Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model

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Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model Md. Ayaz a,∗ , Talib Mansoor b b

a Civil Engineering Section, University Polytechnic, Aligarh Muslim University, Aligarh 202002, India Civil Engineering Department, Zakir Husain College of Engg. & Tech., Aligarh Muslim University Aligarh, 202002, India

Received 6 April 2018; received in revised form 21 August 2018; accepted 2 October 2018

Abstract In the present study, ANN models have been developed to predict the discharge coefficients of oblique sharp-crested weirs for free and submerged flow cases using Borghei et al.’s experimental data. The discharge coefficients predicted by ANN models are then used to predict the discharges. The results so obtained are compared with the traditional regression model analysis performed by Borghei et al. (2003) in which the prediction error in the discharge was found within the range of ±5%. On the other hand, the developed ANN models predict the discharge coefficients as well as discharges within the error range of ±1%. Furthermore, sensitivity analysis of developed ANN models have been carried out for all the parameters (weir height, oblique weir length, head over weir and downstream head over weir) involved in the study and it was found that the weir length (L) is the most and weir height (P) is the least sensitive input variable to ANN-1 model. In the case of ANN-2 model, weir length (L) is the most and downstream head over weir (Hd ) is the least sensitive input variable. © 2018 National Water Research Center. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Artificial neural network; Oblique weir; Discharge coefficient; Free flow; Submerged flow; Discharge measurement

1. Introduction Sharp-crested weirs are widely used flow measuring device especially in the field of irrigation and drainage engineering. Use of oblique weir is advantageous over normal weir in the case of limited channel width when higher discharge is required at relatively lower head. Oblique alignment of weir increases the effective weir length beyond the channel width which consequently increases the efficiency of weir. Weirs having crest thickness equal to or less than 2 mm in the direction of flow are classified as ‘sharp-crested’ weirs. Among the various shapes of sharp-crested ∗

Corresponding author. E-mail addresses: [email protected] (Md. Ayaz), [email protected] (T. Mansoor). Peer review under responsibility of National Water Research Center.

https://doi.org/10.1016/j.wsj.2018.10.002 1110-4929/© 2018 National Water Research Center. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Nomenclature Notations a0 , b0 Numerical constants b Bias weight Channel width B c Crest width Discharge coefficient of oblique weir Cd Discharge coefficient of normal weir CD Ce Elementary discharge coefficient e Vector of network errors at output f Sigmoid activation function Acceleration due to gravity g Hessian matrix H H Head over weir Downstream head over weir Hd I Identity matrix J Jacobian matrix kb , kH , Cp Dimensional quantities Ks Discharge coefficient for submerged flow k0 , k1 , k2 , k3 Positive weir constants L Oblique weir length n1 Number of neurons in the layer Input received at the jth neuron Netj P Oblique weir height Discharge Q q Discharge per unit width of normal weir Discharge per unit width of oblique weir q0 wij Weight between the connection of ith and jth neuron xi Input at the ith neuron Output at jth neuron yj α Slope parameter β Constant ηw (y − P)/P A scalar μ φ 2θ/(π − 2θ) θ Oblique weir angle

weirs, rectangular, triangular, sutro, trapezoidal and circular are the most widely used shapes. The alignment of weir with respect to channel axis is important as it influences the discharge characteristics of weirs significantly. Based on alignment, sharp-crested weirs can be further classified into three categories, viz., normal weir, side weir and oblique weir. In case of normal weir, the channel axis is perpendicular to the weir axis; whereas, in side weirs, the channel axis is parallel to the weir axis. Oblique weir is the most general case in which the weir is aligned at an angle of θ with respect to the channel axis (Fig. 1). Based on downstream depth, flow over weirs can be classified into two categories, namely, free flow and submerged flow. Use of oblique weir is advantageous in the case of limited channel width when higher discharge is required at relatively lower head. Oblique alignment of weir increases the effective weir length beyond the channel width which consequently decreases the required water head and hence increases the efficiency of weir. Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 1. Sectional elevation and plan of oblique weir (free flow).

