Manning's equation on the PE Civil WRE exam

Manning's equation is the workhorse of open-channel hydraulics on the Water Resources & Environmental exam. Ditch capacity, storm-sewer sizing, normal depth for a trapezoid, a flooded channel with overbanks: most of them reduce to one relation and careful geometry. It sits in the Handbook's Steady Uniform Flow section, next to the table of approximate Manning's roughness coefficients.

What you actually need to own

• The equation, both unit systems: Q = (k/n) · A · R^(2/3) · S^(1/2), with k = 1.486 in USCS (often rounded to 1.49) and k = 1.0 in SI. Divide by A for velocity. The same n value works in both systems. Only k changes.
• Hydraulic radius R = A/P. P is the wetted perimeter: the boundary in contact with water. It stops at the free surface. Know A, P and top width for rectangles, trapezoids, triangles and circles without hesitating. The Handbook's Flow in Channels table lists them, but you'll move faster if you already know them.
• S is the slope of the energy grade line. In uniform flow it equals the bed slope, which is why a problem hands you a channel slope and expects you to use it directly.
• Choosing n. Read it from the table by material and condition, such as concrete, corrugated metal or a weedy natural stream. When the problem gives an n value, use it.
• Normal depth by trial. Capacity at a known depth is a direct substitution. Depth at a known flow is not. Rearrange to A·R^(2/3) = Qn / (k·√S), then iterate on y until the section factor on the left matches. Two or three well-chosen trials, or your calculator's solver, is enough.
• Compound sections. Split a main channel with overbanks into subsections at vertical lines. Give each its own A, P and n, compute each flow, and add them. The dividing lines aren't counted in wetted perimeter.
• Partially full circular pipes. Use the Handbook's hydraulic-elements graph for circular pipe, which gives the ratios of A, R, v and Q to their full-pipe values against d/D. Two facts are worth knowing cold. Half-full and full pipes have the same R (D/4), so with constant n they have the same velocity. And with constant n, flow peaks a little below full depth, at about 94% of D.

Where people lose points

• Wrong wetted perimeter. Including the free surface, or dropping an interior wall in a multi-cell channel, changes R. The shortcut R ≈ y is only for wide channels and fails on a narrow section.
• R = D/2 for a full pipe. It's D/4.
• Mixing constants and units. Using 1.486 with SI dimensions, or 1.0 with feet, gives an answer off by that factor. Put the unit system in the margin before you substitute.
• Averaging n across a compound section instead of splitting it into subsections.
• Reading the wrong curve on the partial-flow graph. The graph separates constant-n curves from n-varying-with-depth curves, so check which one the problem intends.
• Solving for the wrong depth. Normal depth comes from Manning. Critical depth doesn't involve n or slope. Comparing the two tells you whether the channel is mild or steep, which is the subject of critical depth and specific energy.

How to study it

Drill the geometry until A, P and R for a trapezoid take seconds. Then run the same channel both ways: capacity at a given depth, then depth at a given flow, so the trial loop becomes routine. Add compound sections and part-full pipes after that. Manning also feeds the friction term in culvert outlet control, and the design flows usually come from the rational method or curve number.

Part of the Hydraulics–Open Channel area of the PE Civil WRE exam. → Start practicing free — the whole practice bank, free with an account.