Centrifugal pumps: characteristics and cavitation

Why it matters

Centrifugal pumps circulate engine coolant, feed boilers, supply water and move fuel and process liquids — they are the most common rotodynamic machine an engineer will select, run and troubleshoot. Choosing a pump means matching its head–flow curve to the system, staying near the best-efficiency point and providing enough suction head to avoid cavitation. All of this follows from the Euler equation and the affinity laws.

Key ideas

• Components. Suction pipe with foot valve and strainer; an impeller with curved vanes (usually backward-curved); a volute casing or diffuser ring that converts kinetic energy into pressure; a delivery pipe with a delivery valve.
• Priming. A centrifugal pump must be filled with liquid before starting. The pressure it develops is proportional to fluid density, so with air in the casing it cannot lift water up the suction pipe.
• Euler head. With radial (whirl-free) entry, V_w1 = 0, the theoretical head is H_e = V_w2·u₂/g.
• Blade outlet angle β₂ (measured from the tangent, opposite to the direction of rotation):
  • Backward-curved (β₂ < 90°): head falls as Q rises; stable, non-overloading power curve; highest efficiency — the usual choice.
  • Radial (β₂ = 90°): theoretical head constant with Q.
  • Forward-curved (β₂ > 90°): head rises with Q; power keeps rising (overloading), less stable.
• Real head. The finite number of blades makes the actual whirl smaller than ideal (slip, slip factor σ_s ≈ 0.8–0.9), and friction and shock losses reduce it further. The manometric head H_m is the head actually measured across the pump: suction and delivery static lifts plus pipe losses plus exit velocity head.
• Efficiencies. Manometric η_mano = g·H_m/(V_w2·u₂). Mechanical η_m = impeller power / shaft power. Overall η_o = ρgQH_m / shaft power.
• Characteristic curves.
  • Main characteristics: H, P and η against Q at constant speed. The best-efficiency point (BEP) is the design point.
  • Operating characteristics at the rated speed; muschel (iso-efficiency) curves over a range of speeds.
  • The system curve H_sys = H_static + k·Q² (static lift plus friction and minor losses). The operating point is where it intersects the pump curve. Throttling the delivery valve steepens the system curve and moves the point to lower Q.
• Affinity laws (same pump, or geometrically similar pumps): Q ∝ ND³, H ∝ N²D², P ∝ N³D⁵. Trimming the impeller diameter in the same casing follows these approximately.
• Specific speed N_s = N·√Q/H^(3/4) characterises impeller shape: low N_s → radial impeller (high head, low flow); medium → mixed flow; high → axial (propeller) pumps.
• Pumps in series and parallel. In series, heads add at the same Q — used for high heads, as in multistage pumps. In parallel, flows add at the same H — used for large or variable demand.
• Minimum starting speed. Flow starts only when the centrifugal head (u₂² − u₁²)/(2g) exceeds the manometric head.
• Cavitation and NPSH. The lowest pressure is at the impeller eye. If it falls to the liquid's vapour pressure, bubbles form and collapse further in, causing noise, vibration, pitting and a sudden drop in head and efficiency.
  • NPSH available = (absolute pressure head at the suction flange + velocity head) − vapour pressure head, which depends on the installation.
  • NPSH required comes from the pump maker's test.
  • Avoid cavitation by keeping NPSH_a > NPSH_r with a margin: reduce the suction lift (or flood the suction), use a short, large suction pipe with few fittings, keep the liquid cool, and avoid running far beyond BEP flow.
  • Thoma's cavitation number σ = NPSH/H.

Formulas

u = π·D·N/60 Blade speed (m/s); D diameter (m), N speed (rpm).

H_e = V_w2·u₂/g V_w2 = u₂ − V_f2/tan β₂ Euler head (m) with radial entry; V_f2 flow velocity at outlet (m/s), β₂ blade outlet angle.

Q = π·D₂·b₂·V_f2 Flow rate (m³/s); b₂ outlet width (m); blade thickness neglected.

η_mano = g·H_m / (V_w2·u₂) P_shaft = ρ·g·Q·H_m/η_o Manometric efficiency; shaft power (W).

H = (p₂ − p₁)/(ρg) + (V₂² − V₁²)/(2g) + (z₂ − z₁) Head from gauge readings at the pump flanges.

Q₂/Q₁ = N₂/N₁, H₂/H₁ = (N₂/N₁)², P₂/P₁ = (N₂/N₁)³ Affinity laws for one pump at a changed speed (D fixed).

