Field-style article image prepared for pump affinity laws.
A water treatment plant engineer receives a call—the 1750 RPM pump motor failed, and the replacement available from stock runs at 1450 RPM. Before swapping it, she needs to know: will the lower speed still deliver the required 380 gpm and 105 ft TDH? Will the smaller motor handle the load?
Pump affinity laws are mathematical relationships that predict how centrifugal pump flow rate, head, and power change when rotational speed or impeller diameter changes. The three core speed-change laws state that flow scales linearly with speed (Q₂/Q₁ = N₂/N₁), head scales with speed squared (H₂/H₁ = (N₂/N₁)²), and power scales with speed cubed (P₂/P₁ = (N₂/N₁)³). Parallel laws exist for impeller diameter changes. These formulas apply to all centrifugal pump types—radial, mixed-flow, and axial—but not to positive displacement pumps.
الوجبات الرئيسية
- Flow rate changes proportionally with speed (Q₂/Q₁ = N₂/N₁) and impeller diameter (Q₂/Q₁ = D₂/D₁).
- Head changes with speed squared (H₂/H₁ = (N₂/N₁)²) and diameter squared (H₂/H₁ = (D₂/D₁)²).
- Power changes with speed cubed (P₂/P₁ = (N₂/N₁)³); a 20% speed reduction cuts power consumption by approximately 49%.
- Affinity laws are most accurate within ±30% speed change and ±15% impeller trim; beyond these ranges efficiency shifts invalidate the constant-efficiency assumption.
- The laws predict the new pump curve, but the actual operating point requires solving for the intersection with the unchanged system curve.
- Always verify motor capacity after speed increases using the cubic power relationship to prevent overload.
The Three Speed-Change Affinity Laws
The speed-change affinity laws for centrifugal pumps are:
**Flow rate:** Q₂/Q₁ = N₂/N₁
**Head:** H₂/H₁ = (N₂/N₁)²
**Power:** P₂/P₁ = (N₂/N₁)³
المكان:
- Q = flow rate (gpm or m³/h)
- N = rotational speed (RPM)
- H = total head (ft or m)
- P = brake horsepower or shaft power (HP or kW)
- Subscript 1 = original operating condition
- Subscript 2 = new operating condition
These relationships apply to all centrifugal pump configurations: radial-flow, mixed-flow, and axial-flow designs. They assume geometrically similar operating conditions and constant pump efficiency across the speed range.
Why Power Changes Cubically
The cubic power relationship stems from the physics of impeller work. Impeller tip velocity is directly proportional to rotational speed (v ∝ N). Head, which represents energy per unit mass, scales with velocity squared because kinetic energy follows KE ∝ v². Power, the rate of energy transfer, equals flow (a volumetric rate proportional to speed) multiplied by head (proportional to speed squared), yielding P ∝ N³.
This cubic relationship is the reason variable frequency drives (VFDs) create dramatic energy savings. According to PumpToolkit (https://pumptoolkit.com/blog-affinity-laws-guide.html), the cubic power relationship enables large reductions in energy consumption when flow demand decreases. A pump running at 80% speed consumes only 51% of full-speed power, even though it still delivers 80% of the flow.
Speed-Change Worked Example: VFD Slowdown
**Initial operating condition:**
- Speed N₁ = 1750 RPM
- Flow Q₁ = 400 gpm
- Head H₁ = 120 ft
- Power P₁ = 25 HP
**New operating condition after VFD slowdown:**
- Speed N₂ = 1400 RPM
**Speed ratio:** N₂/N₁ = 1400/1750 = 0.80
**Calculate new flow:**
Q₂ = Q₁ × (N₂/N₁) = 400 gpm × 0.80 = **320 gpm**
**Calculate new head:**
H₂ = H₁ × (N₂/N₁)² = 120 ft × (0.80)² = 120 ft × 0.64 = **76.8 ft**
**Calculate new power:**
P₂ = P₁ × (N₂/N₁)³ = 25 HP × (0.80)³ = 25 HP × 0.512 = **12.8 HP**
A 20% speed reduction (from 1750 to 1400 RPM) reduces flow by 20%, head by 36%, and power by 49%. The motor load drops from 25 HP to 12.8 HP, cutting energy consumption in half while still delivering 80% of the original flow. This demonstrates why VFD control is cost-effective in variable-demand applications.
Impeller Diameter Change: Trim Laws and Calculations
The affinity laws also apply to impeller diameter changes, typically used when permanent flow reduction is needed or when fine-tuning a pump’s performance curve. The diameter-change formulas mirror the speed-change relationships:
**Flow rate:** Q₂/Q₁ = D₂/D₁
**Head:** H₂/H₁ = (D₂/D₁)²
**Power:** P₂/P₁ = (D₂/D₁)³
Where D = impeller diameter (inches or mm).
