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. 2021 Oct 4;6(41):27183–27199. doi: 10.1021/acsomega.1c03851

Kinetic Study of Product Distribution Using Various Data-Driven and Statistical Models for Fischer–Tropsch Synthesis

Yixiao Wang †, Jing Hu †, Xiyue Zhang †, Abubakar Yusuf †, Binbin Qi ‡, Huan Jin §,*, Yiyang Liu ∥, Jun He †, Yunshan Wang ⊥, Gang Yang ⊥, Yong Sun †,#,*
PMCID: PMC8529696  PMID: 34693138

Abstract

graphic file with name ao1c03851_0011.jpg

Three modeling techniques, namely, a radial basis function neural network (RBFNN), a comprehensive kinetic with genetic algorithm (CKGA), and a response surface methodology (RSM), were used to study the kinetics of Fischer–Tropsch (FT) synthesis. Using a 29 × 37 (4 independent process parameters as inputs and corresponding 36 responses as outputs) matrix with total 1073 data sets for data training through RBFNN, the established model is capable of predicting hydrocarbon product distribution i.e., the paraffin formation rate (C2–C15) and the olefin to paraffin ratio (OPR) within acceptable uncertainties. With additional validation data sets (15 × 36 matrix with total 540 data sets), the uncertainties of using three different models were compared and the outcomes were: RBFNN (±5% uncertainties), RSM (±10% uncertainties), and CKGA (±30% uncertainties), respectively. A new effective strategy for kinetic study of the complex FT synthesis is proposed: RBFNN is used for data matrix generation with a limited number of experimental data sets (due to its fast converge and less computation time), CKGA is used for mechanism selections by the Langmuir–Hinshelwood–Hougen–Watson (LHHW) approach using a genetic algorithm to find out potential reaction pathways, and RSM is used for statistical analysis of the investigated data matrix (generated from RBFNN through central composite design) upon responses and subsequent singular/multiple optimizations. The proposed strategy is a very useful and practical tool in process engineering design and practice for the product distribution during FT synthesis.

1. Introduction

The year 2020 witnessed how weak the supply chain is in how goods (the empty grocery-store shelves and the out of stock of surgical masks) are sourced, distributed, and where they are stored.1 These surely ring the alarm bell for the acceleration of long-term shifts in the supply chain on various aspects.2 Energy security issues are also increasing as crude oil prices are intensively volatile during the pandemic as thousands of airplanes are grounded, usual commuters are working from homes, and popular tourist resorts have become deserted.3,4 Although the energy demand of using crude oil is significantly falling, the petrochemical production from crude oil is still in demand as the pendulum of daily life does not stop and the consumption of fast-moving consumer goods (FMCG) bounces back quickly.5,6 The challenging issues of importing crude oils from oil-rich countries through the weak supply chain still exist.

The alternative technology of using syngas to produce hydrocarbons (both for bulk/fine chemical manufacturing and energy consumption) via one of the most commercial viable process (Fischer–Tropsch, FT synthesis) shows its merit in tackling thee thorny problems for many countries with poor oil reserve.7−10 The kinetic modeling of Fischer–Tropsch (FT) synthesis is the cornerstone for sizing and process design of gas to liquid (GTL) technology.11−13

Many scholars have proposed different modeling approaches to model the kinetics.14−16 These models include (i) data regression by formulating the minimization of the sum of the square of errors through a numerical Levenberg–Marquardt (LM) algorithm, (ii) a genetic algorithm (GA) to model the product distribution of FT synthesis such as using the Anderson–Schulz–Flory (ASF) method, (iii) an explicit model based on a second-order polynomial for correlating the critical operational parameters using a response surface methodology (RSM), and (iv) data-driven soft computation machine learning techniques such as using different artificial neuron networks (ANNs) in directly correlating the product distribution and critical operational parameters.17−24

With the advancement of computation techniques, computational power has become more economical and practical, and this looms new opportunities of combining different modeling techniques to analyze the complicated kinetic system such as FT synthesis.25−27 Until now, there have been very rare efforts in reporting the establishment of a robust paradigm that combines the advanced calculation systems for kinetic modeling of product distribution during FT synthesis.

