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Analytical Investigation of the Impact of Jet Geometry on Aeration Effectiveness Using Soft Computing Techniques
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Analytical Investigation of the Impact of Jet Geometry on Aeration Effectiveness Using Soft Computing Techniques
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  • Diksha Puri
    Diksha Puri
    School of Environmental Science, Shoolini University, Solan, Himachal Pradesh 173229, India
    More by Diksha Puri
  • Raj Kumar*
    Raj Kumar
    Department of Mechanical Engineering, Gachon University, Seongnam 13120, South Korea
    *Email: [email protected]
    More by Raj Kumar
  • Parveen Sihag
    Parveen Sihag
    Department of Civil Engineering, Chandigarh University, Mohali, Punjab 140301, India
  • Mohindra Singh Thakur
    Mohindra Singh Thakur
    Department of Civil Engineering, Shoolini University, Solan, Himachal Pradesh 173229, India
  • Kahkashan Perveen
    Kahkashan Perveen
    Department of Botany & Microbiology, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia
  • Faisal M. Alfaisal
    Faisal M. Alfaisal
    Department of Civil Engineering, College of Engineering, King Saud University, Riyadh 11495, Saudi Arabia
  • Daeho Lee*
    Daeho Lee
    Department of Mechanical Engineering, Gachon University, Seongnam 13120, South Korea
    *Email: [email protected]
    More by Daeho Lee
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ACS Omega

Cite this: ACS Omega 2023, 8, 35, 31811–31825
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https://doi.org/10.1021/acsomega.3c03294
Published August 22, 2023

Copyright © 2023 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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Jet aeration is a commonly used technique for introducing air into water during wastewater treatment. In this investigation, the efficacy of different soft computing models, namely, Random Forest, Reduced Error Pruning Tree, Artificial Neural Network (ANN), Gaussian Process, and Support Vector Machine, was examined in predicting the aeration efficiency (E20) of circular and square jet configurations in an open channel flow. A total of 126 experimental data points were utilized to develop and validate these models. To assess the models’ performance, three goodness-of-fit parameters were employed: correlation coefficient (CC), root-mean-square error (RMSE), and mean absolute error (MAE). The analysis revealed that all of the developed models exhibited predictive capabilities, with CC values surpassing 0.8. Nonetheless, when it comes to predicting E20, the ANN model outperformed other soft computing models, achieving a CC of 0.9748, MAE of 0.0164, and RMSE of 0.0211. A sensitivity analysis emphasized that the angle of inclination exerted the most significant influence on the aeration in an open channel. Furthermore, the results demonstrated that square jets delivered superior aeration compared to that of circular jets under identical operating conditions.

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Copyright © 2023 The Authors. Published by American Chemical Society

