# On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach

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## Abstract

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## 1. Introduction

- The MH approach developed does not make use of gradients and does not call on any previous knowledge (e.g., initial guess, initial approximation, continuity, differentiability and small auxiliary parameters) of the problem. Unlike other deterministic approaches, the ANN-TTA-SQP only requires initial parameter settings (e.g., max. iterations, population size, etc.) and execution stopping criteria.
- A simple method is provided that enables the singularity and non-linearity of complex systems, such as longitudinal porous heat exchangers, to be successfully dealt with.
- Stochastic approaches based on ANN, in contrast to deterministic solvers, are capable of providing a continuous solution across the entirety of the integration domain.

## 2. Mathematical Formulation of Physical Problem

## 3. Proposed Methodology

#### 3.1. Neural Networks Based Differential Equation Models

#### 3.2. Optimization Procedure

#### 3.2.1. Tiki-Taka Algorithm

#### 3.2.2. Sequential Quadratic Programming

## 4. Results and Discussion

## 5. Statistical Analysis

## 6. Conclusions

- A novel unsupervised framework for an intelligent method was designed to construct surrogate solutions for the governing non-linear mathematical model of a fully wetted longitudinal FGM fin. The ANN-TTA-SQP algorithm was implemented to investigate the significance of variations in the dimensionless ambient temperature, parameter for a moist porous medium, convection parameter, in-homogeneity index, radiation parameter, and power index on the temperature distribution of the FGM fin with multiple fluctuations in thermal conductance.
- The approximate solutions obtained were validated by comparing the statistics with state-of-the-art-techniques, including the particle swarm optimization (PSO) algorithm, the cuckoo search algorithm (CSA), the whale optimization algorithm (WOA), the grey wolf optimization (GWO) algorithm and the machine learning algorithm. Minimum values of the mean square errors were observed in the solutions of the proposed technique.
- The thermal distribution in the fin fell when the values of the convective coefficient, radiation coefficient, and parameter for a moist porous medium increased. Increase in the ambient temperature, power index and inhomogeneity parameters caused an increase in the dispersion of the temperature over the heat exchanger.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**A diagrammatic representation of the porous longitudinal fin model, illustrating natural phenomena of radiation and convection.

**Figure 4.**Graphical illustration of the working steps of hybrid technique of the Tiki-Taka algorithm and local search processing of SQP for the training/optimization of neurons in feed-forward architecture of ANN for the minimization of fitness functions in Equation (21).

**Figure 5.**The effect of $\left({\theta}_{a}\right)$ surrounding temperature on the heat dispersion profile of the fully wetted longitudinal porous fin with $Nr=5,Nc=10,\beta =p={m}_{2}=\alpha =1$.

**Figure 6.**Graphical illustration of influence of convection parameter on thermal profile of the linear, quadratic and exponential FGM fin with $Nr=5,{\theta}_{a}=0.2,\beta =p={m}_{2}=\alpha =1$.

**Figure 7.**Demonstration of influence of radiation parameter on temperature distribution of FGM fin with $Nc=10,{\theta}_{a}=0.2,\beta =p={m}_{2}=\alpha =1$.

**Figure 8.**Significance of $\left({m}_{2}\right)$, parameter for a moist porous medium on the heat dispersion profile of linear, quadratic and exponential FGM fins with $Nr=5,Nc=10,\beta =p=\alpha =1,{\theta}_{a}=0.2$.

**Figure 9.**Illustration of variations in power index on temperature distribution of fully wetted longitudinal fin for different thermal conductivities with $Nr=5,Nc=10,\beta ={m}_{2}=\alpha =1,{\theta}_{a}=0.2$. (

**a**) $p=0$, (

**b**) $p=1$, (

**c**) $p=2$.

**Figure 10.**Impact of variations in $\alpha $ and $\beta $ (inhomogeneity index) on thermal profiles of fully wetted longitudinal fin for different thermal conductivities with $Nr=5,Nc=10,{m}_{2}=1,{\theta}_{a}=0.2$. (

**a**) $p=0$, (

**b**) $p=1$, (

**c**) $p=2$.

**Figure 11.**Graphicalillustration of the behaviour of objective function during minimization using the proposed hybrid algorithm for approximate solutions of fully wetted longitudinal fin with (

**a**–

**c**) linear (

**d**–

**f**) quadratic and (

**g**–

**i**) exponential thermal conductivities. Here, red, green and black lines represents the minimum, median and mean values of each case.

