## 3 The Hardy Cross Method; Different Versions
### The Hardy Cross Method; Original Approach  
The pressure drop function for each pipe is listed in Table 1 (for initial flow pattern, in the fourth column). The sign in front of the pressure drop function shown in the fourth column is the same as for flow from the observed iteration. The fifth column of Table 1 includes the first derivatives of the pressure drop function, where the flow is treated as a variable. The column of the function of pressure drops is computed algebraically, while the column of the first derivatives is estimated numerically for each loop. Flow correction \(\Delta Q\) has to be computed for each loop \(x\) (Equation (3)). \[\Delta Q_{x}=\left(\frac{\sum\pm f}{\left|f^{\prime}\right|}\right)_{x}=\left( \frac{\sum\pm 4810.\frac{\rho_{x}\cdot L\cdot Q^{1.82}}{D^{4.82}}}{\sum\left|1.82 -4810.\frac{\rho_{x}\cdot L\cdot Q^{0.82}}{D^{4.82}}\right|}\right)_{x} \tag{3}\]  
For the network from Figure 1, flow corrections for the first iteration in each loop can be calculated using Equation (4). \[\left(\left|-f_{1}^{\prime}\right|+\left|f_{7}^{\prime}\right|+\left|f_{8}^{ \prime}\right|+\left|f_{9}^{\prime}\right|+\left|-f_{10}^{\prime}\right|+\left| -f_{12}^{\prime}\right|\right)\Delta Q_{I}=-f_{1}+f_{7}+f_{8}+f_{9}-f_{10}-f_{ 12} \tag{4}\] \[\left(\left|-f_{2}^{\prime}\right|+\left|-f_{11}^{\prime}\right|+ \left|f_{12}^{\prime}\right|\right)\cdot\Delta Q_{II}=-f_{2}-f_{11}+f_{12}\] \[\left(\left|-f_{3}^{\prime}\right|+\left|f_{4}^{\prime}\right|+ \left|f_{10}^{\prime}\right|+\left|f_{11}^{\prime}\right|+\left|-f_{14}^{ \prime}\right|\right)\Delta Q_{III}=-f_{3}+f_{4}+f_{10}+f_{11}-f_{14}\] \[\left(\left|f_{5}^{\prime}\right|+\left|-f_{9}^{\prime}\right|+ \left|f_{13}^{\prime}\right|+\left|f_{14}^{\prime}\right|\right)\cdot\Delta Q_ {IV}=f_{5}-f_{9}+f_{13}+f_{14}\] \[\left(\left|f_{6}^{\prime}\right|+\left|-f_{8}^{\prime}\right|+ \left|f_{13}^{\prime}\right|\right)\cdot\Delta Q_{V}=f_{6}-f_{8}+f_{13}\]  
In the second iteration, the calculated correction \(\Delta Q\) has to be added algebraically to the assumed gas flow (the first initial flow pattern). Further, the calculated correction \(\Delta Q\) has to be subtracted algebraically from the gas flow computed in the previous iteration. This means that the algebraic operation for the first correction is the opposite of its sign, i.e., add when the sign is minus, and vice versa. A pipe common to two loops receives two corrections simultaneously. The first correction is from the particular loop under consideration, while the second one is from the adjacent loop, which the observed pipe also belongs to. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{6}{c}{**Iteration 1**} \\ \hline \multirow{2}{*}{**Loop**} & \multirow{2}{*}{**Pipe**} & \multicolumn{1}{c}{\({}^{\textbf{a}}\)\(Q\)} & \multicolumn{1}{c}{\({}^{\textbf{b}}\)\(f\)\(=\)\(p_{1}^{2}\)\(-p_{2}^{2}\)} & \multicolumn{1}{c}{\({}^{\textbf{c}}\)\(\left|f\)\(\downarrow\)} & \multicolumn{1}{c}{\({}^{\textbf{d}}\)\(\Delta Q_{1}\)} & \multicolumn{1}{c}{\({}^{\textbf{e}}\)\(\Delta Q_{2}\)} & \multicolumn{1}{c}{\({}^{\textbf{f}}\)\(Q_{1}\)\(=\)\(Q\)} \\ \hline \multirow{5}{*}{I} & 1 & \(-0.3342\) & \(-14451856.8\) & \(787025109.2\) & \(-0.0994\) & & \(-0.4336\) \\  & 7 & \(+0.7028\) & \(+859927106.7\) & \(2226902866.0\) & \(-0.0994\) & & \(+0.6034\) \\  & 8 & \(+0.3056\) & \(+306964191.0\) & \(1828124435.8\) & \(-0.0994\) & \(-0.0532\) & \(+0.1530\) \\  & 9 & \(+0.2778\) & \(+800657172.4\) & \(5245486154.8\) & \(-0.0994\) & \(-0.0338\) & \(+0.1446\) \\  & 10 & \(-0.1364\) & \(-241342976.1\) & \(3220265516.7\) & \(-0.0994\) & \(+0.0142\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\,\)\(\,\)\(\,\)\(\,\,\)\(\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\,\)\(\,\,\)\(\,\,\,\)\(\,\,\,\)\(The upper sign after the second correction in Table 1 is plus if the flow direction in the mutual pipe coincides with the assumed orientation of the adjacent loop, and minus if it does not (Figure 2).