All tools

Manning's Equation for 24-inch Concrete Pipe — Partial-Flow Worked Example

A municipal storm-sewer capacity check on a new 24-inch reinforced concrete pipe (RCP) reach. The upstream Rational Method hydrograph gives a design peak of 6.8 cfs under the 10-year event. Because the pipe will not run full at that flow, we cannot use full-section tables — we need partial-flow geometry (central angle θ, wetted area A, wetted perimeter P, and hydraulic radius R) and iterate on depth ratio y/D until Manning's equation closes. Sources: FHWA HDS-5 (culvert and storm-drain hydraulics), Chow Open-Channel Hydraulics (1959), and standard circular-conduit geometry as summarized in King & Brater.

The reach

A 380-ft run of 24-inch RCP on a 0.30% slope ties a new subdivision trunk into an existing 30-inch main. The city requires minimum self-cleansing velocity of 3 ft/s and a maximum of 10 ft/s at design flow (typical for sanitary/storm combined guidance; storm-only sewers often cite 2–3 ft/s minimum per local standard). As-built survey and design drawings give:

Pipe	24-inch RCP, inside diameter D = 2.0 ft
Manning's n	0.012 (new concrete pipe; see n reference)
Slope S	0.30% = 0.003 ft/ft
Design discharge Q	6.8 cfs (10-yr peak from upstream basin)
Flow regime	Subcritical, uniform, steady (normal depth check)
Velocity limits	3–10 ft/s at design Q

Step 1 · Full-flow benchmark

Capacity ceiling before partial-flow iteration

At full depth (y/D = 1.0), the pipe is a complete circle. This sets the upper bound for any trial depth.

Full section (radius r = D/2 = 1.0 ft):
  A_full = πr² = π(1.0)² = 3.142 ft²
  P_full = πD = 2π = 6.283 ft
  R_full = A/P = 3.142/6.283 = 0.500 ft

Manning's (English units, K = 1.486):
  Q_full = (K/n) · A · R^2/3 · S^1/2
  Q_full = (1.486/0.012) · 3.142 · 0.500^2/3 · 0.003^1/2
  Q_full = 123.8 · 3.142 · 0.630 · 0.0548 = 13.4 cfs

Design Q = 6.8 cfs → Q/Q_full = 0.51 — expect y/D slightly above 0.50 (Chow Fig. 5-3).

Run the same numbers in the Manning's equation tool with D = 24 in, n = 0.012, S = 0.30%, full-flow mode to confirm 13–14 cfs.

Step 2 · Partial-flow geometry from y/D

Circular conduit: θ, A, P, R

For depth y measured from the invert, define the depth ratio y/D. The central angle θ (radians) subtended by the water surface is (Chow Eq. 5-11; HDS-5 circular geometry):

θ = 2 · arccos(1 − 2y/D)
A = (D²/8) · (θ − sin θ)
P = (D/2) · θ = r · θ
R = A/P

We will evaluate these at each trial y/D during iteration. First, work one trial explicitly at y/D = 0.57 (a depth many designers use as a “upper normal” check before surcharging):

Trial at y/D = 0.57 (y = 1.14 ft):
  1 − 2(0.57) = −0.14 → arccos(−0.14) = 1.711 rad
  θ = 2 · 1.711 = 3.422 rad (196°)
  sin θ = sin(196°) = −0.277
  A = (4/8) · (3.422 − (−0.277)) = 0.5 · 3.699 = 1.850 ft²
  P = 1.0 · 3.422 = 3.422 ft
  R = 1.850/3.422 = 0.541 ft

Q = 123.8 · 1.850 · 0.541^2/3 · 0.0548 = 8.1 cfs — too high for design 6.8 cfs

Do not assume y/D = 0.57 carries 6.8 cfs on this slope. At 0.30% and n = 0.012, a depth ratio of 0.57 delivers roughly 8 cfs — about 19% over the stated design peak. This is the most common partial-flow mistake in plan review: using a target depth from a chart without re-running Manning's for the actual n, S, and D.