**Figure 12.**Boxplot analysis of the MAD values obtained during 100 runs of the proposed algorithm. The red lines shows the median value; the upper and lower quartiles represent the maximum and minimum values during the multiple executions.

**Figure 13.**Convergence of TIC values for linear, quadratic and exponential cases of wetted longitudinal porous fin.

**Table 1.**Examination of the differences between the approximated results and mean square errors obtained by the proposed algorithm with PSO, CSA, GWO, and FFNN-BLM algorithms for thermal distribution of fully wetted longitudinal FGM fin with linear thermal conductivity for ${\theta}_{a}=0.6$, $Nr=5,Nc=10,\beta =p={m}_{2}=\alpha =1$.

Approximate Solution | Mean Square Errors | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{X}$ | PSO | CSA | WOA | GWO | FFNN-BLM | ANN-TTA-SQP | PSO | CSA | WOA | GWO | FFNN-BLM | ANN-TTA-SQP |

0 | 0.999711 | 1.000306 | 0.999667 | 0.999504 | 0.999945 | 1.000001 | $2.275\times {10}^{-8}$ | $5.081\times {10}^{-8}$ | $8.432\times {10}^{-9}$ | $3.743\times {10}^{-8}$ | $1.407\times {10}^{-9}$ | $2.148\times {10}^{-11}$ |

0.1 | 0.887736 | 0.888135 | 0.887723 | 0.887571 | 0.887901 | 0.887954 | $8.253\times {10}^{-7}$ | $1.414\times {10}^{-7}$ | $2.609\times {10}^{-7}$ | $1.404\times {10}^{-6}$ | $1.223\times {10}^{-7}$ | $2.037\times {10}^{-10}$ |

0.2 | 0.817149 | 0.817429 | 0.817140 | 0.817001 | 0.817260 | 0.817297 | $4.688\times {10}^{-6}$ | $9.814\times {10}^{-7}$ | $6.690\times {10}^{-7}$ | $7.370\times {10}^{-6}$ | $8.965\times {10}^{-7}$ | $1.702\times {10}^{-9}$ |

0.3 | 0.770009 | 0.770241 | 0.770031 | 0.769846 | 0.770111 | 0.770156 | $2.351\times {10}^{-6}$ | $8.038\times {10}^{-7}$ | $4.663\times {10}^{-7}$ | $3.568\times {10}^{-6}$ | $7.725\times {10}^{-7}$ | $4.539\times {10}^{-9}$ |

0.4 | 0.737502 | 0.737702 | 0.737539 | 0.737317 | 0.737594 | 0.737637 | $1.765\times {10}^{-6}$ | $1.834\times {10}^{-7}$ | $1.238\times {10}^{-7}$ | $2.665\times {10}^{-6}$ | $2.078\times {10}^{-7}$ | $7.022\times {10}^{-10}$ |

0.5 | 0.714736 | 0.714904 | 0.714770 | 0.714520 | 0.714809 | 0.714842 | $6.404\times {10}^{-7}$ | $2.978\times {10}^{-7}$ | $2.256\times {10}^{-7}$ | $1.286\times {10}^{-6}$ | $3.261\times {10}^{-7}$ | $1.051\times {10}^{-9}$ |

0.6 | 0.698784 | 0.698937 | 0.698823 | 0.698519 | 0.698847 | 0.698884 | $2.715\times {10}^{-6}$ | $4.156\times {10}^{-7}$ | $3.335\times {10}^{-7}$ | $4.165\times {10}^{-6}$ | $5.494\times {10}^{-7}$ | $2.095\times {10}^{-11}$ |

