Prediction of Pressure Drop Using Artificial Neural ... - ACS Publications

Ind. Eng. Chem. Res. 2010, 49, 9423–9429

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Prediction of Pressure Drop Using Artificial Neural Network for Gas Non-Newtonian Liquid Flow through Piping Components Nirjhar Bar,† Manindra Nath Biswas,‡ and Sudip Kumar Das*,† Chemical Engineering Department, UniVersity of Calcutta, 92 A. P. C. Road, Kolkata 700 009, India, and GoVernment College of Engineering & Leather Technology, LB Block, Sector III, Salt Lake City, Kolkata 700098, India

The ANN approach proved its worth when rigorous fluid mechanics treatment based on the solution of first principle equations is not tractable. Evaluation and prediction of the frictional pressure drop across different piping components such as orifices, gate and globe valves, elbows, and horizontal pipe in 0.0127 m diameter for gas non-Newtonian liquid flow is manifested in this paper. In this paper, we have used the power-lawmodel (Oswald-de Waele model) liquids only. The experimental data used for the prediction is taken from our earlier work, Bandyopadhyay (Bandyopadhyay, T. K. Studies on non-Newtonian and gas-non-Newtonian liquid flow through horizontal tube and piping components. Ph.D Thesis, University of Calcutta, Kolkata, India, 2002) and the subsequent publications (Banerjee T. K.; Das, S. K. Gas-non-Newtonian liquid flow through globe and gate valves. Chem. Eng. Commun. 1998, 167, 133-146. Samanta A. K.; Banerjee, T. K.; Das, S. K. Pressure loses in orifices for the flow of gas-non-Newtonian liquids. Can. J. Chem. Eng. 1999, 77, 579-583. Bandyopadhyay, T. K.; Banerjee, T. K.; Das, S. K. Gas-non-Newtonian liquid flow through elbows. Chem. Eng. Commun. 2000, 82, 21-33). The proposed approach toward the prediction is done using a multilayer perceptron (MLP with one hidden layer and four different transfer functions), which is trained with backpropagation algorithm. Introduction The problem of predicting pressure losses in piping components is much more uncertain than that for straight pipes because the mechanism of the flow is not clearly defined. At least three types of losses are superposed-skin friction, loss due to change of flow direction, and the constriction of the flow path. There is only a little experimental data available in the literature.5 Twophase gas-liquid flow occurs in many engineering applications, such as in equipment related to the oil, chemical, process, and power generation industries. The hydrodynamics of cocurrent gas-liquid flows have received extensive treatment during last few decades. There are, however, areas that have received little attention; one of these areas is gas-liquid flow through piping components. The gas-liquid flow through piping components is even more complex in nature.2-5 The gas-non-Newtonian two-phase flow is gaining importantce, as is clear from the review of Das et al.6 Mathematical models derived from the physical description and understanding of the gas-liquid flow through piping components is extremely difficult, as the phenomena of momentum transfer between the phases, the wall friction, the shear at the phase interface, change of flow direction, phase separation, and constriction cannot be specified quantitatively. Hence, the empirical equations for pressure losses across the different piping components for gas-non-Newtonian liquid flow have been derived by Das and his co-workers1-3 from their experimental data. In recent years, the concept of artificial neural network (ANN) has gained widespread application in many engineering problems.7 ANN models can learn from examples, incorporate large number of variables, and provide an adequate and quick response to the new information. The advantages of neural networks are as follows: * Corresponding author. E-mail: [email protected]. † University of Calcutta. ‡ Government College of Engineering & Leather Technology.

(1) The ability to represent both linear and nonlinear relationships. (2) The ability to learn these relationships directly from the data used is an advantage of ANN. (3) The advantage of MLP is that this type of network can be used to create a model that correctly maps the input to the output using historical data so that the model can then be used to produce the output when the output is unknown. (4) At least in some cases, if not always, i.e., for prediction by the trained network, we can say that the ANN systems are alternative to experimentation and save a lot of time that may have been consumed. However, ANN technique has some disadvantages such as8 (1) Minimizing overfitting requires a great deal of computational efforts. (2) The individual relations between the input variables and output variables are not developed by engineering judgmen,t so that the model tends to be a black box without analytical basis. Rumelhart and McClelland9 proposed the “multilayer perceptron” (MLP) and showed that it was application for parallel distributed processing. From the end of the 1980s, there has been explosive growth in applying neural networks to various problems in different fields of science and technology such as online color sorting and quality control of apples,10 controlling the fermentation process in penicillin production,11 investigating the drying process of food processing,12 flood forecasting,13 mineral deposit potential mapping,14 etc. Himmelblau7 reviewed the application of the artificial neural networks in chemical engineering field. The ANN approach was used to predict pressure drop of non-Newtonian fluid foods through tubes of 7.51-16.34 mm stainless steel tubes.15 Larachi et al.16 combined the ANN and dimensional analysis to derive correlations for bed porosity and liquid and gas holdups in three phase fluidization. Blanco et al.17 reported the applicability of ANN modeling to determine several linear and nonlinear physical

