Pump affinity laws: speed, diameter, and power

A mechanical engineer at a municipal water treatment plant is replacing a pump motor with a variable-frequency drive. The pump's nameplate says 1,750 RPM, 450 gpm, 120 ft TDH — but the system only needs 360 gpm at peak. How much does head drop? How much energy does the VFD save when the controller dials the speed back to 80%? The answers come from three equations derived from dimensional analysis of rotating machinery, known collectively as the pump affinity laws.

This guide derives all three laws from first principles, applies them to both speed changes (VFD operation) and impeller diameter changes (trimming), quantifies the dramatic cubic energy savings available from VFDs, explains where the laws break down, and walks through two complete worked examples — one commercial HVAC chilled-water system and one agricultural irrigation mainline. Every numeric value traces to ANSI/HI standards, DOE pump system guidance, or ASHRAE 90.1. The pump sizing calculator on this site applies these laws interactively.

What the affinity laws are — and where they come from

The affinity laws are not empirical curve-fits. They follow from dimensional analysis of the Navier-Stokes equations applied to geometrically similar turbomachinery operating at dynamically similar conditions. The key insight is that for a centrifugal pump operating at the same efficiency point (the same specific speed, the same position on its characteristic curve), all the relevant velocity triangles scale uniformly when speed or diameter changes.

The Euler turbomachinery equation gives the theoretical head rise as:

Hth = (u₂ · c₂ᵤ − u₁ · c₁ᵤ) / g

where u₂ is the impeller tip speed (= π · D₂ · N / 60), c₂ᵤ is the tangential component of absolute velocity at the outlet, u₁ is inlet peripheral speed, c₁ᵤ is inlet whirl velocity (often zero for radial inflow), and g is gravitational acceleration. When speed N doubles with diameter held constant, every u doubles. For geometrically similar flow, every c also doubles (the velocity triangles scale). Therefore H scales as the product u · c — which is N². Flow rate scales as the volume swept per revolution times speed, so Q ∝ N. Power equals ρgQH / η; since Q ∝ N and H ∝ N², power scales as N³. The same argument applies when diameter changes at constant speed, because u₂ = π · D · N / 60 makes tip speed proportional to D.

The three speed-change laws (Set 1)

With impeller diameter held constant, the three laws for a speed change from N₁ to N₂ are:

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

Law 1 (flow): Flow is proportional to impeller tip speed, which is proportional to shaft speed. Doubling the speed doubles the flow. Reducing to 80% speed delivers 80% flow.

Law 2 (head): Head is proportional to the square of tip speed, because it represents kinetic energy (½u²) converted to pressure. Reducing speed to 80% drops head to 64% (0.8² = 0.64). This is critical for system curve matching — if your system curve has significant static head, the pump may not be able to deliver any flow at very low speeds.

Law 3 (power): Shaft power is the product of flow and head (times density and a constant), so it scales as N · N² = N³. This cubic relationship is the economic engine behind variable-frequency drives. At 80% speed, shaft power is only 0.8³ = 0.512 — a 49% reduction in energy consumption for a 20% reduction in flow.

The bar chart below illustrates the three laws simultaneously at three operating speeds: 100%, 80%, and 60% of rated speed. Each group shows the normalized ratio. Notice how the power (rightmost group) collapses far faster than flow — the physical reason VFDs pay for themselves in 12–36 months on large HVAC systems.

The three diameter-change laws (Set 2)

With shaft speed held constant, changing impeller diameter D produces analogous scaling:

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

The physics is the same: tip speed u₂ = π · D · N / 60 is proportional to D when N is fixed, so all the same scalings follow. The practical application is impeller trimming: machining the outside diameter of an impeller smaller to reduce its output when the pump is permanently oversized for the duty.

