Static Head Is Not a Constant

The energy equation contains one term that looks like a constant of the site: the elevation the water has to gain. It is not a constant. It is the distance between two water surfaces, and a pumping station is a machine arranged so that one of those surfaces rises and falls all day. Get this term wrong and every number downstream of it — head, flow, velocity, power, energy cost — is wrong by the same amount and in the same direction, forever.

2.3.1Between two surfaces, and nothing else

Static head, or static lift, is one subtraction: the elevation of the water surface the pump delivers to, minus the elevation of the water surface it draws from, both on the same datum. The reason it is surfaces comes straight out of §2.1. The energy equation must be written between two points where you know the state of the water, and a free water surface is the one place you always do: gauge pressure is zero because the atmosphere is on it, and the velocity head is negligible because a reservoir is enormous compared with a pipe. Elevation is the only term left standing. Pick your two points anywhere else and you must already know the pressure and velocity there — which is to say you must already have done the rest of the calculation.

So static lift is not measured between the suction and discharge flanges, not from the impeller centreline, not from the wet well floor, and not from the crown of the pipe leaving the building. A gauge on the discharge flange does not read static lift; it reads what the pump is producing at that instant, which is the static lift, plus every loss beyond the gauge, less the height the gauge stands above the wet well water surface, and less the velocity head in the pipe at the tapping. Write the energy equation from the gauge to the receiving surface and those are the four terms you get. Note which height that is: a difference of two elevations, not the gauge's own elevation on the datum. Subtracting a 96 m benchmark elevation from a 24 m head is the version of this sentence people actually write down, and it is nonsense. Raise the pump 2 m on its plinth and the static lift does not move by a millimetre. Reroute the main on a longer, flatter alignment and it does not move either — the losses do, and that is Module 3.

One worked station carries the rest of the lesson: a sanitary lift station taking a gravity sewer at invert 96.10 m and pumping 1,200 m of 250 mm cement-lined ductile iron to a covered balancing reservoir on a ridge. Elevations are metres above a site datum, the only kind this subject accepts; §5.2 covers what happens when two drawings use two of them.

2.3.2The near surface moves every cycle

The wet well level is a design decision, and the two levels bounding it are set by unrelated constraints. Pump-off is set from below, by the submergence the suction inlet needs so it does not draw a vortex (§8.3) and the suction margin the pump needs so it does not cavitate (§6.5). Lead-pump-on is set from above, by the volume needed between the two so the motor does not start more often than its manufacturer permits (§8.1). Neither is chosen with static head in mind; static head inherits what they decide.

On the worked station they are 94.90 m and 93.20 m, so the well works over a band of 1.70 m (5.58 ft). Hold the receiving surface still at its mid level of 110.70 m and the consequence is already visible: static lift runs from 110.70 − 94.90 = 15.80 m with the well full to 110.70 − 93.20 = 17.50 m once it has drawn down. Same pump, same pipe, same afternoon, and the elevation term has moved 1.70 m in the time it takes to empty the well.

The direction of that move matters, because the hard cases compound rather than cancel. Most static lift occurs at the lowest wet well level — which is also least available suction margin, the surface being pulled from sitting furthest below the impeller. The hardest instant for the discharge side is the hardest instant for the suction side, and it arrives on every cycle for thirty years.

2.3.3The far surface moves too, on a different clock

The receiving surface rarely sits still either, and unlike the wet well it is usually somebody else's decision. Four cases cover nearly everything:

• A reservoir or tank being drawn down and refilled. The operating band between low water and overflow is the whole point of storage — a balancing reservoir at a works, a service reservoir on a potable system, an elevated tank — and it is commonly several metres. On this station it is 4.20 m (13.78 ft).
• A tidal or river receiving water. A coastal outfall's static lift moves with the tide twice a day, and a river outfall's moves with the stage — which is highest exactly when the wet-weather inflow is highest. This is what Crossness bought a reservoir to survive.
• A receiving manhole that surcharges. Discharge into a gravity system and the far surface is the water level in that system. In dry weather it is a few centimetres of flow over an invert; in a storm the trunk main backs up and the surface climbs several metres, taking your static lift with it.
• Another station's wet well. Pump into a downstream station and your discharge surface is their drawdown band, cycling on their control settings, which nobody told you about.

