The Energy Line and the Hydraulic Grade Line

2.2.1Two lines, because there are two quantities

A hydraulic profile carries two lines above the pipe, and many drawings label only one. They are not two opinions about one quantity. They are two quantities, and every conclusion you draw from a profile turns on which you are reading.

§2.1 left the energy at a point as a sum of three lengths: elevation z, pressure head p/ρg, and velocity head V²/2g. Plot the sum against distance along the pipe and you have the energy grade line; plot the first two terms only and you have the hydraulic grade line. That is the whole definition of both, and the rest of this lesson is one subtraction.

EGL(x)  =  z  +  p/ρg  +  V²/2g        total head, metres
HGL(x)  =  z  +  p/ρg                  piezometric head, metres

EGL − HGL  =  V²/2g                    the velocity head, and nothing else

The two lines, and the only thing between them.

The HGL earns its own name because it is the line you can measure. Tap a piezometer into the side of a pipe and the water stands at the elevation of the tap plus its pressure head. Put a gauge there instead, divide by ρg, add the tap's elevation: one point on the HGL. Every gauge in a station locates the hydraulic grade line; nothing reads the EGL unless someone installed a tube facing into the flow.

Two consequences follow at once. The HGL is the water surface wherever there is one: a free surface is at atmospheric pressure, so p/ρg is zero, and its velocity is negligible — both lines sit on it. That is why static lift is measured between water surfaces rather than flanges (§2.3). And in a pipe running part full the HGL is the water surface inside the pipe: the gravity sewer arriving at §1.5's station is not pressurised, so its HGL is the free surface with the EGL riding V²/2g above. One definition, no special case for open channels.

2.2.2The separation: what it is, and how small it is

The gap between the lines is V²/2g. Not approximately, not with a coefficient, not depending on the pipe material: exactly the velocity head, everywhere, by definition. Take §1.5's force main — 250 mm cement-lined ductile iron at its design flow of 60 L/s. The velocity is 1.222 m/s (4.01 ft/s) and the separation is 0.0762 m (0.250 ft), the whole way from the station wall to the discharge.

It is constant wherever the velocity is constant, so on a full pipe of one diameter at steady flow the two lines are parallel and knowing one is knowing the other. It changes only when the velocity changes, and in a closed pipe there are three ways for that: the diameter changes, the flow changes, or flow joins or leaves at a branch. A rougher pipe, a longer pipe, a warmer liquid — at the same flow in the same bore, none of them move it.

The exponent is what makes that table matter. Velocity head goes as Q²/D⁴, so diameter dominates. Step from 250 mm to 150 mm at the same 60 L/s and the separation multiplies by (250/150)⁴ = 7.72, from 0.0762 m to 0.5878 m. The interesting velocity head in a station is never out on the main; it is at the suction bell, the nozzle, the meter, the valve someone left half shut.

And it is small. Against a static lift of 24.55 m and a total dynamic head of 31.58 m, 0.0762 m is 0.24% of what the pump must produce — thinner, at any sane vertical scale, than the line used to draw it. Which is why a profile arrives with a schedule of losses beside it, and why the instrument below has readouts as well as a picture.

2.2.3The slope, and the one place the EGL goes up

The separation tells you the water's speed; the slope tells you its losses. The EGL falls in the direction of flow — always, everywhere between the pump and the discharge — because its slope is the rate at which head is dissipated per metre: the hydraulic gradient. Energy cannot appear in a pipe, so an energy grade line that rises downstream has been drawn rather than computed. Be exact about which line you are checking: the hydraulic grade line is under no such rule, and at the expansion below it does rise. For §1.5's main at 60 L/s the gradient is 0.542 m per 100 m, and the EGL falls 6.50 m over 1 200 m. Where 0.542 comes from is Module 3; here it is measured.

Measured is not a figure of speech. Two gauges on one uniform reach give the slope with no knowledge of the pipe. Put them 300 m apart on a level reach reading 420 kPa and 380 kPa: 40 kPa is 4.08 m of water at 15 °C, where ρg = 9 798 N/m³, so the gradient is 1.36 m per 100 m. Because the reach is uniform the velocity head is identical at both taps and cancels out of the subtraction — so that is the slope of the EGL too. A reach measuring far steeper than design has something in it: tuberculation, a collapsed liner, a valve nobody logged as closed, air.

Against that steady fall, three features are abrupt.

• The pump is a vertical jump in the EGL, and the only one. Its height is the definition of total dynamic head: EGL just downstream minus EGL just upstream. Here at 60 L/s that is 31.58 m — 24.55 m of static lift, 6.50 m of friction in the main, 0.42 m in the station's own discharge pipework, 0.04 m on the suction side, 0.076 m thrown away at the outlet. Add them up: within the centimetre they are rounded to they are the 31.58 m, and a schedule that will not close is missing a term.
• Each fitting is a vertical drop. The swing check valve alone, K = 2.0 on 0.0762 m of velocity head, costs 0.152 m — twice the whole separation. Module 4.
• A submerged outlet pins the HGL to the receiving water surface: the conventional exit loss K = 1.0 discards exactly one velocity head, so the HGL just inside the pipe equals the surface outside it, at every flow. That identity anchors the drawing.

