Aspect ratio 1:1 cylinder: a cylindrical reactor holds 100,000 litres (100 m^3) of liquid and has height:diameter (H:D) = 1:1. What is the approximate liquid height?

Introduction / Context:Quick geometric estimates are common in bioprocess design. For right circular cylinders, knowing volume and aspect ratio allows back-calculating height and diameter. Here, the reactor has H:D = 1:1 and contains 100,000 litres (100 m^3).

Given Data / Assumptions:

• Volume V = 100 m^3.
• H:D = 1:1, so H = D.
• Right circular cylinder: V = π*(D/2)^2*H.

Concept / Approach:With H = D, substitute into the cylinder formula: V = π*(D^2/4)*D = (π/4)*D^3. Solve for D and set H = D. Convert litres to cubic metres already done (100 m^3).

Step-by-Step Solution:1) Write V = (π/4)*D^3.2) Rearrange: D^3 = 4V/π = 400/π.3) Evaluate numerically: 400/π ≈ 127.32; cube root ≈ 5.03 m.4) Since H = D, height ≈ 5.03 m → approximately 5.0 m.

Verification / Alternative check:Sanity check: With D ≈ 5 m and H ≈ 5 m, volume V ≈ π*(2.5^2)*5 ≈ 3.1416*6.25*5 ≈ 98.17 m^3, close to 100 m^3; rounding explains the slight difference.

Why Other Options Are Wrong:1.0 m and 3.0 m are far too small to enclose 100 m^3 at aspect 1:1; 7.0 m and 10.0 m would overshoot the volume for a 1:1 aspect unless diameter also increased beyond the implied value.

Common Pitfalls:Using litres directly without converting to m^3, or forgetting that H = D for a 1:1 aspect ratio.

Final Answer:5.0 m