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  • 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 23 ANN MODEL FOR PREDICTION OF LENGTH OF HYDRAULIC JUMP ON ROUGH BEDS Mujib Ahmad Ansari1 Associate Professor, Department of Civil Engineering, Zakir Husain College of Engineering and Technology, Aligarh Muslim University, Aligarh, 202002, U.P., India. ABSTRACT Hydraulic jumps have immense practical utility in hydraulic engineering and allied fields, such as energy dissipater to dissipate the excess energy of flowing water downstream of hydraulic structures (spillways and sluice gates), efficient operation of flow measurement flumes, chlorinating of wastewater, aeration of streams which are polluted by biodegradable wastes and many other cases. The length of hydraulic jumps is one of the most important parameters in designing the stilling basin, however, it cannot be calculated by mathematical analyses only, experimental and laboratorial results should also be used. In this study, artificial neural network (ANN) technique was used to determine the length of the hydraulic jumps in channel having smooth and rough beds. The selected model can predict the length of jumps with high accuracy and satisfy the evaluation criteria, with root mean square error ܴ‫ܧܵܯ‬ =3.2438, mean absolute percentage error ‫ܧܲܣܯ‬ =6.9231 and coefficient of determination ܴଶ = 0.9596. A comparison between the ANN model and empirical equation of Hughes and Flack (1984) was also done and the results showed that the ANN method is more precise. INTRODUCTION Hydraulic jump is one of the most interesting and useful phenomena in the field of hydraulic engineering. It has immense practical utility in hydraulic engineering and allied fields, such as energy dissipater to dissipate the excess energy of flowing water downstream of hydraulic structures. The length of hydraulic jumps is one of the most important parameters in designing the stilling basin. Many studies have been done with regard to design of a stilling basin, hydraulic jumps, and estimating the length of jumps. A jump formed in a horizontal, wide rectangular channel with a smooth bed is often referred to as the classical hydraulic jump and has been studied extensively (Peterka, 1958; Rajaratnam, 1967; McCorquodale, 1986; Hager, 1992). A study on hydraulic jumps due to submerging conditions in a smooth bed channel was done by Rao and Rajaratnam (1963), and Rajaratnam (1965). INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2014): 7.9290 (Calculated by GISI) www.jifactor.com IJCIET ©IAEME
  • 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 24 A wide range of investigation was conducted to evaluate the effectiveness of roughened beds (Rajaratnam, 1968; Hughes and Flack, 1984; Hager, 1992; Alhamid, 1994) and corrugated beds (Ead and Rajaratnam, 2002; Carollo and Ferro 2004a,b; Izadjoo and Shafai-Bajestan, 2007, Carollo et al.. 2007) on hydraulic jump. A complete description of the hydraulic jumps involves the sequent depths ݄ଵ and ݄ଶ which are the flow depths at the toe and end sections of the jump, and the jump length ‫ܮ‬௝ defined as the distance between these two sections. The length of hydraulic jumps is one of the most important parameters in designing the stilling basin, however, it cannot be calculated by mathematical analyses only and experimental and laboratorial results should also be used. Recent advancement in soft computing techniques and its application in hydraulics engineering have challenged the conventional methods of the analysis. Various hydraulics engineering problems are now being solved using several Artificial Intelligence (AI) techniques like Artificial Neural Networks (ANN). Several researches have shown that soft computing techniques are more accurate and feasible than older techniques and the best part of these techniques is that any of the AI techniques is not confined to a particular problem rather a technique can be applied to solve any problem from any field. ANN has been widely applied in various areas of hydraulics and water resources