| name | positive-displacement-pumps |
| description | Design and analyze gear, piston, and screw pumps with volumetric efficiency |
| category | thinking |
| domain | mechanical |
| complexity | intermediate |
| dependencies | ["numpy"] |
Positive Displacement Pumps
Positive displacement (PD) pumps move fluid by trapping a fixed volume and forcing it into a discharge pipe. Unlike centrifugal pumps, they deliver nearly constant flow regardless of discharge pressure.
Types of Positive Displacement Pumps
Gear Pumps
External Gear Pumps
- Two meshing gears rotate in opposite directions
- Fluid trapped between gear teeth and casing
- Simple, reliable, good for clean, viscous fluids
- Flow proportional to speed
- Typical efficiency: 80-95%
Internal Gear Pumps
- One gear inside another
- Crescent-shaped seal between gears
- Smoother flow, less pulsation
- Good for viscous fluids
- Compact design
Piston Pumps (Reciprocating)
Single-Acting Piston
- Fluid displaced on one stroke only
- High pulsation
- Simple construction
Double-Acting Piston
- Fluid displaced on both strokes
- Reduced pulsation
- Higher efficiency
Multi-Piston (Triplex, Quintuplex)
- Multiple pistons offset in phase
- Smoother flow
- Common in high-pressure applications
- Typical efficiency: 85-95%
Diaphragm Pumps
Air-Operated Double Diaphragm (AODD)
- Two flexible diaphragms
- Air pressure drives operation
- Self-priming, can run dry
- Excellent for slurries and solids
- Lower efficiency (~30-70%)
Mechanically Driven
- Diaphragm actuated by mechanical linkage
- Higher efficiency than AODD
- Good for metering applications
Screw Pumps
Single Screw (Progressive Cavity)
- Rotor rotates within stator
- Continuous, non-pulsating flow
- Excellent for viscous, shear-sensitive fluids
- Self-priming
Twin/Triple Screw
- Two or three intermeshing screws
- Low pulsation
- Good for high-pressure applications
- Typical efficiency: 75-90%
Lobe Pumps
- Two or more lobes rotate in opposite directions
- Gentle handling of product
- Common in food, pharmaceutical industries
- Easy to clean (sanitary designs)
- Typical efficiency: 50-80%
Key Characteristics
Constant Flow Behavior
Ideal Behavior:
- Flow rate independent of discharge pressure
- Flow proportional to speed only
- Q = N × V_d
Where:
- Q = volumetric flow rate
- N = pump speed (rpm)
- V_d = displacement per revolution
Real Behavior:
- Flow decreases slightly with pressure (slip)
- Efficiency varies with operating conditions
Volumetric Efficiency
Volumetric efficiency accounts for internal leakage (slip):
η_v = Q_actual / Q_theoretical
η_v = (Q_theoretical - Q_slip) / Q_theoretical
Factors affecting volumetric efficiency:
- Clearances and wear
- Fluid viscosity (higher = better sealing)
- Differential pressure (higher = more leakage)
- Operating speed
Slip and Leakage
Slip Flow:
Q_slip = C × ΔP / μ
Where:
- C = slip coefficient (depends on clearances)
- ΔP = differential pressure
- μ = dynamic viscosity
Implications:
- Viscous fluids: less slip, higher efficiency
- High pressures: more slip, lower efficiency
- Worn pumps: increased clearances, more slip
Pulsation
Causes:
- Discrete volume displacement
- Reciprocating motion
- Gear tooth engagement/disengagement
Pulsation Index:
PI = (Q_max - Q_min) / Q_avg × 100%
Typical Pulsation Levels:
- Single piston: Very high (100%+)
- Duplex piston: High (~50-60%)
- Triplex piston: Moderate (~10-20%)
- Gear pumps: Low to moderate (~5-15%)
- Screw pumps: Very low (<5%)
Self-Priming Capability
Most PD pumps are self-priming:
- Create vacuum on suction side
- Can lift fluid from below pump
- Can evacuate air from suction line
