9.2 Hydrology for Incidental Drainage: The Rational Method

9.2 Hydrology for Incidental Drainage: The Rational Method

The Board's reference list names "Rational Method Hydrological Calculations" as a state-specific exam resource, linking to the Stormwater Calculations chapter of the NCDEQ BMP Manual. That is your signal: the exam expects you to compute a peak discharge, not merely to describe one.

Recall from Section 9.1 that the Guidelines exclude "hydraulics for drainage systems, and hydrology for storm conditions" in general but expressly permit "that level of hydrology and hydraulics necessary in the performance of 'incidental drainage' ... within a subdivision." The Rational Method is the canonical example of that permitted level.

---

1. The Formula

$Q = C \cdot i \cdot A$

The formula is dimensionally "rational" because of an arithmetic coincidence: one acre-inch per hour equals 1.008 cubic feet per second. Because that is within one percent of unity, the conversion constant is dropped and the product C i A is read directly in cfs.

The assumptions you are accepting

The method is simple because it assumes a great deal. Know these, because exam items often test the limits rather than the arithmetic:

1. Rainfall intensity is uniform over the entire drainage area for the duration of the storm.
2. Rainfall intensity is constant in time over that duration.
3. The peak discharge occurs when the whole area is contributing — that is, at a storm duration equal to the time of concentration.
4. The recurrence interval of the peak discharge equals the recurrence interval of the rainfall.
5. The runoff coefficient is constant for a given surface, independent of storm size and antecedent moisture.

Assumption 1 is what makes the method a small-watershed tool: a uniform storm over a few dozen acres is plausible, over several square miles it is not. That is the honest reason large basins fall outside incidental design, and it aligns with exclusion 6 of the Board Guidelines.

---

2. Runoff Coefficients and the Composite C

Published coefficient tables vary by source. The values below are the ranges commonly used in North Carolina municipal and NCDEQ-referenced practice; on the exam, use whatever table the item supplies.

+-----------------------------------------------------------------------------+
|                 TYPICAL RATIONAL RUNOFF COEFFICIENTS (C)                    |
+----------------------------------------------+--------------------+---------+
| Surface / land use                           | Typical range      | Typical |
+----------------------------------------------+--------------------+---------+
| Watertight roofs, concrete and asphalt paving| 0.85 - 0.95        |  0.90   |
| Commercial and business, high density        | 0.70 - 0.90        |  0.80   |
| Industrial and manufacturing                 | 0.60 - 0.85        |  0.70   |
| Residential single-family subdivision lots   | 0.35 - 0.55        |  0.45   |
| Lawns, heavy clay soil, steep slope (>7%)    | 0.25 - 0.40        |  0.30   |
| Lawns, sandy soil, flat slope (<2%)          | 0.10 - 0.15        |  0.12   |
| Forested timberland, woodlots, dense brush   | 0.10 - 0.25        |  0.18   |
+----------------------------------------------+--------------------+---------+

Where a basin has mixed cover, the composite (area-weighted) coefficient applies:

$C_{comp} = \frac{\sum (C_i A_i)}{A_{total}} = \frac{C_1 A_1 + C_2 A_2 + \cdots + C_n A_n}{A_1 + A_2 + \cdots + A_n}$

[!NOTE] A subdivision is almost always a composite. Roofs and pavement, lawns, and any retained woodland each carry different coefficients, and a design that uses a single "residential" C for the whole plat will understate the peak where impervious cover is concentrated.

---

3. Time of Concentration

Time of concentration (Tc) is the travel time for runoff from the hydraulically most distant point in the watershed to the design point. It sets the storm duration, which in turn sets the intensity, which sets Q — so Tc is where most Rational Method errors originate.

Under NRCS TR-55 the total is the sum of three regimes:

$T_c = T_{sheet} + T_{shallow} + T_{channel}$

1. Sheet flow. Shallow flow over plane surfaces, limited under current TR-55 guidance to a maximum length of about 100 feet (the original 1986 method allowed 300 feet). Manning's kinematic solution:

$T_{sheet} = \frac{0.007\,(n L)^{0.8}}{P_2^{0.5}\, S^{0.4}}$

where n is the sheet-flow roughness, L is flow length in feet, P2 is the 2-year 24-hour rainfall in inches, and S is land slope in ft/ft. Tsheet is in hours.

2. Shallow concentrated flow. Beyond the sheet-flow length, runoff concentrates in rills and swales. Velocity is estimated as $V = k\sqrt{S}$ with k about 16.13 for unpaved and 20.32 for paved surfaces, then $T = L / (60V)$ with T in minutes, L in feet, V in ft/s.

3. Channel or pipe flow. Velocity from Manning's equation (Section 9.3), then $T = L / (60V)$.

[!IMPORTANT] Minimum Tc. Because IDF curves rise steeply as duration shortens, a very small Tc produces an unrealistically large intensity. North Carolina local design manuals commonly enforce a minimum Tc of 5 minutes; some use 10 minutes for larger systems. This is a local design-manual convention, not a state rule — check the ordinance governing the project.

---

4. Rainfall Intensity from NOAA Atlas 14

North Carolina design intensity comes from NOAA Atlas 14, Volume 2 precipitation frequency estimates, queried at the project's latitude and longitude. Read the value for a duration equal to Tc at the chosen recurrence interval. Intensity falls as duration rises: the 10-year 5-minute intensity in the Piedmont is on the order of 7 in/hr, while the 10-year 60-minute value is closer to 2.5 in/hr.

Choosing the recurrence interval

Only the first two are state rules; the rest are set by the local ordinance or NCDOT. On an exam item, if a design storm is named as a rule, it is almost certainly the 10-year storm of 15A NCAC 04B .0108.

---

5. Worked Example

+-----------------------------------------------------------------------------+
|            WORKED CALCULATION - RATIONAL PEAK DISCHARGE                     |
|                                                                             |
|   A subdivision catchment in Wake County contains 12.0 acres:               |
|     3.0 ac  impervious asphalt roads and roofs .............. C = 0.90       |
|     7.0 ac  residential lawn on clay soil ................... C = 0.35       |
|     2.0 ac  undisturbed woodland ............................ C = 0.15       |
|   Computed time of concentration ............................ Tc = 15.0 min |
|   NOAA Atlas 14, 10-year, 15-minute intensity ............... i = 4.80 in/hr |
|                                                                             |
|   Step 1  Composite runoff coefficient                                      |
|     C = [(3.0)(0.90) + (7.0)(0.35) + (2.0)(0.15)] / 12.0                    |
|       = [2.70 + 2.45 + 0.30] / 12.0 = 5.45 / 12.0 = 0.454                   |
|                                                                             |
|   Step 2  Peak discharge                                                    |
|     Q10 = C i A = 0.454 x 4.80 in/hr x 12.0 ac = 26.1 cfs                   |
|                                                                             |
|   Step 3  Sanity check                                                      |
|     A fully impervious 12-acre site at the same intensity would give        |
|     0.95 x 4.80 x 12.0 = 54.7 cfs. The computed 26.1 cfs is about 48% of    |
|     that -- consistent with roughly one quarter impervious cover.           |
+-----------------------------------------------------------------------------+

A second habit worth building: carry the pre-development case. Most North Carolina local ordinances require that the post-development peak not exceed the pre-development peak for one or more design storms. Running the same basin at pre-development C and Tc gives the release rate any detention facility must meet — and if that facility turns out to need a pump or a dam, exclusions 4 and 11 of the Board Guidelines put it beyond incidental design.