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Centrifugal pump performance is governed by three elegant scaling relationships known as the Affinity Laws. Whether you are commissioning a Variable Frequency Drive (VFD), trimming an impeller, or sizing parallel pump arrays, mastering these laws translates directly into energy savings, reduced capital expenditure, and optimised process control.
1. The Three Affinity Laws
For a centrifugal pump operating at changed rotational speed \(N\) (or changed impeller diameter \(D\)) — with the same fluid and geometry — the following dimensionless ratios hold:
Law 1 – Flow Rate (Capacity)
Flow rate scales linearly with speed:
$$\frac{Q_2}{Q_1} = \frac{N_2}{N_1}$$Halving the speed halves the flow. For diameter variation (impeller trimming at constant speed):
$$\frac{Q_2}{Q_1} = \frac{D_2}{D_1}$$Law 2 – Head (Pressure Rise)
Head scales with the square of the speed ratio:
$$\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^2$$Equivalently for impeller diameter:
$$\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^2$$Law 3 – Power Consumption
Power scales with the cube of the speed ratio — the most commercially important law:
$$\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3$$For impeller diameter:
$$\frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^3$$These three laws apply simultaneously; you cannot change speed and have only one of them hold. They assume constant pump efficiency — a reasonable approximation near the Best Efficiency Point (BEP).
2. VFD Power Savings: The Cubic Law in Practice
The cubic relationship between power and speed is why VFDs (Variable Frequency Drives) deliver such dramatic energy savings on system-curve-dominated applications (i.e., where friction head dominates over static head).
The 50% Speed Rule
If speed is reduced to half its rated value:
$$P_2 = P_1 \times \left(\frac{0.5\,N_1}{N_1}\right)^3 = P_1 \times (0.5)^3 = 0.125\,P_1$$Running at 50% speed consumes only 12.5% of full-speed power.
VFD vs. Throttle Valve — Energy Comparison
Consider a pump rated at 30 kW at 2950 RPM delivering 150 m³/h. The process requires only 75 m³/h (50% flow). Two control strategies are compared:
| Control Method | Operating Condition | Motor Power (kW) | Annual Energy (MWh/yr)* |
|---|---|---|---|
| Throttle valve | Full speed, valve partially closed | ~26 kW (pump still near full speed) | ~228 |
| VFD speed reduction | Speed reduced to 50% (1475 RPM) | 3.75 kW | ~33 |
| Saving | — | ~22 kW | ~195 MWh/yr |
*Assuming 8760 operating hours/year and constant reduced-flow demand.
At a typical industrial electricity cost of \$0.10/kWh, the annual saving exceeds \$19,500, often achieving VFD payback in under two years.
3. System Curve and the Operating Point
A pump does not operate in isolation — it works against a system curve defined by the piping network resistance:
$$H_{\text{sys}} = H_{\text{static}} + K \cdot Q^2$$where:
- \(H_{\text{static}}\) — static head component: elevation difference plus pressure difference between source and destination (independent of flow rate)
- \(K\) — system resistance coefficient (depends on pipe diameter, length, fittings, valves)
- \(Q\) — volumetric flow rate
- \(K \cdot Q^2\) — friction (dynamic) head, which grows with the square of flow
Operating Point
The pump operates at the intersection of the pump H-Q curve and the system curve. When a VFD reduces speed, the pump H-Q curve shifts downward — at lower \(N\), both the head and flow at every point scale by the affinity laws. The new intersection with the system curve determines the true operating conditions.
For a pure friction system (\(H_{\text{static}} = 0\)), the operating point tracks the affinity law parabola exactly: speed and flow remain proportional, and the cubic power law applies perfectly. When significant static head is present, the savings are less dramatic but still substantial.
4. Worked Example: Speed Change Calculation
Given conditions at operating point 1:
- Speed: \(N_1 = 2950\) RPM
- Flow rate: \(Q_1 = 150\) m³/h
- Head: \(H_1 = 60\) m
- Shaft power: \(P_1 = 30\) kW
Required: Find \(Q_2\), \(H_2\), \(P_2\) at \(N_2 = 2000\) RPM.
Step 1 — Speed Ratio
$$r = \frac{N_2}{N_1} = \frac{2000}{2950} = 0.6780$$Step 2 — New Flow Rate (Law 1)
$$Q_2 = Q_1 \times r = 150 \times 0.6780 = \mathbf{101.7 \text{ m}^3/\text{h}}$$Step 3 — New Head (Law 2)
$$H_2 = H_1 \times r^2 = 60 \times (0.6780)^2 = 60 \times 0.4597 = \mathbf{27.6 \text{ m}}$$Step 4 — New Shaft Power (Law 3)
$$P_2 = P_1 \times r^3 = 30 \times (0.6780)^3 = 30 \times 0.3117 = \mathbf{9.35 \text{ kW}}$$Reducing speed from 2950 to 2000 RPM reduces power from 30 kW to just 9.35 kW — a 69% power reduction for a 32% reduction in flow.
