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Modeling of Thermal Cracking of Heavy Liquid Hydrocarbon: Application of Kinetic Modeling, Artificial Neural Network, and Neuro-Fuzzy Models Mehdi Sedighi,† Kamyar Keyvanloo,*,‡ and Jafar Towfighi† Department of Chemical Engineering, Tarbiat Modares UniVersity, P.O. Box 14115-143, Tehran, Iran, and Department of Chemical Engineering, Brigham Young UniVersity, ProVo, Utah 84602, United States
To predict the main product yields of thermal cracking of heavy liquid hydrocarbon, four models, kinetic, artificial neural networks (ANN), neuro-fuzzy (NF), and polynomial, were developed. The models investigated the influence of COT, steam ratio, and feed flow rate on product yields at the reactor tube outlet. A semimechanistic kinetic model based on free radical chain reactions was developed using experimental results. This semimechanistic kinetic model contains 148 reactions for 43 species. An objective function was defined to optimize the kinetic parameters. For the artificial intelligence systems, a three-layer perceptron neural network with back-propagation (BP) training algorithm and Sugeno inference system were used. To compare the accuracy of artificial intelligence method, another empirical method based on response surface methodology was also developed. Finally, the models were compared to experimental data, and a comparison between the results of kinetic model, designed ANN, and NF was also carried out by analysis of variance (ANOVA) calculation. The obtained results demonstrate that these four models are in good agreement with experimental data, while the ANN and NF models show better results than do the kinetic model and polynomial model. 1. Introduction Steam cracking is considered to be the main rout of olefin production. Developing the efficient models to predict the product yields in different operating conditions is of primary importance, due to the high demand of ethylene and propylene as the backbone of the petrochemical industry. Modeling of steam pyrolysis of hydrocarbons for production of light olefins has been devoted much endeavor over the last several decades. Recently, Keyvanloo et al.1 investigated the effect of temperature, residence time, and steam-to-naphtha ratio and their interactions on the products yields in naphtha steam cracking. Furthermore, two models for ethylene and propylene were developed on the basis of experimental data by statistical design of experiments. These two models were used for optimization of operating conditions to obtain the maximum yield of light olefins. Belohlav et al.2 developed a model for thermal cracking of ethane, petroleum gases, and primary naphthas. The model of pyrolysis reactions involved free radical reactions and a set of pure and formal molecular reactions. The experimental data set was used for the kinetic model optimization, and verification was assembled from the plant cracking of different feedstock in reactors of various configurations under various reaction conditions. Poutsma3 rationalized fundamental reactions of free radicals relevant to pyrolysis reactions. The fundamental relationships between thermochemical parameters and kinetic parameters were summarized, and the use of these to estimate rate constants for cases where data are not available was demonstrated. Kinetic models for thermal cracking of ethane, propane, n-butane, i-butane, and the simple mixtures of them based on free radical reaction network were developed by Sundaram and Froment.4 Ghassabzadeh et al.5 investigated thermal cracking of kerosene for producing ethylene and propylene in an experimental setup. An applicable kinetic model was also developed to predict yield distribution of products of * To whom correspondence should be addressed. Tel.: (801) 6367882. E-mail:
[email protected]. † Tarbiat Modares University. ‡ Brigham Young University.
the kerosene thermal cracking. Therefore, a reaction mechanism was generated on the basis of major reactions classes in the pyrolysis and feed compounds using some simplification assumptions in the model. They used an objective function to tune the model with experimental data. Depeyre et al.6 studied the effects of temperature, steam to gas oil ratio, and residence time on major products in gas oil thermal cracking. The best yield of ethylene, 27% in mass, was obtained in the quartz reactor at 770 °C, residence time of 0.6 s, and mass ratio of steam to gas oil equal to 1. Finally, a model was developed on the basis of the radical mechanism to predict the yield distribution of products. Towfighi et al.7 developed a mechanistic reaction model for the pyrolysis of LPG to predict the yields of the major products from a given LPG sample with commercial indices. A complete reaction network, using a rigorous kinetic model, for the decomposition of the LPG feed was developed and used for the simulation of industrial LPG crackers. Zahedi et al.8 studied the thermal cracking of atmospheric gas oil. The obtained maximum yield of ethylene was equal to 30.9 wt %, as well as the maximum yield of propylene was 12.2 wt %. A mechanistic model was developed based on experimental results to predict the yield distribution of products and the coke formation in the range of operating conditions. Different authors have also modeled various feedstocks.9-12 One of the simplest ways of fitting data is statistical approaches, which have been done by response surface methodology. It is a collection of data by experimental design and use mathematical and statistical techniques for developing models that best suits for obtaining a response by giving several variables to the system. This method of fitting data has gained popularity in various fields such as hydrocarbon thermal cracking modeling,1,13 catalysis,14,15 and material science.16,17 In recent years, artificial neural network, as another fitting data method, has been very interesting in chemical engineering processes, such as catalyst design,18,19 modeling of complex chemical process,20 and dynamic modeling of chemical process.21,22 Nabavi et al.23 used a three-layer perceptron neural network, with back-propagation (BP) training algorithm to develop modeling of thermal cracking of LPG. The model