To achieve this goal, large experimental data sets (a 29 × 36 matrix with total 1073 data sets, of which inputs were operational parameters such as temperature (T), pressure (P), gas hour space velocity (GHSV), and syngas ratio (SR), outputs were the rate of olefin/paraffin from C1–C15, respectively) were fed into different modeling techniques, namely, a radial basis function neural network (RBFNN), the comprehensive kinetics based on a genetic algorithm (CKGA) for the regional and global optimization using a genetic algorithm (GA) via the objective function (minimization of the sum of the square of errors) for mechanism selection using the Langmuir–Hinshelwood–Hougen–Watson (LHHW) approach, and the second-degree polynomial correlations through a response surface methodology (RSM), and explored and compared. Using additional validation data sets (15 × 36 matrix with total 540 data sets), the comparisons of these three techniques (RBFNN, CKGA, and RSM) were made, and a paradigm of accurate determination and prediction of reaction pathways and optimization of the system is proposed. To the best of the author’s knowledge, this new modeling strategy for product distribution during FT synthesis has been rarely reported before.

2. Theoretical Background

2.1. Radial Basis Function Neural Network (RBFNN)

Artificial neural networks (ANNs) are a well-developed and applied soft computing technique.28 With the advancement of computation technology, these soft computing techniques have recently remerged and begun to attract attention in both academia and industries for complex system modeling.29 Inspired from the neurological system and its basic elements for handling the information, ANNs are principally constructed as an input function x of the formal neuron i corresponding to the incoming activity (e.g., synaptic input) of the biological neuron; weight wi represents the effective magnitude of information transmission between neurons (e.g., determined by synapses), activation function zi = f(x,wi) describes the main computation performed by a biological neuron (e.g., spike rates), and the output function yi = f(zi) corresponds to the overall activity transmitted to the next neuron in the processing stream.30 With the learning phase on prior representative data as a robust training and testing step, this constructed learning technique is recognized to yield good predictions in many complex systems. From configurations of neural topologies and learning procedure perspectives, different ANNs have been constructed and deployed. Among them, the multilayer perceptron (MLP) and radial basis function neural networks (RBFNNs) are one of the well-formulated and rigorous ANNs.31,32 In this work, we choose the RBFNN as the machine learning algorithm for product distribution predictions during FT synthesis. The rationale of using RBF lies in its relative simple neuron architecture and efficient computation. The schematic diagram of RBFNN is shown in Figure 1.

Figure 1.

Figure 1

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Schematic diagram of RBFNN formulated from the Gaussian function.

The RBFNN is unique in its neuron topology, which only possesses one hidden layer representing its nodes Nh (including its bias term Nb); the Gaussian function is implemented in the network, which is formulated based on its center (cc) and the spread coefficient (d2). The general Gaussian function is expressed as follows

2.1. 1

The maximal function (as analytic, and their limit as x → ∞ is 0) value of a Gaussian function is found for zero activation, where the weight is zero. The function is even and defined as f(zi) = f(−zi). The function value decreases with increasing absolute value of activation. This decrease can be controlled by parameter ∑, whose larger values result in a decrease in function values with an increasing distance from the maximum (the function becomes “broader”). In calculations steps, the Euclidian norm is applied to assess the distance of a vector (x) between nominated center cc and the Gaussian function using the following expression

2.1. 2

where n is the dimension of the matrix and ccik is the nominated center in the tensor, and then it is incorporated with the Gaussian function; the Gaussian-RBFNN is defined as follows

2.1. 3

The cross-validation was employed for the evaluation of the statistical performance of the established model. The number of folds was set at 17, which results in exactly one data item being a test item during each fold iteration.33 The detailed procedures for cross-validation could be found from our previous literature reports.34 The mean square error (MSE) and mean average residual relation (MARR) can be calculated as follows

2.1. 4
2.1. 5

where Nexp is the number of experimental conditions and riexp and ri are experimental and calculated values of the rates of CO, paraffin (C1–C15), and olefin (C2–C15). All of the procedures were repeated twice.

2.2. Comprehensive Kinetics with the Genetic Algorithm (CKGA)

In this work, the procedure of model selection was divided into two stages. At a preliminary stage, 10 different mechanisms over 30 different models were explored for CO initiation against experimental data. These models include carbide,35 enolic,36 and CO insertion mechanism,37 respectively. The corresponding elementary steps of different mechanisms are shown in Table 1. Among these models, the direct hydrogenation of the carbide mechanism (M1) is found to be both physically and statistically relevant to the experimental results. Once the initiation mechanism is determined, the selected detailed carbide mechanism M1 was employed to further derive a comprehensive model. To account for the olefin to paraffin ratio (OPR), the 1-olefin readsorption model assuming weak Van der Waal’s interaction between olefin and the surface of the catalyst is deployed. Then, the activation energy for olefin readsorption can be written as16,38

2.2. 6

where Ere0 refers to the readsorption energy that is independent of the chain length (kJ·mol–1), Ere refers to the part of activation energy that is dependent on the chain length (kJ·mol–1), and n refers to the carbon number (n ≥ 4). Then, the rate constant can be expressed as follows39