Impeller trimming—machining down the impeller outer diameter—is a permanent mechanical modification, unlike VFD speed changes which can be adjusted dynamically. Pipe Flow Lab (https://pipeflowlab.com/guides/pump-affinity-laws) notes that diameter changes are most accurate within ±15% of the original diameter. Beyond this limit, efficiency shifts become significant because the impeller geometry no longer maintains true similarity with the volute casing design.
Diameter-Change Worked Example: 12-Inch to 11-Inch Trim
**Initial operating condition:**
- Impeller diameter D₁ = 12.0 inches
- Flow Q₁ = 250 gpm
- Head H₁ = 80 ft
- Power P₁ = 15 HP
**After trim:**
- Impeller diameter D₂ = 11.0 inches
**Diameter ratio:** D₂/D₁ = 11.0/12.0 = 0.917
**Calculate new flow:**
Q₂ = Q₁ × (D₂/D₁) = 250 gpm × 0.917 = **229 gpm**
**Calculate new head:**
H₂ = H₁ × (D₂/D₁)² = 80 ft × (0.917)² = 80 ft × 0.841 = **67.3 ft**
**Calculate new power:**
P₂ = P₁ × (D₂/D₁)³ = 15 HP × (0.917)³ = 15 HP × 0.771 = **11.6 HP**
Trimming from 12 inches to 11 inches (an 8.3% reduction) decreases flow by 8.3%, head by 15.9%, and power by 22.9%. The motor load drops from 15 HP to 11.6 HP. This permanent modification suits applications where the system demand is consistently lower than the original pump design point.
System Curve Interaction: Why the Actual Flow Differs from Affinity Prediction
Affinity laws predict how the pump curve shifts, but they do not directly predict the new operating point. The actual flow and head after a speed or diameter change depend on where the new pump curve intersects the system curve.
The system curve represents the head required to move fluid through the piping, valves, and elevation changes at different flow rates. It follows the relationship H_sys = H_static + k·Q², where k is the system resistance coefficient. When pump speed changes, the system curve remains unchanged—the piping friction and elevation do not change with pump speed.
**Example:** A pump originally operates at 1750 RPM, delivering 400 gpm at 120 ft head. The affinity laws predict that at 1400 RPM (80% speed), the pump curve shifts to deliver 320 gpm at 76.8 ft if the system curve matched perfectly. However, if the system curve requires 85 ft of head at 320 gpm, the actual operating point will be where the new pump curve (scaled by the affinity laws) intersects the system curve at 85 ft. The actual flow will be less than 320 gpm.
Turn2Engineering (https://turn2engineering.com/equations/pump-affinity-laws) emphasizes that affinity laws must be paired with head-loss calculations and system curve analysis to determine the true operating point after a speed change. Affinity laws scale the pump curve; the system curve determines where the pump actually operates.
Practical Checks and Limits
Accuracy Boundaries
Affinity laws assume constant pump efficiency and geometrically similar flow patterns. These assumptions hold well within moderate operating ranges but break down at extremes.
**Speed changes:** Most accurate within ±30% of the original speed. Beyond this range, Reynolds number effects alter friction losses in the impeller passages, shifting efficiency. Pipe Flow Lab (https://pipeflowlab.com/guides/pump-affinity-laws) specifies this ±30% guideline as the practical accuracy limit.
**Diameter changes:** Most accurate within ±15% of the original diameter. Larger trims alter the relationship between impeller and volute geometry, changing flow patterns and efficiency. Trims exceeding 15% often require manufacturer consultation.
KSB (https://www.ksb.com/en-global/centrifugal-pump-lexicon/article/pump-affinity-laws-1117584) notes that speed changes also produce different Reynolds numbers, which affect boundary layer behavior and hydraulic losses. For large speed changes, efficiency can shift by several percentage points, making affinity law predictions optimistic or pessimistic depending on whether speed increases or decreases.
Motor Overload Check Procedure
When increasing pump speed, the cubic power relationship can quickly exceed motor capacity. Follow this procedure to verify motor adequacy:
**Step 1: Calculate new power requirement**
Use P₂ = P₁ × (N₂/N₁)³ to find the brake horsepower at the new speed.
**Step 2: Compare to motor nameplate rating**
Check the motor nameplate horsepower. If P₂ exceeds the nameplate rating, proceed to Step 3.
**Step 3: Check service factor**
Most motors have a service factor (typically 1.15), meaning they can safely operate at 115% of nameplate rating continuously. If P₂ < (nameplate HP × service factor), the motor is acceptable.
**Step 4: Decide on motor upgrade**
If P₂ exceeds the motor’s service factor limit, a larger motor is required before increasing speed.
**Example:** A pump originally runs at 1450 RPM drawing 18 HP, driven by a 20 HP motor (service factor 1.15). If speed increases to 1750 RPM:
P₂ = 18 HP × (1750/1450)³ = 18 HP × 1.594 = 28.7 HP
The motor’s maximum continuous rating is 20 HP × 1.15 = 23 HP. Since 28.7 HP > 23 HP, the motor would overload. A 30 HP motor is required.
