1 Introduction

Due to dwindling petroleum reserves and the fluctuations of the crude oil costs in the past few years, Fischer–Tropsch synthesis (FTS) of syngas (CO and H2 mixtures) has been considered as one of the foremost promising ways to provide ultra-clean fuels at an economically possible cost (Fu et al. 2013; Park et al. 2014). In FTS, which is a kind of polymerization process, a large range of light gases and distillates, including olefins, paraffins, and oxygenated compounds are produced from synthesis gas in a catalytic reaction (Parnian et al. 2014a, b; Liu et al. 2015). The synthetic fuel made from Fischer–Tropsch synthesis is characterized by a prime quality, significantly from the view point of low pollution and a high cetane number (Coronel-García et al. 2015; Rodríguez-Fernández et al. 2009). The spectrum of the FTS product contains the complex mixtures of linear and branched hydrocarbons ranging from light alkenes to heavy waxes and oxygenated products (Derevich et al. 2012; Todic et al. 2018; Najafabadi et al. 2016; van Helden et al. 2017; Khodakov et al. 2007). The most desired products are those with low methane, low alcohol, high alkene/alkane ratio, and high \(C_{5}^{ + }\) content, which can be controlled by a modification of the catalyst, the reactor and the reaction conditions (Fu et al. 2013). Due to the complex nature of the synthesis, CO hydrogenation is faced with a basic problem of controlling the product selectivity, and this is closely related to the reaction mechanism and the behavior of reaction intermediates (van Helden et al. 2017). The most common catalysts for FTS are cobalt- or iron-based catalysts. Some inorganic supports with high surface area, such as silica and alumina, have been used to increase the active phase dispersion (Khodakov et al. 2007; Ralston et al. 2017).

The use of Fe-based catalysts on FTS has been studied by many investigators and was shown to possess satisfactory performances within the production of liquid fuels and waxes, particularly at high pressures. Most studies with iron catalysts have targeted on reaction conversion, overall rates of reaction and also on the product distribution over some catalyst compositions. In many reported literatures, the use of Ni as promoter not only has resulted in increasing light hydrocarbons, but it also has caused heavy hydrocarbons to decrease. The use of bimetallic is more common compared to pure metal catalysts (Li et al. 2014). Ishihara et al. (1987) exhibited the higher activity of bimetallic Fe–Ni in comparison with Fe and Ni catalyst. In a bid to achieve a better knowledge of the FTS, some studies were conducted to understand the effects of operating conditions and different catalyst promoters on the distribution of the FTS products. These studies have demonstrated that iron-based catalysts produced paraffins, particularly paraffins and olefins with low molecular weight, depending on the reaction conditions used and the kind of catalysts and reactors used (Farias et al. 2008; Feyzi et al. 2015; Peña et al. 2018). Although there are a lot of kinetic expressions for CO consumption in literature, the one which focuses on the prediction of hydrocarbon selectivity is hardly found. A challenge facing contemporary FTS research is the development of techniques for controlling process parameters, such as selectivity on liquid products, productivity of the catalytic bed, the yield of specific groups of hydrocarbons, etc. (Zhang et al. 2014; Bashiri et al. 2018; Savost’yanov et al. 2018; Challiwala et al. 2018). Articles about product selectivity are reported qualitatively, while no article has been presented the model which can predict what will happen with manipulating the operating conditions (Sun et al. 2018; Yang et al. 2010; Cheng et al. 2018).

In this work, the methodologies of design of experiment (DOE) and ANN were used for selectivity modeling of the hydrocarbon products and CO conversion. For selectivity modeling by RSM, the simulation of DOE data is of necessity; therefore ANN was used to produce these data from the experimental data. The RSM was used to determine the exact optimum point for maximum \(C_{5}^{ + }\), minimum CH4 distribution in products and maximum CO conversion in the range of the temperature and the pressure. This method can be applied to calculate the product selectivity and CO conversion for any temperature and pressure within the experimental condition ranges. Furthermore this method is very helpful in different industries to increase preferable products and also to decrease undesirable ones by manipulating operating conditions.

2 Experimental

Mirzaei et al. (2012) attempted to obtain the experimental data from fixed bed micro reactor. The catalyst was prepared by co-precipitation method. The molar ratio of the solution Fe/Ni was 40/60. The catalyst (Fe–Ni/Al2O3) was dried at 383.15 K for 16 h and calcined at 873.15 K in flowing air for 6 h at 5 ºC/min the catalyst was used under reaction conditions of Fischer–Tropsch synthesis. Experimental data were obtained in 54 runs (Mirzaei et al. 2012).

3 Modeling method

With a variety of statistical tools available, the response surface methodology is an efficient procedure in scientific studies. This methodology is an assortment of statistical techniques for the experimental design, the building of the models, evaluating the consequences of factors, and searching for the optimum conditions. Using the experimental designs of response surface methodology makes it attainable to use statistical tools for the modeling (Shojaeimehr et al. 2014; Atashi and Rezaeian 2017).

