Mathematical Formula
Critical flow in a prismatic channel is defined by the condition that the Froude number equals one. For a channel of arbitrary cross-section, this is most cleanly written in terms of the discharge Q, the free-surface top width T, the cross-sectional area A, and gravitational acceleration g:
$$ \mathrm{Fr}^2 \;=\; \frac{Q^2\,T}{g\,A^3} \;=\; 1 \qquad\Longleftrightarrow\qquad \frac{Q^2}{g} \;=\; \frac{A^3}{T} $$
Both A and T are functions of the unknown flow depth y, so this is, in general, a nonlinear equation in y. Once the critical depth yc has been found, the critical slope Sc is obtained from Manning's equation rearranged for slope:
$$ S_c \;=\; \left[ \frac{Q\,n}{k\,A\,R^{2/3}} \right]^{2} \qquad R = \frac{A}{P} $$
where k = 1.000 in SI metric (m, m³/s) and k = 1.486 in US customary (ft, ft³/s). Gravitational acceleration g = 9.81 m/s² in SI and g = 32.2 ft/s² in US customary. The same numerical value of n is used in both unit systems by convention — the unit-dependence is absorbed into the factor k.
Area & top-width formulas for each prismatic section
where b is the bottom width, z is the side-slope ratio (horizontal to vertical, i.e. z H : 1 V), D is the pipe diameter, and θ = 2·arccos(1 − 2y/D) is the angle subtended by the water surface at the pipe centre. The wetted perimeter P is, in each case, b + 2y for rectangular, 2y√(1+z²) for triangular, b + 2y√(1+z²) for trapezoidal, and Dθ/2 for circular.
Closed-Form vs. Iterative Computation
Substituting the rectangular A(y) and T(y) into the Froude criterion collapses everything to a simple cube root:
$$ y_c \;=\; \left( \frac{q^2}{g} \right)^{1/3} \qquad\text{where}\quad q = Q/b $$
The triangular case is similar — substituting A = zy² and T = 2zy gives another exact analytical root:
$$ y_c \;=\; \left( \frac{2\,Q^2}{g\,z^{2}} \right)^{1/5} $$
Trapezoidal and circular sections are different. For a trapezoid the substitution produces a quintic in y that mixes by, zy², and their cubes — there is no general algebraic solution. The circular case is worse: the area and top width both involve trigonometric functions of an angle which is itself a transcendental function of y. Engineering tables and nomographs once filled the gap, but on a computer the natural answer is to find the root numerically.
This calculator uses a hybrid bracketing-bisection scheme for both shapes. The algorithm first locates a depth interval [ylo, yhi] over which the function f(y) = Q²·T(y) − g·A(y)³ changes sign, then halves the interval at each step. The loop terminates when the interval width drops below the tolerance (1 × 10⁻⁵) or after 100 iterations — whichever comes first. Bisection is preferred here because the critical-flow function is well-behaved and monotonic on the open-channel branch, and bisection cannot diverge the way Newton-Raphson sometimes does near the singular y → D limit of a circular pipe (where T → 0). For users who prefer Newton-Raphson, the underlying functions are exposed in the source.
One subtlety: for circular pipes, the critical-depth equation has no solution if the required Q exceeds the maximum subcritical capacity of the pipe at full flow — beyond that point, the section pressurises and open-channel mechanics no longer apply. The solver caps the upper bracket at y = 0.99·D and returns a warning if the required Q cannot be matched within the open-channel range.
Manning's n Reference Table
Critical slope is highly sensitive to Manning's roughness coefficient (it varies as n²). Use a value that reflects the as-built or anticipated condition of the channel surface. The table below summarises typical design values from Chow (1959) and the FHWA hydraulic design references for the four most common channel linings.
| Material / Lining |
Design n |
Range |
| Concrete, trowel-finished | 0.013 | 0.011 – 0.015 |
| Concrete, float-finished | 0.015 | 0.013 – 0.017 |
| Concrete, unfinished / formed | 0.017 | 0.014 – 0.020 |
| Concrete, shotcrete / gunite | 0.019 | 0.016 – 0.023 |
| Earth, clean & straight | 0.022 | 0.018 – 0.025 |
| Earth, winding, some weeds | 0.030 | 0.025 – 0.033 |
| Earth, dense weeds / dense aquatic plants | 0.035 | 0.030 – 0.040 |
| Earth, dragline-excavated, with growth | 0.040 | 0.035 – 0.050 |
| Gravel, fine, smooth bed | 0.025 | 0.022 – 0.030 |
| Gravel / rubble, coarse | 0.030 | 0.025 – 0.035 |
| Cobble bed, natural stream | 0.035 | 0.030 – 0.050 |
| Boulder bed, mountain stream | 0.050 | 0.040 – 0.070 |
| CMP, 2⅔ × ½ in annular corrugations | 0.024 | 0.022 – 0.027 |
| CMP, 3 × 1 in annular corrugations | 0.027 | 0.025 – 0.030 |
| CMP, helical corrugations (paved invert) | 0.015 | 0.013 – 0.018 |
| CMP, 6 × 2 in structural-plate | 0.034 | 0.030 – 0.037 |
Note that for natural channels the value of n varies with stage, season, and sediment load — 20–40 % variation is common. Where a nearby stream gauge is available, back-calculating n from a known discharge gives a far more reliable design value than any handbook table.