10.1021/ie1007739  2010 American Chemical Society Published on Web 09/02/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

Figure 1 shows the schematic diagram of the artificial neural network. Four different transfer functions in the hidden layer were used for our network. These are represented below as transfer function 1: y ) f(x) ) tanh βx ) transfer function 2: y ) f(x) ) βx where

eβx - e-βx eβx + e-βx

(1)

βx ) 1 for βx > 1 βx ) -1 for βx < -1

(2)

transfer function 3: y ) f(x) ) βx where

Figure 1. Schematic diagram of neural network.

transfer function 4: y ) f(x) )

Figure 2. Variation in the minimum value of MSE with the number of processing elements in the hidden layer for orifice for four different transfer functions.

properties of bitumens. Shaikh and Al-Dahhan18 reported the prediction of the overall gas holdup in bubble column reactors using artificial neural network using a databank of around 3500 experimental data collected from the open literature. Xie et al.19 used ANN for the prediction of flow regime in gas-liquid-pulp fiber slurry flow. Sablani et al.20 reported an ANN approach to predict the friction factor for flow of bingham plastic. Sablani and Shayya21 developed ANN-based noniterative calculation of the friction factor for power law fluids. Zhao et al.22 reported the use of ANN to calculate the thixotropic stress of waxy crude oil. Torkar et al.23 used ANN to predict the apparent viscosity of alumina-paraffin suspension. Lahiri and Ghanta24 reported the use of ANN for the prediction of holdup in slurry transportation system. In the present paper application of ANN is used to predict the two-phase pressure drop for 0.0127 m horizontal pipe and different piping components based on our earlier experimental data obtained from Bandyopadhyay1 and their subsequent publications.2-4

βx ) 0 for βx < 0 βx ) 1 for βx > 1

1 1 + e-βx

(3) (4)

where, β is the gain and it is used to control the steepness of the transfer function. The value of β is mostly unity. Optimization of the ANN. Initially the total data was randomized. The first 60% of data points are used for training, the next 20% for cross-validation, the next 10% for testing and the rest used for prediction. The synapse that connects the hidden layer to the input layer adjusts the weights and the learning rates. It is always desired that the number of processing elements in the hidden layer must be kept minimum to reduce the complexity of the network. In the hidden layer, the number of processing elements is optimized, varying the number from 1 to 25, and for each case the MSE was calculated. Figure 2 shows the variation in MSE with the number of processing elements for orifice. A similar procedure was followed by Yetilmezsoy and Demirel.25 The optimum number of processing elements is that where the MSE is at a minimum. These optimum numbers of processing elements are used for further subsequent analysis. After the output is generated, the criterion function accepts the output(s) of the network and compares them with the desired output(s). It calculates the error and passes this error to the backpropagation components, which adjust the weights of the network for training. MLPs are trained with the backpropagation (BP) algorithm. The backpropagation process propagates the errors backward through the network and allows adaptation of the hidden processing elements. A closed-loop control system is thus established. The weights are automatically adjusted using a gradient-descent-based algorithm. The MSE is measured that shows the difference between the network output and the output from the training sample. The following standard formula was used for optimization, MSE )

1 N

N

∑ (x

i

- yi)2

(5)

i)1

Results and Discussion Data Collection. Experimental data have been collected from our earlier study.1-4 Table 1 represents the description of the number data used for neural network analysis, and Table 2 represents the range of data used.

Table 1. Description of the Data Used for Neural Network Analysis system type

data considered for training

data considered for cross validation

data considered for testing

data considered for prediction

horizontal pipe elbow orifice gate valve globe valve

76 216 216 432 357

26 72 72 144 119

12 35 36 72 59

12 35 36 72 59

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Table 2. Range of Different Data Sets for All Five Different Systems measurement type

range

Piping Components pipe diameter (m) angle of elbow (deg) radius of curvature of elbow (m) ratio of the valve opening to the full opening of valve (gate and globe) orifice diameter (m) ratio of orifice to pipe diameter

