# Developing Predictive Equations for Water Capturing Performance and Sediment Release Efficiency for Coanda Intakes Using Artificial Intelligence Methods

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}values for both the WCP and the SRE. Results showed outperformance of the empirical equations against those of MNLR. Sensitivity analysis carried out by the ANNs revealed that the geometric parameters of the intake were comparably more sensitive than the flow characteristics.

## 1. Introduction

## 2. Methods and Methodology

#### 2.1. Experimental Setup and Data

#### 2.2. Multicollinearity Analysis

#### 2.3. Genetic Algorithm (GA)

#### 2.4. Artificial Neural Networks (ANNs)

_{i}) are passed on to neurons at the hidden layer as X

_{i}V

_{ij}. These neurons, in turn, sum the weighted input as ${\mathrm{net}}_{\mathrm{j}}={\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{X}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{ij}}\right)$ which is then transferred by a nonlinear transfer function (mostly, the sigmoid or the tangent hyperbolic function) to produce an output. This response can become an input for other neurons located in the next layer. This is continued until the network produces an output (

**Z**). The propagation from input to hidden to output neurons is called a forward pass.

_{model}**V**,

_{ij}**W**) that can produce an output vector

_{ij}**Z**= (z

_{1}, z

_{2},..., z

_{p}) as near as possible to target output vector

**T**= (t

_{1}, t

_{2},...,t

_{p}). For each input pattern (p), the network produces an output and the related error is computed and then the total error (E

_{total}) is obtained by summing each error as follows [25]:

_{p}is the model produced output for p-pattern, t

_{p}is the target (actual) output for p-pattern, and N is the number of training patterns.

_{total}):

#### 2.5. Constructing Empirical Equations

_{1}is a coefficient and a

_{1}, a

_{2}, a

_{3}, a

_{4}, and a

_{5}are exponents whose optimal values are obtained by the GA. The optimal values of c

_{1}, a

_{1}, a

_{2}, a

_{3}, and a

_{4}are searched within the positive range while the search space covered a wide range (positive to negative) for a

_{5}since there is no clear variation pattern between L/R and WCP, as more details are given in the next section.

_{2}is a coefficient and b

_{1}, b

_{2}, b

_{3}, b

_{4}, and b

_{5}are exponents. The optimal values of c

_{2}, b

_{1}, b

_{2}, and b

_{3}are obtained by the GA, searching values within the positive range, while the search space covered a wide range (positive to negative) for b

_{4}and b

_{5}since there is no clear variation pattern between these parameters and SRE. More details are given in the next section.

## 3. Results

#### 3.1. GA-Based Empirical Equations

_{1}and c

_{2}was set to 1–200 and for the exponents of a

_{5}, b

_{4}, and b

_{5}the search space was set to −3 and +3 while it was set to 0–3 for the other exponents. The obtained optimal values of the coefficients are shown in Table 6. Figure 8 and Figure 9 present the validation and calibration stages for WCP and SRE, respectively. As seen, predictions are satisfactory for both cases for which the related error measures are summarized in Table 7.

#### 3.2. MNLR Based Empirical Equations

#### 3.3. ANN Predictions

^{2}values.

#### 3.4. Sensitivity Analysis by ANNs

## 4. Discussion

^{2}values of 0.88 (calibration) and 0.87 (validation). When its success is compared against that of the MNLR equation (Equation (12)), it is clearly seen that the empirical equation outperformed the MNLR one, which produces relatively low R

^{2}= 0.70 and high errors of MAE = 6.95, RMSE = 9.12, at the calibration stage and MAE = 6.28, RMSE = 8.48, R

^{2}= 0.77 at the validation stage (see Table 14). The GA-based empirical model produced comparable results against those of the ANN, which is a very powerful soft computing method for nonlinear problems. Although ANN produced less errors and high R

^{2}values, as seen in Table 14, they do not yield any mathematical equation, as opposed to the empirical one.

