In agitation engineering, what is the correct expression for the impeller Reynolds number (Rei) in a stirred tank, using Di for impeller diameter, N for rotational speed, ρ for fluid density, and μ for fluid viscosity?

Introduction / Context:
The impeller Reynolds number (Rei) is a dimensionless group that indicates the flow regime in a mechanically agitated vessel. It guides decisions on mixing scale-up, power draw correlations, gas dispersion, and mass transfer. This question tests recognition of the standard Rei formula expressed with commonly used variables in bioreactor design.

Given Data / Assumptions:

• Stirred tank with an impeller of diameter Di rotating at speed N (revolutions per second or per minute converted to s^-1).
• Fluid of density ρ and dynamic viscosity μ (Newtonian assumption).
• No baffles or impeller-specific correction factors are included in Rei itself.

Concept / Approach:
Rei compares inertial to viscous forces. In stirred tanks, the characteristic velocity scale is N * Di and the characteristic length scale is Di. Thus Rei = (ρ * N * Di^2) / μ. This form parallels the pipe Reynolds number but replaces the mean velocity with a tip-speed scale and the diameter with the impeller diameter.

Step-by-Step Solution:
Choose velocity scale: U ~ N * Di.Form Reynolds number: Re = ρ * U * L / μ.Use L = Di and U = N * Di ⇒ Re = ρ * (N * Di) * Di / μ.Hence Rei = ρ * N * Di^2 / μ.

Verification / Alternative check:
Dimensional analysis confirms Rei is dimensionless: [ρ]=M L^-3, [N]=T^-1, [Di^2]=L^2, [μ]=M L^-1 T^-1; multiplying and dividing gives M L^-3 * T^-1 * L^2 / (M L^-1 T^-1) = 1.

Why Other Options Are Wrong:
Di * V * ρ / μ: introduces an undefined V; not the standard impeller scaling.Di^2 * V * ρ / μ: again uses V rather than N; incorrect for Rei.Di * N * ρ / μ: missing one factor of Di; wrong scaling.

Common Pitfalls:
Using rpm without converting to s^-1, mixing tip speed (π * Di * N) with Rei formula, or substituting tank diameter T for Di.

Final Answer:
Di^2 * N * ρ / μ