Consider a pump shipped with a 12-inch impeller developing 200 gpm at 80 ft TDH, but the installed system only needs 180 gpm at 65 ft. The diameter ratio needed: D₂/D₁ = √(65/80) = √0.8125 = 0.901, so trim to 0.901 × 12 = 10.81 inches — roughly 10 7/8 inch. The corresponding flow check: 180 / (0.901 × 200) = 0.998 — close enough. This calculation is the routine starting point; verify with the manufacturer's tested trim curve before machining.

Why Set 2 is less accurate than Set 1. When you trim an impeller with a straight radial cut (the only practical machining option), you do not just scale the geometry — you change the vane exit angle and increase the gap between the impeller tip and the volute throat. The Hydraulic Institute (ANSI/HI 14.3-2019 Table 4.3) notes that diameter-law predictions carry increasing error beyond 10% trim; actual head can be 2–5% below the scaled prediction for trims of 15–20%.

How pump and system curves shift together

A centrifugal pump operates at the intersection of its H-Q curve and the system curve. When speed changes, the entire pump H-Q curve shifts: every point (Q₁, H₁) on the original curve maps to a new point (Q₁ · r, H₁ · r²) on the reduced-speed curve, where r = N₂/N₁. This family of curves is called an affinity parabola — all the mapped points lie on the curve H = (H_BEP / Q_BEP²) · Q² passing through the origin.

The system curve has the form H_sys = H_static + R · Q², where H_static is the static lift (elevation + pressure difference) and R is the system resistance coefficient. Two important cases arise:

1. Friction-dominant system (H_static ≈ 0). The system curve starts at the origin. The operating point tracks down the affinity parabola exactly — the pump always operates at the same specific speed position on its curve. VFD control is smooth and the affinity laws are most accurate here. Typical example: a recirculating chilled-water loop with no static head.
2. Static-head-dominant system. The system curve starts well above the origin. At low speed, the pump H-Q curve at reduced speed may fall entirely below the system curve — the pump cannot deliver any flow. This is the minimum speed problem in VFD applications. Calculate the minimum speed at which the pump just overcomes static head: set H_static = H_BEP · (N_min/N_rated)² and solve for N_min = N_rated · √(H_static / H_BEP).

The animation below shows two pump curves — one at rated speed and one at 75% speed — overlaid on a friction-dominant system curve. The operating point slides down the affinity parabola as speed decreases.

VFD energy savings — the cube law in practice

The power cube law is the economic foundation of pump VFD retrofits. A 100 HP pump running continuously at full speed in a system that only needs 80% flow is consuming 100 HP when it could be consuming 51 HP — wasting 49 HP of electricity every hour. At $0.12/kWh and 8,000 operating hours per year, that wasted 49 HP (37 kW) costs approximately $35,500 per year. A VFD retrofit on a 100 HP motor typically costs $8,000–$15,000 installed, giving a payback period under six months on a pump that operates at partial load most of the time.

ASHRAE Standard 90.1-2022, Section 10.4.3.2, mandates variable-speed drives on all HVAC pump motors ≥ 5 HP serving variable-flow systems, precisely because this payback math holds at virtually every installation. The DOE's "Improving Pumping System Performance" guide (2nd edition, 2006, updated 2021) identifies oversized pumps with throttling control as the single largest efficiency opportunity in industrial fluid systems, responsible for 20–50% of pump energy waste across the U.S. industrial base.

Two important qualifications apply in practice:

1. The cube law applies to shaft power, not wire-to-water power. Motor efficiency and VFD efficiency both vary with load. At 50% motor load, motor efficiency may drop 3–5 percentage points from nameplate. Modern premium-efficiency motors and high-quality VFDs maintain efficiency down to about 25% load, but below that the overall system efficiency degrades enough that the actual energy savings are somewhat less than the pure cube law predicts.
2. NPSHr scales with N² too. Reducing speed reduces NPSHr (required NPSH) approximately as the square of the speed ratio, which means cavitation risk decreases at reduced speed. However, if the system has a high suction lift or the fluid is hot, verify NPSHa at reduced flow against the manufacturer's reduced-speed NPSHr curve rather than scaling blindly.