1. Introduction

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Jet aeration is commonly linked to the development of a large air–water contact region as well as minimum energy dissipation, making it a better aerated system, especially in small tanks. There is a general consensus that the facilitation of oxygen transportation is significantly improved when an aeration device is made to create a plunging hollow jet. (1) Aeration is defined as the process of transportation, circulation, and mixing of DO within liquid, and this step is crucial in the secondary treatment of water and wastewater, as it enhances the level of dissolved oxygen, facilitating both chemical and biological oxidation. (2,3) In addition to its many applications in the treatment of wastewater, aeration and other gas transfer activities have a specific place in the management of water quality and play vital roles in both the pollution and self-purification of fresh water. For instance, to increase gas and liquid exchange in environmental sectors, plunging liquid jets can stir up a liquid pool’s surface when they come into contact with it. Larger interfacial regions between gas and liquid can be formed by using this phenomenon. Interfacial shear is one of two complementing processes that often produce air entrainment, which acts at the intermittent region where the air and water mix, shrinks the air boundary layer, and is one of two complementing mechanisms that typically produce air entrainment. The second event, however, occurs when a descending jet collides with a liquid pool, trapping air in the area around it. (4) Air is trapped at the top of the water and squeezed off, creating a huge cavity with a cylindrical bottom. This causes a bubbly plume to flow downstream beneath the impact surface of water. (5) The establishment of a significant air–water interface in the liquid serves as the basis for aeration because it causes atmospheric oxygen to be absorbed from air transported by a high-velocity discharge jet under the free liquid surface. An impinging water jet’s plume of air bubbles interacts with water in a still pool to transfer oxygen mass in the process of oxygenation by plunging jets. (6) The surface area and retention duration of bubbles below the water’s surface affect how much oxygen mass is exchanged to liquid. Smaller bubbles are preferred for oxygen transfer since their biphasic interfacial area per unit volume of gas is inversely related to their size. Oxygen is transferred and distributed in the liquid body through convection and diffusion. Due to their air/gas entrainment properties, plunging jets are common in many circumstances and have been proven to be efficient ways to aerate industrial effluents. (7,8) There are several different aeration techniques, such as surface aeration using eddy jet mixes, mechanical agitators, gas jet aerators, turbine agitators, plunging jet aeration using surface jet aerators, and static tube mixtures. (9) Every technique aims to incorporate oxygen into the healing process as effectively as possible. (10) With up to 50% of the total energy used in a wastewater treatment process, aeration is often the most expensive step. (11) To ensure that the biological wastewater treatment process is effectively aerated, simple basic units are needed. Jet aerators provide these characteristics. (12) Also, compared to conventional systems, the oxygen exchange mechanism using a sputtering jet of water is a better way to add oxygen to a pool of water. (13) Additional uses include cell bioreactor, oxygenation, and mineral-processing flotation cells. (14−16) In another industrial process that involves casting polymers and glass, enhancing the interaction between gas and liquid aeration is crucial for the effective creation of goods. (17) Based on how much electricity is used, an aeration system’s efficiency is measured by how well it can move oxygen into liquid water.
At hydraulic structures like drop shafts, weirs with various forms, stepped channels, chutes, etc., plunging jets are also used for aeration. Such initiatives enhance water quality and oxygen transmission. It was observed in research on vertical plunging hollow jets that KLa20increased with an increase in JV and that thicker jets have higher KLa20 values in all experiments for a given jet velocity. (10) Substantial efforts were also made to discover the influence of the number of jets on KLa20 and OTE when four sets of aerators with 1, 4, 8, and 16 units were investigated. The results indicated that the KLa20 and OTE are notably greater for the multiple plunging jets in air/water systems compared to a single plunging jet with the same flow area and similar conditions. (18) Studies have also been conducted to observe the influence of the nozzle design on oxygenation by plunging jets. The study examined three types of nozzles: circular, elliptical, and rectangular, with rounded edges. Experimental findings indicated distinct flow characteristics and entrainment patterns on the surface of free water jets as well as within the submerged water jet region of the receiving tank. The rectangular nozzle with rounded edges demonstrated higher KLa20 and OTE due to water jet expansion. However, it exhibited a shallower Hp compared with the other nozzles. On the other hand, the ellipse nozzle exhibited the greatest Hp. These results suggest that the ellipse nozzle is suitable for aerating deep pools, while the rectangular nozzle with rounded ends is preferable in applications requiring a high concentration of bubbles. (19) The study has shown that the geometry/shape of a plunging jet has a major impact on the depth of penetration and the total volumetric oxygen transfer coefficient. Although rectangular plunging jets with rounded edges have a significantly higher overall volumetric oxygen transfer coefficient (1.45 times) than circular plunging jets for the same flow area and other comparable conditions, it has been observed that conventional circular plunging jets have the highest penetration depth. (20) Plunging breakers are known to be formed by the action of jets plunging which contribute to wave dissipation and sediment transportation process. (21) In a study, the data was gathered through experiments conducted to examine how the θ of hollow jet aerators affects the oxygen transfer in water. Three different θ values, namely, 30, 45, and 60°, were utilized to generate hollow jets striking the pool water surface. The experimental findings demonstrated a notable influence of the hollow jet θ on the KLa20 at elevated JV. It was observed that the SOTE of 60° hollow jet aerators exhibited better performance with low-powered jets. However, when the kinetic power increased, the 30° hollow jet aerator displayed superior SOTE characteristics. (22) In another study, the impact of several variables such as JT, JV, and JL, and the arrangement of hollow jets to aerate the system have been explored. To acquire data for the KLa20 and SOTE experiments, we employed plunging hollow jets with numerous combinations of parameters. It was found that the KLa20 increases with JT and JV. This is mainly associated with the increased jet momentum which produces high shear and large turbulence in the pool of water. (23) When it comes to oxygenation efficiency, it has been observed that thinner jets are more effective in oxygenation, achieving efficiencies of up to 4.01 kg/kWh. (24,25)
Nowadays, soft computing and multicriteria decision making techniques have generated significant interest in various fields of engineering including hydraulics and water resources. (26−33) Numerous research studies have documented the utilization of soft computing techniques in the field of aeration, highlighting their usefulness in this field. PSO, GA, and hybrid ANN algorithms with HHO models have been extensively employed for studying hydraulic formations, showcasing exceptional efficacy in simulating the scouring depth downstream of a spillway caused by ski-jump flows. (34) A study was also carried out to predict the E20 in Parshall and Venturi flumes using a range of soft computing algorithms, including RF, tree-based M5P, GMDH, and MARS. (35) Their findings demonstrated that the MARS model is the superior predictor. A single plunging jet’s oxygen transfer rate in turbulent crossflow was measured using two distinct models and the complex behavior of the two-phase air–water flow was investigated using the flow visualization technique. (36)
The authors draw the conclusion that soft computing approaches are mostly used in forecasting the aeration features of “closed systems” with nozzle jets, jet thickness, jet length, etc. based on a critical evaluation, as shown in Table 1. However, in the current study, the focus is on creating soft computing-based models to estimate E20 determined by experimental work on various plunging hollow jet aeration system geometries in an open channel. To the best of the authors’ knowledge, α, D, HR, NJ, Fr. No, and jet geometries Jcir and Jsq are not taken into account when relating the E20 characteristics of hollow jets that plunge when these factors are present.
Table 1. Literature Review of Soft Computing Models Applied to Study Jet Aeration
authorsexperimental setupinput parametersjet geometrysoft computing model(s) usedreview
Kumar et al. (37)closed systemJT, JV, JL, WD-MNLR, ANN, ANFIS, MARS, GRNNANFIS using gbellmf and ANN are useful tools for modeling the oxygen transport.
Kumar et al. (38)closed systemJT, JV, JL, WD-MNLR, SVM, GP,SVM with an RBF kernel predicts the tested KLa20 to almost within a 20% scatter and performs well.
Singh et al. (39)closed systemJD, JL,HP, JV, D, P/VCircular, square, rectangular, and rectangular with rounded edge.NN, SVM, GPSVM is the best model.
Bodana et al. (40)closed systemJL, D, HPTapered and cylindrical hollow jets.ANN, GP, MNLRGP_RBF and GP_PUK are the best models for predicting the Hp entrained by the plunging hollow jets.
Onen (41)venturi aerationHP, D, θ GEP, ANN, MNLR, MLRANN is the best model.
Deswal (42)closed systemJV, JD, NJ SVM and GPSVM is the best model.
Kumar et al. (43)closed systemθ, JT, JV, Fr. No, Re. No.conical hollow jets.SVM, MNLR, M5TreeSVM model outperforms the other models to understand the oxygenation by plunging jets.
Present Studyopen channel flowα, JN, D, HR, Fr. No.circle and square hollow jets.RF, REPTree, ANN, GP, and SVMANN is the best model.
Taking into consideration the aforementioned information, the goals of the current investigation are as follows:
  • To assess the efficacy of soft computing models, RF, REPTree, ANN, GP_PUK, and SVM_RBF in understanding the jet aeration having different jet geometries.

