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A vortex tube ejector comprises a tube with a slitted crown that lies flush across the entire width of a channel bed surface. The bed and suspended loads are ejected with minimal flushing water through the slit with the same efficacy as any other alternative extractor. The whirling flow phenomena through the vortex duct are very complex, so ordinary classical models have results contrary to required design guidelines. So, the machine learning (ML) models of the artificial neural network (ANN), deep neural network (DNN), gradient boosting machine (GBM), stacked ensemble (SE), and adaptive neuro-fuzzy inference system (ANFIS) are used to predict vortex tube trapping efficiency (VTE). The input dataset takes the size of the sediment (Sz), concentration (I) of the sediment, the ratio of slit thickness to diameter of the tube (th/dia), and extraction ratio (Extro) while trapping efficiency (TE) is taken as output. Based on statistical assessments, GBM appears to be better than all proposed models. However, other proposed ML models give comparable performance. The classical models, multivariate linear, and nonlinear regression techniques also provide comparatively good results. According to sensitivity analyses, the extraction ratio is the most relevant parameter in evaluating the VTE.

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  • An experimental test of the vortex tube sediment ejector efficiency (VTE) has been studied.

  • Machine learning models are used for estimating the VTE.

  • The VTE by DNN, GBM, SE, NFS, NN, etc., models are compared.

  • The GBM model performs the best among all models.

  • The extraction ratio is the most sensitive parameter.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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deep neural network (DNN), gradient boosting machine (GBM), neural network (NN), stacked ensemble (SE), vortex tube trapping efficiency (VTE)
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Rivers are one of the world's most vital freshwater supplies and the primary source of many civilizations and the planet's existence. They contribute to a country's economic growth by providing irrigation, navigation, flood control, and energy generation. Sediments in Himalayan rivers are a big issue in many nations (UPIRI 1975). Controlling sediments entering irrigation and power canals delivered by river water is one of the primary issues in irrigation and hydraulic system design as canals have a lower carrying capacity than rivers, and the river water conveyed to irrigation canals has a considerable amount of silt, and as a result irrigation canals lose a large amount of carrying capacity and desilting is both expensive and unviable in terms of further affecting continuous irrigation supplies (Tiwari et al. 2020a, 2020b). Another issue arises when power canals with silt feed water to hydroelectric plants, which causes turbine blades to deteriorate and break (Ranga Raju & Kothyari 2004; Raju & Kothyari 2005). Therefore, the excess sediments must be removed from the canal or excluded at the headworks to maintain the canal carrying capacity. Various conventional sediment-controlling hydraulic devices, including either silt ejectors installed downstream of the head regulator in the upstream of the canal or silt excluders provided at the entrance of the off-taking canals, are used to remove and restrict the quantity of sediment that enters the canals. The latter devices act as a preventive method, while the former behave as a curative one.

Controlling sedimentation in the canal head reach with a sediment extractor is usually a cost-effective technique compared with removal manually or by machine by closing the channel. According to the usage, tunnel-type silt ejectors (Tiwari et al. 2020a, 2020b, 2022) and settling basins (Raju et al. 1999; Athar et al. 2002; Singh et al. 2008) are used. A tunnel type has a removal efficiency of about 40% with 15%–30% escape discharge, which is very uneconomical as a sizeable amount of water is lost and it is not suitable for a region where the water problem is acute. A settling basin suffers from multiple drawbacks as it needs relatively large space, long residence time, and frequent interruption during physical cleaning, but vortex settling chambers are free from these difficulties described above. Still, the limitations of the vortex-type settling chamber are that its design and structure are very complex and used for sediment removal for a limited amount of sediment-laden water.

Nevertheless, the vortex tube ejector has the edge over other available alternative structures as it is modest, efficient, and does not undergo the shortcomings that other alternative desilters suffer. It is employed to eject bed and suspended load sediments in the face of an acute water crisis in a region. There is a severe water supply problem; with minimal water loss, around 5% to 10% is required to flush the sediments (Orak & Asareh 2015). In addition, the size of the vortex tube ejector is very small and easy to install when compared with other desilters treating an equal bulk of sediment-laden water. Thus, the construction cost, which includes the installation cost of the vortex tube ejector, is just a smaller part of the cost needed for constructing other desilters to extract similar sediment particles. So, in many cases, the vortex tube ejector is a cost-effective and water-saving substitute compared with other desilter devices (Atkinson 1994a, 1994b).

The vortex tube sediment ejector (Atkinson 1994a, 1994b; Moradi et al. 2013; Dashtbozorgi & Asareh 2015; Orak & Asareh 2015) is a structural fluidic device for trapping or ejecting sediments that enter irrigation and power canals. It works on the principle that sediments lying near the bed level are forced to pass through the vortex tube from the slit, and sediments are ejected with the vortex (spiral flow) formed in it. The sediment-capturing flushing efficiency of a vortex tube ejector is defined as the percentage of the sediment load conveyed through a canal that is removed. The ratio of the quantity of sediment load transported by a canal to that removed is used to calculate the vortex tube silt ejector trapping efficiency. The vortex-chamber-type sediment extractor was studied by Nguyen & Jan (2010) to separate fine sediment from raw water utilized in agriculture and water treatment.

The bulk of bed-material sediments may frequently be removed from a canal at the cost of 10% to 20% of the canal flow. The trapping effectiveness of vortex tube ejectors has been investigated using physical and numerical models (Atkinson 1994a, 1994b). Singh et al. (2021) employed ANFIS, GPR, M5P, RF, and MLR models to predict the trapping efficiency of the vortex tube sediment ejector, while Sharafati et al. (2021) used hybrid neuro-fuzzy models to estimate tunnel sediment ejector efficiency. Asareh & Kamanbedast (2018) did the experimental investigation of a vortex tube orifice for sediment trapping. Athar et al. (2005) found a relation for the sediment removal efficiency of a vortex-chamber-type sediment extractor.

Aims and objectives

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Many researchers have used physical modeling and tried to enhance the effectiveness of the VTE by suggesting several geometrical changes to the vortex tube ejector. However, the results of trapping efficiency are found to be unsatisfactory and remain inconclusive. The vortex tube design in vogue is based on rule of thumb, empirical equations, past experiences, physical model studies, and conventional techniques. Since the flow mechanism through the vortex tube is so complicated and nonlinear, it becomes a challenge for ordinary empirical relations to create a model that can accurately forecast trapping efficiency. Furthermore, several scholars in water resources have also used ML algorithms in recent years (Nayak et al. 2004; Singh et al. 2008; Kumar et al. 2018) to analyze the models.

