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Prediction of shockwave location in supersonic nozzle separation using self-organizing map classification and artificial neural network modeling
Accepted Manuscript Prediction of shockwave location in supersonic nozzle separation using selforganizing map classification and artificial neural network modeling Pouriya H. Niknam, B. Mokhtarani, H.R. Mortaheb PII:
S1875-5100(16)30533-9
DOI:
10.1016/j.jngse.2016.07.061
Reference:
JNGSE 1683
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 15 March 2016 Revised Date:
22 July 2016
Accepted Date: 23 July 2016
Please cite this article as: Niknam, P.H., Mokhtarani, B., Mortaheb, H.R., Prediction of shockwave location in supersonic nozzle separation using self-organizing map classification and artificial neural network modeling, Journal of Natural Gas Science & Engineering (2016), doi: 10.1016/ j.jngse.2016.07.061. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Prediction of shockwave location in supersonic nozzle separation using self-organizing map classification and artificial neural network modeling
Pouriya H. Niknam, B. Mokhtarani*, H.R. Mortaheb*
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Chemistry & Chemical Engineering Research Center of Iran, P.O. Box 14335-186, Tehran, Iran
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Abstract One of the novel technologies for natural gas dehydration and natural gas dew-point conditioning is supersonic separation, which has remarkable features, including compact and maintenance-free design. Due to its complex design and the difficulty of experimental analysis, researchers tend to conduct numerical modeling for behavior investigation of the nozzle focusing on shockwave which is the main phenomena inside the nozzle. The present NN-model outperforms a selection of data and proposes an efficient NN-based algorithm for shockwave position estimation as the key nozzle geometry parameter. Data for the shockwave location was collected from a wide range of results from the literature and then a neural network based self-organizing map was adapted to the dataset. This created a classified dataset and the use of unreal weight and repeated experimental results from different research were avoided. A neural network was employed for modeling the shockwave location through the nozzle using a better quality dataset. Additionally, the one-dimensional inviscid theory was utilized in the recursive approach for comparison to the main proposed model. Simulation results presented in this research reveal the effectiveness of the proposed neural network technique for supersonic nozzle modeling and make it possible to determine the shockwave location from the nozzle pressure boundary conditions. The results showed that the supersonic nozzle separation have capability to be used in both low-pressure applications and high pressure ones. The dimensionless length for shockwave location is predicted in the range of 0.82 to 0.92 for the former and 0.72 to 0.95 for the later, depending on pressure recovery ratio. Keywords: Shockwave location, Supersonic nozzle, Neural network, Self-organizing maps, natural gas separation. *
Corresponding authors. Tel: +98-21-44787751, Fax: +98-21-44787781, E-mail address; [email protected], [email protected].
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1. Introduction
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Extracted natural gas from wells usually contains certain quantities of water that condense and form gas hydrates if the gas pressure is reduced or the gas is cooled below its hydrate formation temperature or water dew point. Water as an undesirable component must be removed to inhibit hydrate and condensation formation. Erosion, corrosion, plugged pipelines, reduced pipeline flow capacity, and operation interruptions are possible consequences of the mentioned phenomena. To prevent these problems, the gas stream must be dehydrated. Specifications typically call for water content to be no more than 110mg/Sm3 in US pipeline systems, 60 mg/Sm3 in Canadian pipeline systems, and 130 mg/Sm3 in Iranian pipeline systems. This requirement puts the maximum water in sales gas at the dew point of -10 ºC at any outlet delivery and line pressure. These values provide protection against water condensation and hydrate formation as a consequence of low ambient temperature. There are several methods of dehydrating natural gas, such as direct cooling of the gas stream, absorption, and adsorption processes (Mokhatab et al., 2012).
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Recent research has introduced a novel technology based on gas expansion in a supersonic nozzle that changes gas stream conditions by lowering temperature and pressure, while part of gas enthalpy transforms to kinetic energy. The basic component of the system is a converging–diverging nozzle with no moving parts where condensation occurs at supersonic velocities. The minimum temperature and pressure are found at the shockwave position. During this process, water and other possible condensations are removed from the gas stream by the supersonic nozzle for dew point control applications and no external work is used as in conventional refrigeration units (Brouwer et al., 2003 & Kalikmanov et al., 2007, Alfyorov, 2005). Fig. 1 shows the structure of the conventional supersonic nozzle which consists of swirling generator, Laval nozzle, and supersonic diffuser. Different scales of the nozzle were used in the literature.
Fig. 1. Schematic of the supersonic nozzle
Over the past decade, important numerical and modeling investigations to study supersonic separation in the nozzle have been performed. These studies can be categorized into three general areas: economic, hydrodynamic, and geometry investigations. Machado et al. (2012) presented an economic analysis for supersonic nozzle technology compared with the common conventional unit (TEG+JT/LTS). Gonzalez et al. proposed an arrangement of supersonic nozzles for the unsteady inlet. Castier et al. (2014) proposed an adapted equation of state for numerical study and modeling, consisting of single-scaling wavelets. Prast et al. (2006) presented a computational fluid dynamic (CFD) model for heavy cut extraction from natural gas stream, which was proven by the experimental results. Vaziri et al. (2015) 2
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used a CFD model to adjust the optimal values of axial, angular, and radial velocity components of entering fluid. Shooshtari et al. (2013) proposed a theoretical study to predict the liquid droplet growth in binary mixtures inside the laval nozzle using related thermodynamic concepts. Jassim et al. (2008b) noted that their CFD modeling could not predict phase change and that gas condensation was neglected in the modeling. Also, Abdi et al. (2010) proposed CFD modeling, but pointed to the incapability of the model to consider two-phase conditions. This is due to that computational methods have drawbacks as multiphase compressible swirling flow entails phase changes in high-pressure conditions and most of them fail to cover all the mentioned aspects. Sforza et al. (2012) proposed a swirling compressible model under a controlled condensation rate based on quasi-one-dimensional equations for syngas purification using a supersonic swirl tube. Significant research has concentrated on supersonic nozzle dynamic parameters (Wen et al., 2012; Haghighi et al., 2013). Vaziri et al. (2013) developed a numerical model using neural networks for geometry and boundary conditions of the supersonic nozzle. The authors validated this model with numerical results and it could not be widely used in similar cases.
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The correct prediction of separation performance by the supersonic nozzle is important for its design. Information on nozzle characteristics is still limited. Moreover, there is a need for further research, especially into where the gas is shocked and where the separation process largely happens. As a consequence, the main aim of the present study is to create a reliable and simple model based on previous numerical and experimental studies, to cover the wide range of design parameters of the supersonic nozzle.
