The idea
A centrifugal pump at fixed speed and impeller diameter is a one-flow machine: for every flow it delivers exactly one head, draws exactly one power, and converts the difference with exactly one efficiency. Plot head against flow and you have the characteristic curve — the pump's fingerprint, the first thing a vendor publishes and the last word on what the pump will actually do once installed.
Three landmarks organize the curve. The shutoff point at zero flow, where the pump only churns liquid; the best efficiency point (BEP), where hydraulic losses bottom out; and the runout end, beyond which vendors stop publishing data — required NPSH climbs steeply and performance is undefined.
Curve shape is not arbitrary. It follows from the impeller's exit geometry, and it evolves through a family of canonical shapes as specific speed changes.
The equation
Strip away slip and losses and the ideal (Euler) head is a straight line that falls with flow:
| Symbol | Meaning | Units |
|---|---|---|
| \(H_e\) | ideal (Euler) head — before slip and losses | ft (m) |
| \(U_2\) | peripheral speed at impeller OD, \(U_2=\pi D_2N/60\) | ft/s (m/s) |
| \(V_{m2}\) | meridional (radial) exit velocity, \(V_{m2}=Q/(\pi D_2 b_2)\) — proportional to \(Q\) | ft/s (m/s) |
| \(\beta_2\) | blade angle at exit from the tangential direction; backward-curved means \(\beta_2 \lt 90°\) | ° |
| \(g\) | gravitational acceleration | 32.17 ft/s² (9.81 m/s²) |
From the straight line to the real curve
Real pumps fall short of the line for three reasons. Finite blade count causes slip: the fluid leaves with less whirl \(V_{u2}\) than the blade angle implies, tilting the line down. Passage friction grows with \(Q^{2}\), sagging the curve further at high flow. Off-design incidence (shock loss) penalizes every flow except the one the inlet vanes were sized for. Near shutoff, recirculation loops liquid back through the impeller and adds churn. The net result is the familiar drooping curve: efficiency is zero at shutoff, peaks at BEP, and power follows one of the canonical shapes below.
The normalized family — shape follows specific speed
Normalize both axes with the BEP values,
and every pump collapses onto one of a handful of canonical curves. The classic seven-band family, in US units of \(N_s=N\sqrt{Q}/H^{3/4}\):
| Curve | \(N_s\) (US) | \(n_q\) / \(\Omega_s\) | Normalized H–Q shape | Power behaviour |
|---|---|---|---|---|
| 1 | ≈ 900 | 17 / 0.33 | rising toward shutoff — unstable droop | power rises steeply with flow |
| 2 | ≈ 1500 | 29 / 0.55 | slight rise toward shutoff | power rises with flow |
| 3 | ≈ 2200 | 43 / 0.80 | flat | power nearly flat |
| 4 | ≈ 3000 | 58 / 1.10 | continuously falling — stable | power nearly flat |
| 5 | ≈ 4000 | 77 / 1.46 | falling, steeper | power flattens toward shutoff |
| 6 | ≈ 5700 | 110 / 2.09 | steeply falling | power peaks near shutoff |
| 7 | ≈ 9200 | 178 / 3.37 | very steep, with a low-flow saddle | power maximum at shutoff |
Curves 1–2 are the rising (unstable) family, 3 is flat, 4–7 are the steeply falling (stable) family. Curves 1–5 are typical of double-suction impellers, 6–7 of single-suction wheels.
