Montana Flume Aeration Performance Evaluation with Machine Learning Models

Montana flume is derived from Parshall flume by eliminating diverging part and throat. The mass transfer of oxygen from the atmosphere into the water is known as aeration. The dissolved oxygen (D.O.) concentration in the water body determines water quality. The experiment was performed on six different Montana flumes fixed in a tilting prismatic rectangular channel. Experimental observations were used to develop classical and machine learning models to predict Montana flume aeration efficiency. The developed models are namely multi nonlinear regression (MNLR), adaptive neuro-fuzzy inference system (ANFIS), and artificial neural network (ANN). The models were tested, and the results show that all these three developed models perform very well. However, ANN gives better results than other models as it has the highest cc and lowest rmse values. According to the sensitivity analysis results, the Reynolds number (Re) was the most crucial input element in determining the aeration efficiency of the Montana flume in the case of dimensionless datasets. However, discharge per unit width (q) is found to be of relative significance in the case of dimensional datasets.

efficiency of the Parshall flume using machine learning models, K nearest neighbor (K.N.N.), Decision Tree Regression (DTR), and Random Forest Regression (R.F.R.). Dursun [7] investigated the oxygenation of Parshall and modified Venturi flumes, concluding that Parshall and Venturi flumes can be utilized to aerate channels with zero or extremely low slopes. Tiwari N.K. [8] investigated the effect of hydraulic jump on aeration performance. Turbulence occurs as a result of a hydraulic jump, which causes aeration. The flume's shape influences oxygen transfer; narrowing the throat and lengthening it improves aeration efficiency. Flow rate also affects the oxygen transfer rate; a higher discharge increases aeration efficiency [7].
The rate of mass transfer of oxygen to the water [9] is proportional to the difference between existing and saturation concentrations.
where C s = saturation concentration, K L = Liquid film coefficient, a = specific surface area.
For simplicity, it is assumed that the product K L a remains constant over a hydraulic structure. Then Eq. 1 is integrated concerning residence time of entrained bubble, which gives deficit ratio, r [9].
where C d = concentration of gas downstream, C u = concentration of gas in upstream, t = bubble residence time.
Eq. 2 can be expressed as aeration efficiency, E Equation 3 is aeration efficiency at actual temperature, and correction is applied to report it at a reference temperature of 20 °C. The equation given by Gulliver et al. [10] is most widely used to account for the effect of temperature on the aeration process.
where E = aeration efficiency at the actual water temperature, E 20 = aeration efficiency at 20 °C.
f is exponent given by Gulliver et al. [10] described as: The foregoing review suggests that little /no work has been carried out related to Montana flume aeration, and no machine learning models have been used in its prediction. Machine learning algorithms have now widely been applied in numerous branches of engineering since they need not require any information on the methods. In addition, they eliminate issues of scale effect and also optimize the cost acquired in fabrications and running the models in the field and laboratory once models were created for the dataset range [11].
The novelty of the work has many folds since the work sums up the prediction of the Montana flume aeration performance by conducting laboratory tests with varying discharge per unit width, depth of flow, and Montana flume geometry. Subsequently, the Montana flume aeration performance is predicted and compared for both dimensional as well as dimensionless parameters with artificial intelligence algorithms of ANN, ANFIS, and classical equations, including MNLR and other relations that exist in the text using experimental observations and their ability, are examined using two performance metrics. Finally, sensitivity analysis was carried out to know the relative implication of the input parameter on Montana flume aeration performance results.  Table 1.

Methodology
Water entered the flume through a headbox, and the upstream section of the flume was furnished with a metal screen to dampen the turbulence in the flow of water. The channel has a closed, re-circulated mechanism that constantly feeds the channel by redrawing water from the aeration 'cum' storage tank. Aeration cum storage tank is used for storing water circulated in the flume for experiment and measuring the D.O. concentrations. The centrifugal motor pump (1HP) was attached with regulating valve to control the discharge and was connected to the pipe of 5.08 cm diameter supplying the headbox. Discharge was measured using an electromagnetic flowmeter with an accuracy of ±0.5% installed in the pipeline. A thermometer (±0.1 • C ) was used to measure the temperature of the water in the aeration tank. A pointer gauge (least count of 0.01 cm) was used to measure water depth in the Montana flume at a 2/3rd distance from the throat in the upstream direction. The flow was gradually allowed to become stabilized before measuring the flow depth to ensure a minimum error.
The aeration tank was filled with tap water. The deoxygenation of water is carried out for each test run by adding sodium sulphite of 7.9 mg/m 3 to drop the D.O. down to 1.0-2.0 ppm and cobaltous chloride of 3.3 g/m 3 as a catalyst to accelerate the reaction [9] such that the D.O. level of the tank would not reach or surpass the saturation level during aeration. A sample from the deoxygenated water in the tank was withdrawn for the measurement of dissolved oxygen (D.O.), which represents the initial D.O. value (upstream D.O. concentration, C u ). The channel was allowed to run for a set time [12], and the final D.O. was determined (downstream D.O. concentration, C d ). Azide modification method  where p i is the constant, x 1 , x 2 , x 3 , x 4 , y 1 , y 2 , y 3 , y 4 , and y 5 are the function coefficients and can be found by reducing error squares in estimation.