This paper investigates proposed ML algorithms, and traditional and existing published models to predict the sediment trapping efficiency of the VTE. The sediment trapping effectiveness of the VTE is affected by several factors, including the diameter-to-slit-thickness ratio, angle of deviation, sediment size, the concentration of sediment, and the amount of flushing discharge. Flow behavior in the vortex tube ejector is very complex, making it challenging to estimate sediment trapping efficiency accurately by ordinary traditional models. To address these issues, the current research focuses on ML approaches such as neural networks (NN), deep neural networks (DNN), stacked ensemble (SE), gradient boosting machines (GBM), and neuro-fuzzy systems (NFS), which can track the nonlinear behavior of the flow. The estimated trapping efficiencies of the VTE using ML methods are compared amongst themselves and with traditional mathematical formulae developed in the present study (multivariate linear regression, MVLR; multivariate nonlinear regression, MVNLR) and existing published models in the texts (Tiwari et al. 2020a, 2020b).

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Machine learning (ML) is a discipline of computer science engineering that investigates and generates computer systems that can perform activities that ordinarily need human intelligence. ML is a vast, parallel-distributed processor with its method of learning and remembering experimental knowledge. These models attempt to attain correct overall performance by interconnecting many essential computational devices or neurons. The net structure, function of nodes, and education or learning set of rules are used to make artificial intelligence models exact. This collection of principles establishes an initial set of weights and explains how weights should be adjusted during the training phase to improve performance.

Artificial neural network (ANN)

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Neural networks (NNs) are computer systems based on the biological neural networks that make up human brains. A neural network (NN) is made up entirely of connected units known as neurons, with each connection (synapse) between neurons capable of transmitting a signal to any other neuron. The signals can be processed by the receiving (postsynaptic) neurons. Real numbers generally represent neurons, usually ranging from 0 to 1. The weights of neurons and synapses may change as learning progresses, depending on the intensity of the signal. The NN is a branch of AI capable of solving nonlinear function estimates, pattern identification, clustering, and simulation. Nonlinear mapping is just a ‘black-box’ modeling approach. The principal elements of its architecture are three layers, viz. the input layer, hidden layer, and output layer, and connecting weights and bias along with activation function and summation node. Its actions are split into two phases: the first is learning (training/calibrating), and the second is generalization (testing/validating). Weights and biases in the training network are employed to acquire the desired results by minimizing the error function. The general structure of the NN is shown in Figure 1.
Figure 1
Structure of the NN.
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Structure of the NN.

Figure 1
Structure of the NN.
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Structure of the NN.

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It replicates the human nervous system, and each layer consists of several nodes or neurons. In this, one input layer contains the number of input variables available and weighted connections between them. After that, it forms a network-like structure that distributes and processes the data in one or more hidden layers. The overall process occurs by multiplying the input variables (nodes) by corresponding weights. After adding up, the total output (Ex) is obtained, which is the final process of the network:

Ex is total output, is the interconnection weight (y to x), and is the input value. The present study uses an NN model's hidden layer of eight neurons.

Deep neural network (DNN)

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The DNN is also known as a deep multilayer neural network as it consists of more than one hidden layer, and the number of nodes generated is large compared with ANN. Neural networks come in diverse shapes and sizes but always have similar essential components. It also works similarly to the human brain and can be trained again to other machine learning algorithms. In this, the number of input variables or nodes multiplied by the weights and biases adjusts and moves forward under a particular activation function, such as sigmoid rectified linear activation unit (ReLU), tanh, etc., randomly initialized. This overall process takes place on various hidden layers, and backpropagation of error is used to adjust bias and weights to activate the hidden neurons shown in Figure 2. It computes their internal factors in forwarding passes. Then it iteratively polishes during backpropagation to extract input data structures. It begins by adding the input values, which are then guided over the activation function to generate the output. Any differential function can be utilized as the activation function. All deal-out nodes in DNN are organized into layers. There are no nodes in the identical layer that are connected. A DNN usually has an input layer that acts as a distribution structure for data fed into the network and is not used for any processing, and is followed by hidden layers with one or more processing levels. The resulting processing layer is the output layer. When an entire training dataset has traveled over the neural network in both forward and backward directions, it is called an epoch. To learn more sophisticated aspects of the data, activation functions bring nonlinearity into the neural network. The rectified linear activation function (RELU) is a piecewise linear function regarded as a landmark in the design of a DNN. If the input value is positive, the RELU activation function outputs that value; otherwise, the output is zero and straightforward to train. With DNN, it has been discovered that using RELU outperforms alternative activation functions. The rectifier activation function is given as:
where is neuron input, the number of input variables or nodes multiplied by the weights and bias, which adjusts and moves forward under a particular activation function such as sigmoid, ReLU, and tanh, randomly initialized. This overall process takes place on various hidden layers, and backpropagation of error is used to adjust biases and weights to activate the hidden neurons shown in Figure 2. The learning rate is the user-defined parameter used to update and alter network weights during the training phase of a DNN. It is chosen randomly, depending on previous experiences and published works.
Figure 2
Structure of the DNN.
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Structure of the DNN.

Figure 2
Structure of the DNN.
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Structure of the DNN.

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The adaptive learning rate approaches allow a DNN's learning rate to be adjusted adaptively throughout the training process. During the training of a DNN, optimum values of numerous user-defined boundaries must be achieved, i.e., the optimization algorithm, activation function, number and kind of hidden layers, number of neurons in the hidden layer, and number of epochs are all parameters to consider.

Gradient boosting machine (GBM)

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The GBM is an ensemble-learning-based ML method from the boosting family. In ensemble techniques, different ML algorithms are used based on averaging the whole model, whereas boosting methods work on a productive approach to ensemble development. To connect with the analytical structure, a gradient descent boosting procedure was obtained (Friedman et al. 2000; Friedman 2001). GBM is a forward machine learning ensemble method (Natekin & Knoll 2013). It combines numerous simple model predictions from multiple decision trees (weak learners), and the resulting final prediction is obtained, as shown in Figure 3. In this, a parallel tree is built so that the regression trees generate a sequent feature of data in a distributed way. The error generated in each layer or iteration or tree is taken into account, and a new tree is generated in such a way that the new tree is superior to previous trees. As a result, all succeeding decision trees are built on earlier trees' accountability. This is how the trees in a GBM algorithm are built sequentially and are created in this order. Its advantage is that it is highly flexible and customizable to any particular modeling having a highly data-driven task. By trial and error, the most appropriate model is generated without difficulties and is easy to implement. Ensemble models are powerful for various prediction tasks since they produce superior accuracy outcomes than single strong machine learning models. This can be used accurately to forecast the vortex tube ejector trapping efficiency (VTE). The models generated from this can also help researchers better understand the effect of various input parameters on trapping efficiency. Other machine learning algorithms, such as ANN and DNN, work on a specific learned pattern distributed among neurons. In contrast, in the case of the boosted ensemble, the base learners play an essential role in learning from previous models and collecting the pattern sequentially, increasing the model's accuracy.
Figure 3
Structure of GBM.
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Structure of GBM.

