# Comparison of Standalone and Hybrid Machine Learning Models for Prediction of Critical Heat Flux in Vertical Tubes

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset Generation

#### 2.2. Methodology

#### 2.2.1. Look-Up Table (LUT) Method

^{2}s), pressures (ranging from 0 to 21 MPa), and local qualities (ranging from −0.50 to 0.90). This comprehensive and standardized database is specifically designed to support safe and reliable decision-making processes in a variety of contexts. The CHF LUT approach offers several benefits, including the ability to cater to a wide range of practical applications, ease of use, and the absence of iterative calculations required to predict CHF [15]. Groeneveld et al. [55] presented a general equation aimed at addressing the diameter correction of the tube. The equation is formulated as follows:

#### 2.2.2. Artificial Neural Network (ANN)

_{kj}are their cross-ponding weights of the input vector

**x**. Z and

_{j}**b**reflect the total weighted inputs and the bias of node k, respectively. Then this information is passed through activation function f [59]. Figure 2 illustrates the architecture of the ANN.

#### 2.2.3. Support Vector Regression (SVR)

#### 2.2.4. Random Forest (RF)

#### 2.2.5. Data-Driven Hybrid Model

_{L}) of the data-driven model is fundamentally a non-linear function of the input variables. During the learning process, as depicted in Figure 4, the error (σ) is calculated by subtracting y

_{L}from the experimental output (y). The error (σ) obtained from the predicted values is utilized to train the ANN, SVR and RF. The predicted errors (σ

_{m}) from ANN, SVR, and RF are compared with the error (σ) using an error/cost function. Typically, the loss function is presented in the form of mean absolute error (MAE) or mean squared error (MSE). The loss function is generally presented by MAE or MSE. The error function is optimized (minimized) during the learning/training process. The final prediction of the hybrid model y

_{h}is the sum of y

_{L}and σ

_{m}. Relative root-mean-square error (rRMSE) is used to measure how well the hybrid model performs when evaluated against experimental results. The rRMSE is described as

## 3. Simulation Settings

## 4. Performance Evaluations

#### 4.1. Standalone ML Models (ANN vs. SVM vs. RF vs. LUT)

#### 4.2. Comparison of Hybrid and Standalone Approaches

#### 4.3. Sensitivity Analysis

## 5. Conclusions

- ○
- The hybrid approach using ANN outperforms both traditional ML techniques and the conventional LUT technique when it comes to predicting accuracy.
- ○
- Although standalone ML-based models performed better than the widely used conventional LUTs, the hybrid model greatly outperforms standalone ML models for prediction of CHF in vertical tubes for diverse set of operating parameters, with lower dispersion and non-biased parametric patterns.
- ○
- ML architecture can be greatly simplified in the hybrid framework as compared to its standalone version to reduce computing costs when working with big databases.
- ○
- From the parametric analysis in this work, it is confirmed that standalone ANN and hybrid (ANN + LUT) models have more suitable regression features between input and output than conventional LUT.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

AI | Artificial Intelligence |

ANN | Artificial neural network |

b | Bias term |

BPN | Backpropagation neural network |

C | Kernel function |

CHF | Critical heat flux |

D | Heated diameter |

DNB | Departure from nucleate boiling |

DNBR | Departure from nucleate boiling ratio |

DNN | Deep neural networks |

DT | Decision tree |

EPRI | Electric Power Research Institute |

f | Unknown function |

FNN | Feed-forward neural network |

G | Mass flux |

HONN | Higher order neural network |

L | Heated length |

LUT | Look-up table |

MAE | Mean absolute error |

MDNBR | Minimum value of DNBR |

MLP | Multi-layer perceptron |

MSE | Mean square error |

ML | Machine learning |

m | Number of data points |

P | Pressure |

PWR | Pressurized water reactor |

RBF | Radial basis function |

ReLU | Rectified Linear unit |

RF | Random Forest |

rRMSE | relative Root mean squared error |

SVR | Support Vector Regression |

w | Weight factor |

x | Local equilibrium/exit quality |

X | Input matrix |

y | Desired output |

y_{h} | Hybrid model output |

ξ | Slack in SVR |

σ | Error |

σ_{m} | ML predicated error |

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**Figure 6.**(

**a**,

**b**). Experimental vs. predicated CHF for standalone best-estimate models (LUT vs. SVM vs. RF vs. ANN).

**Figure 7.**Absolute-relative deviation vs. data fraction of hybrid ANN+LUT (

**a**), RF+LUT (

**b**), SVM+LUT (

**c**), and standalone ML models.

**Figure 10.**CHF relative error distribution vs. tube diameter (

**a**), heated length (

**b**), mass flowrate (

**c**), pressure (

**d**), exit quality (

**e**).

Author | Mass-Flux (G) [kg/m^{2}s] | Pressure (P) [MPa] | Equilibrium Quality (x) [-] | Heated Length (L) [mm] | Heated Diameter (D) [mm] | CHF [MW/m^{2}] | No. of Samples |
---|---|---|---|---|---|---|---|