2.2. 7
2.2. 8
2.2. 9

The rate constants of chain-length-dependent (CLD) readsorption and desorption can be expressed as follows

2.2. 10
2.2. 11
2.2. 12

where εc* and εc are the constants reflecting the part of activation energy of olefin readsorption and desorption with carbon number dependency and R refers to the gas constant (8.314 J·K–1·mol–1). For FT synthesis, the surface fraction of CO, H2, and On (the olefin chain-length-dependent readsorption is considering eqs 13–15, with n ≥ 4) can be expressed as follows

2.2. 13
2.2.
2.2. 14
2.2.
2.2. 15
2.2. 16

Calling quasi-steady-state assumption (QSSA)

2.2. 17

where

2.2.
2.2. 18

where

2.2.
2.2. 19

The QSSA is applied with a chain number longer than 4

2.2. 20

The approximate second-order polynomial expression using θCH2S as variables can be solved as

2.2. 21

where

2.2.
2.2.
2.2.

The corresponding root could be solved as

2.2.

Using QSSA in alkylidene-1-S

2.2. 22

where

2.2.

For the alkylidene-2-S intermediate

2.2. 23

where

2.2.

For alkylidene-3-S

2.2. 24

where

2.2.

For the alkylidene-n-S intermediate

2.2. 25

The following linear algebra will be casted

2.2. 26
2.2.

With assumption of σ4 = σ5 = ... = σn and γ4 = γ5 = ... = γn, alkylidene-n-S can be given as follows (n ≥ 4)

2.2. 27

Thus, the site balance is expressed as the following

2.2. 28

The schematic diagram of model selection using a genetic algorithm for data regression is shown in Figure 2. Finally, the rate expressions of the reagent and different products are then expressed as follows with n ≥ 4

2.2. 29
2.2. 30
2.2. 31
2.2. 32
2.2. 33
2.2. 34
2.2. 35
2.2. 36
2.2. 37

With

2.2.
2.2.
2.2.
2.2.
2.2.
2.2.
2.2.

Table 1. Elementary Steps for Different Mechanisms,16,38,40−42 Where M1 Refers to the Carbide Mechanism, M2 Refers to the Enolic Mechanism, and M3 Refers to the CO Insertion Mechanism (with n ≥ 4), A Refers to Alkylidene and S Refers to the Active Site on the Surface of the Catalyst.

  M1
 M2
 M3
constant elementary constant elementary constant elementary
KH2ad H2 + 2S ↔ 2HS KH2ad H2 + 2S ↔ 2HS KH2ad H2 + 2S ↔ 2HS
KCOad CO + S ↔ COS KCOad CO + S ↔ COS KCO CO + CnH2n+1 – S → CnH2n+1CO – S
KH2Oad H2O + S ↔ H2O – S KH2CO CO + H2 ↔ H2CO – S kw CnH2n+1CO – S + H2 ↔ CnH2n+1C – S + H2O
KOn On + S ↔ On – S kAN H2C – S + CnH2n+1 – S → CnH2n+1CH2 – S + S kAN CnH2n+1C – S + H2 → CnH2n+1CH2 – S
kco CO – S + S → C – S + O – S kCH4 CH3 – S + H – S → CH4 + 2S kCH4 CH3 – S + H2 → CH4 + HS
kc C – S + H – S → CH – S + S kPn CnH2n+1 – S + H – S → CnH2n+2 + S kPn CnH2n+1 – S + H2 → CnH2n+2 + HS
kCH CH – S + H – S → CH2 – S + S kOn CnH2n+1 – S ↔ CnH2n + HS kOn CnH2n+1 – S → CnH2n + HS
kw HO – S + H – S → H2O – S + S        
kIN CH2 – S + H – S → CH3 – S + S        
kAN An – S + CH2 – S → An+1 – S + S        
kCH4 CH3 – S + H – S → CH4 + 2S        
kP2 A2 – S + H – S → P2 + 2S        
kP3 A3 – S + H – S → P3 + 2S        
kPn An – S + H – S → Pn + 2S        
kO2 A2 – S + S ↔ O2 – S + H – S        
kO3 A3 – S + S ↔ O3 – S + H–S        
kOn An – S + S ↔ On – S + H – S        
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Figure 2.

Figure 2

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Schematic diagram of the genetic algorithm (GA) for mechanism discrimination and data regression, where CLD refers to chain-length dependent.

The model parameters were calculated by minimizing the sum of squares of residuals defined by calling the objective function, as follows

2.2. 38

where Nexp is the number of experimental data and ri,jexp and ri,j are experimental and calculated molar rates for species i in j.

