ANN methodology in lumped kinetic modeling of Fischer–Tropsch reaction

Fuel Processing Technology 106 (2013) 631–640

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Fuel Processing Technology journal homepage: www.elsevier.com/locate/fuproc

The application of hybrid DOE/ANN methodology in lumped kinetic modeling of Fischer–Tropsch reaction Mehdi Shiva a,⁎, Hossein Atashi a, Farshad Farshchi Tabrizi a, Ali Akbar Mirzaei b, Akbar Zare b a b

Department of Chemical Engineering, University of Sistan and Baluchestan, 98164-161, Iran Department of Chemistry, University of Sistan and Baluchestan, 98135-674, Iran

a r t i c l e

i n f o

Article history: Received 30 October 2011 Received in revised form 21 June 2012 Accepted 6 September 2012 Available online 6 November 2012 Keywords: Kinetic modeling Fischer–Tropsch Response surface methodology Artificial neural networks Bimetallic catalysis

a b s t r a c t In this article the application of design of experiment coupled with artificial neural networks in kinetic study of CO hydrogenation reaction and a method of data collection/fitting for the experiment are presented. The kinetic experimental data has been collected from two factors of central composite designs, pressure and feed ratio in different temperatures. Response surface and artificial neural network models have been constructed based on the DOE points and used for generating simulated data. Then different data sets were used to fit LHHW kinetic rates of CO disappearance. The application of the neural networks to solve the problem of correlation of the rate equation parameters has been addressed. The results of kinetic modeling with simulated data sets from ANN and RSM models were compared with randomly collected experimental and the DOE data sets. It was observed that three rate equations were able to fit the data sets and kinetic modeling with the proposed DOE-based and simulated data, are very efficient for assessing the quality of the models. However, conducting kinetic modeling with ANN simulated data presented the best results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction As an efficient tool for organization of the experimental studies, designing of experiments, DOE is widely used in various sciences and technology fields as well as designing, optimizing and developing of the catalyst [1,2]. In catalytic kinetic studies, where experimental data is used to estimate the kinetic model parameters, DOE may have several benefits because of time limitation, cost and other technical considerations. Selection of an appropriate experimental design is critical in a successful experimental research. Although non-standard optimal designs have been successfully used in kinetic studies of heterogeneous catalytic reactions [3–5], standard designs haven't been considered so much. Because of constant exponential dependency of rate on the temperature (Arrhenius behavior), kinetic rate equations are believed to be nonlinear in the behavior. Juusola et al. [3] indicated that using factorial or central composite designs to obtain parameter estimates for highly nonlinear-mechanistic models will almost always lead to highly-correlated (and hence unreliable) estimations of these parameters. Providing the experimental points with optimal designs makes it possible to reduce efficiently the experimental points required to fit the model as well as covariance of the model parameters. In optimal designs such as D-optimal, the experimental points are produced with the aim of minimizing the optimality criterion. However, in kinetic studies, one of the requirements for running the D-optimal design is primarily ⁎ Corresponding author. E-mail address: [email protected] (M. Shiva). 0378-3820/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fuproc.2012.09.056

assigning the rate equation [5]. For complicated catalytic reactions such as Fischer–Tropsch with various mechanistic rate equations, providing an initial guess for rate equation is difficult. The other drawback is that the optimal designs also require collecting experimental points step by step. At first, some experimental points must be collected to fit the kinetic model rate equation and estimate the model coefficients, then the new points which must be collected and used to fit the rate equation again are produced by D-optimality criterion and this procedure could be continued until the D-optimality criterion is provided. On the other hand, the benefit of standard designs is the possibility of gathering all of the experimental points together. Applying the standard experimental designs of response surface methodology (Central Composite design or Box Behnken design), makes it possible to use regression/statistical tools for the modeling. It is also possible to use the RSM models to generate the simulating data. The regular distribution of the designed experimental points in all experiment domains makes it possible to develop more general modeling/simulating tools such as artificial neural network (ANN) models via minimum experimental data. The ANN is widely used to model the nonlinear behaviors [6,7]. Some kinetic studies are also conducted with ANN [8–10]. The application of artificial neural networks in prediction of FT products has been investigated by Sharma et al. [11], but its application in kinetic study of Fischer–Tropsch has not been addressed. However, the utilization of standard experimental designs has not been considered in FT kinetic studies [12]. The combination of standard design of experiment with ANN may help kinetic study to be conducted more efficiently from the standpoints of run (kinetic test) minimization and information maximization.

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Data fitting of kinetic rate equations with simulated points would increase the chance of success in kinetic studies. Lumped kinetic modeling of Fischer–Tropsch on single metallic catalysis such as Fe, Co and Re, has been widely investigated [13–18], however, on bimetallic catalysis a few lumped kinetic studies are available. Sachtler and Van Santen [18] noted that the selectivity and stability of bimetallic catalysis are often different from that of their components attributed to the changes in both electronic and geometric structures of catalyst. Ishihara et al. [19] investigated Fe–Co, Co–Ni and Ni–Fe pairs over various supports and found that Fe–Co catalysis had the best alkene products; Hence, Fe–Co catalysis may be considered for industrial practices. In this article a strategy based on coupled DOE and ANN used to collect the experimental points and curve-fitting with linearization and modified form of Arrhenius equation is provided in which the problem of parameter correlation of kinetic rates has been successfully solved. We also presented the capability of response surface methodology (RSM) and artificial neural network in modeling and kinetic studies. 2. Experiments 2.1. Catalyst preparation The catalysts were prepared by incipient wetness impregnation of MgO (surface area 118.93 m2/g, average pore size 19 Å, pore volume 4.1×10−2 cm3/g) with aqueous cobalt nitrate Co(NO3)2·6H2O and iron nitrate Fe(NO3)3·6H2O solutions. The MgO support was first calcined at 600 °C in flowing air for 6 h before impregnation (surface area 127.09 m2/g, average pore size 15 Å, pore volume 3.2×10−2 cm3/g) to ensure morphological stability during subsequent application. Then the calculated amounts of cobalt nitrate and iron nitrate were dissolved in distillate water and directly impregnated into the MgO support using incipient wetness. The obtained suspension was then rotated and aged for 4 h in a rotary-evaporator at 60 °C. The aged suspension was after that filtered, and dried at 120 °C for 16 h to give a catalyst precursor. The precursor was then calcined at 600 °C for 6 h. The final obtained catalyst consists of 5 wt.%Co–5 wt.%Fe/90 wt.%MgO. The catalyst was ground and sieved to obtain very fine powder particles (less than Mesh 120). 2.2. Design of experiments Two Central Composite designs with two factors were employed in two different temperatures; 320 °C and 340 °C. The two factors