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The main drawback of model-based approaches is the requirement for accurate system details, which may not be available in many practical applications. Intelligent NN models are designed to overcome some of these restrictive requirements. The artificial neural network is a novel technique that gives researchers an alternative way to solve these kinds of complex problems. Neural networks and other NN-based hybrid models have been used for a wide range of application for prediction of the outputs. (Valipour, 2012, 2015, 2016)
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In this work an alternative method for shockwave capture that uses a well-trained neural network and a classified dataset is proposed. The main goal is to evaluate whether the NN-based approach is feasible for this kind of application and to compare it with the conventional 1-D supersonic model. In contrast to previous numerical models, the proposed model covers a wide range of boundary conditions revealing the effect of pressure recovery ratio. Also the previously neglected role of the swirling effect is accounted for by using the proper experimental dataset. Also, computational complexity is avoided to a large extent by using the NN as a black-box model. So the NN model considers all physical principles such as multiphase interactions and swirling effect indirectly using the input and output valid data obtained from the experiments.
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2. Method description 2.1. Database Finding the axial position of normal shock wave inside the nozzle has an indirect solution, as the shock stands at a position that enables the desired exit pressure to be achieved.
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The flow proceeding the nozzle consists of two isentropic and one non-isentropic process: one isentropic applied from the nozzle entrance to the shockwave position and a second from the shockwave position to nozzle exit and, in the middle, one non-isentropic across the shockwave that encounters pressure and entropy increase. A selected dataset for supersonic hydrodynamic was used as the main source for training and testing the proposed system. The dataset includes inlet pressure, outlet pressure, or equivalent pressure recovery ratio (Pr), which is defined as the ratio of outlet pressure to inlet pressure, and the nozzle length.
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Using raw data directly without any classification may lead to low level of result reliability. SOM do as a data reconstruction algorithm and a bundling agent to make data classified and effectively do the selection of the most representative data for further analysis which guarantee the robustness of the next stages. The clustering procedure will form sets which will be introduced to the main neural network. The network gets the data and then the common training, test and validation procedures are performed to finally have a black box model for shockwave location prediction.
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Supersonic flow shockwave data was collected from the research results of Karimi et al. (2009), Yang et al. (2014a, 2014b, 2014c), and Jing et al. (2014) as shown in appendix B, which presented numerical and experimental results for supersonic nozzle separation of Natural gas under different boundary conditions. Each of 32 sample sets is based on investigation with different parameters as inlet pressure or temperature in different case studies and then, there are some overlaps while there is no uniformity regarding the range and the number of data. For example, at a specified pressure, pressure recovery ratio and temperature, there are two or more data for the shockwave location based on the literature and in these cases, as shown in Fig3 (a), cluster formation yielded single data value. Table I. includes the minimum and maximum of the available data as feed pressure, pressure recovery ratio, nozzle length, and feed temperature and specify the limits of the final model.
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Table I. The range of parameters for supersonic nozzle separation parameter Min max 7 300 Inlet Pressure, P (bar) 0.14 0.85 Pressure Recovery ratio 274.15 374.15 Inlet temperature, T (K) 0.1 0.6 Nozzle length, L (m)
Most of the literatures follow a fixed geometry based on Ariana et al. (2004) research, but with difference scale. Moreover, dimensionless length, which is defined as the ratio of the shockwave axial position to the total length of the nozzle, is considered to maintain generality.
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2.2. Data Classification The underlying idea of classification is to seamlessly incorporate item information into the main network. The neural network is used to classify the elements of the dataset before the main neural network-based modeling steps.
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The preliminary classification step was applied to the supersonic nozzle raw dataset. Selforganizing map (SOM) was used for classification and to preserve the map topology to simulate the real input features. Similarity-based clustering applied by SOMs learning to generate reduced-dimensional dataset. (Kohonen et al., 2001)
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The winning neuron is the weight () with the closest match to the input pattern (x). The weights change iteratively, as Eq.2, to minimize the distance between the two vectors of node weight and input. (Kohonen et al., 2001): ( + 1) = () + ()ℎ ( − ()) (2)
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α, l, and m are the learning rate on a scale of between 0 and 1, the winning neuron positions, and their neighboring output nodes, respectively. Also, Gaussian function is the selected neighborhood function (ℎ ) at iteration as Eq. 3. (Kohonen et al., 2001): ‖ − ‖
(3) ℎ = exp (− ) 2 ()
− denotes the distance between neurons l and m on the map grid and is the topological neighborhood width. The training sequences are repeated until the convergence is reached, and then with the final SOM network, clusters are identified within the SOM map.
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Based on the well-developed features of the MATLAB package for applying SOMs for classification in different application (Ballabio et al., 2009 & 2012), this environment was selected for the implementation and it was executed on an Intel Core i3- 3.40 GHz PC with 3.40 GB of RAM. To apply the two-step SOM on the supersonic nozzle shockwave data, the size of the Kohonen layer was determined as a five-by-five grid. This size is large enough to ensure that a suitable number of clusters are formed from the training data. The structure of the proposed-NN is depicted in Fig. 2. After creation of the neural network structure and training the network, the clusters on the map are identified (Fig. 3a) and prepared as the inputs of main neural network at the next stage.
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Classified dataset Output 1
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1 1 1 1 1 Fig. 2. Architecture of SOM Classification
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Fig. 3. (a) Hits of classification and (b) SOM neighbor Weight Distances
The graphs in Fig. 3b depict distances of neighbor weight, while neurons are described by dark hexagons. The mapping distances are provided by the color of regions: the lighter the color, the smaller the distance. The distribution of data shows enough observations in each class and ensures that the classified dataset covers all parameters as a neural network model input, and improves the reliability of the model prediction results.
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2.3. Neural Network modeling
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In this section, neural networks (NN) are formally described. A neural network is black box modeling to generate corresponding output for input vectors. The configuration of the proposed NN system is illustrated in Fig. 4. A two-layer feed-forward network, with a sigmoid transfer function in the hidden layer and a linear transfer function in the output layer, is undertaken with varying parameters, and then applied to the classified dataset as the outcome of the previous section. A two-dimensional trial and error procedure is performed to find both the optimal number of hidden layer neurons and optimal training function for the minimum mean square error (MSE). Each model was run ten times for each neuron number to reduce the impact of the initialization of the weights and biases on the results. The results are shown in detail in Table II, including the performance of different network configurations. The bold values represent the lowest value for each approach. The scaled conjugate gradient-based training function with seven neurons in the hidden layer is the best method with the minimum MSE of 3.99e-5. The weights and the bias of the final network are given in appendix A. The implementation was executed on an Intel Core i3- 3.40 GHz PC with 3.40 GB of RAM.
