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In this paper, the sparrow search algorithm is used to predict the discharge coefficient (Cd) of the triangular side orifice for the first time. Dimensionless parameters influencing the size of Cd of side orifices are obtained as input values and discharge coefficient as output values of the model. The results show that the determination coefficient R2 is 0.973, the root means square error RMSE is 0.0122, and the average absolute percentage error is 0.010% in the testing phase. The model has high forecast accuracy, strong generalization ability and higher accuracy than other models and traditional empirical formulas. Quantitative analysis by Sobol's method shows that the ratio W/H of top orifice height to side orifice height, Fr of upstream Froude number, and ratio B/L of channel width to a bottom edge length of side orifice are the main factors influencing the discharge capacity of triangular side orifice. The first-order sensitivity coefficient and global sensitivity coefficient are 0.23, 0.11, 0.17 and 0.41, 0.39, 0.35 respectively.

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  • In this paper, the sparrow search algorithm is used to predict the discharge coefficient of the triangular side orifice for the first time.

  • The importance of dimensionless parameters on discharge coefficient is quantified using the Sobol's method.

  • The flow characteristics of the side orifice are analyzed.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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discharge coefficient, hydraulic characteristics, quantitative analysis, side orifice, Sobol's method, sparrow search algorithm, digitalwatercollection, digitalwatertechniques
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With the increasing shortage of water resources, it is imperative to implement water-saving irrigation. It is particularly important to study measuring equipment with high precision, strong adaptability and good convenience (Wang et al. 2016). As a hydraulic control structure widely used in hydraulic and irrigation works, side orifices in the open channel are used to divert water discharge from the main channel to other channels (Hussain et al. 2011a). Because of the importance of this diversion structure, many experiments, analyses and numerical studies have been carried out on its hydraulic characteristics. The discharge coefficient is the most important hydraulic parameter of the side orifice. Therefore, it is of great significance to calculate the discharge coefficient of the side orifice accurately in the irrigation system and hydraulic engineering.

In recent decades, to measure water accurately, different scholars have carried out in-depth research on hydraulic characteristics of triangular side orifices through model tests and given the calculation formula of discharge coefficient as shown in Table 1. Vatankhah & Mirnia (2018) analyzed hydraulic characteristics of triangular side orifices through model tests and obtained a discharge coefficient relationship based on the classical regression method. Jamei et al. (2021) used multiple linear regression to give the discharge coefficient calculation formulas for both known and unknown upstream channel discharges. Vatankhah (2019) developed the calculation equation of discharge coefficient of side orifice considering the change of orifice centerline. Considering that the discharge coefficient equation fitted by different scholars is difficult to be unified, it is not convenient for people to use. Therefore, it is necessary to develop a systematic and highly accurate discharge coefficient calculation method to solve the problem of discharge measurement in small channels.

Table 1

Calculation formula of discharge coefficient fitted by different scholars

Serial numberEquationsAuthors
1  Vatankhah & Mirnia (2018)  
2  Jamei et al. (2021)  
3  Jamei et al. (2021)  
4  Vatankhah (2019)  
Serial numberEquationsAuthors
1  Vatankhah & Mirnia (2018)  
2  Jamei et al. (2021)  
3  Jamei et al. (2021)  
4  Vatankhah (2019)  

Where Cd is the discharge coefficient of the side orifice; L is the orifice length (m); H is the orifice height (m); B is the main channel width (m); W is the orifice crest height (m); y1 is the flow depth upstream of the channel (m); Fr1 is the upstream Froude number.