The genetic algorithm (GA) was utilized for global optimization and Levenberg–Marquardt (LM) for local optimization using the MATLAB toolbox to approximate reaction rate constants. In addition, all of the parameters that were obtained from the model fit should also be physically relevant to the Arrhenius and Van’t Hoff adsorption laws, as follows

2.2. 39
2.2. 40

where ki (mol·kgcat–1·h–1) and Ki (mol·kgcat–1·h–1) are reaction and adsorption constants, respectively, Ea,i is apparent activation energy (kJ·mol–1), and ΔHi is heat of adsorption for i species (kJ·mol–1).

2.3. Response Surface Methodology (RSM)

In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables.43,44 The main idea of RSM is to use a sequence of designed experiments to obtain an optimal response. In addition, it can also be used to directly correlate the influential operational parameters with responses using the most widely deployed second-degree polynomial expressions to find out the offset, linear, quadratic, and interactive terms as follows

2.3. 41

where Yi is the responded value (in this work, there are 36 different responses including XCO, SCH4, rCO, SCO2, SC2–C4, rPn, rOn with n ≤ 15), XiXj is the binary parameter of the investigated operational parameters (T, P, GHSV, and SR), and b0, bi, and bii (bij) are coefficients from the polynomial expression. Although it does not shed any insightful lights upon the kinetic mechanism, it does provide the useful statistical indications for the significances of responses that are caused by the variations of process variables (either singular or binary). With experimental results (Table 4 in the Experimental Part section), the RSM is capable of calculating the corresponding formation rate of olefin and paraffin with different carbon numbers (Figure 3).

Table 4. Experimental Designs for Kinetics During Fischer–Tropsch Synthesis, Where XCO is CO Conversion, SCH4 is Methane Selectivity, SCO2 is CO2 Selectivity, SC2_C4 is Alkene, TOS is Time on Stream (h) Selectivity, * is Baseline Conditions, T is Temperature (°C), P is Pressure (Bar), GHSV is Gas Hour Space Velocity (L·h–1), and SR is Syngas Ratio, Respectively.

  operational parameter
          
runs T/°C P/bar GHSV/L·h–1 SR experimental XCO experimental SCH4 experimental SCO2 experimental SC2_C4 TOS/h
0* 220 20 3000 1.5 0.3201 0.1238 0.0054 0.2103 48
1 215 20 2790 2.2 0.5560 0.1570 0.0481 0.2540 60
2 215 17 2795 1.5 0.4850 0.1082 0.0183 0.2768 72
3 215 15 1380 1.7 0.5707 0.0865 0.0513 0.1247 84
4 200 17 2796 1.2 0.3371 0.0495 0.0067 0.1024 96
5 200 17 1380 1.7 0.5503 0.0774 0.0484 0.0520 108
6 215 15 2796 1.2 0.4388 0.0958 0.0083 0.2780 120
7* 220 20 3001 1.5 0.3199 0.1238 0.0054 0.2103 132
8 215 17 1380 1.2 0.5146 0.0760 0.0422 0.0443 144
9 215 17 3003 1.7 0.4850 0.1082 0.0183 0.2768 156
10 215 20 1380 1.7 0.5567 0.0792 0.0495 0.0618 168
11 220 20 4202 1.5 0.2214 0.1900 0.0321 0.1361 180
12 230 17 1380 1.7 0.5707 0.0870 0.0515 0.1233 192
13 230 17 2792 2.2 0.5905 0.0763 0.0515 0.2647 204
14 215 15 2791 2.2 0.5841 0.0766 0.0507 0.2701 216
15 200 17 3000 2.2 0.5290 0.1245 0.0374 0.2682 228
16 230 17 2796 1.2 0.4536 0.0958 0.0074 0.2793 240
17 215 17 1380 2.2 0.5705 0.0871 0.0518 0.1186 252
18 230 20 2796 1.7 0.5107 0.1458 0.0345 0.2693 264
19 215 17 2807 1.7 0.4850 0.1082 0.0183 0.2768 276
20 230 15 2797 1.7 0.5608 0.0804 0.0466 0.2766 288
21 215 17 4212 1.2 0.2819 0.0865 0.0173 0.0679 300
22 215 17 2799 1.5 0.4850 0.1082 0.0183 0.2768 312
23 200 20 2796 1.7 0.3543 0.1130 0.0045 0.2317 324
24 200 15 2800 1.5 0.4760 0.0769 0.0094 0.2796 336
25 215 17 4210 2.2 0.3046 0.1592 0.0402 0.2146 348
26 220 17 3000 1.7 0.4850 0.1082 0.0183 0.2768 360
27 200 17 4200 1.5 0.2729 0.0752 0.0195 0.0643 372
28 215 15 4210 1.7 0.1327 0.2579 0.0502 0.1442 384
29 230 17 4212 1.5 0.2222 0.2163 0.0463 0.1851 396
30* 220 20 2989 1.5 0.3027 0.1127 0.0044 0.2055 408
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Figure 3.