  • To analyze the effect of jet geometry on E20.

  • Sensitivity analysis to investigate the impact of each input variable on E20.

2. Overview of Soft Computing

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The following section shows the soft computing techniques employed to predict E20 for the current study. The techniques used are discussed below.

2.1. Random Forest

The RF algorithm was introduced as an ensemble learning technique. (44) This technique trains and forecasts sample data using several decision trees with an identical data distribution. (45) It utilizes the tree structure to represent these rules and solves classification and regression problems. In RF, the Bootstrap approach is data- sets to resample the original data, resulting in the construction of multiple trees, each using a different bootstrap sample. (46,47) The RF approach is widely adopted in hydrology research because of its versatility and outstanding results, particularly in limited data sets. (48,49) The selection of the forest is determined by the votes of each tree. The function of the RF tree model is depicted in Figure 4, illustrating that an RF consists of a collection of classification and regression trees. The RF approach compares favorably with ANN and SVM in terms of simplicity of use and computing speed. (50) The minimal number of model parameters makes it easier to construct, and it is resistant to overfitting and can capture nonlinearity. (51) The working of the RF model tree is illustrated in Figure 1.

Figure 1

Figure 1. Working of the RF-based model.

2.2. Reduced Error Pruning Tree

REPTree begins with the optimal function and then prunes errors using a variance and information gain reduction method. (52) In this method, when a decision tree’s output is enormous, the decision tree is used to streamline the modeling process using training data, and the REPTree is then employed to lessen the complexity of the tree’s structure. As a result of this method, the modeling process is intended to become simpler when large input data are used by reducing the complexity of the modeling process. As it produces an effective subtree, based upon the pruning process, researchers frequently utilize the REPTree approach. (53) Using the information gain ratio, decision trees are split and pruned using two different techniques in REPTree. (54,55) Prepruning, which happens when the tree expansion is stopped during data construction, is the initial stage. This procedure is used when there are fewer occurrences than the training percentage, which shows that this node has not been disaggregated. Postpruning is the second stage, during which the tree is improved until the training set procedure is complete. The prepruning approach yields trees more quickly than the postpruning procedure, according to a comparison of the two. But compared to prepruning, postpruning results in a more accurate tree.

2.3. Artificial Neural Network

ANN, encouraged by human neural networks, is a widely parallelized processor made up of simple processing units known as neurons which are co-operating to address a specific challenge. (56) ANN model attempts to use some “organizational” principles believed to be used in humans. ANN is primarily utilized in sectors connected to information processing because it is their primary role. Numerous ANNs are employed for engineering tasks including pattern recognition, forecasting, and data compression in addition to being utilized to mimic genuine neural networks to investigate behavior and control in animals and machines. Inputs (such as synapses) are essentially multiplied by the weights in these. Weights are the strength of interunit connections that store processing capacity. (57) The weight’s value might be zero, positive, or negative. If the weight is zero, then there is no connection between the two neurons. If the weight is negative, then the signal has been blocked or diminished. To produce the desired output, ANN weights can be adjusted by using algorithms, and this approach of altering weights is called learning. The activation of the neuron is subsequently determined by these weights, which are computed by a mathematical function. This network’s neurons in this network simply add their inputs. The input multiplied by weight will be the output of the input neurons, as they only have one input. Figure 2 depicts W0Wn as the weight and V1Vn as the input variables.

