EVOLV/wiki/concepts/pump-affinity-laws.md

Pump Affinity Laws & Curve Theory

Used by: mechanical-process-engineer agent, rotatingMachine node, pumpingStation node Validation: Verified against Engineering Toolbox, Hydraulic Institute standards, and ScienceDirect

Affinity Laws

The affinity laws describe how centrifugal pump performance scales with changes in rotational speed (N) or impeller diameter (D). They are derived from dimensional analysis under the assumption of geometric similarity (velocity triangles at the impeller remain geometrically similar).

Speed Variation (constant diameter)

Q₂/Q₁ = N₂/N₁
H₂/H₁ = (N₂/N₁)²
P₂/P₁ = (N₂/N₁)³

Diameter Variation (constant speed)

Q₂/Q₁ = D₂/D₁
H₂/H₁ = (D₂/D₁)²
P₂/P₁ = (D₂/D₁)³

Combined Variation

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

Practical Example (validated against Engineering Toolbox)

A pump at 1750 rpm delivering 100 gpm at 100 ft head using 5 bhp:

• At 3500 rpm: 200 gpm flow, 400 ft head, 40 bhp power
• A 10% speed increase yields: +10% flow, +21% head, +33% power

Accuracy Limitations

• Affinity laws are approximate — accuracy decreases with large speed changes (>±30%)
• Efficiency shifts slightly with speed change (not captured by basic affinity laws)
• Trimming impeller diameter >15-20% significantly reduces accuracy
• Laws assume no significant change in Reynolds number effects

Pump Curve Types

Q-H Curve (Flow vs. Head)

• Primary performance curve
• Head decreases as flow increases (for centrifugal pumps)
• Shape depends on specific speed (Ns): flat, steep, or drooping
• Monotonicity: Should be monotonically decreasing for stable operation. Non-monotonic (drooping) curves can cause instability in parallel operation.

Q-P Curve (Flow vs. Power)

• Power consumption as function of flow
• Shape varies by pump type:
  • Radial: power increases with flow (non-overloading possible at shutoff)
  • Mixed flow: relatively flat
  • Axial: power decreases with flow (overload risk at low flow)

Q-η Curve (Flow vs. Efficiency)

• Efficiency peaks at Best Efficiency Point (BEP)
• Falls off on both sides of BEP
• Operating far from BEP causes excessive vibration, cavitation risk, and energy waste

Best Efficiency Point (BEP)

The BEP is the operating point where the pump converts the maximum fraction of input power to useful hydraulic work.

BEP Tracking Under VFD Control

When speed changes via VFD, the BEP shifts along a parabolic path in the Q-H plane:

H_BEP ∝ Q_BEP²

This is because both Q and H scale with speed, but H scales as the square of Q's scaling factor.

Preferred Operating Region

• Continuous operation: 80% – 110% of BEP flow
• Allowable range: 70% – 120% of BEP flow
• Outside this range: increased bearing loads, seal wear, cavitation risk

System Curve Theory

The system curve describes the head required by the piping system as a function of flow:

H_system = H_static + k · Q²

Where:

• H_static = static head (elevation difference + tank pressure difference)
• k = system resistance coefficient (Pa·s²/m⁶ in SI)
• Q = volumetric flow rate

Duty Point

The duty point (operating point) is the intersection of the pump curve and system curve:

H_pump(Q) = H_system(Q)

This is solved numerically — find Q where pump curve equals system curve.

Parallel Pump Operation

Flow Summation Rule

For pumps in parallel at equal head:

Q_total = Q₁ + Q₂ + ... + Qₙ  (at each head value)

The combined curve is constructed by horizontally adding individual pump curves.

Key Considerations

• Each pump must overcome the same system head
• Adding a pump shifts the combined curve right, moving the operating point
• Diminishing returns: each additional pump adds less incremental flow
• Risk of back-flow through stopped pumps (check valves required)
• Unstable operation if pump curves have a drooping characteristic

Series Pump Operation

For pumps in series at equal flow:

H_total = H₁ + H₂ + ... + Hₙ  (at each flow value)

The combined curve is constructed by vertically adding individual pump curves.

Specific Energy

The key energy KPI for pumping systems:

SE = P / Q  [W / (m³/s) = J/m³]

More commonly expressed as:

SE = P / Q  [kWh/m³] (with appropriate unit conversion)

Where:

• P = electrical power input (kW)
• Q = volumetric flow rate (m³/h)
• SE = P / Q · (1/1000) for kWh/m³ when P in W and Q in m³/s

Wire-to-Water Efficiency

η_total = η_motor · η_VFD · η_pump = (ρ · g · Q · H) / P_electrical

Authoritative References

1. Karassik, I.J. et al. "Pump Handbook" 4th ed. — McGraw-Hill (comprehensive pump engineering reference)
2. Europump/Hydraulic Institute (2001). "Pump Life Cycle Costs: A Guide to LCC Analysis for Pumping Systems"
3. Engineering Toolbox — "Affinity Laws for Pumps" (https://www.engineeringtoolbox.com/affinity-laws-d_408.html)
4. Hydraulic Institute Standards (HI 9.6.1 — Rotodynamic Pumps Guideline for NPSH Margin)
5. Gülich, J.F. (2014). "Centrifugal Pumps" 3rd ed. — Springer (theoretical foundation)