Causes and mechanisms — why pumps end up oversized

Understanding why the affinity laws are needed requires understanding why pumps are routinely oversized in the first place. The pattern is consistent across commercial HVAC, industrial process, and municipal water systems:

1. Design-phase safety factors compound. A mechanical engineer adds 10% to calculated flow for "future expansion," the contractor adds 10% for installation uncertainty, and the pump manufacturer's next standard model is 15% larger than the calculated duty. The result: a pump 35–40% oversized before a single pipe fitting is installed.
2. Friction factors are estimated conservatively. Pipe roughness tables give new-pipe C-factors; aged pipe values are more restrictive. Designers often use aged-pipe values for conservative head estimates, but the pump operates for years on new pipe before the system grows into the installed capacity.
3. Peak-demand sizing for partial-time loads. A commercial chilled-water system sized for 100% cooling load at peak outdoor temperatures on the hottest day of the year will operate at 30–60% load for the vast majority of annual hours. Without VFD control, the pump runs at full speed through a throttling valve, destroying energy and wearing the valve seat.
4. Standard pump selection rounds up. Centrifugal pump impeller diameters come in stock sizes (10-inch, 11-inch, 12-inch, 13-inch). If the calculated duty falls between sizes, the engineer almost always rounds up — another systematic upward bias.

The affinity laws provide the tools to correct these problems: VFDs for variable-flow duty, impeller trimming for fixed-duty systems permanently oversized by a small margin.

Symptoms in the field — how engineers recognize affinity law problems

An oversized pump running against a throttling valve produces characteristic field indicators that a knowledgeable engineer can recognize:

1. Control valve nearly closed at design flow. If the throttling valve needs to be 20% or less open to reach the desired flow, the pump is oversized. The pressure drop across the valve equals the excess head the pump is generating — pure wasted energy converted to heat.
2. High pump suction-to-discharge differential at low flows. With a throttling valve closed, the pump deadheads toward shutoff head. The differential pressure across the pump will be significantly higher than the design TDH, a clear sign the pump is running left of BEP.
3. Vibration and noise at low flow. Operation far left of BEP causes recirculation at both the impeller inlet and outlet, producing cavitation-like noise, shaft deflection, and accelerated seal and bearing wear. ANSI/HI 9.6.3 defines the minimum continuous stable flow (MCSF) — typically 70–80% of BEP flow — and operating continuously below MCSF shortens pump life significantly.
4. Amperage lower than nameplate at design conditions. An oversized pump delivering rated flow through a throttled valve consumes less power than nameplate because the effective head (flow × pressure drop through the valve) is the same as designed, but the pump is operating with a poor head-to-efficiency ratio. If measured amperage is 60–70% of nameplate FLA at "design" conditions, the pump and motor assembly are not matched.

Prevention and fixes — ranked by cost

1. Variable-frequency drive (VFD). Most flexible and highest long-term energy savings. Applicable to any variable-flow system. Typical installed cost: $6–18 per HP for motors up to 100 HP. ROI driven directly by the cube law — often under 12 months on continuously operating systems. Bonus: reduces mechanical wear by eliminating valve throttling and allows soft-start.
2. Impeller trimming. Best for fixed-duty systems permanently oversized by 5–15%. One-time cost: $300–$800 for machining a 6–12 inch impeller. No recurring cost. Irreversible — confirm with manufacturer before cutting. Not suitable for variable-flow systems since you cannot restore capacity when demand increases.
3. Impeller replacement. When trimming limit is exceeded or the pump is badly oversized (>20%), a smaller impeller from the same pump family is more accurate than heavy trimming. More expensive than machining but retains accuracy of manufacturer-tested curves. Typical cost: $400–$2,500 depending on impeller size and material.