Now put both bands on at once. The extremes pair the worst of one surface with the worst of the other, because nothing couples them — the reservoir does not know where the wet well level is. Most lift is the highest receiving surface against the drawn-down well, 112.80 − 93.20 = 19.60 m (64.30 ft); least lift is the lowest receiving surface against the full well, 108.60 − 94.90 = 13.70 m (44.95 ft). The operating envelope of static lift is 5.90 m (19.36 ft) wide — 35% of its own mean of 16.65 m — and the high end is 43% above the low end.

That 5.90 m is the sum of 1.70 and 4.20, and the identity deserves stating on its own: the ranges add. They do not average, and the larger does not swallow the smaller. An engineer who correctly identifies the reservoir as the dominant mover and quotes its 4.20 m has still understated the envelope by 29%.

2.3.4Why the envelope is a range of flows, not a range of heads

A pump is not a flow source. Take as given, from §6.2, the one property of a pump curve that matters here: the more head it is asked for, the less flow it delivers. Give the station a pump falling from a shutoff head of 28.5 m as an idealised parabola — H = 28.5 − 2050·Q² in metres, with Q in m³/s, so k is in s²/m⁵ — treat the main as frictionless for one paragraph, and the static envelope becomes a flow envelope directly: Q = √((28.5 − static)/2050) gives 84.97 L/s at the least-lift corner and 65.89 L/s at the most-lift corner. No single operating point, and nothing yet said about friction.

Now admit that the pipe takes something. This module will not tell you how much — Module 3 does that for the pipe, Module 4 for the fittings — but the shape of the dependence is all this lesson needs: the loss grows with flow, roughly as its square. Here it is 7.0 m at 60 L/s (951 gpm, 1.222 m/s, 4.01 ft/s), about 6.50 m of it pipe and 0.46 m fittings. Add that and the crossing moves:

static lift envelope        13.70 m  ..  19.60 m        range 5.90 m
loss term at 60 L/s          7.0 m   (6.50 m pipe + 0.46 m fittings, rounded)
                                     so R = 7.0/0.060² = 1944 s2/m5

                       least lift          most lift        swing
static                   13.70 m             19.60 m        5.90 m
loss at that flow         7.20 m              4.33 m       −2.87 m
total dynamic head       20.90 m             23.93 m        3.03 m   ← narrower
flow                   60.87 L/s           47.20 L/s      13.67 L/s
velocity, 250 mm main   1.240 m/s           0.962 m/s
water power              12.47 kW            11.07 kW      1.40 kW
shaft power at 75%       16.62 kW            14.76 kW
energy per m3 pumped  0.0759 kWh          0.0868 kWh        +14%

Solved at the crossing of the pump curve and the system curve. Both curves are quadratics here, so the flow has the closed form Q = √((H₀ − static)/(k + R)) — which is how the instrument's verifier checks the numerical solution it actually uses. Water power is ρgQH and needs no efficiency; the shaft power and the energy per cubic metre divide it by a placeholder pump efficiency of 75%, because a real efficiency curve moves across the envelope too and that is §6.2 and §7.4.

The line to look at is the total head. The static term swings 5.90 m but the total dynamic head swings only 3.03 m (9.93 ft), and that is not a rounding artefact — it happens every time. More static lift means less flow, less flow means a smaller loss term, so part of the extra lift is repaid out of the losses. The total head range is always narrower than the static range on a real main. Across this envelope the static term is 66% to 82% of the total head, which is why a mostly-static station gets very little turndown from a variable-speed drive: a drive can only attack the part of the head that depends on flow.