Engineering criteria
| Situation | What it means | Verdict |
|---|---|---|
| Shutoff head ≥ 1.15–1.20 × BEP head, no hump anywhere | continuously rising to shutoff — stable at any duty and safe for parallel operation | 🟢 |
| Shutoff head 1.0–1.15 × BEP head, still monotonic | rising but shallow — verify static and dynamic stability against the actual system curve | 🟡 |
| Flat or drooping segment (hump, shutoff head below peak) | operate only to the right of the hump; never run two such pumps in parallel on the drooping branch | 🟡 |
| Duty point in a positive-slope region | flow hunts, and can surge against large gas cushions | 🔴 |
| Pump curve slope ≥ system curve slope at the crossing | statically unstable; humped curves make twin intersections and the low-flow one always collapses or runs away | 🔴 |
| System holds a large gas volume or free surface, curve flat or rising | static stability is not enough — dynamic surge is possible even on a zero-slope curve | 🟡 |
| \(N_s \gtrsim 5000\), single-suction | shutoff power exceeds BEP power — size the motor for shutoff torque and start against an open or bypassed valve | 🟡 |
| \(N_s \lesssim 5000\) | shutoff power below BEP power — closed-valve start is permissible and the driver can be sized for normal duty | 🟢 |
| Operation beyond runout (published curve end) | NPSHr and vibration escalate steeply; performance is undefined | 🔴 |
| Continuous operation below MCSF (minimum continuous stable flow) | suction and discharge recirculation: low-frequency pulsation, cavitation, fatigue — forbidden for continuous duty | 🔴 |
| Pressure pulsation peak-to-peak reaches the stage static pressure rise | signature of deep low-flow damage — check for impeller-OD pitting and shroud bulging | 🔴 |
| Wearing-ring clearance ≥ 2 × as-new value | replace or restore the rings; with abrasive particles the gap grows fast and BEP efficiency bleeds away | 🔴 |
| Semi-open impeller tip clearance > 0.060 in (1.5 mm) | efficiency degrades markedly; tightening from 0.060 to 0.010 in can recover up to 10 points of efficiency | 🟡 |
| Impeller trim beyond 10–20 % of diameter | rarely done; blade overlap is lost and the affinity-style prediction fails — confirm by test | 🔴 |
| Impeller mounted backwards (vanes forward-curved) | BEP efficiency collapses to roughly 47–71 % of normal while power can rise up to 1.8× — a field-diagnosis classic | 🔴 |
| Blade/vane count combination with multiple difference m = 0 or m = 1 | all blades pass the cutwater in phase — torque pulsation (m = 0) or acoustic resonance (m = 1); check to at least the 3rd order | 🔴 |
| Gap B (impeller–diffuser radial gap) < 4 % of impeller radius | pressure-field interaction — blade, vane and cutwater vibration with structural risk; normal band is 4–15 %, high-energy pumps take the upper end | 🔴 |
| Volute discharge cone angle > 7° or exit area > 2 × throat area | diffusion losses grow — the low-loss volute design window is exceeded | 🟡 |
Rules of thumb — quick estimates
| Name | Rule / formula | Applies to |
|---|---|---|
| Shutoff head coefficient | \(\psi_{s/o}=gH_{s/o}/U_2^{2}\approx0.585\); typical radial pumps \(\psi_{s/o}\gt0.5\) | radial impellers at \(Q=0\) |
| Shutoff power ratio | \(P_{s/o}\approx0.44\,P_{BEP}\); grows with \(N_s\) and with \(b_2/D_2\) | radial pumps |
| Shutoff power coefficient | \(\hat P_{s/o}=P_{s/o}/(\rho\omega^{3}r_2^{5})=f(b_2/D_2)\); \(b_2/D_2=0.1325\rightarrow\hat P\approx0.047\) | empirical chart estimate |
| Stability shutoff rise | design for \(H_{s/o}\ge(1.15\text{–}1.20)\,H_{BEP}\) | conservative stable-curve design |
| Slip factor | \(\mu=V_s/U_2\approx0.1\text{–}0.2\) | conventional closed impellers |
| Collector pressure recovery | static rise in volute/diffuser ≈ 1/3 of the impeller's own static rise; collector recovers 20–25 % of the stage total | conventional geometry |
