This document discusses methods for estimating peak runoff and time of concentration using the rational method for small drainage basins with green infrastructure. It presents the rational method equation and variables, then describes equations for estimating runoff coefficient, rainfall intensity, and time of concentration based on factors like soil type, slope, and drainage area. Graphs show relationships between the unified time of concentration equation and the rational method when varying coefficients, rainfall amounts, and basin characteristics. The areas under curve intersections provide insight into limitations of the rational method for runoff calculations.

1. AN EVALUATION OF ESTIMATING PEAK
RUNOFF with Tc for GREEN INFRASTRUCTURE
TC Equation Relationship to
Rational Method Peak Flow

2. Rational Method Equation and Variables:
Q = C i A
Where:
Q = Maximum Rate of Runoff (cfs)
C = Runoff Coefficient
i = Average Rainfall Intensity (in/hr)
A = Drainage Area (in acres)
The EQUATION (Q)

3. Rational Method Equation and Variables:
Where:
C = Runoff Coefficient
R = Total Depth of Runoff (in)
P = Total Depth of Precipitation (in)
RUNOFF COEFFICIENT (C)
Rossmiller (1981) C = 7.210-7 CN3 T0.05 (0.01CN)-0.6s-.02
((0.001CN)1.48 )0.15 – 0.1I ((P+1)/2) 0.7
Where: CN is the Soil Conservation Service Curve Number; T is the recurrence interval (years); S is the average land slope (%);
I is the intensity (in/hr); and P is percent imperviousness.
C =
𝑅 𝑑
𝑃 𝑑

4. Rational Method Equation and Variables:
RAINFALL INTENSITY (i)
The determination of rainfall intensity (i)
for use in the Rational Formula involves
consideration of three factors:
• Average frequency of occurrence
• Intensity-duration characteristics for
a selected rainfall frequency.
• The time of concentration (tc).

5. Rainfall Intensities Based upon I-D-F Curves
As the Tc goes up, the “I” goes down.
Rainfall Intensities are
obtained by entering the
Log-Log nomograph with
the Time of Concentration
along the abscissa,
intersecting the Storm
Return Period Curve, and
proceeding horizontally to
the Rainfall Intensity.
http://hdsc.nws.noaa.gov/hdsc/pfds/index.html

6. Rational Equation with an “R” Runoff Ratio:
Where: q = Maximum Rate of Runoff (cfs), c = Runoff Coefficient, i = Average Rainfall Intensity (in/hr), A = Drainage Area (acres)
𝐑𝐚𝐭𝐢𝐨𝐧𝐚𝐥 Equation 𝐀𝐫𝐫𝐚𝐧𝐠𝐞𝐝 to ‘c’ Coeff. & ‘A’ Area
𝐜 =
𝐪(𝐜𝐟𝐬)
𝐢 ∗ 𝐀
𝐀 =
𝐪(𝐜𝐟𝐬)
𝐢 ∗ 𝐜
𝐑𝐮𝐧𝐨𝐟𝐟 𝐑𝐚𝐭𝐢𝐨 =
𝐝𝐢𝐬𝐜𝐡𝐚𝐫𝐠𝐞 𝐫𝐚𝐭𝐞
𝐢𝐧𝐭𝐞𝐧𝐬𝐢𝐭𝐲
= 𝐑 =
𝐪(cfs)
𝐢 (
in
hr
)
R = 2 cfs/4 in. = 2.0
Or
R= 4 cfs/2 in. = 0.5
c =
R
A
Now A =
R
c
&
Where:

7. Plotting the Rational Equation using
“R” Runoff Ratio with “c” coefficient

8. Plotting the Rational Equation using
“R” Runoff Ratio with “A” area

9. Rational Method Assumptions & Limitations
• Maximum watershed area has a 200 acre limit
• The method is applicable when the time of
concentration (tc) for the drainage area is less
than the duration of peak rainfall intensity.
• The time of concentration (tc) is the time
required for water to travel from the
hydraulically most remote point of the basin to
the point of interest within the basin.

10. • The calculated runoff is directly proportional
to the rainfall intensity.
• Rainfall intensity is uniform throughout the
duration of the storm.
• The frequency of occurrence for the peak
discharge is the same as the frequency of
the rainfall producing that event.
Rational Method Assumptions & Limitations

11. • Rainfall is distributed uniformly over the
drainage area.
• The minimum duration to be used for
computation of rainfall intensity is 10
Minutes. (Several jurisdictions use a 5
minute minimum)
• The rational method does not account for
storage in the drainage area. Available
storage is assumed to be filled.
Rational Method Assumptions & Limitations

12. Rational Method Equation and Variables:
TIME OF CONCENTRATION (tc)
• If the chosen storm duration > tc, then the rainfall
intensity will be less than that at tc (Peak discharge
< optimal value).
• If the chosen storm duration < tc, then the watershed
is not fully contributing runoff to the outlet for that
storm length (i.e. optimal value will not be realized).
• Therefore, use storm duration = tc for peak discharge.