Figure 2

Figure 2. Schematic diagram of ANN technique.

It has been demonstrated that ANN-based approaches can be a useful tool for simulating a variety of engineering systems in the actual world without having to solve intricate algebraic problems. (58,59) The ability to learn and identify the nonlinear relationship among the system inputs and the system outputs makes ANN useful as a black-box model. (60) Instead of calculating intricate mathematical equations, this intelligent technique imitates how the human brain responds to various challenges. (61) ANN has the capacity to generalize since they can handle unseen data faster and simpler than conventional methods after a learning phase with a few measured information sets. As consequently, researchers and scholars in a number of engineering and industrial disciplines, such as modeling and optimization, have been interested in ANN-based methodologies. (62−65)

2.4. Gaussian Process

GP is a nonparametric probabilistic technique based on kernels. It assumes that adjusted information items must share knowledge with one another. (66) Gaussian regression is the generalization of the Gaussian distribution. In the GP regression, the covariance and mean are used to represent a matrix and vectors of a Gaussian distribution. Evaluation of generalization does not need to be performed because functional dependence and data are known in advance. Based on the input test data, the Regression models using GP may distinguish between the predicted dispersion.

2.5. Support Vector Machine

SVM was developed based on statistical learning theory. (67) The primary tenet of the SVM is optimum class separation. From a limitless number of linear classifiers, SVM selects the separable classes with the lowest generalization error, or it establishes a maximum for the error produced by structural risk minimization. As a result, the greatest margin between the two classes may be calculated from the chosen hyperplane by adding the distances of the hyperplane from the points closest to the two classes. Additionally, the SVM approach offers certain advantages over other conventional methods, including a singular solution and the use of a set of multidimensional spaced kernel functions that includes intelligent nonlinear transformation.

2.6. Details on Kernel Function

How to choose a better kernel function is also a study topic because there are several kernel functions in GP and SVM. There are two common kernel functions, nevertheless, that are used for general purposes, which are represented as eqs 1 and 2, respectively.
(1)

RBF

eγ|xixj|2
(1)

(2)

PUK

1[1+(||xixj||2221Ωσ1)2]Ω
(2)
In this case, Gaussian noise, γ, σ, and Ω are kernel parameters. It is generally known that a suitable choice of meta-parameters affects the generalization performance (prediction accuracy) of GP and SVM, Gaussian noise, and γ, σ, and Ω influence the prediction (regression) model complexity.

2.7. Performance Evaluation Parameters

Each model’s performance was evaluated using three statistical metrics: CC, MAE, and RMSE. To assess how tight the linear connection is, the CC value is employed, and it ranges from −1 to 1. (68) The MAE method is used to determine how well the forecast matches the outcomes. It treats each error equally. MAE is used to describe errors that are evenly distributed and have a value between 0 and ∞. (69) When normally distributed errors are described, RMSE is used. It also ranges from 0 to ∞. (70)
This may be computed by using the formulas stated in eqs 35.
CC=i=1N(EiF¯)(EiE¯)i=1N(FiF¯)2i=1N(EiE¯)2
(3)
MAE=1Ni=1N|EiFi|
(4)
RMSE=1Ni=1N(EiFi)2
(5)

3. Methodology

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The models used for this study were trained and validated to assess their ability to predict E20 based on seven input parameters: α, D, NJ, HR, Fr. No, Jsq, and Jcir. Software known as WEKA 3.9 was utilized to create the RF, REPTree, ANN, GP, and SVM employed in this work. For optimizing each model, certain user-defined parameters must be set. The user-defined parameters for each model in the current investigation are listed in Table 2. The experiments for obtaining data sets for E20 as output for each machine learning algorithm were conducted. The data set and experimentation procedure for the investigation are given below.
Table 2. User-Defined Parameters for Applied Soft Computing Models
modelsuser-defined parameters
RF• See d = 2
REPTree• See d = 3
ANN• Learning rate = 0.3
• Hidden Layers = 600
• Momentum = 0.2
• Training Time = 500
• Neurons = 20
GP_PUK• Ω = 4
• σ = 6
• Gaussian Noise = 1
SVM_RBF• γ = 0.008

