Fan and system curves: the operating point, and the cube law

Fan pressure against flow with the system curve, the point they cross at, and the shaft power.

Fan pressure against flow with the system-resistance curve ΔP = K Q², their crossing being the operating point; below it, shaft power against flow, where following the system curve puts power on the cube of speed.

Reference conditions: K = 0.015 Pa·s²/m⁶, illustrative fan curve ΔP = 210 − 0.006 Q² Pa, operating point 100 m³/s at 150 Pa, ηfan 0.70 — 21.4 kW at design, 28.5 kW at 110 % speed, 15.6 kW at 90 %.

Formula: ΔPfan(Q) − ΔPsystem(Q) = 0, P = Q ΔP / ηfan, and the fan laws Q2/Q1 = N2/N1, P2/P1 = (N2/N1)³.

Cooling-tower fan and system curves: fan pressure against flow with the operating point at 100 m³/s and 150 Pa, and shaft power against flow showing the cube law — 21.4 kW at design, 28.5 kW at 110 % speed, 15.6 kW at 90 % speed (illustrative)Illustrative cooling-tower fan/system chart. The rising curve is the system resistance ΔP = K Q² with K = 0.015 Pa·s²/m⁶; the falling curve is the illustrative fan curve ΔP = 210 − 0.006 Q² Pa. They cross at the operating point: 100 m³/s and 150 Pa. Shaft power is P = Q ΔP / ηfan with ηfan = 0.70, so 100 × 150 ÷ 0.70 = 21.43 kW. Because the fan follows the system curve, power scales with the cube of speed: at 110 m³/s (110 %) it is 28.52 kW, about 33 % more, and at 90 % speed it is 15.62 kW, roughly 73 % of the design power. Values are computed from the formulas, not measured; no fan, no rating. (chart-fan-system)Fan curve and system — Illustrative — not a measurementOperating point and the cube law020406080100120140050100150200Volumetric flow Q (m³/s)Fan pressure ΔP (Pa)100 / 150system ΔP = K Q²fan curve020406080100120140010203040Volumetric flow Q (m³/s)Shaft power (kW)21.428.5 (1.1 N)15.6 (0.9 N)Reference: 100 m³/s at 150 Pa · fan efficiency 0.70 · drive 0.95 · motor 0.94Fan curve ΔP = 210 − 0.006 Q² (Pa, illustrative) · system ΔP = 0.015 Q² (Pa)operating point: Q (m³/s) / ΔP (Pa)100 / 150system coefficient K (Pa·s²/m⁶)0.015Pshaft at N1 (kW) = Q ΔP ÷ 0.7021.43at 1.1 N: Q 110, ΔP 181.5, P (kW)28.52at 0.9 N: P (kW) = 0.9³ of N115.62speed increase of 10 %+33 % shaft powerComputed from the formula — not field data.
Illustrative, computed from the formulas above — not a measurement. The rising curve is the system resistance, the falling curve the illustrative fan curve, and their crossing is the operating point the plate prints.

Sources

Water side

What the water balance costs, and what the fill does with the water it is given.