Sizing and spec guidance

When specifying a centrifugal pump for a variable-flow application, the selection sequence recommended by ANSI/HI 9.8 (Pump Intake Design) and the Hydraulic Institute Selection Guide is:

1. Calculate system TDH at maximum design flow, including all fittings and static head.
2. Select a pump that places BEP within ±10% of the design flow point.
3. Specify VFD for variable-flow systems; calculate minimum speed from the static head constraint.
4. For fixed-duty, verify the exact impeller diameter (or trim) needed using Set 2 laws.
5. Check NPSHa ≥ NPSHr + 5 ft margin per ANSI/HI 9.6.1-2017 at maximum flow.

Codes and standards

Pump affinity law application is governed by a stack of standards that a specification engineer should cite by document number and revision year:

1. ANSI/HI 14.3-2019 — Rotodynamic Pumps for Design and Application. Section 4.4 defines Set 1 and Set 2 affinity laws, their derivation, and the accuracy limits for both speed and diameter scaling. This is the primary normative reference for any pump affinity law calculation.
2. ANSI/HI 9.6.1-2017 — Rotodynamic Pumps: Guideline for NPSH Margin. Covers the NPSHr scaling with speed (NPSHr ∝ N²) and specifies minimum NPSH margins for various duty classes. Required reading before applying speed reduction in systems with marginal suction conditions.
3. ANSI/HI 9.6.3-2017 — Rotodynamic Pumps: Guideline for Operating Region. Defines preferred operating region (POR, typically 70–120% of BEP flow) and allowable operating region (AOR). Establishes that minimum continuous stable flow is typically 70–80% of BEP flow — the lower bound on VFD turndown for a given pump.
4. ASHRAE Standard 90.1-2022 — Energy Standard for Sites and Buildings Except Low-Rise Residential Buildings. Section 10.4.3.2 mandates VFDs on pump motors ≥ 5 HP in variable-flow HVAC systems. Appendix G provides the simulation methodology for demonstrating energy savings from VFD retrofits in energy models.
5. DOE "Improving Pumping System Performance" (2nd Ed., 2006 / updated 2021) — published by the U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy. Provides the practical guidance on pump system assessment, impeller trimming procedures, and VFD economics that supplements the ANSI/HI standards. Available free from energy.gov.
6. 10 CFR 431, Subpart Y — DOE federal regulation establishing minimum efficiency standards for commercial and industrial pumps. Clean water pumps ≥ 1 HP are subject to PEI (Pump Energy Index) requirements; VFD control can help meet PEI targets for pumps at the efficiency boundary.

Software and advanced analysis

The affinity laws are first-order approximations. For system design decisions where accuracy matters — commissioning a large chilled-water plant, specifying a pump for a long-distance irrigation mainline, or determining VFD minimum speed for a system with significant static head — supplement the affinity law calculations with:

1. Manufacturer's performance curves at specified speeds. Most major pump manufacturers (Grundfos, Xylem, Wilo, Goulds/ITT, Armstrong) publish factory-tested H-Q curves at multiple impeller diameters and, increasingly, at multiple speeds. Always use manufacturer data when available in preference to affinity-law extrapolation, especially for trims exceeding 10%.
2. Pipe-Flo Professional (KY Pipe). Industry-standard hydraulic modeling software that imports pump curves and computes system curve intersections at multiple speeds. Widely used for industrial and municipal pump system analysis. Incorporates the affinity laws within a full system hydraulic model rather than applying them in isolation.
3. Bentley WaterGEMS / HAMMER. For municipal water distribution or fire suppression systems where pump speed changes interact with a network of pipes, pressure zones, and demand nodes. The affinity laws define the pump boundary condition in the model, but the system response is computed by iterative network solution.
4. CFD (Computational Fluid Dynamics). For impeller redesign, not just trimming. When affinity-law trimming would require a >20% diameter reduction, or when the duty is highly unusual (viscous fluid, abrasive slurry), CFD of the impeller and volute geometry is warranted. Tools include ANSYS Fluent and OpenFOAM.