13. Unified TC Equation for Channelization using ‘C’
Tc =
𝟏−𝑪avg
𝟑
𝒔
𝟏𝟐𝟓
𝑳
+
𝟑 𝒔
𝟏𝟒
• Tc = Time of Concentration (minutes)
• 𝑳 = Length of Flow Path (feet)
• 𝑪avg = Rational method’s average runoff coefficient
• 𝒔 = % Slope of Flow Path (decimal format)
• Equation Limits: 1 to 225 acres for drainage basin
1 to 12 percent slope for flow path
0.10 to 0.95 rational runoff coefficient

14. ‘C’ Relationships for Soil Types
using a 10yr. Storm Event

15. Unified Tc ‘C’ Equation uses an average ‘C‘
coefficient (near B soil type)
Basin weighed ‘C’ value is attained by
adjusting ‘C’ soil types to a ‘Cavg’ type
Cavg =
𝑪𝒕𝒚𝒑𝒆 𝟐𝟏+𝟎.𝟕𝒙+𝟎.𝟏𝟓𝒙 𝟐 −𝒙+𝟏.𝟓
𝟐𝟐.𝟓
Cavg = Average C values used in Kirpich-Velocity Eq.
𝑪𝒕𝒚𝒑𝒆 = Rational method’s runoff coefficient per soil type
𝒙 = NRCS’s soil type factor shown below
Type A Soil: x = 0 Type B Soil: x = 1
Type C Soil: x = 2 Type D Soil: x = 3

16. Mockus (USDA 1973) developed an empirical relationship
between flow length and drainage area using data from
Agricultural Research Service (ARS) watersheds.
Time of Concentration’s Flow Length
and Drainage Area Relationship
National Engineering Handbook
630.1502 Methods for estimating time of concentration
Eq. 15-5 𝒍 = 𝟐𝟎𝟗 𝑨 𝟎.𝟔
Where:
l = length of runoff flow, (ft)
A = drainage area, (acres)

17. Kirpich Equation compared to NRCS
Equation by Area of Drainage Basin

18. Tc =
𝟏−𝑪avg
𝟑
𝒔
𝟏𝟐𝟓
209 𝑨
0.6 +
𝟑 𝒔
𝟏𝟒
Unified TC Equation with ‘C’ for a Basin Area
UTC Equation arranged for Rational Runoff
Coefficient ‘c’ for Basin Areas at 5 & 10 Minutes
𝟓 𝐨𝐫 𝟏𝟎 𝐦𝐢𝐧. 𝐓 𝐜 =
1 − 𝑪
3
𝒔
125
209 𝑨0.6 +
3
𝒔
14
𝒄 = 1 − 5 3
𝒔
125
209 𝑨0.6
+
3
𝒔
14
𝒄 = 1 − 10 3
𝒔
125
209 𝑨0.6
+
3
𝒔
14
5 min. Runoff Coefficient ‘c’ 10 min. Runoff Coefficient ‘c’

19. Total Hydraulic Time Calculations
(TR55, Velocity, or SCS Method)
Sheet Flow Tt = 0.007(nL)0.8/(P2
0.5S0.4)
Shallow Concentrated Flow Tt = L /3600V
Open Channel Flow Tt= (L*n) /(1.49R0.67S 0.5)
(Manning’s Equation)
Where Hydraulic Radius = conveyance flow depth then:
Manning’s equation becomes Tt = L/3600V
Total Watershed Time of Concentration
tc=STt
L= ft., Tt = hr., S= % slope, R= ft., P= in.(2yr.24hr.) , V= ft./sec.

20. Sheet Flow
TR-55 Sheet Flow—The sheet flow time computed for each area of
sheet flow that requires the following input data:
Hydraulic Length—Defined flow length for the sheet flow.
Manning's n—Manning's roughness value of the sheet flow.
Slope— The defined slope of the sheet flow/catchment.
Precipitation
Infiltration
Kinematic Wave Eq.

