How it works — momentum in, pressure out
A jet pump has no moving parts: a high-velocity motive jet shears and drags the suction flow into a throat, where the two streams mix, and a diffuser converts the mixed velocity into discharge pressure. Everything about its performance follows from one momentum balance plus three pressure losses.
1 · The four building blocks
Nozzle (motive jet, loss coefficient \(K_n\approx0.05\)): the jet dynamic pressure \(Z=\rho_1 V_n^{2}/2\) absorbs the motive-to-suction pressure drop:
$$P_i-P_s=Z\,(1+K_n)$$Throat entry (accelerating the suction flow, \(c=\frac{1-b}{b}\), entry loss \(K_{en}\approx0\)):
$$P_s-P_o=\frac{Z\,S\,(1+K_{en})\,M^{2}}{c^{2}}$$Throat mixing (momentum conservation of the two merging streams, mixing loss \(K_{td}\approx0.2\)):
$$P_t-P_o=Z\Bigl[\,2b+\frac{2SM^{2}b^{2}}{1-b}-b^{2}(1+K_{td})(1+M)^{2}\Bigr]$$Diffuser (5–8° included angle, recovery loss \(K_{di}\)): combining the last three with \(K_{td}=K_{th}+K_{di}\) gives the closed-form characteristic — the heart of this tool:
$$\text{Num}=2b+\frac{2SM^{2}b^{2}}{1-b}-b^{2}(1+K_{td})(1+M)^{2}-\frac{S(1+K_{en})M^{2}}{c^{2}}$$ $$N=\frac{P_d-P_s}{P_i-P_d}=\frac{\text{Num}}{1+K_n-\text{Num}}$$2 · Efficiency and the N(M) curve
For a fixed area ratio \(b\), \(N(M)\) falls monotonically while \(\eta=M\cdot N\) rises to a single peak at \(M_{mep}\) and falls again — the jet-pump "N curve". Operating rule of thumb: place the duty at \(M_{op}=\tfrac{2}{3}M_{mep}\) — about 90–95% of peak efficiency with a full cavitation margin. Start designs at \(b=0.25\) (peak-efficiency band \(b=0.2\)–\(0.3\)); small \(b\) trades flow for pressure, large \(b\) the reverse.
3 · Cavitation limit
\(P_v\) is the vapour pressure at the suction temperature and \(\sigma\) the cavitation factor (0.8–1.4 in tests; 1.35 is a conservative design value). Duty with \(M\ge M_L\) chokes the throat — flow will not increase no matter how the discharge valve opens. Screening margin: \(CR=\frac{M_L-M_{op}}{M_{op}}\times100\%\).
4 · Verdict thresholds (🟢🟡🔴)
- 🔴 CR < 0 — duty beyond the cavitation limit: the throat chokes, efficiency collapses. Increase \(b\), reduce \(M\), or improve suction pressure.
- 🟡 CR 0–20% — thin margin: temperature or pressure drift can trigger cavitation.
- 🟢 CR > 20% — healthy cavitation margin.
- 🟢 M within ±15% of ⅔·Mmep — the recommended operating band; 🟡 below it wastes efficiency, 🟡 above it walks toward \(M_L\).
- 🟢 b = 0.2–0.3 — peak-efficiency band; 🟡 outside it (still valid, just less efficient).
5 · Sizes from the duty
with \(\rho_1\) in slug/ft³ (US). Nozzle-to-throat spacing \(sp\approx1\times D_{th}\), mixing-tube length \(L\approx6\times D_{th}\), diffuser 5–8° included angle.