0.0127 45-135 0.011 e Rc e 0.017 0.25 e R e 1.00

every separate system, the optimum result was achieved using 32 000 epochs for training. The cross validation was also done simultaneously. The total data set was randomized before the network was used. Performance of the ANN. The performance of the network was checked using mean square error (MSE) and the following parameters: average absolute relative error (AARE)

0.0059 e Do e 0.0090 0.4646 e DO/Dt e 0.7087

AARE )

Physical Properties of Liquid 3

concentration of SCMC solution (kg/m ) flow behavior index consistency index (Nsn′/m2) density (kg/m3) surface tension (N/m)

0.2-0.8 0.6015 e n′ e 0.9013 0.0142 e K′ e 0.7112 1001.69 e F e 1003.83 0.07834 e σl e 0.0832 1.1614 0.00001846

Flow Rate liquid flow rate Ql × 105(m3/s) gas flow rate Qg × 105(m3/s)

3.75-29.83 1.7860-44.7530

Frictional Pressure Drop (Experimental) (kPa) horizontal pipe elbow orifice gate valve globe valve

σ)

∑|

[|

(yi - xi) - AARE xi

N

i)1

(yi - xi) xi

(6)

N

∑ N -1 1 i)1

]

|

2

(7)

The AARE and standard deviation are kept as small as possible for the better performance of the neural network. Tables 3 and 4 represent the performance of the network for five different piping systems for training and cross-validation, respectively. cross-correlation coefficient (R)

0.5111 e ∆Ptp e 24.3777 0.1333 e ∆Ptp e 7.6000 0.4000 e ∆Ptp e 46.4000 0.2667 e ∆Ptp e 16.2667 1.7333 e ∆Ptp e 45.4667

Input Parameters. Input parameters are the physical and operating variables of the system. The physical variables include the diameter of the tube (D), radius of curvature of the elbow (Reb), angle of the elbow (Reb), ratio of the valve opening to the full opening of valve (R) in the case of globe and gate valve, properties of the non-Newtonian liquid, i.e., flow behavior index (n′), consistency index (K′), surface tension of the liquid (σsl), density of liquid (Fl), density of gas (Fg), viscosity of gas (µg), and acceleration due to gravity (g), whereas the operating variables are liquid flow rate (Ql) and gas flow rate (Qg). The values of the physical properties of gas, i.e., density and viscosity, tube diameter, and acceleration due to gravity, are constant, so they are ineffective for ANN programming. For

|

1 N

standard deviation (σ)

Physical Properties of Air density (kg/m3) viscosity of air (Ns/m2)

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N

∑ (x

i

R)

- jx)(yi - jy)

i)1

∑

(8)

N

N

(xi - jx)

i)1

2

∑ (y

i

- jy)

2

i)1

It has also been verified that the Ccross-correlation coefficient is as close to unity as possible. chi-square test (χ2) N

χ2 )

∑ i)1

(xi - yi)2 yi

(9)

The chi-square test was also performed to find the best-fit network model when the values of cross-correlation coefficients are close to each other. The value closest to zero indicates the best model.

Table 3. Performance of Best Neural Network for Training over 5 Runs system type

measurement type

transfer function 1

transfer function 2

transfer function 3

transfer function 4

horizontal pipe

SD (σ) MSE SD (σ) MSE SD (σ) MSE SD (σ) MSE SD (σ) MSE

0.000050 0.000241 0.000602 0.002200 0.000345 0.001351 0.000021 0.001004 0.000142 0.000704

0.000026 0.000028 0.000385 0.000615 0.000147 0.000525 0.000061 0.000716 0.000063 0.000443

0.000041 0.000127 0.000088 0.001437 0.000141 0.000839 0.000086 0.001026 0.000078 0.000750

0.000211 0.000437 0.000169 0.002682 0.000076 0.001884 0.000027 0.001283 0.000075 0.000979

elbow orifice gate valve globe valve

Table 4. Performance of the Best Neural Network for Cross Validation over 5 Runs system type

measurement type

transfer function 1

transfer function 2

transfer function 3

transfer function 4

horizontal pipe

SD (σ) MSE SD (σ) MSE SD (σ) MSE SD (σ) MSE SD (σ) MSE

0.001301 0.003521 0.000369 0.006549 0.000188 0.001314 0.000052 0.001371 0.000114 0.000952

0.001558 0.004575 0.011327 0.006491 0.000088 0.001637 0.000071 0.001180 0.000087 0.000916

0.001500 0.005771 0.000443 0.006172 0.000186 0.001148 0.000032 0.001482 0.000058 0.001011

0.001038 0.008821 0.000165 0.007163 0.000161 0.001222 0.000047 0.001651 0.000073 0.001036

elbow orifice gate valve globe valve

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Figure 3. Variation in the average MSE for training over 5 different runs vs the number of epochs in case of horizontal pipe for four different transfer functions in the hidden layer.