0.7 | 0.687842 | 0.688008 | 0.687907 | 0.687509 | 0.687915 | 0.687972 | $2.980\times {10}^{-7}$ | $3.983\times {10}^{-10}$ | $1.082\times {10}^{-10}$ | $9.148\times {10}^{-8}$ | $2.936\times {10}^{-9}$ | $2.581\times {10}^{-10}$ |

0.8 | 0.680782 | 0.680978 | 0.680884 | 0.680373 | 0.680877 | 0.680957 | $1.521\times {10}^{-6}$ | $5.003\times {10}^{-7}$ | $4.015\times {10}^{-7}$ | $4.100\times {10}^{-6}$ | $5.490\times {10}^{-7}$ | $4.603\times {10}^{-11}$ |

0.9 | 0.676882 | 0.677108 | 0.677015 | 0.676395 | 0.676995 | 0.677087 | $2.711\times {10}^{-6}$ | $4.119\times {10}^{-7}$ | $3.176\times {10}^{-7}$ | $4.277\times {10}^{-6}$ | $5.757\times {10}^{-7}$ | $6.330\times {10}^{-12}$ |

1 | 0.675655 | 0.675904 | 0.675807 | 0.675080 | 0.675777 | 0.675874 | $1.872\times {10}^{-6}$ | $6.511\times {10}^{-7}$ | $6.444\times {10}^{-7}$ | $6.155\times {10}^{-6}$ | $6.027\times {10}^{-7}$ | $1.375\times {10}^{-10}$ |

**Table 2.**Thepercentages of absolute errors in the solutions that were calculated by the ANN-TTA-SQP method for different values of ${\theta}_{a}$ with $Nr=5,Nc=10,\beta =p={m}_{2}=\alpha =1$.

Linear FGM | Quadratic FGM | Exponential FGM | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{X}$ | $\mathbf{0}.\mathbf{2}$ | $\mathbf{0}.\mathbf{4}$ | $\mathbf{0}.\mathbf{6}$ | $\mathbf{0}.\mathbf{8}$ | $\mathbf{0}.\mathbf{2}$ | $\mathbf{0}.\mathbf{4}$ | $\mathbf{0}.\mathbf{6}$ | $\mathbf{0}.\mathbf{8}$ | $\mathbf{0}.\mathbf{2}$ | $\mathbf{0}.\mathbf{4}$ | $\mathbf{0}.\mathbf{6}$ | $\mathbf{0}.\mathbf{8}$ |

0.00 | 0.00005 | 0.00028 | 0.00046 | 0.00077 | 0.00018 | 0.00057 | 0.00013 | 0.00019 | 0.00077 | 0.00021 | 0.00040 | 0.00035 |

0.10 | 0.00006 | 0.00118 | 0.00143 | 0.00450 | 0.00186 | 0.00219 | 0.00116 | 0.00064 | 0.00035 | 0.00042 | 0.00056 | 0.00317 |

0.20 | 0.00093 | 0.00531 | 0.00413 | 0.00999 | 0.00538 | 0.00309 | 0.00361 | 0.00110 | 0.00020 | 0.00200 | 0.00176 | 0.00963 |

0.30 | 0.00242 | 0.00696 | 0.00674 | 0.00665 | 0.00858 | 0.00528 | 0.00315 | 0.00167 | 0.00169 | 0.00436 | 0.00437 | 0.00989 |

0.40 | 0.00531 | 0.00040 | 0.00265 | 0.00336 | 0.00184 | 0.00312 | 0.00179 | 0.00118 | 0.00057 | 0.00351 | 0.00475 | 0.00268 |

0.50 | 0.00460 | 0.00257 | 0.00324 | 0.00206 | 0.00475 | 0.00372 | 0.00220 | 0.00149 | 0.00320 | 0.00235 | 0.00353 | 0.00651 |