Worked example 1 — commercial HVAC chilled-water pump VFD retrofit

A 75 HP centrifugal pump serves a chilled-water loop in a commercial office building. Rated conditions: Q₁ = 600 gpm, H₁ = 95 ft TDH, N₁ = 1,750 RPM. The building energy management system determines that peak cooling demand requires only 480 gpm. A VFD is being considered.

Step 1 — Required speed for target flow:

N₂ = N₁ × (Q₂ / Q₁) = 1,750 × (480 / 600) = 1,750 × 0.800 = 1,400 RPM

Step 2 — Head at reduced speed:

H₂ = H₁ × (N₂/N₁)² = 95 × 0.8² = 95 × 0.64 = 60.8 ft

Check that the system curve at 480 gpm requires approximately 60.8 ft of head — if the system curve is friction-dominant (no significant static head), this will match by definition of the affinity parabola.

Step 3 — Power at reduced speed:

P₂ = P₁ × (N₂/N₁)³ = 75 HP × 0.8³ = 75 × 0.512 = 38.4 HP

The pump now draws 38.4 HP instead of 75 HP — a saving of 36.6 HP (27.3 kW). At $0.12/kWh and 6,000 hours/year of operation (a typical commercial HVAC schedule), annual savings are approximately $19,600/year. A VFD for a 75 HP motor costs approximately $6,000–$10,000 installed — under six months payback.

Worked example 2 — agricultural irrigation mainline impeller trim

An irrigation district installs a submersible turbine pump for a drip-irrigation mainline. Factory specification: Q₁ = 1,200 gpm, H₁ = 180 ft, 12-inch impeller. After hydraulic modeling of the actual field distribution system, the district determines the true design point is 1,080 gpm at 145 ft — the pump is oversized and VFD is not in the budget.

Step 1 — Required diameter ratio (using head, which is usually more accurate than flow for trim calculations):

D₂/D₁ = √(H₂/H₁) = √(145/180) = √0.806 = 0.898

Step 2 — Target diameter:

D₂ = 0.898 × 12 in = 10.77 in ≈ 10¾ in

Step 3 — Verify predicted flow at trimmed diameter:

Q₂ = Q₁ × (D₂/D₁) = 1,200 × 0.898 = 1,078 gpm

Close to the target 1,080 gpm — the trim is consistent. The diameter reduction is approximately 10.2%, right at the HI accuracy threshold. The district should request the manufacturer's tested trim curve for this pump at 10¾-inch diameter before machining, and budget for a verification flow test after installation. Power reduction from the trim: P₂/P₁ = (10.77/12)³ = 0.898³ = 0.724 — a 27.6% reduction in shaft power, which also reduces motor and starter thermal stress over the life of the installation.

Frequently asked questions

What are the pump affinity laws?

The pump affinity laws are three proportionality rules that describe how flow rate, head, and shaft power change when you alter the speed or impeller diameter of a centrifugal pump. Speed law 1: flow scales linearly with speed (Q₂/Q₁ = N₂/N₁). Speed law 2: head scales with the square of speed (H₂/H₁ = (N₂/N₁)²). Speed law 3: power scales with the cube of speed (P₂/P₁ = (N₂/N₁)³). The same three relationships hold for impeller diameter D when speed is held constant.

Do the affinity laws apply to all centrifugal pumps?

They apply to all dynamic (rotodynamic) pumps — radial-flow, mixed-flow, and axial-flow centrifugal machines — as a first-order approximation. They do not apply to positive-displacement pumps (gear pumps, piston pumps, diaphragm pumps), where flow is set by displacement volume per revolution, not impeller dynamics. For centrifugal pumps, accuracy is best within ±30% speed change and ±15% diameter trim; outside those bands the assumption of constant efficiency breaks down.

How accurate are the affinity laws for impeller trimming?

