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Pipe network analysis methods like the Hardy-Cross method require two conditions to be met. 1) The inflow to a node must equal the outflow from that node. 2) The pressure change along a complete loop of pipes must be 0.

Is there a way to solve a pipe network flow problem without needing to know anything about pipe loops?

I am working on a simulator where the user can add pipes and connect them together as well as change their properties and run a simulation. The problem I'm having is in detecting loops. There are ways of detecting loops in graphs but they seem to take a long time for any realistic complicated network.

Is there a way to solve a pipe network WITHOUT needing to know which pipes are connected to loops?

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The loop and node requirements are consequences of Kirchhoff's Laws for generalized networks. They are conservation laws which any network must observe. Without them, the network becomes unphysical and analysis is then impossible.

There are dynamical system analysis tools that run on computers which detect both nodes and loops and assert those laws for them automatically. ENPORT is one example.

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