| Impeller OD pressure | \(p_{OD}\approx75\text{–}80\,\%\) of the stage pressure rise above inlet | axial-thrust estimates |
| BEP vs shockless flow | \(Q_{BEP}\approx0.9\,Q_{SE}\) — blade blockage pulls BEP about 10 % below the shockless-entry flow | inlet design checks |
| Radial-vane curve shape | \(\beta_2=90°\): H–Q flat from shutoff to ≈ 75 % of \(Q_{BEP}\), steep beyond; single-stage heads to ≈ 8000 ft (2400 m) | small high-speed radial-vane pumps, \(D\lt6\) in, up to 30,000+ rpm |
| Volute throat velocity | \(r_T V_T\approx(0.9\text{–}0.95)\,r_2 V_{u2}\); radial diffuser throat \(\approx0.8\,r_2 V_{u2}\) | BEP collector sizing |
| Volute discharge cone | expansion angle 7°; exit area ≤ 2 × throat area | low-loss volute outlet |
| Blade blockage | vane thickness ≈ 2 % of \(D_2\) (halved near the leading edge); exit passage openness ≈ 85 % (up to 90 % on large wheels) | closed impellers |
| Inlet incidence | positive incidence 2–3° at the rms radius; shroud blade angle ≈ 1° below the flow angle | inlet vane design |
| Inlet deceleration | mean relative velocity \(W\) drops ≥ 10 % entering the blade row — minimizes incidence + friction loss at BEP and lowers NPSHr above BEP | radial impellers |
| Speed-versus-efficiency gain | 15,000 → 30,000 rpm: ≈ +15 efficiency points; 1240 → 1880 rpm on a small pump: ≈ +1 point | high-speed machine trade-off |
Reference data tables
Effect of wearing-ring clearance growth on performance
Values are percentages of the as-new BEP performance at the stated ring clearance (percent of new):
| \(N_s\) (\(\Omega_s\)) | Design head ft (m) | Clearance, % of new | \(Q\) % | \(H\) % | \(P\) % | \(\eta\) % | Shutoff \(H\) % | Shutoff \(P\) % |
|---|---|---|---|---|---|---|---|---|
| 2100 (0.77) | 63 (19.2) | 178 | 100 | 98.3 | 98.9 | 99.4 | 97.0 | 100 |
| 2100 (0.77) | 63 (19.2) | 356 | 100 | 97.5 | 99.0 | 98.5 | 93.6 | 98.2 |
| 2100 (0.77) | 63 (19.2) | 688 | 100 | 96.0 | 98.9 | 97.1 | 91.2 | 94.8 |
| 2100 (0.77) | 63 (19.2) | 1375 | 100 | 94.3 | 97.4 | 96.8 | 88.8 | 92.5 |
| 3500 (1.28) | 65 (19.8) | 354 | 100 | 90.0 | 99.1 | 90.8 | 85.0 | 96.2 |
| 4300 (1.57) | 41 (12.5) | 7270 | 62 | 65.5 | 81.7 | 49.8 | 44.3 | 106 |
| 4800 (1.76) | 26 (7.9) | 5220 | 96 | 78.8 | 89.2 | 84.8 | 78.2 | 83.3 |
The higher the specific speed, the faster clearance growth eats the curve — at \(N_s\approx4300\) a worn ring can cost half the BEP efficiency.
Impeller mounted backwards — measured penalties
Six pumps re-tested with the impeller reversed (vanes effectively forward-curved); values are percentages of normal BEP performance:
| Stages | \(N_s\) (\(\Omega_s\)) | Shutoff head % | Head % | Flow % | Power % | Efficiency % |
|---|---|---|---|---|---|---|
| 2 | 828 (0.303) | 86 | 111 | 65 | 104 | 71 |
| 2 | 1024 (0.375) | 82 | 112 | 88 | 145 | 68 |
| 1 | 1240 (0.454) | 75 | 105 | 38.5 | 68.5 | 59 |
| 1 | 1430 (0.523) | 82 | 106 | 69.7 | 138 | 53.5 |
| 1 | 2570 (0.940) | 74.5 | 117 | 62 | 138 | 52.5 |
| 1 | 2740 (1.003) | 77.5 | 138 | 61.5 | 180 | 47 |
Common mistakes
- Quoting "the pump head" as a single number. A pump has one head per flow; the installed head is wherever its curve crosses the system curve.
- Comparing or stacking curves at different speeds or diameters without affinity scaling — the shapes will not correspond point for point.
- Selecting the motor from BEP power on a high-specific-speed pump, where the shutoff power is the maximum the shaft will ever see.
- Reading wear off the duty point alone: ring clearance growth shows first in the shutoff head and in BEP efficiency, not in flow.
⭐ Deep-data appendix
Field-grade numbers behind the rules above: minimum-flow limits by service, starting and standstill criteria, the bookkeeping formulas for torque, acceleration time and temperature rise, plus measured trim-correction and vane-underfiling data.
