This article is organised as follows: Section 2 presents a theoretical analysis based on the two-film theory to determine the overall oxygen mass transfer rate, standard oxygen transfer rate, and standard aeration efficiency.

Section 3 encompasses materials and methods, including the experimental setup, experimental design, and procedure to evaluate the performance of venturi aeration. In addition, this section discusses the artificial neural network integrated with particle swarm optimisation and genetic algorithm technique and the performance analysis.

A detailed summary of the results and discussions is discussed in Section 4, which includes performance evaluation of the developed models, ANN modelling, PSO, a comparison between the integrated models, and the effects of operating parameters on aeration efficiency.

Section 5 discusses the conclusion of this article, including the authors’ perspectives on the subject of the study and potential directions for further research.

A 220 L volume of an experimental tank with measurements of 90 cm × 55 cm × 45 cm was operated with the venturi for oxygenating the full volume of water (Figure 1). The venturi circulates water with the help of pipe fittings in a closed-loop system with an associated pump, valves, water metre, and pressure gauge. The well-known sections of the venturi are divided into three distinct parts, namely, converging, throat, and diverging. The converging section is connected with the narrow section at the neck of the venture, and it terminates with the diverging section as an outlet of the venturi. Each of the sections of the venturi may be separated from one another. Another extended outlet pipe (EOP) is also connected with the diverging section to supply oxygenated water into the full volume of the water tank. Both converging and diverging sections were fabricated with the same length of 100 mm. The middle section of the venture, i.e., throat consists of five sets of different lengths, namely, 10 mm , 20 mm , 30 mm , 40 mm , and 50 mm . In total, four different extended outlet pipe lengths were designated in the current study to see the performance of aeration efficiency, i.e., 2 m , 4 m , 6 m , and 8 m .

All the experiments were conducted using a 1-HP centrifugal pump, which circulates water by venturi in a closed-loop manner with connecting extended outlet pipes of various lengths. The throat section produces a vacuum and makes it possible for air entry from air holes due to the pressure reduction. As a result, the air bubbles mix with water that flows via the venturi to the outlet pipe and continues to travel in the water tank. The experiments were conducted at three different flow rates (Q) of 0.30 L/s, 0.60 L/s, and 0.90 L/s for all sets of design parameters (throat length, TL; and extended outlet pipe, EOP). Table 1 presents the planned experiments to improve the venturi aeration system’s aeration efficiency. A total of 60 experiments were performed with three replicates to optimise the aeration efficiency of the venturi aerator. With a 5% threshold of significance, substantial model parameters were found using the analysis of variance (ANOVA). In the current study, the AE of the venturi aeration system was analysed and compared using integrated models, i.e., ANN-PSO and ANN-GA. The extended outlet pipe (EOP), throat length (TL), and flow rate (Q) are the independent variables, and the dependent variable is chosen as AE to determine the model response.

In this analysis, pure tap water was initially deoxygenated using non-steady re-aeration experiments (ASCE, 2007). Each experiment utilised clean tap water that was deoxygenated to reduce the DO of each milligramme per litre employing 10 mg of sodium sulphite and 0.1 mg of cobalt chloride as a catalyst per litre volume of water (Boyd, 1998). The EXTECH DO metre 407,510 was used to measure the temperature, and DO level datasets for each set of operations were carried out at 1 min of an identical interval until 80% of DO saturation was attained. For each DO saturation measurement that was acquired, the measurements for the DO deficiencies were also produced. Furthermore, the graphical representation amongst natural log deficits (ordinate) and aeration time (abscissa) represents the best-fit gradient line to assess K L a T . The gradient line was additionally improved by a 20°C temperature correction factor. Moreover, the oxygen transfer ability of the venturi has been estimated by K_La₂₀, OTR, and AE using Equations 5–7, respectively.

The artificial neural network (ANN) is a data processing system that distributes data uniformly and has certain traits and features that are comparable to the neural structure of the human brain (Haykin, 1999; Mohanty et al., 2010; Yadav and Roy, 2023). An ANN includes an input layer (independent variables), a hidden layer, and an output layer (dependent variables); each layer contains a variety of processing components. An increasingly popular solution to conventional modelling methodologies and a more accurate tool to utilise with complicated non-linear interactions is the ANN, which relies on the idea of real-world neurons as well. An average neural system is fully interconnected, meaning that every single neuron in a single layer has a link to all the other neurons in each layer above it (Kumar et al., 2002). The process of the data information modelling approach starts using basic analysing units before scaling up to more complicated ones, modifying balanced inputs to make neural networks more effective. The effectiveness of an ANN multilayer model relies on both the number of neurons and connections and is affected by how the information is received, processed, and subsequently processed in hidden layers (Karri and Sahu, 2018). The available research shows that most researchers utilise a gradient-based optimisation technique called feed-forward neural network development with a conventional back-propagation strategy (Rumelhart et al., 1986).

