The fan laws
Why the curve is the shape it is: three proportionalities govern
every fan (and every pump). They are not catalog conventions — they fall out of the physics of a
spinning wheel. Drag the speed and watch the three quantities separate.
Fan Law 1 — airflow
Q2 = Q1 ·
N2N1
Why linear: the wheel geometry is fixed, so each revolution
moves the same “bite” of air. Twice the revolutions per minute, twice the bites per minute.
Fan Law 2 — pressure
P2 = P1 ·
N2N12
Why squared: pressure is made by flinging air to blade-tip
speed, and kinetic energy goes with velocity². Same reason VP = (V/4005)² — velocity pressure
is a v² phenomenon everywhere in our ductwork.
Fan Law 3 — power
W2 = W1 ·
N2N13
Why cubed: power = flow × pressure. Flow brings one factor of
N; pressure brings two more. N · N² = N³. (In numbers: BHP = Q·P / (6,356·η).)
Airflow — scales with N
100%
= 10,000 CFM on the design system
Pressure — scales with N²
100%
= 3.50 in. w.g.
Power — scales with N³
100%
= 8.3 BHP
One speed change, three different answers
The punchline: at half speed you still move half the air —
but you pay one-eighth of the power. This asymmetry is the entire economic case
for variable-speed drives, and it’s written into ASHRAE 90.1 (heat-rejection fans must hit
≤ 30% power at 50% airflow — a number straight off the aqua line).
The laws are the curve family
Take the tested 1,750 RPM curve and apply the three laws to every
point on it: each point slides along a parabola through the origin (Q ∝ N, P ∝ N² ⇒ P ∝ Q²). The
result is the same curve, re-scaled — the manufacturer tests one speed, the laws print the
rest of the catalog. And because a duct system is also P = C·Q², the operating point rides
one of these parabolas as speed changes: the system curve is a constant-efficiency rail.
What the fan laws do not say
- They re-rate one fan; they don’t move you along its curve. The system still decides where you operate. Laws first, intersection second.
- Same fan, same point of rating only. Never fan-law between two different fans, and never from a point in one flow regime to another (ASHRAE Handbook — HVAC S&E, ch. 21 §5, Table 2).
- Air density is a hidden variable. Catalog curves assume standard air (0.075 lb/ft³). Altitude and temperature re-scale pressure and power directly.
- Motors, belts, and VFDs are not fans. Drive and motor losses ride on top of BHP — the cube law applies to the shaft, not the utility meter (more in demo 4).