Figure 5. Comparison of two-phase pressure drop across the five different piping systems for testing using transfer function number 1 in the hidden layer.

Figure 6. Comparison of two-phase pressure drop across the five different piping systems for prediction using the optimized network for the respective transfer functions in the hidden layer. Figure 4. Variation in the average MSE for cross-validation over 5 different runs vs the number of epochs in case of gate valve for four different transfer functions in the hidden layer.

Figure 3 represents the training curve for horizontal pipe for four different transfer functions used in the hidden layer. The same procedure was followed for the other four different piping systems, namely, elbow, orifice, gate valve, and globe valve, respectively. Figure 4 represents the cross-validation curve for gate valve for four different transfer functions used in the hidden layer. The same procedure was followed for the other four different piping systems, namely, horizontal pipe, elbow, orifice, and globe valve, respectively. Initially the MSE for each epoch for training and cross-validation in all five cases were recorded

for five different runs separately. Each run consists of 32 000 epochs. The average of the MSE of all the five different runs was then calculated for each of the 32 000 epochs for each of the five systems. The gradual decrease in the values of average MSE in all five cases shows that the training was accurate. Figure 5 show the comparison between the experimental to the predicted output for testing for the above-mentioned five different systems respectively for transfer function number 1 used in the hidden layer. Figure 6 show the comparison between the experimental to the predicted output for the above-mentioned five different systems respectively for different transfer functions used in the hidden layer after optimization. This comparison proves the effectiveness of the neural network analysis.


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Table 5. Performance of the Best Neural Network for Testing system type

measurement type

transfer function 1

transfer function 2

transfer function 3

transfer function 4

horizontal pipe

AARE SD (σ) MSE CCC (R) AARE SD (σ) MSE CCC (R) AARE SD (σ) MSE CCC (R) AARE SD (σ) MSE CCC (R) AARE SD (σ) MSE CCC (R)

0.139431 0.159955 1.155644 0.973274 0.198541 0.298257 0.097880 0.986369 0.074940 0.062268 2.473256 0.994096 0.123889 0.241808 0.139428 0.988455 0.091750 0.106653 1.519431 0.993495

0.091344 0.113177 0.786913 0.982891 0.390736 0.888029 0.184884 0.977530 0.146182 0.202334 6.773724 0.978081 0.114436 0.136457 0.168411 0.986104 0.081437 0.087487 1.488616 0.993687

0.141090 0.221266 1.941306 0.954967 0.164034 0.145425 0.118365 0.984129 0.116760 0.129884 5.042553 0.984655 0.104425 0.118239 0.102588 0.991832 0.095528 0.102939 1.756384 0.992771

0.102353 0.146129 1.080488 0.973249 0.217509 0.189990 0.143730 0.980557 0.091214 0.087222 2.915442 0.993024 0.105825 0.142197 0.116460 0.990975 0.090104 0.099928 1.576777 0.993492

elbow

orifice

gate valve

globe valve

Table 6. Performance of the Best Neural Network for Prediction measurement type

transfer function 1

AARE SD (σ) MSE CCC (R) χ2 optimum no. of processing elements in hidden layer

0.049228 0.040359 0.503364. 0.993446 0.506992 23


0.109949 0.095818 0.071158 0.970161 1.142137 14


0.102161 0.135718 1.056509 0.996085 0.097863 15


0.103767 0.110178 0.281146 0.977378 4.784179 18


0.097179 0.177516 3.082688 0.980708 11.387228 19

transfer function 2

transfer function 3

transfer function 4

Horizontal Pipe 0.063092 0.060217 0.417634 0.994157 0.598442 15

0.031665 0.025103 0.355275 0.995086 0.251003 25

0.036619 0.040873 0.613286 0.991108 0.468126 16

0.148015 0.111625 0.069816 0.970622 1.553100 25

0.147962 0.115751 0.066139 0.969900 1.535549 25

0.252263 0.494438 1.306411 0.995488 -12.71403 24

0.108418 0.160257 1.289722 0.995386 5.448912 13

0.102240 0.097182 0.291895 0.976331 5.079541 21

0.106946 0.092749 0.240387 0.980929 3.935746 13

0.097012 0.209676 4.158719 0.973646 13.35638 24

0.093106 0.209544 4.003748 0.974839 12.25122 7

Elbow 0.179042 0.176449 0.075493 0.969416 2.648163 21 Orifice 0.225771 0.443963 1.301362 0.995262 -27.40547 17 Gate Valve 0.106917 0.103566 0.324183 0.973481 5.967776 22 Globe Valve

