Mixing dimensionless groups: the impeller Power number (Po) is defined as which of the following nondimensional ratios (using power P, density ρ, rotational speed N, impeller diameter D)?

Introduction:The Power number Po characterizes the relationship between power draw and flow conditions for a given impeller geometry. It is central to scale-up because it connects dimensional power to speed, fluid density, and size in a dimensionless manner.

Given Data / Assumptions:

• P0 denotes shaft power (P).
• ρ is fluid density, N is rotational speed (s^-1), D is impeller diameter (m).
• Po is dimensionless and typically constant in fully turbulent regimes for a given impeller type.

Concept / Approach:By definition, Po = P / (ρ * N^3 * D^5). This arises from dimensional analysis balancing power with inertial scaling. It enables comparison across scales and fluids using Reynolds number simultaneously.

Step-by-Step Solution:Write Po = P / (ρ N^3 D^5).Match this with the listed forms.Identify the exact expression in option (b).

Verification / Alternative check:Standard mixing references present Po along with Re = ρ N D^2 / μ, both widely used in agitation correlations.

Why Other Options Are Wrong:

• Other exponents on N and D do not reduce to a dimensionless group equal to Po.
• ρ N D^3 / P0 is the reciprocal of part of the intended scaling and is not Po.

Common Pitfalls:Confusing Power number with pumping number; ensure correct exponents for N and D.

Final Answer:P0 / N3Di5ρ