The AE of the venturi aerator was considered as a result of the model with its selected operational parameters, i.e., TL, EOP, and Q, which were used as model inputs. A total of 70 % of the 60 data points were applied for training, whilst the rest of the 30 % were employed for network setting. In the present study, the experimental dataset of a 70:30 splitting ratio was used (Roy et al., 2022; Yadav and Roy, 2023). The optimal network layout for the ANN model was discovered to be 3–6-1 as presented in Figure 2. In the current study, the overfitting issue was resolved by standardising the experimental findings using min–max normalisation before utilising them for ANN training.

The particle swarm optimisation (PSO) modelling integrated with ANN is a cutting-edge, accurate, and experimental approach. A kind of meta-heuristic model called ANN-PSO can be utilised to build realistic models and deal with non-uniform multi-component interactions between dependent and independent variables (Yadav and Roy, 2023). The PSO requires fewer parameter adjustments than any other optimisation approach available and is simpler to implement (Eberhart and Shi, 2001). Furthermore, if the target function includes a single attribute and several local minima/maxima, the ideal value may be reached using the PSO method (Roy et al., 2021b). As PSO is a naturalistic exploration method as opposed to gradient-based optimisation, it has the drawback of being a sluggish procedure (Roy et al., 2021b). On the other hand, complicated non-linear optimisation issues are frequently solved using mathematical techniques such as genetic algorithm-based optimisation (ANN-GA) and ANN modelling. In the present study, ANN was integrated with PSO and GA algorithms separately to generate data of ANN using various amounts of neurons in the middle-hidden layer whilst taking into account the sample estimates for PSO and GA (Ghose et al., 2023). The movement of the points in space may be understood by their velocities, which also have the potential to hold their local and global best values (Jun et al., 2020). Each particle kept its location and modified velocity based on its own flight performance along with that of other elements in the search space to identify the most efficient result. The updated velocities and orientations of every single particle as it travels to the local optimal point (pbest) and global best point (gbest) are computed using the following Equations 10, 11.

where, i = 1 , 2 , … N ; N = population size, v i k = flow rate of the element i at iteration k ; x i k = element position i at iteration k ; pbes t i = personal best of element i ; gbest = best point in the community; w = weight of inertia; c 1 and c 2 = coefficients; and r 1 and r 2 = random numbers uniformly scattered in the range (0, 1) (Figure 3).

Holland (1992) established the genetic algorithm (GA), a meta-heuristic optimisation method that utilises the ideas of the natural selection process and heredity. It is employed to resolve optimisation issues that are both confined and uncontrolled. Using statistical modifications, GA may continually adjust the number of distinctive approaches whilst searching over enormous solution spaces. The GA technique consists of three crucial processes, including the first group creation, genetic operator training (reproduction, crossover, and mutation), and evaluation according to a group’s fitness value (Yang et al., 2020). The genetic algorithm identifies people at random within the existing population at each stage, treating them as a pair of parents to create progeny for the generations to come. Over succeeding generations, the population progresses in the direction of the optimal solution. It is possible to choose an adequate goal function by using a mixed genetic algorithm. The ability of ANN models to estimate the future is improved by the multidirectional searching technique used in GA (Figure 4).

In this study, the highest possible swarm size was set at 10 random particles produced and 50 iterations. The values taken for w , c 1 , and c 2 are 0.8, 1.05, and 1.05, respectively. The programme was run by MATLAB-2013a to execute the ANN-PSO and ANN-GA techniques. Although the MATLAB PSO method employs an optimisation strategy, its goal is to maximise AE, which is expressed as a similar minimising problem, as follows:

where X = X 1 X 2 X 3 T = selected resolution vector and F X = result of the ANN. The limits are specified as 10 ≤ X 1 TL ≤ 50 , 2 ≤ X 2 E O P ≤ 8 , and 0.3 ≤ X 3 Q ≤ 0.9 .

To evaluate the efficacy of the developed ANN and linear regression (LR) models, the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) were used as indicators of effectiveness. The variables used in the produced models were compared, and the algorithm of high coefficients of R² and fewer values of MAE and RMSE was determined as the best fit. The following Equations 13, 14, 15 were used to calculate R², RMSE, and MAE, respectively:

where N is the total amount of observations, Oi and Pi are the observable and projected values, respectively, and the bar represents the variable’s mean.

