Which dimensionless number is most directly calculated from impeller diameter and peripheral (tip) speed in an agitated tank?

Introduction / Context:Dimensionless numbers help generalize mixing behavior across scales. In stirred tanks, the Froude number relates inertial forces to gravity and is especially relevant when free-surface vortexing or surface waves may occur. It can be expressed using impeller diameter and tip speed (peripheral speed).

Given Data / Assumptions:

• Impeller diameter D and rotational speed N define tip speed u_tip = π*D*N.
• Gravity g is constant.
• Liquid properties are otherwise unspecified.

Concept / Approach:The Froude number for mixing is often written as Fr = N^2*D/g, which is dimensionally proportional to (u_tip^2)/(g*D). Thus, Fr depends directly on diameter and peripheral speed. By contrast, the Reynolds number Re = ρ*N*D^2/μ requires viscosity and density; the power number is an empirical function of Re (and geometry), not directly computed from tip speed alone.

Step-by-Step Solution:1) Recognize the formula connection: Fr ∝ u_tip^2/(g*D).2) Since u_tip = π*D*N, Fr can be calculated from D and N (or u_tip) with g.3) Therefore, the number “most directly” tied to diameter and peripheral speed is Froude.

Verification / Alternative check:Free-surface behavior and vortex depth often correlate with Fr, highlighting its practical link to tip speed and diameter.

Why Other Options Are Wrong:Reynolds requires fluid properties; power number is not a stand-alone input calculation; Schmidt relates momentum and mass diffusivity and is unrelated to impeller tip speed.

Common Pitfalls:Interchanging Re and Fr in scale-up; they capture different physics (viscous versus gravitational effects).

Final Answer:Froude number