Fig.4. Neural network structure for shockwave location modeling in supersonic nozzles
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Quasi Newton secant method
2 16.26 3.41 12.59 23.52 0.77 78.18 20.06 35.14 5.52 1.66 5.69
3 9.13 6.65 15.34 1.68 12.51 58.34 12.12 20.61 4.09 6.44 22.40
2.4. 1-D model of supersonic nozzle
4 9.41 10.10 3.08 1.26 22.56 15.13 1.10 15.19 2.39 3.53 12.83
5 2.85 5.97 2.40 1.88 21.94 33.80 9.12 3.28 5.08 10.84 6.97
6 4.56 5.81 2.76 2.77 55.83 94.57 0.48 6.23 1.06 13.98 9.91
7 0.40 17.13 26.42 2.34 6.47 14.50 14.22 8.06 2.01 6.93 9.37
8 2.45 4.71 5.41 8.60 8.70 40.57 1.95 8.87 2.72 1.03 5.90
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Gradient Descent
function trainscg traincgb traincgf traincgp traingda traingdm trainrp traingdx trainlm trainbfg trainoss
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method Conjugate Gradient
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Table II. MSE for feed-forward neural network with varying training function and hidden layer number (*105) 9 7.50 6.04 1.59 2.48 9.69 132.69 2.63 13.62 1.26 8.12 12.98
10 6.66 2.76 7.65 10.81 3.32 56.62 4.40 23.61 0.58 14.37 24.43
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The classical model is based on 1-D flow in the convergent-divergent nozzle. In the axial direction of the nozzle, four pressure values are defined as P1, P2, P3, and P4, representing the inlet pressure, the pressure at the upstream side of the shock, the pressure on the downstream side of the shock, and the exit pressure, respectively. An iterative algorithm is proposed to determine the pressure profile along the nozzle and solve the system of equations to find the location where the shockwave stands. The procedure sequence is implemented for thermodynamic properties estimation.
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The technique will assume a value for P2 and then compute P3 and P4 by Eq. 7 to 9. If the value of P4 matches the exit pressure, then the P2 finding has converged to the correct solution. Otherwise, a new value will be chosen and the process repeated until convergence is achieved. The flowchart of the algorithm is depicted in Fig. 5.
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Fig. 5. Chart for shockwave location finding using 1-D supersonic equations
Governing equation based on the rearrangement of Rayleigh line equation for pressure ratio (Borgnakke, 2009 & Jasim et al., 2013):
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+−' " Area ratio or the relation of local Mach number and the area of shock location evaluated from the following equation: +0'
+−' " "(+*') -" !' ' + " !" = . / - ' !" ' + + − ' ! " ' " where k=k(T,P) is heat capacity ratio preciously considered for natural gas stream.
Using calculated M2, M3 can be derived as Eq. 6:
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&" ' + +!"1 (7) = &1 ' + +!"" where shockwave pressure change is considered. The shock downstream Mach number (M3) is computed after the pressure rise across the shock location. Then, a same equation as Eq. 5 is valid for the second isentropic process between shockwave and nozzle outlet as follows: +0'
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+−' " "(+*') -3 !1 ' + " !3 = . / - 1 !3 ' + + − ' ! " 1 "
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The exit pressure is determined based on the isentropic flow between the shock and the nozzle exit. The derived value for exit pressure (P4) must be compared to the previous assumption for the pressure to determine if the guess for P2 was correct. The relationship between pressure and Mach number at this stage can be expressed mathematically as:
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As shown in Fig. 5, the actual location of the shockwave can be found when iteration is terminated and final convergence is achieved with minimization of the error between real boundary conditions and the calculation results based on the guess values.
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3. Results and discussion
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The simulation was carried out and illustrates the performance of the proposed neural networkbased regression with the classified dataset, and also verifies different supersonic nozzle separation performances under different boundary conditions. According to Yang et al. (2014) the sensitivity of the shockwave location to the pressure boundary values is more than to the temperature boundary values. The current neural network model results also showed considerably higher sensitivity of shockwave location to pressure than temperature boundary values.
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The neural network model results are reasonably good for all target data, with R-value of 0.99428 as shown in Fig. 6. The training set uses 70% of learning data while the testing and validation sets each utilize 15%.
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Fig. 6. Regression of neural network model for the target data (Karimi et al., 2009, Yang et al, 2014a, 2014b, 2014c and Jing et al., 2014)
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Fig. 7. Neural network model prediction for shockwave location in a wide range of the main parameter of inlet pressure and pressure recovery ratio
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Fig. 8. Neural network model prediction for shockwave location versus inlet Pressure
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The final profile of the shockwave position in the nozzle is provided in Fig. 7. Shockwave location ratio increased and the shockwave moved towards the nozzle exit side with pressure recovery ratio reduction. The same results are concluded from the 1-D model as in Fig. 9. There are occasionally some differences between trends obtained from the 1-D model and the results of experimental-based neural-network models. This is more obvious in high pressure and high-pressure recovery values. This condition seems to prevent the shockwave from moving toward the end side of the nozzle. This is highlighted in Fig. 8, which shows that the shockwave occurred sooner in higher inlet pressure and this effect is magnified in higher-pressure recovery ratio cases. In contrast, the 1-D model is probably unable to predict due to simple assumption as neglecting of swirling effect inside the nozzle. This is because that the axial is the only available form of the flow direction in this model, while main velocity components in real supersonic nozzle process are tangential and axial ones. Consequently, forcing high pressure to the same geometry improves the comparable role of axial velocity and this helps to reach the Mach number limit required for shockwaves faster. This is well predicted by the proposed model based on classified experimental data.
Fig. 9. 1-D model prediction for shockwave location in a wide range of the main parameter of inlet Pressure and Pressure recovery ratio Accuracy and validity were investigated using three statistical metrics, namely mean absolute percentage error, mean-square error, and mean absolute deviation.
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:; (<) − ;=>?@ (<): 4567 = (1/9) . / ∗ 100 ; (<) 4C7 = (1/9) DE;F (<) − ;GHIJ (<)EK
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where ypred is the predicted value as model output, yt is the target output, and n is number of observations.
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As data has no zero or near zero, MAPE can give an appropriate picture of the error. MSE shows the common error metric for numerical predictions of the models. Also, MAE helps conceptualize the volume of errors. The obtained evaluation results are presented in Table III, which shows that the NN-based model performs better than the 1-D model prediction. Table III -Statistical error analysis for the applied models for shockwave position prediction model MSE (m2) LMNO (-) LMO (m) 3.91 1.04e-3 3.19e-2 Proposed neural network 15.44 1.01e-2 9.86e-2 Iterative 1-D model
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4. Conclusion
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This model would not be applicable if the fluid density is highly different with natural gas or air. Also in case of fluid injection along the nozzle or presence of external energy source, model prediction results may not be as valid as expected. Using sensor probe, wings or anything that could make a hindrance to the flow are the source of disturbance and cause a shift in shockwave location along the nozzle.