are: pressure (1–7 bar) and feed ratio. In design 1, the volume flow rate of CO was kept constant (VCO = 20 ml/min) and volume flow rate of hydrogen changed from 16 to 44 (feed ratio H2/CO: 0.8–2.2). In design 2, the volume flow rate of hydrogen was kept constant (VH2 = 20 ml/min) and volume flow rate of CO changed from 14 to 45 (feed ratio H2/CO: 0.4–1.56). 13 experimental points were created according to each CC design (total 26 points). Because our designs were conducted in only two temperatures, temperature dependency of the kinetic parameters or estimated values of activation energies might be unreliable. So, we added two runs for 330 °C. The 28 experimental points which were generated from this manner are referred as 28 DOE points, in the text.

2.3. Reactor and intrinsic kinetic tests The kinetic experiments were carried out in a tubular fixed bed micro-reactor with a composite catalyst of 1 g. Schematic diagram of the reactor is shown in Fig. 1. The catalyst was diluted with quartz wool for heat dissipation during the CO hydrogenation process. The reaction was carried out in 1 cm diameter tube. The reaction tube was placed in a furnace equipped with temperature controller (JUMO IMAGO 500 Co.). Prior to the reaction, the catalysis was activated in situ using H2 (30 ml/min) and N2 (30 ml/min) gas mixture at 400 °C for 3 h. After the reduction, purified H2, CO and N2 gases were fed into the reactor. The flow rates of H2, CO and N2 were controlled by three mass flow controllers (BROOKS 5850E). It was possible to divert the feed mixture entering or the products leaving the reactor to the GC for analysis. The feed and product gases were analyzed for CO and C hydrocarbons by an on-line GC model UNICAM Pro GC +(THERMO ONIX Co.). The GC has three channels; hydrocarbon channel equipped with a capillary column type CSAlumina (length 30 m, diameter 0.53 mm), hydrogen channel consisted of two packed columns, Haysep Q, 60– 80 mesh (length 1.5 m, diameter 0.125 in.) and MolSieve 5A (length 2 m, diameter 0.125) and permanent gases channel equipped with two columns, Haysep QS, 60–80 mesh (length 3 m, diameter 0.125) and MolSieve 5A (length 2 m, diameter 0.125). The hydrogen and permanent channels are connected to a TCD and the hydrocarbon channel is connected to an FID. The catalysts were extremely fine particles, so that intra-particle diffusion could be neglected. It was also observed that the conversions remain unchanged during the constant contact time but flow rates of identical feed composition vary, so the external mass transfer resistance was negligible [20,21].

Fig. 1. Schematic of the reactor for FT reaction. 1—gas cylinders, 2—pressure regulators, 3—niddle valves, 4—simple valves, 5—Mass flow controllers, 6—digital pressure gauges, 7—pressure gauges, 8—one-way valve, 9—mixing chamber, 10—ball valve, 11—furnace, 12—reaction zone, 13—temperature controller, 14—resistant temperature detector, 15—exchanger, 16—liquid trap, 17—back pressure regulator, 18—flow meter, dewetter, gas chromatograph, hydrogen generator.

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Table 1 The kinetic data according to the CCD, in two different temperatures collected in a fixed bed micro reactor. VH2, VCO and VN2 are the volume flow rates of H2, CO and N2, respectively. P°CO and P°H2 are the input pressures, XCO represents CO conversion and XH2 is hydrogen conversion. Point

T

P

VH2 (ml/min)

VCO (ml/min)

VN2 (ml/min)

Ratio (H2/CO)

P°CO (bar)

P°H2 (bar)

XCO

XH2

rCO (mol/g s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

320 320 320 320 320 320 320 320 320 320 320 320 320 340 340 340 340 340 340 340 340 340 340 340 340 340 330 330

4 4 6 6 6.8 4 4 4 2 2 1.17 4 4 4 6 4 6.8 1.17 4 4 4 4 2 4 6 2 1 2

16 30 40 20 30 30 30 30 20 40 30 44 30 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 20 20 20 20 20 33 20 20 20 20 20 45 20 33 13 15 14 20 20

39 25 15 35 25 25 25 25 35 15 25 11 25 35 22 35 35 35 35 35 10 35 22 42 40 41 35 35

0.8 1.5 2 1 1.5 1.5 1.5 1.5 1 2 1.5 2.2 1.5 1 0.6 1 1 1 1 1 0.4 1 0.6 1.56 1.4 1.4 1 1

1.067 1.067 1.600 1.600 1.813 1.067 1.067 1.067 0.533 0.533 0.312 1.067 1.067 1.067 2.640 1.067 1.813 0.312 1.067 1.067 2.400 1.067 0.880 0.693 1.200 0.373 0.267 0.533

0.853 1.600 3.200 1.600 2.720 1.600 1.600 1.600 0.533 1.067 0.468 2.347 1.600 1.067 1.600 1.067 1.813 0.312 1.067 1.067 1.067 1.067 0.533 1.067 1.600 0.533 0.267 0.533

2.8 5.5 9.8 4.8 8.5 5.2 4.9 5 1.27 3.85 1.1 7.1 4.6 6.1 8.5 7.1 9.8 3.9 6.5 7.2 5.2 5.8 3.3 7.5 9.2 5.1 2.3 3.4

