## 3 The Hardy Cross Method; Different Versions
### The Multi-Point Iterative Hardy Cross Method  
The here described multipoint method can substitute the Newton-Raphson iterative procedure used in all the above described methods. Recently, we successfully used this multipoint method for acceleration of the iterative solution of the Colebrook equation for flow friction modeling [16; 17]. On the contrary, for the gas network example in Figure 1, the multipoint method requires the same number of iterations as the original Newton-Raphson procedure. For the test, we used the three-point method from [PERSON] et al. [27]. Flow corrections \(\Delta Q_{I}\) from Equations (10) and (11) from the first loop I should be calculated using the three-point procedure (Equation (13)):  
\[\left\{ \begin{array}{l}\Delta Q_{I}=-\frac{(\sum\
eq f_{i})_{i}}{(\sum f_{i })_{i}},\text{ where }i=1;\\ \Delta Q_{I}=-\frac{(\sum\
eq f_{i})_{i}}{(\sum\
eq f_{i})_{i}-2\cdot(\sum \
eq f_{i})_{i+1}}.\frac{(\sum\
eq f_{i})_{i+1}}{(\sum[f_{i}^{\prime}])_{i}}, \text{ where }i=2;\\ \Delta Q_{I}=-\frac{(\sum f_{i})_{i+2}}{(\sum f_{i}^{\prime})_{i}\left[1-2 -\frac{(\sum\
eq f_{i})_{i+1}}{(\sum\
eq f_{i})_{i}}-\frac{(\sum\
eq f_{i})_{ i+1}}{(\sum\
eq f_{i})_{i}}\right]\left[1-\frac{(\sum\
eq f_{i})_{i+2}}{(\sum \
eq f_{i})_{i}}\right]},\text{ where }i>2;\end{array} \right. \tag{13}\]  
Formulas of flow corrections \(\Delta Q_{I}\) depend on the counter \(i\). The algorithm starts from \(i=1\), in which the multipoint method is the same as the original Newton-Raphson procedure (Equation (13a)):  
\[\Delta Q_{I}=-\frac{(\sum\
eq f_{i})_{i}}{\left(\sum[f_{i}^{\prime}])_{i}} \tag{13a}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**a** **Pipe Number** & **Diameter (m)** & **Length (m)** & **b** **Assumed Flows (m3/h)** & **c** **Calculated Flows (m3/h)** & **Gas Velocity (m/s)** \\ \hline
1 & 0.305 & 1127.8 & 1203.2 & 1583.6 & 1.5 \\
2 & 0.203 & 609.6 & 9.2 & 245.2 & 0.5 \\
3 & 0.203 & 853.4 & 841.6 & 899.7 & 1.9 \\
4 & 0.203 & 335.3 & 65.6 & 7.5 & 0.01 \\
5 & 0.203 & 304.8 & 165.6 & 320.2 & 0.7 \\
6 & 0.203 & 762.0 & 65.6 & 322.7 & 0.7 \\
7 & 0.203 & 243.8 & 2530.0 & 2149.6 & 4.6 \\
8 & 0.203 & 396.2 & 1100.0 & 462.4 & 1.0 \\
9 & 0.152 & 304.8 & 1000.0 & 465.0 & 1.8 \\
10 & 0.152 & 335.3 & 491.2 & 813.5 & 3.1 \\
11 & 0.254 & 304.8 & 431.2 & 609.1 & 0.8 \\
12 & 0.152 & 396.2 & 60.0 & 204.8 & 0.8 \\
13 & 0.152 & 548.6 & 100.0 & 4-2.6 & \(-\)0.009 \\
14 & 0.152 & 548.6 & 100.0 & 312.7 & 1.2 \\ \hline \hline \end{tabular}  
* Network from Figure 1 (flows are for normal pressure conditions; real pressure in the network is \(4\times 10^{5}\) Pa abs, i.e., \(3\times 10^{5}\) Pa); b chosen to satisfy [PERSON]’s first law for all nodes (dash arrows in Figure 1); c calculated to satisfy [PERSON]’s first law for all nodes and [PERSON]’s second law for all closed path formed by pipes (full errors in Figure 1); d the minus sign means that the direction of flow is opposite to the initial pattern for assumed flows.  
\end{table} 
Table 2: Pipe diameters and lengths, flows, and velocities of gas within pipes. In the second iteration, \(i=2\), flow corrections \(\Delta Q_{l}\) have a little bit more complicated form (Equation (13b)):  
\[\Delta Q_{l}=-\frac{\left(\sum\pm f_{l}\right)_{i}}{\left(\sum\pm f_{l}\right)_{ i}-2\cdot\left(\sum\pm f_{l}\right)_{i+1}}.\frac{\left(\sum\pm f_{l}\right)_{i+1}}{ \left(\sum\left|f_{l}^{\prime}\right|\right)_{i}} \tag{13b}\]  
The symbol \(\left(\sum\pm f_{l}\right)_{i}\) represents stored values from the first iteration, whereas \(\left(\sum\pm f_{l}\right)_{i+1}\) represents values from the second iteration.