KSCE Journal of Civil Engineering (0000) 00(0):1-7 Copyright ⓒ2015 Korean Society of Civil Engineers DOI 10.1007/s12205-015-0440-y
Water Engineering
pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205
TECHNICAL NOTE
Discharge Prediction of Circular and Rectangular Side Orifices using Artificial Neural Networks A. Eghbalzadeh*, M. Javan**, M. Hayati***, and A. Amini**** Received August 5, 2014/Accepted April 26, 2015/Published Online June 12, 2015
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Abstract A side orifice created in the side of a channel is a structure for diverting some of the flow from the main channel for different purposes. The prediction of the discharge through this side structure is very important in hydraulic and irrigation engineering. In the present study, three artificial neural network models including feed forward back propagation, radial basis function, and Generalized Regression neural networks as well as a multiple non-linear regression method were used to predict the discharge coefficient for flow through both square and circular shapes of sharp-crested side orifices. The discharge coefficient was modeled as a function of five input non-dimensional variables resulted from five dimensional variables, which were the type of orifice shape, the diameter or width of the orifice, crest height, depth and velocity of approach flow. The results obtained in this study indicated that all of the neural network models could successfully predict the discharge coefficient with adequate accuracy. However, according to different performance measures, the accuracy of radial basis function approach was a bit better than two other neural network models. The neural network models predicted the discharge coefficient more accurately than the non-linear regression relation. Keywords: side orifice, discharge coefficient, prediction, artificial neural network, flow ··································································································································································································································
1. Introduction The flow diversion structures such as side orifices, sluice gates and side weirs are provided in the side of a channel to spill or divert water from the main channel. These structures are used in water and wastewater treatment plants, irrigation networks, etc. Open channels with side orifices are applied to convey water from the main channel to the irrigated areas. It is important to determine the quantity of the lateral discharge through the side orifices. Most of the previous experimental researches were focused on side weirs (Borghei et al., 1999; Cosar and Agaccioglu, 2004; Aghayari et al., 2009; and Emiroglu et al., 2010). Borghei et al. (1999) concluded that the approach flow Froude number, ratio of the flow depth in the main channel and sill height, and ratio of the length of weir and width of channel affect the discharge coefficient of flow over sharp-crested side weirs. Side orifices have been experimentally investigated by Ramamurthy et al. (1986, 1987), Gill (1987), Swamee et al. (1994), Ojha and Subbaiah (1997), Bryant et al. (2008) and Hussain et al. (2010, 2011). Artificial Neural Network (ANN) is an attractive alternative
for costly experimental investigations, because it is able to learn complex non-linear relationships between input and output data sets. Therefore, ANN is increasingly becoming a practical tool to predict different phenomena of engineering. This artificial intelligence method has been successfully used to predict the hydraulic engineering applications, such as flow field (Benning et al., 2001), critical submergence of an intake in still water and open channel flow (Kocabas et al., 2008), backwater through bridge constrictions (Seckin et al., 2009 and Pinar et al., 2010), the friction factor of open channel flow (Yuhong and Wenxin, 2009), discharge capacity of straight compound channels (Unal et al., 2010), alluvial channel geometry (Riahi-Madvar et al., 2011), equilibrium scour depth around hydraulic structures (Keshavarzi et al., 2012 and Karami et al., 2012) and the length of hydraulic jumps (Naseri and Othman, 2012). A few studies have been performed on the prediction of the lateral outflow over triangular labyrinth and rectangular side weirs located on straight and curved channels (Emiroglu et al. (2010, 2011); Bilhan et al. (2010, 2011) and Kisi et al., 2012). But no one to the best of our knowledge has been applied ANN models to predict the lateral flow through side orifices. The aim of this study was to examine the ability of ANN
*Assistant Professor, Dept. of Civil Engineering, Faculty of Engineering, Razi University & Water and Wastewater Research Center, Tagh-E-Bostan, Kermanshah-67149, Iran (Corresponding Author, E-mail:
[email protected]) **Assistant Professor, Dept. of Civil Engineering, Faculty of Engineering, Razi University & Water and Wastewater Research Center, Tagh-E-Bostan, Kermanshah-67149, Iran (E-mail:
[email protected]) ***Professor, Dept. of Electrical Engineering , Kermanshah Branch, Islamic Azad University, Kermanshah-Iran (E-mail:
[email protected]) ****Assistant Professor, Kurdistan Agricultural and Natural Resources Research Center, AREEO, Sanandaj, Iran (E-mail:
[email protected]) −1−
A. Eghbalzadeh, M. Javan, M. Hayati, and A. Amini
models in prediction of the discharge coefficient for flow through both square and circular shapes of sharp-crested side orifices located on a straight rectangular channel. For this purpose, three different ANN approaches; Feed Forward Back Propagation (FFBP), Radial Basis Function (RBF), and Generalized Regression (GRNN) neural networks, were employed. The experimental data of Hussain et al. (2010, 2011) were used for training and testing the neural networks. Also, the results of the ANN models were compared with a relation obtained from a Multiple Non-linear Regression (MNLR) method in the present study.