Additional engineering criteria
| Situation | What it means | Verdict |
|---|---|---|
| Minimum flow vs suction-recirculation flow \(Q_{SR}\) | water service, ≤ 2,500 gpm and ≤ 150 ft: \(Q_{min}\ge50\,\%\,Q_{SR}\) continuous / 25 % intermittent; hydrocarbons: 60 % / 25 %; the higher the energy level, the closer \(Q_{min}\) must stay to \(Q_{SR}\) | 🟡 |
| Pumps in series, second-stage stuffing box | the box sees the first stage's discharge pressure — a special high-pressure box or leak-back to the first-stage suction is mandatory | 🔴 |
| Hot-liquid pump start-up | warm the pump to working temperature first (unless designed for fast start); never run a hot pump at shutoff — a metered-orifice bypass is required | 🔴 |
| Restart into backspin (discharge valve left open, no check valve) | the pump may be reverse-running at runaway speed; starting now means a long driver overload and a protection trip — confirm zero reverse rotation first | 🔴 |
| Synchronous-motor drive | starting torque between 90–100 % of rated speed must stay below the pull-in torque; the motor must pull the train into step from ≈ 95 % speed within ≈ 0.2 s (watch siphon-discharge propeller pumps) | 🟡 |
| Pumps in parallel at reduced combined flow | the operating split must never force any single unit below its own anti-recirculation minimum flow | 🔴 |
| Cavitation surge (1–6 Hz oscillation) | a recirculation–vapor-lock cycle at low flow; avoided by keeping the operating flow above \(Q_{min}\) — rare on low-\(N_{ss}\) impellers | 🟡 |
| Double-suction single-stage pump at low flow | axial thrust at part load can become the factor that sets the minimum flow | 🟡 |
Additional rules of thumb
| Name | Rule / formula | Applies to |
|---|---|---|
| Shaft torque | \(M=5252\,P/n\) lb·ft (\(P\) in hp); \(M=9549\,P/n\) N·m (\(P\) in kW); add ≈ 10 % when acceleration torque matters | driver sizing |
| Water horsepower | \(\text{lhp}=Q\cdot H\cdot SG/3960\) (gpm, ft); \(P\,[\text{kW}]=9.797\,Q\cdot H\cdot SG\) (m³/s, m) | hydraulic power |
| Acceleration time | \(\Delta t=I\,\Delta n/[k\,(M_m-M)]\); \(k=307\) (US: lb·ft², lb·ft) / 9.549 (SI: kg·m², N·m); requires \(M_m-M\) ≈ constant over the interval | start-up studies |
| Breakaway torque | ≈ 15 % of rated with sleeve bearings + packing, ≈ 10 % with rolling bearings; falls roughly linearly to near zero at 15–20 % of rated speed | starting-torque curves |
| Parallel combined efficiency | \(\eta=(H\cdot SG/k)\cdot(\sum Q/\sum P)\); \(k=3960\) (US: gpm, hp) / 0.1021 (SI: L/s, W) | parallel operation |
| Series combined efficiency | \(\eta=(Q\cdot SG/k)\cdot(\sum H/\sum P)\); same \(k\) as above | series operation |
| Pump temperature rise | \(\Delta T=gH(1-\eta)/(g_oC_pJ)+\Delta T_c\); US: \(g/g_o=1\), \(J=778\) ft·lbf/Btu; SI: \(J=1\) | minimum-flow / bypass design |
| Compression heating \(\Delta T_c\) | hydrocarbon fuels ≈ 3 °F/1000 psi (0.24 °C/MPa); 350 °F feedwater ≈ 1.6 °F/1000 psi (0.129 °C/MPa); cold water negligible — subtract it when inferring efficiency from temperature rise | thermodynamic efficiency tests |