3.1. Data Set

The research focuses on the examination of E20 of jet formation within individual acrylic sheets positioned in an open channel. Essential variables, including α, ranging from 0 to 3°, D, ranging from 3.41 to 4.75 L/s, and NJ, ranging from 1 to 64, were utilized as input parameters for analysis. Additionally, E20 predictions were provided by utilizing five soft computing strategies: RF, REPTree, ANN, GP_PUK, and SVM_RBF. Utilizing the 14 acrylic sheets, each having NJ of 1,2,4,8,16,32, and 64 (Figure 3a,b), a total data set of 126 experimental readings for E20 (eqs 68) were recorded. The data set is further divided into training (84 readings) and testing (42 readings) sub-data sets. The data set was categorized based on an arbitrary choice. The characteristics of both collections of data sets are listed in Table 3.
Table 3. Statistical Characteristics of Training and Testing Data Sets
 αDNJHRFr. NoJsqJcirE20
Training Data Set
mean1.5001091.10518.1430.6566.0260.5000.5000.221
median1.500453.1258.0000.5225.6200.5000.5000.227
std. dev.1.2321237.20921.3640.4372.1800.5030.5030.071
kurtosis–1.5180.0120.3680.602–1.173–2.049–2.049–0.632
skewness0.0001.1701.3060.7780.2780.0000.000–0.224
minimum0.00053.2811.0000.1732.8310.0000.0000.070
maximum3.0003840.00064.0001.5649.9631.0001.0000.356
Testing Data Set
mean1.5001219.57618.1430.6566.2800.5000.5000.226
median1.500593.7508.0000.5226.5040.5000.5000.227
std. dev.1.2401576.86521.4940.4402.5240.5060.5060.072
kurtosis–1.5381.1500.4700.5630.279–2.103–2.1030.340
skewness0.0001.5771.3300.7921.0550.0000.000–0.082
minimum0.00074.2191.0000.1733.7910.0000.0000.093
maximum3.0004750.00064.0001.56411.8481.0001.0000.370

Figure 3

Figure 3. Models of jet fabricated (a) circular jets and (b) square jets.

3.2. Experimental Procedure

Experiments for the present study were conducted using a tilting flume (Figure 4) with dimensions of 0.45 × 0.25 × 5 m. The channel was equipped with an aeration device, strategically positioned to allow water to pass solely through hollow jets that impinged on the screens. To initiate the experiments, the water tank was filled, and an electric motor with a power of ’2 HP’ was used to circulate the water within the channel. Sodium sulfite (Na2SO3) and a catalyst, cobalt chloride (CoCl2), were mixed in the water tank to deplete oxygen in the water sample. A water sample was then collected from upstream of the aeration device, and the initial concentration of dissolved oxygen (Cn) was determined using the azide modification. (71) Subsequently, water circulation through the imping jets of the aeration device was allowed for a specified duration (time, t = 2 min). After the designated time had elapsed, the aerated/oxygenated water sample was collected from downstream of the aeration device to measure the dissolved oxygen concentration (Cm). Throughout the experiment, the water temperature (T) was measured using a digital thermometer. The value of E20 was calculated using eqs 68. The methodology adopted for this study is depicted in schematic form in Figure 5.

Figure 4

Figure 4. Experimental setup

Figure 5

Figure 5. Flowchart of the methodology.

To determine E20, the following equations were used. (72)
E=CnCmCS*Cn=1CS*CmCS*Cn=11r
(6)
where Cn, Cm, CS*, and r are the oxygen level in upstream, downstream locations, saturated oxygen concentration, and oxygen aeration deficient ratio, respectively.
When all of the oxygen is transported to the water, E20 is one, whereas a zero number means that no oxygen can be transmitted. The findings from various temperature trials are normalized at 20 °C using the subsequent formula to preserve consistency in measured experiments. (73)
1E20=(1E)1/f
(7)
The oxygen aeration efficiency (E) at a water temperature other than 20 °C may be calculated using the following time-independent formula, (73) where E20 is the oxygen transfer efficiency at 20 °C and f is the aeration exponent.
f=1+2.1×102(T20)+8.25×105(T20)2
(8)

4. Results

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The results obtained from the analysis performed by applying various soft computing techniques are described below.

4.1. Assessment of RF- and REPTree-Based Model

REPTree predictive results were obtained from training and testing data sets. The CC, RMSE, and MAE were subsequently determined on the basis of eqs 35. For the RF model, the training data prediction is higher than the testing data set prediction. The results of the evaluation parameters as listed in Table 4 show that CC values for training and testing data are found to be 0.9951 and 0.956, respectively. For the REPTree model, the testing data CC value of 0.8797 shows marginally good results. It is observed that the RF model performs better than REPTree in training and testing data sets. Figures 6 and 7 also display the outcomes of the RF and REPTree model training and testing. It has been found that the RF model predicts E20 more precisely than the REPTree model. Additionally, it can be observed that in the RF model (Figure 6a,b), the points are near the line of agreement; however, in the REPTree model, the points are more dispersed (Figure 7a,b). REPTree is less aggressively competitive than RF.
Table 4. Performance Assessment of the Applied Models
approachesCCRMSEMAE
Training Data Set
RF0.99510.00760.0064
REPTree0.96420.01880.0133
ANN0.97680.02040.0154
GP_PUK0.98070.01390.0097
SVM_RBF0.97440.01590.0114
Testing Data Set
RF0.9560.02310.0187
REPTree0.87970.03470.0309
ANN0.97480.02110.0164
GP_PUK0.95690.02270.0187
SVM_RBF0.96890.01770.136

Figure 6

Figure 6. Scatter plot for the observed and predicted points of E20 using RF model (a) training and (b) testing.

Figure 7

Figure 7. Scatter plot for observed and predicted points of E20 using REPTree Model (a) training and (b) testing.