^{2}values of 0.75 (calibration) and 0.80 (validation). When its success was compared against that of the MNLR equation (Equation (13)), it was seen that the empirical equation outperformed the MNLR one, which had produced high errors and very low R

^{2}values of 0.22 (calibration) and 0.17 (validation) (see Table 15). It may be stated that the MNLR method had produced a worse performance for SRE than for WCP. The GA-based empirical model produced comparable results against those of the ANN, as seen in Table 15. However, as it is pointed out above, ANNs do not accomplish any mathematical relations between the dependent and independent variables. They are, as it is pointed out in the literature, very powerful interpolators but poor extrapolators and they are black box models [25]. The empirical models, on the other hand, can be easily employed for both the interpolation and extrapolation purposes.

## 5. Summary and Conclusions

^{2}values. Yet, the ANN cannot accomplish neither any mathematical relations nor can they be used for the extrapolation purpose, unlike the empirical equations. The sensitivity analysis results carried out by the ANNs reveal that the geometric characteristics parameters of Coanda intakes are comparably more sensitive than the flow ones for the WCP. Similarly, both the geometric and sediment parameters are more sensitive than the flow characteristics in the case of the SRE.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 14.**Summary of statistics of predicted WCP values by Taylor Diagram (black contours: Pearson correlation coefficient; green contours: centered RMSE; blue contours: standard deviation).

**Figure 15.**Summary of statistics of predicted SRE values by Taylor Diagram (black contours: Pearson correlation coefficient; green contours: centered RMSE; blue contours: standard deviation).

Screen Type | Sloth Width (mm) | Curvature Radius (mm) | Void Ratio (e/a) |
---|---|---|---|

Coanda R800 (1) | 1 | 800 | 0.046 |

Coanda R800 (2) | 2 | 800 | 0.092 |

Coanda R800 (3) | 3 | 800 | 0.138 |

Coanda R1200 (1) | 1 | 1200 | 0.046 |

Coanda R1200 (2) | 2 | 1200 | 0.092 |

Coanda R1600 (1) | 1 | 1600 | 0.046 |

Data Sets | WCP (Q_{in}/Q_{div}) | SRE (S_{in}/S_{re)} |
---|---|---|

Maximum | 100 | 90.4 |

Minimum | 38.7 | 0.3 |

Range | 61.3 | 90.0 |

Mean | 70.3 | 52.7 |

St. Deviation | 16.8 | 26.0 |

Dimensionless Parameters | Description |
---|---|

ϴ | Screen Slope (degree) |

L/R | Net screen length/Screen curvature radius |

m = e/a | Bar openings area/Total screen area |

e/R | Bar opening/Curvature radius |

${\mathrm{Fr}}_{\left(\mathrm{R}\right)}=\frac{\mathrm{V}}{\sqrt{\mathrm{gR}}}$ | Froude number based on screen curvature radius |

${\mathrm{We}}_{\left(\mathrm{R}\right)}=\frac{{\mathsf{\rho}\mathrm{V}}^{2}\mathrm{R}}{\mathsf{\sigma}}$ | Weber number based on screen curvature radius |

D_{50}/R | Median of the sediment diameter/Flow depth at beginning of the screen |

D_{50}/e | Median of the sediment diameter/Bar opening |

WCP (Q_{in}/Q_{div}) | Water capturing performance |

SRE (S_{in}/S_{re}) | Sediment release efficiency |

Independent Parameters | VIF Value |
---|---|

ϴ (Screen Slope) | 1.00 |

m = (e/a) | 1.18 |

L/R | 6.67 |

Froude | 5.56 |

Weber | 6.25 |

Independent Parameters | VIF Value |
---|---|

ϴ (Screen Slope) | 1.00 |

L/R | 5.88 |

Froude | 5.88 |

Weber | 6.25 |

D50/e | 1.11 |

Parameter | c_{1} | c_{2} | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

WCP | 151.8 | - | 0.12 | 0.15 | 0.39 | 0.07 | 0.15 | - | - | - | - | - |

SRE | - | 56.5 | - | - | - | - | - | 0.26 | 0.81 | 0.04 | −0.21 | 0.60 |

GA | Calibration Stage | Validation Stage | ||
---|---|---|---|---|

WCP | SRE | WCP | SRE | |

MAE | 4.32 | 10.32 | 4.71 | 10.77 |

RMSE | 5.72 | 13.18 | 6.15 | 12.99 |

R^{2} | 0.88 | 0.75 | 0.87 | 0.80 |

Parameter | c_{1} | c_{2} | c_{3} | c_{4} | c_{5} | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} |
---|---|---|---|---|---|---|---|---|---|---|