Figure 3

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Schematic diagram of response surface methodology (RSM) for significance analysis, where NPR refers to a normal plot of residues, RVP refers to residues versus predictions, BCP refers to the Box–Cox plot for power transform, and CD refers to Cook’s distance.

3. Results and Discussion

3.1. Model Comparisons

Once experimental data (Table 4 from the Experimental Part section) were fed into different models (RBFNN, CKGA, and RSM), model predictions against validation results (Table S1) were used to test the performances of constructed models. Because of a range of differences, the parameters, i.e., XCO, SCH4, rCO, SCO2, and SC2_C4, and the formation rate of olefin and paraffin with different carbon numbers (rPn, rOn with n ≤ 15) are plotted in Figure 4a,b, respectively. Clearly, with different models, data present appreciably different aggregating patterns both in Figure 4a1–c1 and a2–c2, respectively.

Figure 4.

Figure 4

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Comparisons between the experiment and model prediction when the rate is in mole·h–1 for (a) RBFNN model, (b) comprehensive kinetics based on the genetic algorithm (CKGA) model, and (c) RSM-based model, where Exp_rH2 and rCO refer to the experimental hydrogen and CO consumption rate (mole·h–1), Calcd_rH2 and rCO refer to the calculated hydrogen and CO consumption rate (mole·h–1), Exp_rPi and rOi refer to the experimental paraffin and olefin rate with the carbon number i (mole·h–1), and Calcd_rPi and rOi refer to calculated paraffin and olefin rates with the carbon number i (mole·h–1).

Taking the center aggregation (purple circle with the range from 0 to 0.0006 for both axis), for instance, the percentage of data falling out of this range from RBFNN, CKGA, and RSM models is 14.2, 16.6, and 18.9%, respectively. Apparently, the patterns of data distribution using different modeling approaches are not same, which will lead to the variation of relative errors between the experiment and predictions. To further assess the deviations between the experimental results and model predictions, the corresponding standard deviations were calculated as follows

3.1. 42

where N represents the numbers of experimental runs, x̅ represents the averaged values of each parameters (XCO, SCH4, rCO, SCO2, SC2_C4, rPn, rOn with n ≤ 16), and xi refers to corresponding specific value. Then, the standard error (SE) of each experimental value is defined as follows

3.1. 43

The errors between experiment and model predictions using three different models are shown in Figure 5, with the color bar from blue to red indicating the variations of standard errors from ±5 to ±20% in validation tests. The standard errors for majority data are less than ±10% with the presence of some exceptionally high errors (over ±20%) for all of three models (RBFNN, CKGA, and RSM). The RBFNN model approach is found to have relatively fewer standard errors within an experimental data range of 0–0.001 (mol·h–1), while the CKGA model presents the largest deviation among the three investigated models. The possible reasons could be more additional conditions needed to be met for constant regressions during the mechanism selection (both in statistical and physical relevance such as meeting Arrhenius and Van’t Hoff criteria).

Figure 5.

Figure 5

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Error visualization of different models. (a) RBFNN model, (b) CKGA model, (c) RSM model.

In addition, regarding to those data leading to large discrepancies (SE > ±10%), the calculations in RBFNN are found to undershoot, while the calculations from CKGA and RSM overshoot with more larger errors being observed from CKGA model. The comparison of experimental data and model predictions for C1–C15 at the baseline condition are shown in Figure 6 (the run baseline with best data fit was chosen). Different model approaches provide different accuracies during data regression. Because of direct computation (using training data for RBFNN and second-degree polynomial expression for RSM), both RBFNN and RSM tend to provide predictions with relatively higher accuracy and less computational time.

Figure 6.

Figure 6

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Comparisons of experimental data and model predictions (baseline condition) for C1–C15 at the baseline condition: (a) paraffin formation rate, (b) 1-olefin formation rate, (c) olefin to paraffin ratio (OPR), and (d) ASF distribution.