Figure 3
Structure of GBM.
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Structure of GBM.

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Using a decision tree in the GBM algorithm is a computationally practical approach for recording interactions between variables. A decision tree aims to use a tree-based rule system to segment the space of input variables. Each split in the tree corresponds to an if–then rule applied to a single input variable. The interactions between predictor variables are naturally encoded and modeled by the structure of a decision tree. The number of splits, or homogeneously, the interaction depth, is a typical parameter for these trees. It is also feasible to have one of the variables split numerous times in a row. A tree stump is a specific example of a decision tree having just a single split (i.e., the tree with dual terminal neurons). As a result, if one needs to fit an additive model using tree base-learners, the tree stumps can be used. Minor trees and tree stumps produce precise results in various practical applications (Jiang 2002).

Furthermore, there is much indication that even complicated models with a lot of tree structure (interaction depth > 20) are not any better than compact trees (interaction depth 5). One of the essential properties of decision trees is that they are always designed to extrapolate the function with the constant value. As a result, even a primary function, like a straight line with a non-zero angle, cannot be accurately represented with a single decision tree.

Stacked ensemble (SE)

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The SE algorithm, initially projected by Wolpert (1992) and defined by Breiman (1996), is also recognized as a stacked generalization, as it presents the notion of a meta-learner, a unique technique of blending different models. A typical stacking is a dual-layer assembly in which the next layer's meta-model (level-1 model) is used to aggregate the outcome of the first layer's base learners (level-0). Previous research has revealed that the precision and variety of base learners are essential factors in the success of a stacking model (Nath & Sahu 2019). It is expected that applying to stacking to unite numerous diverse base learners that can effectively adjust for each other's flaws will enhance projections when equated to base learners. As a result, choosing suitable foundation learners is a vital aspect of stacking. Most studies have assessed models purely on their precision, whereas diversity has not been efficiently defined (Wang et al. 2020). In the present research, base learners based on precision and variety are chosen, as shown in Figure 4.
Figure 4
Structure of SE.
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Structure of SE.

Figure 4
Structure of SE.
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Structure of SE.

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The data were randomly divided into calibration data (75%) and validation data (25%), and the training data were further subdivided into five folds. Four folds were selected for calibrating base learners for every five iterations, while the residual folds were kept out for trapping efficiency prediction. The meta-features were five-fold cross-validated predictions that acted as input parameters for the meta-learner. The number of parameters for calibrating the meta learner was equal to the number of base learners when original features were not included in the stacking. The GBM model, distributed random forest (DRF), and extremely random trees (XRT) model algorithms were employed as candidate base learners. Furthermore, some research suggests that a few base learners should be layered together rather than all accessible learners, with three or four base learners being optimum. As a result, in the present study, three-stack base learners are evaluated for predicting the performance of trapping efficiency.

Individual learners’ stacking performance was estimated using GBM, DRF, and XRT as base learners, comparable to numerous ensemble methods. The performance of related stacking models in calculating trapping efficiency was evaluated. The meta-model is a simple general linear model (GLM), which can deliver an even explanation of the projections of base models. The primary characteristics were used to train the base learners by characteristic significance values provided by the GBM, DRF, and XRT. Four essential features were used as supplementary inputs of the meta-learner in the stacking models.

Neuro-fuzzy system (ANFIS)

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NFS is a soft computing technique for modeling complex system issues based on input and output parameters frequently used to predict the outcome of a model. It is a hybrid of fuzzy logic and artificial intelligence techniques and behaves the same as our brains work on neurons, i.e., neural networks (Jang et al. 1997). Mamdani and Sugeno are two forms of fuzzy inference system; Sugeno is more commonly employed for mathematical analysis. Figure 5 depicts a schematic structural perspective. Grid partitioning and subtractive clustering are two forms of fuzzy inference system (FIS), and the membership function's result is linked to an input parameter; trimf, trapmf, gaussmf, gbellmf, and more input membership functions are available, and for output, linear and constant membership functions are available.
Figure 5
Structure of ANFIS.
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Structure of ANFIS.

Figure 5
Structure of ANFIS.
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Structure of ANFIS.

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The first-degree Sugeno fuzzy type is made up of four fuzzy sets (if–then):

Rule 1: if m is and n is then
(1)
Rule 2: if m is and n is then
(2)
Rule 3: if m is and n is then
(3)
Rule 4: if m is and n is then
(4)
where P1, P2, and Q1 and Q2 are fuzzy sets of input m and n, fuv (u, v = 1, 2); w is any constant value after the product of normalized firing strength and first-order polynomial.
Fuzzy layer 1: Every neuron is adaptive, and the fuzzification layer assigns membership rates to input and output, which are:
(5)
(6)
where m and n are crisp inputs, Pu and Qv are fuzzy sets, and a small, average, and large size membership function (MF) is applied, which might take any form like a triangular function, trapezoidal function, bell-shaped function, Gaussian function, and so on.
Rule layer 2: All neurons are fixed and marked or designated as ∏, which play a part in a basic multiplier with the following result:
(7)
and are the output of layer 2 and firing strength correspondingly.
Average neurons layer 3: The job of layer 3 is to normalize the calculated firing strengths by dividing each value by the overall firing strength.
(8)
Consequent nodes layer 4: Each adaptive point has a point (neuron) function, and the output is the yield of standardized firing strength and a first-degree polynomial, as shown below:
(9)
Output nodes layer 5: the neuron outcome in the layer is the total output of the structure.
(10)

Choice of membership function

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A membership function (MF) is applied, which could take any shape like a triangular function, trapezoidal function, bell-shaped function, Gaussian function, etc. In the present study the triangular function has been used as it predicts the best model, and it is defined below:
in which J, K, and L are scalar parameters on which the vector function m depends.

Classical models

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Multivariate linear regression (MVLR)

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MVLR is also known as multiple linear regression, and it is applied to analyze or generate a relationship between one or more independent variables and one dependent variable. It can be generalized as follows:
(11)
in which Y is the output, c0 is a constant variable (when all variables = 0), are regression coefficients,
are the independent variables. By using this method, a relationship is developed and is given as below:
(12)
in which TE, Sz, I, th/dia and Extro are the trapping efficiency (%) of the vortex tube, sediment size (mm), the concentration of sediment, the ratio between slit thickness and diameter of the tube, and extraction ratio, respectively.