When Affinity Laws Do Not Apply
**Positive displacement pumps:** Affinity laws are specific to centrifugal pumps. Positive displacement pumps (gear, lobe, piston, diaphragm) deliver a fixed volume per revolution regardless of discharge head. For PD pumps, flow equals displacement times speed (Q = V × N), and power depends primarily on discharge pressure. Neutrium (https://neutrium.net/articles/equipment/pump-affinity-laws/) explicitly states these relationships do not apply to positive displacement pumps.
**Cavitating conditions:** When net positive suction head available (NPSHA) falls below net positive suction head required (NPSHR), cavitation occurs. Affinity laws do not account for vapor bubble formation, which alters flow patterns and drastically reduces head and efficiency. NPSH must be verified separately after any speed or diameter change.
**Large Reynolds number shifts:** Affinity laws assume similar flow regimes. When speed changes move the pump from turbulent to transitional flow (or vice versa), friction factors in the impeller and volute change, invalidating the constant-efficiency assumption. This typically occurs only at very low speeds or with highly viscous fluids.
Affinity Laws Quick Reference Table
المعلمة | Speed Change | Diameter Change |
معدل التدفق | Q₂/Q₁ = N₂/N₁ | Q₂/Q₁ = D₂/D₁ |
الرأس | H₂/H₁ = (N₂/N₁)² | H₂/H₁ = (D₂/D₁)² |
الطاقة | P₂/P₁ = (N₂/N₁)³ | P₂/P₁ = (D₂/D₁)³ |
**Note:** Both sets of laws assume constant efficiency and geometrically similar operating conditions. Accuracy is best within ±30% speed change and ±15% diameter trim.
الأسئلة الشائعة
What are the three pump affinity laws?
The three affinity laws describe how centrifugal pump performance changes with speed or diameter. For speed changes: flow is proportional to speed (Q₂/Q₁ = N₂/N₁), head is proportional to speed squared (H₂/H₁ = (N₂/N₁)²), and power is proportional to speed cubed (P₂/P₁ = (N₂/N₁)³). Equivalent relationships exist for impeller diameter changes, substituting D for N.
Why does power change so much faster than flow?
Power scales with the cube of speed because it combines two squared relationships. Head scales with velocity squared (H ∝ v²), and velocity itself is proportional to speed (v ∝ N). Power equals flow (proportional to N) times head (proportional to N²), yielding P ∝ N³. This cubic relationship means a 20% speed reduction cuts power by 49%, making VFD control highly effective for energy savings in variable-demand systems.
Can I use affinity laws for positive displacement pumps?
No. Affinity laws apply only to centrifugal pumps where head is generated by velocity changes in the impeller. Positive displacement pumps (gear, lobe, piston, screw, diaphragm) deliver a fixed volume per revolution regardless of discharge pressure. For PD pumps, flow equals displacement times speed (Q = V × N), and power depends primarily on discharge pressure, not the cubic speed relationship.
How accurate are affinity laws for large speed changes?
Affinity laws are most accurate within ±30% of the original speed. Beyond this range, Reynolds number effects alter friction losses in the impeller and volute, shifting pump efficiency. The constant-efficiency assumption becomes invalid, and actual performance deviates from affinity law predictions. For speed changes exceeding 30%, consult manufacturer pump curves or request performance testing at the new speed.
Do I need a new pump curve after changing speed?
For speed changes, you can scale the existing pump curve using affinity laws—each point on the original curve shifts according to the Q, H relationships. However, for impeller diameter changes, efficiency shifts are more complex, and manufacturer confirmation is recommended. Many manufacturers provide pump curves for multiple impeller diameters. If trimming beyond 15%, request a revised curve or performance test.
What happens to pump efficiency when I change speed?
Affinity laws assume constant efficiency, which holds well within moderate speed ranges (±30%). In reality, efficiency shifts slightly because Reynolds number changes affect friction losses. Small speed reductions often improve efficiency slightly by moving the operating point closer to the best efficiency point (BEP). Large speed changes—especially increases—can reduce efficiency due to increased hydraulic losses and mismatch with the volute design optimized for the original speed.
الخاتمة
Pump affinity laws provide engineers with fast, reliable formulas to predict how centrifugal pump flow, head, and power change when speed or impeller diameter changes. The cubic power relationship (P ∝ N³) explains why VFD-controlled pumps deliver substantial energy savings in variable-demand applications, while the squared head relationship (H ∝ N²) shows how quickly pumping capacity drops at reduced speeds.
These laws serve as powerful first-pass tools for evaluating motor replacements, VFD installations, and impeller trim decisions. However, they are not substitutes for complete engineering analysis. Before finalizing any speed or diameter change, confirm that the new operating point—determined by the intersection of the scaled pump curve and the system curve—lies within the pump’s acceptable efficiency range. Verify motor capacity using the cubic power formula and service factor limits. Check that NPSH available exceeds NPSH required at the new operating condition to prevent cavitation.
Used within their accuracy boundaries (±30% speed, ±15% diameter) and paired with system curve analysis and motor checks, affinity laws enable confident pump performance predictions without expensive testing or simulation.