PE Exam Exam > PE Exam Notes > Mechanical Engineering for PE > Cheatsheet: Pumps And Turbines
Cheatsheet: Pumps And Turbines
1. Pump Fundamentals
1.1 Key Definitions
| Term | Definition |
|---|---|
| Head (H) | Energy per unit weight of fluid, measured in feet or meters of fluid column |
| Total Dynamic Head (TDH) | Total head pump must develop: TDH = hdischarge + hfriction + hvelocity - hsuction |
| Net Positive Suction Head (NPSH) | Absolute pressure head at pump suction above vapor pressure |
| NPSH Available (NPSHA) | Energy available at pump inlet: NPSHA = hatm + hs - hf - hvp |
| NPSH Required (NPSHR) | Minimum energy required at inlet to prevent cavitation (manufacturer specified) |
| Specific Speed (Ns) | Dimensionless parameter characterizing pump geometry: Ns = N√Q / H0.75 |
| Suction Specific Speed (S) | Cavitation parameter: S = N√Q / NPSHR0.75 |
1.2 Fundamental Equations
| Parameter | Equation |
|---|---|
| Pump Power (Water) | Php = (Q × H × SG) / (3960 × η) where Q in gpm, H in ft |
| Pump Power (SI) | PkW = (ρ × g × Q × H) / (1000 × η) where Q in m³/s, H in m |
| Hydraulic Power | Phydraulic = γ × Q × H = ρ × g × Q × H |
| Pump Efficiency | η = Phydraulic / Pinput = (γ × Q × H) / Pshaft |
| Specific Gravity Correction | Pactual = Pwater × SG |
1.3 Affinity Laws
| Parameter Change | Relationship |
|---|---|
| Flow Rate | Q2/Q1 = (N2/N1) × (D2/D1)3 |
| Head | H2/H1 = (N2/N1)2 × (D2/D1)2 |
| Power | P2/P1 = (N2/N1)3 × (D2/D1)5 |
| Speed Only (Constant D) | Q ∝ N; H ∝ N²; P ∝ N³ |
| Diameter Only (Constant N) | Q ∝ D³; H ∝ D²; P ∝ D⁵ |
2. Pump Types and Classification
2.1 Centrifugal Pumps
| Type | Characteristics |
|---|---|
| Radial Flow | Ns = 500-1500; High head, low flow; Disk-shaped impeller |
| Mixed Flow | Ns = 1500-4000; Medium head and flow; Combined radial and axial flow |
| Axial Flow (Propeller) | Ns = 4000-15000; Low head, high flow; Propeller-type impeller |
| Single Stage | One impeller; Head up to 150 ft per stage |
| Multistage | Multiple impellers in series; Htotal = n × Hstage |
| Single Suction | Inlet on one side of impeller; Simpler design |
| Double Suction | Inlet on both sides; Reduces axial thrust; Q doubles |
2.2 Positive Displacement Pumps
| Type | Characteristics |
|---|---|
| Reciprocating (Piston/Plunger) | High pressure capability; Pulsating flow; Q = A × L × N × ncylinders × ηv |
| Rotary (Gear, Screw, Vane) | Continuous flow; Self-priming; Viscous fluids; Q ∝ N |
| Diaphragm | Handles solids and corrosives; Sealed design |
| Peristaltic | Gentle pumping; Contamination-free; Low flow rates |
2.3 Pump Selection Criteria
- Centrifugal: High flow, low-medium head, clean fluids, variable flow
- Positive Displacement: High pressure, viscous fluids, constant flow, self-priming
- Specific Speed Guide: Ns < 2000="" (centrifugal),="" 2000-5000="" (mixed),=""> 5000 (axial)
- Suction Specific Speed: S > 11000 indicates good suction performance
3. Pump Performance and Curves
3.1 Performance Curve Characteristics
- H-Q Curve: Head decreases as flow increases (centrifugal pumps)
- Efficiency Curve: Peak efficiency at Best Efficiency Point (BEP)
- Power Curve: Power increases with flow for centrifugal pumps
- Operating Point: Intersection of pump curve and system curve
- Shut-off Head: Maximum head at zero flow
- Run-out Flow: Maximum flow at minimum head
3.2 System Curve
