# Modeling and Efficiency Optimization of Steam Boilers by Employing Neural Networks and Response-Surface Method (RSM)

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Fundamentals

## 3. Modeling Results of Boiler Efficiency Using Neural Network

^{2}are obtained according to Table 3.

## 4. Optimization with the Help of the Response-Surface Method

## 5. Sensitivity Analysis of Effecting Variables

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The upper and lower limit of input and output data for the construction of a neural network model.

**Figure 4.**Sample schematic structure of a 2-5-1 neural network model: Two neurons in the input layer, five neurons in the hidden layer, and a neuron in the output layer.

**Figure 13.**The correlation coefficient of training, validation, test, and all for artificial neural network (ANN).

**Figure 14.**Mean squared error diagram in different epochs of training process for estimating the boiler efficiency.

$\dot{\mathit{m}}$ | $\mathit{T}(\xb0\mathrm{C})$ | η | $\dot{\mathit{m}}$ | $\mathit{T}(\xb0\mathrm{C})$ | η |
---|---|---|---|---|---|

5.06277 | 179.8114 | 68.8229 | 10.3826 | 202.5607 | 68.3157 |

11.50799 | 217.4076 | 68.10162 | 8.112843 | 195.3354 | 66.59289 |

10.96721 | 227.7698 | 66.7485 | 12.47747 | 228.7439 | 66.83186 |

11.83226 | 195.8293 | 69.17249 | 15.45171 | 206.8831 | 69.76082 |

8.164 | 211.1887 | 66.49393 | 10.33337 | 222.0078 | 66.69333 |

8.230552 | 195.8363 | 67.13933 | 7.35201 | 199.2193 | 66.00094 |

15.4074 | 189.2565 | 69.76787 | 8.328303 | 202.0249 | 67.16067 |

12.80646 | 218.8824 | 68.03788 | 7.689097 | 220.9679 | 65.33974 |

9.523767 | 215.6007 | 67.21874 | 14.14401 | 229.5426 | 67.54719 |

15.35225 | 192.4996 | 69.97655 | 11.38288 | 205.2453 | 68.95443 |

7.509398 | 208.2864 | 66.27828 | 13.44131 | 226.1227 | 67.40915 |

11.39483 | 199.8923 | 68.57542 | 13.74267 | 216.4291 | 68.86722 |

13.96593 | 182.7876 | 68.82806 | 12.49931 | 207.5064 | 68.88944 |

10.24027 | 179.2494 | 67.70906 | 13.05674 | 228.3764 | 67.44843 |

10.84519 | 203.5396 | 68.7183 | 15.47826 | 220.874 | 68.30319 |

12.06786 | 192.4699 | 69.16695 | 13.38398 | 227.8997 | 67.56851 |

15.6384 | 195.6674 | 69.49326 | 15.45541 | 211.61 | 69.13087 |

14.51184 | 192.9422 | 69.69088 | 11.48975 | 197.7378 | 69.29713 |

7.618852 | 186.887 | 66.61787 | 8.318545 | 202.7819 | 66.87618 |

9.65485 | 198.7617 | 68.26366 | 7.606863 | 225.4183 | 64.83067 |

10.63276 | 214.2778 | 67.86751 | 13.88847 | 178.7724 | 69.11839 |

15.35098 | 188.5912 | 69.38854 | 9.938753 | 186.1481 | 67.56152 |

11.02739 | 212.649 | 68.18121 | 13.46965 | 202.5215 | 69.66543 |

9.276587 | 201.0394 | 67.34079 | 13.28742 | 206.2356 | 69.53684 |

12.75497 | 200.5314 | 69.42711 | 12.10742 | 181.1039 | 68.71523 |

11.60532 | 187.1125 | 68.66235 | 13.89687 | 212.2176 | 68.83852 |

15.68032 | 188.0465 | 69.69601 | 11.85233 | 224.261 | 67.02746 |

16.0319 | 210.0539 | 69.16832 | 11.68888 | 183.7009 | 68.7909 |

14.89612 | 191.9886 | 69.48701 | 11.78454 | 200.7642 | 69.38998 |

13.83662 | 207.1033 | 69.35699 | 16.04421 | 192.572 | 69.86828 |

14.1702 | 227.1288 | 67.70859 | 10.72913 | 229.2328 | 66.56324 |

15.69162 | 215.1525 | 68.53946 | 8.083356 | 209.6555 | 66.52948 |

7.57872 | 213.3194 | 65.88433 | 9.281896 | 191.195 | 67.60546 |

10.21025 | 227.8878 | 65.96942 | 13.60328 | 184.896 | 68.9027 |