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Shockwave is the main concept behind the supersonic nozzle separation and investigation of shockwave location inside the nozzle is main focus of the present study. It is presented how the location of shockwave changes in case of using supersonic nozzle separation for low- or high-pressure feeds. A neural network-based flexible model to find the shockwave location in a supersonic nozzle used for natural gas separation is presented. The conceived method combines the classification power of the SOM method, with characteristics of neural network modeling for an extensive range of shockwave experimental data. A classification stage has been designed to improve the efficiency and interpretability of the neural network scheme. One of the key features of the neural network is the capability of finding a nonlinear generalization of the data, i.e. it classifies the available dataset in patterns. SOM places similar patterns and data to the classes’ previously classified training sets. It means that the shockwave occurred at the specified length along the nozzle, which is predicted by the final proposed model using the hydrodynamic 14
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boundary data. It is capable of modeling dependencies to pressure and temperature boundary values, and nonlinearities in the model response.
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The proposed system has been verified by the experimental data under different nozzle boundary conditions. Since shockwave location is directly obtained by using a neural network-based model, the proposed system does not need complex computational routines. The results show that it is sufficiently accurate in mapping feed pressure to pressure recovery ratio as the input and the shockwave standing position. Both higher inlet pressure and higher-pressure recovery ratio cause the shockwave to be placed closer to the inlet.
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An open research question is to utilize the feature of the neural network to identify the swirling effect on nozzle performance.
Nomenclature
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Pr=Pressure recovery ratio yQRST =Model output yU =Actual value MW =Mach number at different position along the nozzle (-) PW = Absolute pressure at different position along the nozzle (Pa) TW = Temperature at different position along the nozzle (K) x =Input data w =Weight value k= Heat capacity ratio (-) xs= shockwave position along nozzle axis L=nozzle length (m) k = Integer counter t=iteration step h\] = Neighborhood function l = Position of the winning neuron m= Winning neuron’s neighboring nodes
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23. Valipour, M., Banihabib, M. E., & Behbahani, S. M. R. (2012). Parameters estimate of autoregressive moving average and autoregressive integrated moving average models and compare their ability for inflow forecasting. J Math Stat, 8(3), 330-338. 24. Valipour, M., Sefidkouhi, M. A. G., & Eslamian, S. (2015). Surface irrigation simulation models: a review. International Journal of Hydrology Science and Technology, 5(1), 51-70. 25. Vaziri, B. M., & Shahsavand, A. (2013). Analysis of supersonic separators geometry using generalized radial basis function (GRBF) artificial neural networks. Journal of Natural Gas Science and Engineering, 13, 30-41.
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26. Vaziri, B. M., & Shahsavand, A. (2015). Optimal selection of supersonic separators inlet velocity components via maximization of swirl strength and centrifugal acceleration. Separation Science and Technology, 50(5), 752759. 27. Wen, C., Feng, Y., Witt, P., Cao, X., & Yang, Y. (2012). CFD simulation of supersonic swirling separation of natural gas using a delta wing.
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28. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Effect of Inlet and Outlet Flow Conditions on Natural Gas Parameters in Supersonic Separation Process. 29. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Numerical simulation of real gas flows in natural gas supersonic separation processing. Journal of Natural Gas Science and Engineering, 21, 829-836.
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30. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Theoretical and numerical analysis on pressure recovery of supersonic separators for natural gas dehydration. Applied Energy, 132, 248-253.
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Appendix A
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Weights and bias between two steps are shown in the following table. They are used to link the input to hidden layer and hidden layer to the output layer. These are interconnection properties of the selected neural network which is finally found to be the optimal neural network model and have the best performance for the present dataset with the minimum error. The table has seven rows as the optimal network found to have 7 hidden layers.
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Table IV. Weights and bias between input and hidden layer and hidden and output layer Weight to Weight to layer 1 from input 1 Bias layer b{1}, bias to b{2}, bias to Iw {1,1} lw{2,1} layer 1 layer 2 1.5691 -0.6435 -1.1812 -1.1372 0.22329 -2.2033 -0.2246 -1.75 -1.2869 0.63963 0.11658 -0.16075 1.552 -0.50632 -1.6744 -0.13261 -1.4941 0.49318 0.69803 -1.0278 1.4738 -1.3978 -0.44023 0.87051 0.43312 0.73731 -1.0969 1.0598 -1.5244 -0.26439 0.72512 0.33906 -0.29645 -1.7488 -1.4001 0.42936 1.4233 1.3896 0.15244 -1.0442 1.39 0.44484 2.3467
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Appendix B
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The following table includes raw data gathered from literature including temperature and pressure condition and shockwave location in the supersonic nozzle with different scales. The Pressure recovery ratio is calculated for each set. Neural network inputs are four parameters including inlet pressure, inlet temperature, Pressure recovery ratio, and nozzle length. Table V. The literature raw data for the nozzle boundary condition and shockwave location
Jing et al., 2014
Pr
Tin (K)
Shock Location (x/D)
0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.70 0.70 0.70 0.70 0.60 0.60 0.60 0.60 0.60 0.60
300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 700.00 500.00 300.00 100.00 10.00 100.00 200.00 300.00 200.00 200.00 200.00 200.00 100.00 100.00 100.00 100.00 7.00 7.00 7.00 7.00 7.00 7.00
210.00 210.00 210.00 210.00 249.00 240.00 217.20 183.00 145.50 490.00 350.00 210.00 70.00 7.50 75.00 150.00 225.00 150.00 150.00 150.00 150.00 85.00 80.00 75.00 62.00 1.00 1.51 2.00 2.95 3.00 4.00
0.70 0.70 0.70 0.70 0.83 0.80 0.72 0.61 0.49 0.70 0.70 0.70 0.70 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.85 0.80 0.75 0.62 0.14 0.22 0.29 0.42 0.43 0.57
274.15 293.15 313.15 333.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 288.00 288.00 288.00 288.00 283.00 313.00 343.00 373.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00
0.85 0.85 0.91 0.93 0.68 0.75 0.83 0.92 1.00 0.81 0.81 0.84 0.92 0.49 0.45 0.38 0.33 0.40 0.39 0.41 0.41 0.50 0.54 0.60 0.67 0.93 0.88 0.85 0.83 0.77 0.70
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Pout (bar)
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Yang et al, 2014a, 2014b, 2014c
Pin (bar)
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Karimi et al., 2009
nozzle length (m)
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reference
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Research Highlights
A two-stage neural network is proposed to find shockwave position in supersonic nozzle.
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Classification and neural network tuning improve the performance of the model.
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The model is capable of estimating shockwave location in a wide range of inputs.
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The method uses gas feed pressure and pressure recovery ratio as inputs.
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Both higher inlet pressure and pressure recovery move the shockwave toward the inlet.
