Partial Pipe Flow Calculator finds discharge with Q = A × (k/n) × R²⁄³ × √S from pipe diameter, flow depth, slope, and Manning n for storm, sewer, culvert, and drainage pipe checks.
Gravity flow in circular drainage conduits rarely fills the entire cross-section, so a Partial Pipe Flow Calculator determines velocity, discharge, and boundary shear at any depth below the crown.
Manning’s equation for open-channel flow governs the hydraulics, with wetted geometry computed from the central angle subtended by the water surface. Correct interpretation of the results separates a stable, self-cleaning design from one that risks sedimentation or erosive velocities.
Hydraulic Principles for Partially Full Circular Pipes
Uniform flow in a partially filled circular section obeys the same energy balance as any open channel. The driving force is the downslope component of gravity, resisted by boundary friction distributed over the wetted perimeter.
Unlike a rectangular channel, both the flow area and the hydraulic radius vary nonlinearly with depth, producing a unique stage-discharge curve that peaks before the pipe runs full.
The standard steady-state relationship is Manning’s equation.
Q = (k / n) x A x R^(2/3) x S^(1/2)
Q is the volumetric flow rate, k is a unit-dependent constant, n is Manning’s roughness coefficient, A is the cross-sectional flow area, R is the hydraulic radius, and S is the energy slope taken as the pipe slope for uniform flow.
For imperial units where length is in feet, k equals 1.486. For SI units where length is in meters, k equals 1.0. This factor converts the original metric Manning formula to the foot-second system.
Hydraulic radius R equals flow area divided by wetted perimeter. Wetted perimeter is the arc length of the pipe wall in contact with the fluid. As depth rises from zero to full, R increases, peaks near 0.81 of diameter, then drops slightly when the pipe runs full because the water surface contacts the crown and adds no extra perimeter. Velocity follows a similar trend, so maximum discharge occurs at roughly 94 percent of full depth, not at the full condition.
Critical Inputs for a Partial Pipe Flow Calculator
The required parameters are internal pipe diameter, flow depth, longitudinal slope, and Manning’s n. Diameter and depth must share the same linear unit, while slope is entered as a percentage.
Internally, the slope is converted to a decimal fraction for calculation. The code expects strictly positive values and halts with an alert if depth equals or exceeds diameter, because that condition violates the partial-pipe assumption.
Diameter and depth accept inch or millimeter units. Selecting millimeters switches the internal constant k to 1.0 and converts all lengths to meters. Imperial mode with inches converts lengths to feet. The Manning’s n field can be typed directly or populated from a material dropdown that assigns standard roughness values.
Synchronization between the dropdown and manual input keeps both in agreement; if a user types a value not listed, the dropdown clears, but calculation proceeds with the manual entry.
Formula Derivation and Geometric Relationships
The wetted cross-section geometry derives entirely from the pipe radius r and flow depth y. Defining a dimensionless ratio reduces the problem to a single trigonometric expression.
ratio_inner = 1 – (y / r)
Central angle theta (radians) = 2 x acos(ratio_inner), with the argument constrained between –1 and 1.
Area A = (r^2 / 2) x (theta – sin theta)
Wetted perimeter P = r x theta
Hydraulic radius Rh = A / P
Top width T = 2 x r x sin(theta / 2)
Hydraulic depth D_h = A / T
Manning’s equation then yields mean velocity V = (k / n) x Rh^(2/3) x S^(1/2), and discharge Q = A x V. Because area and hydraulic radius are exact for a circular segment, no empirical discharge coefficient is needed.
Shear stress at the boundary is computed as gamma x Rh x S, where gamma is the specific weight of water: 62.4 lb/ft³ for imperial or 9810 N/m³ for SI. Velocity head V²/(2g) uses g = 32.2 ft/s² or 9.81 m/s².
The Froude number Fr = V / sqrt(g x D_h) classifies the flow as subcritical (Fr < 0.98), critical, or supercritical (Fr > 1.02). A result near unity warrants caution because standing waves and unstable surface profiles can develop.
Full-pipe capacity Q_full is calculated with the same Manning formula using full area π r² and full wetted perimeter 2π r. Comparing partial flow Q to Q_full yields the percent capacity utilization, a practical indicator of how close the pipe is to surcharging.
Worked Example: Half-Full Pipe in Imperial Units
A 24-inch diameter concrete pipe carries flow at a depth of 12 inches on a 1.0 percent slope. Manning’s n is 0.013. Convert diameter and depth to feet: D = 2.0 ft, y = 1.0 ft, radius r = 1.0 ft.
ratio_inner = 1 – (1.0 / 1.0) = 0.
theta = 2 x acos(0) = 2 x (π/2) = π = 3.1416 rad.
Area A = (1.0² / 2) x (3.1416 – sin 3.1416) = 0.5 x (3.1416 – 0) = 1.5708 ft².