8.54 9.28 12.21 11.88 13.96 8.76 8.28 8.46 3.09 4.85 1.78 8.07 7.81 15.19 33.43 17.72 24.52 9.45 16.18 17.86 27.83 14.48 13.57 12.56 17.79 9.08 7.34 12.23

1.507E−06 2.960E−06 7.911E−06 3.875E−06 7.776E−06 2.798E−06 2.637E−06 2.691E−06 3.417E−07 1.036E−06 1.732E−07 3.821E−06 2.476E−06 3.283E−06 1.132E−05 3.821E−06 8.966E−06 6.139E−07 3.498E−06 3.875E−06 6.296E−06 3.121E−06 1.465E−06 2.624E−06 5.570E−06 9.606E−07 3.094E−07 9.179E−07

The catalytic test data is represented in Table 1. To assure the stability of the catalyst, the experiments of central points of the designs were repeated in the middle and at the end of the experiments. By using the differential system we can calculate the rate of CO disappearance

Y x

The test variables, in the model equation are coded values according to Eq. (3):

mcat X ¼ CO FCO −rCO

ð1Þ xi ¼

XCO − rCO mcat FCO

conversion of CO rate of CO disappearance (mol/g s) catalyst mass (g) molar stream of CO (mol/s).

2.4. Construction of RSM models The CO and Hydrogen (H2) conversions which were collected according to CCD (Table 1) were used to fit the following equations by means of regression analysis.

n X i¼1

Bi xi þ ∑ Bij xi xj þ ij

n X j¼1

2

Bjj x

j



Xi −Xi ΔXi

ð3Þ

where xi is coded value of the ith independent variable, Xi is the un-coded value of the ith independent variable, X⁎i is the un-coded value of the ith independent variable at the center point and ΔXi is the step change value.

Kinetic equations can be based on the overall synthesis gas consumption (−rH2+CO = −rCO − rH2), which is independent of the WGS equilibrium, or based on CO consumption for hydrocarbon products −rFT = −rCO − rWGS. During the kinetic tests, no significant amount of CO2 was observed which indicates that WGS reaction didn't play a significant role. The results of the other literatures also indicate that the WGS reaction could be negligible on Fe–Co catalysis especially in low conversions and equal to high Co/Fe ratios [22,23].

Y ¼ B0 þ

predicted response (conversions) test variables(total pressure and feed ratio) and B: regression coefficient.

ð2Þ

2.5. Construction of ANN models The feed forward neural networks are common in modeling of complex/nonlinear behaviors [24]. These networks consist of input, hidden and output layers. A layer consists of processing elements called neuron or nodes. Neurons are interconnected to the hidden and output layers. The number of neurons in input layer is equal to the number of input factors (variables); the number of neurons in output layer is equal to the number of responses or targets. The number of neurons in the hidden layer must be selected so that the best fit of experimental data would be achieved. The information contained in the input layer is mapped to the output layer through the hidden layers. The strength of the connections among the neurons is called the weight. A neuron accepts a weighted set of inputs with a bias. Then sum of weighted inputs with a bias is subjected to the activation function [6,11,24]. The experimental points must be employed to train the network with suitable learning algorithms. The learning algorithm adjusts the values of connection weighting coefficients of the processing nodes, by minimizing of the possible errors of the network output

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and target. The training process continues until the network output matches the target (experimental data). The performance of the ANN-based modeling/simulation is evaluated by a regression analysis of the network outputs (predicted parameters) and corresponding targets (experimental values). Several criteria may be used to measure the network performance [24]. In this study, a feed forward one hidden layer neural network was developed with 4 input neurons and one output neuron that the input neurons included pressure, hydrogen flow rate, CO flow rate and temperature. The output neuron includes conversion. The 28 experimental points collected according to the DOE were used to train the network. The number of neurons in the hidden layer was adjusted to create the best fit. To avoid over fitting [24], the 28 experiment points were divided to three sections such as training, validating, and testing. The validation data is used to measure the network generalization and halts training when generalization stops improving. The testing data has no effect on training and so provides an independent measure of network performance. The Levenberg–Marquardt (LM) optimization algorithm was used to train and the performance function was the squared average of output and targets (MSE). The sensitivity analysis, partial derivatives, input perturbation and “weights” methods are widely used in ANN literatures to provide explanatory insight into the contributions of the independent variables in the prediction process of the ANN model [25]. In the present study, the following simple equation proposed by Hattori [26] was used to calculate a contribution of the ith input data to kth output data (Cik) which is expected to represent the relative importance of each input data of the weightings set.       Cik ¼ ∑ aij  bjk   j 

ð4Þ

where, aij and bij are the weightings of connecting links of the ith input and jth hidden unit and jth hidden unit and kith output unit. 2.6. Generation of simulating data The models resulted from response surface methodology referred to the response surface models or RSM, were used to produce 70 simulated points for CO and H2 conversions as follows; The RSM model based on Central Composite Design 1 generates several simulated conversions at a constant temperature of 320 °C, constant VCO of 20 ml/min, and different values of VH2 and reactor pressure in the range of 16–44 ml/min and 1–7 bar respectively. The RSM model based on CCD2 produces several simulated conversions at a constant temperature of 340 °C, constant VH2 of 20 ml/min, and different values of VCO and reactor pressure in the range of 13–45 ml/min and 1–7 bar, respectively. In these conditions, CO conversions in the range of 1–14% were simulated. The purpose is to investigate the potential of RSM simulated data in kinetic study and the capability of response surface methodology in prediction of CO conversions. These conversions were used to obtain the experimental kinetic rates of CO disappearance in the differential reactor system (Eq. (1)) and calculate the pressure of hydrogen and CO in the output of the reactor. The simulated experimental data was then employed in the curvefitting process to estimate the parameters of LH rate equations. The artificial neural network models were also employed to generate 138 simulated data in different reactor pressures and also CO and H2 flow rates. To generate these points several CCD designs were developed in which temperature, reactor pressure and feed ratio (H2/CO) changed in the range of 320–340 °C, 1–7 bar and 0.4–2. In these conditions, CO conversions in the range of 1–13 were simulated. The ANN simulated conversions were then used to obtain the kinetic rates of CO disappearance in CCD points.