Adaptive Neuro-Fuzzy Inference System (ANFIS) and Artificial Neural Network (ANN)
ANFIS is a soft computing technique used widely for modeling complex system problems based on input and output parameters. It amalgamates two artificial intelligence techniques of neural network and neuro-fuzzy. The Sugeno-type fuzzy toolbox is utilized to analyze this study. Further, F.I.S. creation is done either by grid partitioning or subtractive clustering techniques. Input variables are mapped to output with various membership functions(mfs). These membership functions are curves that denote and discuss how each position in the input variable is allocated a membership value ranging from 0 to 1. Numerous input mfs are available like triangular, trapezoidal, Gaussian, bell-shaped, Gaussian combination, sigmoidal, the difference between two sigmoids, product of two sigmoids, Z-shaped, Pie-shaped, and S-shaped. There are two output membership functions available one is linear, and another is constant. Artificial neural networks consist of artificial neurons which are hypothetically inspired by biological nervous systems. A neural network is a regression model with neurons in each layer similar to arranged neurons in the brain and can learn from data. ANN is a well-known soft computing technique widely used to solve water resources problems. The neurons are assigned weight and bias, which get adjusted during training. These weights and bias represent the strength of neurons. The network consists of an input layer having nodes equal to a number of inputs, one or more hidden layers where processing takes place, and an output layer. The input is multiplied by a weight (W), and all the weighted inputs are added together with a bias (b). The sum is pushed through an activation function, and output is obtained. Many activation functions are available, like logsigmoid, tan-sigmoid, and linear. The network trains the data fed as input using training functions like Levenberg-Marquardt, Bayesian Regularization, BFGS Quasi-Newton, etc. Gradient descent optimization algorithm also referred to as backpropagation, is used for optimization. It performs computation backward and updates weight and biases where the performance function decreases rapidly.

Data Set
For the analysis of predictive models, dimensional and dimensionless data sets were used, consisting of 90 observation patterns. Out of these 90 observations, the number of observations used for the training data was 63, while the remaining 27 were selected to test the model. The dimensional input parameters consist of discharge per unit depth (q), depth of flow (H a ), upstream entrance width (D), sidewall length (A), throat width (W), and dimensionless input parameter consisting of Froude number (F r ), Reynolds number (R e ), (A/W) and (D/W). In contrast, the Montana flume's aeration efficiency (E 20 ) was considered the output. The data categorization for testing and training was based on arbitrary and random selection. The summary details of the dataset are given in Table 2.

Appraisal Metrics
Appraisal of proposed algorithms (ANN, ANFIS, MNLR, and applied classical models) used in this work was predicted after completing the required training and testing cycles. The computed output results are appraised by measuring goodness-of-fit: correlation coefficient (cc) and rootmean-square error (rmse). The range of variation of cc lies between −1 and 1. If the cc value is one and the corresponding rmse value is around 0, model accuracy is very high, but if cc approaches 0 and rmse tends to 1, then model accuracy is very low. The cc and rmse values are estimated as X obs = Observed E 20 value, X estd = Estimated E 20 value, n = number of observations.

Result of Classical equation
The following equations are obtained using MNLR: The performance of proposed MNLR is better than that of existing classical equations like Wormleaton and Tsang  Table 4. However, Wormleaton and Tsang [14] and Bostan et al. [15] performed very poorly, but Tiwari N.K. [8] was given, by and large, performing well in training and testing datasets for this data range. This contention was further buttressed by observing Fig. 3, where predicted data points are near the perfect line for dimensional and dimensionless MNLR models compared to other considered models. The advantage of conventional models is that they can be easily interpreted as their coefficients can be computed quickly. However, the disadvantage is that they cannot perform well in the case of stochastic and nonlinear phenomena like the complex aeration process taking place in the Montana flume.