10.1021/ie1015552 2011 American Chemical Society Published on Web 01/04/2011
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011
investigated the influence of the coil outlet temperature, steam ratio (H2O/LPG), total mass feed rate, and composition of feed such as C3H8, C2H6, iC4, and nC4 on the thermal cracking product yields. A comparison between the results of mathematical model and designed neural networks was also conducted. Modeling of thermal cracking of naphtha by artificial neural network (ANN), using back-propagation (BP), was developed by Niaei et al.24 The optimum structure of the neural network was determined by a trial-and-error method. Different structures were tried with several neurons in the hidden layer. A comparison between the results of the mathematical model and the designed ANN was also conducted. Their results showed a better performance of the ANN model than the mathematical model. Ghadrdan et al.25 carried out a neural network modeling for introduction of feed type, which was used as a model of a chemical reactor. The input variables of the neural network to represent the feed type were nine structural increments, the amounts of which were calculated from the compositions of the real components. Istadi et al.26 developed a hybrid artificial neural network-genetic algorithm (ANN-GA) numerical technique to model, to simulate, and to optimize a dielectric barrier discharge (DBD) plasma reactor without catalyst and heating. Artificial neural network (ANN) modeling was chosen to model the complex behavior between input and output in the DBD plasma process. To collect data, the design of the experiment was performed using central composite design (CCD) with full factorial design for designing the training and test data sets. Finally, 19 numbers of experimental data were partitioned into training (15 patterns) and test (4 patterns) sets. The complex reaction network in hydrocarbon thermal cracking is concerned with uncertainties that exist in this process. In such an uncertain situation, the Fuzzy Inference System (FIS) may be used in the estimation of uncertainties in the real situation. The hybrid of ANN and FIS benefits from the advantage of both ANN and FIS, which is called neuro-fuzzy (NF). This method has been applied in several aspects of science such as CO2 capturing,27 wind turbine rotor,28 and environmental engineering.29 Nevertheless, to our knowledge, no work has been reported yet in the literature on the application of neuro-fuzzy in modeling of hydrocarbon thermal cracking. In this Article, a semimechanistic model based on free radical chain reactions, ANN, NF, and polynomial models were developed to predict product distribution of thermal cracking of a heavy liquid hydrocarbon. In addition, the results of the models have been compared. Unlike in the previous works, the benefits of each model also have been investigated. 2. Modeling 2.1. Kinetic Model. 2.1.1. Reactor Model. A one-dimensional plug-flow model is used to simulate the thermal cracking reactor. Radial temperature variation has been neglected, as the reactor diameter is small as compared to the tube length. The pilot plant and the feed characteristics are fully described in our previous work.13 In addition, ideal gas behavior and no hydrodynamic or thermal entrance region effects are assumed. Pressure drop of pilot plant is too low. Therefore, the pressure profile (momentum equation) has been ignored. The mass balances for each species and energy equation are given by:30,31 Mass balance: m dFj πd2t )( sijrri) dz 4 i)1
∑
Energy balance:
(1)
n
∑FC
j Pj
j)1
πd2t m
dT ) Q(z)πdt + dz 4
∑ r (-∆H) ri
( )
Cj )
(2)
i)1
rri ) kiΠC|sj ij| ki ) k0i *exp -
i
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Ei RT
p Fj RT n Ft
(3) (4) (5)
∑ t)1
Q(z) ) Uwg(Tw - T) Uwg )
1 rin rout 1 + ln h ktube rin
(6) (7)
Towfighi et al.7 have represented the detail description of the applied mathematical model. 2.1.2. Kinetic Model Development. Over the last decades, kinetic modeling of thermal cracking for different feedstock has been carried out. Empirical, molecular, and mechanistic models are the tools for thermal cracking modeling, while mechanistic radical kinetic models were widely accepted due to flexibility, accuracy, and convenient inter- and extrapolation ability.32 The feed analysis contains n-paraffins, iso-paraffins, naphthenes, and aromatics. This heavy feed is more complex than lighter feed such as naphtha. Therefore, it is necessary to accept several simplifications and lumping procedure. The proposed kinetic model is a semimechanistic model of radical decomposition based on the simplified theory of radical and pure molecular reactions. However, the proposed kinetic model can be adapted by the regression method to the mechanistic model. To avoid complexity in reaction network, the detected species of heavy liquid hydrocarbon were lumped to four pseudo components as n-C9H20 and i-C9H20 for normal paraffins and iso-paraffins, C9H18 for naphthenes, and C10H14 for aromatics. The lumping procedure is as follows: each hydrocarbon group was considered as a pseudo component, and the average molecular weight of each group was found. For each group, a component was selected according to the point that its molecular weight should be near the average molecular weight of the group. Feed lumping results have been shown in Table 1. Different isomers have been selected due to its weight percent. These four pseudo components were used for feedstock characterization in kinetic modeling. The average length of hydrocarbon molecules in this feed is C9, which is different from other feedstocks published in different articles. Therefore, a new semi mechanistic model should be developed. To establish the reaction network, it is necessary to consider the most important species and reactions. The main reaction classes of heavy liquid hydrocarbon pyrolysis are shown in Table 2. The radical reactions contain chain-initiation reactions, chain-propagation reactions, chain-termination reactions, secondary reaction, and isomerization reaction. Molecular reactions contain dehydrogenation, Diels-Alder molecular reaction, and molecular reaction.6,33 Based on pseudo components, the reaction network includes 148 reactions for 43 species (molecular and radical). For the developed reaction network, which is listed in our previous work,34 the kinetic parameters (activation energy and frequency factor) of each reaction should be specified. They can be
