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Storm Sewer Design with the Rational Method and Manning's Equation

A 2-PDH course on designing a storm sewer trunk line: Rational method peak flow, composite runoff coefficients, time of concentration by the NRCS segmental method, accumulating flow down the line, Manning pipe sizing, partial-flow behavior and the checks a reviewer looks for. Six worked examples, graded quiz, certificate.

Learning objectives

• Compute peak flow with the Rational method, including a composite runoff coefficient and intensity from an IDF curve.
• Estimate time of concentration with the NRCS (TR-55) sheet, shallow concentrated and channel flow segments.
• Accumulate C×A and time of concentration down a trunk line and explain the partial-area pitfall.
• Size a circular pipe with Manning's equation, select a standard size, and check velocity.
• Use partial-flow relationships for circular pipes to find flow depth and velocity below full flow.

1. Scope: what a storm sewer design has to show

A storm sewer collects runoff from inlets and carries it in closed conduits to an outfall. The design problem for each pipe is the same: estimate the peak flow that reaches it in the design storm, then choose a size and slope that carry that flow without surcharging, at a velocity high enough to keep the pipe clean and low enough not to damage it. The designer works from the most upstream inlet down to the outfall, because every pipe's flow depends on everything above it.

This course covers the method used for most small urban systems in the United States: the Rational method for peak flow and Manning's equation for pipe capacity. It does not cover inlet spacing and gutter spread, energy losses at structures, or the hydraulic grade line analysis that should follow the sizing. Those are the subject of a later course, and they matter: a system sized correctly pipe by pipe can still surcharge if junction losses or a high tailwater are ignored.

Before the first calculation, find out the governing criteria. A drainage criteria manual (municipal, county or state DOT) normally fixes: the design storm frequency for storm sewers (commonly 10 years, sometimes 2 to 25 years depending on road class and jurisdiction), the IDF source, the minimum time of concentration, runoff coefficients, Manning n by pipe material, the minimum pipe size, minimum and maximum velocities, and minimum cover. Where no local manual applies, FHWA HEC-22, Urban Drainage Design Manual, is the usual reference and is the basis for most of the practice described here.

2. The Rational method

The Rational method estimates the peak runoff rate from a small drainage area:

Q = C · i · A

where Q is the peak flow (cfs), C the dimensionless runoff coefficient, i the rainfall intensity (in/hr) for a duration equal to the time of concentration, and A the drainage area (acres). The unit conversion from acre·in/hr to cfs is 1.008, close enough to 1 that it is ignored.

The method rests on assumptions that define where it can be used:

• The peak occurs when the whole area contributes. That happens when the storm lasts at least as long as the time of concentration, Tc, the travel time from the hydraulically most distant point to the design point. The design storm duration is therefore set equal to Tc.
• Rainfall intensity is uniform over the area and over the duration. This is reasonable only for small areas. HEC-22 recommends the Rational method for drainage areas up to about 200 acres; many local manuals set a lower limit, and some as low as 20 acres.
• The return period of the peak flow equals that of the rainfall. Accepted for design, though it is not strictly true.
• C is constant for the storm. In reality, losses are a larger share of a small storm than of a large one. Some manuals therefore multiply C by a frequency factor for less frequent storms (for example 1.1 for 25 years, 1.2 for 50 years and 1.25 for 100 years, with the product limited to 1.0). Use it only if your criteria require it.

2.1 Runoff coefficients

C is the fraction of rainfall that appears as peak runoff. Typical design values are about 0.70 to 0.95 for asphalt and concrete pavement and 0.75 to 0.95 for roofs; lawns range from about 0.05 to 0.10 (flat, sandy soil) up to 0.25 to 0.35 (steep, heavy soil). Your criteria manual's table governs. Where a drainage area has several surfaces, use an area-weighted composite C:

C_w = Σ(C_j · A_j) / Σ A_j

2.2 Rainfall intensity

Intensity comes from an intensity–duration–frequency (IDF) relationship for the site, read at a duration equal to Tc. In the United States the source is normally NOAA Atlas 14 (the Precipitation Frequency Data Server), which gives depths for durations from 5 minutes to 60 days; dividing a depth by its duration gives intensity. For a spreadsheet it is convenient to fit the IDF values for the design frequency to an equation such as

i = a / (Tc + b)^c

with Tc in minutes. The coefficients a, b and c are found by fitting the published points (many agencies publish them for their region). In this course we use a = 70, b = 12, c = 0.8, which is roughly the shape of a 10-year curve in the southeastern United States: i = 7.26 in/hr at 5 minutes, 5.01 at 15 minutes, 2.29 at 60 minutes. These are illustrative values only: always use the IDF data for your site.