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In recent years, with the development of artificial intelligence technology, many scholars have successfully applied this technology to solve complex hydraulic engineering problems. Azimi et al. (2017) used the adaptive neuro-fuzzy inference system (ANFIS) and a hybrid of ANFIS and a genetic algorithm (ANFIS-GA) to predict the discharge coefficient of rectangular side orifices. At the same time, the discharge rate of side orifices was simulated by FLOW-3D software. The results show that ANFIS-GA has the highest accuracy. Moghadam et al. (2019) used ANFIS and Firefly algorithm (FA) to predict the discharge coefficient of side orifices. The results show that ANFIS-FA has stable performance. Eghbalzadeh et al. (2016) used an artificial neural network to predict the discharge coefficient of rectangular side orifices. The results show that the predicted results of all neural networks are better than those of non-linear regression, and the results of radial basis neural network (RBNN) are better than those of generalized neural network (GRNN) and feed-forward neural network (FFNN). Jamei et al. (2021) and Qian et al. (2019) carried out prediction research on the discharge coefficient of triangular side orifices through the intelligent model. To explore the sensitivity of dimensionless parameters of the model, they arranged and combined the dimensionless parameters affecting Cd into 26 different input combinations. This method not only increases the calculation amount, but also tends to ignore the interaction of different parameters on the discharge coefficient. Based on the current literature research, most scholars focus on comparing the accuracy of the rectangular side-orifice intelligent model but have little research on triangular side-orifice. In addition, the current research lacks quantitative analysis of model input parameters and ignores the specific contribution of dimensionless parameters to Cd.

Therefore, this study aimed at the prediction of the side orifice discharge coefficient. Through Buckingham-π theorem, the dimensionless parameters that affect the triangular side orifice discharge coefficient were obtained as the input parameters of each model, and the discharge coefficient was taken as the output parameter of each model. Considering that back propagation neural network (BPNN) was easy to fall into local optimum, a sparrow search algorithm was used to optimize its weights and thresholds, and three statistical indexes were used to get the performance and accuracy of the model, and the results were compared with those of the literature. Finally, the input parameters of the model were quantitatively analyzed by Sobol's method to determine the sensitivity of each dimensionless parameter of the Cd and the variation rule after the interaction. This study explored the application of artificial intelligence technology in hydraulic engineering and provided new ideas for the design and application of side orifices for the optimization of water use and farmland water-saving irrigation.

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Experimental datas

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The dataset of this study is from Vatankhah & Mirnia (2018) The experiment was carried out in a horizontal rectangular channel 12 m long, 0.25 m wide, and 0.5 m deep. The equilateral triangular side orifices were made of 0.01-m thick Plexiglas sheets with a crest thickness of about 1 mm, and the downstream edge beveled to a 45° angle. The plan is shown in Figure 1. Five hundred and seventy groups of tests were carried out on the length (L = 30 and 40 cm), height (H = 4, 7 and 10 cm) and bottom height (W = 5, 10 cm) of the orifice under free flow conditions, with each data range as shown in Table 2. Seventy per cent of the datasets were randomly selected as the training set and 30% as the testing set.
Table 2

The range of parameters in the experimental data set

Serial numberHeight H/cmLength L/cmBottom height W/cmChannel discharge Qu/(L/s)Side-orifice discharge Qs/(L/s)Upstream water depth y1/cmWater depth in the center of the orifice yc/cmDownstream water depth y2/cm
Min 4 30 5 13.33 1.77 9.41 10.48 10.82 
Max 10 40 10 34.64 17.58 28.57 28.86 82.80 
Serial numberHeight H/cmLength L/cmBottom height W/cmChannel discharge Qu/(L/s)Side-orifice discharge Qs/(L/s)Upstream water depth y1/cmWater depth in the center of the orifice yc/cmDownstream water depth y2/cm
Min 4 30 5 13.33 1.77 9.41 10.48 10.82 
Max 10 40 10 34.64 17.58 28.57 28.86 82.80 
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Figure 1
Schematic diagram of the side orifice plane structure.
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Schematic diagram of the side orifice plane structure.

Figure 1
Schematic diagram of the side orifice plane structure.
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Schematic diagram of the side orifice plane structure.

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Dimensional analysis

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For triangular side orifices, the discharge can be calculated by Equation (1):
(1)
Based on the Buckingham-π theorem and literature research (Hussain et al. 2010, 2011b, 2016), the functional relation affecting the size of Cd can be expressed as follows:
(2)
According to the dimension analysis, H, g and ρ were selected as variables to obtain:
(3)
Since the Reynolds number (Re) is large in the channel, its effect on the Cd can be neglected (Hussain et al. 2010), and dimensionless parameters affecting the size of Cd can be expressed as:
(4)
where Qs is the discharge of side orifice; Cd is the discharge coefficient of the side orifice; hc is the head above the center line of the orifice; L is the orifice length; H is the orifice height; B is main channel width; W is the orifice crest height; v1 is the main channel upstream velocity; y1 is the flow depth upstream of the channel; ρ is Density of liquid; g is the acceleration of gravity; μ is dynamic viscosity coefficient; Fr is the upstream Froude number.