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Table 1. Feed Lumping Results group
average molecule
pseudocompound
chosen structure
paraffin iso-paraffin naphthene aromatic
C9.3H20.6 C8.9H19.9 C9H18 C9.7H13.5
C9H20 C9H20 C9H18 C10H14
E,2-methylcyclohexane butyl benzene
Table 2. Main Reaction Classes of Heavy Liquid Hydrocarbon Pyrolysis (1) chain-initiation reactions P f R1′ + R2′
(2) chain-propagation reaction R1′ f R2′ + O P1 + R1′ f R2′ + P2
(3) chain-termination reaction R1′ + R2′ f P
(4) secondary reaction O + R1′ f R2′ O + R1′ f Rq′ + P Rq′ f O + R′
(5) olefin isomerization R′ f R′′
(6) dehydrogenation reaction P f O + H2
f(θ) )
(7) Diels-Alder molecular reaction C4H6 + O f aromatic + H2
(8) molecular reaction
obtained from experiments or literature. However, for this feedstock, all kinetic parameters cannot be found in the literature. Therefore, in this Article, those kinetic parameters are optimized and tuned with experimental data. Due to the form of the Arrhenius rate equation (eq 4), it is obvious that the rate equation is more sensitive to activation energy than the frequency factor and activation energy used as parameters of each reaction. To optimize model parameters with experimental values, an objective function subjected to eqs 1-7 should be defined. In this Article, an objective function (eq 8) representing the difference between predicted and experimental data for ethylene, propylene, methane, ethane, and Cs+ output data of model at the reactor tube outlet was defined: set
min Εa
starting at an initial estimate. This is generally referred to unconstrained nonlinear optimization. The customized kinetic model is useful and valid only for this particular pilot plant. The governing continuity and energy equations are highly stiff in numerical simulation due to the fact that concentration of radicals is very low as compared to molecular ones. To solve equations simultaneously, a stiff differential equations solver, Ode 23s, was used. Ode 23s solver is based on a modified Rosenbrock formula of order 2. Finally, simulation results were compared to other experiments results at their operating conditions. 2.2. Artificial Neural Network (ANN)-Based Modeling. Neural network can be trained to perform a particular function by adjusting the values of the connections (weights) between elements. In addition to the weight (w), a bias (b) is included to shift the space of the nonlinearity. Neural networks are adjusted or trained so that a particular input leads to a specific target output. The network is adjusted, based on a comparison of the output and the target, until the network output matches well with target. The feed forward ANN model was used in this model development and trained with a back-propagation training function. The network was trained by LevenbergMarquardt optimization algorithm due to its fast convergence and reliability in locating the global minimum of the mean square error (MSE).35 A neural net is a parallel interconnected structure consisting of (1) an input layer of neuron (independent variables), (2) a number of hidden layers, and (3) an output layer (dependent variables). The number of input and output neurons is determined by the nature of the problem. The Tansig and Purelin functions can be utilized as transfer functions in the hidden and output layers. The transfer function in the hidden layer should be nondecreasing and differentiable. The Tansig transfer function (hyperbolic tangent sigmoid) is given as:
n
∑ ∑W i)1 j)1
j
[
sim (Yexp ij - Yij )
Yexp ij
]
2
(8)
where i,j denote compound j in experiment i condition. The optimization was performed by the FMINSEARCH program. It finds the minimum of a scalar function of several variables,
2 -1 1 + exp(-2θ)
(9)
Before training, it is often useful to scale the inputs and targets so that they always fall within a specified range. The function mapminmax scales inputs and targets so that they fall in the range [-1, 1]. The performance of the ANN model is considered as fitness tests of the model, that is, correlation coefficient (R) and epoch number (epochs). The correlation coefficient matrix represents the normalized measure of the strength of linear relationship between variables. The square error algorithm is an example of supervised training, in which the learning rule is provided with a set of examples of desired network behavior. As each input is applied to the network, the network output is compared to the target. The error is calculated as the difference between the target output and the network output. The goal is to minimize the sum of these errors. Objective function is defined as: objective function )
1 N
N
∑ 1
(
|targeti - outputi | targeti
)
(10)
N is the number of data, target is experimental data, and output is ANN data. 2.3. Neuro-Fuzzy (NF)-Based Modeling. A fuzzy inference system (FIS) can be used to model the qualitative aspects of science and reasoning processes without adopting precise quantitative analyses to investigate nonlinear relationships among input and output variables.36 Each fuzzy system contains
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Figure 1. Structure of the ANFIS network.