| Component | Equation |
|---|---|
| System Head | Hsystem = Hstatic + Hfriction = Hs + K × Q² |
| Friction Losses | hf = f × (L/D) × (V²/2g) + Σ K × (V²/2g) |
| Static Head | Hs = zdischarge - zsuction + (Pd - Ps)/γ |
3.3 Parallel and Series Operation
| Configuration | Result |
|---|---|
| Parallel Pumps | Qtotal = Q1 + Q2 at same head; Increases flow capacity |
| Series Pumps | Htotal = H1 + H2 at same flow; Increases head capacity |
| Parallel Efficiency | Each pump operates left of BEP; Less efficient than single large pump |
| Series Efficiency | Each pump operates near BEP; More efficient arrangement |
4. Cavitation and NPSH
4.1 Cavitation Fundamentals
- Occurs when local pressure drops below vapor pressure
- Vapor bubbles form and collapse, causing noise, vibration, and erosion
- Prevention: Ensure NPSHA > NPSHR with safety margin (3-5 ft recommended)
- Signs: Noise, vibration, performance drop, pitting damage
4.2 NPSH Calculations
| Parameter | Equation |
|---|---|
| NPSH Available | NPSHA = (Patm/γ) ± hs - hf,suction - (Pvp/γ) |
| NPSH Available (Pressure) | NPSHA = (Ps - Pvp)/γ + Vs²/(2g) |
| Atmospheric Pressure Head | hatm = 33.9 ft (water at sea level) = 10.33 m |
| Elevation Correction | Patm decreases ~0.5 psi per 1000 ft elevation increase |
4.3 Improving NPSH Available
- Increase suction head: Lower pump elevation or raise liquid level
- Reduce suction losses: Larger pipe diameter, shorter length, fewer fittings
- Decrease liquid temperature: Lower vapor pressure
- Pressurize suction vessel
- Use inducer on pump inlet
- Select pump with lower NPSHR
5. Turbine Fundamentals
5.1 Key Definitions
| Term | Definition |
|---|---|
| Impulse Turbine | Kinetic energy conversion; constant pressure across runner; Pelton wheel |
| Reaction Turbine | Pressure and velocity change across runner; Francis, Kaplan, propeller types |
| Head (H) | Available energy per unit weight: H = (P₁ - P₂)/γ + (V₁² - V₂²)/(2g) + z₁ - z₂ |
| Net Head | Hnet = Hgross - hlosses |
| Specific Speed (Ns) | Ns = N√P / H1.25 (turbines); characterizes turbine type |
5.2 Turbine Power and Efficiency
| Parameter | Equation |
|---|---|
| Theoretical Power | Ptheory = γ × Q × Hnet = ρ × g × Q × Hnet |
| Shaft Power (Water) | Php = (Q × H × η) / 3960 where Q in gpm, H in ft |
| Shaft Power (SI) | PkW = (ρ × g × Q × H × η) / 1000 where Q in m³/s, H in m |
| Overall Efficiency | η = ηhydraulic × ηmechanical × ηvolumetric |
| Hydraulic Efficiency | ηh = Pshaft / Pwater |
5.3 Euler Turbine Equation
| Form | Equation |
|---|---|
| General Form | P = ρ × Q × (U₁Vt1 - U₂Vt2) |
| Head Form | H = (U₁Vt1 - U₂Vt2) / g |
| Tangential Velocity | U = π × D × N / 60 where N in rpm |
| Torque | T = ρ × Q × (r₁Vt1 - r₂Vt2) |
6. Turbine Types
6.1 Impulse Turbines
| Type | Characteristics |
|---|---|
| Pelton Wheel | Ns < 5;="" high="" head="" (300-6000="" ft),="" low="" flow;="" one="" or="" more="" jets;="" bucket="" efficiency=""> |
| Turgo Turbine | Ns = 5-15; Medium head (150-1000 ft); Jet enters on one side, exits on opposite |
| Crossflow Turbine | Ns = 10-30; Low to medium head (10-650 ft); Flow passes through runner twice |
6.1.1 Pelton Wheel Design Parameters
- Jet velocity: Vj = Cv√(2gH) where Cv = 0.97-0.99
- Bucket velocity: U = φ√(2gH) where φ = 0.43-0.48 (optimum ~0.46)
- Number of jets: n = 1-6 (depends on specific speed)
