## 4 Conclusions  
WDNs are complicated structures consisting of various linked nodes and pipes, which transport water of suitable quality from sources to supply customers in a service area. WDNs, which play a substantial role in water supply systems, are considered as one of the major infrastructure assets of industrial society. The optimization of WDNs supports the decision-making process by determining the optimal trade-off between costs and performance. A major challenge in WDNs is the network design. While the complex nature of WDNs has already been explored with complex network analysis (CNA), the literature is still lacking a CNA of optimal water networks when there are different WDN structures with fixed nodes and inflows. In this study, an original approach for the optimum design of WDNs is presented. The following results are derived from the analysis:  
* Variations of head losses based on the pipe diameters showed that a 2-degree polynomial equation with R squared being 0.9638 can describe the trend well. * Based on a linear trend, the design inflows vary from a minimum of 0.0212 to a maximum of 0.205 m\({}^{3}\)/s in the case of a pipe diameter of 0.3 m, while they vary from a minimum of 0.0589 to a maximum of 0.5694 m\({}^{3}\)/s in the case of a pipe diameter of 0.5 m.
* The design pressures according to the water velocities and pipe diameters were estimated. For this purpose, one of the nodes of the WDN was selected and examined. A pressure-velocity model was considered. It was shown that two regressions can describe the model with high accuracy: exponential and polynomial regression. Polynomial regression outperformed the exponential regression, while in the exponential regression, as the pipe diameter increased, the error calculated for each model decreased. The findings reveal that the cost of WDNs varies significantly based on their configuration, with a single-reservoir, non-looped system demonstrating the lowest cost at USD 26,892, while a two-reservoir, looped system incurs the highest cost at USD 30,861. The  
Figure 17: Reynolds number distribution across pipes. analysis of head losses and design inflows indicates that larger pipe diameters contribute to reduced head losses, thereby enhancing the overall efficiency of the network. Furthermore, the study establishes that the design pressures are closely linked to water velocities and pipe diameters, with polynomial regression models providing high accuracy in predicting these relationships. The analysis and optimization approach described in this study provides a robust framework for real-world water distribution network design. By integrating fundamental hydraulic equations--such as the Hazen-Williams equation for friction losses and additional formulations for local losses and Reynolds number calculation--with heuristic flow distribution models, engineers can quickly evaluate network performance under different configurations. This method allows for a preliminary assessment of critical factors such as pressure losses, cost efficiency, and flow regimes. The developed model facilitates choosing the best combinations of pipes, nodes, and pumps by taking into account a number of design constraints, including pressure, pipe diameters, head losses, and velocity. This helps to create durable and effective urban water distribution systems. For practical applications, engineers should calibrate the heuristic coefficients using field measurements or detailed simulations to tailor the model to the specific characteristics of their systems. It is recommended that practitioners use these models to guide initial design choices, optimize pipe sizing and layout, and identify potential areas for system improvement before committing to more resource-intensive detailed CFD analyses for critical network segments. The study's limitations include its examination of dynamic real-world scenarios, such as the consequences of fluctuating water demand and unforeseen failures.