Multivariate nonlinear regression (MVNLR)

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MVNLR is applied to forecast relationships with the multiple variables varying in a curvilinear way. In this, the relationship is generated between dependent and independent variables. It can be generalized as follows:
(13)
where Y is the output (dependent parameter), q0 is the proportionality constant, x1,x2, and x3 are independent parameters that can be taken as input variables, y1, y2, and y3 are exponential constants.
By using this method, a relationship is developed and is given below:
(14)

All proposed traditional models (derived in the present study and existing in text) are listed in Table 1.

Table 1

Conventional models, proposed for several hydraulic structures used for sediment trapping efficiency

Sr NoModel originModelDescription
1 Singh (2016)   TE of tunnel-type ejector 
2 Tiwari et al. (2022)   TE of tunnel-type ejector 
3 MVLR  Present study 
4 MVNLR  Present study 
Sr NoModel originModelDescription
1 Singh (2016)   TE of tunnel-type ejector 
2 Tiwari et al. (2022)   TE of tunnel-type ejector 
3 MVLR  Present study 
4 MVNLR  Present study 
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Methodology

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Experimental setup details

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The experiment was conducted in the Hydraulic Laboratory of the National Institute of Technology, Kurukshetra, India, in a rigid channel of 30 cm in width, 50 cm in depth, and 1,490 cm in length, having a maximum discharge of 16 l/s with a re-circulating system from the sump to the overhead tank through which water flows under gravitational force. The vortex tube is fixed in the channel at a distance of 3.92 m from the inlet of the main channel in such a way that no particle will settle down before reaching the ejector, to obtain maximum efficiency. The vortex tube is fitted with a control valve at the end, from where the sediment-laden water comes out so that the extraction discharge from the vortex tube is regulated and sediments are collected in the trapping device. The water is conveyed back to the escape channel. The sediments collected in the trapping device are dried, and their weight is determined to estimate the efficiency of the vortex tube ejector. The velocity in the channel is measured with the help of the current-meter, and a pointer gauge of sensitivity of 0.01 cm is used to measure flow depth. A Cipoletti weir is also employed to measure the discharge in the channel. The flow velocity in the main channel varies from 29 to 33 cm/s while depth varies from 9 to 16 cm. The experimental matrix is summarized in Table 2. A schematized view of the experimental setup is demonstrated in Figure 6.
Table 2

Summary of the experimental matrix

Velocity (cm/s)dia (cm)th (cm)th/diaSz (mm)Extro (%)
33.0 4.00 0.500 0.125 0.840, 0.504,
0.424, 0.220 		7.523, 3.44 
33.0 5 0.625 0.125 0.840, 0.504,
0.424, 0.220 		3.125, 1.75 
33.0 1.87 0.23 0.125 0.840, 0.504,
0.424, 0.220 		2.5, 1.247 
29.0 4.40 1.32 0.300 0.840, 0.504,
0.424, 0.220 		7.50, 3.94 
29.0 2.80 0.84 0.300 0.840, 0.504,
0.424, 0.220 		3.75, 2.37 
29.0 1.87 0.56 0.300 0.840, 0.504,
0.424, 0.220 		2.94, 1.56 
Velocity (cm/s)dia (cm)th (cm)th/diaSz (mm)Extro (%)
33.0 4.00 0.500 0.125 0.840, 0.504,
0.424, 0.220 		7.523, 3.44 
33.0 5 0.625 0.125 0.840, 0.504,
0.424, 0.220 		3.125, 1.75 
33.0 1.87 0.23 0.125 0.840, 0.504,
0.424, 0.220 		2.5, 1.247 
29.0 4.40 1.32 0.300 0.840, 0.504,
0.424, 0.220 		7.50, 3.94 
29.0 2.80 0.84 0.300 0.840, 0.504,
0.424, 0.220 		3.75, 2.37 
29.0 1.87 0.56 0.300 0.840, 0.504,
0.424, 0.220 		2.94, 1.56 
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Figure 6
Schematic diagram of the experimental setup.
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Schematic diagram of the experimental setup.

Figure 6
Schematic diagram of the experimental setup.
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Schematic diagram of the experimental setup.

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Data division

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A total of 142 readings were collected from experiments that were performed in the laboratory. This total data is divided into two parts randomly, out of which 75% of data (106) is training (calibrating), and 25% of data (36) is used for testing (validation). Four input parameters are as follows: the size of sediments (Sz) in mm, the concentration of sediment (I) in ppm, the th/dia ratio in which th denotes slit width in cm and dia is the diameter of the vortex tube in cm, Extro is extraction ratio (%), TE denotes trapping efficiency (%), which is the output obtained from the system. The statistical analysis of calibrating and validation data is shown in Table 3.

Table 3

Calibrating and validation data

ParametersUnitsMinMaxMeanStdKurtosisSkewness
Calibrating data (training) 
Sz mm 0.220 0.850 0.493 0.229 −1.018 0.381 
I mg/l 209.000 475.000 333.276 84.217 −1.243 0.122 
Extro % 1.247 7.523 3.412 1.842 −0.082 0.958 
th/dia – 0.125 0.300 0.210 0.087 −2.038 −0.019 
TE % 16.900 85.600 37.711 15.143 0.909 1.089 
Validation data (testing) 
Sz mm 0.220 0.850 0.497 0.229 −0.994 0.423 
I mg/l 209.000 472.000 351.641 82.466 −1.082 −0.049 
Extro % 1.247 7.523 3.162 1.715 2.137 1.502 
th/dia – 0.125 0.300 0.210 0.088 −2.108 0.053 
TE % 17.400 83.200 38.453 16.102 0.749 0.909 
ParametersUnitsMinMaxMeanStdKurtosisSkewness
Calibrating data (training) 
Sz mm 0.220 0.850 0.493 0.229 −1.018 0.381 
I mg/l 209.000 475.000 333.276 84.217 −1.243 0.122 
Extro % 1.247 7.523 3.412 1.842 −0.082 0.958 
th/dia – 0.125 0.300 0.210 0.087 −2.038 −0.019 
TE % 16.900 85.600 37.711 15.143 0.909 1.089 
Validation data (testing) 
Sz mm 0.220 0.850 0.497 0.229 −0.994 0.423 
I mg/l 209.000 472.000 351.641 82.466 −1.082 −0.049 
Extro % 1.247 7.523 3.162 1.715 2.137 1.502 
th/dia – 0.125 0.300 0.210 0.088 −2.108 0.053 
TE % 17.400 83.200 38.453 16.102 0.749 0.909 
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Various modeling approaches are used to evaluate the training and testing datasets using root mean square error (RMSE), coefficient of correlation (CC), mean square root (MSE), and determination coefficient (R2).