- Jet diameter: d = √(4Q / πnVj)
- Pitch diameter: D = 10d to 20d
6.2 Reaction Turbines
| Type | Characteristics |
|---|---|
| Francis Turbine | Ns = 10-120; Medium head (50-1000 ft); Radial inlet, axial outlet; η = 0.90-0.95 |
| Kaplan Turbine | Ns = 120-400; Low head (10-200 ft), high flow; Adjustable propeller blades; η = 0.90-0.95 |
| Propeller Turbine | Ns = 100-300; Low head (10-100 ft); Fixed propeller blades; Less flexible than Kaplan |
| Bulb Turbine | Ns > 200; Very low head (5-60 ft); Generator in bulb in water passage |
6.3 Turbine Selection by Specific Speed
- Ns < 5:="" pelton="" wheel="" (single="">
- Ns = 5-15: Pelton wheel (multi-jet)
- Ns = 10-120: Francis turbine
- Ns = 120-400: Kaplan/Propeller turbine
- Selection: Calculate Ns from site conditions, choose turbine type accordingly
7. Performance and Operating Characteristics
7.1 Turbine Performance Parameters
| Parameter | Description |
|---|---|
| Design Head | Head at maximum efficiency; turbine optimized for this condition |
| Part Load Operation | Efficiency drops below design point; cavitation risk increases |
| Over Load Operation | Power increases but efficiency decreases; mechanical stress increases |
| Runaway Speed | Maximum speed with no load; Nrunaway ≈ 2 × Nrated (varies by type) |
7.2 Draft Tube
- Used with reaction turbines to recover kinetic energy at exit
- Allows installation above tailwater level
- Draft head: hd = (Patm/γ) - (Pvp/γ) - hlosses - safety margin
- Maximum theoretical height: ~10 m at sea level (limited by atmospheric pressure)
- Efficiency gain: 5-15% from kinetic energy recovery
7.3 Cavitation in Turbines
| Parameter | Description/Equation |
|---|---|
| Thoma Cavitation Parameter | σ = NPSH / Hnet = (Patm/γ - Pvp/γ - hs - hf) / Hnet |
| Critical Sigma | σc = manufacturer specified; σ > σc to avoid cavitation |
| Plant Sigma | σplant = (Hatm - Hvp - Hs) / Hnet |
| Maximum Setting Height | Hs,max = Hatm - Hvp - σc × Hnet |
7.4 Governing and Control
- Speed Governor: Maintains constant speed under varying load
- Wicket Gates (Francis): Control flow by changing guide vane angle
- Blade Pitch Control (Kaplan): Adjust runner blade angle for efficiency
- Needle Valve (Pelton): Controls jet diameter and flow rate
- Deflector (Pelton): Diverts jet for emergency shutdown without water hammer
8. Pump and Turbine Similarity
8.1 Dimensional Analysis
| Parameter | Equation |
|---|---|
| Flow Coefficient | φ = Q / (N × D³) |
| Head Coefficient | ψ = g × H / (N² × D²) |
| Power Coefficient | λ = P / (ρ × N³ × D⁵) |
| Reynolds Number | Re = ρ × N × D² / μ |
| Specific Speed (Pumps) | Ns = N√Q / H0.75 (US units: rpm, gpm, ft) |
| Specific Speed (Turbines) | Ns = N√P / H1.25 (US units: rpm, hp, ft) |
8.2 Model Testing and Scaling
- Geometric Similarity: Model and prototype have same shape (Dm/Dp = scale ratio)
- Kinematic Similarity: Velocity triangles similar at corresponding points
- Dynamic Similarity: Force ratios identical (same dimensionless parameters)
- Froude Number: Fr = V / √(gL); important for free surface flows
- Model-Prototype Relations: Use affinity laws with scale factor
8.3 Unit Quantities
| Parameter | Definition |
|---|---|
| Unit Speed | N11 = N × D / √H (speed for 1 m diameter, 1 m head) |
| Unit Discharge | Q11 = Q / (D² × √H) |
| Unit Power | P11 = P / (D² × H1.5) |
| Homologous Operation | Operation at same N11 and Q11 ensures similar performance |