Data preprocessing

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In order to reduce the relative relationships and effects between different quantities, the tests were normalized to between [0,1] by Equation (5):
(5)
where Xmax is the maximum value of sample data and Xmin is the minimum value of sample data.
Also, the model was trained and tested, and the dataset was inversely normalized to the original value by Equation (6):
(6)

Model building

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BPNN

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Back propagation neural network, as a traditional multi-layer FFNN, consists of an input layer, a hidden layer and an output layer. It mainly includes forward propagation of signal and reverses transmission of error. For error back-propagation, the output errors of each neuron layer are calculated step by step through the output layer, and then the weights and thresholds of each layer are adjusted according to the error gradient descent method so the final output of the modified network can approach the expected value. Due to the small amount of calculation and strong parallelism, it has been widely used in the engineering field (Shi et al. 2019; An et al. 2021; Yang et al. 2021).

SSA-BPNN

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Sparrow Search Algorithm (SSA), a new intelligent optimization algorithm to simulate sparrow foraging behaviour and anti-predation behaviour, is also widely used in the engineering field (Song et al. 2021; Wang et al. 2021; Feng et al. 2022). In this algorithm, each sparrow corresponds to one of the solutions (Xue & Shen 2020), and it has high convergence performance and local search ability. The location update for the discoverer is as follows:
(7)
where t is the current number of iterations; j = 1, 2, …, d; is the position information of the ith sparrow in the jth dimension; Cmax is the maximum number of iterations of the algorithm; a ∈ (0,1], R2 ∈ [0,1] and ST ∈[0.5,1] represent a random number, warning value and safety value respectively; Q is a random number subject to normal distribution; L is a 1 × d dimensional matrix. The location updates for enrollees are as follows:
(8)
where Xpt+1 is the best position of the finder in the t + 1 iteration; Xw is the current global worst position; A+ is the 1 × d matrix of random assignment, and each element has a value of 1 or −1. When i > n/2, it indicates that the ith scrounger with the worse fitness value is most likely to be starving. When i ≤ n/2, it is the exact opposite. The location of the vigilant is updated as follows:
(9)
where Xbt is the global optimum position at the tth iteration; β is the step size control parameters and obeys the normal distribution random number with a mean value of 0 and variance of 1. K ∈ [−1,1], fi, fg and fw are the individual fitness values, the global optimum and the global worst fitness values; ɛ is the minimum constant so as to the denominator is not 0. When fi > fg this indicates that the sparrow is at the edge of the group, and fi = fg shows that the sparrows, which are in the middle of the population, are aware of the danger and need to move closer to the others.
The BPNN optimized by SSA can effectively avoid the weight and threshold of the back-propagation algorithm falling into the optimal local solution, and improve the forecast accuracy and stability of the model. The optimized flow chart is shown in Figure 2.
Figure 2
SSA optimization BP discharge chart.
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SSA optimization BP discharge chart.

Figure 2
SSA optimization BP discharge chart.
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SSA optimization BP discharge chart.

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ELM

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The Extreme Learning Machine (ELM) is a single hidden layer FFNN (Huang et al. 2006). For a single hidden layer neural network, ELM can randomly initialize the input weights and biases to get the corresponding output weights. Suppose there are N arbitrary samples ⁠, where ⁠. For a single hidden layer neural network with L hidden layer nodes, it can be expressed as Equation (10):
(10)
where is activation function, is input weight, is output weight, is bias of the i hidden layer unit. is the inner product of Wi and Xj.
For a single hidden layer neural network, the learning goal is to minimize the output error, which can be expressed as Equation (11):
(11)
where and bi and Equation (10) can be expressed as Equation (12):
(12)
expressed as a matrix:
where H is the output of the hidden layer node, is output weight, T is expected output.
(13)

Therefore, ⁠, where is Moore-Penrose generalized inverse matrix of matrix H. Studies have shown that ELM is a general-purpose approximator, and its simplification contributes to rapid calculation and high generalization ability, and is suitable for various situations (Sun et al. 2008; Luo & Zhang 2014; Roushangar et al. 2018).