In order to evaluate the behavior of product selectivity and CO conversion with temperature and pressure and also to determine the optimum range of operational condition (pressure and temperature), in which selectivity of desired products (\(C_{5}^{ + }\)) was maximized and undesired products (CH4) was minimized, modeling was necessary. In the reference experimental work (Mirzaei et al. 2012), CO conversion or products selectivity were reported in limited points such as T: 563, 573 K…, and P: 2, 3, 4 bar…hence, it was impossible to determine the CO conversion or products selectivity from unseen points. Therefore, modeling becomes an asset in determining the exact CO conversion or products selectivity for any temperature or pressure such as 547 K or 3.5 bar, and also to demonstrate the interaction between the parameters. Moreover, in order to develop RSM models, it is necessary to collect experimental data based on the DOE methods. In the reference work (Mirzaei et al. 2012), experimental data were not collected by DOE methods. To compensate for this drawback, ANNs in which original data were trained to produce appropriate input data for RSM were constructed. Then, RSM was used for the selectivity and CO conversion modeling.

3.1 Artificial neural networks and design of experiment

From chemical engineering viewpoint, selectivity studies play a significant role in achieving an optimum performance of FTS process. As an effective tool for organization of the experimental studies, designing of experiments (DOE) is widely used in various science and technology fields, as well as designing, optimizing and developing of the catalyst (Atashi and Rezaeian 2017; Atashi et al. 2015). In the last few years ANN based modeling helped to develop empirical models and also achieve better statistical analysis on experimental data (Nasr et al. 2013). ANN is a colossal structure of interconnected networks consisting of numerous individual elements called neurons, capable of performing parallel computations for data processing. Hence without having any prior knowledge about individual functional relationships, ANN can handle multiple independent and dependent variables simultaneously (Pirdashti et al. 2013).

In this work, three feed-forward networks with two input–single output neuron structures were constructed. The reactor operating conditions represented by the two inputs are temperature (523–573 K) and pressure (2–10 bar). The single output neuron of each of the three networks, respectively represented the following quantities, Network I: CO conversion, Network II: CH4 selectivity and Network III: \(C_{5}^{ + }\) selectivity. The number of neurons in the hidden layer must be selected in order for the best fit of experimental data to be achieved. Therefore for the first network, the optimal number of hidden neurons was found to be 8 whereas for both Networks II and III, the corresponding numbers were 6.

3.2 Producing simulated data

In the range of operating condition, a Central Composite Design (CCD) in software with two factors, pressure and temperature, was employed to create DOE points. With the use of the designed points from CCD and the trained networks, simulated outputs were generated in MATLAB. The simulated data were used to obtain the CO conversion and selectivity models by the adoption of response surface methodology.

3.3 Response surface methodology

The CO conversion, CH4, and \(C_{5}^{ + }\) selectivities data which were generated from MATLAB by using the neural networks and DOE points from CCD (Table 1), were analyzed using the RSM. The quadratic equation for the variable is as follows:

$$Y = \beta {}_{0} + \sum {\beta_{i} } X_{i} + \sum {\beta_{ii} } X_{i}^{2} + \sum\limits_{i} {\sum\limits_{j} {\beta_{ij} } } X_{i} X_{j}$$
(1)

where Y is the predicted response; β0 is a constant; βi is the first-order model coefficient; βii is the squared coefficient for the factor i, and βij is the linear model coefficient for the interaction between factors i and j. Xi is the value of the main effect according to the following:

$$X_{i} = \frac{{x_{i} - x_{i}^{*} }}{{\Delta x_{i} }}$$
(2)

where Xi is the 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.

Table 1 DOE points and simulated outputs
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All statistical analysis was done by DESIGN EXPERT software. The purpose was to study the potential of ANN simulated data in selectivity study and the capability of RSM in prediction of CO conversions, CH4 and \(C_{5}^{ + }\) selectivities, as well as optimum condition.

The following statistics were applied for selectivity modeling of evaluating the quality of linear and nonlinear regression.

The correlation coefficient,

$$R^{2} = 1 - \frac{{\sum\nolimits_{i = 1}^{n} {(y_{{i,{\text{obs}}}} - y_{{i, {\text{model}}}} )^{2} } }}{{\sum\nolimits_{i = 1}^{n} {(y_{{i,{\text{obs}}}} - \mathop y\limits^{ - } )^{2} } }}$$
(3)

Adjusted R2:

$$R_{\text{adj}}^{ 2} = 1 - \frac{{(1 - R^{2} )(n - 1)}}{n - m}$$
(4)

Mean Absolute Relative Residual, MARR:

$${\text{MARR}} = \sum\limits_{i}^{n} {\left| {\frac{{y_{{i,{\text{obs}}}} - y_{{i,{\text{model}}}} }}{{y_{i,obs} }}} \right|\frac{1}{n}} \times 100$$
(5)