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S1875-5100(16)30533-9
DOI:
10.1016/j.jngse.2016.07.061
Reference:
JNGSE 1683
To appear in:
Journal of Natural Gas Science and Engineering
Received Date: 15 March 2016 Revised Date:
22 July 2016
Accepted Date: 23 July 2016
Please cite this article as: Niknam, P.H., Mokhtarani, B., Mortaheb, H.R., Prediction of shockwave location in supersonic nozzle separation using self-organizing map classification and artificial neural network modeling, Journal of Natural Gas Science & Engineering (2016), doi: 10.1016/ j.jngse.2016.07.061. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Prediction of shockwave location in supersonic nozzle separation using self-organizing map classification and artificial neural network modeling
Pouriya H. Niknam, B. Mokhtarani*, H.R. Mortaheb*
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Chemistry & Chemical Engineering Research Center of Iran, P.O. Box 14335-186, Tehran, Iran
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Abstract One of the novel technologies for natural gas dehydration and natural gas dew-point conditioning is supersonic separation, which has remarkable features, including compact and maintenance-free design. Due to its complex design and the difficulty of experimental analysis, researchers tend to conduct numerical modeling for behavior investigation of the nozzle focusing on shockwave which is the main phenomena inside the nozzle. The present NN-model outperforms a selection of data and proposes an efficient NN-based algorithm for shockwave position estimation as the key nozzle geometry parameter. Data for the shockwave location was collected from a wide range of results from the literature and then a neural network based self-organizing map was adapted to the dataset. This created a classified dataset and the use of unreal weight and repeated experimental results from different research were avoided. A neural network was employed for modeling the shockwave location through the nozzle using a better quality dataset. Additionally, the one-dimensional inviscid theory was utilized in the recursive approach for comparison to the main proposed model. Simulation results presented in this research reveal the effectiveness of the proposed neural network technique for supersonic nozzle modeling and make it possible to determine the shockwave location from the nozzle pressure boundary conditions. The results showed that the supersonic nozzle separation have capability to be used in both low-pressure applications and high pressure ones. The dimensionless length for shockwave location is predicted in the range of 0.82 to 0.92 for the former and 0.72 to 0.95 for the later, depending on pressure recovery ratio. Keywords: Shockwave location, Supersonic nozzle, Neural network, Self-organizing maps, natural gas separation. *
Corresponding authors. Tel: +98-21-44787751, Fax: +98-21-44787781, E-mail address; [email protected], [email protected].
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1. Introduction
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Extracted natural gas from wells usually contains certain quantities of water that condense and form gas hydrates if the gas pressure is reduced or the gas is cooled below its hydrate formation temperature or water dew point. Water as an undesirable component must be removed to inhibit hydrate and condensation formation. Erosion, corrosion, plugged pipelines, reduced pipeline flow capacity, and operation interruptions are possible consequences of the mentioned phenomena. To prevent these problems, the gas stream must be dehydrated. Specifications typically call for water content to be no more than 110mg/Sm3 in US pipeline systems, 60 mg/Sm3 in Canadian pipeline systems, and 130 mg/Sm3 in Iranian pipeline systems. This requirement puts the maximum water in sales gas at the dew point of -10 ºC at any outlet delivery and line pressure. These values provide protection against water condensation and hydrate formation as a consequence of low ambient temperature. There are several methods of dehydrating natural gas, such as direct cooling of the gas stream, absorption, and adsorption processes (Mokhatab et al., 2012).
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Recent research has introduced a novel technology based on gas expansion in a supersonic nozzle that changes gas stream conditions by lowering temperature and pressure, while part of gas enthalpy transforms to kinetic energy. The basic component of the system is a converging–diverging nozzle with no moving parts where condensation occurs at supersonic velocities. The minimum temperature and pressure are found at the shockwave position. During this process, water and other possible condensations are removed from the gas stream by the supersonic nozzle for dew point control applications and no external work is used as in conventional refrigeration units (Brouwer et al., 2003 & Kalikmanov et al., 2007, Alfyorov, 2005). Fig. 1 shows the structure of the conventional supersonic nozzle which consists of swirling generator, Laval nozzle, and supersonic diffuser. Different scales of the nozzle were used in the literature.
Fig. 1. Schematic of the supersonic nozzle
Over the past decade, important numerical and modeling investigations to study supersonic separation in the nozzle have been performed. These studies can be categorized into three general areas: economic, hydrodynamic, and geometry investigations. Machado et al. (2012) presented an economic analysis for supersonic nozzle technology compared with the common conventional unit (TEG+JT/LTS). Gonzalez et al. proposed an arrangement of supersonic nozzles for the unsteady inlet. Castier et al. (2014) proposed an adapted equation of state for numerical study and modeling, consisting of single-scaling wavelets. Prast et al. (2006) presented a computational fluid dynamic (CFD) model for heavy cut extraction from natural gas stream, which was proven by the experimental results. Vaziri et al. (2015) 2
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used a CFD model to adjust the optimal values of axial, angular, and radial velocity components of entering fluid. Shooshtari et al. (2013) proposed a theoretical study to predict the liquid droplet growth in binary mixtures inside the laval nozzle using related thermodynamic concepts. Jassim et al. (2008b) noted that their CFD modeling could not predict phase change and that gas condensation was neglected in the modeling. Also, Abdi et al. (2010) proposed CFD modeling, but pointed to the incapability of the model to consider two-phase conditions. This is due to that computational methods have drawbacks as multiphase compressible swirling flow entails phase changes in high-pressure conditions and most of them fail to cover all the mentioned aspects. Sforza et al. (2012) proposed a swirling compressible model under a controlled condensation rate based on quasi-one-dimensional equations for syngas purification using a supersonic swirl tube. Significant research has concentrated on supersonic nozzle dynamic parameters (Wen et al., 2012; Haghighi et al., 2013). Vaziri et al. (2013) developed a numerical model using neural networks for geometry and boundary conditions of the supersonic nozzle. The authors validated this model with numerical results and it could not be widely used in similar cases.
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The correct prediction of separation performance by the supersonic nozzle is important for its design. Information on nozzle characteristics is still limited. Moreover, there is a need for further research, especially into where the gas is shocked and where the separation process largely happens. As a consequence, the main aim of the present study is to create a reliable and simple model based on previous numerical and experimental studies, to cover the wide range of design parameters of the supersonic nozzle.
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The main drawback of model-based approaches is the requirement for accurate system details, which may not be available in many practical applications. Intelligent NN models are designed to overcome some of these restrictive requirements. The artificial neural network is a novel technique that gives researchers an alternative way to solve these kinds of complex problems. Neural networks and other NN-based hybrid models have been used for a wide range of application for prediction of the outputs. (Valipour, 2012, 2015, 2016)
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In this work an alternative method for shockwave capture that uses a well-trained neural network and a classified dataset is proposed. The main goal is to evaluate whether the NN-based approach is feasible for this kind of application and to compare it with the conventional 1-D supersonic model. In contrast to previous numerical models, the proposed model covers a wide range of boundary conditions revealing the effect of pressure recovery ratio. Also the previously neglected role of the swirling effect is accounted for by using the proper experimental dataset. Also, computational complexity is avoided to a large extent by using the NN as a black-box model. So the NN model considers all physical principles such as multiphase interactions and swirling effect indirectly using the input and output valid data obtained from the experiments.
