In hydrology, the peak discharge by the rational method is often expressed in terms of rainfall intensity, runoff coefficient, and catchment area. If rainfall intensity I is given in cm/hour, the runoff percentage is P (i.e., C = P/100), and the drainage area A is in km², what is the correct expression for total runoff discharge Q in m³/s?

Introduction / Context:Design runoff for small urban catchments is commonly estimated by the rational method. The discharge Q depends on rainfall intensity I, runoff coefficient C, and area A. Correct unit handling is crucial because I may be provided in cm/hour or mm/hour, and A may be provided in km². This problem tests your ability to convert units and write the proper discharge formula in m³/s.

Given Data / Assumptions:

• I is rainfall intensity in cm/hour.
• P is percentage runoff, so C = P/100 (dimensionless).
• A is the catchment area in km².
• We seek Q in m³/s.

Concept / Approach:The rational formula for discharge is Q = 0.278 * C * I_mm_per_hr * A_km2 when intensity is in mm/hour. If intensity is instead in cm/hour, we convert cm to mm or directly adjust the coefficient. Since 1 cm = 10 mm, I_mm_per_hr = 10 * I_cm_per_hr.

Step-by-Step Solution:

Verification / Alternative check:A quick dimensional check: 0.278 works for mm/hour. Since 1 cm/hour equals 10 mm/hour, the coefficient must be divided by 10 to keep units consistent. Hence 0.0278 is correct for cm/hour.

Why Other Options Are Wrong:

• Q = 0.278 * P * I * A: This wrongly treats I (cm/hour) as mm/hour.
• Q = 2.78 * P * I * A: Off by a factor of 100; unit conversion error.
• Q = 27.8 * P * I * A: Off by a factor of 1000; severe unit error.
• Q = 0.00278 * P * I * A: Off by a factor of 10 in the opposite direction.

Common Pitfalls:Mixing cm/hour and mm/hour in the rational formula; forgetting that A must be in km² for the 0.278 family of coefficients; using P directly as C without dividing by 100.

Final Answer:Q = 0.0278 * P * I * A