Pumps, Compressors & Fans

7% of exam

Affinity laws, pump and fan power and efficiency, NPSH and cavitation, operating point, parallel/series operation, and compressor work.

5 concepts

Exam areas

E. Pumps, Compressors, and Fans

Pump and Fan Affinity Laws and the Operating Point

The affinity laws (flow ∝ N, head ∝ N², power ∝ N³), riding the pump/fan curve, and why a VFD beats a throttling valve on energy.

Almost every HVAC pump and fan problem on the exam turns on one idea: a centrifugal machine does not pick its own flow and head — the system it serves does. The machine offers a curve of what it ·can· do, the piping or ductwork imposes a curve of what it ·demands·, and the steady-state operating point is where those two curves cross. Change the speed or the impeller and you slide that curve, and the affinity laws tell you exactly how flow, head, and power follow. This is also where the biggest energy savings in a building hide: dialing a fan back to 70% speed cuts its power to roughly a third, and a surprising number of exam questions are really asking you to prove that. The relations are tabulated in the NCEES PE Mechanical Reference Handbook — §3.7.6 Pump Affinity Laws and §9.3.8.3 Fan Affinity Laws.

What the affinity laws actually say

For a fixed impeller running at variable speed

N

, a centrifugal pump or fan obeys three proportionalities: volume flow tracks speed linearly, head (or pressure) tracks speed squared, and shaft power tracks speed cubed. The cube on power is the whole story behind variable-speed savings, and it follows directly from the other two — power is proportional to flow times head, and

N \times N^2 = N^3

. Hold these as ratios so units never bite you; the laws are dimensionless scaling rules, valid as long as efficiency stays roughly constant between the two operating states.

\frac{Q_2}{Q_1}=\frac{N_2}{N_1}, \qquad \frac{h_2}{h_1}=\left(\frac{N_2}{N_1}\right)^{2}, \qquad \frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^{3}

The diameter forms

Trimming an impeller has the same effect as slowing it, because both reduce blade tip speed. With speed fixed, flow scales with diameter

D

, head with

D^2

, and power with

D^3

— the identical exponents. The handbook also lists a specific-gravity column: head produced (in feet of fluid) is independent of density, but the pressure and the brake horsepower scale linearly with

SG

, since a denser fluid weighs more to lift at the same head. Impeller trimming is a permanent, one-time fix used when a pump is oversized for a fixed system; a VFD is the adjustable equivalent.

\frac{Q_2}{Q_1}=\frac{D_2}{D_1}, \qquad \frac{h_2}{h_1}=\left(\frac{D_2}{D_1}\right)^{2}, \qquad \frac{bhp_2}{bhp_1}=\left(\frac{D_2}{D_1}\right)^{3}\frac{SG_2}{SG_1}

Riding the curve to a new operating point

Here is the trap the affinity laws set: the laws scale a ·single point· along a parabola through the origin, not a whole operating point on a real system. A real system curve has static head — an elevation lift or a fixed pressure — so it does ·not· pass through the origin. When you slow a pump on such a system, the true new operating point is where the ·new· pump curve meets the ·same· system curve, and that point does not lie on the affinity parabola through the old point. The clean exam approach: scale a family of points with the laws to draw the new pump curve, then intersect it with the unchanged system curve. Only when the system is pure friction (no static head,

h \propto Q^2

straight through the origin) do the affinity laws land you exactly on the new operating point.

The system curve sets the exponent you can claim

Because the power law is cubic only when you move along a parabola through the origin, the energy you actually save with a VFD depends on how much of the head is static. On an all-friction loop (variable-flow chilled water, a VAV duct system at a fixed damper position), reducing flow to a fraction

f

of design drops power to about

f^3

— go to 70% flow and you spend roughly

0.7^3 \approx 34\%

of design power. Add a large static lift and the curve flattens, the cube weakens toward a square or less, and the savings shrink. Knowing which regime a problem lives in is the difference between a right and wrong number.

\frac{P_2}{P_1}\approx\left(\frac{Q_2}{Q_1}\right)^{3}\quad\text{(friction-dominated system, }h\propto Q^2\text{)}

VFD versus throttling

The two ways to reduce flow are not equal. A throttling valve (or an outlet damper) adds resistance — it steepens the system curve so its intersection with the ·unchanged· pump curve slides to lower flow at ·higher· head; the pump still spins at full speed and still draws most of its design power, with the throttled pressure simply burned across the valve as waste. A VFD instead lowers the pump curve to meet the ·unchanged· system curve at lower flow and lower head, and the cube law collapses the power. That gap is the savings, and it is exactly why ASHRAE 90.1 pushes variable-speed drives on larger pumps and fans.

