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Apparatus
Model checks
The science
What is modelled
Water runs from a reservoir by gravity through pipes that form closed loops. The flows must add up at every junction, and around every loop the head losses must cancel. The Hardy Cross method starts from any flow that satisfies the first rule and repeatedly corrects each loop until the second rule holds.
Equations
- Hazen-Williams head loss
h = r Q1.852,r = 10.67 L / (C1.852 D4.87)with Q in m3/s. - Loop correction
ΔQ = − Σ h / Σ (1.852 r |Q|0.852). - Pressure head at a junction = reservoir head minus the losses along a path minus the ground height.
Checks run in the page
- Continuity at the four junctions.
- The head losses around both loops.
- An independent Newton solver on the junction heads.
- The power balance between reservoir, demands and friction.
Honest simplifications
Demands are constant, the pipes are full and the minor losses of bends and valves are ignored. The Hazen-Williams law is an empirical formula for water at ordinary temperatures, and an ancient system would have had open channels, siphons and uneven roughness as well.
What to try
- Press Cross-link closed off and see the flows reroute.
- Press Fire flow at node 4 and watch the lowest pressure fall.
- Press Old encrusted pipes and see the pressure drop with lower C.