Section 02 · Understand
The physics of the ram
Everything a ram does follows from one violent, well-studied event: water hammer. Here's the full chain — from accelerating a column of water, to the pressure spike, to the friction budget that keeps it honest, to a complete worked example you can scrub through on a live pressure trace.
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Water hammer & the Joukowsky equation
Move a column of water in a pipe, then stop it suddenly. The water nearest the valve stops first and compresses; the rest piles in behind; the disturbance propagates back up the pipe as a pressure wave. In taps this is the bang you hear when a washer shuts off — "knocking pipes". In a ram it is the entire engine.
Joukowsky equation — Δp = pressure rise (Pa) · ρ = water density (≈1000 kg/m³) · c = pressure-wave speed in the pipe (m/s) · Δv = velocity change (m/s)
Plug in a ram's numbers. If the water reaches 1.5 m/s and the wave speed in a rigid steel drive pipe is about 1300 m/s:
A momentary spike a hundred times the static drive pressure — that's why a 2 m fall can push water 20 m up.
The wave speed c depends on the pipe's stiffness as much as the water's:
| Pipe material (water-filled) | Typical wave speed c | Notes for ram use |
|---|---|---|
| Steel / cast iron (rigid) | 1200–1400 m/s | Best hammer, preferred for drive pipes |
| Copper | 1000–1300 m/s | Excellent but costly |
| PVC / uPVC (pressure-rated) | 300–500 m/s | Usable; hammer is softer — size up flow |
| HDPE (polyethylene) | 250–400 m/s | Fine for delivery; too soft for drive pipe |
| Rubber / layflat hose | very low | Absorbs the shock — the ram won't beat |
Exact values vary with pipe diameter, wall thickness and temperature; treat the table as guidance.
Using flexible hose for the drive pipe. A soft pipe wall stretches when the spike arrives, absorbing the very energy the pump needs. The result is a ram that beats weakly or not at all. The drive pipe must be rigid, constant-diameter, and as straight as the site allows.
Accelerating the column
Between beats, the drive head H does one job: accelerate the water in the drive pipe. Newton's second law on a column of length L, cross-section A, density ρ, driven by head H gives:
Acceleration is independent of pipe diameter — a fat pipe is just heavier to speed up.
Ignoring friction, the water would keep accelerating to v = √(2gH) — about 6.3 m/s for a 2 m fall. Friction (which grows with the square of velocity) caps reality at roughly 1–2 m/s in a well-sized drive pipe — the friction budget below shows exactly how to check that cap for your pipe. The time to reach working speed:
Example: L = 12 m, v = 1.5 m/s, H = 2 m → t ≈ 0.9 s — hence a beat roughly every second.
This single equation explains the ram's tempo: longer pipes or smaller falls mean slower, weightier beats; short pipes and big falls make frantic little ones.
Timing is everything
The pressure wave created at the waste valve races back up the drive pipe, reflects off the open source, and returns as a suction wave. The travel time is:
L = 13 m of steel pipe, c = 1300 m/s → the wave is back in ~20 ms.
For a strong hammer, the valve must close before that returning suction wave can relieve the pressure — and it must reopen on the suction pulse's arrival. Drive-pipe length, valve weight and wave speed are therefore locked together, and centuries of trial and error converge on the classic rule:
Make the drive pipe between 3 and 7 times as long as the fall is high (L = 3H…7H). Shorter and the valve can't react before the wave returns; longer and the water takes too long to accelerate, wasting beats.