5. Parallel and Series Pump Configurations
When a single pump cannot meet the system requirement, pumps are combined in parallel or series.
Parallel Pumps (Flow Adds at Same Head)
Two identical pumps in parallel: at any given head, the combined flow is the sum of individual flows:
$$Q_{\text{total}} = Q_A + Q_B \quad \text{at constant } H$$The combined H-Q curve is constructed by horizontal addition of individual curves. Parallel operation is used when high flow at moderate head is required.
Series Pumps (Head Adds at Same Flow)
Two identical pumps in series: at any given flow, the combined head is the sum of individual heads:
$$H_{\text{total}} = H_A + H_B \quad \text{at constant } Q$$The combined H-Q curve is constructed by vertical addition. Series operation is used when high head at moderate flow is required.
| Feature | Parallel Pumps | Series Pumps |
|---|---|---|
| Curve construction | Horizontal addition (flow) | Vertical addition (head) |
| Best for | High flow, moderate head | High head, moderate flow |
| System curve type | Low static head, high friction | High static head systems |
| Flexibility | One pump can serve partial load | Both pumps usually required |
| Turndown risk | Lead pump may surge at low flow | Overpressure if downstream blocked |
6. Impeller Trimming
Instead of a VFD, a permanent reduction in duty can be achieved by machining the impeller to a smaller diameter — a low-cost, irreversible modification that reduces Q, H, and P proportionally to the diameter ratio.
Trimming Example
Original impeller diameter: \(D_1 = 280\) mm → Trimmed to \(D_2 = 250\) mm.
Diameter Ratio
$$r_D = \frac{D_2}{D_1} = \frac{250}{280} = 0.8929$$New Flow
$$Q_2 = Q_1 \times 0.8929 = 150 \times 0.8929 = \mathbf{133.9 \text{ m}^3/\text{h}}$$New Head
$$H_2 = H_1 \times (0.8929)^2 = 60 \times 0.7973 = \mathbf{47.8 \text{ m}}$$New Power
$$P_2 = P_1 \times (0.8929)^3 = 30 \times 0.7119 = \mathbf{21.4 \text{ kW}}$$Trimming vs. VFD — Decision Guide
| Criterion | Impeller Trimming | VFD |
|---|---|---|
| Cost | Very low (machining only) | High (drive + installation) |
| Reversibility | Irreversible (must replace impeller) | Fully reversible |
| Variable demand | Not suitable | Ideal |
| Fixed duty reduction | Ideal (<20% diameter change) | Overkill |
| Energy savings | Permanent reduction at new duty | Adjustable, up to 87.5% saving |
| Max recommended trim | ~15–20% of diameter | N/A |
Industry guidance: if the trim ratio \(D_2/D_1 < 0.80\), consider replacing the impeller entirely or selecting a different pump frame size, as efficiency losses become significant below 80%.
7. Specific Speed — Pump Type Classification
Specific Speed (\(N_s\)) is a dimensionless (or semi-dimensional) index that characterises the shape and type of a centrifugal pump impeller. It is evaluated at the BEP:
$$N_s = \frac{N \sqrt{Q}}{H^{3/4}}$$where \(N\) is in RPM, \(Q\) in m³/s (or m³/min depending on convention), and \(H\) in metres.
Note: SI and US conventions use different unit bases, producing numerically different values. Always confirm which convention applies.
| Pump Type | Specific Speed \(N_s\) (metric, RPM, m³/min, m) | Impeller Shape | Typical Application |
|---|---|---|---|
| Radial flow | 10 – 70 | Narrow, high-blade-angle | High head, low flow |
| Francis / Mixed radial | 70 – 160 | Mixed radial/axial blades | Medium head & flow |
| Mixed flow | 160 – 300 | Diagonal discharge | Moderate head, higher flow |
| Axial flow (propeller) | 300 – 800+ | Propeller-type | Low head, very high flow |
Specific speed guides impeller geometry selection during pump design. Radial flow pumps suit high-pressure boiler feed service; axial flow pumps excel in low-head irrigation or cooling water circulation. Operating a pump far outside its design \(N_s\) range causes efficiency penalties and vibration problems.
8. Key Takeaways
- The three affinity laws (Q∝N, H∝N², P∝N³) govern all centrifugal pump speed changes.
- The cubic power law makes VFDs extremely attractive: 50% speed = 12.5% power.
- The system curve \(H = H_{\text{static}} + KQ^2\) determines where the pump actually operates; always analyse the system, not just the pump curve.
- Parallel pumps add flow; series pumps add head — match the configuration to the system curve shape.
- Impeller trimming is a low-cost permanent solution for fixed duty reductions under ~20%.
- Specific speed classifies impeller geometry and guides pump selection for new designs.
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