three main parts, fuzzifier, fuzzy database, and defuzzifier. Its basic structure is a model that maps input characteristics to input membership functions, input membership function to rules, rules to a set of output characteristics, output characteristics to output membership functions, and the output membership function to a single-valued output or a decision associated with the output. The adaptive neuro-fuzzy inference system (ANFIS) is the best manufacturer of FIS.32 Neuro-fuzzy is soft computing aimed at an accommodation with the pervasive imprecision of the real world. The guiding principle of soft computing is: exploit the tolerance for imprecision, uncertainty, and partial truth to achieve tractability, robustness, and low solution cost. Neuro-fuzzy utilizes neural network techniques to automatically generate rule bases and membership functions from sample data. Within fuzzy logic, such systems play a particularly important role in the induction of rules from observations. Using a given input/output data set, ANFIS constructs a fuzzy inference system (FIS) whose membership function parameters are tuned (adjusted) using either a backpropagation algorithm alone, or in combination with a least-squares type of method. Therefore, unlike the multilayer perceptron (MLP), where weights are tuned, in ANFIS, fuzzy language rules or conditional (if-then) statements are determined to train the model.37 There are several fuzzy inference engines, of which Sugeno and Mamdani are the two most well-known ones. The Sugeno method is similar to the Mamdani method in many respects. The first two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy operator, are exactly the same. The main difference between Mamdani and Sugeno is that the Sugeno output membership functions are either linear or constant. The ANFIS system is functionally equivalent to the Sugeno first-order fuzzy model.38 It sould have a single output, obtained using weighted average defuzzification. All output membership functions should be the same type and either linear or constant. Also, it includes unity weight for each rule. The optimization methods train the membership function parameters to emulate the training data. In this work, the hybrid optimization method, which is a combination of least-squares and backpropagation gradient descent method, is employed to train ANFIS. 2.3.1. Architecture of ANFIS. Without loss of generality, we assume two inputs, x1 and x2, and one output, y. Assume a first-order Sugeno type of rule base with the following four rules: If x1 is A1 and x2 is A2, then y1 ) a0,1 + a1,1x1 + a2,1x2. If x1 is A1 and x2 is B2, then y2 ) a0,2 + a1,2x1 + a2,2x2. If x1 is B1 and x2 is A2, then y3 ) a0,3 + a1,3x1 + a2,3x2. If x1 is B1 and x2 is B2, then y4 ) a0,4 + a1,4x1 + a2,4x2.
Here, A1, B1 and A2, B2 are the membership functions for inputs x1 and x2, respectively; a0,1, a1,1, a2,1 and a0,2, a1,2, a2,2 and a0,3, a1,3, a2,3 and a0,4, a1,4, a2,4 are the parameters of the output function. The architecture of ANFIS network is represented in Figure 1 by using the if-then rule format with a twodimensional input.39 In Figure 1, there are two subgroups of nodes in layer 1. The first subgroup includes nodes of A1 and B1, which are linked by x1, and the second subgroup includes nodes of A2 and B2, which are linked by x2. These nodes are equal to the linguistic variables in the original Sugeno inference system, and they serve for the partition of the input space. A description of the layers in the network follows. Layer 1. Each node in this layer is adaptive with a parametric activation function. Its output is the grade of membership in which the given input satisfies the membership function. Usually a Gaussian function is used as a membership function.40 For instance, the membership function for A1 is defined as:
[
µA1(x1) ) exp -
(x1 - C1)2 2σ21
]
(11)
where c1 and σ1 are premise parameters of the membership function, known as mean and the variance, respectively. Layer 2. Each node in this layer is a fixed node whose output is the product of all incoming signals. The outputs wj, j ) 1, ..., 4 represent the firing strength. Thus, w1 ) µA1(x1), µA2(x2)
(12)
w2 ) µA1(x1), µB2(x2)
(13)
w3 ) µB1(x1), µA2(x2)
(14)
w4 ) µB1(x1), µB2(x2)
(15)
and
Layer 3. Each node in this layer is a fixed node, which calculates the ratio of the jth rule’s firing strength relative to the sum of all rule’s firing strengths: w¯j )
wj w1 + w2 + w3 + w4
(16)
The result is a normalized firing strength. Layer 4. Each node in layer 4 is an adaptive node with a node output:
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w¯jyj ) w¯j(a0,j + a1,jx1 + a2,jx2)
(17)
where wjj is the normalized firing strength from layer 3, and {a0,j, a1,j, a2,j} is the parameter set of this node. Parameters in this layer are called consequent parameters. Layer 5. The single node in this layer calculates the overall output of the ANFIS as
∑ w¯ y
j j
j
∑wy
j j
)
j
(18)
∑w
j
j