The HGL is under no obligation to fall. Where the pipe expands the water slows and the velocity head it gives up reappears as pressure: the HGL rises across the expansion while the EGL falls. Widen this main from 250 mm to 400 mm at 60 L/s and the HGL steps up 0.042 m while the EGL steps down 0.023 m — two lines moving opposite ways across one fitting. That is pressure recovery, and what a diffuser is for.

Which gives the procedure, for a profile handed to you at a stated flow:

1. Anchor at the discharge and work upstream: both lines meet on the receiving water surface, and that level is set by the structure, not by your arithmetic.
2. Add each loss as you pass it. Every point stands higher than the discharge by the head still to be spent getting there.
3. Draw the HGL as the EGL less V²/2g — one subtraction per diameter, not per station.
4. Jump at the pump, then continue up the suction side to the wet well surface, where the lines must meet again. If they do not, a term is wrong.
5. Check the HGL against the crown at every local maximum, at the lowest flow you will run.

2.2.4When the HGL crosses the pipe

The vertical distance from the pipe to the HGL is the gauge pressure head inside it — the definition rearranged, p/ρg = HGL − z. So the HGL crossing the crown is the moment the pressure at the top of the pipe passes through zero, and below the crown the water is below atmospheric.

Take §1.5's summit. The crown at 0+760 is at 28.80 m, the discharge water surface at 27.30 m. With the pumps off the main is static and the HGL dead level at 27.30 m, so the crown stands 1.50 m above it — 1.50 m of vacuum, −14.7 kPa gauge — and not only at the peak: the pipe stands above the grade line from about 0+720 until it has fallen clear at 0+841, on its way into the swale at 0+900. Start pumping and the 440 m below the summit costs friction, which lifts the HGL there; at 60 L/s the summit HGL is 29.68 m and the crown is 0.88 m under pressure. The crossing is at 47 L/s.

And 0+760 is not the only place to look. §1.5's alignment crests again at 1+050, where the crown is 27.20 m — 0.10 m below the static grade line, so at pump-off it is barely clear. But it has only 150 m of main beneath it, so its grade line lifts by 150i where the summit's lifts by 440i, and above about 61 L/s the shorter high point becomes the governing one, a metre and a half taller notwithstanding. Which is the rule: check every local maximum, at every flow you intend to run. The instrument's crown readout names the station it is reporting for that reason.

The HGL at a high point is the discharge water surface plus the friction still to be spent downstream of it, so less flow means a lower grade line. The pressure at a summit is worst at pump-off — inverted from friction, power, pipe stress and surge, where peak flow governs.

Air bites first. Water in equilibrium with the atmosphere carries roughly 2% of its own volume of dissolved air (AWWA Manual M51, Air-Release, Air/Vacuum, and Combination Air Valves). Lower the pressure and some comes out of solution, collecting at local maxima because that is the only place it can go. The pocket takes bore with it: less area, a steeper gradient, the operating point pushed back up the pump curve (§7.2). A station that has lost a third of its output with no fault indicated is usually air, and the air valve that answers it is sited by reading this crossing off a profile.

Contamination matters most in potable work: a joint that weeps outward under pressure draws inward under vacuum, and what it draws is whatever is in the trench. The Recommended Standards for Water Works — the Ten State Standards — asks that a distribution system be designed to hold at least 20 psi (about 140 kPa, or 14 m of water) at ground level everywhere under all conditions of flow. That is a floor under the HGL, written as a pressure.

Column separation is the hard limit. Drop the absolute pressure to the vapour pressure and the water boils; below that there is no liquid to sustain tension. At this summit's elevation and 15 °C the limit is 10.13 m below atmospheric, and the two figures behind it are worth separating: at 28.80 m of elevation the atmosphere is worth 10.306 m of 15 °C water rather than the 10.33 m of the sea-level, 4 °C constant, and the vapour pressure at 15 °C eats 0.174 m of that. Warm the water to 40 °C and the limit falls to 9.62 m. Steady operation does not reach it; a pump trip moves the HGL tens of metres in seconds, and when the cavity collapses the columns rejoin at speed (§8.5).

2.2.5What these two lines do not yet contain

Given the losses, the two lines are determined, and every number above follows from the definition and one subtraction. What is not here is the size of the losses. The 0.542 m per 100 m came from Darcy-Weisbach with a Colebrook friction factor (Module 3) — the same pipe by Hazen-Williams at C = 130 gives 0.606 m per 100 m, 12% steeper, the normal disagreement between two loss families rather than an error in either. Of the 0.42 m inside the station, 0.35 m is ΣK·V²/2g over a check valve, a gate valve, three elbows, a tee and a meter (Module 4) and the remaining 0.07 m is ordinary friction along the 12 m of pipe between the pump nozzle and the wall — two different methods in one line of a schedule, which is exactly why a line of a schedule needs to say which. The suction side adds 0.04 m more. The wet well will not stay at 2.75 m (§2.3), and the 60 L/s was asserted rather than derived (§7.2).

Two habits outlast the arithmetic. Whenever you write down a pressure, write down the elevation it was measured at — a pressure without an elevation is not a point on either line. And whenever someone shows you a grade line, ask what flow it is drawn at, then ask for the lowest flow the station will run. That is the drawing that finds the summit.