Pipe Network Solver

The Hardy-Cross method solves looped pipe networks where water can flow through multiple paths simultaneously. Unlike branched systems with unique solutions, looped networks require iterative balancing to satisfy two fundamental hydraulic laws: continuity (mass conservation at nodes) and energy conservation (head loss balance around loops). This algorithm, developed by Hardy Cross in 1936, remains the foundation for modern water distribution analysis software.

Network Topology: A pipe network consists of nodes (junctions, reservoirs, tanks) connected by pipes. Junctions have demands (water consumption), reservoirs provide unlimited supply at fixed head, and tanks have finite volume with variable head. Pipes have properties: length, diameter, material (affecting roughness), and minor loss coefficients for fittings. The network forms a graph where loops create multiple flow paths.

Continuity Equation: At every junction, inflow must equal outflow: $\Sigma Q_{in} = \Sigma Q_{out} + D$, where D is node demand. This ensures mass conservation - water entering a node equals water leaving plus consumption. Reservoir nodes automatically supply whatever flow the network demands.

Energy Conservation: Around any closed loop, the sum of head losses must equal zero: $\Sigma h_f = 0$. Head loss in the clockwise direction must equal head loss in the counter-clockwise direction. This principle drives the Hardy-Cross iteration - if loop head loss isn't zero, flows need adjustment.

Hardy-Cross Iteration: The algorithm starts with initial flow guesses satisfying continuity. For each loop, it calculates: $\Delta Q = -\Sigma h_f / (n \times \Sigma |h_f/Q|)$, where n=1.85 for Hazen-Williams or n=2 for Darcy-Weisbach. This correction adjusts all pipe flows in the loop simultaneously. Iterations continue until corrections fall below tolerance.

Head Loss Equations: Hazen-Williams: $h_f = 10.67 \times L \times Q^{1.852} / (C^{1.852} \times D^{4.871})$ meters, where C is roughness coefficient (150 for smooth pipes, 100 for rough). Darcy-Weisbach: $h_f = f \times (L/D) \times (V^2/2g)$, where f is friction factor from Moody diagram based on Reynolds number and relative roughness.

Pressure Calculation: Once flows converge, pressure head at each node is calculated from the energy equation: $H_j = H_{source} - \Sigma h_f(path)$, where the path traces from source to junction. Pressure in meters of water head converts to kPa by multiplying by 9.81.