Limitations:
- Maximum suction lift ~8m (limited by atmospheric pressure)
- Requires reasonable seal condition
- May need priming for high-viscosity fluids
Design Calculations
Displacement Per Revolution
Gear Pump:
V_d = 2 × π × b × (D_o² - D_i²) / 4
Where:
- b = gear width
- D_o = outer diameter
- D_i = inner (root) diameter
Piston Pump:
V_d = (π × d² / 4) × L × n
Where:
- d = piston diameter
- L = stroke length
- n = number of pistons (single-acting) or 2n (double-acting)
Screw Pump:
V_d = 4 × A_c × p
Where:
- A_c = cavity area
- p = pitch
Theoretical Flow Rate
Q_theoretical = V_d × N / 60
Where:
- Q_theoretical in m³/s or L/min
- V_d in m³ or L
- N in rpm
Actual Flow Rate (Accounting for Slip)
Q_actual = η_v × Q_theoretical
Q_actual = Q_theoretical - Q_slip
Slip as function of pressure and viscosity:
Q_actual = Q_theoretical - (C × ΔP / μ)
For design:
- Specify desired flow at operating pressure
- Account for expected volumetric efficiency
- Select pump with adequate theoretical capacity
Power Requirements
Hydraulic Power:
P_hydraulic = Q × ΔP
Where:
- P in Watts
- Q in m³/s
- ΔP in Pa
Brake Power (Shaft Power):
P_brake = P_hydraulic / η_overall
η_overall = η_v × η_m
Where:
- η_v = volumetric efficiency
- η_m = mechanical efficiency (bearings, seals)
Typical Overall Efficiencies:
- Gear pumps: 70-85%
- Piston pumps: 80-90%
- Screw pumps: 70-85%
- Diaphragm pumps: 30-70%
Motor Power (with safety factor):
P_motor = P_brake × SF
SF typically 1.15-1.25
NPSH Requirements
PD pumps generally require lower NPSH than centrifugal pumps:
NPSH_required = P_atm/ρg - h_suction - h_friction - P_vapor/ρg - safety_margin
Typical NPSH_required:
- 0.5-2 m for most PD pumps
- Higher for high-speed pumps
- Lower for slow-speed pumps
When to Use PD vs Centrifugal Pumps
Choose Positive Displacement When:
-
High-Pressure Applications
- ΔP > 10-20 bar
- PD pumps maintain efficiency at high pressure
- Centrifugal pumps become impractical
-
Viscous Fluids
- μ > 100 cP
- PD efficiency improves with viscosity
- Centrifugal efficiency drops dramatically
-
Constant Flow Required
- Metering and dosing
- Flow independent of pressure variations
- Predictable delivery
-
Low Flow, High Pressure
- Centrifugal pumps inefficient at low flow
- PD pumps excel in this range
-
Self-Priming Required
- Suction lift needed
- Air entrainment possible
- Dry-run capability
-
Shear-Sensitive Fluids
- Food products, polymers
- Use lobe or progressive cavity pumps
- Gentle handling
Choose Centrifugal When:
-
High Flow, Low Pressure
- Q > 100 m³/h, ΔP < 10 bar
- More economical
- Simpler maintenance
-
Low Viscosity Fluids
- μ < 50 cP (water-like)
- Centrifugal pumps efficient
- Less expensive
-
Continuous, Smooth Flow
- No pulsation acceptable
- Variable flow needed
- Throttling control
-
Particulate Handling
- Large solids
- PD pumps can jam
- Centrifugal more forgiving
-
Lower Initial Cost
- Simple installation
- Standard motors
- Lower maintenance
Pulsation Dampening
Pulsation can cause:
- Vibration and noise
- Inaccurate flow measurement
- Pressure spikes
- System fatigue
Dampening Methods
1. Pulsation Dampener (Accumulator)
Gas-Charged Bladder Type:
- Bladder separates gas and fluid
- Gas compresses during pressure peaks
- Gas expands during pressure valleys
- Smooth flow output
Sizing:
V_dampener = (Q_theoretical × C_d) / (η_p × f)
Where:
- C_d = dampening coefficient (typically 5-10)
- η_p = pulsation reduction efficiency (0.9-0.95)
- f = pump frequency (Hz)
Gas Pre-Charge Pressure:
P_precharge = 0.6 × P_operating (typical)
2. Multiple Pistons
Triplex Pump (3 pistons at 120°):
