name	pump-performance-curves
description	Generate and interpret H-Q curves, apply affinity laws, and predict off-design performance
category	thinking
domain	mechanical
complexity	intermediate
dependencies	["numpy","matplotlib","scipy"]

Pump Performance Curves

Overview

This skill provides comprehensive methods for generating, interpreting, and manipulating pump performance curves. Performance curves are fundamental to pump selection, system design, and operational analysis. The skill covers the four primary curve types (H-Q, η-Q, P-Q, and NPSH-Q), affinity laws for scaling performance, curve fitting techniques, and operating point determination.

Performance Curve Components

1. Head-Flow (H-Q) Curve

The H-Q curve shows the relationship between total dynamic head and volumetric flow rate.

Characteristics:

Shutoff Head (H₀): Head at zero flow (Q = 0)
Design Point: Nominal operating condition (Q_design, H_design)
Runout Flow: Maximum flow at zero head
Curve Shape: Typically descending, influenced by impeller design

Mathematical Representation:

H(Q) = H₀ - a·Q - b·Q²

Or polynomial form:

H(Q) = c₀ + c₁·Q + c₂·Q²

Physical Interpretation:

Steep curve: Radial impeller, low specific speed
Flat curve: Axial impeller, high specific speed
Rising curve (unstable): Poor design, avoid

Stability Criteria:

dH/dQ < 0  (stable operation)
dH/dQ > 0  (unstable - avoid)

2. Efficiency-Flow (η-Q) Curve

The efficiency curve shows how effectively the pump converts shaft power to hydraulic power.

Key Points:

Best Efficiency Point (BEP): Peak of efficiency curve
Preferred Operating Range: 70-120% of BEP flow
Acceptable Range: 50-150% of BEP flow
Avoid: Operation below 40% or above 150% of BEP

Efficiency Definition:

η = (ρ·g·Q·H) / P_shaft

Typical Efficiency Curve Shape:

η(Q) = η_max · (1 - k₁·(Q/Q_BEP - 1)² - k₂·(Q/Q_BEP - 1)⁴)

Efficiency by Pump Size:

Small pumps (P < 20 kW): η = 50-75%
Medium pumps (20-200 kW): η = 70-85%
Large pumps (P > 200 kW): η = 80-92%

3. Power-Flow (P-Q) Curve

The power curve shows shaft power consumption vs. flow rate.

Power Calculation:

P_shaft = (ρ·g·Q·H) / η

Curve Shapes by Impeller Type:

Backward-Curved Blades (most common):

Power increases with flow
Non-overloading characteristic
Motor sized for maximum flow

Radial Blades:

Steep power increase
Can overload motor
Requires careful motor selection

Forward-Curved Blades (rare):

Very steep power curve
Highly overloading
Generally avoided

Motor Selection:

P_motor = P_max / η_motor · SF

Where SF = safety factor (typically 1.10-1.25)

4. NPSH-Flow (NPSH_req-Q) Curve

Net Positive Suction Head Required curve shows minimum inlet pressure needed to avoid cavitation.

NPSH Definition:

NPSH_req = (P_inlet - P_vapor)/(ρ·g) + v²/(2g)

Typical Behavior:

NPSH_req(Q) = NPSH₀ + k·(Q/Q_BEP)^n

Where n ≈ 1.5-2.5

Safety Margin:

NPSH_available > NPSH_required · 1.3  (minimum)
NPSH_available > NPSH_required · 2.0  (recommended)

Thoma Cavitation Parameter:

σ = NPSH_req / H

Typical values:

Radial pumps: σ = 0.06-0.12
Mixed flow: σ = 0.10-0.20
Axial pumps: σ = 0.15-0.35

Curve Characteristics and Interpretation

Shutoff Head

Definition: Maximum head at zero flow.

Typical Ratios:

H₀/H_BEP = 1.10-1.35  (centrifugal pumps)

Significance:

Protection against deadhead operation
Check valve selection
System pressure relief sizing

Best Efficiency Point (BEP)

Definition: Operating point with maximum hydraulic efficiency.

Significance:

Minimum wear and vibration
Longest bearing life
Lowest energy consumption
Reference for operating range

Design Guidelines:

Size pump so normal duty is 70-110% of BEP
Avoid continuous operation far from BEP
Maximum deviation: ±30% of Q_BEP

BEP Identification:

dη/dQ = 0  (at BEP)
d²η/dQ² < 0  (maximum, not minimum)

Operating Range

Preferred Operating Range (POR):

0.7·Q_BEP ≤ Q ≤ 1.2·Q_BEP

Acceptable Operating Range (AOR):

0.5·Q_BEP ≤ Q ≤ 1.5·Q_BEP

Continuous Operation Limits:

Minimum: 40% Q_BEP (avoid recirculation)
Maximum: 150% Q_BEP (avoid cavitation and vibration)

Stable vs. Unstable Regions

Stable Curve:

dH/dQ < 0  for all Q

Single intersection with system curve
Predictable operation
Desirable characteristic

Unstable Curve:

dH/dQ > 0  for some Q range

Multiple intersections possible
Hunting and oscillation
Parallel pump problems
Avoid in design

Dip in H-Q Curve:

Common at low specific speeds
Caused by recirculation
Creates unstable region
May require minimum flow bypass

Affinity Laws

The affinity laws allow scaling pump performance for speed or diameter changes.