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2. Method description 2.1. Database Finding the axial position of normal shock wave inside the nozzle has an indirect solution, as the shock stands at a position that enables the desired exit pressure to be achieved.
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The flow proceeding the nozzle consists of two isentropic and one non-isentropic process: one isentropic applied from the nozzle entrance to the shockwave position and a second from the shockwave position to nozzle exit and, in the middle, one non-isentropic across the shockwave that encounters pressure and entropy increase. A selected dataset for supersonic hydrodynamic was used as the main source for training and testing the proposed system. The dataset includes inlet pressure, outlet pressure, or equivalent pressure recovery ratio (Pr), which is defined as the ratio of outlet pressure to inlet pressure, and the nozzle length.
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Using raw data directly without any classification may lead to low level of result reliability. SOM do as a data reconstruction algorithm and a bundling agent to make data classified and effectively do the selection of the most representative data for further analysis which guarantee the robustness of the next stages. The clustering procedure will form sets which will be introduced to the main neural network. The network gets the data and then the common training, test and validation procedures are performed to finally have a black box model for shockwave location prediction.
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Supersonic flow shockwave data was collected from the research results of Karimi et al. (2009), Yang et al. (2014a, 2014b, 2014c), and Jing et al. (2014) as shown in appendix B, which presented numerical and experimental results for supersonic nozzle separation of Natural gas under different boundary conditions. Each of 32 sample sets is based on investigation with different parameters as inlet pressure or temperature in different case studies and then, there are some overlaps while there is no uniformity regarding the range and the number of data. For example, at a specified pressure, pressure recovery ratio and temperature, there are two or more data for the shockwave location based on the literature and in these cases, as shown in Fig3 (a), cluster formation yielded single data value. Table I. includes the minimum and maximum of the available data as feed pressure, pressure recovery ratio, nozzle length, and feed temperature and specify the limits of the final model.
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Table I. The range of parameters for supersonic nozzle separation parameter Min max 7 300 Inlet Pressure, P (bar) 0.14 0.85 Pressure Recovery ratio 274.15 374.15 Inlet temperature, T (K) 0.1 0.6 Nozzle length, L (m)
Most of the literatures follow a fixed geometry based on Ariana et al. (2004) research, but with difference scale. Moreover, dimensionless length, which is defined as the ratio of the shockwave axial position to the total length of the nozzle, is considered to maintain generality.
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2.2. Data Classification The underlying idea of classification is to seamlessly incorporate item information into the main network. The neural network is used to classify the elements of the dataset before the main neural network-based modeling steps.
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The preliminary classification step was applied to the supersonic nozzle raw dataset. Selforganizing map (SOM) was used for classification and to preserve the map topology to simulate the real input features. Similarity-based clustering applied by SOMs learning to generate reduced-dimensional dataset. (Kohonen et al., 2001)
(1)
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The winning neuron is the weight () with the closest match to the input pattern (x). The weights change iteratively, as Eq.2, to minimize the distance between the two vectors of node weight and input. (Kohonen et al., 2001): ( + 1) = () + ()ℎ ( − ()) (2)
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α, l, and m are the learning rate on a scale of between 0 and 1, the winning neuron positions, and their neighboring output nodes, respectively. Also, Gaussian function is the selected neighborhood function (ℎ ) at iteration as Eq. 3. (Kohonen et al., 2001): ‖ − ‖
(3) ℎ = exp (− ) 2 ()
− denotes the distance between neurons l and m on the map grid and is the topological neighborhood width. The training sequences are repeated until the convergence is reached, and then with the final SOM network, clusters are identified within the SOM map.
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Based on the well-developed features of the MATLAB package for applying SOMs for classification in different application (Ballabio et al., 2009 & 2012), this environment was selected for the implementation and it was executed on an Intel Core i3- 3.40 GHz PC with 3.40 GB of RAM. To apply the two-step SOM on the supersonic nozzle shockwave data, the size of the Kohonen layer was determined as a five-by-five grid. This size is large enough to ensure that a suitable number of clusters are formed from the training data. The structure of the proposed-NN is depicted in Fig. 2. After creation of the neural network structure and training the network, the clusters on the map are identified (Fig. 3a) and prepared as the inputs of main neural network at the next stage.
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Classified dataset Output 1
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1 1 1 1 1 Fig. 2. Architecture of SOM Classification
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Fig. 3. (a) Hits of classification and (b) SOM neighbor Weight Distances
The graphs in Fig. 3b depict distances of neighbor weight, while neurons are described by dark hexagons. The mapping distances are provided by the color of regions: the lighter the color, the smaller the distance. The distribution of data shows enough observations in each class and ensures that the classified dataset covers all parameters as a neural network model input, and improves the reliability of the model prediction results.
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2.3. Neural Network modeling
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In this section, neural networks (NN) are formally described. A neural network is black box modeling to generate corresponding output for input vectors. The configuration of the proposed NN system is illustrated in Fig. 4. A two-layer feed-forward network, with a sigmoid transfer function in the hidden layer and a linear transfer function in the output layer, is undertaken with varying parameters, and then applied to the classified dataset as the outcome of the previous section. A two-dimensional trial and error procedure is performed to find both the optimal number of hidden layer neurons and optimal training function for the minimum mean square error (MSE). Each model was run ten times for each neuron number to reduce the impact of the initialization of the weights and biases on the results. The results are shown in detail in Table II, including the performance of different network configurations. The bold values represent the lowest value for each approach. The scaled conjugate gradient-based training function with seven neurons in the hidden layer is the best method with the minimum MSE of 3.99e-5. The weights and the bias of the final network are given in appendix A. The implementation was executed on an Intel Core i3- 3.40 GHz PC with 3.40 GB of RAM.
Fig.4. Neural network structure for shockwave location modeling in supersonic nozzles
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Quasi Newton secant method
2 16.26 3.41 12.59 23.52 0.77 78.18 20.06 35.14 5.52 1.66 5.69
3 9.13 6.65 15.34 1.68 12.51 58.34 12.12 20.61 4.09 6.44 22.40
2.4. 1-D model of supersonic nozzle
4 9.41 10.10 3.08 1.26 22.56 15.13 1.10 15.19 2.39 3.53 12.83
5 2.85 5.97 2.40 1.88 21.94 33.80 9.12 3.28 5.08 10.84 6.97
6 4.56 5.81 2.76 2.77 55.83 94.57 0.48 6.23 1.06 13.98 9.91
7 0.40 17.13 26.42 2.34 6.47 14.50 14.22 8.06 2.01 6.93 9.37
8 2.45 4.71 5.41 8.60 8.70 40.57 1.95 8.87 2.72 1.03 5.90
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function trainscg traincgb traincgf traincgp traingda traingdm trainrp traingdx trainlm trainbfg trainoss
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method Conjugate Gradient
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Table II. MSE for feed-forward neural network with varying training function and hidden layer number (*105) 9 7.50 6.04 1.59 2.48 9.69 132.69 2.63 13.62 1.26 8.12 12.98
10 6.66 2.76 7.65 10.81 3.32 56.62 4.40 23.61 0.58 14.37 24.43
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The classical model is based on 1-D flow in the convergent-divergent nozzle. In the axial direction of the nozzle, four pressure values are defined as P1, P2, P3, and P4, representing the inlet pressure, the pressure at the upstream side of the shock, the pressure on the downstream side of the shock, and the exit pressure, respectively. An iterative algorithm is proposed to determine the pressure profile along the nozzle and solve the system of equations to find the location where the shockwave stands. The procedure sequence is implemented for thermodynamic properties estimation.