WCP | 140.09 | 10.39 | 7.54 | 215.73 | 19.96 | −30.15 | 7.83 | −0.77 | −0.21 | 2.47 |

Parameter | d_{1} | d_{2} | d_{3} | d_{4} | d_{5} | b_{1} | b_{2} | b_{3} | b_{4} | b_{5} |
---|---|---|---|---|---|---|---|---|---|---|

SRE | 1.83 | 21.00 | −217.90 | 51.67 | 0.82 | 0.85 | 20.00 | −9.81 | 0.47 | 0.43 |

MNLR | Calibration Stage | Validation Stage | ||
---|---|---|---|---|

WCP | SRE | WCP | SRE | |

MAE | 6.95 | 18.04 | 6.28 | 20.14 |

RMSE | 9.12 | 24.00 | 8.48 | 25.98 |

R^{2} | 0.70 | 0.22 | 0.77 | 0.17 |

ANN | Training Stage | Testing Stage | ||
---|---|---|---|---|

WCP | SRE | WCP | SRE | |

MAE | 0.75 | 1.47 | 3.39 | 4.32 |

RMSE | 1.08 | 2.37 | 4.30 | 5.37 |

R^{2} | 0.99 | 0.99 | 0.94 | 0.96 |

Independent Parameters | MAE | RMSE | R^{2} |
---|---|---|---|

θ (Screen Slope) | 4.259 | 5.475 | 0.892 |

M = (e/a) | 4.670 | 6.132 | 0.865 |

L/R | 1.220 | 1.826 | 0.988 |

Froude | 1.353 | 1.886 | 0.987 |

Weber | 1.209 | 1.666 | 0.990 |

Independent Parameters | MAE | RMSE | R^{2} |
---|---|---|---|

θ (Screen Slope) | 7.408 | 9.304 | 0.871 |

L/R | 2.857 | 3.464 | 0.982 |

Froude | 2.511 | 3.249 | 0.984 |

Weber | 2.134 | 2.853 | 0.988 |

D50/e | 9.033 | 11.24 | 0.812 |

WCP | Calibration Stage | Validation Stage | ||||
---|---|---|---|---|---|---|

GA | MNLR | ANN | GA | MNLR | ANN | |

MAE | 4.32 | 6.95 | 0.75 | 4.71 | 6.28 | 3.39 |

RMSE | 5.72 | 9.12 | 1.08 | 6.15 | 8.48 | 4.30 |

R^{2} | 0.88 | 0.70 | 0.99 | 0.87 | 0.77 | 0.94 |

SRE | Calibration Stage | Validation Stage | ||||
---|---|---|---|---|---|---|

GA | MNLR | ANN | GA | MNLR | ANN | |

MAE | 10.32 | 18.04 | 1.47 | 10.77 | 20.14 | 4.32 |

RMSE | 13.18 | 24.00 | 2.37 | 12.99 | 25.98 | 5.37 |

R^{2} | 0.75 | 0.22 | 0.99 | 0.80 | 0.17 | 0.96 |

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**MDPI and ACS Style**

Hazar, O.; Tayfur, G.; Elçi, S.; Singh, V.P.
Developing Predictive Equations for Water Capturing Performance and Sediment Release Efficiency for Coanda Intakes Using Artificial Intelligence Methods. *Water* **2022**, *14*, 972.
https://doi.org/10.3390/w14060972

**AMA Style**

Hazar O, Tayfur G, Elçi S, Singh VP.
Developing Predictive Equations for Water Capturing Performance and Sediment Release Efficiency for Coanda Intakes Using Artificial Intelligence Methods. *Water*. 2022; 14(6):972.
https://doi.org/10.3390/w14060972

**Chicago/Turabian Style**

Hazar, Oğuz, Gokmen Tayfur, Sebnem Elçi, and Vijay P. Singh.
2022. "Developing Predictive Equations for Water Capturing Performance and Sediment Release Efficiency for Coanda Intakes Using Artificial Intelligence Methods" *Water* 14, no. 6: 972.
https://doi.org/10.3390/w14060972