engineering (Liriano and Day, 2001; Nagy et al., 2002; Raikar et al., 2004 ,Azmathullah et al., 2005, Ansari and Athar, 2013 and Ansari, 2014). Recently Naseri and Othman, 2012, used ANN for the determination of length of hydraulic jump on smooth beds. Omid et al., 2005 developed ANN models for estimating sequent depth and jump length of gradually expanding hydraulic jumps of rectangular and trapezoidal sections. The aim of this study is to develop a model for estimation of the length of hydraulic jump in rectangular section with a horizontal apron having rough beds. The artificial neural networks (ANN) technique is employed for this purpose. ANN networks are able to calculate the length of jumps involving different parameters. METHODS AND MATERIALS (i) Length of hydraulic jump On the basis of scrutiny of relations for length of hydraulic jump, ‫ܮ‬௝ on rough beds the length of hydraulic jump, ‫ܮ‬௝ can be expressed by following functional relationship: ‫ܮ‬௝ ൌ ݂ሺܸଵ,݄ଵ, ݃, ߩ, υ, ݇௦ሻ (1) The variables of Eq. (1) can be easily arranged into the following non-dimensional form using Buckingham ߨ theorem. ‫ܮ‬௝ ݄ଵ⁄ ൌ ݂ ቀܸଵ ඥ݄݃⁄ ଵ , ݇௦ ݄ଵ⁄ , ܸଵ݄ଵ υ⁄ ቁ (2) where ‫ݎܨ‬ଵ is the Froude number and ܴ݁ is the Reynolds number ‫ݎܨ‬ଵ ൌ ܸଵ ඥ݄݃⁄ ଵ , ܴ݁ ൌ ܸଵ݄ଵ υ⁄ For large ܴ݁ values if viscous effect is neglected (Rajaratnam, 1976; Hager and Bremen, 1989), then the Eq. 2 can be written as: ‫ܮ‬௝ ݄ଵ⁄ ൌ ݂ሺ‫ݎܨ‬ଵ, ݇௦ ݄ଵ⁄ ሻ (3) Such a functional relationship will be used to develop an ANN model for the length of the hydraulic jump on rough beds.
  • 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 25 (ii) Experimental Data The data gathered from various studies of hydraulic jump on rough beds (Hughes and Flack, 1984; and Ead and Rajaratnam, 2002) were used in the present work. The experiments have a wide range in parameters that are listed in Table 1. According to preceding researcher's works, graphs and empirical equations have been provided for determining the length of hydraulic jump on rough beds. (iii) Artificial Neural Network Model An ANN is an information processing system composed of many nonlinear and densely interconnected processing elements or neurons, which is organized as layers connected via weights between layers. An ANN usually consists of three layers: the input layer, where the data are introduced to the network; the hidden layer or layers, where data are processed; and the output layer, where the results of given input are produced. The main advantage of the ANN technique over traditional methods is that it does not require information about the complex nature of the underlying process under consideration to be explicitly described in mathematical form. In order to map the causal relationship related to the hydraulic jump an input-output scheme was employed, where causal non-dimensional parameters were used. The model thus takes the input in the form of causative factors namely, ‫ݎܨ‬ଵ and ݇௦ ݄ଵ⁄ and yields the output, the length of hydraulic jump as ‫ܮ‬௝ ݄ଵ⁄ . Thus the model is: ‫ܮ‬௝ ݄ଵ⁄ ൌ ݂ሺ‫ݎܨ‬ଵ, ݇௦ ݄ଵ⁄ ሻ (4) The present study used the data considered above for prediction of the length of hydraulic jumps. The training of the above model was done using 80 % of the data selected randomly. Validation and testing for proposed model was made with the help of the remaining 20 % of observations, which were not involved in the derivation of the model. In the present work the usual feed forward type of network was considered. It was trained using both back propagation as well as cascade correlation algorithms with a view to ensure that proper training is imparted. Further, in order to see if advanced training schemes provide better learning than the basic back propagation, a radial basis function network was also used. The resulting neural network models are called Feed Forward Back Propagation (FFBP), Cascade Forward Back Propagation (CFBP), and Radial Basis Function (RBF). In this study neural network models with single hidden layer were developed. The task of identifying the number of neurons in the input and