4.2. Assessment of ANN-Based Model

A multilayer perceptor layer directs the iterative process of creating an ANN model. To obtain the best predictive model, three distinct performance assessment criteria were used, as shown in Table 4. Findings from the table indicate that ANN-based models perform best among other models with CC values of 0.9748, RMSE values of 0.0211, and MAE values of 0.0164. Figure 8a,b shows the training and testing phases of ANN-based models with a scatter graph showing actual and anticipated values. Most of the points on these graphs are concentrated on the line of perfect agreement, which reflects the closest match between the characteristics of the actual and projected outcomes and denotes increased dependability. Table 2 displays the user-defined parameters used to evaluate the ANN model.

Figure 8

Figure 8. Scatter plot for observed and predicted points of E20 using ANN model (a) training and (b) testing.

4.3. Assessment of GP_PUK-Based Model

Developing a GP model is a trial-and-error process with setting user-defined parameters to obtain the best model. The user-defined parameters set for the current study are shown in Table 2. Table 4 shows the results (CC, RMSE, and MAE) obtained by the GP_PUK model for predicting E20 with different jet geometries in an open channel. These assessment criteria values show that model GP_PUK performs relatively better than other models (Figure 9a,b) which depicts a scatter plot for the GP_PUK model using the training and testing data sets, respectively. Observing Table 4 and Figure 9, it is found that the GP_PUK model is appropriate for estimating E20 with the current data set.

Figure 9

Figure 9. Scatter plot for observed and predicted points of E20 using GP_PUK model (a) training and (b) testing.

4.4. Assessment of SVM_RBF-Based Model

After a trial-and-error procedure, the best user-defined parameters were chosen based on the problem’s various characteristics. After the experiment with various training functions, the best training function for the problem was determined. The values of user-defined parameters were determined by trial and error. In the present study, the RBF has been used as the kernel function to forecast E20 provided by jets having different geometries in an open channel. The optimal quantities of these parameters are assessed for the suggested approach and presented in Table 2 since they influence the accuracy of the SVM model. For the SVM_RBF forecasting model, the majority of the points are along the slanted line (Figure 10a,b). The performance of SVM_RBF appears to be fairly acceptable according to Table 4. With the data set from the current investigation, it is the second-best model for predicting jet aeration.

Figure 10

Figure 10. Scatter plot for observed and predicted points of E20 using SVM_RBF model (a) training and (b) testing.

4.5. Comparison of Applied Models

In this section, E20 of plunging hollow jets having different geometries (circle and square) in an open channel is examined by implementing five machine learning techniques, i.e., RF, REPTree, ANN, GP_PUK, and SVM_RBF. Seven input parameters, α, D, NJ, HR, Fr. No, Jsq, and Jcir jet geometry, as well as E20 as an output parameter for the prediction of the performance of jet geometries to achieve higher efficiency are assessed by performance evaluation parameters as given in eqs 68. Table 4 shows a comparison of all of the models for both training and testing stages. Out of the models applied, ANN exhibits outperforming results in the testing stage with the highest CC value of 0.9748 and least errors (RMSE = 0.0211, and MAE = 0.0164). The performance evaluation of all of the models for both training and testing stages suggests that the ANN-based model is near to actual data during the testing stage (Figure 11). The predicted relative error by the applied models is shown in Figure 12. Table 5 shows quartile values of actual and predicted E20 to compare the predicted and actual values. The best estimated model ANN has an IQR of 0.113. Figure 13 shows a boxplot which depicts that all models can estimate E20.

Figure 11

Figure 11. Comparison graph of applied soft computing models.

Figure 12

Figure 12. Error values of applied soft computing models in the training and testing stages.

Figure 13

Figure 13. Boxplot with all applied models using the testing stage.

Table 5. Quartile Values Using Actual and Predicted of All Applied Models for the Testing Stage
statisticactualRFREPTreeANNGP_PUKSVM_RBF
minimum0.0930.1040.0840.0620.1030.083
maximum0.370.3360.3350.4030.3360.37
first quartile0.1870.17950.160.161750.181750.1765
mean0.2262140.2204050.2201430.2191190.222190.224333
third quartile0.275250.2660.274250.274750.269250.27525
IQR0.088250.08650.114250.1130.08750.09875