Speed Changes (Constant Diameter)

Flow Rate:

Q₂/Q₁ = N₂/N₁

Head:

H₂/H₁ = (N₂/N₁)²

Power:

P₂/P₁ = (N₂/N₁)³

Efficiency:

η₂ ≈ η₁  (approximately constant)

NPSH Required:

NPSH₂/NPSH₁ = (N₂/N₁)²

Validity:

Accurate for ±20% speed change
Assumes Reynolds number independence
Efficiency slightly improves at higher speeds

Example Application:

# Original: 1750 RPM, Q=100 m³/h, H=50m, P=20 kW
# New speed: 1450 RPM

N_ratio = 1450/1750  # = 0.829
Q_new = 100 * 0.829 = 82.9 m³/h
H_new = 50 * 0.829² = 34.4 m
P_new = 20 * 0.829³ = 11.4 kW

Impeller Diameter Changes (Constant Speed)

Flow Rate:

Q₂/Q₁ = (D₂/D₁)

Head:

H₂/H₁ = (D₂/D₁)²

Power:

P₂/P₁ = (D₂/D₁)³

Efficiency:

η₂ ≈ η₁ - Δη

Where Δη accounts for increased clearance effects (typically 2-5% loss)

Practical Limits:

Maximum trim: 75% of original diameter
Below 75%: Efficiency penalty increases significantly
Minimum recommended: 85% of original diameter

Trimming Guidelines:

D_min = 0.75·D_original  (absolute minimum)
D_practical = 0.85·D_original  (recommended minimum)

Example Application:

# Original: D=300mm, Q=200 m³/h, H=80m, P=60 kW
# Trimmed to: D=270mm (90% of original)

D_ratio = 270/300  # = 0.90
Q_new = 200 * 0.90 = 180 m³/h
H_new = 80 * 0.90² = 64.8 m
P_new = 60 * 0.90³ = 43.7 kW
η_penalty = 0.02  # 2% efficiency loss

Combined Changes

For simultaneous speed AND diameter changes:

Q₂/Q₁ = (N₂/N₁) · (D₂/D₁)
H₂/H₁ = (N₂/N₁)² · (D₂/D₁)²
P₂/P₁ = (ρ₂/ρ₁) · (N₂/N₁)³ · (D₂/D₁)³

Density Effects

For different fluids or temperatures:

H₂ = H₁  (head is independent of density)
P₂/P₁ = ρ₂/ρ₁  (power scales with density)
NPSH₂ = NPSH₁  (head-based, not pressure)

Important: Specific gravity affects power, not head!

Curve Fitting from Test Data

Polynomial Curve Fitting

H-Q Curve (2nd Order):

H = a₀ + a₁·Q + a₂·Q²

Fitting Methods:

Least squares regression
Three-point method (Q=0, Q=Q_BEP, Q=Q_max)
Weighted least squares (emphasize BEP region)

Implementation:

import numpy as np
from scipy.optimize import curve_fit

# Polynomial model
def head_model(Q, a0, a1, a2):
    return a0 + a1*Q + a2*Q**2

# Fit to data
params, _ = curve_fit(head_model, Q_data, H_data)

Quality Metrics:

R² = 1 - SS_res/SS_tot  (coefficient of determination)
RMSE = √(Σ(H_predicted - H_measured)²/n)

Target: R² > 0.99 for good fit

Efficiency Curve Fitting

Gaussian-like Model:

η(Q) = η_max · exp(-k·(Q - Q_BEP)²)

Polynomial Model:

η(Q) = b₀ + b₁·Q + b₂·Q² + b₃·Q³

Constraints:

η(0) = 0
η(Q_BEP) = η_max
η → 0 as Q → ∞

Power Curve Fitting

Direct Calculation:

P(Q) = ρ·g·Q·H(Q) / η(Q)

Or Polynomial Fit:

P(Q) = c₀ + c₁·Q + c₂·Q² + c₃·Q³

NPSH Curve Fitting

Power Law Model:

NPSH_req(Q) = NPSH₀ + k·Q^n

Where n ≈ 1.5-2.0

Exponential Model:

NPSH_req(Q) = a·exp(b·Q) + c

Uncertainty Analysis

Measurement Uncertainties:

Flow: ±2-5%
Head: ±1-3%
Power: ±1-2%
Efficiency: ±2-5% (compound error)

Propagation:

