In a stirred-tank bioreactor, compute the impeller Reynolds number for the following operating conditions (use SI units and standard mixing definition Re = (rho * N * D^2) / mu):\r \r Fluid density = 2.5 g/mL (i.e., 2500 kg/m^3)\r Dynamic viscosity = 0.002 kg/(m·s)\r Impeller diameter D = 50 cm (i.e., 0.50 m)\r Agitation speed N = 120 rpm (i.e., 2.0 s^-1)\r \r What is the Reynolds number of the liquid in the reactor?

Introduction:
The impeller Reynolds number is a key dimensionless group in mixing that indicates the flow regime (laminar, transitional, or fully turbulent). Correctly calculating Re from density, viscosity, impeller size, and rotational speed helps decide correlations for power number, mass transfer, and scale-up in bioprocess engineering.

Given Data / Assumptions:

• rho (fluid density) = 2.5 g/mL = 2500 kg/m^3.
• mu (dynamic viscosity) = 0.002 kg/(m·s).
• D (impeller diameter) = 50 cm = 0.50 m.
• N (rotational speed) = 120 rpm = 2.0 s^-1.
• Re definition for stirred tanks: Re = (rho * N * D^2) / mu.

Concept / Approach:
For agitated vessels, the Reynolds number incorporates rotational speed and impeller diameter squared. Unit consistency is essential: convert g/mL to kg/m^3, cm to m, and rpm to s^-1 before substitution.

Step-by-Step Solution:
Step 1: Convert units: rho = 2500 kg/m^3; D = 0.50 m; N = 120 rpm = 120/60 = 2.0 s^-1; mu = 0.002 kg/(m·s).Step 2: Apply Re = (rho * N * D^2) / mu.Step 3: Compute D^2 = 0.50^2 = 0.25 m^2.Step 4: Numerator = 2500 * 2.0 * 0.25 = 1250.Step 5: Re = 1250 / 0.002 = 625000.

Verification / Alternative check:
Magnitude check: with high density and moderate viscosity, Re in the 10^5–10^6 range is expected for a 0.5 m impeller at 120 rpm, confirming turbulent flow.

Why Other Options Are Wrong:

• 2500: Off by two orders of magnitude; ignores D^2 and/or unit conversions.
• 1000000 and 1250000: Overestimates from arithmetic/rounding errors.
• 100: Severe underestimate; inconsistent with dimensions and speed.

Common Pitfalls:
Mixing units (rpm vs s^-1), forgetting to square D, or using kinematic viscosity instead of dynamic viscosity without proper conversion (nu = mu/rho).

Final Answer:
625000