5. Comparison of the Sensitivity and t-Test Analysis

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According to the findings, the ANN model is the most accurate among all of the models used to forecast aeration effectiveness in an open channel flow with various jet shapes. As a result, a sensitivity analysis was done to determine the most important input parameter. For sensitivity analysis, seven input parameters are employed, as indicated in Table 6. According to the results of the sensitivity study, α is the input parameter that is the most sensitive, having a smaller CC and hence more errors. This study’s main objective is to evaluate the effectiveness of circular and square jet geometry for aeration. Table 6 also reveals that square jets are more responsive than circular geometry jets, suggesting that square jets can transfer more DO under the same experimental circumstances. Hence, achieving good E20 can be accomplished by deploying square jets in an open channel.
Table 6. Results for Comparison of Sensitivity and t-Test Analysis
sensitivity analysis with best predictive model (ANN)
variables combinationparameter eliminatedCCRMSEMAE
E20 = f (HR, α, NJ, Fr. No, DJsqJcir) 0.97480.02110.0164
E20 = f (HR, NJ, Fr. No, DJsqJcir)α0.91520.03560.0273
E20 = f (HR, α, NJDJsqJcir)Fr. No0.9470.03860.0316
E20 = f (α, NJD, Fr. No, JsqJcir)HR0.96620.02410.0195
E20 = f (HR, α, NJ, Fr. No, JsqJcir)D0.97120.02130.0174
E20 = f (HR, α, Fr. No, DJsqJcir)NJ0.9720.02410.0188
E20 = f (HR, α, NJ, Fr. No, DJcir)Jsq0.97370.02010.0157
E20 = f (HR, α, NJ, Fr. No, DJsq)Jcir0.97430.02020.0163
experimental t-test
variableT statP value
Fr. No28.82.50 × 10–57
α11.635.24 × 10–22
HR10.981.19 × 10–20
NJ9.431.39 × 10–16
D9.391.75 × 10–16
Jsq8.211.03 × 10–13
Jcir7.828.80 × 10–13
Additionally, Table 6 also displays the t-test outcomes for the comparison of two samples, assuming variances that are not equal. The results obtained from the experimental t-test align with the sensitivity analysis acquired through the use of the modeling technique known as ANN. The input parameter α shows a high degree of sensitivity, as indicated in the sensitivity table derived from the machine learning approach. On the other hand, the parameter Fr. No exhibits the highest t-statistic value. This can be justified by acknowledging that the Fr. No is not an independent variable and does not possess direct input significance. The Fr. No is a function of the discharge rate and jet area and is calculated as:
Fr.No=DvgHR
(9)
where Dv is the discharge velocity (which is 110.89 cm/s for 3.41, 124.88 cm/s for 3.84, and 154.47 cm/s for 4.75 l/s), g denotes accelerated gravity (g/cm3), and HR is the hydraulic radius for each jet.
HR=Jetareaπ×NJ×DJn2;NJ=1,2,4,8,16,32,64
(10)
where Jet area is 30.75 cm2, NJis the number of jets, and DJn is the diameter/sides of each jet geometry.
Nevertheless, it is noteworthy that in both examinations, the variables α, Fr. No, and HR consistently exhibited an identical level of significance. Moreover, upon contrasting the two jet configurations, it became evident that square jets outperformed circular jets, displaying a substantially superior performance.

6. Discussion

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As opposed to classical computing, soft computing uses approximation models to address complicated real-world issues. In a sense, soft computing takes its cues from the human mind. Examples of soft computing methods include fuzzy logic, GA, ANN, machine learning, and expert systems. (74) Prediction of E20 is a highly prioritized study, especially for water management resources that are heavily polluted. In order to estimate the Jsq and Jcir aeration in an open channel flow, this study explores and tests the effectiveness of the soft computing models: RF, REPTree, ANN, GP_PUK, and SVM_RBF. For model development and validation, 126 data points were obtained experimentally. The developed models’ performance is evaluated using three different goodness-of-fit parameters: CC, RMSE, and MAE. Seven different input parameter angles of α, D, HR, NJ, Fr. No, and jet geometries such as Jcir and Jsq were utilized to analyze E20. Various statistical criteria have been utilized to compare the efficiency of models. It has been observed that RF was the best-performing model with a CC value of 0.9951 during the training stage but drops to 0.956 during the calibration of model. When comparing the performances of models exhibited during the testing stage, it is observed that ANN outperforms other models with the highest CC value of 0.9748 and comparatively low RMSE of 0.0211 and MAE of 0.0164. The REPTree exhibits considerable inaccuracy during testing which is the least performing model for the current data set.
Certain studies have documented the effectiveness of employing soft computing approaches in the aeration field. ANFIS and LS-SVM were effectively applied in research related to the air entrainment rate and E20 derived from downward falling jets passing over triangular weirs. The jets’ input variables comprised discharge, drop height, and weir crest angle. The performance of these modeling methods was evaluated by comparing them to predictive equations derived from both multiple linear and multiple nonlinear regression. (75) In another study, the predictive capabilities of computational techniques (ANN, SVM, and GP) in relation to estimating total KLa20 were examined. Various configurations of plunging jets were investigated, including circular, square, rectangular, and rounded rectangle designs. (76) In another study on modeling of infiltration rate using soft computing techniques such as GEP, GRNN, and ANN. It was found that ANN outperforms other models. (77) Subsequently, three types of hybrid intelligent systems were created: an imperialist competitive algorithm (ICA)- ANN, a GA-ANN, and a PSO-ANN. These systems were specifically designed to estimate the tensile strength of rocks. The findings indicated that while all of the established predictive models demonstrate satisfactory accuracy in predicting the strength, the PSO-ANN predictive model exhibited superior performance. Consequently, it was deduced that, owing to the enhanced capability of the PSO-ANN model, it could be proposed as a novel approach for forecasting the tensile strength of rocks. (78) In another study, a pair of soft computing techniques, specifically the ANN and the Co-active Neuro-Fuzzy Inference System (CANFIS), was utilized alongside MLR to formulate a model for estimating the daily pan evaporation (Ep). Results indicated that the ANN model, incorporating all input variables and a single hidden layer, proved to be the most effective in simulating Ep. (79) In a study to compare the performance of ANFIS and ANN to predict the cement-based compressive strength, it was found that ANN was the best-performing model rather than ANFIS. (80)