σ_η = η·√((σ_Q/Q)² + (σ_H/H)² + (σ_P/P)²)

System Curve Intersection (Operating Point)

System Curve Definition

System Head Equation:

H_system = H_static + K·Q²

Where:

H_static = elevation difference + pressure head
K = friction coefficient = (8·f·L)/(π²·g·D⁵)

Components:

H_static = (z₂ - z₁) + (P₂ - P₁)/(ρ·g)
H_friction = (f·L·v²)/(D·2g) = K·Q²
H_minor = Σ(K_i·v²/(2g))

Operating Point Determination

Graphical Method:

Plot H-Q curve (pump)
Plot H_system curve
Intersection = operating point

Analytical Method:

H_pump(Q) = H_system(Q)
a₀ + a₁·Q + a₂·Q² = H_s + K·Q²
Q_op = (-a₁ + √(a₁² - 4·(a₂-K)·(a₀-H_s)))/(2·(a₂-K))

Numerical Method (Newton-Raphson):

def find_operating_point(pump_curve, system_curve, Q_initial=1.0):
    f = lambda Q: pump_curve(Q) - system_curve(Q)
    Q_op = newton(f, Q_initial)
    H_op = pump_curve(Q_op)
    return Q_op, H_op

Multiple Pump Configurations

Pumps in Series:

H_total = H₁ + H₂
Q_total = Q  (same flow through both)

Use: High head applications

Pumps in Parallel:

Q_total = Q₁ + Q₂
H_total = H  (same head for both)

Use: High flow applications

Parallel Operation Stability:

Requires H-Q curves with dH/dQ < 0
Unequal flow distribution if curves differ
One pump may operate at shutoff

Variable Speed Operation

Operating Point Trajectory:

H/H₀ = (Q/Q₀)²  (system curve)
H/H₀ = (N/N₀)²  (affinity law)

Combined:

Q/Q₀ = N/N₀

Energy Savings:

P₂/P₁ = (Q₂/Q₁)³  (for system curve flow)

Reducing flow by 20% saves ~50% power!

Design Guidelines and Best Practices

Curve Selection

Select pump with BEP near duty point
- Target: Q_duty = 0.8-1.1·Q_BEP
- Margin for fouling and wear
Verify operating range
- All expected flows within 50-150% Q_BEP
- Consider seasonal variations
- Account for future expansion
Check curve stability
- Ensure dH/dQ < 0 throughout
- Avoid dips or humps
- Important for parallel operation
Validate NPSH margin
- NPSH_avail > 1.3·NPSH_req (minimum)
- Higher margin at elevated temperatures
- Consider transient conditions

Common Mistakes to Avoid

1. Ignoring Efficiency:

❌ Selecting pump based only on H-Q intersection
✅ Verify operating point near BEP
✅ Calculate lifecycle energy costs

2. Neglecting Operating Range:

❌ Designing for single duty point
✅ Consider full operating envelope
✅ Verify performance at extremes

3. Misapplying Affinity Laws:

❌ Extrapolating beyond ±20% speed change
❌ Ignoring efficiency changes with trimming
✅ Use manufacturer curves when available
✅ Account for Reynolds number effects

4. System Curve Errors:

❌ Using only friction losses (forgetting static head)
❌ Assuming constant K with control valves
✅ Include all system resistances
✅ Account for changing conditions

5. Parallel Pump Design:

❌ Assuming 2 pumps = 2× flow
❌ Ignoring individual pump operating points
✅ Plot combined H-Q curve
✅ Verify stable operation for all combinations

Verification Checklist

Before finalizing pump selection:

Advanced Topics

Variable Frequency Drive (VFD) Operation

Speed Control Advantage:

Power savings = 1 - (N₂/N₁)³

Considerations:

Bearing minimum speed limits
Cooling at reduced flow
VFD efficiency (typically 95-97%)

Specific Speed Effects

Curve Shape vs. Specific Speed:

Ns = N·√Q / H^0.75

Low Ns (10-30): Steep H-Q curve, flat efficiency
Medium Ns (30-80): Moderate slope, peak efficiency
High Ns (80-150): Flat H-Q curve, narrow efficiency peak

Temperature Effects

Viscosity Impact:

Reynolds number: Re = ρ·v·D/μ
Head reduction: H_viscous < H_water
Efficiency penalty increases at low Re
Use HI correction charts for Re < 10⁶

Vapor Pressure:

NPSH_req increases with temperature
Critical for hot liquids
May require booster pumps

Summary

Pump performance curves are essential tools for:

Pump selection and sizing
Operating point prediction
Energy consumption analysis
System optimization
Troubleshooting and diagnostics

Mastery requires understanding:

Physical meaning of each curve type
Proper application of affinity laws
System-pump interaction
Operating range considerations
Impact of off-design operation

References

See reference.md for detailed standards, equations, and literature references. See plotter.py for verified computational examples.