output layer is normally simple, as is indicated by the input and output variables considered in the model of physical process. Whereas, the appropriate number of hidden layer nodes for the models are not known for which a trial and error method was used to find the best network configuration. The optimal architecture was determined by varying the number of hidden neurons. Optimal configuration was based upon minimizing the difference between neural network predicted value and the desired output. In general, as the number of neurons in the layer is increased, the prediction capability of the network increases in beginning and then becomes stationary. The performance of all neural network model configurations was based on the Coefficient of determination of the linear regression line between the predicted values from neural network model and the desired output (ܴଶ ), Mean Absolute Percentage Error (‫,)ܧܲܣܯ‬ Root Mean Square Error (ܴ‫,)ܧܵܯ‬ Average Absolute Deviation (‫.)ܦܣܣ‬ The training of the neural network models was stopped when either the acceptable level of error was achieved or when the number of iterations exceeded a prescribed maximum. The neural network model configuration that minimized the MAPE and optimized the 2 R was selected as the optimum and whole analysis was repeated several times. Sensitivity tests were conducted to determine the relative significance of each of the independent parameters (input neurons) on the length of hydraulic jump (output) in the model. In the
  • 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 26 sensitivity analysis, each input neuron was in turn eliminated from the model and its influence on prediction of length of hydraulic jump on rough beds was evaluated in terms of the (ܴଶ ), (‫,)ܧܲܣܯ‬ (ܴ‫)ܧܵܯ‬ and (‫)ܦܣܣ‬ criteria. RESULTS AND DISCUSSIONS All patterns were normalized within range of 0.0 to 1.0 before their use. Similarly all weights and bias values were initialized to random numbers. While the numbers of input and output nodes are fixed, the hidden nodes were subjected to trials and the one producing the most accurate results (in terms of the Correlation Coefficient) was selected. Fig.1 shows the change in error as a function of the number of hidden nodes for present ANN model. The training of neural network models was stopped after reaching the minimum mean square error of 0.0001 between the network yield and true output over all the training patterns. The information on number of nodes required to achieve minimum error taken in the training scheme used (i.e. RBF) is shown in Table 2 for present model. The network architecture of the model as given by Eq. (4), is shown in Fig.2. The error estimation parameters ( 2 R , MAPE , RMSE and ‫)ܦܣܣ‬ on the basis of which the performance of a model is assessed are given in Tables 3 for the network details shown in Table 2. The predicted values of length of hydraulic jump have been plotted against its observed values in Figs. 3 obtained from RBF training scheme. The most suitable network, RBF has the highest 2 R =0.9596 lowest MAPE = 6.9231 and RMSE=3.2438. All the ANN models featured small MAPE and RMSE during training, however, the value was slightly higher during validation. The models showed consistently good correlation throughout the testing. In the end therefore the network configuration (RBF) is recommended for general use in order to predict the length of hydraulic jump on rough and smooth beds. i) Comparison of ANN model with existing empirical model for length of hydraulic jump on rough beds To evaluate the accuracy of the ANN model in prediction of the length of hydraulic jump on smooth and rough beds, a comparison was made using the same data set between the ANN model with an empirical equation developed by Hughes and Flack. (1984): ‫ܮ‬௝ ݄ଶ ൌ ൫଼ி௥భା଺ඥி௥భିଵସ൯൫ଵି଴.ଷହ√௥൯ ሺଵ.ସி௥భି଴.ସሻሺଵି଴.