# Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Network Piping System; Flow Distribution Calculation

#### 2.1. Topology of the Network

#### 2.2. A Hydraulic Model

^{3}/s), and $p$ is pressure (Pa).

## 3. The Hardy Cross Method; Different Versions

#### 3.1. The Hardy Cross Method; Original Approach

#### 3.2. A Version of the Hardy Cross Method from Russian Practice

_{x}for the first iteration is (Equation (8)).

#### 3.3. The Modified Hardy Cross Method

#### 3.4. The Multi-Point Iterative Hardy Cross Method

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\rho}_{r}$ | relative gas density (-); here ${\rho}_{r}=0.64$ |

$\mathsf{\rho}$ | density of air (kg/m^{3}); here $\mathsf{\rho}$ = 1.2 kg/m^{3} |

$L$ | length of pipe (m) |

$D$ | diameter of pipe (m) |

$Q$ | flow (m^{3}/s) |

$\Delta Q$ | flow correction (m^{3}/s) |

$p$ | pressure (Pa) |

$\Delta p$ | pressure correction (Pa) |

$f$ | function of pressure |

${f}^{\prime}$ | first derivative of function of pressure |

$\lambda $ | Darcy (Moody) flow friction factor (dimensionless) |

$Re$ | Reynolds number (dimensionless) |

$\frac{\epsilon}{D}$ | relative roughness of inner pipe surface (dimensionless) |

${C}_{d}$ | flow discharge coefficient (dimensionless) |

$A$ | area of ventilation opening (m^{2}) |

$\pi $ | Ludolph number; $\pi \approx $3.1415 |

$i$ | counter |

## Appendix A. Hydraulic Models for Water Pipe Networks and for Ventilation Systems

## Appendix B. The Life and Work of Hardy Cross

## References

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**Figure 2.**Rules for the upper and lower sign (correction from the adjacent loop; second correction).

**Figure 4.**Rules for terms from Lobačev equations, which do not exist in the Hardy Cross method applied for the network from Figure 1.

**Figure 5.**The number of required iterations for the solution using the original vs. the improved Hardy Cross method.

**Table 1.**Procedure for the solution of the flow problem for the network from Figure 1 using the modified Hardy Cross method (first two iterations)—First iteration.

Iteration 1 | |||||||
---|---|---|---|---|---|---|---|

Loop | Pipe | ^{a}$\mathit{Q}$ | ^{b}$\mathit{f}={\mathit{p}}_{1}^{2}-{\mathit{p}}_{2}^{2}$ | ^{c}$\left|{\mathit{f}}^{\prime}\right|$ | ^{d}$\mathbf{\Delta}{\mathit{Q}}_{1}$ | ^{e}$\mathbf{\Delta}{\mathit{Q}}_{2}$ | ^{f}${\mathit{Q}}_{1}=\mathit{Q}$ |

I | 1 | −0.3342 | −144518566.8 | 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 ‡ | −0.2217 | |

12 | −0.0167 | −6238747.4 | 679911398.4 | −0.0994 | +0.0651 ‡ | −0.0511 | |

$\Sigma $ | ${f}_{I}$ = +1575448179.8 | 13987715480.9 | |||||

II | 2 | −0.0026 | −80628.9 | 56440212.4 | −0.0651 | −0.0677 | |

11 | −0.1198 | −14582531.0 | 221537615.9 | −0.0651 | +0.0142 ‡ | −0.1707 | |

12 | +0.0167 | +6238747.4 | 679911398.4 | −0.0651 | +0.0994 $\mp $ | +0.0511 | |

$\Sigma $ | ${f}_{II}$ = −8424412.4 | 957889226.7 | |||||

III | 3 | −0.2338 | −406110098.1 | 3161336093.1 | −0.0142 | −0.2480 | |

4 | +0.0182 | +1530938.1 | 153093808.5 | −0.0142 | +0.0040 | ||

10 | +0.1364 | +241342976.1 | 3220265516.7 | −0.0142 | +0.0994 $\mp $ | +0.2217 | |