The discharge over a normal sharp-crested weir can be expressed as: Q=

 2 CD BH 2gH 3

(1)

where B = channel width; CD = discharge coefficient of normal weir; H = head over weir and g = acceleration due to gravity. The general formula for the discharge coefficient of normal and full width weir in terms of weir height, P and head over the weir, H can be expressed as: CD = a0 + b0

H P

(2)

where a0 and b0 are numerical constants to be found experimentally. The numerical values of constants a0 and b0 were proposed by many researchers on the basis of their experimental findings. Rehbock (1929) proposed the following equation for discharge coefficient in which the values of a0 and b0 are 0.61 and 0.08 respectively. CD = 0.61 + 0.08

H P

(3)

Kindsvater and Carter (1957) studied the effect of viscosity and surface tension for rectangular sharp-crested weir and proposed the following equation: 3

Q = Cp (B + kb ) (H + kH ) 2

(4)

where Q = discharge; Cp is a dimensional constant having dimension L1/2 T−1 ; b + kb = effective width of weir; H + kH = effective head; B = channel width; kb and kH are dimensional quantity on account of viscosity and surface tension. Kandaswamy and Rouse (1957) studied the discharge characteristics of sharp-crested weir having low weir height and proposed the following equation for discharge coefficient:   P 3/2 (5) for H/P > 15 CD = 1.06 1 + H Swamee (1988) used the experimental data of Kandaswamy and Rouse (1957) and proposed the following full-range equation for discharge coefficient of sharp-crested weir:  10  15 −0.1 14.14P H CD = 1.06 + (6) 8.15P + H H +P A substantial amount of work is available in the literature for normal sharp-crested weir. However, there have been very few studies conducted on oblique sharp-crested weirs. Aichel (1953) proposed the following equation to relate Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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the discharge q0 per unit width of crest across oblique weir to the discharge q per unit width of crest across normal weir of identical geometry working under identical hydraulic conditions: q0 H =1− β q P

(7)

where β = constant which depends on oblique angle θ. The results of this study were presented in tabular form. Swamee et al. (2011) converted these tabular results in the following mathematical equation:   H 1 Cd = CD 1 − (8) 1 + 3.7φ1.17 P where Cd = discharge coefficient of oblique weir; and φ = 2θ/(π − 2θ). In this study, authors reported that the discharge coefficient of oblique weir (Cd ) increases with the increase in weir angle θ. Ganapathy et al. (1964) studied the behavior of broad-crested oblique weir and plotted curves for Cd versus H with weir angle θ as third parameter. In this study also, authors reported that Cd increases with θ. Muralidhar (1965) conducted experiments on broad-crested oblique weir of crest width c and presented the results in the form of curves Cd /CD versus H/c with θ as third parameter. Swamee et al. (2011) proposed the following equation to convert these curves into mathematical formulation:   0.4 + 1.73φ c 0.2 + 1.7φ2 H Cd = (9) + CD 1 + 1.73φ c + H 1 + 1.7φ2 H + c Like previous studies, in this study also it was observed that Cd increases with θ. Borghei et al. (2003) proposed the following equation for discharge coefficient of sharp-crested oblique weir: Cd = (0.701 − 0.121 sinθ) + (2.229 sinθ − 1.663)

H P

(10)