On the other hand, CKGA does provide elementary mechanism (using the Langmuir–Hinshelwood–Hougen–Watson approach for elementary reaction pathway derivation) that sheds insightful lights on the CO hydrogenation mechanism at the investigated experimental conditions, even though the accuracy of the CKGA model tends to be lower during predictions at some experimental runs (over ±30%). Therefore, RBFNN is found to be the most robust (computational time and accuracy) to make predictions among the investigated models, if sufficient training data were fed. The relatively big uncertainties C5 in Figure 6a might lie in the vaporization and loss of C5 during trap swap. With the CKGA model, the kinetic expressions (constants in the kinetic expression Table 2) could be obtained through data regressions. The obtained constants can be compared with reported values from literature reports. For instance, the values of kP3 and kPn are quite close to each other, suggesting the reactivity of alkylidene to paraffin being stable as the carbon number increases over 4. The FT activation energy is close to the typical cobalt-based catalyst indicating the validity of constructed reaction pathways.38,39,45

Table 2. Estimation of Kinetic Parameters, Where $ Represents Lower and Upper Bounds for 95% Confidence Interval, ∧ Marked Constant Are Assumed to be Temperature Independent, with n ≥ 4.

parameter kinetic parameters
 energy barrier
 literature
description pre-exponential factor unit confidence limit$ symbol value kJ·mol–1 confidence limit$ kJ·mol–1
KH2ad 2.1 × 10–6 bar–1 2.6 × 10–7 ΔHH2 –11.8 7.9 × 10–4 –1546
kFT 3.6 × 104 mol·gcat–1 h–1 bar–1 2.1 × 103 ΔEFT 90 4.5 × 10–2 80–13038
kCOKCOad∧ 6.1 × 10–2 mol·gcat–1 h–1 bar–1 2.8 × 10–3        
kIN∧ 3.1 × 101 mol·gcat–1 h–1 bar–1 2.1 × 100        
kA∧ 4 × 103 mol·gcat–1 h–1 bar–1 1.0 ×102        
kCH4 8 × 103 mol·gcat–1 h–1 bar–1 7.0 × 102 ΔECH4 76 2.1 × 10–2 8247
kPn∧ 2.2 × 10–2 mol·gcat–1 h–1 bar–1 4.3 × 10–1        
kP3∧ 4.1 × 10–2 mol·gcat–1 h–1 bar–1 1.1 ×10–1        
kP2∧ 1.1 × 10–4 mol·gcat–1 h–1 bar–1 2.5 ×10–3        
kOn 4.4 × 10–2 mol·gcat–1 h–1 bar–1 3.4 × 100 ΔEOn 58 3.6 × 10–2 100–13045
kOnrevKOn∧ 2.9 × 10–1 mol·gcat–1 h–1 6.4 × 10–2        
kO3 4.9 × 102 mol·gcat–1 h–1 bar–1 2.6 × 101 ΔEO3 71 2.1 × 10–2  
kO3revKO3∧ 4.1 × 10–4 mol·gcat–1 h–1 1.5 × 10–4        
kO2 1.7 × 101 mol·gcat–1 h–1 bar–1 4.5 × 100 ΔEO2 68 1.1 × 10–2 50–13048
kO2revKO2∧ 1.4 × 10–3 mol·gcat–1 h–1 5.1 × 10–4        
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One of major challenges of using CKGA lies in compromising the computational time and accuracy using GA for both regional and global optimization to converge the objective functions. As a dimension of tensor increase, the demand for computational time and internal memories increases significantly. Clearly, it might not be the best choice of obtaining kinetic constants via direct data regression through the elementary mechanisms using quasi-steady state assumption (QSSA) if the goal is for process optimization. Instead, CKGA might be more meaningful in exploring possible reaction pathways during mechanism selections. With regard to optimizing and sizing the reaction system, the RSM surely shows its inherent advantage, even though it does not provide any insightful descriptions between process parameters and the responses. With RSM, the polynomial correlations are able to provide a very accurate correlation with those investigated critical process parameters. For instance, effects of reaction temperature and pressure versus the propane (C3) formation rate and statistical analysis of residues from all validation runs for propane (C3) formation rate are shown in Figure 7.

Figure 7.

Figure 7

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RSM analysis of process parameters toward response, where P_3 refers to propane (C3 paraffin). (a) Temperature and pressure versus the C3 paraffin (propane) formation rate, (b) normal plot residues of all experimental runs for the C3 paraffin (propane) formation rate, (c) residues versus predictions for all experimental runs of the C3 paraffin (propane) formation rate, and (d) Cook’s distance analysis for all experimental runs of the C3 paraffin (propane) formation rate.

