## 3 The Hardy Cross Method; Different Versions
### The Modified Hardy Cross Method  
The Hardy Cross method can be noted in matrix form. The gas distribution network in Figure 1 has five independent loops (Equation (10)).  
\[\left[\begin{array}{cccc}\sum\big{|}f^{\prime}_{I}\big{|}&0&0&0&0\\ 0&\sum\big{|}f^{\prime}_{II}\big{|}&0&0&0\\ 0&0&\sum\big{|}f^{\prime}_{III}\big{|}&0&0\\ 0&0&0&\sum\big{|}f^{\prime}_{IV}\big{|}&0\\ 0&0&0&0&\sum\big{|}f^{\prime}_{V}\big{|}\end{array}\right]\,x\,\left[\begin{array} []{c}\Delta Q_{I}\\ \Delta Q_{II}\\ \Delta Q_{III}\\ \Delta Q_{III}\\ \Delta Q_{IV}\\ \Delta Q_{V}\end{array}\right]=\left[\begin{array}{c}\sum\pm f_{I}\\ \sum\pm f_{II}\\ \sum\pm f_{III}\\ \sum\pm f_{IV}\\ \sum\pm f_{V}\end{array}\right] \tag{10}\]  
Equation (4) provides for each particular loop in the network the same corrections as Equation (10) using matrix calculation. [PERSON] and [PERSON] (1999) improved the original Hardy Cross method ([PERSON] _et al._, 2009) by replacing some of the zeroes in the non-diagonal terms of Equation (10). For example, if pipe 8 is mutual for loop I and V, the first derivative of the pressure drop function for the observed pipe, where flow treated as a variable, will be put with a negative sign in the first column and the fifth row, and also in the fifth column and the first row (Equation (11)).  
\[\left[\begin{array}{cccc}\sum\big{|}f^{\prime}_{I}\big{|}&-f^{\prime}_{II}&-f^ {\prime}_{II}&-f^{\prime}_{9}&-f^{\prime}_{8}\\ -f^{\prime}_{IZ}&\sum\big{|}f^{\prime}_{II}\big{|}&-f^{\prime}_{II}&0&0\\ -f^{\prime}_{II}&-f^{\prime}_{II}&\sum\big{|}f^{\prime}_{III}\big{|}&-f^{ \prime}_{14}&0\\ -f^{\prime}_{9}&0&-f^{\prime}_{14}&-f^{\prime}_{13}\\ -f^{\prime}_{8}&0&0&-f^{\prime}_{13}&\sum\big{|}f^{\prime}_{V}\big{|}\end{array} \right]\,x\,\left[\begin{array}{c}\Delta Q_{I}\\ \Delta Q_{II}\\ \Delta Q_{III}\\ \Delta Q_{IV}\\ \Delta Q_{V}\end{array}\right]=\left[\begin{array}{c}\sum\pm f_{I}\\ \sum\pm f_{II}\\ \sum\pm f_{II}\\ \sum\pm f_{IV}\\ \sum\pm f_{V}\end{array}\right] \tag{11}\]  
In the modified Hardy Cross method, corrections for the first iterations are shown in Equation (12), and the solutions are listed in Table 1.  
\[\left[\begin{array}{cccc}+13987715480.9&-69911398.4&-3220265516.7&-5245486154.8 &-1828124435.8\\ -69911398.4&+957889226.7&-221575195.9&0&0\\ -322026516.7&-221575195.9&+8186058014.8&-1429824980.5&0\\ -5245486154.8&0&-1429824980.5&+84028108128.&-1429824980.5\\ -1828124435.8&0&0&-1429824980.5&+3605869136.3\\ \end{array}\right]\,x\,\left[\begin{array}{c}\Delta Q_{I}\\ \Delta Q_{I}\\ \Delta Q_{II}\\ \Delta Q_{IV}\\ \Delta Q_{IV}\end{array}\right]=\left[\begin{array}{c}+1575481279.8\\ -70109380.6\\ -70109380.6\\ -749435158.7\\ -3252052177.5\end{array}\right] \tag{12}\]  
This procedure significantly reduces the number of iterations required for the solution of the problem (Figure 5).  
Figure 5: The number of required iterations for the solution using the original vs. the improved Hardy Cross method.  
The first two iterations for the network in Figure 1 are shown in Table 1. Pipe diameters and lengths, as well as the first, assumed, and the final calculated flow distributions for the network in balance are shown in Table 2.  
The gas velocity in the network is small (can be up to 10-15 m/s). The network can be the subject of diameter optimization (as in [4]), which can also be done by using the Hardy Cross method (diameter correction \(\Delta D\) should be calculated for known and locked flow, where the first derivative of the Renouard function has to be calculated for diameter as a variable). The network should stay unchanged, even if planned gas consumption on nodes 5, 6, 8 and 10 increases, as pipes 4 and 13 will be useful thanks to increased gas flow.  
Similar examples, but for water flow, can be seen in [26]. Optimization of pipe diameters in a water distributive pipe network using the same approach can be seen in [6].