| Temperature-rise limits | ΔT ≤ 100 °F (56 °C) general-purpose cold-liquid pumps; ≤ 50 °F (28 °C) modern boiler feed pumps — beyond that a minimum-flow bypass is mandatory | low-flow protection |
| Best-efficiency specific-speed band | highest BEP efficiency at \(N_s\approx2000\text{–}3000\) (\(\Omega_s\approx0.73\text{–}1.10\)); total-loss minimum near \(N_s\approx2500\) (\(\Omega_s\approx0.91\)) | double-suction single-stage, ≥ 12 in discharge |
| Exit-vane underfiling | filing the back of the exit vanes adds up to ≈ 10 % flow and often improves peak efficiency; overfiling changes almost nothing | capacity tuning |
| Two half-size pumps, one motor | two pumps in parallel on one driver can run ≈ 40 % faster than one double-capacity pump — the cheaper high-speed motor offsets much of the second pump | driver-cost trade-offs |
Impeller trim — empirical corrections vs the bare ratio
Worked corrections for a radial impeller starting at \(D=16\) in: the target diameter from the bare \(Q\propto D,\ H\propto D^{2}\) ratio is corrected upward by the empirical trim-correction chart, and an independent chart (\(k\approx0.6\) at \(N_s\approx1950\)) lands between the two. Prediction error against test grows with the depth of the cut (BEP flow error −2.9 % at the first trim to +7.3 % at the deepest):
| Target \(D'\) | Target \(D'/D\) | Corrected \(D'/D\) | Corrected \(D'\) | Independent chart \(D'\) |
|---|---|---|---|---|
| 15.125 in (384 mm) | 0.927 | 0.935 | 15.25 in (387 mm) | 15.60 in (396 mm) |
| 14.000 in (356 mm) | 0.858 | 0.874 | 14.26 in (362 mm) | 14.93 in (379 mm) |
Mixed-flow wheels behave far worse: trimming \(D\) 16 → 15⅝ → 15 in (mean diameter \(D_m=\sqrt{(D_o^2+D_i^2)/2}\): 13.17 → 12.89 → 12.62 in) produced measured flows off the prediction by −11.3 % to +17.3 %, and the second cut removed most of the blade overlap — \(N_s\cdot D_m\) is not conserved on mixed-flow impellers.
Exit-vane underfiling — measured results on nine pumps
Changes relative to the pre-filing test, after filing the back (low-pressure) side of the exit vanes (\(d_F/d\) = diameter ratio after/before filing):
| \(N_s\) (\(\Omega_s\)) | Stages | \(D_2\) in (cm) | \(d_F/d\) | Shutoff head Δ% | BEP head Δ% | BEP flow Δ% | BEP power Δ% | BEP η Δ pts |
|---|---|---|---|---|---|---|---|---|
| 862 (0.315) | 1 | 11.5 (29.2) | 1.13 | +4 | +1.5 | +4 | 0 | +9 / −5* |
| 945 (0.346) | 2 | 8⅜ (21.3) | 1.05 | +5.5 | 0 | +3 | +4 | −0.5 / +7.5* |
| 1000 (0.366) | 4 | 9⅛ (23.2) | 1.055 | +5.5 | 0 | +5.5 | 0 | +3 / +2.5* |
| 1080 (0.395) | 2 | 9⅞ (24.9) | 1.08 | +10 | +2.5 | +10 | 0 | +6.8 / +3* |
| 1525 (0.558) | 2 | 10½ (26.7) | 1.035 | +3.2 | +3 | +0.5 | +4.5 | +3.5 / +1.2* |
| 1950 (0.714) | 1 | 22 (55.9) | 1.02 | +1.5 | +1 | +1.5 | 0 | +1.5 / 0* |
| 3300 (1.208) dbl. | 1 | 12 (30.5) | — | +7.8 | +2.5 | +7.8 | 0 | +8.5 / −0.6* |
| 3450 (1.262) dbl. | 1 | 30½ (77.5) | — | +6.5 | +0.5 | +6.5 | 0 | +7.8 / −2.2* |
| 4300 (1.573) | 1 | 41¼ (104.8) | — | +5 | +5 | 0 | +0.5 | +4.5 / —* |
*Where two values appear, the pump was tested at two speeds. The consistent pattern: flow and shutoff head gain a few percent, efficiency usually improves — and the gains fade as \(N_s\) rises past about 3000.