2. Neural Networks The ANN approach is a mathematical methodology raised from biological neural networks. An ANN consists of many simple processing elements, known as neurons, which are similar to the human brain. The use of neural networks has been specified the useful properties and capabilities of the nonlinearity, input-output mapping, adaptivity, evidential response, contextual information, fault tolerance, VLSI implement ability, Neurobiological Analogy and analysis and design uniformity (Haykin, 1999). The computational intelligence technique of the ANN achieves dense interconnection via good performance of complex non-linear relationships between input and output data sets. A network, activation rule and learning rule are three components of the ANN (Doszkocs et al., 1990). ANN methods used in this study were Feed Forward Back Propagation (FFBP), Radial Basis Function (RBF) and Generalized Regression (GRNN) neural networks. These methods are introduced and discussed separately.
which contains a forward and backward passes, inside of the different layers of the network. The synaptic weights are fixed and adjusted with an error correction rule during the forward and backward passes, respectively (Haykin, 1999). Throughout the study, a MLP network with one hidden layer has been used and a back propagation algorithm was applied in training stage of the network. 2.2 Radial Basis Function In this study, a RBF network was also used to examine the ability of advanced training schemes in comparison with the basic back propagation in the prediction of the discharge coefficient of the sharp-crested circular and rectangular side orifices located on a straight rectangular channel. A feed forward network with a hidden layer is called Radial Basis Function (RBF) network. The basic form of a RBF network is shown in Fig. 2. During the learning and classification stages (normal operation), there is no feedback among input, hidden and output layers in the feed forward networks. The RBF networks train much faster than the multi-layer perceptron networks. A Gaussian or some other basis kernel function is used in the hidden nodes of the RBF networks (Fig. 2). Because of the radial basis function in the hidden nodes, feed forward networks are less sensitive with non stationary inputs. A nonlinear transformation is applied
2.1 Feed Forward Back-propagation A multi-layer FFBP, also known as the Multi-layer Perceptron (MLP) network, is by far the most popular in hydraulic engineering. Fig. 1 shows the structure of a multi-layer FFBP. A typical FFBP consists of three layers of neurons: input layer, hidden layer and output layer. Three characteristics of a FFBP are a nonlinear activation function in the each neuron of the network, existence of one or more layers in the hidden layer and a high connectivity degree in the network (Haykin, 1999). MLP networks are always trained by back propagation algorithm
Fig. 1 Configuration of a FFBP Neural Network
Fig. 2. The Basic form of a RBF Network
Fig. 3. Configuration of a GRNN −2−
KSCE Journal of Civil Engineering
Discharge Prediction of Circular and Rectangular Side Orifices using Artificial Neural Networks
from the input space to the hidden space in RBF network. The training stage of the RBF network is done by using a learning algorithm of the Orthogonal Least Squares (OLS). 2.3 General Regression Neural Network Specht (1991) proposed General Regression Neural Network (GRNN), which could be added to RBF neural networks category. The GRNN is applied for finding decision boundaries among patterns categories and estimating values of continuous dependent variables. The GRNN advantage compared to the other nonlinear regression methods is continuous functions estimation by non iterative training algorithms. The convergence of the GRNN based on kernel regression, is underlying regression surface. Schematic diagram of a GRNN is shown in Fig. 3. The GRNN consists of input, pattern, summation, and output layers (Fig. 3). The output of each neuron in pattern layer is the measured distance of the input from the stored patterns. Each unit in input layer is connected to all neurons of the pattern layer, but each neuron of the second layer is connected to the two neurons of the summation layer.
