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Hydraulics, open channels and fluid power
Sudden enlargement mixing loss
Head lost when a flow expands abruptly from a smaller to a larger pipe, from the speeds U₁ just upstream and U₂ well downstream (Borda–Carnot model).
- Ref
- HYD-11
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- hL [m]
When a flow expands abruptly, the stream mixes with slower fluid and part of the kinetic energy is lost. With the pressure on the shoulder equal to the upstream pressure, momentum and energy give the Borda–Carnot loss hL = (U₁ − U₂)²/(2g). It is zero when the speeds are equal and largest for a large expansion, where it approaches the full upstream velocity head. It matches a pipe discharging into a tank (U₂ = 0).
Head lost when a flow expands abruptly from a smaller to a larger pipe, from the speeds U₁ just upstream and U₂ well downstream (Borda–Carnot model). It is zero when the two speeds are equal.
Calculator
- Upstream velocity head, U₁²/(2g)0.4587 m
- Loss coefficient referred to U₁, (1 − U₂/U₁)²0.4444
- Downstream velocity head, U₂²/(2g)0.05097 m
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.
- Slowing from 3 to 1 m/s loses 0.2039 m of head, 44.4 % of the upstream velocity head.
- The rest of the velocity head change, 0.2039 m, is recovered as a rise in pressure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U1 Speed upstream of the enlargement | 2.5 to 50 m/s | 0.1147 | 122.4 | 122.3 | +10% | +32.3 % |
| g Gravitational acceleration | 1 to 30 m/s² | 0.06667 | 2 | 1.933 | +10% | −9.09 % |
| U2 Speed downstream of the enlargement | 0 to 2.5 m/s | 0.01274 | 0.4587 | 0.446 | +10% | −9.75 % |
Sweep of Speed downstream of the enlargement
| Speed downstream of the enlargement (m/s) | Head loss (m) |
|---|---|
| 0 | 0.4587 |
| 0.3125 | 0.3681 |
| 0.625 | 0.2875 |
| 0.9375 | 0.2168 |
| 1.25 | 0.1561 |
| 1.563 | 0.1053 |
| 1.875 | 0.06451 |
| 2.188 | 0.03365 |
| 2.5 | 0.01274 |
Sweep one variable
Variables and how each one contributes
Head loss
What the equation gives you. Everything else feeds it.
Speed upstream of the enlargement
The mean speed just upstream of the enlargement, in m/s. A faster stream upstream raises the loss with the square of the speed difference. Raising it by 10 % (to 3.3) moves the result by +32.3 %, from 0.2039 to 0.2696 m.
Speed downstream of the enlargement
The mean speed far downstream of the enlargement, in m/s. A faster downstream speed means a smaller expansion, so the loss falls. Raising it by 10 % (to 1.1) moves the result by −9.75 %, from 0.2039 to 0.184 m.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the head loss in inverse proportion. Raising it by 10 % (to 10.79) moves the result by −9.09 %, from 0.2039 to 0.1853 m.
Worked example with the values above
- Speed difference: U₁ − U₂ = 3 − 1 = 2 m/s; squared 4.
- Divide by 2g = 19.62.
- hL = 0.20387 m.
Assumptions and limits
- Steady incompressible flow, with the friction factor and loss coefficients held at their duty-point values.
- Head losses proportional to the square of the flow, with positive flow in each pipe.
Common mistakes
- Swapping the speeds, which gives the same loss but hides a contraction, where the loss is different.
- Using the Borda–Carnot loss for a gradual diffuser, where the loss is smaller.
- Using volume flow in place of speed.
Statistics and uncertainty
Loss coefficients and friction factors are empirical and are usually known to within 10 % to 20 %, so network results carry that uncertainty. The quadratic forms assume the friction factor stays fixed as the flow changes.
Where it is used
- Building system curves for pumps and checking the effect of fittings.
- Balancing flows in looped water networks, and simplifying branches into one resistance.
Maths you need
- Head loss: energy lost per unit weight, in metres
- Darcy friction factor: f, as in hL = f (L/D) V²/2g
- Continuity: flow into a node equals flow out