Water distribution networks are engineered to deliver potable water to end consumers at required flow rates and adequate residual pressures while maintaining water quality. Head loss through pressurized pipe networks is primarily evaluated using the empirical Hazen-Williams Equation : $$v = 0.849 \cdot C_{hw} \cdot R^{0.63} \cdot S^{0.54}$$ Where $v$ is flow velocity, $C_{hw}$ is the Hazen-Williams roughness coefficient, $R$ is hydraulic radius, and $S$ is the hydraulic slope ($\frac{h_f}{L}$). Expressed directly in terms of head loss ($h_f$) for a pipe of diameter $D$ and length $L$: $$h_f = \frac{10.67 \cdot Q^{1.852} \cdot L}{C_{hw}^{1.852} \cdot D^{4.87}}$$ For complex looped pipe networks, flow distribution is solved iteratively using the Hardy Cross Method . The method relies on two fundamental hydraulic principles: mass conservation at each junction ($\sum Q = 0$) and energy conservation around any closed loop ($\sum h_f = 0$). The flow correction factor ($\Delta Q$) a...