2.7. Derivation of kinetic rate equations based on LHHW approach According to Adesina [27], the review of kinetic studies of FTS on monometallic catalysis shows that the rates of CO disappearance are generally expressed by the Langmuir–Hinshelwood (LH) expression of the type: rCO ¼

kPγH2 PλCO : n X c d 2 ½1 þ ai PH2 PCO 

ð5Þ

i¼1

The LH rate equations are usually derived from the sequence of assumed elementary adsorption/surface reactions by several critical assumptions such as quasi-equilibrium, rate determining steps (RDS), and most abundant reactive intermediate (MARI) [28]. In the Eley– Rideal type rate equations it is assumed that there is one reactant in the gas phase [29]. For complicated catalytic reactions such as Fischer–Tropsch with various kinds of intermediates and elementary surface reactions, derivation of LH models is a challenging subject. Therefore, providing information about the reactions as well as adsorption processes on catalytic surface may be extremely helpful. UBI–QEP methodology appeared to be an efficient approach which is capable of calculating the binding and activation energies on transition metals, through relatively simple algebraic relations [30]. The input data is the surface structure, atomic binding energies and gasphase energy bonds which are available from experiments or DFT/ ab initio calculations. The theory of UBI–QEP and formulations are available in the reference publication [30]. The formulation of some bimetallic catalysts such as Fe–Co needs some modifications [30]. We used these relations to calculate the binding energies of species and activation barriers of surface reactions involved in the FT reaction. The calculations were conducted in zero coverage. In derivation of kinetic model rates, the elementary reactions with very high and unreal energy barriers were eliminated in the kinetic studies and were not used. The RDS assumption was used for ones with sufficient high activation barriers. Therefore, LH (and ER) approach was conducted more efficiently and the number of candidate rate equations reduced impressively. From the set of the candidate rate equations, the following rate equations were able to fit the experimental data sets and our UBI–QEP criterions: rCO ¼

k0 e−E=RT PCO PH2 ð1 þ aPCO Þ2

ðIÞ

rCO ¼

k0 e−E=RT PCO PH2 : 1 þ aPCO

ðIIÞ

The pre-exponential factor k0 and activation energy of the lumped models are apparent values. Eq. (I) is the Yates–Satterfield model [31] equation. From a mechanistic point of view to the LH rate equations, Eq. (I) may be derived from the following elementary reaction sequences: molecularly adsorption of CO and dissociative adsorption of hydrogen molecule. Then, hydrogen atom (H*) reacts with CO* to give CHO*. The CHO* then reacts with another H* to give CH* and OH* or CH2* and O*. Assuming that CO adsorption, the hydrogen adsorption/dissociation and formyl (CHO*) formation steps to be in quasi-equilibrium and surface reaction of CHO*+H* to be RDS, we may derive the rate Eq. (I). The details of derivation of the LH rate equations are presented in Appendix A (Scheme 1). Eq. (II) is the Eley–Rideal type. The proposed mechanism for the rate Eq. (II) may follow as: CO adsorbs molecularly. H2 from gas phase reacts directly with CO* to give CH2* and O*. The direct reaction of gas phase hydrogen with adsorbed CO has been proposed by Dry et

M. Shiva et al. / Fuel Processing Technology 106 (2013) 631–640

13.3

Table 2 RSM models from regression analysis of CO and H2 conversions in two different temperatures, in 320 °C ratio (VH2/VCO) = 0.8–2, pressure = 1–7 bar, VCO = 20 ml/min, V = 75 ml/min. In 340 °C, ratio (VH2/VCO) = 0.4–1.6, pressure = 1–7 bar, VH2 = 20 ml/min, V = 75 min/min.

XCO (320 XH2 (320 XCO (340 XH2 (340

°C) °C) °C) °C)

R2

R2adj

F-Regression (P-value)

F-lack of fit (P-value)

98.9 98.4 96.3 97.4

98.1 97.2 93.7 95.6

123.8 (0.000) 85.08 (0.000) 36.59 (0.000) 53.46 (0.000)

1.16 1.25 0.16 0.94

y = 10832x - 5.0363

13.2

R2 = 0.9969

13.1 13

-lnk

Model

635

12.9

(0.429) (0.403) (0.919) (0.502)

12.8 12.7

RSM Models (coded units) used for simulation: XCO(320 oC) = 5.0217 + 3.5181P + 2.3977Ratio − 0.1995P2 − 0.0661Ratio2 + 1.1922P × Ratio XH2(320 oC)= 8.4902 + 5.8876P + 0.2597Ratio − 0.6643P2 − 0.2605Ratio2 − 0.7016P × Ratio XCO(340 oC) = 6.4894 + 3.1297P + 1.0403Ratio − 0.2734P2 − 0.2140Ratio2 − 0.5612P × Ratio XH2(340 oC)= 16.489 + 8.993P − 7.564Ratio0.799P2 + 3.690Ratio2 − 5.686P × Ratio

12.6 0.00162 0.00163 0.00164 0.00165 0.00166 0.00167 0.00168 0.00169

1/T Fig. 2. The calculation of activation energy of Eq. (I) using the Arrhenius law. The activation energy has been obtained from k values in 5 different temperatures available from ANN simulated data.

