Question

The Rational Method formula for estimating peak runoff is given as \(Q = \frac{CiA}{360}\). For the result \(Q\) to be in cubic meters per second (\(m^3/s\)), what must be the units of rainfall intensity (\(i\)) and catchment area (\(A\))?

• A. \(i\) in \(cm/hr\), \(A\) in \(km^2\)
• B. \(i\) in \(mm/hr\), \(A\) in \(km^2\)
• C. \(i\) in \(mm/hr\), \(A\) in hectares
• D. \(i\) in \(m/hr\), \(A\) in hectares

Hint:

The Rational Method formula appears in different forms depending on the units used. The two most common forms are: 1. \(Q = \frac{CiA}{360}\) for \(Q\) in \(m^3/s\), \(i\) in \(mm/hr\), and \(A\) in hectares. 2. \(Q = \frac{CiA}{3.6}\) for \(Q\) in \(m^3/s\), \(i\) in \(mm/hr\), and \(A\) in \(km^2\). Remembering these two standard forms can help you quickly solve such unit-based problems.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question requires us to perform a dimensional analysis on the Rational Method formula, \(Q = \frac{CiA}{360}\). We need to find the specific units for rainfall intensity (\(i\)) and catchment area (\(A\)) that will yield the peak runoff (\(Q\)) in \(m^3/s\), given the constant denominator of 360. The runoff coefficient (\(C\)) is a dimensionless quantity.
Step 2: Detailed Explanation:
We will analyze the units for each option to see which combination results in the correct conversion factor. The goal is to make the units on the right side of the equation equal to \(m^3/s\).
Let's convert the units of \(i\) and \(A\) to meters and seconds for each option.
(A) \(i\) in \(cm/hr\), \(A\) in \(km^2\):
\(i\) in \(\frac{cm}{hr} = \frac{10^{-2} \, m}{3600 \, s}\)
\(A\) in \(km^2 = (10^3 \, m)^2 = 10^6 \, m^2\)
\(i \times A = \left(\frac{10^{-2}}{3600}\right) \frac{m}{s} \times (10^6) m^2 = \frac{10^4}{3600} \frac{m^3}{s} = \frac{1}{0.36} \frac{m^3}{s}\)
So, \(Q = \frac{CiA}{0.36}\). This does not match the given formula.
(B) \(i\) in \(mm/hr\), \(A\) in \(km^2\):
\(i\) in \(\frac{mm}{hr} = \frac{10^{-3} \, m}{3600 \, s}\)
\(A\) in \(km^2 = 10^6 \, m^2\)
\(i \times A = \left(\frac{10^{-3}}{3600}\right) \frac{m}{s} \times (10^6) m^2 = \frac{10^3}{3600} \frac{m^3}{s} = \frac{1}{3.6} \frac{m^3}{s}\)
So, \(Q = \frac{CiA}{3.6}\). This is a common form of the Rational Method but does not match the formula given in the question.
(C) \(i\) in \(mm/hr\), \(A\) in hectares:
\(i\) in \(\frac{mm}{hr} = \frac{10^{-3} \, m}{3600 \, s}\)
\(A\) in hectares \(= 10^4 \, m^2\)
\(i \times A = \left(\frac{10^{-3}}{3600}\right) \frac{m}{s} \times (10^4) m^2 = \frac{10}{3600} \frac{m^3}{s} = \frac{1}{360} \frac{m^3}{s}\)
So, \(Q = C \times (i \times A) = C \times i \times A \times \frac{1}{360} = \frac{CiA}{360}\).
This perfectly matches the formula and units given in the question.
(D) \(i\) in \(m/hr\), \(A\) in hectares:
\(i\) in \(\frac{m}{hr} = \frac{1 \, m}{3600 \, s}\)
\(A\) in hectares \(= 10^4 \, m^2\)
\(i \times A = \left(\frac{1}{3600}\right) \frac{m}{s} \times (10^4) m^2 = \frac{10000}{3600} \frac{m^3}{s} = \frac{1}{0.36} \frac{m^3}{s}\)
So, \(Q = \frac{CiA}{0.36}\). This does not match the given formula.
Step 3: Final Answer:
The only combination of units that satisfies the formula \(Q = \frac{CiA}{360}\) for \(Q\) in \(m^3/s\) is when rainfall intensity \(i\) is in \(mm/hr\) and catchment area \(A\) is in hectares.