# Multigene Genetic-Programming-Based Models for Initial Dilution of Laterally Confined Vertical Buoyant Jets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Analysis of a Laterally Confined Vertical Buoyant Jet

^{2}u

_{j}, M = u

_{j}Q, and B = g’Q, where D is the port diameter, u

_{j}is the initial jet velocity, and g’ is the modified gravitational acceleration. Following a dimensional analysis, characteristic variables can be typically defined as functions of D and the densimetric Froude number, Fr [1,10,19,30,31]. Fr characterizes the relative importance of inertia over buoyancy as:

_{a}is the density of ambient water, and ρ

_{j}is the initial jet density.

_{a})/(C

_{j}− C

_{a}), where C is the concentration at a point, C

_{a}is the concentration of the ambient water, and C

_{j}is the initial jet concentration. Because the dilution properties are mainly affected by Fr and the confinement, the dimensionless concentration can be expressed as a function of the dimensionless confinement diameter, D

_{r}/D, dimensionless confinement height, H

_{r}/D, and dimensionless cross section, Z/D, as:

_{c}is the centerline concentration at a cross section, r is the radial distance of a point from the centerline, and b

_{gc}is the half concentration width. Therefore, the concentration field can be easily estimated if C

_{c}is known, and thus the key parameter characterizing the dilution properties is the jet centerline concentration. In order words, developing an MGGP model for C

_{c}is significant because estimating C

_{c}from the flow and geometrical variables is quite challenging, but in this case developing an MGGP model for the concentration distribution is not necessary because it can be efficiently estimated using Equation (4) after C

_{c}is calculated. Therefore, the present study selected the jet centerline concentration as the output variable.

_{r}/D and H

_{r}/D, can be combined, and the combined parameter is referred to as the confinement index, β, which is defined by β = (H

_{r}/D)/[(D

_{r}/D)−1]. Therefore, the dimensionless jet centerline concentration can be expressed as:

#### 2.2. Experimental Data Sets

#### 2.3. The MGGP Method

_{1}and Gene

_{2}in the first generation shown in Figure 2 are

_{1}and x

_{2}are the input variables.

_{2}) and 2.1x

_{1}of the parent genes are exchanged, and child genes (the second generation) are formed, which can be expressed as:

_{1}is replaced by cos(x

_{2}), forming a new form of Gene 1 (in the third generation) as:

_{1}and α

_{2}are the weights of Genes 1 and 2, respectively; and γ is the bias term. These weights and the bias term are determined using an ordinary least-squared method.

#### 2.4. MGGP Modeling Setup

## 3. Results and Discussion

#### 3.1. Pareto-Optimal MGGP Models

_{1}-α

_{5}denote the weighting coefficients of the genes, and γ is the bias term.

#### 3.2. General Observations

#### 3.3. Quantitative Assessment of Model Performances

^{2}) values are also shown in the same plots. The training data points were used to evolve and select the MGGP models, and the testing data points were employed to serve as unseen data points for assessing the predictive capacity of the evolved models. For the training period, the results shown in Figure 5 are consistent with the observations in Figure 3a; namely, Model A (RMSE = 0.037, R

^{2}= 0.968) had the best whereas Model C (RMSE = 0.094, R

^{2}= 0. 798) had the worst performance among the models on the Pareto front in fitting the training data sets for dimensionless jet centerline concentration.

^{2}= 0.956) and Model B (RMSE = 0.039, R

^{2}= 0.957) were almost identical. Because of the lower complexity, Model B could be more favorable than Model A if model simplicity is of great importance. The fitting indices of Models A and B for the testing data sets were only slightly different from those for the training data sets, demonstrating that the risks of over-fitting were well controlled. Model C is much simpler than Models A and B, as it only has two genes, but its performance in predicting the dimensionless jet centerline concentration was much poorer at the same time (RMSE = 0.082, R

^{2}= 0.808). This confirms that Model C was under-trained and thus showed consistently poor performance in both the training and testing periods.

^{2}values indicated that both Models A and B had less than 5% error in predicting the data sets. Figure 6c shows that the symbols for Model C are distributed in a wider range and the model had about 20% error in the predictions, revealing the poor performance of Model C.

#### 3.4. Comparison with the SGGP Model

^{2}values are also reported. For the training data sets, the predictions of the best SGGP model (RMSE = 0.068, R

^{2}= 0.895) were obviously less accurate than the MGGP Models A and B but more accurate than the MGGP Model C. In terms of the testing data sets, the RMSE and R

^{2}values in the SGGP predictions were 0.063 and 0.888, respectively, which also demonstrated that the performance of the SGGP model was better than the MGGP Model C but worse than the MGGP Models A and B.

^{2}values compared with the MGGP Model C indicate that an SGGP-based model could have higher prediction capability than an MGGP model that is over simplified.

