Primary sedimentation design: What is the detention time of a circular tank of diameter d (m) and water depth H (m) receiving sewage at Q (m^3 per hour)?

Introduction / Context:

Detention time (also called hydraulic retention time) is a basic check in the design of primary sedimentation tanks. It represents the average time wastewater spends in the tank, influencing suspended solids removal and settling performance.

Given Data / Assumptions:

• Circular tank of diameter d (m) and uniform water depth H (m).
• Inflow rate Q is in m^3 per hour.
• Completely mixed, steady-state volume calculation.

Concept / Approach:

The detention time equals tank volume divided by flow rate. For a circular tank, the volume is the area of the plan multiplied by the depth. No correction is needed for sludge storage in this simple formula; it is a geometric check.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency: numerator (m^3) divided by Q (m^3/h) yields hours, confirming the formula's units. This aligns with standard design practice in primary clarifier sizing.

Why Other Options Are Wrong:

• Expressions lacking the 1/4 factor ignore the circle's area formula.
• Using d instead of d^2 or mixing minutes/hours leads to unit inconsistency.
• Formulas with π * d * H are dimensionally incorrect for volume.

Common Pitfalls:

• Forgetting that diameter must be squared in the area formula.
• Mixing flow units (e.g., m^3/s vs m^3/h) and producing wrong detention times.

Final Answer:

t = (π * d^2 * H) / (4 * Q) hours