Mean square error (MSE)

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For estimating the excellence of the numerical matrix, MSE is used and is evaluated as:
(15)
where = mean observed values, = mean predicted values, p = number of observations.

Root mean square error (RMSE)

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RMSE is calculated by the square root of MSE and is evaluated as:
(16)

Coefficient of determination (R2)

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For estimation of the excellence of numerical prediction, R2 is used and is evaluated as:
(17)

Coefficient of correlation (CC)

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For estimation of the excellence of numerical prediction, CC is used and is evaluated as:
(18)

Mean absolute deviance (MAD)

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For estimation of the excellence of numerical prediction, MAD (%) is used and is evaluated as:
(19)
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Optimization of user-defined parameters produced by applying several trials on the training and corresponding testing data set is used to implement DNN, ANN, GBM, SE, and NFS to predict the best model because the tuning stage of the primary parameters is a crucial part of creating an effective soft computing model. The optimum values of user-defined parameters attained in the prediction of vortex tube sediment trapping efficiency are shown in Table 4. Various statistical evaluation criteria, including the root mean square error (RMSE), mean average deviation (MAD), coefficient of correlation (CC), and coefficient of determination (R2), were used to determine the performance of the proposed models. Lower RMSE and MAD values indicate the best model estimation, while higher CC and R2 values indicate a good relation between input and output variables. The DNN, ANN, GBM, SE, and NFS models were chosen as the most appropriate models for building the vortex tube silt ejector. Singh (2016) and Tiwari et al. (2022) examined the gathered data with the mathematical regression formula. Also, they compared the outcomes with the produced multivariate linear and nonlinear regressions to determine the best model. The performance evaluation of all proposed models is listed in Table 5.

Table 4

Optimal values of user-defined parameters

ML-based modelsUser-defined parameters
ANFIS Number of input mf = 4, MF type = constant, fuzzy system = Sugeno-type, optimization = constant, and epochs = 20 
ANN Hidden layer = 1, number of neurons = 8, and functions used = traincgf 
SE Meta-model = GLM, nfolds = 5, base models = DRF, XRT, GBM 
GBM Nfolds = 5, number of decision trees = 111, depth of decision trees = 4, distribution = Gaussian, learning rate = 0.1 
DNN Nfolds = 5, rho = 0.99, epochs = 10,000, activation function = rectifier with dropout (RELU), hidden layer = 3 with 50 nodes each 
ML-based modelsUser-defined parameters
ANFIS Number of input mf = 4, MF type = constant, fuzzy system = Sugeno-type, optimization = constant, and epochs = 20 
ANN Hidden layer = 1, number of neurons = 8, and functions used = traincgf 
SE Meta-model = GLM, nfolds = 5, base models = DRF, XRT, GBM 
GBM Nfolds = 5, number of decision trees = 111, depth of decision trees = 4, distribution = Gaussian, learning rate = 0.1 
DNN Nfolds = 5, rho = 0.99, epochs = 10,000, activation function = rectifier with dropout (RELU), hidden layer = 3 with 50 nodes each 
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Table 5

Performance evaluation using various conventional and non-conventional methods

ModelsCCR2MSE (%)RMSE (%)MAD (%)
Training 
Tiwari et al. (2022)  0.390 0.152 319.782 17.882 13.178 
Singh (2016)  0.710 0.501 443.429 21.057 17.916 
MVLR 0.881 0.777 50.552 7.109 6.010 
MVNLR 0.903 0.816 96.013 9.798 8.573 
ANFIS 0.959 0.920 18.059 4.249 2.957 
ANN 0.880 0.775 53.368 7.305 5.721 
DNN 0.976 0.952 11.344 3.368 0.549 
SE 0.984 0.969 7.153 2.674 1.614 
GBM 0.994 0.988 2.780 1.667 1.103 
Testing 
Tiwari et al. (2022)  0.537 0.288 320.857 17.912 13.631 
Singh (2016)  0.678 0.461 514.699 22.686 19.224 
MVLR 0.952 0.907 33.615 5.797 4.709 
MVNLR 0.945 0.880 9.382 8.539 0.893 
ANFIS 0.864 0.747 70.459 8.394 6.686 
ANN 0.943 0.890 28.914 5.377 4.467 
DNN 0.988 0.977 5.713 2.390 1.371 
SE 0.996 0.992 2.513 1.585 1.070 
GBM 0.996 0.992 2.046 1.430 0.878 
ModelsCCR2MSE (%)RMSE (%)MAD (%)
Training 
Tiwari et al. (2022)  0.390 0.152 319.782 17.882 13.178 
Singh (2016)  0.710 0.501 443.429 21.057 17.916 
MVLR 0.881 0.777 50.552 7.109 6.010 
MVNLR 0.903 0.816 96.013 9.798 8.573 
ANFIS 0.959 0.920 18.059 4.249 2.957 
ANN 0.880 0.775 53.368 7.305 5.721 
DNN 0.976 0.952 11.344 3.368 0.549 
SE 0.984 0.969 7.153 2.674 1.614 
GBM 0.994 0.988 2.780 1.667 1.103 
Testing 
Tiwari et al. (2022)  0.537 0.288 320.857 17.912 13.631 
Singh (2016)  0.678 0.461 514.699 22.686 19.224 
MVLR 0.952 0.907 33.615 5.797 4.709 
MVNLR 0.945 0.880 9.382 8.539 0.893 
ANFIS 0.864 0.747 70.459 8.394 6.686 
ANN 0.943 0.890 28.914 5.377 4.467 
DNN 0.988 0.977 5.713 2.390 1.371 
SE 0.996 0.992 2.513 1.585 1.070 
GBM 0.996 0.992 2.046 1.430 0.878 
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The dataset was derived from experimental observations, and modeling strategies included the suggested conventional methods (MVLR, MVNLR), existing prediction equations, and soft computing techniques (DNN, ANN, ANFIS, GBM, SE). A total of 142 observed datasets were used in this experiment. The datasets were randomly divided into two groups: training data (106) and testing data (36).

Results of the ANN model

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The present study develops the ANN model using the trial-and-error method. The ANN is made up of multiple layers, each with its own set of neurons (nodes). A weighted connection is used to connect the layers. In a typical artificial neural network, three layers are formed; the first layer represents input, the hidden (middle) layer evaluates input weights, and the last layer is the output layer. The ANN was built in three stages: the first stage required the preparation of training data, the second stage required varied placement and assembly of effective network topologies, and the third stage required testing (validating).