Step 3 · Iterate y/D to match Q = 6.8 cfs

Bisection on depth ratio

Bracket the solution: y/D = 0.45 gives ~5.6 cfs (low); 0.57 gives 8.1 cfs (high). Refine:

Trial y/D	θ (rad)	A (ft²)	R (ft)	Q (cfs)	Bracket
0.45	2.941	1.372	0.466	5.58	Low
0.50	3.142	1.571	0.500	6.73	Low
0.52	3.204	1.631	0.509	7.03	High
0.51	3.162	1.601	0.506	6.88	High (close)
0.505	3.152	1.586	0.503	6.80	Match

Converged normal depth: y/D ≈ 0.505 (depth y ≈ 1.01 ft below crown soffit clearance ≈ 0.99 ft)

At y/D = 0.505:
θ = 3.152 rad, A = 1.586 ft², R = 0.503 ft
Q = 123.8 · 1.586 · 0.503^2/3 · 0.0548 = 6.80 cfs ≈ 6.8 cfs ✓

For reference, if the pipe were flowing at y/D = 0.57 (8.1 cfs), that would represent roughly a 19% capacity margin above the 10-yr peak — acceptable if the reviewer accepts sustained depths that high, but it is not the normal depth for 6.8 cfs on this slope.

Step 4 · Velocity check

Self-cleansing range 3–10 ft/s

Velocity is mean cross-sectional velocity V = Q/A (or directly from Manning's with V = (K/n)R^2/3S^1/2). Check at the converged depth and at the y/D = 0.57 trial for comparison:

At design normal depth (y/D = 0.505):
  V = Q/A = 6.8/1.586 = 4.29 ft/s
  Manning's check: V = 123.8 · 0.503^2/3 · 0.0548 = 4.28 ft/s ✓

At y/D = 0.57 (8.1 cfs scenario):
  V = 8.1/1.850 = 4.38 ft/s — still within 3–10 ft/s

Result: 4.3 ft/s > 3 ft/s minimum — adequate for grit and organic solids transport per typical municipal criteria. Well below 10 ft/s — no liner scour concern in RCP.

Maximum velocity is not at maximum depth. In a circular pipe, V peaks near y/D ≈ 0.81 (Chow). At design normal depth (~0.50) you are on the rising limb of the velocity curve — a good place for self-cleansing without excessive head loss. Do not confuse “half full” (maximum Q efficiency myth) with “half depth” (where y/D = 0.50).

Step 5 · What we did NOT check, and when you would

Beyond uniform-flow Manning's

No backwater / HGL routing. This is a uniform-flow normal-depth check on one reach. If the 30-inch main tailwater backs up during the 10-yr event, normal depth is invalid — you need a backwater curve (standard step method) from downstream control.
No surcharge at inlets. If a sag inlet ponds above the crown, pressure flow begins and Manning's open-channel form understates capacity. HDS-5 pressure-flow charts apply.
No 25-yr or 100-yr verification. At 2× design Q (~13.6 cfs), the pipe runs essentially full (13.4 cfs capacity). The 25-yr peak should be routed to confirm brief surcharge is acceptable.
No construction tolerance. Out-of-roundness, joint offsets, and biofilm can push effective n to 0.013–0.014 within 5–10 years. Re-run at n = 0.013: Q at y/D = 0.505 drops to ~6.3 cfs — still passes, but margin thins.

For a full storm-sewer network with automatic HGL and multi-pipe sizing, see HydroComplete.

Tools used in this example

Reproduce every step in the live PE-Calc tools: Manning's equation (partial-flow / circular pipe mode). Manning's n came from the Manning's n reference card — “concrete pipe” at 0.012. For the upstream 6.8 cfs design peak, see Rational Method or NRCS curve number depending on basin size.

Buying the lot? Check the drainage before you buy.

Storm-sewer capacity is only one line on a subdivision feasibility review. Easements, floodplain limits, and downstream main capacity all affect whether a 24-inch tie-in is even permitted. SitePrior screens FEMA, NWI, NRCS soils, and USGS context for $29 in about 60 seconds. Run it before the pre-application meeting, not after the easement is recorded.