Eq. (10) is valid for 260 ≤ θ ≤ 610 and 0.08 ≤ H/P ≤ 0.2. Tuyen (2006) conducted experiments on trapezoidal sharpcrested, broad-crested oblique weirs with weir angle 450 and reported that discharge for oblique weir is more than that of normal weir of identical geometry. Swamee et al. (2011) introduced elementary discharge coefficient (Ce ) and proposed the following equation:  10  k3 −0.1 k1 ηw C e = k0 + (11) k2 + η w ηw + 1 where ηw = (y − P)/P; y = flow depth; k0 –k3 = positive weir constants. Brater and King (1976) studied the rectangular sharp-crested weir for submerged flow case and proposed the following equation for discharge coefficient:   1.5 0.385 Hd Ks = 1 − (12) H where Ks = discharge coefficient for submerged flow; Hd = downstream water head over weir crest for submerged flow. Wu and Rajaratnam (1996) proposed the following equation for discharge coefficient:   Hd Hd − 1.331 sin−1 (13) Ks = 1 + 1.162 H H Emiroglu et al. (2011) developed ANN model for estimation of the discharge capacity of triangular labyrinth side-weir located on a straight channel. Abbas Parsaie (2016) developed the Multilayer Perceptron and Radial Basis Function Neural Network model to predict the discharge coefficient of side weir. A limited number of literatures exit associated with the applications of ANN based methodologies that too not for oblique weirs. However, ANN based methodologies are widely used now a days in different fields of engineering and technology. Khan et al. (2015) used nature inspired computing approach for solving non-linear problem arising in electromagnetic theory. Raja et al. (2016) used bio-inspired computing technique in the field of nanotechnology. Sabir et al. (2018) developed neuro-heuristic scheme to solve non-linear Thomas–Fermi equation. On the basis of above reviewed literatures, it can be concluded that most of the researchers proposed empirical equations for calculation of discharge coefficient Cd . In these empirical equations, the discharge coefficient Cd is Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Table 1 Ranges of geometric and flow variables (Borghei et al., 2003). Variables

Oblique angle (θ) Discharge (Q) Weir length (L) Channel width (B) Weir height (P) Water head (H) Downstream water head (Hd ) Number of runs

Unit

◦

l/s mm mm mm mm mm –

Values Free flow

Submerged flow

0–64 8–37 520–1175 520 460–511 40–106 – 95

0–60 8–37 520–1130 520 505–511 6.5–87 81

Fig. 2. Sectional elevation of oblique weir (submerged flow).

non-linearly related with the independent variables namely, H, Hd , P and θ. As far as the incorporation of non-linearity is concerned, the empirical equations generally have certain limitations. On the other hand, artificial neural networks (ANNs) are capable of fitting relatively more complex non-linear relationships with greater accuracy. ANN with at least one hidden layer is also known as universal approximator (Hecht-Neilson, 1987). In the present study, ANN models have been developed to predict the discharge coefficient of oblique sharp-crested weir for free and submerged flow cases. Hopefully, this study will add the following contributions to the knowledge in this subject area: • This work is an attempt to incorporate the applications of soft computing skills into the field of hydraulics and water resources engineering. • The novelty of this work is that the performance results of developed ANN models show a significant improvement in terms of prediction errors than that of traditional regression analysis models used by Borghei et al. (2003). • This work is capable to examine the relative sensitivity order of all the input parameters of ANN models. 2. Data set used The experimental data set collected by Borghei et al. (2003) has been used in the present study. Experiments were carried out in a rectangular concrete channel of length 6600 mm, width 520 mm and height 800 mm. The bed slope of the channel was taken as 0.005. Standard sharp-crested weir made up of Plexiglas of different height and length was used. Discharge was kept between the range 8.45 l/s and 37 l/s. The experimental data set comprises of a total number of 95 runs for free flow and 84 runs for submerged flow. Experiments for free flow cases were conducted on weirs of different weir lengths (520, 595, 672, 728, 801, 876, 1030, 1175 mm) in eight sets. Each set corresponds to a weir length mentioned above and weir height ranging between 460 and 511 mm. In the case of submerged flow, seven different weir lengths (595, 672, 728, 801, 876, 1030, 1175 mm) have been used to conduct experiments in seven sets. Ranges of different geometric and flow variables for free and submerged flow are given in Table 1. Plans and sectional elevations of free and submerged flow cases are shown in Figs. 1 and 2 respectively. 3. Artificial neural networks Artificial neural networks (ANNs) are widely used machine learning algorithm. ANNs are inspired by the central nervous system of human brain. ANN models are generally used to simulate the complex non-linear processes and Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 3. Typical structure of a feed-forward ANN model.