Exam strategy

Write the ratio you need before plugging numbers, and confirm which variable is independent: speed-change problems use

N

, impeller problems use

D

. Flow gets exponent 1, head exponent 2, power exponent 3 — memorize the 1-2-3 ladder cold. For a power question, never apply the cube unless the system is friction-only (no static head); if there is a static lift, slow-then-intersect. When a problem contrasts a VFD with a valve or damper, the VFD answer is always the smaller power because it rides the cube law down while throttling does not. Finally, keep head in feet of fluid (density-independent) but remember pressure and bhp pick up the specific-gravity factor.

Key equations

Affinity laws — speed change

\frac{Q_2}{Q_1}=\frac{N_2}{N_1},\quad \frac{h_2}{h_1}=\left(\frac{N_2}{N_1}\right)^{2},\quad \frac{bhp_2}{bhp_1}=\left(\frac{N_2}{N_1}\right)^{3}

Fixed impeller, variable speed $N$ (rpm). Flow linear, head/pressure square, power cube. Valid at constant efficiency.

Affinity laws — impeller diameter

\frac{Q_2}{Q_1}=\frac{D_2}{D_1},\quad \frac{h_2}{h_1}=\left(\frac{D_2}{D_1}\right)^{2},\quad \frac{bhp_2}{bhp_1}=\left(\frac{D_2}{D_1}\right)^{3}

Fixed speed, trimmed impeller diameter $D$ . Same 1-2-3 exponents as speed change.

Specific-gravity scaling

\frac{bhp_2}{bhp_1}=\frac{SG_2}{SG_1}\quad(\text{at fixed }Q,h)

Head in feet of fluid is density-independent; pressure (psi) and brake horsepower scale linearly with $SG$ .

Fan law form (pressure)

\frac{P_2}{P_1}=\left(\frac{N_2}{N_1}\right)^{2}\left(\frac{\rho_2}{\rho_1}\right)

$P$ = fan total, static, or velocity pressure (in. wg). Fan pressure also scales with air density $\rho$ — important at altitude or elevated temperature.

Fan law form (power)

\frac{\dot{W}_2}{\dot{W}_1}=\left(\frac{N_2}{N_1}\right)^{3}\left(\frac{\rho_2}{\rho_1}\right)

Fan shaft power scales with speed cubed and linearly with air density. Per NCEES §9.3.8.3 (dry standard air, 14.696 psia, 70°F unless stated).

Friction-system power reduction

\frac{P_2}{P_1}\approx\left(\frac{Q_2}{Q_1}\right)^{3}

VFD savings estimate on a friction-dominated system ( $h\propto Q^2$ through the origin). Overstates savings if static head is present.

Head as a parabola through the origin

h_2 = h_1\left(\frac{Q_2}{Q_1}\right)^{2}

Combines the flow and head laws; eliminating $N$ shows affinity-scaled points lie on a parabola $h\propto Q^2$ through the origin.

Operating point definition

h_{pump}(Q)=h_{system}(Q)

Steady-state flow is the intersection of pump/fan curve and system curve. Affinity scaling moves the machine curve; the system curve is unchanged.

Worked examples

Slowing a pump with a VFD

Problem. A circulating pump at $1750\,\text{rpm}$ delivers $400\,\text{gpm}$ at $80\,\text{ft}$ of head and draws $9.0\,\text{bhp}$ . The loop is all friction (no static lift). A VFD slows it to $1400\,\text{rpm}$ . Find the new flow, head, and brake horsepower.

Solution. Speed ratio

N_2/N_1 = 1400/1750 = 0.800

. Flow (exponent 1):

Q_2 = 400\,(0.800) = 320\,\text{gpm}

. Head (exponent 2):

h_2 = 80\,(0.800)^2 = 80\,(0.640) = 51.2\,\text{ft}

. Power (exponent 3):

bhp_2 = 9.0\,(0.800)^3 = 9.0\,(0.512) = 4.61\,\text{bhp}

. Final:

Q_2 = 320\,\text{gpm}

h_2 = 51.2\,\text{ft}

bhp_2 = 4.61\,\text{bhp}

(3 s.f.). Sanity check: a 20% speed cut dropped power by nearly half — exactly the cube law at work, and the new point lies on

h\propto Q^2

since the system is friction-only.

bhp_2 = 9.0\left(\frac{1400}{1750}\right)^{3}=4.61\ \text{bhp}

VFD savings versus an outlet damper

Problem. A supply fan delivers $10{,}000\,\text{cfm}$ at $6.0\,\text{in. wg}$ drawing $12.0\,\text{bhp}$ on a friction-dominated VAV duct system. The load drops and only $7{,}000\,\text{cfm}$ is needed. Estimate the fan power if flow is cut with (a) a VFD and (b) an outlet damper that throttles to $7{,}000\,\text{cfm}$ while the fan stays at full speed (assume the damper holds the fan near $11\,\text{bhp}$ ).

