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Hydraulics, open channels and fluid power
Hardy Cross loop flow correction
Flow correction applied once around a loop of three links, for trial flows that satisfy continuity at the nodes.
- Ref
- HYD-15
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- ΔQ [m³/s]
Hardy Cross balances a looped network by adjusting the flows round each loop. For a trial flow distribution that satisfies continuity at the nodes, the loop correction is ΔQ = −Σ R q|q| / Σ 2R|q|, with the flows signed along one chosen direction around the loop. Add ΔQ to every link in the loop, then repeat for the other loops until the corrections are small. One pass is only one iteration. This calculator takes three links.
Flow correction applied once around a loop of three links, for trial flows that satisfy continuity at the nodes. Flows are signed along one loop direction, and the correction is one iteration only.
Calculator
- Loop head imbalance, Σ R q|q|1 m
- Denominator, Σ 2R|q|100 s/m²
- Corrected flow in link 1, q₁ + ΔQ0.19 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.
- The loop is out of balance by 1 m, and a correction of −0.01 m³/s is added to every link in the loop.
- After the update, recheck continuity at the nodes and the loop imbalance, and repeat until the correction is a small fraction of the flows.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m³/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| q1 Trial flow in link 1 | −5 to 5 m³/s | −2.356 | 2.361 | 4.717 | +0.2 m³/s | +829 % |
| q3 Trial flow in link 3 | −5 to 5 m³/s | −2.363 | 2.354 | 4.717 | +0.2 m³/s | +733 % |
| q2 Trial flow in link 2 | −5 to 5 m³/s | −2.315 | 2.315 | 4.63 | +0.2 m³/s | +543 % |
| R3 Resistance of link 3 | 1 to 1,000,000 s²/m⁵ | −0.08212 | 0.09997 | 0.1821 | +10% | −42.3 % |
| R1 Resistance of link 1 | 1 to 1,000,000 s²/m⁵ | −0.09998 | 0.04901 | 0.149 | +10% | +34.6 % |
| R2 Resistance of link 2 | 1 to 1,000,000 s²/m⁵ | −0.04998 | −1.247e-4 | 0.04986 | +10% | +7.84 % |
Sweep of Trial flow in link 1
| Trial flow in link 1 (m³/s) | Loop flow correction (m³/s) |
|---|---|
| −5 | 2.361 |
| −3.75 | 1.74 |
| −2.5 | 1.121 |
| −1.25 | 0.5137 |
| 0 | 0.05 |
| 1.25 | −0.4944 |
| 2.5 | −1.111 |
| 3.75 | −1.732 |
| 5 | −2.356 |
Sweep one variable
Variables and how each one contributes
Loop flow correction
What the equation gives you. Everything else feeds it.
Resistance of link 1
The resistance of link 1, in s²/m⁵. A larger R₁ raises the weight of link 1 in both sums. Raising it by 10 % (to 110) moves the result by +34.6 %, from −0.01 to −0.01346 m³/s.
Resistance of link 2
The resistance of link 2, in s²/m⁵. A larger R₂ raises the weight of link 2 in both sums. Raising it by 10 % (to 110) moves the result by +7.84 %, from −0.01 to −0.01078 m³/s.
Resistance of link 3
The resistance of link 3, in s²/m⁵. A larger R₃ raises the weight of link 3 in both sums. Raising it by 10 % (to 110) moves the result by −42.3 %, from −0.01 to −0.005769 m³/s.
Trial flow in link 1
The trial flow in link 1, in m³/s, signed along the loop direction. Raises the head imbalance, so the correction changes. Raising it by 10 % (to 0.22) moves the result by +76.9 %, from −0.01 to −0.01769 m³/s.
Trial flow in link 2
The trial flow in link 2, in m³/s, signed along the loop direction. Raises the head imbalance, so the correction changes. Raising it by 10 % (to 0.11) moves the result by +18.6 %, from −0.01 to −0.01186 m³/s.
Trial flow in link 3
The trial flow in link 3, in m³/s, signed along the loop direction. Raises the head imbalance, so the correction changes. Raising it by 10 % (to −0.22) moves the result by −84.6 %, from −0.01 to −0.001538 m³/s.
Worked example with the values above
- Head imbalance: Σ R q|q| = 4 + 1 + −4 = 1 m.
- Denominator: 2 Σ R|q| = 100.
- ΔQ = −(1) / 100 = −0.01 m³/s.
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
- Using flows that do not satisfy continuity at the nodes.
- Forgetting to apply the same correction to every link in the loop, with the right sign.
- Stopping after one pass.
Statistics and uncertainty
One Hardy Cross pass only improves the flows, and with fixed friction factors the final answer is only as good as the resistances, which come from friction factors known to 10 % to 20 %.
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