Results of Adaptive Neuro-Fuzzy Inference System (ANFIS)
The ANFIS model was developed for two types of data sets which include dimensional and dimensionless parameters. The developed model was Sugeno F.I.S. as it is computationally efficient and well suited for mathematical analysis. A fuzzy inference system can be generated using sub-clustering or grid partitioning. However, grid partitioning was used in this study to create a fuzzy inference system as it gives better results. It was found that the model results for the train data set and corresponding unknown test values of models by ANFIS were close to a perfect agreement line. There were many membership functions available for input and output; however, input mfs; trimf, trapmf, gbellmf, gaussmf, and gauss2mf were used to predict the aeration efficiency of Montana flume, which gave satisfactory results. In contrast, the constant output membership function was applied for data sets since it produced excellent results compared to linear functions. The 'AND' rule was employed for this modeling instead of 'OR' and 'NOT,'. The hybrid optimization technique was preferred over the backpropagation technique for the training model since it produced closer to desired results. Agreement diagrams for the actual aeration efficiency and the predicted aeration efficiency for the defined train and test data set are shown in Fig. 4 for dimensional and dimensionless, respectively. The model's performance for different membership functions of dimensional and dimensionless data sets can be referred to in Table 5. From the perusal of Table 5 and also Fig. 4 for the test dataset, it is evident that gbellmf (dimensional) and trimf (dimensionless)-based ANFIS is performing well with the highest value of cc and lowest value of rmse; however, all proposed mfs-based ANFIS was performing well and could be utilized for Montana flume aeration performance evaluation. The ANFIS draws advantages of both ANN and fuzzy logic. i.e., the ANFIS model has the advantage of numerical and linguistic knowledge. The ANFIS model can be trained without relying solely on expert knowledge sufficient for a fuzzy logic model. Consequently, several advantages of the ANFIS exist, including its adaptation capability, nonlinear ability, and rapid learning capacity. The disadvantage is that its behavior is like a black box, as the input variables and processes are not visible to the user.

Results of Artificial Neural Network (ANN)
In the present analysis cascade-forward neural network was used, similar to a feed-forward neural network. Each layer is connected to the input and previous layer in Cascadeforward neural network. It is a multi-layer perceptron, and in this study, the network is developed for dimensional and dimensionless data sets. An optimized, developed network for dimensional data set has three layers: an input layer with five neurons, a hidden layer with 12 neurons, and an output layer with one neuron. An optimized, developed network for dimensionless data set has three layers consisting of an input layer with four neurons, two hidden layers with 12 neurons Test Predicted E20 Actual E20 Fig. 4 Performance of ANFIS for a Dimensional, b Dimensionless data set each, and an output layer with one neuron. The training algorithm for the dimensional data set used was resilient backpropagation (trainrp), log-sigmoid (logsig) as a transfer function was used for hidden layers. The output layer utilized the hyperbolic tangent sigmoid (tansig) transfer function. In contrast, to that of the dimensionless data set, the training algorithm was levenberg-Marquardt (trainlm), hyperbolic tangent sigmoid (tansig) as a transfer function for the hidden layer, and linear (purelin) transfer function for the output layer. The training algorithm and transfer function used for the developed model gives the best result among all available training and activation functions. An agreement diagram between actual aeration efficiency and predicted aeration efficiency of Montana flume for train and test dataset is shown in Fig. 5 for both dimensional and dimensionless cases. It has been observed that predicted data points for train and test datasets were near the perfect line for both dimensional and dimensionless cases. This aspect was further reinforced by going through Table 5, where the  Dimensionless Predicted E 20 cc value was higher, and the rmse value was less for dimensional and dimensionless cases. A comparison between observed aeration efficiency and the predicted aeration efficiency for the defined test and train data set is shown in Fig. 5, which was used to develop the ANN model. It was found that the results of the defined test and train values of ANN for the dimensional data set were close to the perfect agreement line, while for the dimensionless testing data set, they were slightly away from the perfect line. The CC value (0.982) for dimensional testing data, (0.9631) for dimensionless testing data of ANN was very high, and the RMSE value (0.0235) for dimensional testing data, (0.0363) for dimensionless testing data was very low. The primary advantages of ANN models are that their learning capability is fast and rapid. They can model a nonlinear and complex relationship. However, model results interpretation is not easy as conventional models as they behave like the black box because the inputs and operations are not visible to the user.