11 | +0.1198 | +14582531.0 | 221537615.9 | −0.0142 | +0.0651 $\mp $ | +0.1707 | |

14 | −0.0278 | −21840183.8 | 1429824980.5 | −0.0142 | −0.0338$\pm $ | −0.0757 | |

$\Sigma $ | ${f}_{III}$ = −170493836.7 | 8186058014.8 | |||||

IV | 5 | +0.0460 | +7523646.2 | 297674697.0 | +0.0338 | +0.0798 | |

9 | −0.2778 | −800657172.4 | 5245486154.8 | +0.0338 | +0.0994 ‡ | −0.1446 | |

13 | +0.0278 | +21840183.8 | 1429824980.5 | +0.0338 | −0.0532 = | +0.0084 | |

14 | +0.0278 | +21840183.8 | 1429824980.5 | +0.0338 | +0.0142 $\mp $ | +0.0757 | |

$\Sigma $ | ${f}_{IV}$ = −749453158.7 | 8402810812.8 | |||||

V | 6 | +0.0182 | +3479197.2 | 347919720.0 | +0.0532 | +0.0714 | |

8 | −0.3056 | −306964191.0 | 1828124435.8 | +0.0532 | +0.0994 ‡ | −0.1530 | |

13 | −0.0278 | −21840183.8 | 1429824980.5 | +0.0532 | −0.0338 $\pm $ | −0.0084 | |

$\Sigma $ | ${f}_{V}$ = −325325177.5 | 3605869136.3 | |||||

Iteration 2 | |||||||

Loop | Pipe | ${\mathit{Q}}_{1}=\mathit{Q}$ | $\mathit{f}={\mathit{p}}_{1}^{2}-{\mathit{p}}_{2}^{2}$ | $\left|{\mathit{f}}^{\prime}\right|$ | $\mathbf{\Delta}{\mathit{Q}}_{1}$ | $\mathbf{\Delta}{\mathit{Q}}_{2}$ | ${\mathit{Q}}_{2}=\mathit{Q}$ |

I | 1 | −0.4336 | −232172997.6 | 974431560.7 | −0.0058 | −0.4394 | |

7 | +0.6034 | +651439280.6 | 1965036192.1 | −0.0058 | +0.5976 | ||

8 | +0.1530 | +87112249.4 | 1036457217.8 | −0.0058 | −0.0178 = | +0.1294 | |

9 | +0.1446 | +243990034.4 | 3070921097.1 | −0.0058 | −0.0098 = | +0.1290 | |

10 | −0.2217 | −584137977.5 | 4795666298.0 | −0.0058 | +0.0018 ‡ | −0.2257 | |

12 | −0.0511 | −47725420.6 | 1700518680.1 | −0.0058 | −2.1·10^{−5} $\pm $ | −0.0569 | |

$\Sigma $ | ${f}_{I}$ = +118505168.7 | 13543031045.9 | |||||

II | 2 | −0.0677 | −30372941.9 | 816962908.0 | +2.1·10^{−5} | −0.0676 | |

11 | −0.1707 | −27780459.9 | 296182372.8 | +2.1·10^{−5} | +0.0018 ‡ | −0.1689 | |

12 | +0.0511 | +47725420.6 | 1700518680.1 | +2.1·10^{−5} | +0.0058 $\mp $ | +0.0569 | |

$\Sigma $ | ${f}_{II}$ = −10427981.2 | 2813663960.8 | |||||

III | 3 | −0.2480 | −451970989.4 | 3317464222.8 | −0.0018 | −0.2497 | |

4 | +0.0040 | +99061.2 | 44589235.4 | −0.0018 | +0.0023 | ||

10 | +0.2217 | +584137977.5 | 4795666298.0 | −0.0018 | +0.0058 $\mp $ | +0.2257 | |

11 | +0.1707 | +27780459.9 | 296182372.8 | −0.0018 | −2.1·10^{−5} = | +0.1689 | |