- Pulsation reduced ~90%
- Common in high-pressure applications
Quintuplex Pump (5 pistons at 72°):
- Pulsation reduced ~95%
- Smoother than triplex
3. Air Chambers
Simple expansion chamber on discharge:
- Gas cushion absorbs pulsation
- Requires regular air charging
- Lower cost than bladder type
4. Flexible Discharge Line
- Hose instead of rigid pipe (short section)
- Elasticity absorbs pulses
- Simple, low cost
- Limited effectiveness
5. Flow Stabilizer
- Restrictor orifice
- Creates back pressure
- Dampens pressure fluctuations
- Energy loss
Design Considerations
Critical Frequencies:
Avoid resonance with system natural frequency:
f_pump = N × n / 60
Where:
- N = pump speed (rpm)
- n = number of pistons or pumping chambers
Dampener Location:
- Close to pump discharge
- Before flow meter (if smooth flow required)
- Consider accessibility for maintenance
Maintenance:
- Check bladder integrity
- Verify gas pre-charge pressure
- Inspect for leaks
Application Selection Guide
High-Pressure Chemical Injection
Recommended: Plunger pump (triplex)
- High pressure capability (up to 1000+ bar)
- Good metering accuracy
- Pulsation dampener required
Viscous Oil Transfer
Recommended: Gear pump or screw pump
- Handles high viscosity well
- Self-priming
- Relatively smooth flow
Slurry and Solids
Recommended: Diaphragm pump or progressive cavity
- Can handle solids
- Won't jam easily
- Gentle action
Food and Pharmaceutical
Recommended: Lobe pump or sanitary diaphragm
- Hygienic design
- Easy to clean
- Gentle product handling
Metering and Dosing
Recommended: Diaphragm metering pump or plunger pump
- Excellent accuracy
- Adjustable stroke
- Handles chemicals
General Water Transfer
Recommended: Centrifugal pump
- Lower cost
- Lower maintenance
- Adequate for low viscosity
Design Example Workflow
-
Define Requirements:
- Flow rate (actual, at operating pressure)
- Differential pressure
- Fluid properties (density, viscosity)
- Operating conditions
-
Select Pump Type:
- Based on application guide above
- Consider fluid compatibility
- Evaluate cost constraints
-
Calculate Theoretical Capacity:
- Account for volumetric efficiency
- Q_theoretical = Q_actual / η_v
-
Size Displacement:
- V_d = Q_theoretical × 60 / N
- Choose operating speed N
-
Calculate Power:
- P_hydraulic = Q × ΔP
- P_brake = P_hydraulic / η_overall
- P_motor = P_brake × SF
-
Check NPSH:
- NPSH_available > NPSH_required + margin
- Size suction piping appropriately
-
Evaluate Pulsation:
- Calculate pulsation index
- Determine if dampening needed
- Size dampener if required
-
Verify Operating Range:
- Minimum/maximum speed
- Pressure limitations
- Viscosity range
Performance Monitoring
Key Parameters to Track:
- Flow rate vs. speed (check for increased slip)
- Discharge pressure
- Power consumption (increased = wear)
- Vibration levels
- Temperature
Indicators of Wear:
- Reduced flow at same speed
- Increased power consumption
- Increased noise/vibration
- Reduced volumetric efficiency
Maintenance Planning:
- Replace seals/gaskets per schedule
- Monitor clearances in gear pumps
- Check valve seats in piston pumps
- Inspect diaphragms regularly
Summary
Positive displacement pumps are essential for:
- High-pressure applications
- Viscous fluid handling
- Metering and constant flow
- Self-priming requirements
Key design considerations:
- Account for volumetric efficiency (slip)
- Size for actual flow needed
- Consider pulsation dampening
- Match pump type to application
Trade-offs vs. centrifugal:
- Higher pressure capability
- Better viscosity handling
- Pulsating flow
- Higher initial cost
- More maintenance
