In fluid dynamics, pipe network analysis is the analysis of the fluid flow through a hydraulics network, containing several or many interconnected branches. The aim is to determine the flow rates and pressure drops in the individual sections of the network. This is a common problem in hydraulic design.
Description To direct water to many users, municipal water supplies often route it through a water supply network. A major part of this network will consist of interconnected pipes. This network creates a special class of problems in hydraulic design, with solution methods typically referred to as pipe network analysis. Water utilities generally make use of specialized software to automatically solve these problems. However, many such problems can also be addressed with simpler methods, like a spreadsheet equipped with a solver, or a modern graphing calculator.
Deterministic network analysis Once the friction factors of the pipes are obtained (or calculated from pipe friction laws such as the Darcy-Weisbach equation), we can consider how to calculate the flow rates and head losses on the network. Generally the head losses (potential differences) at each node are neglected, and a solution is sought for the steady-state flows on the network, taking into account the pipe specifications (lengths and diameters), pipe friction properties and known flow rates or head losses. The steady-state flows on the network must satisfy two conditions:
At any junction, the total flow into a junction equals the total flow out of that junction (law of conservation of mass, or continuity law, or Kirchhoff's first law) Between any two junctions, the head loss is independent of the path taken (law of conservation of energy, or Kirchhoff's second law). This is equivalent mathematically to the statement that on any closed loop in the network, the head loss around the loop must vanish.
If there are sufficient known flow rates, so that the system of equations given by (1) and (2) above is closed (number of unknowns = number of equations), then a deterministic solution can be obtained. The classical approach for solving these networks is to use the Hardy Cross method. In this formulation, first you go through and create guess values for the flows in the network. The flows are expressed via the volumetric flow rates Q. The initial guesses for the Q values must satisfy the Kirchhoff laws (1). That is, if Q7 enters a junction and Q6 and Q4 leave the same junction, then the initial guess must satisfy Q7 = Q6 + Q4. After the initial guess is made, then, a loop is considered so that we can evaluate our second condition. Given a starting node, we work our way around the loop in a clockwise fashion, as illustrated by Loop 1. We add up the head losses according to the Darcy–Weisbach equation for each pipe if Q is in the same direction as our loop like Q1, and subtract the head loss if the flow is in the reverse direction, like Q4. In other words, we add the head losses around the loop in the direction of the loop; depending on whether the flow is with or against the loop, some pipes will have head losses and some will have head gains (negative losses). To satisfy the Kirchhoff's second law (2), we should end up with 0 about each loop at the steady-state solution. If the actual sum of our head loss is not equal to 0, then we will adjust all the flows in the loop by an amount given by the following formula, where a positive adjustment is in the clockwise direction.
Δ Q = − ∑ head loss c − ∑ head loss c c n ⋅ ( ∑ head loss c Q c + ∑ head loss c c Q c c ) , {\displaystyle \Delta Q=-{\frac {\sum {\scriptstyle {\text{head loss}}_{c}}-\sum {\scriptstyle {\text{head loss}}_{cc}}}{n\cdot (\sum {\frac {{\text{head loss}}_{c}}{Q_{c}}}+\sum {\frac {{\text{head loss}}_{cc}}{Q_{cc}}})}},}
where
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