# Numerical Simulation and Analytical Prediction of Residual Strength for Elbow Pipes with Erosion Defects

^{*}

## Abstract

**:**

## 1. Introduction

_{2}corrosion at high partial pressure [16]. Khalaj et al. presented some results of the research connected with the development of a new approach based on the artificial neural network (ANN) of predicting the ultimate tensile strength of the API X70 steels after thermomechanical treatment [17]. In the study of Khalaj et al., bainite fraction results of continuous cooling of high-strength low alloy steels had been modeled by artificial neural networks [18].

## 2. Erosion Rate Analysis of 90° Elbow

#### 2.1. Predictive Model for Erosion Wear

#### 2.2. Model Parameters and Boundary Conditions

^{3}and a gas viscosity of 1.79 × 10

^{−5}Pa·s, which flows in at the inlet at a speed of 34.1 m/s and flows out freely at the outlet. X100 pipeline steel is used as the material for the pipes. The particle size is 182 µm, and the particle-to-gas phase mass flow rate ratio is 0.013.

#### 2.3. Mesh Division and Mesh Independence Verification

#### 2.4. Validation of Numerical Methods

## 3. Erosion Rate Analysis of Defective Elbow Pipes

#### 3.1. Model Parameters

#### 3.2. Analysis of Simulation Results

## 4. Analysis of Residual Strength of Defective Elbow Pipes

#### 4.1. Model Parameters and Boundary Conditions

- (1)
- The coupling effect of pipe gas and pipe soil (such as the friction between pipe gas and pipe soil, the velocity of the medium in the pipe, etc.) during the operation of the pipelines is not considered; the thermal expansion and cold contraction due to changes in ambient temperature are not considered The thermal stress generated in the pipelines when the phenomenon is limited by the internal and external constraints of the pipelines; the force generated by the anti-erosion protection measures of the pipelines are not considered.
- (2)
- Only the effect of the internal pressure on the pipelines during the operation of the pipelines is considered, and the direction of the force is perpendicular to the inner surface of the pipelines; the forces generated by the pipelines’ own weight, bending moment, and seismic load are not considered.
- (3)
- Displacement constraints are imposed on the left and right ends of the model; that is, the ends are completely fixed. The elbow model is shown in Figure 10.

#### 4.2. Meshing and Mesh Independence Verification

#### 4.3. Numerical Method Validation

#### 4.4. Analysis of Simulation Results

## 5. Prediction of Erosion Life and Residual Strength

#### 5.1. Extreme Learning Machine

- (1)
- Set the hidden layer activation function $g\left(x\right)$ and the number of hidden layer nodes L;
- (2)
- Randomly generate hidden layer node parameters $\left(a,{b}_{i}\right)$, $i=1,2,\cdots L$ ;
- (3)
- Calculate the hidden layer output matrix H;
- (4)
- Calculate the output weight $\stackrel{\wedge}{\beta}={H}^{+}T$, where ${H}^{+}$ is the generalized inverse of Moore–Penrose of the matrix H.

#### 5.2. Genetic Algorithm

#### 5.3. Prediction Results and Analysis of Test Data

#### 5.4. Prediction Results and Analysis of Simulation Data

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$E(\theta )$ | corrosion rate (kg/m^{2}·s) |

$g(\theta )$ | function of impact angle |

${E}_{90}$ | reference wear rate (kg/m^{2}·s) |

${H}_{v}$ | Vickers hardness of the eroded material |

${u}_{p}$ | relative velocity between particle and wall (m/s) |

${d}_{p}$ | particle size of erosion particles (μm) |

${u}_{pref}$ | particle reference velocity constant |

${d}_{pref}$ | reference particle size |

${\sigma}_{\theta}$ | Von Mises equivalent stress (MPa) |

${\sigma}_{1}$ | the stress along the X-axis (MPa) |

${\sigma}_{2}$ | the stress along the Y-axis (MPa) |

${\sigma}_{3}$ | the stress along the Z-axis (MPa) |

$l$ | defect length (mm) |

$w$ | defect width (mm) |

$d$ | defect depth (mm) |

$\mathrm{GA}-\mathrm{ELM}$ | extreme learning machine optimized by genetic algorithm |

$n$ | number of input layers |

$L$ | number of hidden layer neurons |

${a}_{i}$,${b}_{i}$ | hidden layer node parameters |

${\beta}_{i}$ | outer weight between the i-th hidden layer node and the network output |

$g$ | activation function |

$H$ | hidden layer output matrix |

$J$ | squared loss function |

$N$ | determine population size |

${P}_{c}$ | cross probability |

${P}_{m}$ | mutation probability |

$X\left(0\right)$ | initial population |

$k$ | current evolutionary algebra |

$Y$ | maximum evolutionary algebra |

$h$ | safe allowable wall thickness of the pipe wall (m) |

$\rho $ | density of X100 steel (kg/m^{3}) |

$T$ | erosion life (s) |

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**Figure 5.**Defect location and partial local enlarged map of the elbow. (