A hybrid algorithm adjusts the consequent parameters ai,j (i ) 0,1,2 and j ) 1,2,3,4) in a forward pass and the premise parameters {Ci, σi} in a backward pass.38 In the forward pass, the network inputs propagate forward until the fourth layer where the consequent parameters are identified by the leastsquares method. In the backward pass, the error signals propagate backward, and the premise parameter is updated by gradient descent. More information for ANFIS and hybrid algorithm can be found in related literature.38 2.3.2. ANFIS Modeling. In this Article, one fuzzy inference system was developed as a result of each ANFIS modeling process, of which the membership functions were adjusted and the rules were generated. Hence, five inference systems were developed for ethylene, propylene, methane, ethane, and C5+, each of which contains three inputs (temperature, flow rate, steam ratio), 18 rules, and one output. The structure of a sample inference system for each product is shown in Figure 2. The rule base consists of 18 Sugeno-type rules. The premise part of each rule is a conjunction of qualitative labels of the input variables; the consequent part of a Sugeno-type rule is a linear function between the output variable and all the input variables. For example, the 22th rule is given as follows. If temperature is high and flow rate is medium and steam ratio is nearly low, then product is mf15. mf15 is a linear function of three input variables, which can be expressed as follows: mf15 ) a0,15 + a1,15 × (temperature) + a2,15 × (flow rate) + a3,15 × (steam ratio) (19) where mf15 is the output (product) in the 15th rule, and a0,15, a1,15, a2,15, a3,15 are the coefficients for the constant and three input variables, respectively. Generally, if temperature is Al and flow rate is Bl and steam ratio is Cl, then product is mfj. mfj ) a0,j + a1,j × (temperature) + a2,j × (flowrate) + a3,j × (steam ratio) (20) where mfj is the output in the jth rule and a1,j, a2,j, a3,j are coefficients for three input variables and a0,j is the constant coefficient and Al (temperature), Bl (flowrate), Cl (steam ratio) are the linguistic labels of membership functions for each input variables. Figure 3 shows the structure of ANFIS for thermal cracking of heavy liquid hydrocarbon, which consists of three intermediate layers and 18 rules. 3. Results and Discussion 3.1. Application of Kinetic Model. The developed model is utilized to determine the product yields distribution in the entire range of operating conditions. Figure 4 shows the scatter diagram of ethylene, propylene, methane, and C5+. It shows a good agreement between the predicted and experimental data. Maximum error as compared to experimental data for products
used in the objective function are 15%. It should be noted that the majority of dispersal is devoted to C5+. Figure 5 shows the scatter diagram of ethane, hydrogen, and propane. It also indicates overall favorable agreement between the predicted and the experimental data. Maximum error as compared to experimental data for products that were used in the objective function was 18%. As shown in the figure, ethane caused the majority of dispersal. The main product yields versus the length of the reactor are shown in Figure 6. This figure obtained at COT, residence time, and steam ratio of 850 °C, 0.3 s, and 0.6 (gr/gr), respectively. In general, the yields of ethylene, methane, ethane, and hydrogen products increase continuously along the reactor. Because of the secondary reactions, the yield of propylene increases and reaches a maximum value and then decreases. Temperature profile has been shown in Figure 7. It shows the furnace temperature (set point), tube outside skin temperature, and gases temperatures. It indicates that in the design setup, approximately 15 cm of the tube has been used to heat the gas temperature to the primary cracking temperature (550 °C). Figure 8 shows the yields of main cracking products against the severity index (C3H6 /C2H4, g/g). The product yield distribution of thermal cracking of specific feedstock, in different reactors with different scales, is the same at a specific severity.8,41 In industrial units, severity index is the appropriate way to show the reaction improvement. In general, increasing severity causes increased yield of propylene and decreased yield of thermally stable components such as methane, ethylene, and aromatic. In our experimental data and calculated values from the kinetic reaction network, the above-mentioned trend has also been observed. 3.2. Application of ANN and NF Models. Artificial intelligence models are effective approaches to handle nonlinear and noisy data without any prior knowledge about the relationships among physical processes. One of the problems that occurred during neural network training is called overfitting. The network has memorized the training examples, but it has not learned to generalize to new situations. However, to solve this problem, data were partitioned to training (used to estimate model parameter), validation (used to check the generalization ability of new samples), and test (monitoring) sets. A system with three inputs and five outputs of an artificial neural network model was developed for modeling of heavy liquid hydrocarbon thermal cracking reactor. Three independent variables, that is, coil outlet temperature (°C), steam ratio (gr/gr), and feed flow rate (gr/min), are used as input of the network, and five dependent variables or objectives, that is, C2H4 (Y1), C3H6 (Y2), C2H6 (Y3), CH4(Y4), and C5+ (Y5), are selected as outputs of the network. There is no general guideline for choosing the number of hidden nodes on hidden layer, but theoretically the multilayer perceptron (MLP) with one hidden layer is enough to solve any problem that MLP with two hidden layers can solve.42 There are two approaches to model thermal cracking reactor. In the first approach, threelayer MLP was considered with multi neurons in output layer, but in the second approach, an individual three-layer MLP was considered for any components.23 We have compared two approaches (not included in the text). It was found that the second approach has better performance than the first approach by comparing their RMSE and R2. Our approach for modeling of thermal cracking reactor was to consider an individual three-layer multilayer MLP for any components. So in this approach, there is only one neuron in the output layer for any network. On the basis of this approach,
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Figure 2. Structure of inference system for each product.