- Data set 4(138ANNsim): 138 simulated rate data based on ANN model. al. [32]. Assuming that the former reaction is in equilibrium and the later reaction is RDS, the rate Eq. (II) is obtained. The details of the derivation are also presented in the appendix (Scheme 2). 2.8. Curve-fitting and estimation of the parameters of CO disappearance in rate equation Four data sets were used to fit the LHHW type rate Eqs. (I) and (II) by means of nonlinear regression: - Data set 1(40RND): 40 experimental rate data collected without design of experiments in the range of P = 1–7 bar and feed ratio of 0.4–2.2 in temperatures of 320 °C, 330 °C and 340 °C. From these experimental points, the CO conversions in the range of 2– 13% were obtained. - Data set 2(28DOE): 28 experimental rate data collected with Central Composite Designs, CCD, and two additional points. From the DOE points, the CO conversions in the range of 2–10% were obtained. - Data set 3(70RSMsim): 70 simulated rate data based on RSM models. Table 3 The details of neural network which have been used for simulation.

The goal was to compare the efficiency of these data sets in the curve-fitting process and to investigate the capability of RSM and ANN models for accurate demonstration of conversions. In addition, the data set 4 was used to solve the correlation problem of the model rate parameters. In the process of data fitting it was observed that there is a high correlation between pre-exponential factor (frequency factor) and activation energy. To avoid this problem we employed the ANN simulated data and obtained the rate constant, k, in several temperatures (320, 325, 330, 335 and 340 °C). Then the activation energy was calculated from a linear form of the Arrhenius equation and this value was used as initial guess in rate equation of the curve-fitting software. 2.9. Assessing the quality of linear/nonlinear regression The following statistics were used in kinetic study of assessing the quality of linear (RSM) and nonlinear regression as well as comparing the adequacy of the models and data sets. The correlation coefficient, n  2 X yi;obs −yi;moldel 2

2

R : R ¼ 1−

i¼1

Number of neurons in hidden layer: 5, performance function: MSE, learning algorithm: LM Sample size: training = 18, validation = 4, testing = 4 Hidden layer weights 0.1711 −0.1580 −0.0198 1.1630 0.8825 2.2448 −0.4696 −0.7284 1.5802 1.3993

−1.0479 −1.2829 0.9115 −0.9829 −0.4435

:

ð6Þ

i¼1

Adjusted R 2:

2.5020 1.7033 −1.9041 0.5441 −0.4393

2 Radj

¼ 1−

  2 1−R ðn−1Þ n−m

:

ð7Þ

Mean Absolute Relative Residual, MARR:

Hidden layer bias 2.2726 −1.6465 0.4434 0.1297 2.3258 Output layer weights 0.1163

n  2 X yi;obs −y

    n yi;obs −yi;model  1 MARR ¼ ∑i    100:  n yi;obs

ð8Þ

Relative Variance, Srel: 0.2457

0.0235

−1.0246

0.4384

Output layer bias −0.0207 MSE (training) = 7.86715E−2 R2 = 0.992987 MSE (validation) = 2.00731E−1 R2 = 0.984422 MSE (testing)= 3.44769E−1 R2 = 0.977126 R2 (total) = 0.9894 Relative importance (RI). C (temperature): 1.2098; C (Pressure): 1.6800; C (VH2): 0.3969; C (VCO): −0.0854.

Srel ¼

n X yi;obs −yi;model yi;obs i

!2

!0:5 1  100: n−m

ð9Þ

Root mean of standard deviation, Rmsd, Rmsd ¼

n  2 1 X 2 ½ yi;obs −yi;model  n i

ð10Þ

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Table 4 The estimation of model parameters (coefficients) for CO rate Eqs. (I) and (II) (C.L.: confidence limits).

40RND (Data Set 1) 28DOE (Data Set 2) 70RSMsim (Data Set 3) 138ANNsim (Data Set 4)

Eq. Eq. Eq. Eq. Eq. Eq. Eq. Eq.

(I) (II) (I) (II) (I) (II) (I) (II)

k0 (C.L.)

E (J/mol) (C.L.)

a (C.L.)

R2

MARR

Srel

Rmsd

172.6989 ± 9.913119 182.7999 ± 1.288872 101.9995 ± 0.724609 101.9993 ± 0.722425 172.6959 ± 4.20551 182.7994 ± 0.1780235 102 ± 0.431582 102.9992 ± 1.221123

9.067E+04 ± 286.7227 9.092E+04 ± 35.3538 8.80E+04 ± 35.57585 8.80E+04 ± 35.46876 9.075E+04 ± 121.3093 9.099E+04 ± 4.887481 8.80E+04 ± 21.18007 8.80E+04 ± 59.00538

0.0437986 ± 0.0142668 0.0962512 ± 0.0046904 0.04369 ± 0.002049 0.095829 ± 0.004497 0.0391564 ± 0.0051618 0.0866778 ± 0.0005605 0.039277 ± 0.001328 0.085777 ± 0.008041

0.9894048 0.9894436 0.9917432 0.991793 0.9957945 0.9958643 0.979761 0.979827

24.58 24.58 13.3 13.29 16.95 17.01 13.6 13.53

48.00 48.18 21.8 21.8 55.24 55.54 30.81 30.94

4.425E−08 4.417E−08 4.73E−08 4.72E−8 2.102E−08 2.084E−08 2.8E−08 2.79E−08

Table 5 The estimated model parameters (coefficients) for CO rate Eq. (III)—power law obtained from curve-fitting with data set 2. The same results have been obtained from other data sets. Eq. (III) Power law

k0 (C.L.)

E (J/mol) (C.L.)

α (C.L.)

β (C.L.)