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The technique will assume a value for P2 and then compute P3 and P4 by Eq. 7 to 9. If the value of P4 matches the exit pressure, then the P2 finding has converged to the correct solution. Otherwise, a new value will be chosen and the process repeated until convergence is achieved. The flowchart of the algorithm is depicted in Fig. 5.
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Fig. 5. Chart for shockwave location finding using 1-D supersonic equations
Governing equation based on the rearrangement of Rayleigh line equation for pressure ratio (Borgnakke, 2009 & Jasim et al., 2013):
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+−' " Area ratio or the relation of local Mach number and the area of shock location evaluated from the following equation: +0'
+−' " "(+*') -" !' ' + " !" = . / - ' !" ' + + − ' ! " ' " where k=k(T,P) is heat capacity ratio preciously considered for natural gas stream.
Using calculated M2, M3 can be derived as Eq. 6:
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&" ' + +!"1 (7) = &1 ' + +!"" where shockwave pressure change is considered. The shock downstream Mach number (M3) is computed after the pressure rise across the shock location. Then, a same equation as Eq. 5 is valid for the second isentropic process between shockwave and nozzle outlet as follows: +0'
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The exit pressure is determined based on the isentropic flow between the shock and the nozzle exit. The derived value for exit pressure (P4) must be compared to the previous assumption for the pressure to determine if the guess for P2 was correct. The relationship between pressure and Mach number at this stage can be expressed mathematically as:
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As shown in Fig. 5, the actual location of the shockwave can be found when iteration is terminated and final convergence is achieved with minimization of the error between real boundary conditions and the calculation results based on the guess values.
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3. Results and discussion
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The simulation was carried out and illustrates the performance of the proposed neural networkbased regression with the classified dataset, and also verifies different supersonic nozzle separation performances under different boundary conditions. According to Yang et al. (2014) the sensitivity of the shockwave location to the pressure boundary values is more than to the temperature boundary values. The current neural network model results also showed considerably higher sensitivity of shockwave location to pressure than temperature boundary values.
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The neural network model results are reasonably good for all target data, with R-value of 0.99428 as shown in Fig. 6. The training set uses 70% of learning data while the testing and validation sets each utilize 15%.
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Fig. 6. Regression of neural network model for the target data (Karimi et al., 2009, Yang et al, 2014a, 2014b, 2014c and Jing et al., 2014)
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Fig. 7. Neural network model prediction for shockwave location in a wide range of the main parameter of inlet pressure and pressure recovery ratio
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Fig. 8. Neural network model prediction for shockwave location versus inlet Pressure
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The final profile of the shockwave position in the nozzle is provided in Fig. 7. Shockwave location ratio increased and the shockwave moved towards the nozzle exit side with pressure recovery ratio reduction. The same results are concluded from the 1-D model as in Fig. 9. There are occasionally some differences between trends obtained from the 1-D model and the results of experimental-based neural-network models. This is more obvious in high pressure and high-pressure recovery values. This condition seems to prevent the shockwave from moving toward the end side of the nozzle. This is highlighted in Fig. 8, which shows that the shockwave occurred sooner in higher inlet pressure and this effect is magnified in higher-pressure recovery ratio cases. In contrast, the 1-D model is probably unable to predict due to simple assumption as neglecting of swirling effect inside the nozzle. This is because that the axial is the only available form of the flow direction in this model, while main velocity components in real supersonic nozzle process are tangential and axial ones. Consequently, forcing high pressure to the same geometry improves the comparable role of axial velocity and this helps to reach the Mach number limit required for shockwaves faster. This is well predicted by the proposed model based on classified experimental data.
Fig. 9. 1-D model prediction for shockwave location in a wide range of the main parameter of inlet Pressure and Pressure recovery ratio Accuracy and validity were investigated using three statistical metrics, namely mean absolute percentage error, mean-square error, and mean absolute deviation.
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:; (<) − ;=>?@ (<): 4567 = (1/9) . / ∗ 100 ; (<) 4C7 = (1/9) DE;F (<) − ;GHIJ (<)EK
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where ypred is the predicted value as model output, yt is the target output, and n is number of observations.
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As data has no zero or near zero, MAPE can give an appropriate picture of the error. MSE shows the common error metric for numerical predictions of the models. Also, MAE helps conceptualize the volume of errors. The obtained evaluation results are presented in Table III, which shows that the NN-based model performs better than the 1-D model prediction. Table III -Statistical error analysis for the applied models for shockwave position prediction model MSE (m2) LMNO (-) LMO (m) 3.91 1.04e-3 3.19e-2 Proposed neural network 15.44 1.01e-2 9.86e-2 Iterative 1-D model
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4. Conclusion
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This model would not be applicable if the fluid density is highly different with natural gas or air. Also in case of fluid injection along the nozzle or presence of external energy source, model prediction results may not be as valid as expected. Using sensor probe, wings or anything that could make a hindrance to the flow are the source of disturbance and cause a shift in shockwave location along the nozzle.
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Shockwave is the main concept behind the supersonic nozzle separation and investigation of shockwave location inside the nozzle is main focus of the present study. It is presented how the location of shockwave changes in case of using supersonic nozzle separation for low- or high-pressure feeds. A neural network-based flexible model to find the shockwave location in a supersonic nozzle used for natural gas separation is presented. The conceived method combines the classification power of the SOM method, with characteristics of neural network modeling for an extensive range of shockwave experimental data. A classification stage has been designed to improve the efficiency and interpretability of the neural network scheme. One of the key features of the neural network is the capability of finding a nonlinear generalization of the data, i.e. it classifies the available dataset in patterns. SOM places similar patterns and data to the classes’ previously classified training sets. It means that the shockwave occurred at the specified length along the nozzle, which is predicted by the final proposed model using the hydrodynamic 14
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boundary data. It is capable of modeling dependencies to pressure and temperature boundary values, and nonlinearities in the model response.