Sobol's sensitivity analysis

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Sobol's method (Sobol 1990) is a global and model-independent sensitivity analysis method based on variance decomposition, which can handle non-linear, non-monotone functions and models. The core of the Sobol method is to decompose the objective function into an incremental sum of dimensions (Nossent et al. 2011), as shown in Equation (14). Generally, the sensitivity of a single input parameter to the output is reflected by calculating the first-order sensitivity, while the global sensitivity reflects the sensitivity to the output after the input parameters interact. To solve the high dimensional and high order integration problems involved in the calculation process, this method uses the Latin hypercube sampling method to generate samples, which reduces the sample size of Monte Carlo simulation and improves the operation efficiency. Therefore, this study uses this method as an effective method of model quantitative analysis to get the sensitivity of different dimensionless parameters to the Cd of triangular side orifices:
(14)
where f (X1, … , Xp) is the output of the objective function of the model; f0 is a constant in the objective function; i, j are the variable number; p = 1,2,3,…,n.

Evaluation index

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To validate the model performance proposed in this study, three statistical methods are used in this paper, root mean square error (RMSE), mean absolute percentage error (MAPE) and determination factor (R2) to determine whether each model meets the criteria under training and testing conditions. All the methods are defined in the following Equations (15)–(17):
(15)
(16)
(17)
where Oi is the size of the test value, is the average value of the test value; Pi is the size of the predicted value, is the average value of the predicted value, and N is the total number of data.
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Model comparison

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In this study, the best parameters of the BPNN and SSA-BPNN were obtained by trial and error method. For the BPNN, as shown in Figure 3, when the number of neurons is 12, the model has better performance, in which the functions of the hidden layer and output layer are tansig and trainlm respectively. For the SSA-BPNN, the population size is 30 and the maximum evolutionary algebra is 50. The scatter plots of BPNN and SSA-BPNN in the training and testing phases are shown in Figure 4. The performance and precision of the model can be judged by the trend line equation y = ax + b and the determination coefficient R2 in the graph. The closer a and b are to 1 and 0 respectively, and the closer R2 is to 1, the more efficient the model is. As shown in Figure 3, the trend line equations a and b of SSA-BPNN are 0.968, 0.016 and 0.971, 0.020 in the training and testing stages, respectively, which are better than 0.910, 0.048 and 0.878, 0.063 of BPNN. Also, the R2 of the SSA-BPNN is increased by 9.25% compared with the BPNN in the training stage and 9.45% in the testing stage, which indicates that the precision of the SSA algorithm has been significantly improved after optimizing BPNN.
Figure 3
BP and SSA-BP structure diagram.
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BP and SSA-BP structure diagram.

Figure 3
BP and SSA-BP structure diagram.
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BP and SSA-BP structure diagram.

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Figure 4
Scatter fit plots of two models in the testing and training phases. (a) Training stage. (b) Testing stage.
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Scatter fit plots of two models in the testing and training phases. (a) Training stage. (b) Testing stage.

Figure 4
Scatter fit plots of two models in the testing and training phases. (a) Training stage. (b) Testing stage.
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Scatter fit plots of two models in the testing and training phases. (a) Training stage. (b) Testing stage.

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Table 3 shows the performance indicators of the two models at different stages. The values of RMSE and MAPE are closer to 0, the smaller the model error and average deviation. Table 3 shows the RSME of SSA-BPNN decreases by 45.16 and 37.11% respectively, based on BPNN in the training and testing stages, indicating that the measured value of the Cd of the SSA-BPNN deviates less from the predicted value. Compared with BPNN, MAPE of SSA-BPNN decreases by 42.86 and 37.5% respectively, which indicates that SSA-BPNN has a better prediction effect and higher accuracy. Therefore, after using the SSA algorithm, the performance of BPNN is improved effectively, and it is enhanced for the global search ability of the neural network.