3. Time of concentration

Tc controls the intensity, and the intensity curve is steep at short durations, so Tc matters: in the curve above, cutting Tc from 15 to 10 minutes raises i by 18%. The NRCS segmental method in TR-55 (Urban Hydrology for Small Watersheds) divides the longest flow path into three kinds of segment and adds their travel times.

3.1 Sheet flow

Shallow, unconcentrated flow over a plane surface at the top of the flow path, computed with the kinematic-wave approximation:

T_t = 0.007 (n L)^0.8 / (P₂^0.5 s^0.4)

with T_t in hours, n the sheet-flow Manning coefficient (TR-55 Table 3-1: 0.011 for smooth surfaces such as pavement, 0.15 for short-grass prairie, 0.24 for dense grasses, 0.40 for light woods underbrush), L the flow length in feet, P₂ the 2-year, 24-hour rainfall depth in inches, and s the slope in ft/ft. TR-55 limits sheet flow to 300 ft; many agencies now cap it at 100 ft, since flow concentrates well before that on most real ground.

3.2 Shallow concentrated flow

After sheet flow, water collects in small rills and swales. TR-55 gives the average velocity as a function of slope:

Unpaved: V = 16.1345 √s     Paved: V = 20.3282 √s   (V in ft/s)

3.3 Channel, gutter and pipe flow

Once flow is in a defined channel, gutter or pipe, its velocity comes from Manning's equation for that conveyance, and T_t = L / V. For pipes already designed upstream, use the pipe velocity found in the design.

Most manuals set a minimum Tc, commonly 5 minutes, because the IDF data and the method's assumptions break down for very short durations. If the computed Tc for a paved inlet area is 2 minutes, use the minimum.

4. Manning's equation for circular pipes

For uniform flow,

Q = (1.486 / n) · A · R^2/3 · S^1/2

with A the flow area (ft²), R = A/P the hydraulic radius (ft), S the slope of the energy grade line, taken equal to the pipe slope for design, and n the Manning roughness (about 0.012 to 0.013 for concrete and smooth-interior plastic pipe, about 0.024 for corrugated metal with 2⅔ × ½-in corrugations; your criteria manual governs). For a circular pipe flowing full, A = πD²/4 and R = D/4, which gives

Q_full = (0.4632 / n) · D^8/3 · S^1/2   and so   D_req = [Q n / (0.4632 S^1/2)]^3/8

with D in feet. Design practice is to compute D_req and select the next larger standard size, never a smaller one; the next standard size up is also checked against the agency's minimum diameter (often 15 in for storm mains).

4.1 Partially full pipes

A pipe carrying less than its full capacity flows partly full, and its depth and velocity follow from the geometry of the circular segment. With a constant n, the ratios to full-flow values depend only on the depth ratio d/D:

Three facts in this table are worth remembering. At half depth, the pipe carries half its full flow at exactly the full-flow velocity. The velocity peaks at about 1.14 × V_full near d/D = 0.81, and the discharge peaks at about 1.076 × Q_full near d/D = 0.94: a pipe running just below full carries slightly more than it does full, because the last bit of depth adds wetted perimeter faster than area. Designers nevertheless size to Q_full, since the extra few percent is unstable and disappears with any surcharge. (In real pipes n increases somewhat at low depth, so measured velocities at small d/D are a little lower than the constant-n values.)

4.2 Velocity limits and slopes

Two velocity checks follow sizing. The minimum keeps sediment moving: HEC-22 recommends a full-flow velocity of about 3 ft/s, and many manuals accept 2.5 ft/s. On flat sites this sets a minimum slope for each size: from Manning, S_min = [V_min n / (1.486 (D/4)^2/3)]². The maximum, often 10 to 15 ft/s, limits abrasion of the pipe invert and scour at the outlet; steep runs may need drop structures or energy dissipation at the outfall.

5. Working down the trunk line

A trunk line is designed run by run, from upstream to downstream, in a tabular form that most agencies expect to see in the drainage report. For each run:

1. Add the C·A of the area entering at the run's upstream inlet to the total C·A from upstream.
2. Find the Tc for the run: the larger of the inlet Tc of the area entering there and the Tc of the upstream run plus the travel time in the upstream pipe.
3. Read i at that Tc and compute Q = i × ΣC·A.
4. Size the pipe for Q, then compute its full-flow velocity and the travel time L/V for use in the next run.

Note step 3: the intensity is applied to the accumulated C·A, not to each sub-area separately. Adding the peak flows of the sub-areas would overstate the flow, because their peaks do not arrive at the same time; the method instead assumes every contributing area produces runoff at the intensity for the longest Tc.

