HVAC Energy Retrofits, Variable Frequency Drives (VFDs) & Turbomachinery Fan Laws
Centrifugal Fan Affinity Laws & VFD Cubic Power Formula
Calculate VFD Turndown Airflow Q2 (m³/h), Quadratic Static Pressure Delta_P2 (Pa), Cubic Shaft Power P2 (kW), and Annual Fan Energy Savings (kWh/yr).
Slowing an AHU supply fan to 80% speed via VFD cuts airflow 20%—but slashes shaft power by nearly 49%!
Throttling an air-handling unit (AHU) or industrial baghouse fan with a discharge damper wastes electricity fighting artificial static drop! Under the three Turbomachinery Fan Affinity Laws on a fixed parabolic system curve (zero static-pressure reset), changing impeller speed from `N_1` to `N_2` (RPM) scales: (1) Volumetric Airflow linearly `Q_2 = Q_1 * (N_2 / N_1)`, (2) Fan Total/Static Pressure quadratically `Delta_P_2 = Delta_P_1 * (N_2 / N_1)^2`, and (3) Impeller Shaft Power CUBICALLY `P_2 = P_1 * (N_2 / N_1)^3`! That cubic exponent (`0.80^3 = 0.512`) makes Variable Frequency Drive (VFD) retrofits the highest-ROI electrical efficiency upgrade in commercial HVAC and industrial dust collection!
Verify on a 24,000 m³/h Supply Fan VFD Retrofit—Rated_Airflow `Q_1` = 24,000.0 m³/h (B2), Rated_Static_Pressure `Delta_P_1` = 750.0 Pa (B3), Rated_Shaft_Power `P_1` = 6.80 kW (B4), Nameplate_Speed `N_1` = 1,450 RPM (B5), VFD_Operating_Speed `N_2` = 1,160 RPM (`80%` speed, B6), and Annual_Run_Hours = 6,000 hr/yr (B7): VFD Airflow equals `19,200.0` m³/h (E2), Static Pressure drops to `480.00` Pa (E3), Shaft Power plunges to `3.48` kW (E4), saving `19,910.4` kWh/year (E5)!
| Fan Affinity Law KPI (24,000 m³/h, 750 Pa, 6.80 kW @ 1,450 -> 1,160 RPM) | Formula | Verified Output |
|---|---|---|
| VFD Turndown Airflow Q2 (m³/h) | Q1 * (RPM2 / RPM1) | 19,200.0 m³/h (14,400.0 @ 870 RPM) |
| VFD Static Pressure dP2 (Pa) | dP1 * ((RPM2 / RPM1) ^ 2) | 480.00 Pa (270.00 Pa @ 870 RPM) |
| VFD Cubic Shaft Power P2 (kW) | P1 * ((RPM2 / RPM1) ^ 3) | 3.48 kW (1.47 kW @ 60% speed!) |
| Annual Electricity Saved (kWh/yr) | (P1 - P1 * ((RPM2/RPM1)^3)) * Hours | 19,910.4 kWh (31,987.2 kWh @ 870 RPM) |
Linear-Quadratic-Cubic Fan Affinity Laws & Annual VFD kWh Savings Formula
Validate `Q_1 > 0`, `Delta_P_1 > 0`, `P_1 > 0`, `RPM_1 > 0`, `RPM_2` in `(0, 1.2 * RPM_1]` (note: running above `100%` synchronous speed rapidly overloads motor amps by `(N_2/N_1)^3`!), and `Annual_Hours` in `(0, 8,760]`!
=LET(qf_r, B2, dp_r, B3, pw_r, B4, nr_p, B5, nv_p, B6, hr_y, B7, IF(OR(qf_r <= 0, dp_r <= 0, pw_r <= 0, nr_p <= 0, nv_p <= 0, nv_p > 1.2 * nr_p, hr_y <= 0, hr_y > 8760), "Invalid Fan Law Input", LET(sr_t, nv_p / nr_p, ROUND((pw_r - (pw_r * (sr_t ^ 3))) * hr_y, 1))))Part-Load 60% Night/Off-Peak Setback Check (Set VFD Speed N2 = 870 RPM)
Step the VFD down to `60%` speed (`N_2` B6 = 870 RPM): Airflow E2 scales linearly to `14,400.0` m³/h, Static Pressure E3 drops to `270.00` Pa (`0.36x`), Shaft Power E4 collapses by `78.4%` down to `1.47` kW (`0.216x`), and Annual Energy Saved E5 jumps to `31,987.2` kWh/year!
=IF(ISTEXT(E5), E5, ROUND(B4 * ((B6 / B5) ^ 3), 2))Try the example
Use the original values or change them to test your understanding.