Library / Dimensional analysis and similarity / DIM-18
Dimensional analysis and similarity
Pump head at a new speed and diameter
Affinity law for pump head: for geometrically similar pumps at corresponding operating points, head is proportional to the square of the speed and the square of the impeller diameter, at the same gravity.
- Ref
- DIM-18
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- H2 [m]
With s = n2/n1 and d = D2/D1, the head at the corresponding operating point scales as H2 = H1 s² d², at the same gravity. For a pump the total pressure rise is ρgH. 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. For a speed change alone, see Pump head after a speed change.
Affinity law for pump head: for geometrically similar pumps at corresponding operating points, head is proportional to the square of the speed and the square of the impeller diameter, at the same gravity.
Calculator
- Speed ratio, s1.2
- Diameter ratio, d1
- Head multiplier, s² d²1.44
- Pressure rise ρgH2 for water141.3 kPa
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.
- A speed ratio of 1.2 and a diameter ratio of 1 multiply the head by 1.44, from 10 to 14.4 m.
- The head depends on the product of speed and diameter, which is proportional to the impeller tip speed: a pump with the same tip speed develops the same head.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| n1 Speed at point 1 | 50 to 10,000 rpm | 0.144 | 5,760 | 5,760 | +10% | −17.4 % |
| D1 Impeller diameter at point 1 | 20 to 3,000 mm | 0.144 | 3,240 | 3,240 | +10% | −17.4 % |
| D2 Impeller diameter at point 2 | 20 to 3,000 mm | 0.064 | 1,440 | 1,440 | +10% | +21 % |
| n2 Speed at point 2 | 50 to 10,000 rpm | 0.025 | 1,000 | 1,000 | +10% | +21 % |
| H1 Head at point 1 | 0.5 to 500 m | 0.72 | 720 | 719.3 | +10% | +10 % |
Sweep of Speed at point 2
| Speed at point 2 (rpm) | Head at point 2 (m) |
|---|---|
| 50 | 0.025 |
| 1,294 | 16.74 |
| 2,538 | 64.39 |
| 3,781 | 143 |
| 5,025 | 252.5 |
| 6,269 | 393 |
| 7,513 | 564.4 |
| 8,756 | 766.7 |
| 10,000 | 1,000 |
Sweep one variable
Variables and how each one contributes
Head at point 2
What the equation gives you. Everything else feeds it.
Head at point 1
The pump head at the original operating point, in m. Raises the new head in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 14.4 to 15.84 m.
Speed at point 1
The original rotational speed, in rpm. Lowers the new head with the square of the speed ratio. Raising it by 10 % (to 1,100) moves the result by −17.4 %, from 14.4 to 11.9 m.
Speed at point 2
The new rotational speed, in rpm. Raises the new head with the square of the speed, so speed is a strong lever. Raising it by 10 % (to 1,320) moves the result by +21 %, from 14.4 to 17.42 m.
Impeller diameter at point 1
The original impeller diameter, in mm. Lowers the new head with the square of the diameter ratio. Raising it by 10 % (to 330) moves the result by −17.4 %, from 14.4 to 11.9 m.
Impeller diameter at point 2
The new impeller diameter, in mm, for a geometrically similar pump. Raises the new head with the square of the diameter. Raising it by 10 % (to 330) moves the result by +21 %, from 14.4 to 17.42 m.
Worked example with the values above
- Ratios: s = 1.2 and d = 1.
- Head multiplier: s² d² = (1.2 × 1)² = 1.44.
- H2 = 10 × 1.44 = 14.4 m.
Assumptions and limits
- Geometric similarity and corresponding operating points, with the same gravity.
- Non-cavitating flow and comparable efficiency.
Common mistakes
- Expecting a system with static head to follow the same scaling: only the friction part of the system head grows with the square of the flow.
- Applying the full-scale diameter law to a trimmed impeller.
- Confusing head in metres with pressure in Pa.
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 (H1 × 1, n2 × 2, n1 × −2, D2 × 2, D1 × −2). Speed and diameter errors count double, and real pumps depart from the law by a few per cent, more at low flow.
Where it is used
- Predicting the head of a pump after a speed change or in a geometrically similar pump of another size.
- Checking that a variable-speed pump can still meet the system head at a lower speed.
Maths you need
- Pump head: energy added to the fluid per unit weight, in metres
- Tip speed: the speed of the impeller edge, proportional to nD