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Hydraulics, open channels and fluid power
Outlet whirl velocity from the impeller velocity triangle
Tangential component of the absolute velocity leaving the impeller, from the tip speed U₂, the outward meridional velocity Vm and the blade angle β measured from the backward tangent (degrees), for ideal blade-guided flow.
- Ref
- HYD-40
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- Vθ2 [m/s]
In the exit velocity triangle, the absolute tangential component is the blade speed less the relative velocity’s tangential part: Vθ₂ = U₂ − Vm₂ cot β₂, where Vm₂ is the radial (meridional) velocity and β₂ the blade angle from the backward tangent. With a radial blade (β₂ = 90°) Vθ₂ = U₂, and backward-leaning blades give less. Real flow has slip, so the whirl is less than this ideal value.
Tangential component of the absolute velocity leaving the impeller, from the tip speed U₂, the outward meridional velocity Vm and the blade angle β measured from the backward tangent (degrees), for ideal blade-guided flow.
Calculator
- Head coefficient, Vθ₂/U₂0.8
- Euler head with no inlet whirl, U₂Vθ₂/9.8132.62 m
- Absolute exit speed, √(Vθ₂² + Vm²)16.49 m/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.
- With a tip speed of 20 m/s and 4 m/s of radial flow, the exit whirl is 16 m/s, 80 % of the tip speed.
- Higher flow gives a lower whirl and so a lower head, which is the falling pump curve of a backward-curved impeller.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U2 Blade tip speed | 1 to 200 m/s | −3 | 196 | 199 | +10% | +12.5 % |
| β Blade angle from the backward tangent | 5 to 175 ° | −25.72 | 65.72 | 91.44 | +10% | +3.65 % |
| Vm Outward meridional velocity | 0 to 50 m/s | −30 | 20 | 50 | +10% | −2.5 % |
Sweep of Blade angle from the backward tangent
| Blade angle from the backward tangent (°) | Outlet whirl velocity (m/s) |
|---|---|
| 5 | −25.72 |
| 26.25 | 11.89 |
| 47.5 | 16.33 |
| 68.75 | 18.44 |
| 90 | 20 |
| 111.3 | 21.56 |
| 132.5 | 23.67 |
| 153.8 | 28.11 |
| 175 | 65.72 |
Sweep one variable
Variables and how each one contributes
Outlet whirl velocity
What the equation gives you. Everything else feeds it.
Blade tip speed
The blade tip speed, in m/s. Raises the whirl in direct proportion. Raising it by 10 % (to 22) moves the result by +12.5 %, from 16 to 18 m/s.
Outward meridional velocity
The outward meridional velocity, in m/s. More flow lowers the whirl, so the head falls with flow. Raising it by 10 % (to 4.4) moves the result by −2.5 %, from 16 to 15.6 m/s.
Blade angle from the backward tangent
The blade angle from the backward tangent, in degrees. A more radial blade (larger angle) raises the whirl. Raising it by 10 % (to 49.5) moves the result by +3.65 %, from 16 to 16.58 m/s.
Worked example with the values above
- Cotangent: cot 45° = 1 / tan 45° = 1.
- Flow term: Vm cot β = 4 × 1 = 4 m/s.
- Vθ₂ = 20 − 4 = 16 m/s.
Assumptions and limits
- Steady incompressible flow with the pump curve and system curve both of quadratic form.
- Ideal blade-guided flow for the velocity-triangle results, so the real head is lower.
Common mistakes
- Measuring the blade angle from the tangent in the forward direction in place of the backward tangent.
- Forgetting that the real whirl is lowered by slip.
- Using degrees in a calculator set to radians.
Statistics and uncertainty
Pump curves are known only to within a few percent and wear them down in service, and available suction head changes with the level, temperature and flow, so use a safety margin on the requirement.
Where it is used
- Choosing pumps and finding the duty point on a pipe system.
- Checking cavitation margin and estimating head from impeller geometry.
Maths you need
- Pump curve: head against flow for a pump at a fixed speed
- System curve: head needed by the pipe system against flow
- Velocity triangle: the vector diagram of blade and fluid velocities