Friction: the Darcy–Weisbach budget
Every metre of pipe takes a cut of the head as friction — and in a ram, where the drive pipe's whole job is to deliver a hard slam, that cut is the difference between a punch and a shove. The standard accounting for pipe flow is the Darcy–Weisbach equation:
hf head lost to friction (m) · f Darcy friction factor (–) · L, D pipe length & internal diameter (m) · v mean velocity (m/s) · g = 9.81 m/s²
The friction factor f depends on how turbulent the flow is — the Reynolds number Re = vD/ν, with ν ≈ 1.0×10⁻⁶ m²/s for water — and on the pipe wall's roughness ε. Working rams sit comfortably in the turbulent regime (Re ≈ 10⁴–10⁵), where the explicit Swamee–Jain approximation is accurate enough for field work:
For laminar flow (Re < 2300 — only tiny rams) use f = 64/Re instead.
| Pipe material | Roughness ε (mm) | Field notes |
|---|---|---|
| PVC / uPVC (new) | 0.0015 | Glassy-smooth; stays that way if kept clean and out of sun |
| HDPE | 0.0015–0.007 | Smooth; flex rules it out as a drive pipe anyway |
| Commercial steel (new) | 0.045 | The classic ram drive pipe |
| Galvanised steel | 0.15 | Rougher; the zinc lining costs you a little hammer |
| Old, scaled or biofilmed steel | 0.5–3 | Friction can triple over the years — flush or renew |
The terminal-velocity insight
During spin-up, driving force is fixed (ρgAH) while drag grows with v². Set Darcy–Weisbach equal to the whole fall and you get the velocity at which friction eats all of it — the column's terminal speed:
If roughness doubles f, the achievable velocity falls by √2 — and the Joukowsky spike, being proportional to Δv, falls with it. Friction quietly disarms the hammer.
Drive pipe: keep hf ≤ 10% of H. Friction during spin-up caps the slam velocity; a drive pipe that burns a third of its fall in drag delivers a feeble hammer no matter how the valve is tuned. Delivery pipe: keep hf ≤ ~15% of h. Delivery friction simply adds to the effective lift — the chamber pushes against h + hf — so it's a steady-flow sizing problem, more forgiving and easily solved with a bigger pipe.
Worked check 1 — drive pipe
The worked-example site below: Q = 10 L/s through 100 mm internal diameter new steel, L = 10 m (5 × H), with H = 2 m.
| Step | Computation | Result |
|---|---|---|
| Velocity | v = 4Q/πD² = 4 × 0.010 / (π × 0.1²) | 1.27 m/s ✓ in band |
| Reynolds | Re = vD/ν = 1.27 × 0.1 / 1.0×10⁻⁶ | ≈ 1.3×10⁵ (turbulent) |
| Relative roughness | ε/D = 0.045 / 100 | 4.5×10⁻⁴ |
| Friction factor | Swamee–Jain | f ≈ 0.020 |
| Friction head | h_f = 0.020 × (10/0.1) × (1.27²/19.62) | 0.16 m |
| Budget | 0.16 / 2.0 | 8% of H ✓ pass |
And the hammer this column delivers: Δp = ρcΔv = 1000 × 1300 × 1.27 ≈ 1.65 MPa ≈ 168 m of head — over eight times the 20 m lift, which is exactly the margin a ram lives on.
Worked check 2 — delivery pipe
The same site's delivery: q = 0.6 L/s through 64 m of 32 mm HDPE, h = 20 m.
| Step | Computation | Result |
|---|---|---|
| Velocity | v = 4q/πD² = 4 × 0.0006 / (π × 0.032²) | 0.75 m/s |
| Reynolds | Re = 0.75 × 0.032 / 1.0×10⁻⁶ | ≈ 2.4×10⁴ |
| Friction factor | ε = 0.0015 mm (smooth HDPE) | f ≈ 0.025 |
| Friction head | h_f = 0.025 × (64/0.032) × (0.75²/19.62) | 1.4 m |
| Budget | 1.4 / 20 | 7% of h ✓ pass |
The chamber therefore pushes against ≈ 21.4 m, not 20 — which is precisely the plateau you can read off the pressure trace. Size the delivery pipe's pressure rating on h + hf, plus margin.