Solution. (a) VFD — flow ratio

7000/10000 = 0.700

, friction system so power follows the cube:

\dot{W}_2 = 12.0\,(0.700)^3 = 12.0\,(0.343) = 4.12\,\text{bhp}.

(b) Damper — the fan stays at full speed and rides up its own curve to a higher static pressure; power barely falls,

\approx 11\,\text{bhp}

. The VFD draws

4.12\,\text{bhp}

, a

65.7\%

reduction from the

12\,\text{bhp}

design point (

1-0.7^3 = 0.657

); that is about

11 - 4.12 = 6.9\,\text{bhp}

less than the damper. Final: VFD

\approx 4.12\,\text{bhp}

vs damper

\approx 11\,\text{bhp}

. Sanity check: the cube law alone predicts a 65.7% power cut at 70% flow — consistent with the figure.

\dot{W}_{VFD}=12.0\,(0.700)^3=4.12\ \text{bhp}

Impeller trim to hit a duty point

Problem. A pump with a $10.0\,\text{in}$ impeller produces $90\,\text{ft}$ at $250\,\text{gpm}$ but the friction-only system needs only $57.6\,\text{ft}$ at that reduced flow. Find the trimmed diameter (assume the duty point stays on the affinity parabola) and the new flow.

Solution. On the affinity parabola, head scales with diameter squared. Required head ratio

= 57.6/90 = 0.640

, so

D_2/D_1 = \sqrt{0.640} = 0.800

. Trimmed diameter:

D_2 = 0.800\times 10.0 = 8.00\,\text{in}

. Flow scales with diameter:

Q_2 = 250\,(0.800) = 200\,\text{gpm}

. Final: trim to

8.00\,\text{in}

, delivering

200\,\text{gpm}

57.6\,\text{ft}

. Sanity check:

(0.8)^2 = 0.64

recovers the head ratio, confirming the trim. Trimming below about 75–80% of full diameter is not advised because efficiency falls off, so

8.0\,\text{in}

is near the practical limit.

\frac{D_2}{D_1}=\sqrt{\frac{h_2}{h_1}}=\sqrt{0.640}=0.800

Common pitfalls

•Applying the cube power law on a system with static head. The cube holds only along a parabola through the origin; with an elevation lift or fixed back-pressure the savings are smaller — slow the curve and re-intersect the system curve instead.
•Confusing the affinity parabola with the system curve. Affinity scaling moves the pump/fan curve; the system curve does not change. The new operating point is their intersection, not the affinity-scaled old point (unless the system is friction-only).
•Forgetting density on fan laws. Fan pressure and power both carry a $\rho_2/\rho_1$ factor — at altitude or in hot air, a fan at the same rpm makes less pressure and less power than at standard 70°F sea-level air.
•Scaling pressure (psi) or bhp with head ratio while ignoring specific gravity. Head in feet is density-independent, but pressure and brake horsepower scale linearly with $SG$ .
•Treating a throttling valve or damper as energy-equivalent to a VFD. Throttling keeps the machine at full speed and burns the excess head; a VFD removes it via the cube law. The VFD always wins on power.
•Using the laws across a big speed change as if efficiency were constant. Affinity scaling assumes the efficiency island doesn't move much; large turndowns drift off best-efficiency and the laws become approximate.

References

NCEES PE Mechanical Reference Handbook — §3.7.6 Pump Affinity Laws
NCEES PE Mechanical Reference Handbook — §9.3.8.3 Fan Affinity Laws
ASHRAE Handbook—HVAC Systems and Equipment, Ch. 44 (Centrifugal Pumps) and Ch. 21 (Fans)
ASHRAE 90.1 — Energy Standard (variable-speed drive provisions) — Why VFDs are mandated on larger pumps and fans.

Pump/Fan Power, Efficiency and NPSH

Hydraulic vs brake vs input power, the GPM·ft/3960 and CFM·in.wg/6356 work shortcuts, and NPSH available vs required to stop cavitation.

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System Curve, Operating Point and Parallel/Series Operation

The system resistance curve H ∝ Q², where it meets the pump/fan curve, throttling vs speed control, and why parallel adds flow while series adds head.

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Fan Types, Curves and Selection

Centrifugal (FC/BI/airfoil) vs axial fans, reading the fan curve and its stable region, class and arrangement, and fan total vs static efficiency.

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Compressor Work, Stages and Volumetric Efficiency

Isentropic vs polytropic compression work, pressure ratio and staging with intercooling, volumetric efficiency and clearance, and reciprocating vs centrifugal/screw machines.

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