The linear regression analysis utilises a total of 60 observed responses; the model was developed using the least square method. A linear regression model was developed for several responses of AE using the investigated dataset. The linear relationship between AE and independent operative variables, i.e., TL, EOP, and Q, is depicted in the regression Equation 16.

The linear regression Equation 16 specifies the hypothesis model that fits well in between the independent variables and AE. The linear regression equation is a second-order polynomial, which has a high coefficient of correlation (R²) value, demonstrating that a significant relationship is feasible. The p-value and the R² value were estimated as 0.972 and 0.005, respectively. The above findings are supported by the value of the modified coefficient of correlation (R² adj.), 0.976. The 97.25% probability from the analysis indicates that the independent variables (TL, EOP, and Q) of the LR model are highly significant.

According to the statistical results (Table 2), every component of the linear model (TL, EOP, and Q), as well as the quadratic model term (TL2, EOP2, and Q2), has statistical significance at the 97.25% level of confidence. There is no statistical significance (p > 0.05) for the interacting model variables (EOP × Q), (EOP × TL), and (Q × TL).

The results of the p values are an easy approach for evaluating the importance of various coefficients. The significance of the associated coefficient increases with decreasing P. The correlation between the variables is extremely significant (F = 196.60; p = 0.000), according to the ANOVA (Table 3) of the best-fitted estimated model. According to the results of the present study, the model term TL (p = 0.000) is significant (p < 0.05), whereas, the other model terms, EOP (p = 0.716) and Q (p = 0.040) are not significant (p > 0.005). The quadratic terms EOP × EOP (p = 0.000) are significant (p < 0.05), whilst the other quadratic terms Q × Q (p = 0.647) and TL × TL (p = 0.292) have a poor fit and are considered non-significant (p > 0.05). Additionally, the interaction between the independent variables (EOP × Q, EOP × TL, and Q × TL) has no apparent impact on the AE (p > 0.005). Consequently, it may be concluded that the proposed model satisfactorily matches the results of the experiments (Pareek et al., 2023a,b).

Artificial neural network (ANN), which gets their inspiration from the way the human brain functions, may be used to predict outcomes for a variety of activities, including analytical and numerical modelling involving a variety of data types. The ANN depends on an awareness of the statistical fundamentals of the events underlying the procedure, and it has significant benefits over traditional modelling approaches (Figure 5). The development of the best back-propagation training technique, the establishment of the appropriate number of neurons, and verifying and testing the model, all three processes were employed to develop the perfect ANN modelling structure (Kedia et al., 2022). The ANN aims to predict available capabilities using real data by replicating the organic nervous system, which replicates data on somebody’s cognitive processes. The ANN often results in an inundation of previous experience data. Once the model was successfully constructed using the trial and error method, the minimum MSE was found between the target value and the ANN model’s forecast value of AE (Yadav and Roy, 2023). The neural network correlation network indicates the projected and targeted datasets, i.e., the training dataset, the testing dataset, and the complete experimental dataset for the ANN model as shown in Figure 5. The ANN model’s coefficient value of R² for the training dataset, the validation dataset, the testing dataset, and the overall prediction was observed to be 0.98855, 0.97891, 0.98320, and 0.98545, respectively (Figure 5). From Figure 5, the R² value between the measured and anticipated responses for the data analysis phases (training, validation, and testing) is close to 1. Hence, the implementation of the ANN was deemed to be very satisfying and robust to estimate the AE values. The highest possible AE value of 0.1050 kg O₂/kWh was found using the expected TL, EOP, and Q values of 50 mm, 6 m, and 0.6 L/s, respectively.

The MSE versus the number of epochs is displayed and is presented in Figure 6. In epoch 8, the value 0.000016445 represents the prime substantial performance. During the training phase, the neural network divides the source and target dataset into three individual data, i.e., training, validation, and testing samples. Using training samples, the network is instructed, and variables are changed in response to errors. The network’s ability to make predictions is assessed using validation samples, and training ceases when it hits the point of saturation. Furthermore, network evaluation is performed before, during, and after training using testing data. Figure 6 shows a green, blue, and red line for validation, training, and testing, respectively. The green colour of the ring circle denotes the network’s peak validation effectiveness. On the other hand, the generalisation comes to an end and the training is ended after 19 epochs. It changes throughout retraining as a result of the network regularly splitting the source and target datasets.