3. Data Presentation In the present study, the experimental data of Hussain et al. (2010, 2011) were used to train and test the different neural networks. Hussain et al. (2010, 2011) experimentally studied the discharge through both square and circular rectangular sharpcrested side orifices (Fig. 4). These experiments were carried out in a rectangular open channel, with a length of 9.15 m, a width of 0.5 m, and a depth of 0.6 m. An orifice was located in the left side at 5.18 m from the upstream end of the main channel. The whole experimental data set consisting of 387 data points was divided into two parts randomly; a training set consisting of 290 data points (75% of total data), and a testing set consisting of 97 data points (25%), which were used for developing a regression
Table 1. The Range of Data in the Experiments of Hussain et al. (2010, 2011) Parameter Qm(m3/s) Q(m3/s) L(m) Ym(m) W(m)
Circular side orifices Min. Max. 0.0281 0.1467 0.0009 0.0288 0.044 0.133 0.154 0.590 0.05 0.20
Rectangular side orifices Min. Max. 0.01654 0.15647 0.00144 0.02942 0.05 0.15 0.1716 0.5938 0.05 0.25
and ANN models. Table 1 shows the range of data in the experiments of Hussain et al. (2010, 2011). In this Table, Qm is the main channel discharge, Q is the discharge through side orifice, L is width and diameter of square and circular orifices, respectively, Ym is the depth of the approach flow in main channel, and W is the crest height.
4. Dimensional Analysis The functional relationship between the discharge through a side orifice and the variables expected to influence it, can be written as: Q = f1( Num, Vm, Y m, L, W, ρ, µ, g)
(1)
where, Num is the type of the orifice (1 for square and 2 for circular), Vm is approach velocity in the main channel, ρ is the water density, µ is the dynamic viscosity and g is the gravity acceleration. By using the dimensional analysis, the discharge coefficient for flow through the orifice is related to five non-dimensional variables as: Vm ⎞ ρVm L Y W -, Fr = -----------Cd = f2 ⎛ Num, -----m-, -----, Re = -----------⎝ µ L L gY ⎠
(2)
m
where, Cd is the discharge coefficient and Re and Fr are the approach flow Reynolds and Froude number, respectively. The discharge coefficient relationships proposed by Hussain et al. (2010) and (2011) are mainly depend on the approach flow Froude number and ratio of the orifice size and the main canal width. Ojha and Subbaiah (1997) suggested the following equation for the discharge through small orifices, which the pressure distribution is constant over the flow area of the orifice: Q = Cd A 2gH
(3)
where, A is the area of the orifice (L for square and π--- L 2 for 4 2
circular one) and H is the head of water above the orifice centerline as shown in Fig. 4.
5. Model Development and Performance Measures The FFBP, RBF and GRNN neural networks were performed
Fig. 4. Definition Sketch of the Main Channel and Side Orifices Vol. 00, No. 0 / 000 0000
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A. Eghbalzadeh, M. Javan, M. Hayati, and A. Amini
by program codes written in MATLAB 7.13.0.564 software. According to Eq. (2), the discharge coefficient through the side orifice (Cd) was output parameter, while the Froude and Reynolds numbers of the approach flow ( Fr = Vm ⁄ gYm and Re = ρVm L ⁄ µ ), the dimensionless approach flow depth (Ym/L), the dimensionless weir height (W/L) and the type of the orifice (Num) were used as five input parameters (Fig. 5). After predicting of the discharge coefficient by the neural networks, the lateral discharge was calculated using Eq. (3). To compare different models, some factors are defined such as Mean Absolute Error (MAE), Root Mean Square Error (RMSE), correlation coefficient (R), and Mean Relative Error (MRE): n
1 MAE = --- ∑ Si observed – Si predicted ni = 1 n
1 RMSE = --- ∑ ( Si ni = 1
observed
(4)
– Si predicted )
2
(5)
where n is the data set number and Si is the discharge coefficient of the ith data. In this study, the best performance was selected based on the values of the MAE.