Fig. 4. Schematic representation of an artificial neuron.

fit the relationships between inputs and outputs. These models are also used in classification problems. The basis processing unit of ANN model is known as artificial neuron. The schematic representation of an artificial neuron is shown in Fig. 4. ANNs are usually comprises of an Input layer, hidden layer and an output layer. The typical structure of a feed-forward artificial neural network is sown in Fig. 3. ANNs may have one or more hidden layers. The number of artificial neurons in the hidden layers of ANN model depends on the complexity of problem being studied. Each neuron of a particular layer is connected to all the neurons of the next adjacent layer. All such connections between two neurons are associated with weights. In feed-forward neural networks, neurons at input layer receive information as input signals. These input signals are then multiplied by associated weights between connections and transmitted in forward direction to the next adjacent layer until the output layer. A detail description on neural computing and basic concepts of ANN is available in Zurada (1990) and Schalkoff (1997). The process of fixing the associated weights between the connections according to some learning rule using input–output data set is known as training. The back-propagation algorithm is the most effective and widely used learning technique in the training of ANN models. The training of ANN comprises of two steps, viz., feed-forward step and back-propagation step. In the feed-forward step, input layer receives the information and pass it through the hidden layers to the output layer. Each neuron of a particular layer receives signals from all the neurons of the previous layer. All the incoming inputs received at a particular neuron are summed up using associated weights. Mathematically, it can be represented as: Netj =

n1 

wij xi + b

(14)

i=1

where Netj = Input at neuron j; wij = associated weight between the connection of ith and jth neuron; n1 = number of neurons in the layer; xi = input at the ith neuron and b = bias weight. The input received (Netj ) at jth neuron is then transformed using a non-linear sigmoid activation function to get the output yj for each neuron.  yj = f Netj =

1  1 + exp −αNetj

(15)

where f is sigmoid activation function; yj = output at jth neuron and α is a slope parameter. In the back-propagation step, weights between the connections wij are initialized and the total error function is calculated for training data Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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set at the output layer. The total error function is then minimized using a suitable back-propagation algorithm. Levenberg–Marquardt algorithm is the most effective and widely used back-propagation algorithm. A brief description of Levenberg–Marquardt algorithm is given below. 3.1. Levenberg–Marquardt algorithm Levenberg-Marquardt (LM) algorithm solves nonlinear least square problems without computing Hessian matrix (H). In such problems Hessian matrix can be approximated in terms of Jacobian matrix (J). H = JT J

(16)

The Levenberg–Marquardt algorithm interpolates between Gauss–Newton and Gradient Descent algorithms. The iterative update of LM algorithm can be represented as: xk + 1 = xk − [JT J + μI]−1 JT e

(17)

where I = identity matrix; e = vector of network errors at output and μ is a scalar. For μ = 0, Eq. (17) becomes exactly the Gauss–Newton’s algorithm. For larger values of μ, LM algorithm behaves like Gradient Descent algorithm. Gauss–Newton algorithm has faster local convergence property near the minimum while the Gradient Descent algorithm is excellent in global convergence. LM algorithm starts with larger values of μ to ensure global convergence and quickly shifts towards Gauss–Newton algorithm for faster local convergence. LM algorithm is considered as the robust training algorithm. The disadvantage of using LM algorithm is that it can minimize only least square problems. A detailed description about the training of neural networks using Levenberg–Marquardt algorithm is available in Hagan et al. (1996) and Hagan and Menhaj (1994). 4. ANN model development Coefficient of discharge (Cd ) over sharp-crested oblique weir primarily depends on weir length (L), weir height (P) and water head (H). The main objective of this study to develop artificial neural network based model to estimate the discharge coefficient (Cd ) for sharp-crested oblique weir. In this study, two different feed forward back-propagation ANN models namely, ANN-1 and ANN-2 have been developed corresponding to two different cases of free and submerged flow. Both ANN-1 and ANN-2 models comprises of three layers – input, hidden and output layer. Input layer has three neurons and the output layer has one neuron. Weir length (L), weir height (P) and water head (H) have

Fig. 5. ANN-1 model (free flow).

Fig. 6. ANN-2 model (submerged flow).

Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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been used as inputs corresponding to three neurons of input layer while the discharge coefficient (Cd ) as output to ANN-1 model. ANN-2 model is developed for submerged flow case. In this model, Weir length (L), water head (H) and downstream water head for submerged flow (Hd ) have been used as inputs while the discharge coefficient for submerged flow (Ks ) as output corresponding to neurons of input and output layers. In order to non-dimensionalize the input parameters of ANN models, all the input parameters (L, P, H and Hd ) have been divided by channel width (B). The output parameter of ANN model is discharge coefficient (Cd or Ks ), which is already a non-dimensional number. Schematic representations of ANN-1 and ANN-2 models are show in Figs. 5 and 6 respectively. A block diagram explaining the flow and execution of work is shown below.

Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Start

Data collecon (Borghei et al (2003) experimental data used

Selecng Input parameters for ANN Model (L, P, H)

Non-dimensionalizing Input parameters of ANN model (L/B, P/B, H/B)

Selecng Output parameter for ANN Model (Cd )

Inialize the weights (Wij ) and train the ANN model

Find opmum number of hidden neurons by minimizing MSE

Once ANN architecture is fixed, use trained ANN model for predicon of Cd

End

Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 7. Regression plot for training (ANN-1 model).

5. Performance evaluations of ANN models Feed forward back-propagation ANN models have been trained to evaluate the performances of these models. Experimental data collected for free and submerged flow cases were used to train the input–output pattern of ANN models. The associated weights between the connections of artificial neurons were initialized randomly and total error function was calculated at the output layer following the feed-forward step. In the back-propagation step, Levenberg–Marquardt algorithm was used for suitable adjustment of weights by minimizing the total error function calculated at the output layer. Mean squared error (MSE) was used as performance evaluation parameter to select the best-performing ANN model by comparing the relative performances of different ANN models. The correlation coefficient (R) was used as another statistical parameter to measure the linear dependence between the actual and the predicted discharge coefficients. MSE and R can be defined as follows: 2 1  Cda,i − Cdp,i n i=1 

n  i=1 Cda,i − Cda Cdp,i − Cdp R =   2  n  2 n C − C da,i da i=1 i=1 Cdp,i − Cdp n

MSE =

(18)

(19)

where n = total number of observations; Cda,i = actual ith discharge coefficient; Cda = actual sample mean of discharge coefficients; Cdp,i = predicted ith discharge coefficient; Cdp = predicted sample mean of discharge coefficients. Trial and error method has been used to fix the number of neurons in the hidden layer of ANN model by comparing the mean squared error (MSE) for different number of hidden neurons. Performances of ANN models were evaluated using a total number of 95 and 81 sets for free and submerged flow respectively. Out of total available data sets, randomly selected 70% data have been used in training while 15% in testing and remaining 15% in validation of ANN models. The performance results of ANN-1 model for training, testing and validation are shown in Figs. 7–11. The performance results of ANN-2 model are shown in Figs. 12–16. Values of performance evaluation parameters obtained after training for ANN-1 and ANN-2 models are summarized in Tables 2 and 3 respectively. 6. Discussion on performance results Performances of developed ANN models have been evaluated using experimental data for two different cases of free flow and submerged flow. After the complete training of ANN models, performance results were obtained and presented with the help of different types of plots. Values of statistical parameters MSE and R were obtained after training the ANN models which are summarized in Tables 3 and 4. Figs. 7–9 represent the scatter plots of ANN-1 Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 8. Regression plot for validation (ANN-1 model).

Fig. 9. Regression plot for testing (ANN-1 model).

Fig. 10. Regression plot altogether (training, validation, testing, ANN-1 model).

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Fig. 11. Plot of MSE with epochs (ANN-1 model).

Fig. 12. Regression plot for training (ANN-2 model).

Fig. 13. Regression plot for validation (ANN-2 model).

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Fig. 14. Regression plot for testing (ANN-2 model).

Fig. 15. Regression plot altogether (training, validation, testing, ANN-2 model).

Fig. 16. Plot of MSE with epochs (ANN-2 model).