7. Conclusions

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The investigation aimed to predict how different jet geometries affect aeration in an open system. Jsq and Jcir were used for this purpose. Several soft computing techniques, namely, RF, REPTree, ANN, GP_PUK, and SVM_RBF, were employed. The results showed that all of these methods displayed strong predictive abilities, surpassing a CC of 0.8. Among them, ANN proved to be the most appropriate method for predicting E20 based on the provided data set, achieving a CC value of 0.9748. Additionally, the predicted values for the RMSE and MAE were determined to be 0.0211 and 0.0164, respectively. Sensitivity analysis using the optimized ANN model confirmed that the tilt angle of the flume bed had a significant impact on E20. Furthermore, the findings indicated that Jsq geometry outperformed Jcir in terms of E20, a result further supported by the t-test and sensitivity analysis.

Author Information

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  • Corresponding Authors
  • Authors
    • Diksha Puri - School of Environmental Science, Shoolini University, Solan, Himachal Pradesh 173229, India
    • Parveen Sihag - Department of Civil Engineering, Chandigarh University, Mohali, Punjab 140301, India
    • Mohindra Singh Thakur - Department of Civil Engineering, Shoolini University, Solan, Himachal Pradesh 173229, India
    • Kahkashan Perveen - Department of Botany & Microbiology, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia
    • Faisal M. Alfaisal - Department of Civil Engineering, College of Engineering, King Saud University, Riyadh 11495, Saudi Arabia
  • Author Contributions

    D.P.: Writing and preparing the manuscript. R.K.: Writing and preparing the manuscript, methodology. P.S.: Reviewing, editing. M.S.T.: Reviewing, editing, supervision. K.P.: Reviewing, editing, supervision. F.M.A.: Reviewing, editing, supervision D.L.: Reviewing, editing, revision, supervision.

  • Notes
    The authors declare no competing financial interest.

    Statements of ethics The information has been properly cited and has been collected from published works that are in the public domain.

Acknowledgments

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The authors acknowledge the support provided by Researchers Supporting Project Number RSP2023R297, King Saud University, Riyadh, Saudi Arabia.

Abbreviations

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ANFIS

adaptive neuro-fuzzy inference system

ANNs

artificial neural networks

CC

correlation coefficient

D

discharge

DO

dissolved oxygen

E20

aeration efficiency

Fr. No

Froude number

GA

genetic algorithm

GEP

gene expression programing

GMDH

group method of data handling

GP

Gaussian process

GRNN

generalized neural network

HHO

Harris Hawk’s optimization

Hp

bubble penetration depth

HR

hydraulic radius

IQR

interquartile range

Jcir

circular jets

JD

jet diameter

JL

jet length

Jsq

square jets

JT

jet thickness

JV

jet velocity

KLa20

volumetric oxygen transfer coefficient

LS-SVM

least square support vector machine

MAE

mean absolute error

MARS

multivariate adaptive regression splines

MLR

multiple linear regression

MNLR

multiple nonlinear regression

NN

neural network

NJ

number of jets

OTE

oxygen transfer efficiency

PSO

particle swarm optimization

P/V

jet power per unit volume

PUK

Pearson VII function kernel

RBF

radial basis kernel

Re. No

Reynolds number

REPTree

reduced error pruning tree

RF

random forest

RMSE

root-mean-square error

SOTE

standard oxygen transfer efficiency

Std. Dev.

standard deviation

SVM

support vector machine

α

angle of inclination of channel bed

γ

γ

θ

jet plunging angle

σ

sigma

Ω

omega

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  • Abstract

    Figure 1

    Figure 1. Working of the RF-based model.

    Figure 2

    Figure 2. Schematic diagram of ANN technique.

    Figure 3

    Figure 3. Models of jet fabricated (a) circular jets and (b) square jets.

    Figure 4

    Figure 4. Experimental setup

    Figure 5

    Figure 5. Flowchart of the methodology.

    Figure 6

    Figure 6. Scatter plot for the observed and predicted points of E20 using RF model (a) training and (b) testing.

    Figure 7

    Figure 7. Scatter plot for observed and predicted points of E20 using REPTree Model (a) training and (b) testing.

    Figure 8

    Figure 8. Scatter plot for observed and predicted points of E20 using ANN model (a) training and (b) testing.

    Figure 9

    Figure 9. Scatter plot for observed and predicted points of E20 using GP_PUK model (a) training and (b) testing.

    Figure 10

    Figure 10. Scatter plot for observed and predicted points of E20 using SVM_RBF model (a) training and (b) testing.

    Figure 11

    Figure 11. Comparison graph of applied soft computing models.

    Figure 12

    Figure 12. Error values of applied soft computing models in the training and testing stages.

    Figure 13

    Figure 13. Boxplot with all applied models using the testing stage.

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