The visualization of the C3 paraffin formation rate responding to the variations of pressure and temperature is shown in Figure 7a. In addition, the residues (Figure 7b), residues versus predictions (Figure 7c), and Cook’s distance (Figure 7d) can be analyzed to find out the range that is out of normal values. For instance, experimental data of run 10 and run 28 (from Table 1) tend to yield a relatively larger Cook’s distance (>0.4). The corresponding coefficients from a second-order quadratic expression for the propane formation rate together with the p-value are shown in Table 3.

Table 3. Coefficients of the Quadratic Polynomial Expression for the C3 Rate of Formation and the Corresponding p-Value.

coefficient rP3/mol·h–1 p-value
b0 –7.534 × 10–3 0.0427
b1 8.496 × 10–5 0.0080
b2 –3.503 × 10–4 0.5818
b3 –8.224 × 10–7 0.5908
b4 2.593 × 10–3 0.0009
b5 –4.270 × 10–7 0.8335
b6 2.074 × 10–9 0.5414
b7 –9.799 × 10–6 0.2917
b8 9.023 × 10–9 0.8609
b9 5.861 × 10–6 0.7056
b10 1.824 × 10–8 0.4259
b11 –1.472 × 10–7 0.4225
b12 1.235 × 10–5 0.0820
b13 4.499 × 10–11 0.9457
b14 –1.456 × 10–4 0.4094
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More importantly, the RSM offers the opportunities for both singular and multiple optimization, which means that not only does hydrocarbons’ selectivity could be maximized but also the selectivity of unwanted byproducts such as CO2 and CH4 selectivity could be simultaneously minimized by manipulating the multiple objective functions. For example, one ideal scenario is that the selectivities of CO2 and CH4 were kept at the minimum, while alkene selectivity (C2–C4) was set at the maximum; the desirability of the binary parameters toward desirability (for convenience of presentation, we only list the most significant parameters) is shown in Figure 8. The corresponding optimal operational conditions were achieved at T_224 (°C), P_17 (bar), GHSV_2801 (L·h–1), and SR_1.2, with the selectivity: SCH4_0.05, SCO2_0.011, and SC2_C4_0.01, respectively. Therefore, a useful second-order polynomial expression is constructed as follows

3.1. 44

For instance, the established coefficients for all (C1–C15) paraffin formation rates can be found from Table S2 with less than ±10% experimental uncertainties.

Figure 8.

Figure 8

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Desirability of multiple optimization using RSM (a) pressure and temperature versus desirability, (b) GHSV and temperature versus desirability, (c) SR and temperature versus desirability, and (d) SR and GHSV versus desirability.

3.2. Strategy for Kinetic Modeling and Process Optimization

For sizing and designing a reactor system, the key challenge still lies in constructing accurate and reliable kinetic expressions.47,49,50 Albeit, the LHHW approach has been successfully and widely employed to derive kinetic models for FT synthesis; the inherent limitations such as yielding the debatable mechanistic elementary reactions and tedious computational times still need to be substantially improved in the practical applications.17,18,51−58 Because of the flexibility for learning the information and a pertinence for pattern recognition in the complex system, the applied soft computing technique such as using RBFNN has been widely used for modeling and control tools for a complex system.59

However, quite a few research studies either focus on kinetic constant correlations by simply using ANNs or generate the central composite design (CCD) matrix through the RBFNN simulations, followed by statistical analysis through RSM.60−62 The critical limitation lies in its lacking understanding of kinetic mechanisms.61 As discussed before, the best strategy is to maximize the merits of each model (RBFNN, CKGA, and RSM); the hybrid approach of using a small number of data points to cost-effectively generate more representative data sets through RBFNN, followed by CKGA (for mechanism selection) and RSM (performance evaluation to scrutinize the effects of process parameters upon the investigated responses) will show more promising prospects. The schematic diagram of the strategy of kinetic modeling is shown in Figure 9. With the advancement of computation technologies, the consideration of a hybrid paradigm of using RBFNN, CKGA, and RSM has become more attractive. The necessity of combining inherent merits of RBFNN, CKGA (mechanistic kinetic), and RSM (statistical analysis) secures both accuracy of predictions and insightful understanding of the reactions (Table 4).

Figure 9.

Figure 9

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Schematic diagram of the strategies of kinetic modeling from different models (RBFNN, CKGA, and RSM).