0.60 | 0.00243 | 0.00220 | 0.00046 | 0.00516 | 0.00208 | 0.00098 | 0.00277 | 0.00072 | 0.00085 | 0.00272 | 0.00333 | 0.00497 |

0.70 | 0.00680 | 0.00163 | 0.00161 | 0.00362 | 0.00192 | 0.00347 | 0.00005 | 0.00017 | 0.00320 | 0.00320 | 0.00484 | 0.00193 |

0.80 | 0.00265 | 0.00210 | 0.00068 | 0.00343 | 0.00431 | 0.00159 | 0.00258 | 0.00118 | 0.00302 | 0.00112 | 0.00091 | 0.00535 |

0.90 | 0.01247 | 0.00690 | 0.00025 | 0.00864 | 0.00267 | 0.00162 | 0.00190 | 0.00288 | 0.00119 | 0.00409 | 0.00593 | 0.00196 |

1.00 | 0.00668 | 0.00535 | 0.00117 | 0.00651 | 0.00460 | 0.00030 | 0.00227 | 0.00382 | 0.00353 | 0.00231 | 0.00432 | 0.00367 |

**Table 3.**Statistical analysis of the performance indicators (i.e., fitness value, MAD, TIC, ENSE) for different cases obtained during multiple executions of the designed approach.

FGM | Objective Value | MAD | TIC | RMSE | ENSE | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\theta}}_{\mathit{a}}$ | Min. | Avg. | Min. | Avg. | Min. | Avg. | Min. | Avg. | Min. | Avg. | |

Linear | 0.2 | $3.15593\times {10}^{-8}$ | $5.23053\times {10}^{-5}$ | $3.48355\times {10}^{-6}$ | $9.05907\times {10}^{-5}$ | $2.23690\times {10}^{-6}$ | $3.84823\times {10}^{-5}$ | $5.69178\times {10}^{-6}$ | $9.78890\times {10}^{-5}$ | $4.52603\times {10}^{-9}$ | $8.41693\times {10}^{-6}$ |

0.4 | $1.73192\times {10}^{-8}$ | $2.63399\times {10}^{-5}$ | $7.26234\times {10}^{-7}$ | $8.60390\times {10}^{-5}$ | $4.51628\times {10}^{-7}$ | $3.19952\times {10}^{-5}$ | $1.31699\times {10}^{-6}$ | $9.32698\times {10}^{-5}$ | $3.16279\times {10}^{-10}$ | $1.52964\times {10}^{-5}$ | |

0.6 | $8.69232\times {10}^{-9}$ | $4.39994\times {10}^{-6}$ | $1.10382\times {10}^{-6}$ | $5.75917\times {10}^{-5}$ | $4.36155\times {10}^{-7}$ | $1.94033\times {10}^{-5}$ | $1.43176\times {10}^{-6}$ | $6.36855\times {10}^{-5}$ | $1.35244\times {10}^{-9}$ | $9.66485\times {10}^{-6}$ | |

0.8 | $3.48950\times {10}^{-8}$ | $1.52920\times {10}^{-5}$ | $3.46206\times {10}^{-6}$ | $6.72115\times {10}^{-5}$ | $1.13085\times {10}^{-6}$ | $2.11378\times {10}^{-5}$ | $4.22472\times {10}^{-6}$ | $7.89558\times {10}^{-5}$ | $4.36061\times {10}^{-8}$ | $4.49682\times {10}^{-5}$ | |

Quadratic | 0.2 | $5.72905\times {10}^{-9}$ | $1.77032\times {10}^{-5}$ | $3.31079\times {10}^{-6}$ | $3.96153\times {10}^{-5}$ | $1.64992\times {10}^{-6}$ | $1.81133\times {10}^{-5}$ | $4.12421\times {10}^{-6}$ | $4.52704\times {10}^{-5}$ | $3.80597\times {10}^{-9}$ | $1.76396\times {10}^{-6}$ |

