Library / Dimensional analysis and similarity / DIM-19
Dimensional analysis and similarity
Fan pressure rise at a new speed, diameter and density
Affinity law for fan pressure rise: for geometrically similar fans at corresponding operating points, pressure is proportional to the inlet density, the square of the speed and the square of the impeller diameter.
- Ref
- DIM-19
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Δp2 [Pa]
With s = n2/n1, d = D2/D1 and z = ρ2/ρ1 for the inlet density, the pressure rise at the corresponding operating point scales as Δp2 = Δp1 z s² d². Compare the same pressure definition at both points, for example total with total. 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.
Affinity law for fan pressure rise: for geometrically similar fans at corresponding operating points, pressure is proportional to the inlet density, the square of the speed and the square of the impeller diameter.
Calculator
- Speed ratio, s1.2
- Diameter ratio, d1
- Density ratio, z1
- Pressure multiplier, z s² d²1.44
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 fan running 1.2 times as fast, with a diameter ratio of 1 and a density ratio of 1, raises its pressure from 500 to 720 Pa.
- A fan moves a fixed volume of air, so in thinner air (lower density) it produces proportionally less pressure and absorbs proportionally less power at the same speed.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρ2 Inlet density at point 2 | 0.1 to 2,000 kg/m³ | 60 | 1,200,000 | 1,200,000 | +10% | +10 % |
| n1 Speed at point 1 | 50 to 10,000 rpm | 7.2 | 288,000 | 288,000 | +10% | −17.4 % |
| D1 Impeller diameter at point 1 | 20 to 3,000 mm | 7.2 | 162,000 | 162,000 | +10% | −17.4 % |
| D2 Impeller diameter at point 2 | 20 to 3,000 mm | 3.2 | 72,000 | 72,000 | +10% | +21 % |
| Δp1 Pressure rise at point 1 | 10 to 50,000 Pa | 14.4 | 72,000 | 71,990 | +10% | +10 % |
| n2 Speed at point 2 | 50 to 10,000 rpm | 1.25 | 50,000 | 50,000 | +10% | +21 % |
| ρ1 Inlet density at point 1 | 0.1 to 2,000 kg/m³ | 0.432 | 8,640 | 8,640 | +10% | −9.09 % |
Sweep of Speed at point 2
| Speed at point 2 (rpm) | Pressure rise at point 2 (Pa) |
|---|---|
| 50 | 1.25 |
| 1,294 | 836.9 |
| 2,538 | 3,219 |
| 3,781 | 7,149 |
| 5,025 | 12,630 |
| 6,269 | 19,650 |
| 7,513 | 28,220 |
| 8,756 | 38,340 |
| 10,000 | 50,000 |
Sweep one variable
Variables and how each one contributes
Pressure rise at point 2
What the equation gives you. Everything else feeds it.
Pressure rise at point 1
The pressure rise at the original operating point, in Pa. Raises the new pressure rise in direct proportion. Raising it by 10 % (to 550) moves the result by +10 %, from 720 to 792 Pa.
Speed at point 1
The original rotational speed, in rpm. Lowers the new pressure rise with the square of the speed ratio. Raising it by 10 % (to 1,100) moves the result by −17.4 %, from 720 to 595 Pa.
Speed at point 2
The new rotational speed, in rpm. Raises the new pressure rise with the square of the speed. Raising it by 10 % (to 1,320) moves the result by +21 %, from 720 to 871.2 Pa.
Impeller diameter at point 1
The original impeller diameter, in mm. Lowers the new pressure rise with the square of the diameter ratio. Raising it by 10 % (to 330) moves the result by −17.4 %, from 720 to 595 Pa.
Impeller diameter at point 2
The new impeller diameter, in mm. Raises the new pressure rise with the square of the diameter. Raising it by 10 % (to 330) moves the result by +21 %, from 720 to 871.2 Pa.
Inlet density at point 1
The inlet density at the original point, in kg/m³. Lowers the new pressure rise in inverse proportion. Raising it by 10 % (to 1.32) moves the result by −9.09 %, from 720 to 654.5 Pa.
Inlet density at point 2
The inlet density at the new point, in kg/m³: it falls in hot or high-altitude air. Raises the new pressure rise in direct proportion. Raising it by 10 % (to 1.32) moves the result by +10 %, from 720 to 792 Pa.
Worked example with the values above
- Ratios: s = 1.2, d = 1 and z = 1.
- Pressure multiplier: z s² d² = 1 × (1.2 × 1)² = 1.44.
- Δp2 = 500 × 1.44 = 720 Pa.
Assumptions and limits
- Geometric similarity, corresponding operating points and low-compressibility flow, with the same pressure definition (static or total) at both points.
- Comparable efficiency and a Reynolds number high enough that its effect is small.
Common mistakes
- Comparing a total pressure rise at one point with a static pressure rise at the other.
- Using the outdoor density instead of the inlet density at the fan.
- Applying the full-scale diameter law to a trimmed impeller.
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 (Δp1 × 1, ρ2 × 1, ρ1 × −1, n2 × 2, D2 × 2). Speed and diameter errors count double, and the density must be taken at the fan inlet, including humidity and temperature, since it shifts with both.
Where it is used
- Predicting the pressure of a fan after a speed change, a density change or a size change.
- Correcting a fan rated at standard air density for use at altitude or at high temperature.
Maths you need
- Fan pressure: total or static pressure rise across a fan
- Air density: mass per volume at the fan inlet