Comparison of Models
The models developed using dimensional and dimensionless data sets of Montana flume were compared using the statistical parameters shown in Tables 4 and 5. All three models, viz. MNLR, ANFIS, and ANN models are efficient in making predictions of the aeration efficiency of the Montana flume. However, the ANN model developed outperformed all the proposed models in predicting the aeration efficiency of the Montana flume. ANN model has the highest cc value and lowest rmse value for both dimensional and dimensionless types of data set. In addition, Fig. 6 provides a graph between actual and predicted values of Montana aeration efficiency using ANN and ANFIS with different mfs for dimensional and dimensionless datasets. It could be observed from Fig. 6 that, by and large, the majority of predicted results for the E 20 fall around the perfect line (i.e., 45-degree line). Four more error lines in the domain of ±25% and ±10% are also plotted between the predicted and actual values of the E 20 . Figure 6a depicts that most of the predicted values of the E 20 by ANN were lying well within ±10% error line from the perfect agreement line in both training and testing cases for a dimensional dataset, including ANFIS_gaussmf values as well. However, barring some predicted points, all predicted values by the machine learning algorithm lie in the range of ±25% error line for training and testing dataset for the dimensional data. Further, the inference from Fig. 6b could be drawn that for the dimensionless data set, ANN and ANFIS_trimf are performing well as their predicted points lie within the ±10% error line for both training and testing datasets. However, all other considered machine learning models give values in the range ± 25% error line, similar to dimensional datasets.

Sensitivity Analysis
The sensitivity analysis identifies the most critical input parameter relative to the other utilized parameters influencing output results. The ANN model outperformed all proposed models for the given data set in this study. Therefore, this algorithm was used for sensitivity analysis by eliminating an input variable while retaining other input variables.
The corresponding impact was observed on the output results assessed in appraisal metrics cc and rmse for a specific set of both test and train data. The corresponding amount of variance in cc and rmse values are illustrated in Table 6, how much an input variable influenced the output results of aeration efficiency. For the dimensional dataset, discharge per unit width, q, appeared to be the most influencing parameter. In contrast, in the case of the dimensionless dataset, the Reynolds number (R e ) was found to be the most relevant in the assessment of aeration.

Dependency of Parameters
The sensitive analysis only gives the sensitivity of a particular input parameter to the output. However, it does not give the nature of trend (increasing or decreasing) on the corresponding output. However, the parametric study would reply to this answer as ANN outperformed all the proposed models for the prediction of E 20 . An ANN-based parametric analysis is conducted to explore the influence of input parameters on the aeration efficiency (E 20 ). For this study, imaginary testing data are created by the varying value of one input parameter, holding the rest of the input parameter intact, and employing this training data with the ANN model generated by utilizing training data patterns. The effect of the two most effective parameters, i.e., q, and R e , on the value of the E 20 was plotted and shown in Fig. 7. From   Fig. 7, it is evident that the value of E 20 was increasing with the increase in the values of q and R e .

Conclusions
Two data sets were prepared from experimental results: dimensional and dimensionless data sets. These data sets were used to develop models to predict the aeration efficiency of the Montana flume. Two types of models were developed, which include the classical model viz. MNLR and machine learning model including ANFIS with various mfs and ANN. The aeration efficiency was also predicted using other proposed mathematical models. The following major conclusions can be derived from the study: • The result of MNLR was far better than other proposed classical models except that of Tiwari [8] for this dataset.
The cc and rmse of MNLR for dimensional parameters were 0.9757 and 0.0276, respectively. In contrast, the cc and rmse for the dimensionless dataset were 0.9488 and 0.046, respectively. Therefore, the MNLR model in the dimensional dataset performs better than the dimensionless dataset. However, both dimensional and dimensionless MNLR models could be utilized to predict the Montana flume's aeration efficiency for this dataset. • In the dimensional dataset, ANFIS_gbellmf outperformed the proposed dimensional models. But in the case of the dimensionless dataset, ANFIS_trimf was giving the best results among all considered dimensionless mfs-based ANFIS. Although, all proposed mfs-based ANFIS of dimensional and dimensionless models give good results in both training and testing dataset and could be employed to predict Montana aeration efficiency for the current data range. • In the case of a dimensional dataset, the ANN, with one hidden layer having 12 neurons, was giving good results, while for a dimensionless dataset, the ANN with two hidden layers with 12 neurons is performing well. • In general, the results of the proposed machine learning models were better than that of considered conventional models. Among all models, machine learning and conventional ANN performed best. • Sensitivity analysis shows that the Reynolds number was the most affecting parameter in predicting the aeration efficiency of the Montana flume in the dimensionless dataset. However, discharge per unit width, q input parameter was influenced the most for a dimensional dataset.
• Parametric analysis shows that aeration efficiency (E 20 ) increases with an increase in discharge per unit width (q) and Reynolds number (R e ).
Author contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by AT, NKT, CSPO and SR. The first draft of the manuscript was written by AT and all authors commented on previous versions of the manuscript. All authors approved the final manuscript.
Funding The author declares that no funds, grants, or other support were received during the preparation of this manuscript.