Inasaka [46] | 4300–6700 | 0.31–0.64 | −0.11 to −0.05 | 100 | 3 | 7.3–12.8 | 6 |

Williams [47] | 325–4683 | 2.7–15.2 | −0.02 to 0.92 | 1840 | 9.5 | 0.39–4.1 | 129 |

Kim [48] | 20–277 | 0.11–0.95 | 0.32 to 1.2 | 300–1770 | 6–12 | 0.12–1.6 | 512 |

Becker [49] | 100–5450 | 0.22–9.9 | 0 to 0.99 | 400–3750 | 3.9–25 | 0.28–7.5 | 3473 |

Lowdermilk [50] | 60–597 | 3.4 | 0.71 to 0.94 | 152 | 3 | 0.47–3.3 | 21 |

Clark [51] | 28–102 | 3.4–13.8 | 0.66 to 0.99 | 239 | 4.6 | 0.23–1.2 | 67 |

Reynold [52] | 1166–2889 | 3.6–10.7 | 0 to 0.47 | 229 | 4.6 | 3.6–9 | 67 |

Peskov [53] | 750–5361 | 10–20 | −0.23 to 0.13 | 400–1650 | 10 | 0.9–4.3 | 17 |

Thompson [54] | 542–7975 | 0.1–20.7 | −0.86 to 0.21 | 25–3048 | 1–37.5 | 1–19.3 | 1585 |

Total | 20–7975 | 0.1–20.7 | −0.86 to 1.2 | 25–3750 | 1–37.5 | 0.12–19.3 | 5877 |

Data-Driven Model | LUT |
---|---|

ML Approach | ANN, SVR, RF |

Best-estimate ANN approach | |

- ▪
- Network architecture
| 5/50/50/50/1 (5/100/100/100/50/1 if standalone ANN) |

- ▪
- Weight optimization algorithm
| Adam |

- ▪
- Hidden layers activation function
| ReLU (Rectified Linear unit) |

- ▪
- Learning rate
| 0.001 |

Best-estimate SVR approach | |

- ▪
- Parameters
| Kernel: Rbf, C: 100, Nu: 0.9 (Kernel: Rbf, C: 100, Nu: 1 if standalone SVR) |

Best-estimate RF approach | |

- ▪
- Number of estimators
| 100 (300 if standalone RF) |

Approach | Test rRMSE (%) | Data-Points within ±10% Error | Data-Points within ±20% Error |
---|---|---|---|

LUT | 15.8 | 68% | 85% |

SVM | 15.5 | 72% | 86% |

RF | 14.7 | 80% | 89% |

ANN | 12.2 | 87% | 95% |

Hybrid SVM + LUT | 11.8 | 87% | 97% |

Hybrid RF + LUT | 10.7 | 89% | 99% |

Hybrid ANN + LUT | 9.3 | 91% | 100% |

Sensitivity Analysis Technique | Test RMSE (%) | Test Samples within ±10% Error |
---|---|---|

80% train + 20% test | 9.30 | 91% |

5-fold cross-validation | 9.15 | 89% |

10-fold cross-validation | 8.90 | 91% |

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**MDPI and ACS Style**

Khalid, R.Z.; Ullah, A.; Khan, A.; Khan, A.; Inayat, M.H. Comparison of Standalone and Hybrid Machine Learning Models for Prediction of Critical Heat Flux in Vertical Tubes. *Energies* **2023**, *16*, 3182.
https://doi.org/10.3390/en16073182

**AMA Style**

Khalid RZ, Ullah A, Khan A, Khan A, Inayat MH. Comparison of Standalone and Hybrid Machine Learning Models for Prediction of Critical Heat Flux in Vertical Tubes. *Energies*. 2023; 16(7):3182.
https://doi.org/10.3390/en16073182

**Chicago/Turabian Style**

Khalid, Rehan Zubair, Atta Ullah, Asifullah Khan, Afrasyab Khan, and Mansoor Hameed Inayat. 2023. "Comparison of Standalone and Hybrid Machine Learning Models for Prediction of Critical Heat Flux in Vertical Tubes" *Energies* 16, no. 7: 3182.
https://doi.org/10.3390/en16073182