Worked example (anchor case)
Motive 73.96 gpm at 133.5 psi gauge, suction 50 gpm at 0 psi (atmospheric), discharge 40 psi, cold water (\(P_v\approx0.5\) psia), \(b=0.25\). The tool returns:
| Quantity | Value |
|---|---|
| \(M=Q_2/Q_1\) | 0.676 |
| \(N=(P_d-P_s)/(P_i-P_d)\) | 0.4280 |
| \(\eta=M\cdot N\) | 28.93% |
| \(M_{mep}\) at b = 0.25 | 1.014 → ⅔·Mmep = 0.676 ✓ duty sits exactly on the rule |
| \(M_L\) (σ = 1.35) | 0.863 → CR = 27.6% 🟢 |
| Nozzle / throat diameter | 0.469 in / 0.938 in |
Compare the same duty across area ratios: b = 0.10 gives η = 27.3% but CR = −5.3% (cavitating 🔴); b = 0.40 gives 26.9% at CR = 57%; b = 0.60 gives 21.0% at CR = 92% — the peak-efficiency band is also where the cavitation margin begins to be usable.
Standard 1-in eductor capacity lookup
Below the calculator: pick suction lift, motive pressure and discharge pressure to read the standard 1-inch water-jet eductor capacity (suction flow / motive flow), then scale it with the size capacity ratios (½ in 0.36 · ¾ in 0.64 · 1 in 1.00 · 1½ in 2.89 · 2 in 4.00 · 2½ in 6.25 · 3 in 9.00 · 4 in 16 · 6 in 36). Required size = capacity ratio just above (required suction ÷ 1-in capacity).
Where jet pumps win
- Deep wells — the jet sits down-hole with no moving parts; a surface centrifugal pump feeds it (two-curve matching, both near BEP).
- Tank mixing / agitation — sparger eductors entrain ≈ 3× the motive flow at ≈ 20 psi nozzle drop.
- Slurry & solids transfer — hopper eductors handle sand, lime, fly ash; nothing clogs because nothing rotates.
- Priming & dewatering — sump draining, well pointing, cheap standby dewatering.
- Accept 25–35% efficiency and motive-fluid dilution of the product; if energy cost dominates, a centrifugal pump wins.
Symbol table
| Symbol | Meaning | Units |
|---|---|---|
| \(Q_1,\,Q_2\) | motive / suction (entrained) flow | gpm / m³/h |
| \(P_i,\,P_s,\,P_d\) | motive, suction, discharge pressure | psi / kPa |
| \(M\) | entrainment ratio \(=Q_2/Q_1\) | — |
| \(N\) | head ratio \(=(P_d-P_s)/(P_i-P_d)\) | — |
| \(\eta\) | efficiency \(=M\cdot N\) | % |
| \(b\) | area ratio nozzle/throat \(=A_n/A_t\) | — |
| \(c\) | \((1-b)/b\) | — |
| \(S\) | density ratio \(\rho_2/\rho_1\) | — |
| \(Z\) | jet dynamic pressure \(\rho_1V_n^2/2\) | psi / kPa |
| \(K_n,\,K_{td},\,K_{en}\) | nozzle / throat+diffuser / entry loss coefficients | — |
| \(M_{mep},\,M_L\) | max-efficiency entrainment ratio, cavitation limit | — |
| \(P_v,\,\sigma\) | vapour pressure, cavitation factor | psi, — |
| \(D_n,\,D_{th}\) | nozzle, throat diameter | in / mm |
Engineering criteria applied
- Fixed loss coefficients — \(K_n=0.05\), \(K_{td}=0.20\), \(K_{en}=0\), same-density liquids \(S=1\), retracted nozzle (full jet loss); the values behind the characteristic table used for validation.
- Operating point ⅔·Mmep — about 90–95% of peak efficiency while preserving cavitation margin; running at \(M_mep\) itself leaves zero margin.
- Cavitation factor σ = 1.35 — conservative; improved nozzle/entry profiles reach 1.0, which rescues small-b designs.
- Discharge throttling warning — raising \(P_d\) above design to "hold" the point wastes energy (red); running below design \(P_d\) pushes \(M\) toward \(M_L\) (amber).
- Viscosity ≤ 20 cP — efficiency factor drops only ≈5%; above that, use test data.