Cavitation and suction-side criteria — extended
| Situation | What it means | Verdict |
|---|---|---|
| \(NPSH_A \ge NPSH_i\) (inception NPSH) | no cavitation at all — cavity length is zero; inception sits far above the 3 % point, typically \(NPSH_i \approx 2\text{–}5 \times NPSH_{3\%}\) with most observations near 5× | 🟢 |
| \(NPSH_{3\%} \lt NPSH_A \lt NPSH_i\) | extensive cavities already exist even though head has not dropped 3 % — erosion and pulsation are already at work between the two limits | 🟡 |
| \(NPSH_A \le NPSH_{3\%}\) | head drop begins — performance is no longer guaranteed | 🔴 |
| Suction specific speed \(S = N\sqrt{Q}/h_{sv}^{3/4}\) (double-suction: half the total flow) | \(S \le 6000\) (\(\Omega_{ss} \le 2.2\)) is the conservative band for large machines; 6,000–8,500 normal service; the standard recommendation is 8,500 and most pumps fall in 7,500–11,000 (\(\Omega_{ss}\) 2.7–4.0); above 11,000 (\(\Omega_{ss} \gt 4\)) only inducers or special designs belong | 🟢 ≤ 6000 / 🟡 mid / 🔴 > 11,000 |
| High-energy pump (high inlet-tip speed, high head per stage) | the 3 % point says almost nothing about damage — specify the damage NPSH that guarantees 40,000 h of impeller life (\(NPSH_{40,000h}\)), or "no / minimal visible cavity" on an inspection test; life ends when erosion has eaten 75 % of the blade thickness: \(\text{Life} = 0.75\,t/\text{MDPR}\), \(t\) = blade thickness (mm), MDPR = mean penetration rate (mm/yr) | 🟡 |
| Entrained gas at the impeller eye (GVF) | commercial industrial pumps tolerate only a gas volume fraction \(\lesssim 0.03\); beyond it head, flow and power sag until suction is lost (only special aerospace designs go higher) | 🟢 ≤ 0.03 / 🔴 above |
| Inducer + impeller running above rated flow | the combination's \(NPSH_r\) climbs steeply past the rated flow — constant-pitch inducers must not be run oversize on flow; variable-pitch inducers relax the limit | 🔴 |
| NPSH reduction credited for hot water / hydrocarbons | credit at most \(\min(50\,\%\times\text{cold-water }NPSH_r,\; 10\text{ ft} / 3.0\text{ m})\); forbidden outright when dissolved gas can come out of solution at low suction pressure, when the system sees strong transient pressure/temperature swings, or for liquids outside the tested reduction chart — add margin instead | 🟢 within cap / 🔴 if any disqualifier |
| Suction piping layout | two 90° elbows set at 90° to each other seed bad inlet flow and cavitation damage; 3–4 radial anti-swirl fins extending ≈ ¼ of the inlet diameter suppress preswirl at far less NPSH cost than full-length vanes | 🔴 bad layout / 🟢 fins |
| High-Δp labyrinth wearing ring | can excite large-amplitude, long-period shaft vibration; enlarging the second (inboard) clearance relative to the first calms it — at the price of more leakage | 🟡 |
Cavitation physics — quick numbers
| Name | Rule / formula | Applies to |
|---|---|---|
| Sixth-power erosion law | erosion rate \(\propto U_{t,1}^{\,6}\) (inlet blade-tip speed) — equivalently \(\propto NPSH^{3}\) at a fixed cavitation number; the higher the head per stage, the faster the damage | risk screening |
| Bubble lifecycle | life cycle ≈ 0.003 s; collapse pressure of order 10⁴ atm — no common material survives indefinite exposure | why cavitation destroys metal |
| Water-temperature peak | cavitation damage in water peaks at 100–120 °F (38–49 °C) | controlled tests |
| Inception vs 3 % point | \(NPSH_i \approx 2\text{–}5 \times NPSH_{3\%}\); the zero-head-drop band is very wide, so the 3 % figure is a performance convention, not a no-damage guarantee | acceptance testing |
| Typical installed \(NPSH_A\) | ≈ 60 % of barometric head ≈ 20 ft (6 m) in conventional installations — the basis for condensate-pump speed ceilings | condensate service |
