10.1 Continuity, Energy, and HGL/EGL
Key Takeaways
- Continuity links flow, area, and velocity in both pipes and channels; most PE WRE hydraulics errors start with an area or unit mistake.
- The hydraulic grade line is elevation head plus pressure head in a pressurized pipe, while the energy grade line adds velocity head.
- Energy equations must keep pump head, turbine head, friction loss, minor loss, and velocity-head changes on the correct side of the balance.
- The energy grade line drops through passive losses, jumps up across a pump, and should never fall below the hydraulic grade line.
- Open-channel energy problems use depth, velocity head, and channel bottom elevation rather than pipe pressure head.
Energy Bookkeeping Before Formula Hunting
The April 2024 NCEES PE Civil WRE specification names energy, continuity, grade-line analysis, and momentum under closed-conduit hydraulics, and the same ideas reappear in open-channel flow. Treat this section as the foundation for every pipe, pump, culvert, spillway, and channel problem that follows. The exam rarely rewards memorizing an equation without first identifying what each head term represents.
Core Head Terms
| Term | Closed-conduit meaning | Open-channel meaning |
|---|---|---|
| Q | Flow rate, usually cfs, gpm, m^3/s, or L/s | Same flow rate, often tied to channel geometry |
| A | Full pipe area unless partly full is stated | Wetted flow area at the given depth |
| V | Average velocity, Q/A | Average velocity, Q/A |
| z | Pipe centerline or datum elevation | Channel bottom or section datum elevation |
| p/gamma | Pressure head above the pipe centerline | Usually atmospheric at the free surface |
| V^2/(2g) | Velocity head | Velocity head in specific energy |
| HGL | z + p/gamma | Approximately the water surface for free-surface flow |
| EGL | HGL + V^2/(2g) | Water surface plus velocity head |
Continuity is the first check: Q = VA. If a pipe diameter doubles, area increases by four, so velocity drops to one-fourth for the same flow. In an open channel, area changes with depth, so normal-depth and critical-depth calculations are usually iterative or geometry dependent.
General Energy Setup
Use one consistent datum and write every term as length of water:
- Start with z1 + p1/gamma + V1^2/(2g).
- Add pump head if a pump adds energy between the sections.
- Subtract turbine head if energy is extracted.
- Account for the downstream z2, p2/gamma, and V2^2/(2g).
- Add all friction and minor losses in the direction of flow.
A reliable form is: upstream total head + pump head = downstream total head + losses + turbine head. Reservoir surfaces usually have negligible velocity head and atmospheric pressure. Pipe sections usually need pressure head and velocity head. Channel sections use depth and velocity head in the specific-energy term.
HGL and EGL Sanity Checks
The hydraulic grade line in a pressurized pipe shows the level water would rise to in a piezometer. If the HGL is 30 ft above the pipe centerline, the gauge pressure head is 30 ft. If the HGL falls below the pipe centerline, gauge pressure is negative. Negative gauge pressure is not automatically impossible, but it is a warning near high points, siphons, and pump suctions.
The energy grade line equals the HGL plus velocity head. Because velocity head is nonnegative, the EGL cannot be below the HGL. Through a constant-diameter pipe with only friction loss, the HGL and EGL are parallel. Through an expansion, velocity head drops and pressure head may rise, but the EGL still drops by the loss. Across a pump, the EGL jumps upward by the pump head, and the HGL usually jumps too.
Open-channel problems often hide the same logic. Specific energy is depth plus velocity head measured from the channel bottom. Critical flow occurs at minimum specific energy for a given flow and geometry. A hydraulic jump converts supercritical shallow, fast flow into subcritical deeper, slower flow while dissipating energy, so the downstream EGL is lower even though depth rises.
Exam Workflow
- Draw the sections and datum before substituting numbers.
- Convert pressure to pressure head: 1 psi is about 2.31 ft of water.
- Convert diameter to area carefully; diameter errors are squared.
- Put pump head in as a gain and losses as losses.
- Sketch the expected HGL/EGL profile before selecting an answer.
For PE WRE, the best answer often follows from grade-line behavior before a full calculation. If a passive pipe answer shows pressure rising with no pump, or an EGL below an HGL, it is not hydraulically consistent.
At a point in a 12-inch water main, the pipe centerline elevation is 640 ft, the gauge pressure is 52 psi, and the average velocity is 5.0 ft/s. Approximately what are the hydraulic grade line and energy grade line elevations at that point?
Which statement is correct for flow through a constant-diameter pressurized pipe with no pumps or turbines between two points?