6.986591 | 218.3174 | 65.18013 | 15.10385 | 206.3403 | 69.70184 |

14.99537 | 209.6018 | 69.78638 | 16.37641 | 221.0458 | 68.45829 |

11.87784 | 227.2961 | 66.81758 | 9.036371 | 221.5243 | 66.38769 |

10.50436 | 181.1014 | 67.63481 | 14.33667 | 221.3341 | 68.1886 |

9.147886 | 191.973 | 67.25517 | 12.62322 | 188.5932 | 69.35013 |

12.1442 | 229.3074 | 67.04037 | 10.86203 | 206.3097 | 68.49007 |

9.758881 | 218.1548 | 67.08144 | 11.90317 | 223.497 | 67.82707 |

7.609162 | 202.7184 | 66.5345 | 11.76321 | 184.292 | 68.53394 |

7.770549 | 213.4068 | 65.85192 | 13.27719 | 222.5302 | 67.67154 |

7.608856 | 199.6806 | 66.64859 | 14.17492 | 224.7884 | 67.86074 |

10.92877 | 197.7678 | 68.71765 | 9.880103 | 189.329 | 67.77672 |

8.143986 | 189.0901 | 66.894 | 7.815672 | 182.0029 | 66.12302 |

11.25155 | 197.9128 | 69.15146 | 14.97466 | 202.6592 | 69.45441 |

15.68414 | 179.5427 | 68.899 |

Variable | Definition | Description |
---|---|---|

${\eta}_{\mathit{N}\mathit{N}}$ | Output of the neural network model | Boiler efficiency, the final output of the neural network model |

$\dot{\mathit{m}}$ | 1st input variable | Steam mass flow rate, kg/s |

$\mathit{T}$ | 2nd input variable | Temperature, $\xb0\mathrm{C}$ |

${\left[\mathit{I}\mathit{W}\right]}_{5\times 2}$ | Edge matrix between 1st and 2nd layers | Each of the edges between layer 1 and 2 is assigned a weight |

${\left[\mathit{L}\mathit{W}\right]}_{1\times 5}$ | Edge matrix between 2nd and 3rd layers | Each of the edges between layer 2 and 3 is assigned a weight |

${\left[{\mathit{b}}_{1}\right]}_{5\times 1}$ | Bias matrix of the 2nd layer | After multiplying the weight matrix in the input signal, the results are summed with the bias. |

${\left[{\mathit{b}}_{2}\right]}_{1\times 1}$ | Bias matrix of the 3rd layer | After multiplying the weight matrix in the input signal, the results are summed with the bias. |

Train (70%) | Validation (15%) | Test (15%) | |
---|---|---|---|

Number of data | 67 | 14 | 14 |

Mean squared error (MSE) | 0.043 | 0.079 | 0.053 |

Regression coefficient (R^{2}) | 0.986 | 0.981 | 0.979 |

Constants | C0 | C1 | C2 | C12 | C11 | C22 |
---|---|---|---|---|---|---|

Optimized Value | −31.372 | 1.690 | 0.8977 | −0.00047 | −0.05112 | −0.00227 |

Variable | Objective Function | 1st Independent Variable | 2nd Independent Variable |
---|---|---|---|

Unit | Efficiency (%) | Flow rate (kg/s) | Temperature ($\xb0\mathrm{C}$) |

Optimal value from RSM | 69.8 | 15.7 | 195.9 |

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## Share and Cite

**MDPI and ACS Style**

Maddah, H.; Sadeghzadeh, M.; Ahmadi, M.H.; Kumar, R.; Shamshirband, S.
Modeling and Efficiency Optimization of Steam Boilers by Employing Neural Networks and Response-Surface Method (RSM). *Mathematics* **2019**, *7*, 629.
https://doi.org/10.3390/math7070629

**AMA Style**

Maddah H, Sadeghzadeh M, Ahmadi MH, Kumar R, Shamshirband S.
Modeling and Efficiency Optimization of Steam Boilers by Employing Neural Networks and Response-Surface Method (RSM). *Mathematics*. 2019; 7(7):629.
https://doi.org/10.3390/math7070629

**Chicago/Turabian Style**

Maddah, Heydar, Milad Sadeghzadeh, Mohammad Hossein Ahmadi, Ravinder Kumar, and Shahaboddin Shamshirband.
2019. "Modeling and Efficiency Optimization of Steam Boilers by Employing Neural Networks and Response-Surface Method (RSM)" *Mathematics* 7, no. 7: 629.
https://doi.org/10.3390/math7070629