R2

R2

MARR

Srel

Rmsd

28DOE (Data Set 2)

96.03335 ± 53.5225

8.786E+04 ± 2790.373

0.77

1

0.9887848

0.9887848

15.77

28.8

5.479E−08

where, yi,obs and yi,model are experiment and calculated points, respectively. n is the number of observations and m is the number of parameters. F-regression statistic (analysis of variance): F ¼ MSR=MSE:

ð11Þ

F-lack of fit (analysis of variance): F ¼ MSLF=MSPE

ð12Þ

where MSR, MSE, MSLF and MSPE are Mean sum of regression, Mean square error, Mean Square Lack of Fit and Mean Square Pure Error, respectively. 3. Results and discussion 3.1. RSM and ANN models The RSM models for CO and hydrogen conversions are presented in Table 2. The important statistics are also included in the table. The high value of R2 and R 2adj for CO and H2 conversion models in two different temperatures shows that the models are surely capable of fitting experimental data. Table 2 also presents the ANOVA results of RSM models. The ANOVA results show that the regression is significant but the lack-of-fit is not significant because the calculated F-regression is greater than the F-statistic value (P value lower than 0.05), whereas the F-ratio of lack of fit to pure error is less than the F-statistic value (P values higher than 0.05) at a 95% confidence level.

Fig. 3. Surface plot of CO conversion (XCO) versus temperature and feed ratio.

The ANOVA results indicate that the second order polynomial equations are statistically significant and adequate to represent the actual relationship between inputs (pressure and feed ratio) and outputs (conversions). It is important to note that the estimated parameters for each polynomial equation (Table 2) are different and depend to the temperature. The details of the ANN model are presented in Table 3. The regression R 2 values measure the correlation between outputs and targets. The value 1 means close relationships. The high values of R 2 (0.99) have been obtained for the network. It is possible to determine the relative importance (RI) of temperature, pressure, CO and hydrogen flow rates on the conversions, through Eq. (4). The relative importance values (C) are presented at the bottom of Table (3). The relative importance (C) values clearly show that temperature, pressure and hydrogen flow rate have an increasing effect on CO conversion while CO flow rate has a decreasing effect. The higher value of C for pressure indicates its higher effect on CO conversion. The results clearly show that the ANN model may not be referred as a black box. Similar to RSM models, this neural network was used to generate the simulated data. It is important to note that the ANN model can simulate the conversions in different temperatures, while the RSM can only give simulations in two temperatures (320 and 340 °C). 3.2. Kinetic study and comparison of data set efficiency/accuracy In the process of data fitting, it was observed that there is a high correlation between pre-exponential factor and activation energy. To avoid this problem in the case of Eq. (I), we used 138 simulated ANN data set 4

Fig. 4. Surface plot of CO conversion (XCO) versus temperature and pressure.

M. Shiva et al. / Fuel Processing Technology 106 (2013) 631–640

637

Table 6 The estimation of model parameters (coefficients) for CO rate Eq. (IV) (modified Arrhenius) from 28DOE set and 138ANNsim data set (C.L.: confidence limits). Tref (K) 28DOE (Data Set 2) 138ANNsim (Data Set 4)

kTref (C. L.)

E (J/mol) (C. L.)

k0 (C. L.)

a (C. L.)

R2

MARR

Srel

Rmsd

600

2.25E−6 ± 1.79E−07

9.28E+04 ± 7723.321

269.8024 ± 21.44737

0.050062 ± 0.024554

0.992241

12.8824

21.18

4.58E−08

600

2.22E−06 ± 1.06E−07

8.95E+04 ± 5580.297

136.6472 ± 6.519008

0.040738 ± 0.015722

0.9798

10.95

27.64

2.8E−08

to calculate the rate constant in several temperatures. Then the activation energy was determined via the Arrhenius law (k= k0e−E/RT) and plot of ln(k) versus −E/RT (Fig. 2). The estimated k values in the five temperatures are 1.79E−06 (320 °C), 2.07E−06 (325 °C), 2.49E−06 (330 °C), 2.80E−06 (335 °C) and 3.24E−06 (340 °C). The calculated activation energy was 90.06 kJ/mol. In the next step, the four data sets (Section 2.8) have been used to fit the LH rate Eq. (I) while it was attempted to keep the value of activation energy about 90 kJ/mol. The results are presented in Table 4 which involves the estimated model parameters and confidence limits for each data set. The same procedures were employed for Eq. (II). The calculated k values from the simulated ANN data in the five temperatures of 320, 325, 330, 335 and 340 °C were 1.79E−06, 2.08E−06, 2.49E−06, 2.80E−06 and 3.25E−06, respectively. The calculated activation energy from the Arrhenius relation was 89.95 kJ/mol. Then, the experimental and simulated data (four data sets, Section 2.8) were used to fit the rate equation. The results are presented in Table 4. It is important to note that the kinetic parameters (kinetic constant and activation energy) are apparent values. The high values of R2 in rate Eqs. (I) and (II) indicate their capability of fitting the data sets. The calculated activation energies from all four data sets are similar; 88–91 kJ/mol for rate Eqs. (I) and (II). The nearly similar adsorption coefficients and pre-exponential factors have been obtained for all data sets. Any confidence limits are acceptable. However, the smaller values of MARR and Srel are obtained for 28DOE and 115ANNsim data sets. Important conclusions: - The presented RSM models are able to generate accurate simulated data to conduct kinetic study. - The constructed neural network can simulate the experimental domains very well and ANN simulated data are more effective than RSM simulated data, according to the Srel and MARR values. - The 28DOE data set is sufficient to conduct curve-fitting and its efficiency is equal to the 138ANN simulating data, because the similar values of R 2, Srel and MARR have been obtained from both data sets. However, as mentioned above, the DOE must couple with ANN to avoid correlation between model parameters. - The curve-fitting with simulated data presented closer values of confidence limits.

