Scale-up design problem (Rushton turbine): A cylindrical bioreactor holds 100,000 L of liquid (100 m^3) and has an aspect ratio of 2:1 (liquid height to tank diameter). Estimate the appropriate impeller diameter for a Rushton turbine using the common rule D_impeller ≈ one-third of tank diameter.

Introduction / Context:Estimating impeller diameter is a routine early step in bioreactor specification and scale-up. For Rushton turbines in baffled cylindrical tanks, a common heuristic is D_impeller ≈ 0.33 * D_tank. To use the rule, we must first infer the tank diameter from liquid volume and aspect ratio (height/diameter).

Given Data / Assumptions:

• Liquid volume V = 100,000 L = 100 m^3.
• Aspect ratio AR = H / D_tank = 2 : 1, so H = 2 * D_tank.
• Cylindrical working volume: V = (π * D_tank^2 / 4) * H.
• Standard Rushton diameter rule: D_impeller ≈ D_tank / 3.

Concept / Approach:Compute D_tank from the cylinder volume formula using AR = 2. Then apply the one-third rule to obtain the impeller diameter. Finally, map the numerical result to the closest option.

Step-by-Step Solution:

Verification / Alternative check:Check volume using D_tank = 4.0 m and H = 8.0 m: V_calc = π/4 * 4.0^2 * 8.0 ≈ 100.5 m^3 (close to 100 m^3; rounding explains the small difference). The rule-of-thumb then yields D_impeller ≈ 1.33 m, consistent with the selected answer.

Why Other Options Are Wrong:

• 3.3 m and 5.3 m: these are closer to full tank diameter choices, not one-third.
• 7.3 m: exceeds the tank diameter implied by the volume.
• 0.6 m: far below typical one-third sizing; would under-mix.

Common Pitfalls:Forgetting to convert litres to cubic metres; misinterpreting aspect ratio (height-to-diameter vs diameter-to-height); applying the one-third rule before computing D_tank.

Final Answer:1.3 m.