**a**) Schematic diagram of point defect; (

**b**) Schematic diagram of trench defect; (

**c**) Schematic diagram of double trench defect.

**Figure 13.**Stress diagram. (

**a**) Overall stress diagram; (

**b**) Schematic diagram of stress at the defect.

**Figure 14.**Variation of equivalent stress with internal pressure load under different defect conditions. (

**a**) Different defect depths; (

**b**) Different defect lengths; (

**c**) Different defect widths.

**Figure 15.**Relationship between defect geometry parameters and residual strength. (

**a**) Schematic diagram of the relationship between residual strength and defect depth; (

**b**) Schematic diagram of the relationship between residual strength and defect length; (

**c**) Schematic diagram of the relationship between residual strength and defect width.

**Figure 19.**Comparison of the accuracy of the two prediction models. (

**a**) Schematic diagram of erosion rate prediction; (

**b**) Schematic diagram of residual life prediction; (

**c**) Schematic diagram of residual strength prediction.

Minimum Yield Strength (MPa) | Minimum Tensile Strength (MPa) | Ultimate Tensile Strength (MPa) | Young’s Modulus (MPa) | Poisson’s Ratio |
---|---|---|---|---|

690 | 760 | 886 | 210,000 | 0.3 |

C | Si | Mn | P | S | Cr | Ni | Mo | Fe |
---|---|---|---|---|---|---|---|---|

0.065 | 0.95 | 1.69 | 0.015 | 0.002 | 0.04 | 0.03 | 0.27 | allowance |

**Table 3.**Prediction and comparison of residual strength of pipeline steel under different notch conditions.

Number | Pipe Diameter (mm) | Wall Thickness (mm) | Defect Length (mm) | Defect Depth (mm) | Residual Strength (MPa) (Literature Test Value) | Residual Strength (MPa) (Simulation Results) | Error (%) |
---|---|---|---|---|---|---|---|

1 | 1320 | 22.9 | 200 | 4.58 | 27.79 | 27.87 | 0.3 |

2 | 1320 | 22.9 | 1000 | 11.45 | 16.59 | 16.64 | 0.3 |

3 | 1320 | 22.9 | 514.98 | 11.45 | 17.9 | 17.72 | 1.0 |

4 | 1320 | 22.9 | 1109.93 | 11.45 | 15.4 | 16.01 | 4.0 |

5 | 1320 | 22.9 | 556.88 | 11.36 | 18.1 | 17.3 | 4.4 |

6 | 1320 | 22.9 | 1012.74 | 11.45 | 15 | 15.97 | 6.4 |

7 | 1320 | 22.9 | 800 | 41.2 | 18.76 | 16.97 | 9.5 |

Pipe Diameter (mm) | Wall Thickness (mm) | Defect Length (mm) | Defect Depth (mm) | Defect Width (mm) | Yield Strength (MPa) | Tensile Strength (MPa) | Residual Strength (MPa) |
---|---|---|---|---|---|---|---|