According to the R 2, Srel and MARR, there are no significant differences between the capability of rate Eqs. (I) and (II) to fit data sets. 28DOE data set and 138ANNsim data set 4 were also used to estimate the kinetic parameters of a power law rate equation described by expression (III): −E=RT α β PCO PH2 :

rCO ¼ k0 e

ðIIIÞ

The values 0.75 and 1 were obtained for α and β, respectively. The estimated kinetic constants (activation energy and preexponential factor) are presented in Table 5. The activation energy is 87.86 kJ/mol. The value of α = 1 (linear dependency on H2 partial pressure) is common in FT kinetic rate equations. However, β value is relatively larger than conventional powers of CO partial pressure in rate equations (− 1 to 0.5) for Fe and Co catalysis. The results of this study show that our DOE is very efficient in kinetic studies. However, this DOE must be coupled with an ANN model to estimate reliable kinetic parameters. It was also possible to employ the central composite designs with three factors (T, Ratio and Pressure) which led to the fewer runs. With three factors, a central composite design generates 20 runs (16 runs + 4 replications).To develop an artificial neural network, the training test must be divided into 3 parts; training, validating and testing (to avoid over fitting). Assuming that 30% of the data are employed for validating and testing, only 70% remain for training. So, according to the replicated points only about 10–12 different points are used for network training and the chance of success in training of the network is reduced. However, it is important to note that our study has been conducted in a differential reactor and high CO conversions must be avoided in this kinetic study approach. So, searching the suitable range for the three variables is more difficult rather than the two variables when the variable levels are dictated by experiment designs and in the presence of differential reactor limitations. Moreover, one of the purposes of the present study was to investigate the capability of simulated points through RSM models in kinetic studies. Indeed, development of an RSM model requires very careful collection of experimental design points and only one incorrect data can frustrate the RSM modeling. Finally, we were not sure whether the temperature dependency of CO conversion could be modeled successfully by polynomial equations of RSM methodology. To demonstrate the effect of temperature on CO conversion, we used a combined DOE/ANN. A central composite design with three factors

Table 7 Activation barriers which are obtained from UBI–QEP approach on Fe–Co catalyst model in zero coverage with the assumptions of three fold hollow binding of H, C and O on the surface, L = number of bonds to Co, L′ = number of bonds to Fe, and the activation energy values have unit: kJ/mol. The binding energies of C, O and H on Co and Fe are taken from reference [26] that are in broad agreement with experiment.

1 2 3 4 5 6 7

Surface reaction

Ef (kJ/mol) (forward) L = 2;L′= 1

Er (kJ/mol) (reverse) L= 2;L′ = 1

Ef (kJ/mol) (forward) L = 1;L′ = 2

Er (kJ/mol) (reverse) L = 1;L′= 2

CO(g) + * = CO* H2 + 2* = 2 H* CO* + H* = CHO* CHO* + H* = CH* + OH* CHO* + H* = CH2* + O* CO* + * = C* + O* H2 + CO* = CH2* + O*

0 8.34 132.33 133.88 106.5 152.8 116.24

167 91.7 0 41.13 103.2 143 155.5

0 2 135 127.3 106.5 137.77 109.1

188.8 102.2 0 57.8 117.9 172.25 157.7

638

M. Shiva et al. / Fuel Processing Technology 106 (2013) 631–640

Table 8 The estimation of model parameters (coefficients) for CO rate Eq. (VI) (C.L.: confidence limits). ′

Data set

k0

Eapp (J/mol)

k0

E′ app (J/mol)

a

R2

Rmsd

28DOE

18.12 ± 0.144

8.041E+04 ± 0.04

1.01E+04 ± 559.57

1.198E+05 ± 0.277

0.0167 ± 0.0019

0.9927

4.446E−08

(reactor temperature, pressure and feed ratio) in the range of the under study variables was prepared and the CO conversion of the design points was obtained from the ANN model. The surface plots are presented in Figs. 3 and 4. The plots clearly show the dependency of CO conversions on temperature, pressure and feed ratio. So, the DOE/ ANN approach may be useful for efficient demonstration of the behaviors. 3.3. Application of the modified form of the Arrhenius equation While the linearization form of the Arrhenius relation and combined UBI_QEP/LHHW that we used in our previous study [33] would give suitable value of activation energy as initial guess in the Levenberg– Marquardt algorithm, there is still some uncertainty in the process of data fitting specially in the value of pre-exponential factor (k0). Recently, Schwaab has collected some modified forms of the Arrhenius equation [34]. Here, we used one of them to estimate the kinetic parameters of the LH rate Eq. (I):

rCO ¼

   PCO PH2 kTref exp − RE 1T − T1 ref

ð1 þ aPCO Þ2

ðIVÞ

where,   E kTref ¼ k0 exp − RTref

′

rCO ¼

and, Tref may be obtained from the following relation proposed by Veglio et al. [35]: NE 1X 1

1 ¼ Tref n n

i¼1

Ti

However, it is important to note that the activation barriers (Table 7) have been calculated in zero surface coverage. The elementary reactions involved in the derivation of the rate Eq. (II) are also energetically possible. The assumption of RDS may be acceptable, because the calculated UBI–QEP activation barrier is high (see Table 7–reaction 7). It is important to note that in our previous study [33] we have proved that the rate Eq. (II) is able to satisfy our LHHW/UBI_QEP criterion. In recent years, the LH rate equations with assumption of two RDS steps have been developed for Fe and Fe–Co FT catalysis [13,33]. We investigated some of these type rate equations. The high correlation of the parameters was the main problem. To overcome this problem, a new approach has been conducted based on UBI–QEP calculations in different CO coverage. The coverage would significantly affect the binding energies and activation barriers and must be included in UBI/QEP calculations. It's possible to deliver an initial guess for apparent activation barriers of the LH rate equations from coverage-modified UBI–QEP calculations and to overcome the difficulties associated with the parameter correlations. This subject has been studied elsewhere [33]. In our previous kinetic study we showed that the rate Eq. (I) and also the following rate equation are able to fit data set 28DOE and also satisfy our LHHW/UBI_QEP criterion to some extent:

ð13Þ

The number of experimental points, and Ti the temperature of each of experimental point.

With 28DOE data set we have Tref = 600 K. The estimated parameters are presented in Table 6. Using this form of the modified Arrhenius relation, it was observed that the parameter correlation is completely eliminated. The results in Table 6 indicate that 138ANNsim data set 4 presented closer confidence limits and also smaller MARR and Rmsd values.