#### 3.5. Comparison with the Existing Empirical Equation

^{2}values of the empirical results (RMSE = 0.068, R

^{2}= 0.895) were almost identical to those of the SGGP (RMSE = 0.067, R

^{2}= 0.894) results for the entire data sets. A closer examination of the evolved SGGP model and the empirical formulation shows that the two models have a very similar model structure. These observations demonstrated that the SGGP algorithm can easily evolve an explicit model that is as accurate as a regression-based empirical model which normally requires extensive analyses and efforts. The mean absolute errors (MAEs) were also calculated, and the values corresponding to the MGGP, the SGGP, and the empirical models were 0.027, 0.044, and 0.046, respectively. The lower MAE and RMSE and higher R

^{2}values of the MGGP predictions also demonstrated the generalization capacity of the MGGP model.

#### 3.6. Prediction Confidence Analysis

## 4. Conclusions

^{2}= 0.968) and testing data sets (RMSE = 0.039, R

^{2}= 0.956). Another candidate and less complex MGGP model (Model B) was also found to be accurate in predicting the experimental data (training: RMSE = 0.041, R

^{2}= 0.962; testing: RMSE = 0.039, R

^{2}= 0.957) and may be preferable when model simplicity is of major importance. The best MGGP model had lower errors and higher correlations in fitting the entire data sets (MAE = 0.027, RMSE = 0.038, R

^{2}= 0.966) than the best SGGP model (MAE = 0.044, RMSE = 0.067, R

^{2}= 0.894) and the existing empirical model (MAE = 0.046, RMSE = 0.068, R

^{2}= 0.895). The results of nonlinear regression prediction confidence interval analysis revealed that the mean 95% confidence interval half-width of Model A was 0.093. These results and observations are encouraging. Therefore, the MGGP technique will be applied to other effluent mixing problems in further work.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The Pareto front plot of the evolved MGGP-based models and convergence of the solutions: (

**a**) Pareto front plot; and (

**b**) convergence of the solutions.

**Figure 4.**Actual and predicted concentration profiles: (

**a**) Fr = 2.8; β = 7.7; (

**b**) Fr = 4.0; β = 5.7; (

**c**) Fr = 6.5; β = 4.3; (

**d**) Fr = 7.9; β = 3.7; (

**e**) Fr = 9.9; β = 12.3; and (

**f**) Fr = 12.7; β = 9.2.

**Figure 6.**Scatter plot of the actual and MGGP predicted results: (

**a**) Model A; (

**b**) Model B; and (

**c**) Model C.

**Figure 7.**The Pareto front plot of the evolved SGGP-based models and convergence of the solutions: (

**a**) Pareto front plot; and (

**b**) convergence of the solutions.

**Figure 8.**The actual and SGGP results: (

**a**) results at all the data points; and (

**b**) scatter plots of the actual and predicted results.

Model | Equations | ||
---|---|---|---|

A (best) | α_{1}Gene1 | = | (7.61 × 10^{14}log(0.114Fr^{β} β^{2}))/(7.04 × 10^{13} Fr + 3.52 × 10^{13} (Z/D)^{2} + 7.91 × 10^{15}) |

α_{2}Gene2 | = | (3.31 × 10^{12} β)/(5.0 × 10^{11} Fr + 2.5 × 10^{11} (Z/D)^{2} + 2.81 × 10^{13}) | |

α_{3}Gene3 | = | −0.0536 cos(log(β exp(β))) | |

α_{4}Gene4 | = | −0.0193 log(β^{(β − 2.0)} exp(−β)) | |

α_{5}Gene5 | = | 0.04 cos(β^{2}) | |

γ | = | −0.0775 | |

B | α_{1}Gene1 | = | (1.86 × 10^{15} (log(Fr^{β}β^{3})−2.17))/(2.11 × 10^{14}Fr + 7.04 × 10^{13} (Z/D)^{2} + 1.58 × 10^{16}) |

α_{2}Gene2 | = | −0.0495 cos(log(β exp(β))) | |

α_{3}Gene3 | = | 0.0351 cos(β^{2}) | |

α_{4}Gene4 | = | −0.015 log(β^{β}exp(−β)) | |

γ | = | −0.0409 | |

C | α_{1}Gene1 | = | (8.94 × 10^{32} β)/(3.75 × 10^{31} Fr + 2.5 × 10^{31} (Z/D)^{2} + 2.89 × 10^{33}) |

α_{2}Gene2 | = | 0.0101 cos(β^{2}) | |

γ | = | −0.0111 |

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**MDPI and ACS Style**

Yan, X.; Mohammadian, A. Multigene Genetic-Programming-Based Models for Initial Dilution of Laterally Confined Vertical Buoyant Jets. *J. Mar. Sci. Eng.* **2019**, *7*, 246.
https://doi.org/10.3390/jmse7080246

**AMA Style**

Yan X, Mohammadian A. Multigene Genetic-Programming-Based Models for Initial Dilution of Laterally Confined Vertical Buoyant Jets. *Journal of Marine Science and Engineering*. 2019; 7(8):246.
https://doi.org/10.3390/jmse7080246

**Chicago/Turabian Style**

Yan, Xiaohui, and Abdolmajid Mohammadian. 2019. "Multigene Genetic-Programming-Based Models for Initial Dilution of Laterally Confined Vertical Buoyant Jets" *Journal of Marine Science and Engineering* 7, no. 8: 246.
https://doi.org/10.3390/jmse7080246