Figure 7 displays training state plots for the gradient values, mutation (mu), and validation cheques for ANN. A gradient depicts the tangent’s slope on a function graph. It suggests that there is a rapid rate of increase in the direction that the parameter under consideration is moving. Error convergence is directly influenced by the back-propagation neural network that was constructed to use the controlling parameter called mutation (mu). A validation cheque halted the learning process of the neural network. A variety of validation tests will be carried out based on how frequently the neural network is iterated (Roy et al., 2023). The gradient’s magnitude and the number of validation cheques needed to end the training are displayed in Figure 7A. As training approaches the performance threshold, the gradient becomes very small. The training phase will end if the gradient’s magnitude is less than 0.00001, which can be modified by adjusting the parameter. The mutation is shown against the increasing number of iterations in Figure 7B. This graphic demonstrates how training improves and the network error decreases. Based on Figure 7C, the number of validation cheques indicates how many rounds the validation performance fails to decline in succession. The training will end once this number hits 8 in the current scenario.

The design parameters (TL and EOP) and operational parameters (Q) of the venturi aeration system were optimised using the most reliable prediction model and PSO in order to obtain the highest acceptable AE. The PSO algorithm scatters its elements all over a hyperdimensional search area. Every cluster member serves as an alternate solution to the optimisation problem by evaluating the desired function at that member’s present location. Engelbrecht (2007) defined that the positioning of elements within the search region is altered in accordance with people’s social-psychological propensity to imitate others’ accomplishments. Every single particle in a cluster is therefore driven for movement by fusing some aspect of its previous existence with the expertise of those around it.

The developed ANN-PSO and ANN-GA models were predicted and compared based on statistical characteristics datasets of training and testing by model parameters, i.e., R², RMSE, and MAE. Table 4 displays these model evaluation indicators of created models. The high R² value for training and testing and the low value of RMSE and MAE indicate the significant performance of two values between experimental AE values.

The ANN-PSO model has a high R² value for training and testing and a low value of RMSE and MAE compared to the ANN-GA model indicating a better prediction of AE values. Therefore, it can be concluded that the performance of ANN-PSO is better than ANN-GA. Table 5 displays and compares the optimal parameters produced by the combined ANN-PSO and ANN-GA methods. The developed models ANN-PSO and ANN-GA are used to select the suitable operating conditions and aids to find out the AE of the venturi aerator. A maximum deviation of 5.714% of the anticipated and actual values of AE from ANN-PSO is presented in Table 5.

Table 6 displays the optimum parameters using the combined ANN-PSO method, which has been compared with the results of the earlier experimental research study (Yadav and Roy, 2023). The created ANN-PSO model may be used to estimate the accessible aeration effectiveness of the venturi aeration system and to determine the proper operating parameters.

The combined effect of operating parameters as EOP and TL on AE is shown in Figure 8, the combined effect of EOP and Q on AE is shown in Figure 9, and the combined effect of Q and TL on AE is shown in Figure 10. It can be clearly seen from Figure 8 that the AE steadily rises with the increase in the extended outlet pipe (EOP) from 2 m to 6 m and then drops again to 8 m of EOP. It can also be found that the AE rises rapidly with increasing throat length (TL) up to 50 mm. This is due to the fact that by increasing the throat length (TL = 10 mm to 50 mm), more vacuum is created that facilitates the maximum pressure reduction resulting in more oxygen transfer rate and aeration efficiency. When the effective outlet pipe increases, the air bubbles get dispersed resulting in the reduction of aeration efficiency. The maximum AE was found at 6 m EOP and 50 mm TL.

Figure 9 shows a relation between EOP and Q on AE. It can be seen from Figure 9 that AE increases with the increasing EOP up to 6 m. However, after 6 m of EOP, the AE values decline when it reaches to the maximum EOP of 8 m. In addition, it was found the AE values are influenced by varying the flow rate (Q). The AE value increases with an increase in Q value up to 0.6 L/s. For further increase in the Q from 0.6 L/s to 0.9 L/s, the AE starts to decrease mainly at the EOP of 8 m. This is due to the fact that the increasing discharges create more turbulence and aid in enhancing the transfer rate of oxygen. Furthermore, at the maximum 8 m EOP, inserted air bubbles get dispersed without affecting the oxygen transfer, which reduces the AE of the venturi. Therefore, it can be concluded that the Q is primarily responsible for transferring air into the water (Yadav et al., 2021).

Furthermore, the effect of Q and TL on AE is presented in Figure 10. Figure 10 indicates very clearly that with an increase in the TL of the venturi, the AE increases at every increasing Q. The maximum AE value was found at 50 mm of TL for all increasing flow rates (Q = 0.3 L/s, 0.6 L/s, and 0.9 L/s). The reason behind that the increasing TL provides a high air entrainment rate, that obvious improves the oxygen transfer capability of the venturi with the increasing Q. The maximum SAE value was determined to be 0.105 kg O₂/kWh, when operating at 0.6 L/s flow rate with a 6 m effective output pipe and a 50 mm throat length of the venturi.