ðVÞ

To complete the previous work, we have developed a software to calculate the apparent activation energies of different lumped kinetic rates (with one or two RDS) from combined LHHW/UBI_QEP formulations. The input variable of the software is the binding energies of C, O and H to the Fe and Co model catalyst in different coverage and the output is the binding energies of different FT species and activation energies of different FT elementary reactions as well as lumped apparent activation energies on Fe–Co model catalyst. The apparent activation energies are then used as the initial guess in curve fitting. Here, we complete our previous study with development of a novel kinetic rate equation that completely satisfies our LHHW/UBI_QEP criterion (i.e. the estimated apparent activation energies from nonlinear regression is similar to that obtained from the software):

3.4. Check the accuracy of the lumped models with UBI–QEP observations In Table 7, the calculated UBI–QEP activation barriers for elementary reactions used in derivation of the lumped kinetic models are presented. From the activation barriers, it is clear that the elementary reactions involved in derivation of rate Eq. (I) are energetically possible. We also presented the activation energy of CO dissociation. Zero activation barriers of molecular CO adsorption and its high value for dissociation of adsorbed CO suggest that to assume CO* as MARI is logical. Energy barriers for hydrogen dissociation on the surface are also significantly low (2–8 kJ/mol in Table 7). However, the presence of several routs for H* consumption on the catalyst surface indicates that H* may not be MARI [36]. Assumption of RDS for the reaction CHO* + H* = CH* + OH* or CHO* + H* = CH2* + O* is logical due to the high activation barriers of the reactions (see Table 7).

ke−E=RT PCO PH2 þ k′ e−E =RT PCO : ð1 þ aPCO Þ2

′

rCO ¼

ke−E=RT PCO PH2 þ k′ e−E =RT PCO 0:5 PH2 0:5 : ð1 þ aPCO Þ2

ðVIÞ

The details of mechanism and rate derivation have been presented in the appendix (Scheme 3). In the proposed mechanism, the adsorbed CO* also undergoes to dissociate into C* and O*. Then C* reacts with H* to give CH* and this step is assumed to be RDS beside CHO* + H* reaction. The estimated parameters of the LH rate Eq. (VI) are presented in Table 8. Again, we can see that our DOE is efficient in parameter estimation of rate equations which are based on two RDS reaction and would lead to the reliable estimated parameters in the lumped kinetic study.

M. Shiva et al. / Fuel Processing Technology 106 (2013) 631–640

4. Conclusions A new method based on coupling DOE with ANN simulating model in kinetic study of CO hydrogenation process was addressed. While the experimental designs conducted in two different temperatures (320 °C, 340 °C) and only two runs in 330 °C, it was possible to construct and train an artificial neural network based on the data and to employ the network to simulate not only more points in 330 °C but also a several addition points in different temperatures, pressures and ratios. With the aid of the ANN model it was possible to obtain the rate constants in several temperatures and estimate the activation barriers by the Arrhenius relation. This approach would overcome the correlation problem that exists when the experimental points are collected according to CCD. Our study showed that employment of DOE/ANN would decrease the experimental points required for kinetic studies. The very successful curve-fitting with data set 4(115ANN) also shows the power of the artificial neural network in demonstrating CO conversion dependency on the pressure, feed ratio and also temperature. The estimated apparent activation energies of the LH and ER rate equations were close to each other, 88–90 kJ/mol for all data sets. However, the apparent activation energy of the LH rate equation is able to meet micro kinetic criterion developed in the previous work. Finally, it was shown that 28DOE is able to estimate correctly the kinetic parameters of the rate equation which are developed based on two RDS step and the estimated apparent activation energies are in close agreement with that obtained from micro kinetic (UBI_QEP/LHHW) calculations. From a mechanistic view it was shown that the H-assisted CO dissociation may be the dominant pathway for monomer formation on Fe–Co catalysis and direct CO dissociation may also occur.

Nomenclature a adsorption constant in rate equations aij weightings of connecting links in Eq. (4) bjk weightings of connecting links in Eq. (4) B regression coefficients of Eq. (2) c,d constants in Eq. (6). Cik relative importance in Eq. (4) E, E′ activation energy of reaction (J/mol) Ef,Er activation energy of forward and reverse elementary reaction F statistics in ANOVA FCO molar stream of CO (mol/s) k rate constant in Eqs. (6), (I), (II), (II) and (IV) k′ rate constant in Eq. (IV) k0 preexponential factor L,L′ number of bonds to Fe and Co in Fe–Co model catalysis m number of parameters mcat catalyst mass (g) P total pressure in the reactor (bar) PCO CO pressure (bar) PH2 hydrogen pressure (bar) − rCO rate of CO appearance (mol/g s) V volume flow rates (ml/min) R2 correlation coefficient R 2adj adjusted correlation coefficient Srel relative variance Tref reference temperature in Eq. (13) XCO CO conversion (%) xi coded value of the ith independent variable Xi uncoded value of the ith independent variable uncoded value of the ith independent variable at the center X⁎i point ΔXi step change value Y predicted response

yi,obs yi,model

639

experiment (observed) points calculated points

Greek letters α power of CO pressure in power law form of rate equation (Eq. (III)) β power of H2 pressure in power law form of rate equation (Eq. (III)) γ,λ constants in Eq. (4)

Abbreviation ANN artificial neural network ANOVA analyze of variance CCD Central Composite Design DOE design of experiments MARR mean absolute relative residual MARI most abundant reactive intermediate MSE mean square error RDS rate determining step RI relative importance Rmsd Root mean of standard deviation RSM response surface methodology DFT density functional theory UBI–QEP unity bond index quadratic exponential potential

Acknowledgments The authors would like to acknowledge the financial and instrument support from the University of Sistan and Baluchestan, Iran. Appendix A Derivation of LH/ER model rates.

Scheme 1. Derivation of kinetic model.

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Scheme 2. Derivation of kinetic model.

Scheme 3. Derivation of kinetic model.

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