458.8 | 8.1 | 39.6 | 5.39 | 31.9 | 601 | 684 | 22.68 |

459.4 | 8 | 40.05 | 3.75 | 32 | 589 | 730.5 | 24.2 |

323.9 | 9.8 | 255.6 | 7.08 | 95.3 | 452 | 542 | 14.4 |

323.9 | 9.66 | 305.6 | 6.76 | 95.3 | 452 | 542 | 14.07 |

508 | 14.6 | 500 | 10.35 | 97 | 478 | 600 | 14.6 |

508 | 14.3 | 500 | 10.3 | 97 | 478 | 600 | 13.4 |

76.2 | 2 | 75 | 1.4 | 16 | 391 | 458 | 9.4 |

76.2 | 2.04 | 75 | 1.44 | 16 | 260 | 309 | 5.45 |

762 | 17.5 | 200 | 9 | 200 | 474.1 | 556.6 | 22.64 |

426 | 6.95 | 160 | 2.7 | 25 | 240 | 390 | 10.8 |

… | … | … | … | … | … | … | … |

426 | 7 | 150 | 3.8 | 21 | 240 | 390 | 9.81 |

529 | 9 | 350 | 4.7 | 25 | 285 | 415 | 8.83 |

529 | 9 | 160 | 4.7 | 25 | 285 | 415 | 15.7 |

720 | 8 | 320 | 4.4 | 26 | 425 | 535 | 8.83 |

720 | 8 | 180 | 6.2 | 26 | 425 | 535 | 7.55 |

304.8 | 6.35 | 26 | 4.95 | 20 | 351 | 543 | 15.36 |

304.8 | 6.35 | 33 | 4.25 | 21 | 382 | 570 | 16.29 |

323.9 | 9.74 | 527.8 | 7.06 | 95.3 | 422.5 | 589.6 | 11.3 |

1422.4 | 19.25 | 180 | 10.4 | 0.5 | 740 | 774 | 15.35 |

914.4 | 16.4 | 450 | 6 | 0.5 | 739 | 813 | 24.02 |

… | … | … | … | … | … | … | … |

304.8 | 6.35 | 37 | 4.64 | 30 | 351 | 463 | 14.29 |

324 | 6.01 | 19.35 | 3.6 | 19 | 382 | 570 | 16.22 |

324 | 10.3 | 243 | 5.15 | 154.5 | 380 | 514 | 23.2 |

324 | 10.3 | 243 | 5.15 | 30.9 | 380 | 514 | 22 |

508 | 6.6 | 381 | 2.62 | 35.4 | 443.4 | 598.9 | 11.25 |

508 | 6.35 | 900 | 3.43 | 25.4 | 429.6 | 572.5 | 8 |

323.9 | 9.79 | 500 | 6.99 | 95.3 | 452 | 542 | 11.99 |

323.9 | 9.74 | 527.8 | 7.14 | 95.3 | 452 | 542 | 11.3 |

762 | 17.5 | 200 | 8.4 | 100 | 474.1 | 556.6 | 23.42 |

762 | 17.5 | 200 | 9 | 200 | 474.1 | 556.6 | 22.64 |

Number | Test Values in Literature (MPa) | Predicted Value of GA-ELM (MPa) | Error (%) |
---|---|---|---|

1 | 15.36 | 15.7528 | 2.56 |

2 | 24.3 | 23.7243 | 2.37 |

3 | 11.91 | 12.5701 | 5.55 |

4 | 11.99 | 12.4956 | 4.22 |

5 | 22.64 | 23.2093 | 2.51 |

6 | 24.3 | 23.7668 | 2.19 |

7 | 11.3 | 12.0015 | 6.21 |

8 | 8 | 8.7686 | 9.61 |

9 | 11.25 | 11.4471 | 1.75 |

10 | 5.45 | 6.2013 | 13.78 |

11 | 14.4 | 15.0982 | 4.85 |

Defect | Velocity of Flow (m/s) | Mass Flow Rate (10 ^{−4} kg/s) | Grain Size (µm) | Maximum Erosion Rate (10 ^{−5} kg∙m^{−2}∙s^{−1}) | Residual Life Span (10 ^{5} s) |
---|---|---|---|---|---|