Speed scaling (Set 1) is more accurate than diameter scaling (Set 2). For diameter changes up to about 10%, the laws predict head and flow within 2–3% of tested performance. Beyond a 15% diameter reduction, two accuracy penalties appear: the vane exit geometry changes because a straight radial cut does not follow the vane curvature, and the impeller-to-volute area ratio shifts, increasing diffusion losses. DOE guidance (based on Hydraulic Institute testing) recommends using manufacturer tested curves rather than simple affinity scaling for trims exceeding 10% of full diameter.

How much energy does a VFD save on a pump?

Energy savings follow the cube law: at 80% of full speed, a pump consumes only 0.8³ = 51.2% of full-speed power — a 49% reduction for a 20% flow reduction. At 70% speed the power drops to 34%. Real-world savings are slightly less because motor and VFD efficiency vary with load, and the cube law applies to hydraulic shaft power, not wire-to-water power. ASHRAE 90.1-2022 mandates VFDs on pumps ≥ 5 HP serving variable-flow systems precisely because this cubic relationship makes part-load operation dramatically efficient.

Can I use the affinity laws for positive-displacement pumps?

No. In a positive-displacement pump (gear, piston, diaphragm, lobe), flow is set by the displacement volume swept per revolution, so Q ∝ N is approximately true — but head does not follow N² and power does not follow N³. Head in a PD pump is set by the downstream system resistance, not impeller dynamics. Applying centrifugal affinity laws to a PD pump will give wrong answers for head and power.

What happens to pump efficiency when I change speed with a VFD?

The best efficiency point (BEP) shifts with speed, but BEP efficiency stays nearly constant over a moderate speed range (±30%). The affinity laws assume constant efficiency, and for a well-designed centrifugal pump this is a reasonable approximation. At very low speeds (below ~40% of rated), efficiency drops significantly because the pump enters a low-Reynolds-number regime where viscous friction dominates, and at very high speeds volumetric losses and mechanical losses increase. Operating continuously below 30% rated speed is generally not recommended for centrifugal pumps.

What is the difference between Set 1 and Set 2 affinity laws?

Set 1 laws vary speed (N) with impeller diameter held constant; Set 2 laws vary impeller diameter (D) with speed held constant. Set 1 is more accurate and is the basis for VFD analysis. Set 2 (diameter scaling) introduces additional error because changing D changes the vane geometry and the impeller-to-volute clearance, not just a simple geometric scaling. ANSI/HI 14.3-2019 covers both sets, noting that Set 2 predictions should be verified against manufacturer tested trim data for diameter reductions beyond 10%.

Why does head drop by the square of speed ratio?

Head in a centrifugal pump is proportional to the kinetic energy imparted to the fluid by the rotating impeller, which goes as the square of impeller tip velocity: H ∝ u₂², where u₂ = π · D · N / 60. Since u₂ is proportional to N (for constant D), head is proportional to N². This derivation comes directly from the Euler turbomachinery equation: the theoretical head equals (u₂·c₂ᵤ − u₁·c₁ᵤ)/g, where c_u are the tangential components of absolute velocity. When speed scales, both u and c_u scale proportionally, so their product — and thus the head — scales with N².

Can affinity laws predict cavitation risk at reduced speed?

Yes, with care. NPSHr (required NPSH) scales approximately with the square of speed: NPSHr₂ ≈ NPSHr₁ · (N₂/N₁)². This means reducing pump speed also reduces cavitation risk at the pump — a major secondary benefit of VFD operation. However, at very low speeds the NPSH curve shape can change because the pump enters regions where the Thoma sigma relationship breaks down. Always verify reduced-speed NPSHr against manufacturer data or the ANSI/HI 9.6.1 margin guidelines before relying solely on scaled predictions.

How do I apply affinity laws to select a pump for variable flow duty?

Start with the maximum-flow design point (Q_max, H_max). Select a pump that operates at or near BEP at that point. Then apply the affinity laws to verify the pump stays on its curve at minimum flow: Q_min / Q_max = N_min / N_max, and H_min = H_max · (N_min/N_max)². Check that H_min exceeds the system curve at Q_min (including static head). Confirm the VFD can regulate speed across the range, and verify the pump remains above minimum continuous stable flow (typically 70–80% of BEP flow per ANSI/HI 9.6.3) at lowest speed.