ଶ௥ሻ ൗ (5) Where ‫ݎ‬ ൌ ݇௦ ݄ଵ⁄ It is observed that the ANN result fits the data with high accuracy than Hughes and Flack empirical equation. For Hughes and Flack empirical equation (Hughes and Flack, 1984), the 2 R = 0.9556, compared to 0.9596 for ANN model. Corresponding values of MAPE are 7.5342 and 6.9231 for Hughes and Flack and ANN model respectively. Similarly respective values of RMSE are 3.4732 for Hughes and Flack model and 3.2438 for ANN model. The ANN model give better predictions as shown in Table 4. (ii) Sensitivity Analysis Sensitivity tests were conducted to ascertain the relative significance of each of the independent parameters (input neurons) on the length of hydraulic jump(output) in the present ANN model by Eqs. (4). In the sensitivity analysis, each input neuron was in turn eliminated from the model and its influence on prediction of length of hydraulic jump was evaluated in terms of 2 R ,
  • 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 27 MAPE , ‫ܦܣܣ‬ and RMSE criteria. The network architecture of the problem considered in the present sensitivity analysis consists of one hidden layer with thirty two neurons and the value of epochs has been taken as 1000. Comparison of different neural network models, with one of the independent parameters removed in each case is presented in Table 5. The results in Table 5 show that ‫ݎܨ‬ଵ is the most significant parameter for the prediction of length of hydraulic jump. The variables in order of decreasing level of sensitivity for ANN model are ‫ݎܨ‬ଵ and ݇௦ ݄ଵ⁄ .These findings are consistent with existing understanding of the relative importance of the various parameters on length of hydraulic jump on rough beds. CONCLUSIONS An attempt was made to assess the performance of the artificial neural networks (ANN) (with FFBP, FFCC and RBF training algorithms) prediction models over the regression method (RM) using adequate size of laboratory data for length of hydraulic jump on smooth and rough beds. In general, this study proved that the ANN can be used as a powerful soft computing tool for prediction of length of hydraulic jump on smooth and rough beds and is better in performance as compared to regression model. In the case of ANN models, the Radial Basis Function Neural Networks (RBF) was found to be the best among the other training algorithms of ANN. The neural network with one hidden layer was selected as the optimum network to predict the length of hydraulic jump on smooth and rough beds. The network configuration of present ANN model with RBF is recommended for general use in order to predict the length of hydraulic jump on smooth and rough beds. On the basis of the sensitivity analysis, it is observed that ‫ݎܨ‬ଵ is the most significant parameter. Table 1 Details of data used S.No. Investigator Variable Range From To 1. Hughes and Flack (1984) ݄ଵ (cm), flow depth at the toe of the jump 1.07 3.35 ݄ଶ (cm), flow depth at the end section of the jump 7.80 14.20 ݇௦ (cm), roughness height 0.00 1.13 ‫ܮ‬௝ (cm), length of jump 39.62 88.39 ‫ݎܨ‬ଵ, initial Froude number 2.34 8.40 ܳ (l/s), discharge 9.905 14.716 ݇௦ ݄ଵ⁄ , relative roughness 0.00 0.90 ݄ଶ ݄ଵ⁄ , sequent depth ratio 2.71 11.83 ‫ܮ‬௝ ݄ଵ⁄ 12.73 68.29 2. Ead and Rajaratnam (2002) ݄ଵ (cm), pre jump depth 2.54 5.08 ݄ଶ (cm), post jump depth 10.40 31.00 ݇௦ (cm), roughness height 1.30 2.20 ‫ܮ‬௝ (cm), length of jump 41.00 129.00 ‫ݎܨ‬ଵ, initial Froude number 4.00 10.00 ܳ (l/s), discharge 22.75 92.32 ݇௦ ݄ଵ⁄ , relative roughness 0.25 0.50 ݄ଶ ݄ଵ⁄ , sequent depth ratio 4.09 6.10 ‫ܮ‬௝ ݄ଵ⁄ 16.14 42.91
  • 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 28 Table 2 Details of various ANN prediction models ANN Models ANN Architecture ANN parameters I H O FFBP 2 4 1 Mu = 0.001 CFBP 2 4 1 Mu = 0.001 RBF 2 32 1 Spread = 1.0 Note: I, H, O indicate number of input, hidden and output nodes respectively, FFBP = Feed Forward Back Propagation, CFBP = Cascade Forward Back Propagation: and RBF = Radial Basis Function. Table 3 (a): Performance of ANN prediction models in training ANN Models ࡾ૛ ࡹ࡭ࡼࡱ ࡾࡹࡿࡱ ࡭࡭ࡰ FF BP 0.9352 8.7825 4.0806 8.3375 CF BP 0.9442 8.1728 3.7968 7.9300 RBF 0.9596 6.9231 3.2438 6.5831 Table 3 (b): Performance of ANN prediction models in validation ANN Models ࡾ૛ ࡹ࡭ࡼࡱ ࡾࡹࡿࡱ ࡭࡭ࡰ FF BP 0.916852 11.00981 4.776527 