Figure 3. The ANFIS modeling systems for heavy liquid hydrocarbon thermal cracking with 18 rules.
Figure 4. Scatter diagram of C2H4, C3H6, C5+, and CH4.
Figure 5. Scatter diagram of C2H6, C3H8, and H2.
different numbers of neurons are tested in the hidden layer. It is highly important for network performance to select the best number of neurons in hidden layer. In this Article, for ANN and NF models, the 100 numbers of experimental data were used as training data. Validation and test set contain 6 and 14 samples, respectively. In each network training iteration, the training set was utilized to adjust the weight matrix set, and
the validation set was used to check the generalization ability of new samples. Comparison of the ANN model performance for various topologies is presented in Table 3. In this table, the network structure of 3-9-1 (for example) consists of 3 input neurons, 9 hidden layer neurons, and 1 output. For each output, a related correlation coefficient has been shown. As mentioned, our approach is to consider an individual three-layer multilayer MLP for any component. Due to the fitness tests of the model, it was found that the structure of neural network with 10 neurons in the hidden layer had the best performance for prediction of main product yield in thermal cracking of heavy liquid hydrocarbon feed. The results of ANFIS modeling for the product at data set (training data) for ethylene and propylene are shown in Figure 9. It can be seen that the magnitudes of product fluctuate significantly with index (data values). The figures also indicate that the nonlinearity behavior of product profile is well reproduced by the ANFIS. As shown in Figure 9, there is very good agreement between the output ANFIS behavior and training data. Initial membership functions for each input changes are optimized by training process to represent the input and output mappings. The linguistic labels of membership functions for temperature (°C) and flow rate (gr/min) input variables are low, medium, and high and for the steam ratio input variable are nearly low and nearly high. Gaussian function is used as a membership function, and the center and width of each membership function are initialized by ANFIS. During the training process, the parameter associated with the membership function is adjusted. 4. Comparison of the Developed Models 4.1. Comparison of Kinetic Model with ANN and NF. As was proposed by Legates et al.,43 a perfect way of evaluating a model should contain a combination of relative error measure
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Figure 6. Compound yield profiles among tube length at residence time ) 0.3 s, COT ) 850 °C, steam ratio ) 0.6 g/g.
Figure 7. Temperature profile along tube length.
Figure 8. Ethylene and propylene yield versus severity (experimental data/simulation results).
