definition ensures consistent design from lab to plant for blending, suspension, and gas–liquid contacting.

Given Data / Assumptions:

• P is power (W), rho is fluid density (kg/m^3), mu is viscosity (Pa·s), N is rotational speed (s^-1), and D is impeller diameter (m).
• Thin (low-viscosity) liquids at high turbulence are assumed for the “2–3 hp per gallon” rule-of-thumb; however, Po is general.

Concept / Approach:By dimensional analysis, P has units kg·m^2/s^3. The term rho * N^3 * D^5 also has units kg·m^2/s^3, so their ratio is dimensionless. Thus Po = P / (rho * N^3 * D^5). In turbulent regimes for a given impeller geometry, Po becomes nearly constant, enabling direct horsepower prediction at scale.

Step-by-Step Solution:Write Po candidate forms and check dimensional consistency.Confirm rho * N^3 * D^5 has the same dimensions as P.Select the dimensionally correct and standard definition: Po = P / (rho * N^3 * D^5).

Verification / Alternative check:Impeller datasheets list constant Po values (e.g., Rushton disk turbine Po ≈ 5–6 in turbulent flow), which are used with the above formula to size motors.

Why Other Options Are Wrong:N^2 * D^3 or N * D^2 denominators do not render a dimensionless group equal to Po; those break dimensional consistency.Using viscosity (mu) in Po is incorrect; mu appears in Re, not Po.

Common Pitfalls:Confusing Po with torque number (Np = T / (rho * N^2 * D^5)) or applying laminar correlations in turbulent regimes without checking Re.

Final Answer:Po = P / (rho * N^3 * D^5)