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Table 2 Results of performance parameters for ANN-1 model. Parameters

MSE

R

Data used

Training Validation Testing

8.939e-07 1.629e-06 1.944e-06

0.9992 0.9983 0.9991

67 14 14

Table 3 Results of performance parameters for ANN-2 model. Parameters

MSE

R

Data used

Training Validation Testing

9.720e-06 2.995e-05 1.302e-05

0.9985 0.9961 0.9985

57 12 12

Table 4 Results of sensitivity analysis for perturbed input parameters (ANN-1 model). Perturbed parameter

Performance parameter

Training

Validation

Testing

L

MSE R

5.264e-05 0.9903

3.3439e-05 0.9764

4.9195e-05 0.9723

P

MSE R

9.2571e-07 0.9993

5.6511e-06 0.9979

4.7905e-06 0.9974

H

MSE R

4.8952e-06 0.9959

1.1262e-05 0.9879

1.6541e-05 0.9911

Fig. 17. Comparison of actual and predicted discharge (ANN-1 model).

model for training, validation and testing. Scatter plots of ANN-2 model are represented in Figs. 12–14. In these scatter plots, actual (experimental) discharge coefficients are plotted against the corresponding discharge coefficients predicted by ANN models along with a line of agreement. It was observed that the predicted discharge coefficients are in good agreement with actual discharge coefficients and the prediction error stays within the range of ±1%. Furthermore, the values of correlation coefficient (R) for these plots are found to be close to 1. The discharge coefficients predicted from ANN models are then used to find the discharges for free and submerged flow cases. Figs. 17 and 18 represent the scatter plots for actual and predicted discharges through sharp-crested oblique weir. In these plots also, the prediction Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 18. Comparison of actual and predicted discharge (ANN-2 model).

Fig. 19. Stem-plot between data points and prediction errors (ANN-1 model).

error stays within the range of ±1%. Borghei et al. (2003) used regression analysis with same data set and reported the error in discharge prediction within the range of ±5%. The MSE and R provide only the average values of model performance parameters. In order to show the error distribution in the predicted discharge coefficients, stem-plots have been used. Stem-plots between the data points and corresponding prediction errors for ANN-1 and ANN-2 models are shown in Figs. 19 and 20 respectively. In these plots, it was observed that most of data points fall within smaller error range of the order of 10−3 . Figs. 11 and 16 represent the plot of MSE with number of epochs. The best validation performance was obtained at epoch 93 for ANN-1 model and for ANN-2 model it was obtained at epoch 7. The mean square errors (MSE) corresponding to the best validation performances were found to be 1.629e-06 and 2.995e-05 for ANN-1 and ANN-2 models respectively. Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 20. Stem-plot between data points and prediction errors (ANN-2 model).

Fig. 21. Plot of MSE with number of neurons in hidden layer (ANN-1 model).

Performance results presented above are in good agreement with experimental results, which demonstrate the capability and practical applicability of proposed ANN models in predicting the discharge coefficients. 7. Finding the optimal number of neurons in hidden layer Number of neurons in the hidden layer of ANN model generally depends on the complexity of problem being studied. It has always been a challenging task to find the optimal number of neurons in the hidden layer of ANN model. In this study, trial and error method has been used to find the optimal number of neurons in the hidden layer. In this method, mean square errors (MSE) obtained after training are plotted against the number of hidden neurons which were used in training. Plots of MSE against the number of hidden neurons are shown in Figs. 21 and 22 for ANN-1 and ANN-2 models respectively. The optimal numbers of hidden neurons that give the minimum mean square errors (MSE) Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 22. Plot of MSE with number of neurons in hidden layer (ANN-2 model).

Fig. 23. Comparison of discharges predicted by ANN-1 model and Borghei et al.’s model.

were found to be 15 and 20 for ANN-1 and ANN-2 models respectively. Therefore, the optimal ANN architectures were fixed as 3-15-1 and 3-20-1 for ANN-1 and ANN-2 models respectively. 8. Comparison with Borghei et al.’s model Borghei et al. (2003) studied the discharge over oblique sharp-crested weir and proposed the following equations to predict the discharge coefficients Cd and KS for free and submerged flow respectively. h w 

Cd = (0.701 − 0.121 sinθ) + (2.229 sinθ − 1.663)  KS =

  L L 0.008 + 0.985 + 0.161 − 0.479 B B

(20) Hd H

3  2 (21)

Eq. (20) is valid for 1.14 ≤ L/B ≤ 2.26, H/P < 0.2 and H > 40 mm. Figs. 23 and 24 show the comparison of discharge coefficients predicted by ANN models and the models proposed by Borghei et al. (2003). It is quite clear that the discharge coefficients predicted by ANN models are closer to the line of agreement than that of predicted by Borghei Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 24. Comparison of discharges predicted by ANN-2 model and Borghei et al.’s model.