(e.g., R2) and absolute error measure (e.g., RMSE). In this Article, R2 and RMSE were used to evaluate the performance of the kinetic model, ANN, NF, and polynomial models. R2 and RMSE are defined as below:
Table 3. Performance Comparison of Various ANN Structures for One Hidden Layer Network correlation coefficient (R)
SSE R )1SST RMSE ) √SSE
2
where
(21)
topology
epochs
Y1
Y2
Y3
Y4
Y5
(22)
3-5-1 3-7-1 3-8-1 3-10-1 3-12-1
460 425 360 235 135
0.948 0.953 0.957 0.961 0.961
0.962 0.967 0.973 0.981 0.981
0.972 0.979 0.986 0.985 0.985
0.931 0.935 0.947 0.944 0.945
0.925 0.932 0.936 0.948 0.949
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Figure 9. The results of modeling using ANFIS for (a) ethylene and (b) propylene by training data set. Table 4. Comparison of Performance of Kinetic Modeling with NF Model
C2H4 C3H6 CH4 C2H6 +
C5
sum of squares
df
mean square
F
significance
72.598 413.672 8.306 41.439 25.346 144.132 2.635 27.056 56.744 890.636
1 26 1 26 1 26 1 26 1 26
72.598 15.910 8.306 1.594 25.346 5.544 2.635 1.041 56.744 34.255
4.563
0.042
5.211
0.031
4.572
0.042
2.532
0.124
1.656
0.209
between groups within groups between groups within groups between groups within groups between groups within groups between groups within groups n
SST )
∑ (Y
- Y¯exp)2
(23)
- Ymodel,i)2
(24)
exp,i
i)1
n
SSE )
∑ (Y
exp,i
i)1
The checking performances of all three models for (a) C2H4, (b) C3H6, (c) CH4, (d) C2H6, and (e) C5+ were shown in Figure 10. It could be found that all three models can get high R2 for all components. However, the R2 of the NF model is higher than those of the other two models, which indicates the higher accuracy of the NF model. It was found that the values of R2 and RMSE for ANN (ethylene) were 0.941 and 3.669, respectively, while they are 0.982 and 1.834 for the NF model. Moreover, the above-mentioned parameters are, respectively, 0.881 and 5.23 for the kinetic model. It was observed that the RMSE values for ANN and NF models are also smaller than that for the kinetic model. To investigate whether the significant difference between kinetic, ANN, and NF models prediction exists, ANOVA
calculations were conducted. Tables 4 and 5 showed ANOVA results between kinetic and NF models and between kinetic and ANN models, respectively. It indicated that differences between the models for prediction of C2H4, C3H6, and CH4 were significant. The source of variation has been categorized into two groups: between group and within group. Between group means calculations between ANN or NF model and kinetic model, and within group means calculations in ANN or NF model and kinetic model separately. It should be noted that the kinetic model has some advantages that ANN or NF does not. A mechanistic model based on free radical chain reactions is realistic. It provides tube outside skin temperature, and gases temperature can be specified along the reactor, and on the basis of this concept, setup can be improved to optimize energy saving and construction cost. It also shows yields profile along the tube length. Kinetic model is not limited to ranges of which experiments have been carried out so it is good at extrapolation. Nevertheless, mechanistic model based on free radical chain reactions model takes many times even using simplification assumptions. To achieve a
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Figure 10. Results of the models as compared to the experimental data for (a) ethylene, (b) propylene, (c) methane, (d) ethane, and (e) CS+. Table 5. Comparison of Performance of Kinetic Modeling with ANN Model
C2H4 C3H6 CH4 C2H6 C5+
between groups within groups between groups within groups getween groups within groups between groups within groups between groups within groups
sum of squares
df
mean square
F
significance
70.171 418.107 7.582 41.603 24.591 146.558 2.532 27.480 53.932 890.646
1 26 1 26 1 26 1 26 1 26
70.171 16.081 7.582 1.600 24.591 5.637 2.532 1.057 53.932 34.256
4.364
0.047
4.738
0.039
4.362
0.047
2.396
0.134
1.574
0.221
more precise kinetic model, a detailed mechanistic reaction network involving thousands of reactions and more than a hundred molecular and radical species should be specified. However, it is so complicated. In our work, the average run
time for ANN was 51 s, whereas the run time of kinetic model is 1825 s for 14 cases. The ANN, NF, and mathematical algorithm were developed in a MATLAB 7.7 environment to model and simulate the thermal cracking reactor.
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011 Table 6. Model Summary Statistics R2 model
C2H4
C3H6
CH4
C2H6
C5+
linear 2FI quadratic cubic
0.67 0.73 0.86 0.88
0.49 0.50 0.88 0.89
0.70 0.71 0.89 0.89
0.75 0.78 0.85 0.85
0.77 0.78 0.82 0.83
4.2. Comparison of Artificial Intelligence with RSM. Response surface methodology (RSM) was used to determine the optimum process parameters for the prediction of main products in thermal cracking. A full factorial experimental design for three process variables was used. Full factorial method can give better insight into the quadratic interactions and higher orders in comparison with the other experimental design methods such as central composite design and boxbehnken. The polynomial relationship of the independent variables and the response was calculated by eq 25: k
Y ) β0 +
∑
k
βiXi +
i)0
∑
k-1
βiiXiXi +
i)0
k
∑ ∑ β XX
ij i j
i)1 j)i+1