N_s = N·√Q / H^(3/4) Pump specific speed; N (rpm), Q (m³/s), H (m) per stage.

NPSH_a = (p_atm − p_v)/(ρg) − z_s − h_fs Available NPSH (m) for a pump drawing from an open sump; z_s suction lift (pump above liquid level, m), h_fs suction-line losses (m), p_v vapour pressure (Pa).

Worked examples

Example 1 (standard) — impeller head and power. Given: impeller outer diameter D₂ = 0.4 m, outlet width b₂ = 30 mm, N = 1450 rpm, blade outlet angle β₂ = 30°, outlet flow velocity V_f2 = 2.5 m/s, radial entry, manometric efficiency 80 %, overall efficiency 72 %. Find the manometric head, discharge and shaft power.

1. u₂ = π × 0.4 × 1450/60 = 30.37 m/s.
2. V_w2 = u₂ − V_f2/tan β₂ = 30.37 − 2.5/tan 30° = 30.37 − 4.33 = 26.04 m/s.
3. Euler head H_e = 26.04 × 30.37/9.81 = 80.61 m.
4. H_m = 0.80 × 80.61 = 64.49 m.
5. Q = π × 0.4 × 0.03 × 2.5 = 0.09425 m³/s.
6. P = ρgQH_m/η_o = 1000 × 9.81 × 0.09425 × 64.49/0.72 = 82 810 W. Answer: H_m ≈ 64.5 m, Q ≈ 0.094 m³/s, shaft power ≈ 82.8 kW.

Example 2 (GATE level) — speed change and NPSH check. Given: a pump at 1450 rpm delivers 0.05 m³/s against 30 m and absorbs 20 kW. (a) Find Q, H and P at 1750 rpm and the specific speed at 1450 rpm. (b) It lifts water at 30 °C (p_v = 4.25 kPa) from an open sump (p_atm = 101.325 kPa) with the pump 4 m above the water level and suction losses of 1.2 m. NPSH_r = 5 m. Will it cavitate, and what is the maximum allowable suction lift?

1. Speed ratio r = 1750/1450 = 1.2069.
2. Q = 0.05 × 1.2069 = 0.0603 m³/s; H = 30 × 1.2069² = 43.70 m; P = 20 × 1.2069³ = 35.16 kW.
3. N_s = N·√Q/H^(3/4) = 1450 × √0.05/30^0.75 = 1450 × 0.2236/12.82 = 25.3 → a radial-flow impeller.
4. (b) (p_atm − p_v)/(ρg) = (101 325 − 4250)/(1000 × 9.81) = 9.896 m.
5. NPSH_a = 9.896 − 4 − 1.2 = 4.70 m < NPSH_r = 5 m → it will cavitate.
6. Maximum suction lift for NPSH_a = 5 m: z_s = 9.896 − 1.2 − 5 = 3.70 m. Answer: 0.0603 m³/s, 43.7 m, 35.2 kW at 1750 rpm; N_s ≈ 25; NPSH_a ≈ 4.70 m — lower the pump to at most about 3.7 m above the sump (in practice, keep a further safety margin).

Common mistakes

• Measuring β₂ from the radial direction when the formula uses the tangential direction, or vice versa.
• Applying affinity laws to the system (they apply to the pump curve; the operating point moves along the system curve).
• Using gauge pressure and forgetting vapour pressure in NPSH — NPSH uses absolute pressures.
• Mixing pump specific speed (√Q) with turbine specific speed (√P).
• Adding heads for pumps in parallel or flows for pumps in series.
• Assuming a centrifugal pump can self-prime with air in the casing.

For GATE ME

Expect: Euler and manometric head from outlet triangles, power and efficiencies, affinity-law scaling, specific speed, operating point of a pump with a given system curve (often a quadratic), pumps in series or parallel, minimum starting speed, and NPSH or maximum suction lift. Practise sketching the H–Q, η–Q and system curves — many conceptual MCQs are read off that sketch.

Quick check

1. Which blade shape gives a non-overloading power characteristic?
2. If speed is doubled, what happens to the head and power of a pump?
3. A pump delivers 4 kW of hydraulic power with 5 kW shaft input. What is its overall efficiency?
4. Gauge readings across a pump are 150 kPa and 350 kPa (same level and pipe size). What is the head, in metres of water?
5. Why must a centrifugal pump be primed? Answers: 1. Backward-curved. 2. H rises 4 times, P rises 8 times. 3. 80 %. 4. 200 000/9810 = 20.4 m. 5. The head developed is the same in metres of any fluid, so in air the pressure rise is ~800 times too small to lift water.