Assembling Total Dynamic Head

2.5.1The number a pump is bought against

A pump does not know where it is. It knows one thing: how much head it is being asked to produce at the flow it happens to be passing. Everything the station is — its lift, its pipe, its valves, the level in its wet well this minute — reaches the impeller as one number, and that number is the total dynamic head. Assembling it is the whole of this module's bookkeeping, and getting it wrong is how a station ends up unable to deliver what it was built for.

The extended energy equation of §2.1 already contains the answer. Write it between the two water surfaces the station works between and almost everything cancels: both are at atmospheric pressure, so the pressure terms go, and both are near enough still, so the velocity terms go. Cancelling those two is not the same as saying no kinetic energy is spent: the water is moving in between, and whatever brings it to rest again has to be paid for. What is left is the elevation difference, the losses in between — including that last one — and the pump term that pays for all of it:

TDH  =  static lift  +  friction  +  minor losses  +  velocity head

        static lift    = discharge water surface − suction water surface
        friction       = loss along the pipe walls          (Module 3)
        minor losses   = loss at every fitting and valve    (Module 4)
        velocity head  = v²/2g, spent once, at the far end

The bookkeeping form. Every term is in metres of water (or feet). One velocity head is always spent at the far end; §2.5.4 is which of the last two columns it belongs in.

Two habits of language prevent real errors. First, TDH is dynamic: it is the head required at a stated flow. "The TDH of this station is 33 m" is not a complete sentence; "33 m at 60 L/s with the well drawn down" is. Second, static lift is measured between water surfaces, never between flanges or pump centrelines (§2.3). Measure to the discharge flange and you have deleted the depth of water standing above it, and specified a pump that cannot fill the receiving structure.

2.5.2One term is fixed in flow. Three are not.

Sort the four terms by what they respond to, and the shape of the whole subject appears.

The static term does not care how hard you push. It is elevation, and elevation is indifferent to flow: a station lifting 25.95 m (85.14 ft) lifts 25.95 m at 10 L/s and at 100 L/s. It does move — with level, not with flow — which is why it produces a band rather than an error, and why the wet well's working depth ends up in the pump specification.

The other three grow with flow, and grow fast. Minor losses and velocity head go exactly as Q², because both are multiples of v²/2g and velocity is proportional to flow in a fixed bore. Friction is nearly so: over this station's working band the exponent is about 1.9, not 2, because the friction factor drifts slowly with flow. Module 3 owns that drift. What matters here is structural — the loss terms scale together and the static term does not scale at all — and that asymmetry generates the system curve, the reason parallel pumps disappoint, and the reason a variable-speed drive helps a friction-dominated station and barely touches a static one.

2.5.3One duty, assembled

Keep the alignment and boundary elevations from the station §1.5 read off a drawing, but make the training well 0.60 m deeper so its floor is at elevation 0.00 m: arriving invert at 3.20 m, high water alarm at 2.75 m, discharge water surface at 27.30 m, and 1 200 m (3 937 ft) of 250 mm (9.84 in) cement-lined ductile iron force main between. Two levels this lesson adds, because TDH cannot be assembled without them: pumps stopping at 1.35 m and starting at 2.60 m. Where those two come from is not hydraulics — submergence over the suction bell sets the stop level and permitted motor starts set the volume between them, §8.1 and §8.3. The duty: 60 L/s (951 gpm, 1.369 MGD).

Velocity first, because three of the four terms hang off it. A 250 mm bore has an area of 0.0491 m², so 60 L/s is 1.222 m/s (4.010 ft/s) and the velocity head is v²/2g = 0.076 m (0.25 ft). Note what that needed: no coefficient, no roughness, no chart. Flow and bore determine it completely — the only loss-side term you can never be wrong about.

Static lift next, and here you must choose a condition. With the well drawn down to the stop level the lift is 27.30 − 1.35 = 25.95 m (85.14 ft). That is the worst case, and worst case is what a pump is sized against.

Friction is a black box in this module, so take it the way an engineer in an existing station takes it — from measurement. The drawing's loss schedule states 6.50 m (21.33 ft) for this main at 60 L/s, and a pump test recovers the same figure — two gauges, one at each end of the main, and a flow meter give it directly (§2.2). A single gauge on the discharge flange does not: it gives you TDH at best, out of which friction comes only after the static lift and the fittings are subtracted back off. One measured point carries the term to other flows, because the shape is known: hf = R·Q², with R = 6.50/0.060² = 1 806 s²/m⁵. That shortcut sits within about 4% of a full Darcy-Weisbach calculation across the 40–80 L/s band, and is worth nothing at 10 L/s. Module 3 turns the measurement into a prediction you can make before the pipe exists.