(25) Y is the predicted response; β0 is a constant; βi is the linear coefficient; βii is the squared coefficient; βij is the crossproduct coefficient; and k is the number of factors. Different polynomials were fitted with experimental data. Model summary statistics were given in Table 6. According to the results, the F value of the cubic model is the highest one, which shows higher R2 among linear, 2FI, and quadratic models. Therefore, it was selected for comparison with artificial intelligence systems. There was a better model fit for propylene; however, a poorer fit was found for the C+5 product. This depends on the nonlinearity of the system. The NF and RSM models were compared for their fitness accuracy and predictive capability. The predicted values by the NF as well as the RSM model are tabulated in Table 7. The predictive capability can be best evaluated using a completely new data set (data that were not used for model creation). The NF model was fitted to the experimental data for all outputs with excellent accuracy. Most of the R-squared of the products are above 0.95 with the NF model; however, none of the products could achieve the R-squared above 0.89 by the statistical approach (RSM). Hence, the RSM model shows greater deviation in fitting to the measured responses than does the NF. RMSE was also measured for comparing the predictive capability of the two models. As is indicated in Table 7, neurofuzzy is able to predict the properties with reasonably low prediction error. For the set of data used for validating the models, the RMSE are comparatively lower in the neuro-fuzzy model than in the statistical regression model. Artificial intelligence systems offer an alternative to the polynomial regression method as a modeling tool. Conventional RSM requires the specification of a polynomial function such as linear, first-order interaction, or second-order quadratic, to be regressed. In addition, the number of experimental design points limits the number of terms in the polynomial, and the
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selection of the appropriate polynomial equation can be extremely cumbersome because each response requires its own individual polynomial equation.44 The intelligence methodology provides the modeling of complex relationships, especially nonlinear ones, that may be investigated without complicated equations.45 Intelligence analysis is quite flexible with respect to the number and form of the experimental data, which makes it possible to use more informal experimental designs than with statistical approaches. Also, neural network and neuro-fuzzy models might have better predictive power than regression models. Regression analyses are dependent on predetermined statistical significance levels, and less significant terms are usually not included in the model. If a term is possibly significant, that is, a bit larger than the predetermined probability value, it should be replicated to investigate whether that term is actually significant or not. With the ANN and NF methods, all data are used, and a connection between input, hidden, and output layers is found, potentially making the models more accurate.44,46 Another advantage of the ANN and NF over the RSM can be stated as its ability to calculate multiresponses in a single process. To obtain a multiresponse optimization, the RSM model must be run several times, equal to the number of the parameters to be predicted, and it would be really time-consuming.47 5. Conclusion In the present study, a semimechanistic kinetic model, artificial neural network, neuro-fuzzy model, and polynomial model were developed to predict the main product yields of heavy liquid thermal cracking. Parameters of kinetic model were optimized by an objective function. It was found that the NF model had the best accuracy. Nonetheless, the ANN model could also improve greatly the accuracy over the kinetic modeling. The NF model showed an improvement of R2 and RMSE values of the ANN model by 4.07% and 50.01%, respectively, for ethylene production. The results were 2.25% and 28.99% in propylene production. Similarly, the ANN model could improve R2 and RMSE values of the kinetic model about 6.37% and 29.84%, respectively, for ethylene production and 8.90% and 40.97% in propylene production. Furthermore, the results indicated that the NF and ANN models could reasonably estimate methane, ethane, and C5+ yields. The results also showed that a cubic polynomial model that was developed by response surface methodology could not catch the nonlinearity of the system; therefore, it exhibited lower accuracy as compared to artificial intelligence method. Although the kinetic modeling provided some reasonable results due to predicting the yield distribution and temperature along the reactor, in comparison to the intelligence models, it was not an accurate model. This may refer to simplification of the reaction network, whereas in ANN and NF models nonlinear characteristics can help the models to detect the nonlinear features of hydrocarbon thermal cracking phenomenon without the need of any prior knowledge of the physical and chemical laws that govern the process. In general, an empirical model has limited applicability; it cannot be used for another feed
Table 7. Comparison of the Performance of RSM and NF for Five Model Outputs C2H4
C3H6
CH4
C5+
C2H6
parameter
RSM
NF
RSM
NF
RSM
NF
RSM
NF
RSM
NF
RMSE R2
5.87 0.88
1.83 0.98
4.66 0.89
0.92 0.98
3.42 0.89
0.12 0.96
5.37 0.85
0.85 0.98
6.12 0.83
1.34 0.93
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composition and industrial naphtha crackers. However, due to simplicity and accuracy of artificial intelligence method, the results can be used to recommend the optimum operating conditions for further experimental works especially in kinetic studies. Acknowledgment Financial support from Chemical Engineering Center of Excellence at Tarbiat Modares University is highly appreciated. Nomenclature Cj ) molar concentration (kmol m-3) CPj ) heat capacity of compound j (kJ kmol-1 K-1) dt ) inside diameter of tube reactor (m) Ei ) activation energy of reaction i (kJ kmol-1) Fj ) molar flow of compound j (kmol-1 s-1) h ) heat transfer coefficient (kW m-2 K-1) Ktube ) tube wall conductivity (kW m-1 K-1) ∆Hi ) heat of reaction (kJ mol-1) P ) total pressure (kPa) Q ) heat flux (kW m-2) rin ) inner radius of tube (m) rout ) outer radius of tube (m) R ) gas constant (kJ kmol-1 k-1) sij ) stochiometery factor T ) temperature (K) Tw ) outside tube skin temperature (K) Uwg ) overall heat transfer coefficient (kW m-2 K-1) z ) reactor length (m)
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ReceiVed for reView March 10, 2010 ReVised manuscript receiVed November 29, 2010 Accepted November 29, 2010 IE1015552