The number of neurons and hidden layers are chosen using hit and trial approaches. A total of 12 models are tried for prediction by the ANN, such as trainlm, trainer, trainbfg, trainscg, trainrp, traincgb, traincgf, traincgp, trainoss, traingdx, traingdm, and trained, and the best one of them is selected, which is the traincgf model, which gives the best-desired results nearest to the actual value of trapping efficiency for the vortex tube ejector. This model consists of one hidden layer in which there are eight neurons presented. The optimal values of the user-defined parameters of the ANN are described in Table 4. The random data division is carried out at epoch 23 and six validation checks. Figure 7 depicts the ANN-based model scattered plot between actual trapping efficiency (TE) and its predicted values for the training and testing datasets. It is observed that predicted points for the testing datasets are scattered near the ideal line. Further, by observing Table 5, it is clear that the ANN is performing well and could be used in predicting the VTE. The value of CC is higher, and error values are smaller.
Figure 7
Performance by ANN.
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Performance by ANN.

Figure 7
Performance by ANN.
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Performance by ANN.

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Results of the DNN model

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The vortex tube ejector trapping efficiency (VTE) is determined using the DNN technique. Here, also by the hit and trial method, various models are generated with different distributions of data in training and testing. The first stage in the deep learning approach is partitioning the data into training and testing, as previously described, and then the optimal value of nfolds is selected, and the number of epochs necessary to forecast the model with the lowest computing cost is determined. The primary optimized user-defined parameters are described in Table 4.

Figure 8(a) shows the scour history deviance for the testing and training datasets between deviance and the number of epochs. It can be seen from the graph that the deviation is asymptotic to the x-axis as the epoch reaches 10,000. Further, the scatter plot between observed and predicted VTE by DNN for the training and testing datasets is shown in Figure 8(b). It is evident from Figure 8(b) that all predicted values in both testing and training lie closely around the ideal line, implying that the DNN model is performing well. This claim is further supported by Table 5, which shows that in this model, the value of CC is higher, and the error values are lower.
Figure 8
(a) Scoring deviance of DNN, (b) performance of DNN.
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(a) Scoring deviance of DNN, (b) performance of DNN.

Figure 8
(a) Scoring deviance of DNN, (b) performance of DNN.
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(a) Scoring deviance of DNN, (b) performance of DNN.

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Results of GBM

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The data is analyzed using a gradient boosting machine for regression. The total dataset is divided into two parts, i.e., calibration (training) and validation (testing), in different proportions using the hit and trial method. The first optimized value nfolds has been found, and corresponding ntrees are calculated, which gives the desired results. The primary optimized parameters are shown in Table 4.

Figure 9(a) depicts the plot between the deviance and the number of trees for the testing and training. It is clear from the graph that the deviation is parabolic, and it becomes asymptotic at the value of 111 about the abscissa. Further, a scatter plot between actual and forecast values of the VTE by GBM for the training and testing datasets is shown in Figure 9(b). It is evident from Figure 9(b) that all predicted points in testing and training lie exactly around the ideal line barring a few exceptional points, which implies that the GBM model performs equally well. This claim is substantiated by observing from Table 5 that the value of CC is on the higher side and also the error values are also on the lower side.
Figure 9
(a) Scoring deviance of GBM, (b) performance of GBM.
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(a) Scoring deviance of GBM, (b) performance of GBM.

Figure 9
(a) Scoring deviance of GBM, (b) performance of GBM.
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(a) Scoring deviance of GBM, (b) performance of GBM.

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Results of the stacked ensemble

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In this section, to predict the VTE, the data are analyzed using a stacked ensemble. For the stacked ensemble, various base models are generated randomly, such as gradient boosting machine (GBM) and distributed random forest (DRF). An extremely randomized tree (XRT) and the training model are generated. After that, this training model is run from the meta-model, generalized linear model (GLM), and the predicted values are obtained. The optimal values of the principal parameters are shown in Table 4. Figure 10 depicts the ensemble-based model scattered plot between the actual VTE and its predicted values for the training and testing datasets. It is observed that all predicted points for the testing datasets lie near the ideal line while a few predicted training datasets are a little bit scattered. Further, by observing Table 5, it is clear that the stacked ensemble is performing well.
Figure 10
Performance of SE.
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Performance of SE.

Figure 10
Performance of SE.
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Performance of SE.

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Results of the ANFIS

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The current study developed the ANFIS model using the fuzzy Sugeno and Takagi methods. The ANFIS model is developed using trial-and-error procedures. The ANFIS model uses grid partition to test a variety of models for prediction. Input membership functions such as the trapezoidal function (trapmf), triangular function (trimf), generalized bell-shaped (gbellmf), the difference between two sigmoids (dsigmf), product of two sigmoids (psigmf), and gaussian2function are subdivided by grid partition (gauss2mf), and the Gaussian function (gaussmf) and a pi-shaped membership function (pimf) are used as input functions.

The model is trained using the FIS hybrid optimization approach with a zero-error tolerance. The logical operation constructs the model's 'AND' rule. For the ANFIS model, the output membership function is employed as a ‘constant’. The triangular function (trimf) is found to perform well among all-input-type membership functions (input mfs type), whereas other shaped mfs perform poorly and are ignored and not considered. The optimal values of the user-defined parameters are presented in Table 4.

Figure 11 shows the agreement diagram for both the training and testing datasets between observed values of the trapping efficiency of the vortex tube silt ejector and corresponding predicted values triangular membership function based on ANFIS. According to this diagram, the trapping efficiency training points are nearer the ideal line than the testing points. This contention is supported by Table 5, which shows that the value of CC in testing is lower than in training, and also its error values are higher in testing than in training.
Figure 11
Performance of ANFIS.
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Performance of ANFIS.

Figure 11
Performance of ANFIS.
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Performance of ANFIS.

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Results of MVLR, MVNLR, and conventional models

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The multivariate linear regression (MVLR) and multivariate nonlinear regression (MVNLR) models are developed, as shown in Equations (12) and (14), and their results are compared with equations given by researchers such as Singh (2016) and Tiwari et al. (2022). Scatter plots are drawn between the VTE's actual value and predicted values for the training and testing datasets presented in Figure 12. A perusal of Figure 12 shows that predicted data points by the MVLR and MVNLR models lie near the ideal line compared with the Singh (2016) and Tiwari et al. (2022) models. This implies that the MVLR and MVNLR models perform better than the Singh (2016) and Tiwari et al. (2022) models. This fact is also substantiated from a perusal of Table 5. Further, if it is compared between MVLR and MVNLR, the MVLR performs better than the MVNLR because the predicted points by the MVLR lie nearer the ideal line than those of the MVNLR. Further, the MVLR has fewer errors and more correlation values than the MVNLR. Both the models of Singh (2016) and Tiwari et al. (2022) under-predict as most of the predicted points lie below the ideal line, which is too far away.
Figure 12
Performance of MVLR, MVNLR, and other conventional models.
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Performance of MVLR, MVNLR, and other conventional models.