9.754572 CF BP 0.922979 9.784498 4.647935 8.971553 RBF 0.914814332 10.84756 4.942778 10.1596 Table 3 (c): Performance of ANN prediction models in all data ANN Models ࡾ૛ ࡹ࡭ࡼࡱ ࡾࡹࡿࡱ ࡭࡭ࡰ FF BP 0.9313 9.2216 4.2268 8.6222 CF BP 0.9395 8.4897 3.9790 8.1393 RBF 0.9497 7.6967 3.6419 7.3017 Table 4: Performance of Regression prediction model Model ࡾ૛ ࡹ࡭ࡼࡱ ࡾࡹࡿࡱ ࡭࡭ࡰ Hughes and Flack(1984) All 0.9540 7.6779 3.5556 7.4644 Training 0.9556 7.5342 3.4732 7.3176 Validation 0.9477 8.2631 3.8730 8.0458 Table 5 Sensitivity Analysis for ANN model with radial basis function (RBF) Input variables ࡾ૛ ࡹ࡭ࡼࡱ ࡾࡹࡿࡱ ࡭࡭ࡰ With all (Eq.4) 0.9497 7.6967 3.6419 7.3017 All 0.9596 6.9231 3.2438 6.5831 Training 0.9148 10.8476 4.9428 10.1596 Validation With no ࡲ࢘૚ 0.6060 22.3917 9.2353 19.3759 All 0.6243 21.9324 9.0040 18.6471 Training 0.5446 24.2624 10.1239 22.2748 Validation With no ࢑࢙ ࢎ૚⁄ 0.8634 12.2389 5.8523 11.9545 All 0.8733 11.6983 5.6151 11.5389 Training 0.8242 14.4412 6.7328 13.6027 Validation
  • 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 29 (a) (b) (c) Fig. 1: Variation of performance parameters with number of hidden nodes by RBF Fig. 2: General structure of ANN model 0.75 0.85 0.95 1.05 0 15 30 45 60 R2 Number of hidden nodes All Training Validation RBF 5 8 11 14 0 15 30 45 60 MAPE Number of hidden nodes All Training Validation RBF 5 8 11 14 0 15 30 45 60 MAPE Number of hidden nodes All Training Validation RBF 1 b a r Output layer Hidden layerInput layer 2 n
  • 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 5, Issue 3, March (2014), pp. 23-31 © IAEME 30 (Lj / h1 )Predicted= 0.920( (Lj / h1 )Observed+2.846 (Lj / h1 )Predicted= 0.815( (Lj / h1 )Observed+5.640 (a) (b) Fig. 3: Comparison between observed and computed values of ‫ܮ‬௝ ݄ଵ⁄ by RBF REFERENCES [1]. Alhamid, A.A. (1994). Effective roughness on horizontal rectangular stilling basins. Trans. Eco. Environ Vol. 8 pp.39-46. [2]. Ansari M.A. (2014). Sediment removal efficiency computation in vortex settling chamber using artificial neural networks, Water and Energy International, Central Board of Irrigation and Power (CBIP), New Delhi, Vol. 71, No.1, pp. 54-67. [3]. Ansari M.A. and Athar M. (2013). Artificial neural networks approach for estimation of sediment removal efficiency of vortex settling basins, ISH Journal of Hydraulic Engineering Taylor & Francis Vol. 19, No.1, pp. 38-48. [4]. Azmathullah, H. M., Deo, M. C. and Deolalikar, P. B. (2005). Neural Networks for Estimation of Scour Downstream of a Ski-Jump Bucket. JHE, ASCE, Vol. 131, No. 10, pp. 898-908. [5]. Carollo, F. G., and Ferro, V. (2004a). Contributo allo studio della lunghezza del risalto libero su fondo liscio e scabro. Rivista di Ingegneria Agraria, Vol. 354, pp. 13–20 in Italian. [6]. Carollo, F. G., and Ferro, V. (2004b). Determinazione delle altezze coniugate del risalto libero su fondo liscio e scabro. Rivista di Ingegneria Agraria, Vol. 35, No.4, pp.1–11 in Italian. [7]. Carollo, F. G., Ferro, V. and Pampalone V. (2007). Hydraulic jumps on rough beds. J Hydraul Eng, ASCE, Vol. 133, No.9, pp.989–999. [8]. Ead, S. A., and Rajaratnam, N. (2002). Hydraulic jumps on corrugated beds. J. Hydraul. Eng., Vol. 128, No. 7, pp.656–663. [9]. Hager W.H. (1992). Discussion of force on slab beneath hydraulic jump. J Hydraul Eng, ASCE, Vol.118, No.4 pp.666–8. [10]. Hughes, W. C., and Flack, J. E. (1984). Hydraulic jump properties over a rough bed. J. Hydraul. Eng., Vol. 110 No.12 pp.1755–1771. [11]. Liriano, S.L., and Day, R. A. (2001). Prediction of Scour depth at Culverts Using Neural Networks. J.Hydroinformatics, Vol. 3, No.4, pp. 231-238. [12]. McCorquodale J. A. (1986). Chapter 8: Hydraulic jumps and internal flows. Encyclopedia of fluid mechanics, N. P. Cheremisinoff, ed., Vol. 2, Gulf Publishing, Houston, 120–173. R2 = 0.9592 0 20 40 60 80 0 20 40 60 80 PredictedLj/h1 Observed Lj / h1 Training Line of Agreement Best Linear Fit RBF R2 = 0.9143 0 20 40 60 80 0 20 40 60 80 PredictedLj/h1 Observed Lj / h1 Validation Line of Agreement Best Linear Fit RBF
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