21. Technical Paper No. 40 Rainfall Frequency Atlas of the
U.S. for a 2 yr. Return Period with 24 hr. Storm Duration

22. Kirpich Equation compared to NRCS
Equation with a P2 Sheet Flow of 2 inches
Equations Confluence

23. Kirpich Equation compared to NRCS
Equation with a P2 Sheet Flow of 4 inches
Equations Confluence

24. Kirpich Equation compared to NRCS
Equation with a P2 Sheet Flow of 6 inches
Equations Confluence

25. Unified TC ‘C’ Equation compared to
Kirpich & NRCS’s P2 Sheet Flow of 2”
UTC & Kirpich
Confluence

26. Unified TC ‘C’ Equation compared to
Kirpich & NRCS’s P2 Sheet Flow of 4”
UTC & Kirpich
Confluence

27. Unified TC ‘C’ Equation compared to
Kirpich & NRCS’s P2 Sheet Flow of 6”
UTC & Kirpich
Confluence

28. Unified TC Equation Plotted to Rational
Equation Runoff Ratio ‘R’ per Basin Area

29. Unified TC Equation Plotted to Rational
Equation Runoff Ratio ‘R’ per ‘c’ Coefficient

30. UTC Eq. for 2% Slope Plotted to Rational Eq.
Runoff Ratio ‘R’ per ‘c’ Coefficient

31. UTC Eq. for 10% Slope Plotted to Rational
Eq. Runoff Ratio ‘R’ per ‘c’ Coefficient

32. UTC Equation Plotted to Rational Runoff
Ratio ‘R’ per a ‘c’ Coefficient Variable

33. A Graphical Representation & Significance of a 5 Minute
UTC Equation Intersection to a Rational Ratio‘R’ Equation

34. A Graphical Representation & Significance of a 10 Minute
UTC Equation Intersection to a Rational Ratio‘R’ Equation

35.  Area #1: The rational equation can be used without
checks. No verification of rainfall intensity or time of
concentration reassessment.
 Area #2: The rational equation is limited only by
the Tc time. Time of concentration verification is
necessary to confirm the total time is less than Tc.
 Area #3: The rational equation is limited by rainfall
intensity. There is an iterative process to verification
rainfall intensity or a Tc reassessment.
These Areas Offer Insight to the Rational Equation’s
Restrictions on Runoff Calculations in the Following:
What are the 3 Areas Under the Curves?

36. Rational ‘c’ Equivalent Values for the
Rational Equation & 5 minute Tc Equation
Where:
C = Runoff Coefficient
s = Average flow path slope (ft./ft.) (decimal)
(with Slope limits from 2% to 12%)
R = Rational Ratio 𝐑 = 𝐪(cfs) / 𝐢 (in./hr.)
(with R limit ratios from 0.2 to 10)
An Equation for a 5 minute Balance ‘c’
𝒄 =
0.82 𝑹
4.8 𝒔 + 𝑹

37. Rational ‘c’ Equivalent Values for the
Rational Equation & 10 minute Tc Equation
An Equation for a 10 minute Balance ‘c’
𝒄 =
0.76 𝑹
13.7 𝒔 + 𝑹Where:
C = Runoff Coefficient
s = Average flow path slope (ft./ft.) (decimal)
(with Slope limits from 2% to 12%)
R = Rational Ratio 𝐑 = 𝐪(cfs) / 𝐢 (in./hr.)
(with R limit ratios from 0.2 to 10)

38. Rational ‘c’ Equivalent Values for the Rational
Equation & a Calculated UTC Tc
A Balance ‘c’ Equation for UTC and Rational Eq.
Where:
C = Runoff Coefficient
S = Average flow path slope (ft./ft.) (decimal)
(with Slope limits from 2% to 12%)
R = Rational Ratio 𝐑 = 𝐪(cfs) / 𝐢 (in./hr.)
(with R limit ratios from 0.2 to 10)
Tc = Time of Concentration (minutes) (5-20 min.)
𝒄 =
)(𝒕 𝒄
−0.12
𝑹
0.5 𝒕 𝒄
1.46 𝒔 + 𝑹

39. Previous Equations are Established for
Green Infrastructure on Small Basins
• The TC comparisons are graphed to the rational
empirical equation that convey runoff with a related
surface rational ‘c’ coefficient.
• These empirical equations provide runoff estimations
for homogenous rainfall intensities and a uniform
watershed surface coefficient.
• Runoff time calculations are estimated for small
drainage basins of 60 acres or less and maintain a
primary pattern of a dominate “surface” flow attribute.

40. Ken Kagy, P.E., CFM, CPSWQ, CPESC (678) 242-2543 ken.kagy@cityofmiltonga.us
 Understand each required variable in the time of concentration equation.
 Understand the limits to each time of concentration equation’s variables.
 Understand the basin’s flow path with its flow type, length, depth, and slope.
 Apply acceptable surface roughness Tc coefficients that correlate to equivalent
hydrology’s surface roughness conditions used to calculate the hydrograph.
Time of Concentration Equations Improve Accuracy with the following:
The End