Figure 12
Performance of MVLR, MVNLR, and other conventional models.
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Performance of MVLR, MVNLR, and other conventional models.

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Comparison of results

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The models developed using the datasets of the vortex tube silt ejector are compared using the statistical appraisal parameters shown by Equations (15)–(19). All the models, viz. the MVLR, MVNLR, ANFIS, ANN, GBM, SE, and DNN models, efficiently predicted the trapping efficiency of the vortex tube. However, the GBM model outperformed all the proposed models in predicting the trapping efficiency of the vortex tube silt ejector, as the GBM model has the highest CC value and lowest error values, as shown in Table 5.

All proposed soft computing models performed better than conventional models in training, but in the case of testing, MVLR and MVNLR performed better than ANFIS. Figure 13(a) illustrates a scatter plot comparing the actual and predicted trapping efficiency (TE) of vortex tube ejector values utilizing soft computing models such as ANFIS, ANN, GBM, SE, and DNN. The predicted values for TE lie around the ideal line, as shown in Figure 13(a). Between the predicted and actual values of TE of the vortex tube silt ejector, four other error lines in the domains of ±20% and ±10% are also drawn. In both training and testing situations, most of the predicted values of TE by GBM and DNN are well within the ±10% error band from the perfect agreement line. However, some predicted values by the ANN, ANFIS, and SE models are beyond the ±10% error band. But all predicted points by the soft computing algorithms lie in the range of the ±20% band, except for a few predicted points for both the training and testing datasets. To reinforce the above statement, another graph, the Taylor diagram, is plotted, as shown in Figure 13(b). The Taylor diagram is employed for further investigating the precision of the proposed models, displaying the degree of agreement between the actual and predicted data points summarizing three distinctive statistics simultaneously, including CC, RMSE, and the stdv of the predicted and actual data points. From the perusal of Figure 13(b), the GBM model is very close to the actual value compared with the other employed models, followed by the SE and DNN models in both the training and testing phases. The above claim is further bolstered by observing Table 5, which shows that the estimated values of the GBM, SE, and DNN models are close to the actual values while the ANN and ANFIS models come after that. Table 6 shows the summary statistics of all proposed model predicted results for the training and testing datasets.
Table 6

Statistical summary details of predicted values for the proposed models

ApproachesMinMaxMeanStdvKurtosisSkewness
Training data 
Actual 16.800 85.100 37.711 15.143 0.909 1.089 
Tiwari et al. (2022)  14.977 41.047 26.428 6.129 −0.592 0.305 
Singh (2016)  7.343 45.694 19.794 7.688 0.718 0.829 
MVLR 12.619 75.957 37.711 13.352 0.198 0.780 
MVNLR 8.831 101.436 34.142 19.872 1.353 1.267 
ANFIS 18.226 85.099 37.711 14.529 1.446 1.222 
ANN 13.561 76.383 38.466 14.694 −0.076 0.791 
DNN 16.874 84.369 37.996 15.536 0.883 1.118 
SE 16.953 82.299 37.521 15.246 0.833 1.101 
GBM 17.591 80.089 37.564 14.647 0.698 1.051 
Testing data 
Actual 17.200 83.200 38.453 16.102 0.749 0.909 
Tiwari et al. (2022)  14.746 38.582 26.889 5.901 −0.643 0.057 
Singh (2016)  7.865 39.203 19.229 7.900 −0.074 0.683 
MVLR 14.904 73.605 35.613 13.825 0.870 0.910 
MVNLR 9.132 95.810 32.174 19.940 1.574 2.68 
ANFIS 18.064 95.506 37.831 16.443 3.214 1.595 
ANN 15.106 84.087 39.563 15.451 1.505 1.158 
DNN 17.486 83.263 38.390 15.619 1.012 1.080 
SE 16.395 82.840 39.053 16.517 0.601 0.866 
GBM 17.048 80.617 38.720 15.808 0.599 0.866 
ApproachesMinMaxMeanStdvKurtosisSkewness
Training data 
Actual 16.800 85.100 37.711 15.143 0.909 1.089 
Tiwari et al. (2022)  14.977 41.047 26.428 6.129 −0.592 0.305 
Singh (2016)  7.343 45.694 19.794 7.688 0.718 0.829 
MVLR 12.619 75.957 37.711 13.352 0.198 0.780 
MVNLR 8.831 101.436 34.142 19.872 1.353 1.267 
ANFIS 18.226 85.099 37.711 14.529 1.446 1.222 
ANN 13.561 76.383 38.466 14.694 −0.076 0.791 
DNN 16.874 84.369 37.996 15.536 0.883 1.118 
SE 16.953 82.299 37.521 15.246 0.833 1.101 
GBM 17.591 80.089 37.564 14.647 0.698 1.051 
Testing data 
Actual 17.200 83.200 38.453 16.102 0.749 0.909 
Tiwari et al. (2022)  14.746 38.582 26.889 5.901 −0.643 0.057 
Singh (2016)  7.865 39.203 19.229 7.900 −0.074 0.683 
MVLR 14.904 73.605 35.613 13.825 0.870 0.910 
MVNLR 9.132 95.810 32.174 19.940 1.574 2.68 
ANFIS 18.064 95.506 37.831 16.443 3.214 1.595 
ANN 15.106 84.087 39.563 15.451 1.505 1.158 
DNN 17.486 83.263 38.390 15.619 1.012 1.080 
SE 16.395 82.840 39.053 16.517 0.601 0.866 
GBM 17.048 80.617 38.720 15.808 0.599 0.866 
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Figure 13
(a) Scatter plot and (b) normalized Taylor diagram.
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(a) Scatter plot and (b) normalized Taylor diagram.

Figure 13
(a) Scatter plot and (b) normalized Taylor diagram.
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(a) Scatter plot and (b) normalized Taylor diagram.

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The relatively poor performance of traditional models compared with the proposed soft computing models is because these models do not have enough capacity to deal with all facets that are significant for a nonlinear and complex phenomenon that occurs during the flow through a vortex tube sediment ejector. In contrast, soft computing techniques employ ML algorithms that do not require any limiting assumptions on the form of the model, and further there is the fact that they can generalize, track and detect complex nonlinear relationships between dependent and independent variables.

