Questions: Pump Affinity Laws and Geometric Similarity Scaling

Question 1

A centrifugal pump running at 1200 RPM consumes 100 kW. The operator installs a variable-speed drive and reduces speed to 900 RPM to lower flow. What is the new power consumption?

• A. 75 kW — power scales linearly with speed
• B. 56.25 kW — power scales as the square of the speed ratio
• C. 42.2 kW — power scales as the cube of the speed ratio
• D. 30 kW — power scales as the fourth power due to combined head and flow effects

Answer: (C)

Power follows the cubic affinity law: P₂/P₁ = (N₂/N₁)³. The speed ratio is 900/1200 = 0.75, so P₂ = 100 × 0.75³ = 100 × 0.422 = 42.2 kW. This is the famous result that makes variable-speed drives so energy-effective: a 25% speed reduction cuts power nearly in half. The linear and square-law options represent common misconceptions — mistaking power for flow or head scaling respectively. The cubic relationship arises because power equals ρgQH, and both Q (∝ N) and H (∝ N²) scale with speed, giving P ∝ N × N² = N³.

Question 2

When a pump's speed is varied while its diameter stays constant, the operating points at different speeds in the Q-H diagram:

• A. Fall on a horizontal line, since head depends only on speed and not flow
• B. Fall on a parabola through the origin — the affinity parabola — because H ∝ Q² at each speed ratio
• C. Fall on a straight line through the origin with slope proportional to the speed ratio
• D. Are unpredictable without knowing the system curve, since it shifts with speed

Answer: (B)

Since Q ∝ N and H ∝ N², eliminating N gives H ∝ Q² — the relationship between head and flow at corresponding operating points is parabolic. As speed changes, the operating point traces this parabola, called the affinity parabola, through the Q-H diagram. This geometric insight is practically useful: you can predict where the pump will operate at any speed by finding the intersection of the affinity parabola with the pump curve at the new speed. The system curve does not shift when speed changes — it is a property of the piping, not the pump.

Question 3

Reducing pump speed by 20% reduces power consumption by approximately 49%, because power scales as the cube of the speed ratio.

• T. True
• F. False

Answer: True

With N₂/N₁ = 0.8, P₂/P₁ = 0.8³ = 0.512 — power drops to about 51% of its original value, a savings of ~49%. This is the core engineering argument for variable-speed drives in systems that frequently operate at partial flow: the energy savings are disproportionately large compared to the speed reduction. A pump that runs at 80% speed for half its operating hours saves nearly half the energy during those hours.

Question 4

The affinity laws apply equally to any two pumps regardless of their shape, size, or specific speed, as long as they are both centrifugal pumps.

• T. True
• F. False

Answer: False

The affinity laws require geometric similarity — the pumps must be the same shape (scaled versions of each other) and must operate at the same specific speed (the same dimensionless point on their performance curves). They do not hold between geometrically dissimilar pumps, nor between a pump operating near its best-efficiency point and one operating at the extremes of its range. When significant impeller trimming is applied, viscous scaling effects also cause the efficiency to shift, further limiting the laws' accuracy.

Question 5

Why does power scale as the cube of rotational speed rather than linearly, and what practical implication does this have for systems that frequently operate at partial flow?

Model answer: Power equals ρgQH. Under affinity scaling with constant diameter, Q scales as N (first law) and H scales as N² (second law), so P ∝ ρg × N × N² = ρgN³ — the cubic relationship. Because both flow and head decrease together as speed falls, the power reduction is multiplicative. Practically, this means variable-speed drives are highly effective for systems that frequently operate below design flow: reducing speed by just 20% cuts power nearly in half. Systems with seasonally variable demand — HVAC chilled water loops, building pressure-boosting, irrigation — capture enormous energy savings by matching pump speed to actual demand rather than throttling with control valves (which waste energy by dissipating excess head across the valve).

The contrast with throttle control is instructive: a throttle valve reduces flow by increasing system resistance, forcing the pump to work against higher head. The pump still runs at full speed and consumes most of its rated power, but most of that energy is wasted as heat across the valve. Variable-speed control instead reduces the speed, shifting the entire pump curve downward. The operating point moves along the system curve at lower Q and lower H, with dramatically lower power — the affinity laws ensure all three quantities decrease together.