Fig. 25. Regression plot for perturbed ‘L’ (ANN-1 model).

et al. (2003). Therefore, it can be concluded that the performance of ANN models are significantly better than that of proposed by Borghei et al. (2003). 9. Sensitivity analyses of ANN models Sensitivity analysis is generally performed to test the robustness of the results of a model in the presence of uncertainties. This technique is used to determine how the small perturbations in different independent variables impact the dependent variable of a model. In order to know the relative impact of different inputs (L, H, P and Hd ) to the predicted discharge coefficient (Cd ), sensitivity analysis of ANN models have been carried out in this study. To perform this analysis, input variables are perturbed one at a time by embedding uniformly distributed random noises within the range of ±10% of the actual values. ANN models are then trained with these perturbed input data sets to get the outputs. Scatter plots of ANN-1 model for small perturbations in L, P and H are shown in Figs. 25–27 respectively. Figs. 28–30 show the scatter plots of ANN-2 model for perturbations in L, Hd and H respectively. It was observed that the ANN-1 model predicts the discharge coefficients within the ranges of ±3%, ±1% and ±2% for perturbations in L, P and H respectively. Also, the ANN-2 model is capable of predicting discharge coefficients within the ranges of Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 26. Regression plot for perturbed ‘P’ (ANN-1 model).

Fig. 27. Regression plot for perturbed ‘H’ (ANN-1 model). Table 5 Results of sensitivity analysis for perturbed input parameters (ANN-2 model). Perturbed parameter

Performance parameter

Training

Validation

Testing

L

MSE R

8.2665e-05 0.9981

1.1002e-04 0.9914

1.6408e-04 0.9834

Hd

MSE R

2.6348e-06 0.9960

1.5450e-05 0.9927

2.5439e-05 0.9926

H

MSE R

5.7638e-05 0.9904

6.010e-05 0.9908

7.1956e-05 0.9901

±3%, ±2% and ±2% for perturbations in L, Hd and H respectively. Performance parameters R and MSE obtained after training are summarized in Tables 4 and 5 for ANN-1 and ANN-2 models respectively. Performance results indicate that weir length (L) is the most and weir height (P) is the least sensitive input variable to ANN-1 model. In the case of ANN-2 model, weir length (L) seems to be the most and downstream head over weir (Hd ) is the least sensitive input Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 28. Regression plot for perturbed ‘L’ (ANN-2 model).

Fig. 29. Regression plot for perturbed ‘Hd ’ (ANN-2 model).

variable. Based on the results of sensitivity analysis, it can be concluded that the performances of ANN models are reasonably good and models appear to be robust even for large perturbation of 10%. 10. Conclusions In this study, two different ANN models have been developed to estimate the discharge coefficient of oblique sharpcrested weir for free and submerged flow. The performances of developed ANN models have been evaluated using Borghei et al.’s experimental data and results were compared with traditional regression model analysis performed by Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002

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Fig. 30. Regression plot for perturbed ‘H’ (ANN-2 model).

