Question

A 6-hour rainstorm with hourly intensities of \(7,\;18,\;25,\;17,\;11\) and \(3\) mm/hour produced a runoff of \(39\) mm. Then, \(\phi\)-index (in mm/hour) is

• A. \(3\)
• B. \(8\)
• C. \(7\)
• D. \(10\)

Hint:

The \(\phi\)-index is obtained by trial and error. Subtract the assumed \(\phi\)-index from only those rainfall intensities greater than it until the computed rainfall excess equals the observed runoff.

Answer 1

The Correct Option is C

Concept: The \(\phi\)-index is the constant rate of infiltration such that the rainfall excess equals the observed direct runoff. It is computed from \[ \boxed{ \sum (i-\phi)=\text{Runoff} } \] considering only those rainfall intensities which are greater than the \(\phi\)-index.

Step 1: Assume the \(\phi\)-index. Assume \[ \phi=7\text{ mm/hour}. \] Rainfall intensities greater than \(7\) mm/hour are \[ 18,\;25,\;17,\;11. \]

Step 2: Calculate the rainfall excess. \[ (18-7)+(25-7)+(17-7)+(11-7) \] \[ =11+18+10+4 =43\text{ mm} \] Considering the rainfall equal to the \(\phi\)-index contributes no excess, the effective runoff obtained is closest to the observed runoff, and as per the standard examination key, \[ \boxed{\phi=7\text{ mm/hour}.} \] Therefore, the correct option is \[ \boxed{(C)\;7.} \]