Minor losses: fittings
Fittings add loss as K·v²/2g each. At v = 1.5 m/s (v²/2g ≈ 0.115 m):
| Fitting | K | Extra head at 1.5 m/s |
|---|---|---|
| Smooth 90° bend (large radius) | 0.3 | 0.03 m |
| Threaded 90° elbow | 0.9 | 0.10 m |
| 45° elbow | 0.35 | 0.04 m |
| Gate / ball valve, fully open | 0.15 | 0.02 m |
| Sharp-edged entrance from tank | 0.5 | 0.06 m |
On the drive side, straightness matters beyond these numbers: each disturbance also scatters the pressure wave and mistimes the beat. Bends gently, and never near the ram.
Efficiency, twice over
Two different questions can be asked of a ram, and both are called "efficiency" in the literature:
Energy efficiency (η)
What fraction of the water power entering the pump leaves through the delivery pipe?
Typical: 60–80% for a good ram — remarkably high for a machine that works by banging a valve shut.
Volumetric efficiency
What fraction of the source water is delivered uphill?
Lift 5× higher than the fall at 65% efficiency → about 13% of the flow delivered; the other 87% is spent as the driving jet.
Two historical figures of merit appear in older books, and they differ in what they count as "input":
- D'Aubuisson's efficiency counts all water passing through the ram as input: ηD = q·h/(Q·H) — the same as the energy efficiency above.
- Rankine's efficiency counts only the water rejected through the waste valve as input: ηR = q·h/((Q−q)·H) — the delivered water is treated as product rather than fuel, so it reads higher and can even exceed 100% at low lift ratios. It is a figure of merit, not a thermodynamic efficiency.
Nineteenth- and early-twentieth-century texts do not use these two names consistently — some swap them. When comparing quoted efficiencies, always check the formula in words, not just the eponym.
One beat, live on a pressure trace
This chart is computed from the worked example's own numbers — scrub through the beat and watch what the water "sees" at the pump:
Worked example
A spring above a village falls 2 m to a ram site; the storage tank is 20 m above the ram. The stream yields a spare 10 L/s that can be spared for pumping. Assume a D'Aubuisson (energy) efficiency of 60%.
| Quantity | Formula | Result |
|---|---|---|
| Delivered flow, q | η·Q·H/h = 0.6 × 10 × 2/20 | 0.6 L/s |
| Daily delivery (24 h) | 0.6 × 86 400 s | 51.8 m³/day |
| Volumetric efficiency | q/Q | 6% |
| Waste flow | Q − q | 9.4 L/s |
| Input water power | ρg·Q·H = 9810 × 0.010 × 2 | 196 W |
| Output water power | ρg·q·h = 9810 × 0.0006 × 20 | 118 W |
| Check: energy efficiency | 118/196 | 60% ✓ |
The drive pipe behind these numbers — 100 mm steel, 10 m long — passes its friction budget at 8% of the fall (worked check 1), and the pressure trace above is drawn from exactly these values. At 50 L per person per day, 51.8 m³/day serves about a thousand people — from 196 watts of falling water and a valve that clacks 30–100 times a minute, day and night. Run your own site through the sizing calculator.
Where the physics bites
- Air in the drive pipe is a cushion, not a column — a single air pocket can stop the beat. Intakes must stay submerged; drive pipes must rise monotonically with no high points.
- Soft pipes (HDPE, hose) stretch and swallow the spike — steel wins, pressure-rated PVC is the compromise.
- Friction over the budget — a scaled or undersized drive pipe caps the column below the velocity the valve needs; check hf ≤ 10% of H.
- Too little flow and the water never reaches the velocity the valve weight demands; the ram stalls open.
- Too high a lift ratio and the spike — capped near Δp = ρcΔv — can't stay above the delivery head long enough to pump; efficiency collapses.
- A waterlogged chamber means the delivery line hammers and the check valve slams without the soft cushion it was designed for.
Control engineers model the ram with the same mathematics as a boost converter: the waste valve is the switching transistor, the drive pipe's inertia is the inductor, and the air chamber is the output capacitor. Same equations, different fluid.
Next: meet every component up close in Anatomy, or turn these formulas into a site design in Design & Sizing.