Borghei et al. (2003). The prediction errors in discharge for both the cases of free and submerged flow stay within the range of ±1%. Borghei et al. (2003) reported the prediction error for same experimental data set within the range of ±5%. The performance results of developed ANN models show a significant improvement in the prediction error than that of reported by Borghei et al. (2003). Also, the sensitivity analyses of developed ANN models have been performed and it was concluded that ANN models appear to be robust even for significantly large perturbations of 10%. Based on the performance results, it can be concluded that the proposed ANN model is an effective and robust tool in the prediction of discharge coefficients of oblique sharp-crested weirs. References Aichel, O.G., 1953. Discharge ratio for oblique weirs. Zeitschrift des Vereins Deutscher Ingenieure 95 (1), 26–27 (in German). Borghei, S.M., Vatannia, Z., Ghodsian, M., Jalili, M.R., 2003. Oblique rectangular sharp-crested weir. Water Marit. Eng. 156 (WM2), 185–191. Brater, E.F., King, H.W., 1976. Handbook of Hydraulics, sixth ed. McGraw-Hill, New York. Emiroglu, M.E., Bilhan, O., Kisi, O., 2011. Neural networks for estimation of discharge capacity of triangular labyrinth side-weir located on a straight channel. Expert Syst. Appl. 38, 867–874. Ganapathy, K.T., Raj, A.N., Ramanathan, V., 1964. Discharge characteristics of oblique anicuts. New Irrig. Era Irrig. Branch India 9 (1), 7–12. Hecht-Neilson, R., 1987. Komogorov’s mapping neural network existence theorem. Proceedings of the Internal Conference on Neural Networks, IEEE Press, New York 3, 11–13. Hagan, M.T., Demuth, H.B., Beale, M.H., 1996. Neural Network Design. PWS Publishing, Boston, MA. Hagan, M.T., Menhaj, M., 1994. Training feed-forward networks with the Marquardt algorithm. IEEE Trans. Neural Netw. 5 (6), 989–993. Kandaswamy, P.K., Rouse, H., 1957. Characteristics of flow over terminal weirs and sills. J. Hydraul. Eng. 83 (4), 286–298. Khan, Junaid Ali, et al., 2015. Nature-inspired computing approach for solving non-linear singular Emden–Fowler problem arising in electromagnetic theory. Connect. Sci. 27 (4), 377–396. Kindsvater, C.E., Carter, R.W., 1957. Discharge characteristic of rectangular thin plate weirs. J. Hydraul. Div. 83 (HY6), 1453/1–1453/36. Muralidhar, D., 1965. Some Studies on Weirs of Finite Crest Width. PhD Thesis. Indian Institute of Science, Bangalore, India. Parsaie, A., 2016. Predictive modeling the side weir discharge coefficient using neural network. Model. Earth Syst. Environ. 2 (2), 1–11. Raja, M.A.Z., Khan, M.A.R., Mahmood, T., Farooq, U., Chaudhary, N.I., 2016. Design of bio-inspired computing technique for nanofluidics based on nonlinear Jeffery–Hamel flow equations. Can. J. Phys. 94 (5), 474–489. Rehbock, T., 1929. Discussion of “Precise weir measurements” by Schoder EW, Turner KB. Trans. Am. Soc. Civil Eng. 93, 1143–1162. Sabir, Z., Manzar, M.A., Raja, M.A.Z., Sheraz, M., Wazwaz, A.M., 2018. Neuro-heuristics for nonlinear singular Thomas–Fermi systems. Appl. Soft Comput. 65 (April), 152–169. Schalkoff, R.J., 1997. Artificial Neural Networks. The McGraw Hill Companies, Inc., New York. Swamee, P.K., 1988. Generalized rectangular weir equations. J. Hydraul. Eng. 114 (8), 945–949. Swamee, P.K., Ojha, C.S.P., Mansoor, T., 2011. Discharge characteristics of skew weirs. J. Hydraul. Res. 49 (6), 818–820. Tuyen, N.B., 2006. Flow Over Oblique Weirs. M.Sc. Thesis. Delft University of Technology, Delft, the Netherlands. Wu, S., Rajaratnam, N., 1996. Submerged flow regimes of rectangular sharp-crested weirs. J. Hydraul. Eng. 122 (7), 412–414. Zurada, J.M., 1990. Introduction to Artificial Neural Systems. Jaico Publishing House, Mumbai, India.

Please cite this article in press as: Ayaz, Md., Mansoor, T., Discharge coefficient of oblique sharp crested weir for free and submerged flow using trained ANN model. Water Sci. (2018), https://dx.doi.org/10.1016/j.wsj.2018.10.002