| Pulsation peak | cavitation pressure-pulsation amplitude peaks near \(R \approx 2\) (\(R = NPSH_A/NPSH_{3\%}\)); the frequency rises as \(R\) grows | vibration diagnostics |
| Inducer benefit | runs at ≈ 2 × the suction specific speed of a plain impeller; inducer + impeller needs only ≈ 50 % of the impeller-alone \(NPSH_r\) (up to rated flow); the inducer contributes ≤ 5 % of the total head | high-\(S\) retrofits |
| Backflow recirculator | a passive recirculating-vane device removes inducer cavitation instability from shutoff clear out to runout | unstable-suction cures |
| Inception cavitation coefficient | \(\tau_i \approx 1\) at BEP for conventional blades, ≈ 0.5 for aerodynamically optimized blades; \(\tau_i\) rises at \(Q \lt Q_{BEP}\); \(NPSH_A \ge NPSH_i\) (i.e. \(\tau_A \le \tau_i\)) means zero cavity length | design targets |
| Double-suction convention | compute \(S\) with half the total flow (one eye) | \(S\) bookkeeping |
Material ranking against cavitation erosion
Mass loss in a 2 h magnetostrictive vibration test — the relative ranking of common pump materials (lower = more resistant):
| Material | Mass loss, mg / 2 h | Relative resistance |
|---|---|---|
| Rolled stellite | 0.6 | best (costly, hard to machine) |
| Welded aluminum bronze | 3.2 | excellent |
| Cast aluminum bronze | 5.8 | excellent |
| Welded stainless (17Cr-7Ni, 2 layers) | 6.0 | excellent |
| Hot-rolled stainless (26Cr-13Ni) | 8.0 | very good |
| Tempered rolled stainless (12Cr) | 9.0 | very good |
| Cast stainless (18Cr-8Ni) | 13.0 | good |
| Cast stainless (12Cr) | 20.0 | good |
| Cast manganese bronze | 80.0 | poor |
| Welded mild steel | 97.0 | poor |
| Steel plate | 98.0 | poor |
| Cast steel | 105.0 | poor |
| Aluminum | 124.0 | very poor |
| Brass | 156.0 | very poor |
| Cast iron | 224.0 | worst |
Read the ranking in orders of magnitude: each step down multiplies the erosion life roughly tenfold. Combine it with the 40,000-hour damage criterion when choosing material for a high-energy stage.
NPSH reduction for hot water and hydrocarbons — worked cases
Walking the reduction chart for a pump whose cold-water \(NPSH_r\) is 16 ft (4.9 m); the credit is capped at half the cold-water value, i.e. 8 ft (2.4 m):
| Liquid and temperature | Vapor pressure | Chart reduction | Credited | Resulting \(NPSH_r\) |
|---|---|---|---|---|
| Water at 100 °F (38 °C) — chart-walking demo | 30 psi abs (207 kPa) | ≈ 2.3 ft (0.70 m) | 2.3 ft (0.70 m) | — |
| Propane at 55 °F (13 °C) | ≈ 105 psi abs (724 kPa) | ≈ 9.5 ft (2.9 m) — above the 8 ft cap | 8 ft (2.4 m) | 16 − 8 = 8 ft (2.4 m) |
| Propane at 14 °F (−10 °C) | ≈ 50 psi abs (345 kPa) | ≈ 5.7 ft (1.7 m) — below the cap | 5.7 ft (1.7 m) | 16 − 5.7 ≈ 10 ft (3.0 m) |
How cavitation grows — head-drop anchor points
Cavitation coefficient \(\tau = 2g\cdot NPSH/U_e^{2}\) (\(U_e\) = eye-tip speed), walked down on a boiler-feed test stage; \(\tau_{available} = 0.41\):
| \(\tau\) | Cavity development stage |
|---|---|
| 0.61 | near inception (\(\tau_i\)) |
| 0.39 | cavity lengthening along the blade suction side |
| 0.29 | cavity closing on the adjacent-blade throat |
| 0.20 | 3 % head drop (head = 97 % of the cavitation-free value) — the cavity has reached the throat between adjacent blades |
High-energy pump suction stage — two eye designs
Same duty, two designs of eye flow coefficient \(f_e = V_e/U_e\); on the damage side the comparison reverses the 3 % comparison:
| Parameter | Small eye, \(f_e = 0.30\) | Large eye, \(f_e = 0.25\) |
|---|---|---|
| Eye diameter | 13.37 in (340 mm) | 13.92 in (354 mm) |
| Inlet-tip speed \(U_{t,1}\) | 274 ft/s (83.5 m/s) | 286 ft/s (87.2 m/s) |
| \(NPSH_{3\%}\) | slightly higher | slightly lower |
| Damage NPSH ratio \(R\) (40,000 h basis) | 1.69 | 2.06 |
For high-energy stages the small eye wins: the 3 % penalty is marginal, while the margin to the 40,000-hour damage NPSH is materially better (\(R = 1.69\) vs 2.06).