6. Results and Discussion The characteristics of the most appropriate ANN models were determined using trial and error. The best FFBP model consists of one hidden-layer with 19 neurons in which the logarithmic tansig and linear activation functions were used for the hidden and output layers, respectively. A RBF architecture with 172 neurons and spread parameter equal to 66 was selected as the most appropriate RBF network. The value of spread variable for the best testing performance of the GRNN was obtained equal to 0.15. The multiple non-linear regression relation derived using the same data set as in the training set of the ANN models is written as follows:
n
∑ Xi Yi
Cd = 1.108Num
i=1 R = --------------------------n
2
n
(6)
2
∑ X i ∑ Yi
i=1
i=1
n
∑ Si observed
i=1
Xi = Si observed – -------------------------n
(7)
n
∑ Si predicted i=1 -------------------------Yi = Si predicted – n
(8)
Fig. 5. Input and Output Parameters for the Proposed ANN Models
–0.043
Y m⎞ ⎛ ----⎝L⎠
0.054
⎛W -----⎞ ⎝ L⎠
–0.012
Re
–0.051
Fr
0.019
(9)
It is evident that the other 25% data were used to test the Eq. (9), as done for the ANN models. Table 2 shows the MAE, RMSE, R and MRE of the nonlinear regression relation and the FFBP, RBF and GRNN neural networks for training and testing stages. The values of MAE obtained from different ANNs in the testing stage changed from 0.0081 to 0.0092; those of RMSE varied from 0.0119 to 0.0136 while the R values changed from 0.92 to 0.94. The values of MRE, which is a performance measure used in many engineering applications were between 1.29% and 1.48%. According to Table 2, all of the ANN models predicted discharge coefficient with good accuracy, better than non-linear regression relation. However, the suggested non-linear regression relation yielded the values of R lower and those of MAE, RMSE and MRE higher than the ANN models, significantly. Thus, the present study shows that it is better to apply ANN models instead of traditional regression methods for predicting the discharge coefficient through the lateral orifices. Among different ANNs employed in this study, the RBF model predictions for the discharge coefficient were slightly better than two other models based on all of the performance measures. As seen in Figs. 6-8, ANNs all tended to overestimate high
Table 2. The Error Values for the Predicted Discharge Coefficient Obtained from the Different Models Model FFBP RBF GRNN Regression
Training MAE 0.0055 0.0019 0.0003 0.024
RMSE 0.0084 0.0035 0.0007 0.0358
Testing R 0.9788 0.9963 0.9998 0.4808
MRE% 0.916 0.301 0.044 4.094 −4−
MAE 0.009 0.0081 0.0092 0.0248
RMSE 0.0121 0.0119 0.0136 0.0324
R 0.9369 0.9418 0.9213 0.443
MRE% 1.418 1.291 1.479 4.006
KSCE Journal of Civil Engineering
Discharge Prediction of Circular and Rectangular Side Orifices using Artificial Neural Networks
Table 3. Computational Time Model FFBP RBF GRNN
Fig. 6. Comparison of Measured and Predicted Results of Discharge Coefficient for: (a) Training and, (b) Testing Data using FFBP Model
Fig. 7. Comparison of Measured and Predicted Results of Discharge Coefficient for: (a) Training and, (b) Testing Data using RBF Model
Fig. 8. Comparison of Measured and Predicted Results of Discharge Coefficient for: (a) Training and, (b) Testing Data using GRNN Model
Fig. 9. Comparison of Measured and Predicted Results of Discharge Coefficient for: (a) Training and, (b) Testing Data using Regression Relation Vol. 00, No. 0 / 000 0000
Computational Time (sec) 19.33 15.84 14.59
(>0.7) and low (
