Library / Dimensional analysis and similarity / DIM-17
Dimensional analysis and similarity
Pump or fan flow rate at a new speed and diameter
Affinity law for volume flow rate: for geometrically similar rotodynamic machines at corresponding operating points, flow is proportional to speed and to the cube of the impeller diameter.
- Ref
- DIM-17
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Q2 [m³/s]
With s = n2/n1 and d = D2/D1, the flow at the corresponding operating point scales as Q2 = Q1 s d³. These affinity laws compare corresponding operating points of geometrically similar rotodynamic machines. They assume low-compressibility fan flow or non-cavitating pump flow. The diameter powers describe a fully scaled machine; trimming an impeller in an existing casing changes the geometry ratios and needs a trim curve or manufacturer data. The actual installed operating point is where the new pump or fan curve meets the system curve. For a speed change alone (d = 1) see the speed-only version, Pump flow rate after a speed change.
Affinity law for volume flow rate: for geometrically similar rotodynamic machines at corresponding operating points, flow is proportional to speed and to the cube of the impeller diameter.
Calculator
- Speed ratio, s = n2/n11.2
- Diameter ratio, d = D2/D11
- Flow multiplier, s d³1.2
- Flow at 80 % of the new speed0.048 m³/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Running at 1.2 times the speed with a diameter ratio of 1 multiplies the flow by 1.2, from 0.05 to 0.06 m³/s.
- Flow is linear in speed but cubic in diameter, so a modest increase in size gives a large increase in flow; the head and power change much faster than the flow.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D1 Impeller diameter at point 1 | 20 to 3,000 mm | 6e-5 | 202.5 | 202.5 | +10% | −24.9 % |
| Q1 Flow rate at point 1 | 0.001 to 100 m³/s | 0.0012 | 120 | 120 | +10% | +10 % |
| D2 Impeller diameter at point 2 | 20 to 3,000 mm | 1.778e-5 | 60 | 60 | +10% | +33.1 % |
| n1 Speed at point 1 | 50 to 10,000 rpm | 0.006 | 1.2 | 1.194 | +10% | −9.09 % |
| n2 Speed at point 2 | 50 to 10,000 rpm | 0.0025 | 0.5 | 0.4975 | +10% | +10 % |
Sweep of Impeller diameter at point 2
| Impeller diameter at point 2 (mm) | Flow rate at point 2 (m³/s) |
|---|---|
| 20 | 1.778e-5 |
| 392.5 | 0.1344 |
| 765 | 0.9949 |
| 1,138 | 3.271 |
| 1,510 | 7.651 |
| 1,883 | 14.82 |
| 2,255 | 25.48 |
| 2,628 | 40.31 |
| 3,000 | 60 |
Sweep one variable
Variables and how each one contributes
Flow rate at point 2
What the equation gives you. Everything else feeds it.
Flow rate at point 1
The volume flow rate at the original operating point, in m³/s. Raises the new flow in direct proportion. Raising it by 10 % (to 0.055) moves the result by +10 %, from 0.06 to 0.066 m³/s.
Speed at point 1
The original rotational speed, in rpm (any unit works if n2 uses the same). Lowers the new flow in inverse proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 0.06 to 0.05455 m³/s.
Speed at point 2
The new rotational speed, in rpm. Raises the new flow in direct proportion. Raising it by 10 % (to 1,320) moves the result by +10 %, from 0.06 to 0.066 m³/s.
Impeller diameter at point 1
The original impeller diameter, in mm. Lowers the new flow with the cube of the diameter ratio: a larger reference size means a smaller increase. Raising it by 10 % (to 330) moves the result by −24.9 %, from 0.06 to 0.04508 m³/s.
Impeller diameter at point 2
The new impeller diameter, in mm, for a geometrically similar machine. Raises the new flow with the cube of the diameter, so diameter is the strongest lever. Raising it by 10 % (to 330) moves the result by +33.1 %, from 0.06 to 0.07986 m³/s.
Worked example with the values above
- Ratios: s = 1,200 / 1,000 = 1.2 and d = 300 / 300 = 1.
- Flow multiplier: s d³ = 1.2 × 1³ = 1.2.
- Q2 = 0.05 × 1.2 = 0.06 m³/s.
Assumptions and limits
- Geometric similarity, and corresponding operating points with the same flow coefficient Q/(nD³).
- Low-compressibility flow, with no cavitation in a pump and a Reynolds number high enough that its effect is small.
Common mistakes
- Scaling the diameter of an impeller trimmed inside the same casing as if the machine were fully scaled.
- Mixing the units for the two speeds or the two diameters.
- Treating the result as the new operating point: the operating point is where the machine curve meets the system curve.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (Q1 × 1, n2 × 1, n1 × −1, D2 × 3, D1 × −3). The two diameters dominate, so measure them carefully. The law is itself approximate: efficiency shifts with size and speed, and real machines depart from it by a few per cent.
Where it is used
- Predicting the flow of a pump or fan after a speed change or in a geometrically similar machine of another size.
- Sizing a variable-speed drive and checking a new machine against test data.
Maths you need
- Geometric similarity: same shape with all lengths scaled by one factor
- Operating point: where the machine curve meets the system curve