0.4 | $8.44298\times {10}^{-9}$ | $4.85792\times {10}^{-6}$ | $7.60630\times {10}^{-7}$ | $4.73085\times {10}^{-5}$ | $3.37606\times {10}^{-7}$ | $1.93106\times {10}^{-5}$ | $9.72378\times {10}^{-7}$ | $5.56096\times {10}^{-5}$ | $3.22981\times {10}^{-10}$ | $4.01069\times {10}^{-6}$ | |

0.6 | $5.54459\times {10}^{-9}$ | $2.70528\times {10}^{-6}$ | $1.10470\times {10}^{-6}$ | $4.14064\times {10}^{-5}$ | $4.27882\times {10}^{-7}$ | $1.48472\times {10}^{-5}$ | $1.39391\times {10}^{-6}$ | $4.83595\times {10}^{-5}$ | $1.26696\times {10}^{-9}$ | $7.29760\times {10}^{-6}$ | |

0.8 | $3.34359\times {10}^{-9}$ | $1.10636\times {10}^{-5}$ | $1.64112\times {10}^{-6}$ | $4.71864\times {10}^{-5}$ | $5.47044\times {10}^{-7}$ | $1.53394\times {10}^{-5}$ | $2.03774\times {10}^{-6}$ | $5.71307\times {10}^{-5}$ | $9.30882\times {10}^{-9}$ | $2.82836\times {10}^{-5}$ | |

Exponential | 0.2 | $4.92075\times {10}^{-9}$ | $5.62125\times {10}^{-5}$ | $7.47090\times {10}^{-6}$ | $5.71984\times {10}^{-5}$ | $3.94914\times {10}^{-6}$ | $2.47741\times {10}^{-5}$ | $1.00903\times {10}^{-5}$ | $6.32870\times {10}^{-5}$ | $2.14259\times {10}^{-8}$ | $3.41590\times {10}^{-6}$ |

0.4 | $8.20482\times {10}^{-9}$ | $2.82506\times {10}^{-6}$ | $1.66266\times {10}^{-6}$ | $3.57714\times {10}^{-5}$ | $8.53873\times {10}^{-7}$ | $1.37292\times {10}^{-5}$ | $2.49717\times {10}^{-6}$ | $4.01484\times {10}^{-5}$ | $1.70630\times {10}^{-9}$ | $2.04150\times {10}^{-6}$ | |

0.6 | $1.47040\times {10}^{-8}$ | $2.24315\times {10}^{-6}$ | $8.75080\times {10}^{-7}$ | $3.81599\times {10}^{-5}$ | $3.23961\times {10}^{-7}$ | $1.29419\times {10}^{-5}$ | $1.06533\times {10}^{-6}$ | $4.25542\times {10}^{-5}$ | $8.73426\times {10}^{-10}$ | $4.80864\times {10}^{-6}$ | |

0.8 | $3.24693\times {10}^{-8}$ | $1.69856\times {10}^{-5}$ | $1.74177\times {10}^{-6}$ | $4.71163\times {10}^{-5}$ | $5.11451\times {10}^{-7}$ | $1.52643\times {10}^{-5}$ | $1.91190\times {10}^{-6}$ | $5.70505\times {10}^{-5}$ | $1.12716\times {10}^{-8}$ | $3.52692\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Alhakami, H.; Khan, N.A.; Sulaiman, M.; Alhakami, W.; Baz, A.
On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach. *Entropy* **2022**, *24*, 1280.
https://doi.org/10.3390/e24091280

**AMA Style**

Alhakami H, Khan NA, Sulaiman M, Alhakami W, Baz A.
On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach. *Entropy*. 2022; 24(9):1280.
https://doi.org/10.3390/e24091280

**Chicago/Turabian Style**

Alhakami, Hosam, Naveed Ahmad Khan, Muhammad Sulaiman, Wajdi Alhakami, and Abdullah Baz.
2022. "On the Computational Study of a Fully Wetted Longitudinal Porous Heat Exchanger Using a Machine Learning Approach" *Entropy* 24, no. 9: 1280.
https://doi.org/10.3390/e24091280