14 | −0.0757 | −135261698.0 | 3251481942.9 | −0.0018 | −0.0098 $\pm $ | −0.0873 | |

$\Sigma $ | ${f}_{III}$ = +24784811.3 | 11705384072.0 | |||||

IV | 5 | +0.0798 | +20483898.1 | 467437803.0 | +0.0098 | +0.0896 | |

9 | −0.1446 | −243990034.4 | 3070921097.1 | +0.0098 | +0.0058 ‡ | −0.1290 | |

13 | +0.0084 | +2454799.0 | 534076127.2 | +0.0098 | −0.0178 = | +0.0004 | |

14 | +0.0757 | +135261698.0 | 3251481942.9 | +0.0098 | +0.0018 $\mp $ | +0.0873 | |

$\Sigma $ | ${f}_{IV}$ = −85789639.2 | 7323916970.2 | |||||

V | 6 | +0.0714 | +41857166.9 | 1067095933.1 | +0.0178 | +0.0892 | |

8 | −0.1530 | −87112249.4 | 1036457217.8 | +0.0178 | +0.0058 ‡ | −0.1294 | |

13 | −0.0084 | −2454799.0 | 534076127.2 | +0.0178 | −0.0098 $\pm $ | −0.0004 | |

$\Sigma $ | ${f}_{V}$ = −47709881.5 | 2637629278.1 |

^{a}Pipe lengths, diameters and initial flow distribution are shown in Table 2 and Figure 1;

^{b}$f$ calculated using the Renouard Equation (1);

^{c}${f}^{\prime}$ calculated using the first derivative of the Renouard, Equation (2), where flow is variable;

^{d}calculated using the matrix Equation (10) and entering $\Delta Q$ with the opposite sign (using the original Hardy Cross method for Iteration 1: $\Delta {Q}_{I}$ = +0.1126; $\Delta {Q}_{II}$ = −0.0088; $\Delta {Q}_{III}$ = −0.0208; $\Delta {Q}_{IV}$ = −0.0892; $\Delta {Q}_{V}$ = −0.0902; using the Lobačev method for Iteration 1: $\Delta {Q}_{I}$ = −0.1041; $\Delta {Q}_{II}$ = −0.0644; $\Delta {Q}_{III}$ = −0.0780; $\Delta {Q}_{IV}$ = +0.1069; $\Delta {Q}_{V}$ = −0.1824);

^{e}$\Delta {Q}_{2}$ is $\Delta {Q}_{1}$ from the adjacent loop;

^{f}the final calculated flow in the first iteration is used for the calculation in the second iteration, etc.;

^{g}if $Q$ and ${Q}_{1}$ have a different sign, this means that the flow direction is opposite to that in the previous iteration, etc. (this occurs with the flow in pipe 13 between Iteration 3 and 4).

^{a} Pipe Number | Diameter (m) | Length (m) | ^{b} Assumed Flows (m^{3}/h) | ^{c} Calculated Flows (m^{3}/h) | Gas Velocity (m/s) |
---|---|---|---|---|---|

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 | ^{d}−2.6 | −0.009 |

14 | 0.152 | 548.6 | 100.0 | 312.7 | 1.2 |

^{a}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 Kirchhoff’s first law for all nodes (dash arrows in Figure 1);

^{c}calculated to satisfy Kirchhoff’s first law for all nodes and Kirchhoff’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.

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**MDPI and ACS Style**

Brkić, D.; Praks, P.
Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks. *Appl. Sci.* **2019**, *9*, 2019.
https://doi.org/10.3390/app9102019

**AMA Style**

Brkić D, Praks P.
Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks. *Applied Sciences*. 2019; 9(10):2019.
https://doi.org/10.3390/app9102019

**Chicago/Turabian Style**

Brkić, Dejan, and Pavel Praks.
2019. "Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks" *Applied Sciences* 9, no. 10: 2019.
https://doi.org/10.3390/app9102019