Uncertainty study

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A quantitative assessment of the uncertainty in estimating the vortex tube trapping efficiency with the considered models is presented. The uncertainty study is used for the data of 142 observations employed in this study, which is utilized to create the suggested models. The uncertainty study describes the individual estimation error as ei = ni − mi. Computed estimation errors for the whole data are utilized to estimate mean, and standard deviation, of the prediction errors. A negative mean establishes that the prediction model underestimated the actual values, and a positive value shows that the equation overestimated the actual values. To quantify the uncertainty connected with the suggested models, the confidence band of prediction is achieved by the following equation (Najafzadeh et al. 2016; Ahmadianfar et al. 2022):
(20)

is the standard normal parameter at the level of significant level. A negative mean value exhibits that the forecast model underestimated the actual results, and a positive value shows that the estimated model overpredicted the actual values. By utilizing and Sde values, a confidence limit can be explained around the estimated error values with the Wilson score technique without continuity correction. Using ±1.96Sde yields a roughly 95% confidence band (Najafzadeh et al. 2016; Ahmadianfar et al. 2022). For comparing the uncertainty of the proposed models, all proposed models are presented in Table 7. It is evident from Table 7 that the GBM and ANFIS models in the soft computing technique provide underestimated predictions while the SE, DNN, and ANN models offer overestimated values of vortex tube sediment trapping efficiency. However, all the proposed traditional models MVLR, MVNLR, Tiwari et al. (2022), and Singh (2016) deliver underestimated prediction values of trapping efficiency. The amount of uncertainty band value determines the degree of accuracy. The lowest value of the prediction uncertainty band indicates the best-performing model, and accordingly, the model's performance accuracy has been ranked. The lowest prediction uncertainty (6.319) is observed in the results obtained by the GBM, so it is ranked first (1). In contrast, the Tiwari et al. (2022) model provides the highest uncertainty (54.369) and is ranked last (9). The remaining proposed models lie between ranks 1 and 9, as shown in Table 7.

Table 7

Statistical features of prediction errors of proposed models

ModelsemeanSdeUncertainty-band (⁠⁠)Ranking
GBM −0.035 1.610 3.120 −3.195 6.319 1 
SE 0.024 2.437 4.800 −4.752 9.552 2 
ANFIS −0.168 5.696 10.996 −11.333 22.330 4 
ANN 0.851 6.808 14.194 −12.491 26.686 6 
DNN 0.190 3.139 6.342 −5.961 12.304 3 
MVLR −0.769 6.759 12.479 −14.017 26.497 5 
MVNLR −4.303 8.554 12.463 −21.069 33.533 7 
Tiwari et al. (2022)  −11.359 6.759 15.825 −38.544 54.369 9 
Singh (2016)  −18.271 11.393 4.060 −40.602 44.662 8 
ModelsemeanSdeUncertainty-band (⁠⁠)Ranking
GBM −0.035 1.610 3.120 −3.195 6.319 1 
SE 0.024 2.437 4.800 −4.752 9.552 2 
ANFIS −0.168 5.696 10.996 −11.333 22.330 4 
ANN 0.851 6.808 14.194 −12.491 26.686 6 
DNN 0.190 3.139 6.342 −5.961 12.304 3 
MVLR −0.769 6.759 12.479 −14.017 26.497 5 
MVNLR −4.303 8.554 12.463 −21.069 33.533 7 
Tiwari et al. (2022)  −11.359 6.759 15.825 −38.544 54.369 9 
Singh (2016)  −18.271 11.393 4.060 −40.602 44.662 8 
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This finding is compatible with those achieved from statistical performance modeling criteria, scatter plots and Taylor diagrams that the GBM model outperforms all considered models, followed by the SE and DNN models, while the ANN model performance is poor amongst soft computing models. However, all proposed soft computing models, along with the MVLR and MVNLR models, perform well and can be utilized for the prediction of the trapping efficiency of the vortex tube ejector.

Sensitivity study

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Figure 14 shows the relative relevance of the input parameters using the two best-performing models of DNN and GBM. The extraction ratio (Extro) is found to be the most sensitive parameter as both the DNN and GBM models read out a maximum value on the scale of importance, whereas sediment concentration (I) is found to be the least sensitive parameter since it indicates a minimum value on the same scale of importance.
Figure 14
Sensitivity study of input parameter.
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Sensitivity study of input parameter.

Figure 14
Sensitivity study of input parameter.
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Sensitivity study of input parameter.

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The modeling of trapping efficiency of the vortex tube silt ejector is examined using proposed conventional approaches such as MVLR, MVNLR, existing empirical relations, and soft computing techniques such as ANN, DNN, GBM, SE, and ANFIS. The following vital conclusions are drawn from the above studies:

  • 1.

    The correlation coefficient (CC), root mean square error (RMSE), mean square error (MSE), and mean absolute deviation (MAD) are used to evaluate the performance of the five proposed soft computing techniques. These five soft computing approach models used experimental datasets to forecast the vortex tube silt ejector's trapping efficiency. Compared with the other models, the GBM model performed the best with CC = 0.994 and the lowest RMSE = 1.6675, MSE = 2.78, and MAD = 1.103 for training, and the maximum value CC = 0.996 and lowest values of RMSE = 1.43, MSE = 2.046 and MAD = 0.878 for testing. This study further showed that both the SE and DNN models have an adequate potential for the trapping efficiency of the vortex tube silt ejector. They are the second best-performing models after the GBM model for the present dataset.

  • 2.

    The ANN and ANFIS models can be used to estimate the trapping efficiency of the vortex tube silt ejector. However, they are found to be the least effective models compared with the other suggested soft models. The type of function selected and the number of neurons in the ANN model are significant user-defined parameters in the prediction of trapping efficiency. The selection of membership functions is a key tuning parameter in deciding the predicted accuracy of the ANFIS model for this dataset.

  • 3.

    The MVNLR and MVLR models do well, but the MVLR model performs better than the MVNLR model in forecasting trapping efficiency. Nevertheless, the Singh (2016) and Tiwari et al. (2022) models perform very poorly as both have high errors and low correlations.

  • 4.

    The uncertainty analysis also suggests that the GBM is superior to the other considered models. According to the uncertainty band, all proposed models are ranked according to their performances.

  • 5.

    The sensitivity analysis reveals by the two best-performing models of the DNN and GBM that extraction ratio (Extro) is the most sensitive parameter. At the same time, concentration (I) is the least sensitive.

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The first author (Shubham Kumar) is thankful to the Ministry of Human Resources, Government of India, and the Director, National Institute of Technology Kurukshetra (Haryana), for monitoring and supporting the present work of the Master Degree (MTech) scholarship (32012517).

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Data cannot be made publicly available; readers should contact the corresponding author for details.

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The authors declare there is no conflict.

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