9. Special Topics and Applications
9.1 Water Hammer
| Parameter | Equation |
|---|---|
| Wave Speed | a = √(K/ρ) / √(1 + (K×D)/(E×t)) where K = bulk modulus, E = pipe modulus |
| Pressure Rise (Sudden Closure) | ΔP = ρ × a × ΔV (Joukowsky equation) |
| Pressure Rise (Head) | ΔH = a × ΔV / g |
| Wave Period | T = 4L / a where L = pipe length |
9.1.1 Water Hammer Mitigation
- Slow valve closure: Extend closure time beyond 2L/a
- Surge tanks: Absorb pressure fluctuations
- Air chambers: Provide compressible volume
- Pressure relief valves: Limit maximum pressure
- Flywheels: Maintain turbine speed during load changes
9.2 Pump-Turbine (Reversible)
- Used in pumped storage hydroelectric plants
- Francis-type design operates both as pump and turbine
- Pump mode: Low head, high efficiency required
- Turbine mode: Generate power during peak demand
- Round-trip efficiency: 70-85%
- Transition time: 1-3 minutes between modes
9.3 Pump and Turbine Losses
| Loss Type | Description |
|---|---|
| Hydraulic Losses | Friction in passages, shock losses, flow separation; 5-10% of input |
| Volumetric Losses | Leakage past seals and clearances; 2-5% of flow |
| Mechanical Losses | Bearing friction, seal friction; 1-3% of power |
| Disc Friction | Fluid friction on rotating surfaces; increases with speed |
| Recirculation Losses | Flow reversal at off-design conditions; reduces efficiency |
9.4 Pump Testing Standards
- Hydraulic Institute Standards (HI): ANSI/HI 1.6 Centrifugal Pump Tests
- ISO 9906: Rotodynamic pumps - Hydraulic performance acceptance tests
- Test tolerance grades: Grade 1 (±5% efficiency), Grade 2 (±8% efficiency)
- Required measurements: Flow, head, power, speed, temperature
- Test loop: Calibrated instrumentation, steady-state operation
9.5 Turbine Testing Standards
- IEC 60041: Field acceptance tests to determine hydraulic performance
- ASME PTC 18: Performance test codes for hydraulic turbines
- Model testing: IEC 60193 for laboratory testing of scale models
- Accuracy: ±0.2% efficiency for precision testing
- Index testing: Periodic checks to monitor performance degradation
10. Important Constants and Conversions
10.1 Physical Properties of Water
| Property | Value |
|---|---|
| Density (ρ) at 68°F | 62.4 lb/ft³ = 1000 kg/m³ |
| Specific Weight (γ) | 62.4 lb/ft³ = 9810 N/m³ |
| Atmospheric Pressure | 14.7 psi = 33.9 ft H₂O = 101.3 kPa = 10.33 m H₂O |
| Vapor Pressure at 68°F | 0.339 psi = 0.78 ft H₂O |
| Vapor Pressure at 100°F | 0.949 psi = 2.19 ft H₂O |
| Vapor Pressure at 212°F | 14.7 psi = 33.9 ft H₂O (boiling point) |
10.2 Unit Conversions
| Parameter | Conversion |
|---|---|
| Flow Rate | 1 gpm = 0.002228 ft³/s = 0.0631 L/s = 3.785 L/min |
| Head | 1 ft H₂O = 0.433 psi = 2.989 kPa; 1 m H₂O = 9.81 kPa |
| Power | 1 hp = 550 ft·lb/s = 0.746 kW = 2545 Btu/hr |
| Pressure | 1 psi = 2.31 ft H₂O = 6.895 kPa; 1 bar = 14.5 psi = 100 kPa |
| Speed | 1 m/s = 3.281 ft/s; 1 ft/s = 0.6818 mph |
10.3 Quick Reference Formulas
| Application | Formula (US Units) |
|---|---|
| Pump hp (Water) | hp = (gpm × ft × SG) / (3960 × η) |
| Velocity in Pipe | V (ft/s) = 0.408 × Q (gpm) / D² (in²) |
| Friction Loss (Darcy-Weisbach) | hf = f × (L/D) × (V²/2g) |
| Reynolds Number | Re = V × D / ν = 50.6 × Q (gpm) / (D (in) × ν (cSt)) |
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