4. Conclusions

Three different types of modeling frameworks, namely, RBFNN, CKGA, and RSM, were explored and compared to model the product distribution for FT synthesis using a Ru-promoted Co/Al2O3 catalyst in a conventional packing bed reactor (PBR). A new strategic modeling paradigm for olefin/paraffin rate expressions during FT synthesis was proposed by considering both accuracy and intrinsic understanding of FT kinetics. The proposed new strategy possesses appealing features of modeling the rates of different hydrocarbons (olefin, paraffin with C2–C15), olefin to paraffin ratio (OPR), and statistical analysis of the key operational parameters upon responses (selectivity, olefin, and paraffin with different C2–C15 carbon numbers). The accuracy and data patterns of using different models (RBFNN, CKGA, and RSM) were rigorously compared by an error visualization approach. By combining the inherent advantages such as fast converge and higher accuracy (a hybrid approach of coupling both semiempirical or empirical soft computation technique RBFNN), visualization, direct statistical analysis (RSM), and insightful understanding of mechanistic kinetics (kinetic modeling using genetic algorithm CKGA), a new modeling strategy of kinetic study for complex FT synthesis is proposed.

4.1. Experimental Part

The preparation of cobalt-based catalysts is followed by a conventional wet-impregnation approach with a constitution of 20 wt % Co, and 0.5 wt % Ru on γ-alumina.63,64 The catalyst with a sieve fraction of 37–75 μm was employed and loaded into the reactor (a fixed-bed stainless steel reactor with 12.7 mm ID and 400 mm in length). The detailed experimental operations for collecting kinetic data, catalyst reduction, and product analysis could be found from our previous reports.38,39,61 The CO conversion and alkene (C2–C4) selectivity were defined as follows

4.1. 45
4.1. 46
4.1. 47

where FCO,in is the CO molar flow rate from the outlet (mol·h–1), XCO is the CO conversion (%), FCO,out is the CO molar flow rate from the outlet (mol·h–1), SAi is the molar flow rate of light olefin C2–C4 (mol·h–1), and FCO2 is the CO2 molar flow rate from the outlet (mol·h–1). The outlet concentrations were measured using an online refinery gas analyzer. The following differential equation is used as a reactor model

4.1. 48

The boundary conditions (BCs) were set at

4.1.
4.1. 49

where Nr is the number of reactions involved; αij is the stoichiometric coefficient of the ith compound, these detailed sets of coefficients can be found from previous reports;39Rij is the reaction rate of formation (mol·h–1) of the ith component, refers to molar flow rate of species i (mol·h–1), and refers to the mass of catalyst (g). The partial pressure of the ith compound

4.1. 50

where Nc refers to the total numbers of components, Pi refers to partial pressure of the ith compound (bar), mi refers to the mass of components (g), and PT refers to the total pressure of the system (bar). The gases and liquid (both cold trap setting at 10 °C and hot trap setting at 110 °C) were collected and analyzed by gas chromatography (GC) and gas chromatography-mass spectrometry (GC–MS), respectively. Before systematically exploring the different experimental conditions, the baseline condition (0*) was maintained for about 48 h. At the end of the experiment, the condition was adjusted back to the baseline condition (30*) for about 12 h and deactivation was relatively less than 6%.

It must be noted that FT synthesis yields a wide range of products (C1–C50+); for the simplicity of kinetic modeling, C2–C15 olefin and n-paraffin (ignoring all branched hydrocarbons) were usually taken into account due to the fact that the rate of the hydrocarbons with a larger carbon number was not simply governed by the kinetics at the experimental conditions. To validate the established model, additional 15 runs were conducted, which is shown in Table S1. Therefore, the overall procedure for model establishment and validation are as follows: (i) data in Table 1 were used to establish the model (RBFNN—data train, CKGA—data regression, RSM—data design in the matrix) and (ii) data in Table S1 were used to validate three constructed models (RBFNN, CKGA, RSM).

Acknowledgments

This work was funded by the School of Engineering, Edith Cowan University, Australia, for open access publishing, the University of Nottingham Ningbo China (FoSE New Researchers Grant I01210100011), the Faculty Inspiration Grant of University of Nottingham (FIG. 2019), the Qianjiang Talent Scheme (QJD1803014), the National Key R&D Program of China (Grant: 2018YFC1903500), the Ningbo Science and Technology Innovation 2025 Key Project (Grant 2020Z100), the Ningbo Municipal Commonweal Key Program (Grant 2019C10033 and 2019C10104), and the Key Laboratory of Carbonaceous Wastes Processing and Process Intensification of Zhejiang Province (2020E10018).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c03851.

  • Tables S1–S3 for additional information such as validation experimental results for kinetics during Fischer–Tropsch synthesis, kinetic expressions for detailed product distribution derived from the CKGA model, and coefficients of the quadratic polynomial expression for paraffin formation rate from C1 to C15 (PDF)

Author Contributions

∇ Y.W. and J.H. contributed equally to this work.

The authors declare no competing financial interest.

Supplementary Material

ao1c03851_si_001.pdf (82.7KB, pdf)

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