The cubic-law sales pitch
Every VFD vendor leads with the same equation:
P₂ / P₁ = (N₂ / N₁)³Power scales with the cube of speed. Slow a pump 20% and shaft power drops 49%. That's the headline number on every brochure. It is also, for most real pumping systems, dramatically optimistic.
The cubic law is exact only when the system curve is purely friction — a closed loop, no static head. Add 30 ft of static lift to the same loop and the savings collapse, because the pump still has to make 30 ft regardless of how slowly it spins.
A useful design sequence:
1. Plot your system curve. 2. Compute the static-head fraction of TDH at design flow. 3. Apply the right savings model for that fraction.
Three regimes
Pure friction (static = 0). The system curve passes through the origin. Power scales exactly as the cubic law predicts. VFD savings are real and large.
Mostly static (static > 70% of TDH). The system curve is nearly flat. Slowing the pump barely reduces flow but cuts head against the static lift. The pump moves up its curve toward shutoff and efficiency collapses. VFD savings are negligible or negative.
Mixed (static 20-70%). Reality for most water/wastewater systems. Savings are partial. A correct estimate requires combining the new pump curve at reduced speed with the actual system curve and reading the new operating point's hydraulic + shaft power.
Worked example: 70% friction, 30% static
A booster pump delivers 1,000 gpm at 100 ft TDH. Static lift = 30 ft, friction at 1,000 gpm = 70 ft. Pump efficiency at the design point is 78%.
Shaft power at full speed:
BHP = (Q · TDH · SG) / (3960 · η)
= (1000 · 100 · 1.0) / (3960 · 0.78)
≈ 32.4 hpDemand drops to 700 gpm. Friction drops with the square of flow:
H_friction(700) = 70 · (700/1000)² = 34.3 ft
H_total(700) = 30 + 34.3 = 64.3 ftThe new operating point is 700 gpm at 64.3 ft. Apply the affinity laws to find the pump speed that lands here. From the original curve at 1,000 gpm / 100 ft, the new operating point at 700/64.3 lies on a parabola through the origin only if static = 0 — so we need the actual reduced-speed curve. Approximately:
N₂ / N₁ ≈ Q₂ / Q₁ = 0.70 (affinity flow law)Shaft power at the new operating point:
BHP₂ = (700 · 64.3 · 1.0) / (3960 · ~0.74) [efficiency drops slightly]
≈ 15.4 hpSavings: 32.4 → 15.4 hp = 53%. Not 65% as the cubic law would predict, but still very real.
When throttling beats a VFD
In two situations:
1. Static-dominated systems (static > 70%). A throttle valve absorbs the excess head; a VFD wouldn't have done much better, and you avoid the harmonic + bearing-fluting issues VFDs introduce. 2. Constant-duty systems running > 90% of the time near their design point. The capital cost of a VFD never amortizes.
Throttling burns shaft horsepower, but if shaft horsepower at the throttled point is the same as the VFD-controlled point (which it often is for static-heavy systems), throttling wins on capital cost.
Bypass control: rarely the right answer
Bypass control (recirculating excess flow back to the suction sump) keeps the pump at full speed and full flow. It wastes 100% of the bypassed energy. Only justifiable when:
- A minimum pump flow must be maintained for cooling or seal flushing
- Process variability is small enough that the bypass is rarely active
- VFD installation is structurally impossible (e.g., explosion-proof environments where the drive enclosure cost is prohibitive)
What the VFD doesn't fix
- Pump curves don't shift uniformly. At reduced speed, BEP shifts to lower flow. Operating below BEP at reduced speed risks recirculation cavitation just like operating below BEP at full speed.
- NPSHr also scales with speed² (some references cite ~speed^1.8). At full speed your margin may be fine; at 60% speed your NPSHr drops, but so does NPSHa if the pump is drawing from a vapor-pressure-sensitive fluid. Re-check NPSH at the slowest operating speed.
- Motor cooling. Standard TEFC motors are self-cooled by their own fan. Below ~50% speed the fan no longer cools adequately. Spec an inverter-duty motor with separate cooling, or limit minimum speed to ~30 Hz on a 60-Hz system.
- Harmonics. PWM drives inject harmonics back into the supply. Plants with sensitive electronics need line reactors or harmonic filters. Often forgotten until commissioning.
Sizing the drive
The drive is sized for motor amps, not pump hp. Use the motor nameplate full-load amps. Add 10-15% headroom for the inverter-duty current rating. Don't undersize on the assumption that "we'll run at 60%" — startup, fault recovery, and over-temperature derating all need full-rated capacity.
Energy savings estimate (rule-of-thumb)
For mixed-system pumps spending most of their hours below design flow, expect 20-50% energy savings from a VFD vs. a throttled or fixed-speed alternative. The full cubic-law 60-70% number applies only to closed-loop HVAC and similar pure-friction systems.
For a 50-hp pump running 6,000 hours/year at $0.10/kWh:
- 30% savings = ~$6,700/year
- Drive + install = $8,000-15,000 depending on enclosure rating
- Simple payback = 1-3 years for a friction-dominant duty
Static-dominant duty (e.g., a deep-well submersible lifting 200 ft) often pays back in 8-15 years, which is a "no" on most utility budgets.
What to compute before buying
1. System curve with static / friction split clearly labeled 2. Pump curve at full speed + at the slowest operating speed 3. Operating-point efficiency at every flow you'll actually run 4. Annual operating hours by flow band 5. Annualized energy cost difference vs. throttled / fixed-speed alternative
That five-line analysis lets you walk into a VFD purchase decision with the right number, not the brochure number.
References
- Hydraulic Institute. *ANSI/HI 9.6.5 — Pump Application Guidelines for Variable Speed Pumping.*
- DOE Pump System Assessment Tool (PSAT) — free reference workbook for VFD savings estimates.
- Karassik, I. J., et al. *Pump Handbook,* 4th ed. McGraw-Hill — chapters on variable speed operation.