19.2 Hydraulics Energy and Head Workbook

Build the energy line before calculating

Every hydraulics problem is the energy (Bernoulli) equation between two stations:

z1 + p1/gamma + V1^2/2g + hp = z2 + p2/gamma + V2^2/2g + hL

where z is elevation head, p/gamma is pressure head, V^2/2g is velocity head (with g = 32.2 ft/s2), hp is pump head added, and hL is total head loss. In open-channel flow the same energy idea applies but the free surface, channel slope, and critical-depth control change the setup. The PE WRE exam rewards candidates who recognize the control rather than memorize isolated equations.

Head term checklist

Label station 1 upstream and station 2 downstream, then decide what is known at each. Equal diameters at steady flow let velocity head cancel. A reservoir surface station has negligible velocity (treat V approximately 0). A nozzle or free outfall station has velocity as the central term.

HGL, EGL, and the headloss models

The hydraulic grade line (HGL) is z + p/gamma. The energy grade line (EGL) adds V^2/2g, so EGL always plots above HGL by the velocity head. In a constant-diameter pipe the two lines run parallel; at a pump both jump up; through losses both fall. In open channels the water surface is the HGL, but jumps and transitions need care.

Pressure-pipe headloss

The WRE default is Hazen-Williams (US units): hf = 10.44 L Q^1.85 / (C^1.85 d^4.87), with L and d in ft, Q in gpm, and C the roughness coefficient (C approximately 130-140 for new cast/ductile iron, 150 for PVC). Use Darcy-Weisbach hf = f (L/d)(V^2/2g) only when the friction factor f or a Moody-chart path is given.

Worked example: 1,000 ft of 12-in (d=1.0 ft) ductile iron, C=130, Q=1,400 gpm. hf = 10.44(1000)(1400^1.85)/(130^1.85 x 1.0^4.87). With 1400^1.85 approximately 815,000 and 130^1.85 approximately 7,600, hf approximately 11.2 ft. A magnitude check: a few feet to low tens of feet per 1,000 ft is reasonable for a transmission main.

Open-channel workflow

First classify: normal depth, critical depth, gradually varied flow, hydraulic jump, culvert, weir, or orifice. Manning Q = (1.49/n) A R^(2/3) S^(1/2) handles uniform normal flow, with R = A/P (hydraulic radius) and n approximately 0.013 for concrete, 0.024 for natural channels. Critical depth occurs where specific energy is minimum (Froude number = 1).

Pump power and the brake-horsepower trap

A frequent WRE distractor confuses pump head with pump power. Once you have total dynamic head (TDH), water horsepower is:

WHP = Q (gpm) x TDH (ft) / 3,960

and brake horsepower divides by efficiency: BHP = WHP / eta. Example: 1,400 gpm against a TDH of 120 ft gives WHP = 1,400 x 120 / 3,960 = 42.4 hp; at a wire-to-water efficiency of 0.70, BHP = 42.4/0.70 = 60.6 hp. The 3,960 constant collapses 8.34 lb/gal, 60 min/hr, and 33,000 ft-lb/min per hp. If the question asks for the motor size, the answer is BHP (or the next standard motor up), never WHP -- a classic four-choice trap that lists both.

Mini energy-sketch drill

For every practice problem make a ten-second sketch: upstream station, downstream station, datum, pressure condition, velocity change, pump or control, and the major loss reach. If no energy is added, total energy cannot rise downstream. If the sketch shows a profile climbing through a loss, reject it. This habit is decisive when answer choices mix pressure, head, power, and elevation values that all look numerically plausible.

Reasonableness anchors

Keep target ranges in mind to filter answers. Gravity sewer velocities should be roughly 2-10 ft/s (about 2 ft/s minimum for self-cleansing, 10 ft/s upper bound to limit scour). Force-main and water-main velocities typically run 3-8 ft/s. A negative pressure where the line should stay pressurized signals an error or a real cavitation/air-entrainment concern. A computed Froude number above 1 means supercritical flow (steep, fast, shallow); below 1 means subcritical (mild, deep, slow), and a hydraulic jump transitions from supercritical to subcritical with energy loss.

When a velocity, pressure, or depth lands far outside these bands, re-check the unit basis and the chosen control before committing to an answer choice.

Weirs, orifices, and minor losses

Control structures have their own forms. A rectangular sharp-crested weir follows Q = C L H^(3/2) (with C approximately 3.33 in US units, L crest length, H head over the crest); a V-notch (90-degree) weir follows Q = 2.5 H^(2.5). An orifice follows Q = Cd A sqrt(2gH) with Cd approximately 0.6. Minor losses sum as hL = (sum of K) V^2/2g, with K approximately 0.5 for a sharp entrance, 1.0 for a sudden exit, and roughly 10 for a fully open gate-to-globe range of valves.

When a transmission problem gives both pipe friction and several fittings, add friction and minor losses before applying the energy equation -- omitting a large valve K is a frequent off-by-a-few-feet error that still changes which discrete answer choice is correct.