The shape of the pump curve decides how much of the head swing becomes a flow swing, and that is the one selection decision this lesson can settle alone. Three pumps that all sit at essentially the same nominal duty — 54.2 to 54.5 L/s at 22.4 m, at the mid-envelope lift — differing only in steepness:

• Flat — H = 25.0 − 900·Q², a shutoff 12% above the duty head. Flow across the envelope: 43.57 to 63.03 L/s, a swing of 19.46 L/s. The low end fails the scour figure by a wide margin at 0.888 m/s.
• Medium — H = 28.5 − 2050·Q², shutoff 27% above duty. Flow 47.20 to 60.87 L/s, a swing of 13.67 L/s.
• Steep — H = 32.5 − 3400·Q², shutoff 45% above duty. Flow 49.13 to 59.31 L/s, a swing of 10.18 L/s, and the low end reaches 1.001 m/s — still not scouring, but close.

A steep curve is the stabiliser: it holds flow nearly constant while the levels wander, which is what you want when a minimum velocity binds or a downstream process needs a steady feed. It is not free — it is often a smaller, faster impeller with less tolerance for solids, and it carries more head than the easy end of the envelope needs. §7.2 does the matching properly.

Which leaves the last question: which corner do you check? Not one of them. Each quantity has its own worst case, at a different end of the envelope.

1. Head — most lift. Reservoir at overflow, wet well drawn down: 19.60 m static, 23.93 m total. The pump must still deliver at this condition, and a pump whose shutoff head sits under it delivers nothing at all while running perfectly.
2. Flow, and therefore minimum velocity — also most lift. 47.20 L/s, 0.962 m/s. This is the trap: a minimum velocity requirement binds at the minimum flow, which is the maximum-head corner. The habit of checking "the worst case" as though it meant the biggest number sends people to the other end of the envelope, where the main scours comfortably and the check passes for the wrong reason. See self-cleansing velocity.
3. Power — least lift on this station, and the condition is what to remember, not the corner. 60.87 L/s at 20.90 m is 12.47 kW into the water against 11.07 kW at the other end, because flow rises 29% while head falls 13% and water power is their product. What guarantees that ordering is not the corner but a condition: ρgQ(H₀ − kQ²) rises to an interior maximum at Q = √(H₀/3k) — 68.07 L/s on the medium curve, 56.4 L/s on the steep one — and while the least-lift flow is still short of that peak, as this station's 60.87 L/s is, both corners sit on the rising limb and more flow must mean more power. Past the peak you have to compare, and the comparison does turn over: take the loss term away and the frictionless case two paragraphs up runs 84.97 L/s at 13.70 m, 11.41 kW, against 65.89 L/s at 19.60 m, 12.65 kW — the maximum has moved to the most-lift corner. On this main the crossover is at about 1.7 m of loss at 60 L/s, so it is the nearly-static station that catches you. Size the motor from a sweep of the whole range, never from a rule about corners.
4. Suction margin — lowest wet well level. Independent of the receiving surface entirely, because the reservoir is not on the suction side of anything. §6.5.
5. Surge — the highest flow, hence least lift. The velocity that has to be stopped when a check valve slams is the velocity at the easy corner. §8.5.

So the station has no duty point; it has a locus of them, and every claim about it is a claim about a region. That is why a hydraulic profile (§5.1) is drawn at two conditions rather than one, and why "the station delivers 55 L/s" cannot be checked. What can be checked is: 47 to 61 L/s across a static envelope of 13.70 to 19.60 m, velocity checked at 47 and the motor sized at 61.

What this lesson still owes you is the 7.0 m, which arrived as a magnitude and not a method. Module 3 derives it from the pipe and its age, Module 4 from the fittings — which in a station's own pipework are not minor at all — and Modules 6 and 7 replace the idealised parabola with a manufacturer's curve. None of that disturbs what is established here: the static term is a range, the two bands making it add, and the worst case is a different corner for every quantity you care about.