Tables 5 and 6 represent performance of neural network for testing and prediction. It is clear from these tables that the crosscorrelation coefficient value is more than 0.97 for each of the five cases for four different transfer functions used in the hidden layer. The low value of the average absolute relative error

0.086814 0.159675 2.906793 0.981658 10.051990 25

(AARE) also shows the accuracy of the result in the different systems. This result indicates that the performance of the network output is excellent. Because the cross-correlation coefficient value is more than 0.97 for all the best networks, so the chi-square test was performed to

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find the best result. Table 6 contains the result for the chi-square test. The chi-square test confirms that the best network for prediction of pressure drop for horizontal pipe is the one that has the transfer function 3 with 25 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for elbow is the one that has the transfer function 1 with 14 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for orifice is the one that has the transfer function 1 with 15 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for gate valve is the one that has the transfer function 4 with 13 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for globe valve is the one that has the transfer function 2 with 25 processing elements in the hidden layer.

σl σ

surface tension of liquid (N/m) standard deviation (dimensionless)

Subscripts

c g o t tp

curvature gas orifice tube two phase

Note Added after ASAP Publication: The version of this paper that was published online September 2, 2010 contained an error in eq 1. The revised version was published September 7, 2010. Literature Cited

Conclusions A neural network-based model was developed for the prediction of pressure drop for gas-non-Newtonian liquid flow through different piping components, i. e., horizontal pipe, elbow, orifice gate valve, and globe valve. A multilayer perceptron with backpropagation algorithm was used for this analysis with four different transfer functions used in the hidden layer and a linear function in the output layer. The ANN model accurately predicts the pressure drop across the horizontal pipe and different piping components, as is evident from the cross-correlation coefficient, which is greater than than 0.97. The chi-square test confirms that the best network for prediction of pressure drop for horizontal pipe is the one that has the transfer function 3 with 25 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for elbow is the one that has the transfer function 1 with 14 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for orifice is the one that has the transfer function 1 with 15 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for gate valve is the one that has the transfer function 4 with 13 processing elements in the hidden layer. The chi-square test confirms that the best network for prediction of pressure drop for globe valve is the one that has the transfer function 2 with 25 processing elements in the hidden layer. Nomenclature D K′ N n′ ∆P R R Q x y

diameter (m) consistency index (Nsn′ m-2) total number of data sets flow behavior index (dimensionless) pressure drop (N m-2) radius (m) cross-correlation coefficient (dimensionless) flow rate (m3 s-1) experimental value of pressure drop (N m-2) predicted value of pressure drop (N m-2)

Greek Letters

R β F

ratio of the valve opening to the full opening of valve (dimensionless) gain (dimensionless) density (kg m-3)

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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 (20) Sablani, S. S.; Shayya, W. H.; Kachimov, A. 2003. Explicit Calculation of the friction factor in pipeline flow of Bingham plastic fluids: A neural network approach. Chem. Eng. Sci. 2003, 58 (1), 99–106. (21) Sablani, S. S.; Shayya, W. H. Neural network based non-iterative calculation of the friction factor for power law fluids. J. Food Eng. 2003, 57 (4), 327–335. (22) Zhao, Z. C.; Zhang, X. D.; Wang, D.; Xu, J. M. Calculation of thixotropic stress of waxy crude oil by BP artificial neural network. Dalian Ligong Daxue Xuebao/J. Dalian UniV. Technol. 2005, 45 (2), 181–185. (23) Torkar, D.; Novak, S.; Novak, F. Apparent viscosity prediction of Alumina-paraffin suspensions using artificial neural networks. J. Mater. Process. Technol. 2008, 203 (1-3), 208–215.

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(24) Lahiri, S. K.; Ghanta, K. C. Development of an artificial neural network correlation for prediction of hold-up of slurry transport in pipelines. Chem. Eng. Sci. 2008, 63, 1497–1509. (25) Yetilmezsoy, K.; Demirel, S. Artificial neural network (ANN) approach for modeling of Pb(II) adsorption from aqueous by Antep pistachio (Pistacia Vera L.) shells. J. Hazard. Mater. 2008, 153, 1288–1300.

ReceiVed for reView March 31, 2010 ReVised manuscript receiVed July 31, 2010 Accepted August 17, 2010 IE1007739

Prediction of Pressure Drop Using Artificial Neural ... - ACS Publications

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