## 2 Methodology
### Flow Distribution  
In this study, a heuristic approach was adopted to approximate the progressive reduction in flow along the network. Starting with the full inflow \(Q_{total}\) at the source, the first pipe (with pipe ID 1) is assumed to carry the entire inflow, i.e., no reduction is applied \((1-\alpha\times(1-1)=1)\). For subsequent pipes, we assume a linear reduction in flow given by:  
\[Q_{pipe}=Q_{total}\times(1-\alpha\times(pipe_{ID}-1)) \tag{12}\]  
where \(\alpha\) represents the assumed fractional reduction in the flow per successive pipe. A higher coefficient means that with each additional pipe, a larger fraction of the flow is removed or diverted (simulating higher losses or greater demand along the branch). Conversely, a lower coefficient indicates that the reduction in flow from one pipe to the next is less severe, so more of the original flow is maintained as the water is distributed through the network.  
For WDN 1 (the centralized radial network) and WDN 2 (the looped network), the coefficient \(\alpha\) is assumed to be 0.03, which implies that each additional pipe in the sequence receives 3% less of the original inflow than its predecessor, reflecting the expected cumulative effect of distributed demand and branch flow losses. For the hybrid network (WDN 3), the coefficient is reduced to 0.02:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pipe** & **Pipe Type** &
\begin{tabular}{c} **Approximate** \\ **Average Cost** \\ **(USD/m)** \\ \end{tabular} & **Merits** & **Demerits** & **Approximate** \\ \hline
1 & \
\begin{tabular}{c} Chlorinated polyvinyl \\ chloride (CPVC) \\ \end{tabular} & 2.467 & (1) Inexpensive, (2) Less corrosion &
\begin{tabular}{c} (1) Possible leaks, \\ (2) Lower quality \\ \end{tabular} & 50-75 \\ \hline
2 & \begin{tabular}{c} Cross-linked \\ polyethylene (PEX) \\ \end{tabular} & 3.947 & \
\begin{tabular}{c} (1) Less corrosion, leak, burst etc., \\ (2) In-floor heating \\ \end{tabular} &
\begin{tabular}{c} (1) Less integration \\ possibility \\ \end{tabular} & 80–100 \\ \hline
3 & Copper & 16.447 & \
\begin{tabular}{c} (1) Proven utilization history, \\ (2) Compatible with building codes, \\ (3) Recyclable, (4) UV resistant, \\ (5) Unfailing in natural disasters \\ \end{tabular} &
\begin{tabular}{c} (1) Expensive, \\ (2) Possibility of bursting, \\ freezing, and corrosion \\ \end{tabular} & 50–100 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cost classifications of various pipes.  
\[Q_{pipe}=Q_{total}\times(1-0.02\times(pipe_{ID}-1)) \tag{13}\]  
WDN 3 is designed as a hybrid network, combining elements of both radial and looped configurations. Unlike a purely radial network--where water flows from a single source through a sequence of pipes (resulting in significant cumulative flow reductions)--or a fully looped network, the hybrid design includes localized loops in high-demand zones. These loops provide alternate paths for water to reach peripheral nodes, effectively \"preserving\" higher flow rates downstream. In this scenario, the flow reduction per pipe is less drastic because the loops help maintain pressure and supply. Thus, when formulating the heuristic model for WDN 3, a lower coefficient (0.02) is selected to reflect that each pipe loses a smaller percentage of the total inflow compared with the other configurations.  
In practice, determining the exact value of the coefficient would involve calibrating the model against detailed hydraulic measurements. The chosen values (0.03 for the radial/looped systems and 0.02 for the hybrid system) are approximations that capture the qualitative differences in network design: networks with additional supply paths (localized loops) can maintain higher flow downstream, justifying a lower reduction factor. This approach is common in preliminary network analyses where a balance is struck to capture the dominant trends in flow distributions.