What are the limits of impeller trimming?

Practical limits are 80–85% of full diameter (a 15–20% reduction) for most radial-flow pumps. Beyond that limit, the vane discharge angle is effectively changed by the straight radial cut, the impeller-to-volute gap increases, and recirculation losses grow. Some high-specific-speed mixed-flow impellers cannot be trimmed at all without severe efficiency loss. Axial-flow (propeller) pumps should never be trimmed — their vane profiles depend on the full-span geometry. Always confirm with the pump manufacturer before trimming more than 10%.

Is it better to use a VFD or trim the impeller to reduce flow?

A VFD is almost always the better long-term choice for systems with variable flow demand. Impeller trimming is a permanent, one-time adjustment — once trimmed, the pump cannot deliver more flow without a new impeller. A VFD lets you operate anywhere along the affinity law curve in real time, enables automated control, and recovers energy at part load via the cube law. Impeller trimming is the right choice for fixed-duty systems that are permanently oversized by a small margin (< 10%), where VFD cost cannot be justified and the operating point will never need to change.

Sources and further reading

1. ANSI/HI 14.3-2019 — Rotodynamic Pumps for Design and Application. Hydraulic Institute, Parsippany, NJ. Normative definition of Set 1 and Set 2 affinity laws, accuracy limits, and impeller trimming guidance. Primary reference for all calculations in this guide.
2. ANSI/HI 9.6.1-2017 — Rotodynamic Pumps: Guideline for NPSH Margin. Hydraulic Institute. Covers NPSHr scaling with speed and minimum NPSH safety margins by duty class.
3. ASHRAE Standard 90.1-2022 — Energy Standard for Sites and Buildings Except Low-Rise Residential Buildings. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta. Section 10.4.3.2 mandates VFDs on variable-flow HVAC pump systems ≥ 5 HP.
4. DOE "Improving Pumping System Performance: A Sourcebook for Industry," 2nd Ed. U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, 2006 (reissued 2021). Provides cube-law economics, impeller trimming procedures, and the pump system assessment methodology used by industrial energy auditors. Available free at energy.gov.
5. Crane Technical Paper 410 (TP-410), 2011 edition. Crane Co. Provides K-factor data for fittings referenced in system curve calculations that feed affinity law sizing. Section 2 covers the Darcy-Weisbach equation used to establish R in the system curve H = H_static + R·Q².
6. Gülich, J.F., "Centrifugal Pumps," 3rd Ed. Springer, 2014. Chapter 3 derives the affinity laws from the Euler turbomachinery equation and quantifies accuracy degradation at large speed and diameter changes via Reynolds number correction factors. The standard graduate-level reference on pump hydraulics.

Open the tools

Apply the affinity laws directly in the calculators below:

• Pump sizing calculator — TDH, NPSH-available, motor selection, and energy-cost estimator; apply affinity law speed ratios to see power savings in real time
• Pressure-drop calculator — compute the system curve R coefficient (friction head vs flow) that defines where the pump affinity parabola intersects the system
• Velocity check — confirm pipe velocities stay in the 0.5–3 m/s range at both full and reduced VFD speeds; runout flow at high speed can cause water hammer
• Darcy-Weisbach calculator — for non-water fluids and temperature-adjusted viscosity; affects both the system R coefficient and the pump's Reynolds number correction
• Fitting equivalent lengths — accurate system curve calculation requires all elbows, tees, and valves; Crane TP-410 K-factors
• Pump curves explained — how to read an H-Q curve, identify BEP, and interpret the efficiency island before applying affinity law scaling
• NPSH and cavitation guide — understand NPSHr and NPSHa before applying reduced-speed NPSHr scaling to a VFD application
• Methodology — formulas, density and viscosity tables, and all source references used across this site