Table 3

Performance indicators of BP and SSA-BP models at different stages

ModelTraining stage
Testing stage
RMSEMAPE (%)R2RMSEMAPER2
BPNN 0.0186 0.007 0.873 0.0194 0.016 0.881 
SSA-BPNN 0.0102 0.004 0.962 0.0122 0.010 0.973 
ModelTraining stage
Testing stage
RMSEMAPE (%)R2RMSEMAPER2
BPNN 0.0186 0.007 0.873 0.0194 0.016 0.881 
SSA-BPNN 0.0102 0.004 0.962 0.0122 0.010 0.973 
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Table 4 shows the performance index of ELM model in the training and testing stages. When the number of neurons in the hidden layer is 20, the model achieves the best performance (shown in bold in Table 4). In the training stage, RMSE = 0.011, MAPE = 0.004, R2 = 0.955. In the testing stage, RMSE = 0.012, MAPE = 0.010, R2 = 0.953. Obviously, the predictive and fitting ability of SSA-BPNN is better than ELM. Therefore, SSA-BP is the best intelligent model to predict the discharge capacity of triangular side orifices.

Table 4

Performance indicators of ELM models at different stages

Number of neuronsTraining stage
Testing stage
RMSEMAPE (%)R2RMSEMAPE (%)R2
5 0.019 0.007 0.869 0.018 0.016 0.894 
10 0.016 0.006 0.911 0.016 0.014 0.910 
15 0.012 0.004 0.945 0.014 0.012 0.940 
20 0.011 0.004 0.955 0.012 0.010 0.953 
25 0.011 0.004 0.950 0.012 0.010 0.945 
Number of neuronsTraining stage
Testing stage
RMSEMAPE (%)R2RMSEMAPE (%)R2
5 0.019 0.007 0.869 0.018 0.016 0.894 
10 0.016 0.006 0.911 0.016 0.014 0.910 
15 0.012 0.004 0.945 0.014 0.012 0.940 
20 0.011 0.004 0.955 0.012 0.010 0.953 
25 0.011 0.004 0.950 0.012 0.010 0.945 
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Comparison with literature research

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To further validate the performance of SSA-BPNN. The results of this study were compared with the optimal results of different scholars, and the results are shown in Table 5. The RMSE is smaller than GA-SVR and the empirical equation, which indicates that SSA-BPNN has higher correlation and accuracy in the current research. The MAPE is smaller than the empirical formula to further verify this; meanwhile, R2 is larger than GA-SVR and the empirical equation, which indicates that the model has a stronger fitting ability. For improved Cd calculation equations by Vatankhah (2019) (Equation (4)), the maximum error of the equation is 9.5%, and the average error is 2%. The error distribution of SSA-BPNN in the testing stage is shown in Figure 5. The maximum error and average errors are 6.03 and 0.91% respectively, which are 36.53 and 54.5% lower respectively than that of the improved Cd calculation equation. Therefore, SSA-BPNN can be used as an effective method to predict the Cd of triangular side orifices.
Table 5

Comparison with previous models

ModelRMSEMAPE(%)R2
SSA-BP 0.0122 0.010 0.9730 
GA-SVR (Hussain et al. 2010) 0.0135 / 0.9205 
Empirical (Vatankhah & Mirnia 2018) 0.0150 2.370 0.9216 
ModelRMSEMAPE(%)R2
SSA-BP 0.0122 0.010 0.9730 
GA-SVR (Hussain et al. 2010) 0.0135 / 0.9205 
Empirical (Vatankhah & Mirnia 2018) 0.0150 2.370 0.9216 
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Figure 5
SSA-BP error scatter plot.
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SSA-BP error scatter plot.

Figure 5
SSA-BP error scatter plot.
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SSA-BP error scatter plot.

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Sensitivity analysis

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From the above analysis, it can be seen that SSA-BPNN has high accuracy in estimating the Cd of triangular side orifices. Therefore, this study further used the Sobol's method to quantify the dimensionless parameters of the model. It can be seen from Figure 6 that the first-order sensitivity coefficients S1 of W/H, B/L, B/H, y1/H and Fr are 0.23, 0.11, 0.03, 0.05 and 0.17, respectively, which indicates that the discharge coefficients are greatly influenced by each parameter under the separate action of the W/H, Fr and B/L. The y1/H and B/H have little influence on discharge coefficient. The global sensitivity coefficients Si of each dimensionless parameter are 0.41, 0.39, 0.21, 0.31 and 0.35 respectively, which indicates that the influence of each parameter on the Cd increases after the interaction. W/H, Fr and B/L are the important factors influencing the Cd of the triangular side orifice. Therefore, W/H, Fr and B/L should be considered the primary consideration when designing triangular side orifices for optimum discharge capacity.
Figure 6
First-order sensitivity coefficient S1 and global versus sensitivity coefficient Si of different parameters.
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First-order sensitivity coefficient S1 and global versus sensitivity coefficient Si of different parameters.