5.1 The partial-area check

Example 6 needs one more check. At the outlet, Tc = 16.0 min comes from the large, slow lawn area. But the paved upstream areas reach the outlet sooner, and at their shorter Tc the intensity is higher. If the upstream areas alone, at their own Tc of 12.8 min, produce more flow than the whole area at the longer Tc, the "whole area" answer is not the peak. Here the upstream C·A of 3.04 ac at i = 5.36 in/hr gives 16.3 cfs, less than the 23.1 cfs of the whole area, so the whole-area flow governs. Whenever a large, slow, pervious area joins near the outlet of a fast, impervious system, check the partial area.

6. Checking the hydraulic grade line

Sizing each pipe for uniform flow assumes that every pipe can run at its own normal depth, with nothing downstream holding the water up. That is true only if the outfall is free. When the receiving channel, pond or tide is high, or when losses at structures are large, the water surface in the system, the hydraulic grade line (HGL), rises above the pipe crowns and the pipes flow under pressure. The design is acceptable only if the HGL stays far enough below the inlet grates and manhole rims; a common criterion is that it stay at least 1 ft below the rim or the gutter flow line, though your manual governs.

The check starts at the outfall and works upstream, the opposite direction from sizing:

1. Start at the tailwater. At the outlet, the HGL is the higher of the tailwater elevation and the pipe's critical-depth (or normal-depth) water surface. A tailwater above the outlet crown means the last pipe is surcharged.
2. Add friction loss along each surcharged pipe. In a pipe flowing full, the friction slope is found by solving Manning's equation for S with the full-pipe area and hydraulic radius: S_f = [Q n / (1.486 A R^2/3)]². The HGL rises by S_f × L from the downstream end to the upstream end.
3. Add the loss at each structure as a coefficient times the velocity head of the outgoing pipe, h_j = K V²/2g. K depends on the structure: roughly 0.1 to 0.2 for a straight-through manhole with a shaped invert, rising to about 1 or more for a 90-degree bend, a junction with several inflows, or an inlet where flow plunges in from above. HEC-22 gives a detailed method that builds K from the structure geometry; many local manuals tabulate simplified values.
4. Compare with the rim. Where the HGL in a structure comes within the required margin of the rim or grate, the system needs larger pipes, smaller losses (better-shaped inverts, fewer bends) or a lower tailwater condition.

Where the HGL computed this way falls below a pipe's crown, the pipe is not surcharged, and the HGL in it is set by its own normal depth instead; the upstream calculation then continues from that water surface.

7. Layout checks a reviewer will make

• Pipe size never decreases downstream, even if a steeper downstream run could carry the flow in a smaller pipe. A smaller pipe downstream is a debris trap.
• Crowns (or the 0.8D points) are matched at junctions, rather than inverts, so the larger downstream pipe does not back water into the smaller one. A minimum drop across each structure, often 0.1 ft, accounts for some of the structure loss.
• Minimum cover is met under traffic (often 2 ft, or the pipe manufacturer's requirement for the loading).
• The outfall condition is known. Tailwater in the receiving channel, a downstream pond or a tide can surcharge the whole system. Size by the method above, then check the hydraulic grade line from the outfall upstream, including losses at junctions and bends, and keep it below the inlet grates by the margin your criteria require.
• Bypass and overflow paths exist for storms larger than the design storm: water that cannot enter the system must have a safe surface route.

8. Common mistakes

• Using the time of concentration of each sub-area to compute its own peak and adding the peaks.
• Forgetting the 2 in "2g" or the 1.486 in Manning's equation, or mixing inches and feet in D.
• Rounding the required diameter to the nearest standard size instead of up.
• Using a sheet-flow length of several hundred feet on a site where the flow plainly concentrates in a swale.
• Taking the intensity from the wrong duration, for example the 24-hour depth divided by 24, for a 15-minute Tc.
• Skipping the partial-area check where a large pervious area enters near the outlet.
• Stopping at pipe capacity without checking the hydraulic grade line against the outfall tailwater.

9. Summary

Storm sewer design by the Rational method is a disciplined bookkeeping exercise: accumulate C·A, carry the longest time of concentration forward with pipe travel time, read the intensity at that time, and size each pipe with Manning's equation to the next standard size up. The method is simple; the judgment lies in the inputs (runoff coefficients that match the drainage pattern, a realistic flow path for Tc, the right IDF data) and in the checks that follow (velocity, partial areas, and the hydraulic grade line).

References: FHWA HEC-22, Urban Drainage Design Manual, 3rd ed. (2009, rev. 2013), Ch. 3 and 7; USDA NRCS TR-55, Urban Hydrology for Small Watersheds (1986), Ch. 3; ASCE/WEF MOP 77, Design and Construction of Urban Stormwater Management Systems (1992); NOAA Atlas 14 Precipitation Frequency Data Server.

Final quiz

22 questions. You need 70% to pass. You can retake it as many times as you need.