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The proposed system has been verified by the experimental data under different nozzle boundary conditions. Since shockwave location is directly obtained by using a neural network-based model, the proposed system does not need complex computational routines. The results show that it is sufficiently accurate in mapping feed pressure to pressure recovery ratio as the input and the shockwave standing position. Both higher inlet pressure and higher-pressure recovery ratio cause the shockwave to be placed closer to the inlet.
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An open research question is to utilize the feature of the neural network to identify the swirling effect on nozzle performance.
Nomenclature
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Pr=Pressure recovery ratio yQRST =Model output yU =Actual value MW =Mach number at different position along the nozzle (-) PW = Absolute pressure at different position along the nozzle (Pa) TW = Temperature at different position along the nozzle (K) x =Input data w =Weight value k= Heat capacity ratio (-) xs= shockwave position along nozzle axis L=nozzle length (m) k = Integer counter t=iteration step h\] = Neighborhood function l = Position of the winning neuron m= Winning neuron’s neighboring nodes
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References Abdi, M. A., Jassim, E., Haghighi, M., & Muzychka, Y. (2010). Applications of CFD in natural gas processing and transportation. INTECH Open Access Publisher.
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14. Jassim, E., Abdi, M. A., & Muzychka, Y. (2008). Computational fluid dynamics study for flow of natural gas through high-pressure supersonic nozzles: part 2. nozzle geometry and vorticity. Petroleum Science and Technology, 26(15), 1773-1785. 15. Karimi, A., & Abdi, M. A. (2009). Selective dehydration of high-pressure natural gas using supersonic nozzles. Chemical Engineering and Processing: Process Intensification, 48(1), 560-568. 16. Kohonen, T., Schroeder, M. R., Huang, T. S., & Maps, S. O. (2001). Springer-Verlag New York. Inc., Secaucus, NJ, 43. 17. Machado, P. B., Monteiro, J. G., Medeiros, J. L., Epsom, H. D., & Araujo, O. Q. (2012). Supersonic separation in onshore natural gas dew point plant.Journal of Natural Gas Science and Engineering, 6, 43-49.
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23. Valipour, M., Banihabib, M. E., & Behbahani, S. M. R. (2012). Parameters estimate of autoregressive moving average and autoregressive integrated moving average models and compare their ability for inflow forecasting. J Math Stat, 8(3), 330-338. 24. Valipour, M., Sefidkouhi, M. A. G., & Eslamian, S. (2015). Surface irrigation simulation models: a review. International Journal of Hydrology Science and Technology, 5(1), 51-70. 25. Vaziri, B. M., & Shahsavand, A. (2013). Analysis of supersonic separators geometry using generalized radial basis function (GRBF) artificial neural networks. Journal of Natural Gas Science and Engineering, 13, 30-41.
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26. Vaziri, B. M., & Shahsavand, A. (2015). Optimal selection of supersonic separators inlet velocity components via maximization of swirl strength and centrifugal acceleration. Separation Science and Technology, 50(5), 752759. 27. Wen, C., Feng, Y., Witt, P., Cao, X., & Yang, Y. (2012). CFD simulation of supersonic swirling separation of natural gas using a delta wing.
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28. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Effect of Inlet and Outlet Flow Conditions on Natural Gas Parameters in Supersonic Separation Process. 29. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Numerical simulation of real gas flows in natural gas supersonic separation processing. Journal of Natural Gas Science and Engineering, 21, 829-836.
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30. Yang, Y., Wen, C., Wang, S., & Feng, Y. (2014). Theoretical and numerical analysis on pressure recovery of supersonic separators for natural gas dehydration. Applied Energy, 132, 248-253.
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Appendix A
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Weights and bias between two steps are shown in the following table. They are used to link the input to hidden layer and hidden layer to the output layer. These are interconnection properties of the selected neural network which is finally found to be the optimal neural network model and have the best performance for the present dataset with the minimum error. The table has seven rows as the optimal network found to have 7 hidden layers.
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Table IV. Weights and bias between input and hidden layer and hidden and output layer Weight to Weight to layer 1 from input 1 Bias layer b{1}, bias to b{2}, bias to Iw {1,1} lw{2,1} layer 1 layer 2 1.5691 -0.6435 -1.1812 -1.1372 0.22329 -2.2033 -0.2246 -1.75 -1.2869 0.63963 0.11658 -0.16075 1.552 -0.50632 -1.6744 -0.13261 -1.4941 0.49318 0.69803 -1.0278 1.4738 -1.3978 -0.44023 0.87051 0.43312 0.73731 -1.0969 1.0598 -1.5244 -0.26439 0.72512 0.33906 -0.29645 -1.7488 -1.4001 0.42936 1.4233 1.3896 0.15244 -1.0442 1.39 0.44484 2.3467
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Appendix B
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The following table includes raw data gathered from literature including temperature and pressure condition and shockwave location in the supersonic nozzle with different scales. The Pressure recovery ratio is calculated for each set. Neural network inputs are four parameters including inlet pressure, inlet temperature, Pressure recovery ratio, and nozzle length. Table V. The literature raw data for the nozzle boundary condition and shockwave location
Jing et al., 2014
Pr
Tin (K)
Shock Location (x/D)
0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.70 0.70 0.70 0.70 0.60 0.60 0.60 0.60 0.60 0.60
300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 700.00 500.00 300.00 100.00 10.00 100.00 200.00 300.00 200.00 200.00 200.00 200.00 100.00 100.00 100.00 100.00 7.00 7.00 7.00 7.00 7.00 7.00
210.00 210.00 210.00 210.00 249.00 240.00 217.20 183.00 145.50 490.00 350.00 210.00 70.00 7.50 75.00 150.00 225.00 150.00 150.00 150.00 150.00 85.00 80.00 75.00 62.00 1.00 1.51 2.00 2.95 3.00 4.00
0.70 0.70 0.70 0.70 0.83 0.80 0.72 0.61 0.49 0.70 0.70 0.70 0.70 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.85 0.80 0.75 0.62 0.14 0.22 0.29 0.42 0.43 0.57
274.15 293.15 313.15 333.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 288.00 288.00 288.00 288.00 283.00 313.00 343.00 373.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00
0.85 0.85 0.91 0.93 0.68 0.75 0.83 0.92 1.00 0.81 0.81 0.84 0.92 0.49 0.45 0.38 0.33 0.40 0.39 0.41 0.41 0.50 0.54 0.60 0.67 0.93 0.88 0.85 0.83 0.77 0.70
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Pout (bar)
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Pin (bar)
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nozzle length (m)
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Research Highlights
A two-stage neural network is proposed to find shockwave position in supersonic nozzle.
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Classification and neural network tuning improve the performance of the model.
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The model is capable of estimating shockwave location in a wide range of inputs.
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The method uses gas feed pressure and pressure recovery ratio as inputs.
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Both higher inlet pressure and pressure recovery move the shockwave toward the inlet.
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