Minor losses are a count of velocity heads. Every fitting costs K velocity heads, so the whole term is ΣK · v²/2g and the work is arithmetic on a schedule:

So the fittings cost 7.35 × 0.076 m = 0.560 m (1.84 ft). Those K values are this course's table, in src/core/hydraulics.js; a real design uses the manufacturer's figure for the specific valve, and Module 4 is where they are chosen rather than quoted. Notice that 0.560 m is only 8.6% of the friction term, and that the ratio is an accident of having 1 200 m of pipe: shorten the main to 406 m — 2.2 m of friction, because friction is linear in length — and the fittings are 25% of it, while inside the station's own pipework they are all of it. "Minor" describes a long force main, not a pump station.

Add the column: 25.95 + 6.50 + 0.560 + 0.076 = 33.09 m (108.55 ft). Static lift is 78.4% of it, friction 19.6%, fittings 1.7%, velocity head 0.23%. That last figure is the honest reason the fourth term gets so little respect; §2.5.4 is why it has to be there anyway.

2.5.4One velocity head, counted exactly once

The fourth term is the one that produces double counting, and the cure is to say where the system ends before you add anything up.

If the main ends under water in the receiving structure, the jet dissipates into still water and the whole velocity head is destroyed there: book it as an exit loss with K = 1.0, take the receiving water surface as the endpoint, and there is no residual, because the water has stopped. If the main ends in free discharge, spilling into a manhole above the water, it leaves at 1.222 m/s and takes that kinetic energy with it. Nothing recovers it, so the pump must supply it: book a residual velocity head, with the outlet as the endpoint.

Those are two descriptions of the same duty, and they give the same answer — with the outlet at the receiving water surface, both come to 33.09 m. One velocity head is spent either way; the only question is which column it is written in. Write it in both, and TDH is over-stated by exactly v²/2g.

Then name the case where the term is not decoration. Velocity head goes as v², so it is negligible in a high-head force main at 1.2 m/s and it is not negligible where velocity is high or lift is low. A drainage station lifting 3.0 m into an adjacent manhole at 3.0 m/s (9.84 ft/s) spends 0.459 m (1.51 ft) — 15% of its static lift. Round that away and you have mis-specified the pump by 0.459 m: four fifths of the entire fittings budget of §2.5.3, on a station lifting a ninth as far. The rule is not "the velocity head is small"; it is "compute it, because it is the cheapest term you will ever compute".

2.5.5Why TDH is a curve, and what is still missing

Now push the same station harder. At 90 L/s (1 427 gpm) the friction term is not 1.5 × 6.50 = 9.75 m but 6.50 × (90/60)² = 14.63 m (47.98 ft), the fittings reach 1.26 m and the velocity head 0.171 m — while the static lift sits exactly where it was. TDH goes from 29.96 m at 45 L/s to 42.01 m at 90 L/s: double the flow for a 40% rise in head. Since the loss terms scale as Q² and the static term does not scale at all, TDH(2Q) = 4·TDH(Q) − 3·static.

So TDH is not a number. It is a curve — head demanded against flow — and a station has as many TDHs as it has operating conditions. Two of those belong on every specification. The first is flow, because that is what the curve is drawn against. The second is the wet well level: between start at 2.60 m and stop at 1.35 m the static term travels 1.25 m (4.10 ft), so at 60 L/s this station demands anywhere from 31.84 m to 33.09 m depending on nothing but what minute it is. The band is exactly the working depth, because nothing else follows the level. That is why a pump is selected against a range, and §7.2 is where the range meets the pump's own curve and settles into a flow.

Which term dominates is the first question to ask about any station, and this one answers it differently depending on where you stand. At 60 L/s it is 78% static, so a variable-speed drive can trim very little and a bigger main buys almost nothing. At 100 L/s it is 57% static and 39% friction, and the bigger main starts to pay. Shorten the main to 406 m and it is 90% static; let it tuberculate until it measures 11.9 m at 60 L/s and it is 67% static. That ratio is worth more to you than any number in this lesson.

Four things this lesson has deliberately borrowed and not paid for:

1. The friction term arrived as a measurement. Predicting it from length, bore, roughness and temperature — and choosing between the two standard methods, which disagree by 12% on this very main at the Hazen-Williams C = 130 that §2.2 compares them at, and by anything from 2% under to 30% over as C moves from 140 to 120, so the coefficient matters more than the method — is §3.2 and §3.4.
2. The K values arrived from a table with no defence. Where they come from, and why a swing check valve costs ten times a gate valve, is Module 4.
3. The 60 L/s arrived asserted — from population, peaking and firm capacity, §1.4 — and the station will spend most of its life elsewhere on the curve.
4. TDH says nothing about whether a pump can survive the duty. Suction margin and motor power are separate questions: §6.5 and §6.2. At 60 L/s and 33.09 m this station puts 19.5 kW into the water, whatever the motor costs to do it.

Take the form from Module 2, and the habit of writing it as a column. Static lift, friction, minor losses, one velocity head; the condition beside it; the flow above it. Every module after this one argues about the size of one of those entries. None of them adds a fifth.