Water Resources and Environmental
This area covers moving and managing water: closed-conduit and open-channel hydraulics, hydrology and runoff, and the unit processes of water and wastewater treatment. The PE tests continuity and energy, Manning's and the Darcy-Weisbach equations, the Rational Method, and mass-balance treatment calculations. The recurring theme is conservation — of mass and of energy.
Fluid Properties and Hydrostatics
Start with the fluid itself. Water's specific weight is γ = 9.81 kN/m^3 (62.4 lb/ft^3) and its density is 1,000 kg/m^3; kinematic viscosity ν = μ/ρ, so with μ = 1×10^-3 Pa·s, ν = 1×10^-6 m^2/s. Hydrostatic (gauge) pressure grows with depth: p = γ·h, so 8 m of water gives about 78.5 kPa. On a submerged vertical surface the resultant force is F = γ·h_c·A, where h_c is the depth to the centroid; that resultant acts at the center of pressure, which for a vertical rectangle with its top at the surface sits at two-thirds of the depth (one-third of the height above the base). Buoyancy is Archimedes' principle: the upward force equals the weight of displaced fluid, F_b = γ·V_displaced.
Continuity and the Energy Equation
Conservation of mass for steady incompressible flow gives continuity: Q = V·A is constant, so a pipe that narrows speeds the flow (V_1·A_1 = V_2·A_2). When a 0.30 m pipe reduces to 0.15 m, the area drops by four and the velocity quadruples. Conservation of energy is Bernoulli's equation, expressed in heads: p/γ + V^2/2g + z = constant along a streamline (plus head added by pumps, minus losses). Where velocity rises, pressure falls — the basis of venturi meters and the reason the energy grade line (EGL) sits above the hydraulic grade line (HGL) by exactly the velocity head V^2/2g. Pump hydraulic power is P = γ·Q·H; dividing by efficiency gives the larger brake (shaft) power the motor must supply.
Pipe Flow, Reynolds Number, and Head Loss
The Reynolds number Re = V·D/ν distinguishes regimes: laminar below about 2,000, turbulent above about 4,000. In laminar pipe flow the Darcy friction factor is simply f = 64/Re. Friction head loss uses Darcy-Weisbach: h_f = f·(L/D)·(V^2/2g). Minor (local) losses from fittings are h_L = K·(V^2/2g), where K is the fitting's loss coefficient. For water-distribution design many engineers use Hazen-Williams, where a higher C coefficient means a smoother pipe carrying more flow for the same head. The takeaway hierarchy: pick the equation that matches the problem — Darcy-Weisbach for general pipe friction with a known f, Hazen-Williams for pressurized water mains, and Manning for open channels.
Open-Channel Hydraulics
Open channels flow under gravity with a free surface. Manning's equation gives velocity V = (1/n)·R^(2/3)·S^(1/2) (SI; use 1.49/n in US units), where R is the hydraulic radius A/P and S the channel slope. Because velocity scales with the square root of slope, doubling the slope raises velocity by only about 41%; because it scales inversely with n, doubling roughness halves the velocity. The Froude number Fr = V/sqrt(g·D) classifies the flow: Fr < 1 subcritical (tranquil), Fr = 1 critical, Fr > 1 supercritical (rapid). At critical flow specific energy is minimized; a hydraulic jump is the abrupt transition from supercritical to subcritical flow that dissipates energy. The most hydraulically efficient section — maximum flow for a given area — is the one closest to a semicircle, which is why the half-hexagon trapezoid and semicircle are 'best' sections.
Hydrology and the Rational Method
Hydrology converts rainfall to runoff. The Rational Method, Q = C·i·A, estimates peak flow for small watersheds; in US customary units, with i in in/hr and A in acres, the answer comes out in cfs directly (a convenient near-unity conversion). The runoff coefficient C reflects imperviousness (about 0.9 for pavement, 0.3 for lawn) and is area-weighted for mixed cover. Peak flow occurs when the storm duration equals the time of concentration, the moment the entire watershed contributes at once. For larger or more detailed studies the NRCS (SCS) curve-number method applies — a higher CN means more impervious, higher runoff — and the unit hydrograph represents the direct runoff from one unit of excess rainfall over a set duration. Return period ties to risk: for an n-year event, the annual exceedance probability is 1/T, and the probability it occurs at least once in N years is 1 - (1 - 1/T)^N.
Water and Wastewater Treatment
Environmental questions are mostly mass balances. Detention time is volume divided by flow, t = V/Q. A clarifier's surface overflow rate is flow per unit surface area, Q/A, and the weir loading rate is flow per length of effluent weir. Chemical dosing uses mass = flow × concentration, so a chlorine feed rate is Q·dose with unit-conversion care; the disinfection CT value is the product of residual concentration and contact time. Removal efficiency is (in - out)/in × 100%, and blending two streams is a concentration-weighted average by flow. Know the process train: primary treatment (sedimentation) removes settleable solids; secondary treatment is biological — the activated-sludge process is the classic suspended-growth system that removes dissolved and colloidal BOD. Discrete non-flocculating particles settle per Type I (Stokes) settling, the basis for grit-chamber and clarifier sizing.
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