## 5 Numerical Example Results  
Both of the gas networks modelled include three supplier nodes of natural gas at a 70-barg pressure set point, which are shown in Figure 2 and Figure A1 (Nodes 1, 2, and 3). No constraint was placed on the flow of gas, which can be supplied from any of the supply nodes. Details and the boundary conditions of Network 1 and Network 2 are presented in Table 1 and Appendix D. The load at the demand nodes was weighted according to approximated town populations at the given node. These networks were then modelled as isothermal systems at 288.15 K using MATLAB (modelling the pipe flow equation using the Newton-Raphson method), SAINT (an energy system modelling software), and the new approach proposed in this paper. Results for Network 1 are presented in Tables 3 and 4, while those for Network 2 can be found in Appendix D. In both networks, there are three supply nodes, which represent the current sources of NG for the Island of Ireland Node 1, Moffat the Irish-Great Britain Interconnector, Node 2 the Corrib Gas field, and Node 3 the Kinsale Gas Field. Using the estimated gas consumption for each demand node, along with reported NG supplies [4], with the ZRT term representing the isothermal squared speed of sound in a pipeline, which was nearly 341 m/s in this case, the model was solved for pressure at each node and flow rate in each pipe. Using Equations (14) and (15), the parameters \"\(a_{ij}\) and \(b_{ij}\)\" for each pipe have been calculated, for which \(a_{ij}\) relates to the friction factor, pipe length, ZRT, normal density of gas, and diameter, while \(b_{ij}\) relates to the pipe angle, pipe length, and ZRT. The numbers shown in Table 2 for the smaller network and for the bigger network can be found in the Appendix D. In the pipe flow Equation (13), \(b_{ij}\) is a secondary important term. To supply the estimated overall demand of Network 1, the Moffat interconnectors supply nearly 53.5 m\({}^{3}\)/s, and gas flows from Nodes 2 and 3 are 38 m\({}^{3}\)/s and 88 m\({}^{3}\)/s, respectively. Based on the picture of Ireland's gas network (Figure 2), distributing this flow among the demand nodes resulted in an maximum pressure drop of 4 barg at Node 10 and and minimum pressure drop of 0.6 barg at Node 4 when the inlet pressure was 70 barg using SAINT. It is important to note that the maximum and minimum pressures did not occur at the minimum and maximum elevations, which was 130 m at Node 1 and 25 m at Node 11, respectively; therefore, the major source of pressure loss in the system was due to gas flow in the pipelines. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pipe Number** & **From** & **To** & **Pipe Length,** & **Pipe Angle** & **Pipe Fraction** & **Pipe Diameter,** \\  & **Node** & **Node** & **km** & **Pice Factor** & **m** \\ \hline
1 & 1 & 4 & 550 & \(-\)0.009 & 0.01 & 0.76 \\
2 & 1 & 10 & 350 & \(-\)0.016 & 0.01 & 0.6 \\
3 & 9 & 10 & 50 & \(-\)0.063 & 0.01 & 0.6 \\
4 & 6 & 9 & 15 & 0.095 & 0.01 & 0.6 \\
5 & 6 & 7 & 20 & 0.063 & 0.01 & 0.6 \\
6 & 7 & 8 & 18 & 0.003 & 0.01 & 0.6 \\
7 & 8 & 9 & 8 & 0.014 & 0.01 & 0.6 \\
8 & 3 & 5 & 25 & \(-\)0.069 & 0.01 & 0.6 \\
9 & 5 & 6 & 70 & \(-\)0.025 & 0.01 & 0.6 \\
10 & 5 & 11 & 65 & \(-\)0.004 & 0.01 & 0.6 \\
11 & 4 & 5 & 150 & \(-\)0.004 & 0.01 & 0.6 \\
12 & 2 & 4 & 135 & \(-\)0.008 & 0.01 & 0.76 \\
13 & 11 & 12 & 15 & 0.019 & 0.01 & 0.6 \\
14 & 11 & 13 & 13 & 0.331 & 0.01 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the gas flow equation in the small network with 13 nodes and 14 branches. A summary of the results from the numerical solutions is shown in Tables 3 and 4, and the results of Network 2 are shown in Appendix D. The pipelines' and nodes' details including pipe diameter, length, and inclination are shown. In Figure 2, the designed network is illustrated.