Figure 6
First-order sensitivity coefficient S1 and global versus sensitivity coefficient Si of different parameters.
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First-order sensitivity coefficient S1 and global versus sensitivity coefficient Si of different parameters.

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Figure 7 shows the relationship between the most important hydraulic parameters and predicted Cd. It can be seen from the contour diagram that the discharge capacity of side orifices is the largest when 0.63 < B/L < 0.68 and 0.50 < W/H < 0.65; When 0.80 < B/L < 0.83 and 2.40 < W/H < 2.50, the discharge capacity of side orifices is the smallest. Particularly, when 0.63 < B/L < 0.65, the side orifice discharge capacity under the range of 1.10 < W/H < 1.35 is less than 1.40 < W/H < 1.90.
Figure 7
Variation of the most influential inputs predicted discharge coefficient.
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Variation of the most influential inputs predicted discharge coefficient.

Figure 7
Variation of the most influential inputs predicted discharge coefficient.
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Variation of the most influential inputs predicted discharge coefficient.

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To obtain the relationship between hydraulic parameters W/H, B/L, Fr and Cd, the Cd changes with the W/H, Fr and the B/L as shown in Figures 7 and 8. It can be seen from Figure 8 that when W/H is fixed, the Cd decreases with the increase of Fr, and the larger the value of B/L, the greater the decrease. It can be seen from Figure 9 that when B/L is fixed, the Cd decreases with the increase of Fr, and the greater the value of W/H, the greater the decrease. Where Fr = 0.37, W/H = 1.250 and W/H = 1.428 have the same discharge capacity.
Figure 8
Variation trend of Cd with Fr when W/H = 1.25.
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Variation trend of Cd with Fr when W/H = 1.25.

Figure 8
Variation trend of Cd with Fr when W/H = 1.25.
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Variation trend of Cd with Fr when W/H = 1.25.

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Figure 9
Variation trend of Cd with Fr when B/L = 0.833.
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Variation trend of Cd with Fr when B/L = 0.833.

Figure 9
Variation trend of Cd with Fr when B/L = 0.833.
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Variation trend of Cd with Fr when B/L = 0.833.

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In this study, artificial intelligence technology was applied to solve the complex hydraulic characteristics of triangular side orifices. Firstly, dimensionless parameters influencing the Cd of side orifices were obtained by dimension analysis of the Buckingham-π theorem as the input value and Cd as the output value of the model. The predicted value of the Cd was obtained through the algorithm learning principle. To further obtain the variation rule of influence of dimensionless parameters on Cd, Sobol's method was used to quantify the parameters, and the following conclusions were obtained:

  • (1)

    SSA-BPNN can be used to predict the Cd of triangular side orifices, where maximum error and average error are 6.56 and 1.73%, respectively. The model has high accuracy, strong generalization ability and higher accuracy than other intelligent models and empirical equations.

  • (2)

    The first-order sensitivity of W/H, B/L, B/H, y1/H and Fr is 0.23, 0.11, 0.03, 0.05 and 0.17 respectively, and the global sensitivity is 0.41, 0.39, 0.21, 0.31 and 0.35 respectively. To optimize the discharge capacity of the side orifices, W/H, Fr and B/L should be considered as the primary factors in the design of similar projects.

  • (3)

    When 0.63 < B/L < 0.68 and 0.50 < W/H < 0.65, the discharge capacity of the side orifice is the largest; when 0.80 < B/L < 0.83 and 2.40 < W/H < 2.50, the discharge capacity of side orifices is the smallest.

  • (4)

    Considering the complexity of hydraulic characteristics of side orifices, an intelligent method not only provides a new method to solve similar engineering problems but also further improves the accuracy and efficiency of the Cd of triangular side orifices, which provides an important reference basis for accurate water consumption for farmland irrigation.

Finally, the flow regime of the upstream of the side orifice is measured by the size of Fr, and the flow characteristics of the side orifice are further analyzed in the future research.

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All relevant data are available from an online repository or repositories (https://doi.org/10.1061/(ASCE)IR.1943-4774.0001343).

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The authors declare there is no conflict.

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