Wetted perimeter P = 1.0 x 3.1416 = 3.1416 ft.
Hydraulic radius Rh = 1.5708 / 3.1416 = 0.5000 ft.
S^(1/2) = sqrt(0.01) = 0.1000. Rh^(2/3) = 0.50^(2/3) = 0.62996.
Velocity V = (1.486 / 0.013) x 0.62996 x 0.1000 = 114.308 x 0.62996 x 0.1000 = 7.20 ft/s.
Discharge Q = 1.5708 x 7.20 = 11.31 ft³/s.
Full pipe area A_full = π x 1.0² = 3.1416 ft², Rh_full = A_full / (π x 2.0) = 0.50 ft. Velocity is identical because Rh does not change, so V_full = 7.20 ft/s and Q_full = 22.62 ft³/s. Capacity used is exactly 50 percent.
Hydraulic depth D_h = A / T. Top width T = 2 x 1.0 x sin(π/2) = 2.0 ft. D_h = 1.5708 / 2.0 = 0.7854 ft. Froude number Fr = 7.20 / sqrt(32.2 x 0.7854) = 7.20 / 5.029 = 1.43, indicating supercritical flow.
Shear stress = 62.4 x 0.50 x 0.01 = 0.312 lb/ft². Velocity head = 7.20² / (2 x 32.2) = 51.84 / 64.4 = 0.805 ft.
Metric Example: Identical Proportional Depth
For a 600 mm concrete pipe at 300 mm depth with the same 1.0 percent slope and n = 0.013, the geometry scales directly.
D = 0.600 m, y = 0.300 m, r = 0.300 m. ratio_inner = 0, theta = π rad. A = (0.30² / 2) x π = 0.14137 m². P = 0.30 x π = 0.94248 m. Rh = 0.14137 / 0.94248 = 0.1500 m. Rh^(2/3) = 0.1500^(2/3) = 0.28231. k = 1.0. V = (1.0 / 0.013) x 0.28231 x 0.10 = 2.17 m/s. Q = 0.14137 x 2.17 = 0.3068 m³/s.
T = 2 x 0.30 x sin(π/2) = 0.600 m. D_h = 0.14137 / 0.60 = 0.2356 m. g = 9.81 m/s², sqrt(g x D_h) = 1.520 m/s. Fr = 2.17 / 1.520 = 1.43. Shear = 9810 x 0.15 x 0.01 = 14.72 Pa. Full pipe area = π x 0.30² = 0.28274 m², Q_full = 0.6136 m³/s, capacity used 50 percent. The Froude number matches the imperial result, confirming consistent physics across unit systems.
Manning’s n and Material Selection Decisions
Choice of roughness coefficient directly scales velocity and discharge inversely with n. A 30 percent reduction in n yields roughly a 30 percent increase in capacity for the same depth and slope. Designers therefore face a real material selection decision where long-term surface condition matters as much as the new-pipe specification.
Concrete pipe typically carries an n between 0.011 and 0.013, per FHWA HDS-5 and ASTM C14. Smooth-wall plastic pipe such as PVC or HDPE falls lower, commonly 0.009 to 0.010.
Corrugated metal pipe, depending on corrugation depth, ranges from 0.022 to 0.024. In rehabilitation design, a cured-in-place liner may drop n from 0.013 to 0.010, boosting capacity without changing diameter.
However, a lower n also elevates velocity and Froude number, which may push a previously subcritical reach into supercritical flow and require energy dissipation at outlets.
Slope choice interacts with roughness through self-cleaning criteria. Many municipal standards require a minimum velocity of 2.0 ft/s (0.6 m/s) at peak flow to prevent solids deposition.
For a 12-inch pipe flowing one-third full, achieving that velocity often demands a slope near 0.5 to 1.0 percent with n=0.013. If the available cover limits slope, switching to a lower-n material may be the only way to satisfy the velocity minimum without increasing diameter.
Pipe diameter selection also hinges on partial-flow analysis. A designer may size a storm sewer to run no more than 80 percent full for the design storm, leaving freeboard to account for air entrainment and transient surging. The calculator’s percent capacity output translates that criterion into a measurable margin between design Q and the full-pipe Q_full.
When capacity exceeds roughly 90 percent, a slight increase in roughness from aging or debris can force the pipe into pressurized flow, altering upstream hydraulics.
Because the shear stress result quantifies the tractive force available to move sediment, it guides material selection where abrasion is a concern. Concrete pipes in high-velocity sanitary sewers may face hydrogen-sulfide corrosion; specifying a sacrificial thickness or a protective liner addresses that observed deterioration.
None of these decisions emerge from a single calculation, but the numerical breakdown of area, velocity, shear, and Froude number gives engineers the specific numbers needed to weigh alternatives against code minimums and whole-life cost.