0 | 34.1 | 2.60 | 152 | 5.10 | 18.5 |

0 | 34.1 | 2.80 | 152 | 5.49 | 17.1 |

0 | 33.1 | 2.60 | 152 | 4.74 | 19.9 |

0 | 33.1 | 2.60 | 152 | 5.10 | 18.5 |

0 | 35.1 | 2.60 | 152 | 5.44 | 17.3 |

… | … | … | … | … | … |

0 | 37.1 | 2.20 | 202 | 5.56 | 17.0 |

0 | 37.1 | 2.40 | 202 | 6.06 | 15.5 |

0 | 38.1 | 2.60 | 202 | 6.99 | 13.5 |

0 | 38.1 | 2.80 | 202 | 7.53 | 12.5 |

0 | 38.1 | 3.00 | 202 | 8.06 | 11.7 |

1 | 33.1 | 2.60 | 142 | 83.0 | 1.13 |

1 | 34.1 | 2.60 | 142 | 35.8 | 2.63 |

1 | 35.1 | 2.60 | 142 | 28.7 | 3.28 |

1 | 36.1 | 2.60 | 142 | 54.8 | 1.72 |

1 | 33.1 | 2.80 | 142 | 27.6 | 3.41 |

… | … | … | … | … | … |

1 | 37.1 | 2.00 | 202 | 35.4 | 2.66 |

1 | 37.1 | 2.20 | 202 | 39.0 | 2.42 |

1 | 37.1 | 2.40 | 202 | 42.5 | 2.22 |

1 | 38.1 | 2.80 | 202 | 52.6 | 1.79 |

1 | 38.1 | 3.00 | 202 | 56.4 | 1.67 |

2 | 35.1 | 2.60 | 142 | 47.3 | 1.99 |

2 | 36.1 | 2.60 | 142 | 72.8 | 1.29 |

2 | 33.1 | 2.60 | 142 | 59.3 | 1.59 |

2 | 34.1 | 2.60 | 142 | 66.2 | 1.42 |

2 | 34.1 | 2.80 | 142 | 71.3 | 1.32 |

… | … | … | … | … | … |

2 | 37.1 | 3.20 | 202 | 45.9 | 2.05 |

2 | 37.1 | 3.00 | 202 | 43.0 | 2.19 |

2 | 38.1 | 2.80 | 202 | 44.4 | 2.12 |

2 | 38.1 | 2.40 | 202 | 38.1 | 2.48 |

2 | 38.1 | 2.20 | 202 | 34.9 | 2.70 |

Pipe Diameter (mm) | Wall Thickness (mm) | Defect Length (mm) | Defect Depth (mm) | Defect Width (mm) | Yield Strength (MPa) | Tensile Strength (MPa) | Residual Strength (MPa) |
---|---|---|---|---|---|---|---|

1320 | 22.9 | 100 | 13 | 26 | 690 | 886 | 25.2 |

1320 | 22.9 | 120 | 13 | 26 | 690 | 886 | 24 |

1320 | 22.9 | 140 | 13 | 26 | 690 | 886 | 22.8 |

1320 | 22.9 | 160 | 13 | 26 | 690 | 886 | 21.7 |

1320 | 22.9 | 180 | 13 | 26 | 690 | 886 | 20.7 |

… | … | … | … | … | … | … | … |

1320 | 22.9 | 200 | 14 | 26 | 690 | 886 | 18.4 |

1320 | 22.9 | 200 | 15 | 26 | 690 | 886 | 17.5 |

1320 | 22.9 | 200 | 16 | 26 | 690 | 886 | 16.8 |

1320 | 22.9 | 200 | 17 | 26 | 690 | 886 | 16.1 |

1320 | 22.9 | 200 | 18 | 26 | 690 | 886 | 14.5 |

1422.4 | 20.1 | 100 | 11 | 22 | 795 | 840 | 15.7 |

1422.4 | 20.1 | 120 | 11 | 22 | 795 | 840 | 14.67 |

1422.4 | 20.1 | 140 | 11 | 22 | 795 | 840 | 13.8 |

1422.4 | 20.1 | 160 | 11 | 22 | 795 | 840 | 12.78 |

1422.4 | 20.1 | 180 | 11 | 22 | 795 | 840 | 12.3 |

… | … | … | … | … | … | … | … |

1422.4 | 20.1 | 200 | 12 | 22 | 795 | 840 | 11.2 |

1422.4 | 20.1 | 200 | 13 | 22 | 795 | 840 | 10.01 |

1422.4 | 20.1 | 200 | 14 | 22 | 795 | 840 | 9.31 |

1422.4 | 20.1 | 200 | 15 | 22 | 795 | 840 | 8.57 |

1422.4 | 20.1 | 200 | 16 | 22 | 795 | 840 | 8.2 |

914.4 | 16.4 | 100 | 10 | 20 | 739 | 813 | 15.4 |

914.4 | 16.4 | 120 | 10 | 20 | 739 | 813 | 14.1 |

914.4 | 16.4 | 140 | 10 | 20 | 739 | 813 | 13.21 |

914.4 | 16.4 | 160 | 10 | 20 | 739 | 813 | 12.5 |

914.4 | 16.4 | 180 | 10 | 20 | 739 | 813 | 12.2 |

… | … | … | … | … | … | … | … |

914.4 | 16.4 | 200 | 11 | 20 | 739 | 813 | 10.2 |

914.4 | 16.4 | 200 | 12 | 20 | 739 | 813 | 9.4 |

914.4 | 16.4 | 200 | 13 | 20 | 739 | 813 | 8.7 |

914.4 | 16.4 | 200 | 14 | 20 | 739 | 813 | 7.2 |

914.4 | 16.4 | 200 | 15 | 20 | 739 | 813 | 5.83 |

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**MDPI and ACS Style**

Sun, C.; Wang, Q.; Li, Y.; Li, Y.; Liu, Y.
Numerical Simulation and Analytical Prediction of Residual Strength for Elbow Pipes with Erosion Defects. *Materials* **2022**, *15*, 7479.
https://doi.org/10.3390/ma15217479

**AMA Style**

Sun C, Wang Q, Li Y, Li Y, Liu Y.
Numerical Simulation and Analytical Prediction of Residual Strength for Elbow Pipes with Erosion Defects. *Materials*. 2022; 15(21):7479.
https://doi.org/10.3390/ma15217479

**Chicago/Turabian Style**

Sun, Chao, Qi Wang, Yuelin Li, Yingqi Li, and Yuechan Liu.
2022. "Numerical Simulation and Analytical Prediction of Residual Strength for Elbow Pipes with Erosion Defects" *Materials* 15, no. 21: 7479.
https://doi.org/10.3390/ma15217479