Relative Variance, Srel:

$$S_{\text{rel}} = \left( {\sum\limits_{i}^{n} {\left( {\frac{{y_{{i,{\text{obs}}}} - y_{{i,{\text{model}}}} }}{{y_{{i,{\text{obs}}}} }}} \right)^{2} \frac{1}{n - m}} } \right)^{0.5} \times 100$$
(6)

Root mean of standard deviation (RMSD):

$$RMSD = \frac{1}{n}\left( {\sum\limits_{i}^{n} {\left( {y_{{i,{\text{obs}}}} - y_{{i,{\text{model}}}} } \right)^{2} } } \right)^{2}$$
(7)

where, yi,obs is experimental point, yi,model is calculated point, n is the number of observations and, m is the number of parameters.

F-regression statistic and F-lack of fit are as follows:

$$F-regression = \frac{{{\text{MSR}}}}{{{\text{MSE}}}}$$
(8)
$$F-lack\,of\,fit = \frac{{{\text{MSLF}}}}{{{\text{MSPE}}}}$$
(9)

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 (Atashi and Rezaeian 2017; Atashi et al. 2015).

4 Results and discussion

4.1 Artificial neural networks (ANN)

The temperature and pressure as inputs and CO conversion and product selectivity as targets were presented to the system. Estimation was made and the results were compared with the corresponding desired value. In order to test and validate the trained network, another two sets of data were used and the outputs were obtained. The outputs of the ANNs were compared with the experimental data for the trained, test and validation data sets, respectively. The results for the CO conversion, CH4, and \(C_{5}^{ + }\) selectivity networks are shown in Figs. 1, 2 and 3, respectively.

Fig. 1
figure 1

Trained network for CO conversion data

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Fig. 2
figure 2

Trained network for CH4 selectivity data

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Fig. 3
figure 3

Trained network for \(C_{5}^{ + }\) selectivity data

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Figures 1, 2 and 3 are indicative of the relationship between targets (experimental data) and outputs (ANN data). As can be seen, the prediction of ANN was almost fitted to the experimental value. The high values of R2 (0.96–0.99) were obtained, which indicated that the constructed neural networks can simulate the experimental domains very well. The excellent estimation performances obtained by using the trained networks demonstrated that the trained networks were reliable, accurate and hence could be employed further in the study.

4.2 Response surface methodology (RSM)

On the basis of RSM, the effect of temperature and pressure on CO conversion, methane, and \(C_{5}^{ + }\) selectivity was studied and interactions between parameters were achieved. The regression equations for each variable were obtained after applying RSM, according to the temperature and pressure. The significance of each coefficient was determined by F and P values, and the statistically important terms presented in Table 2. The second order of regression models were reduced by omitting the terms which are unimportant. The analysis of variance was employed to test the significance of the ratio of mean square due to regression and mean square due to residual error. Generally, P values lower than 0.05 indicated that the model was considered to be significant at the 95% confidence level.

Table 2 Analysis of variance and regression coefficient of the RSM modeling
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Table 2 presents the RSM models for CO conversion, CH4, and \(C_{5}^{ + }\) selectivity as well as their R2 values. The R2 values of the predicted model equations were quite high (98%–99%) and these values indicated that the RSM was sufficient enough in properly explaining the selectivity modeling of products and CO conversion.

After determination of the RSM models, 3D surface and contour plots were depicted to show the behavior trend of the targets and then multi-component optimization was established, considering the interactions between parameters in all the range of operating conditions to gain optimum point where \(C_{5}^{ + }\) and CO conversion were maximized and CH4 was minimized.

Also, the RSM models in coded version can be used. The larger regression coefficient for the independent parameters demonstrated the stronger effect they possess on the responses. The equations presented in Table 3 cannot be used instead of the equations of Table 2. In other words, in order to calculate the CO conversion, CH4 or \(C_{5}^{ + }\) selectivity at any point in the range of experimental conditions, only the RSM models in Table 2 can be applied.

Table 3 The obtained RSM models in coded version
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According to the statistical analysis, temperature and pressure and their quadratic terms demonstrated the statistically significant effect on CO conversion. It is evident from the RSM models in Table 3 that the pressure has the greatest effect on CO conversion and \(C_{5}^{ + }\) selectivity. However, the effect of the temperature on CH4 selectivity is stronger than the pressure. As the models indicate, the interaction between temperature and pressure is dimmed due to negligible impact on the model. The interaction between temperature and pressure did not display any statistical significance.

The contour plots (Figs. 4, 5, 6a) and 3D surface plots (Figs. 4, 5, 6b) clearly showed the dependency of CO conversions and product selectivity on temperature and pressure. The red dots on the contour plots showed the DOE points. Therefore, the ANN/RSM approach may be useful for efficient demonstration of the behaviors.

Fig. 4
figure 4

Contour (a) and 3D surface (b) plots for CO conversion

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Fig. 5
figure 5

Contour (a) and 3D surface (b) plots for CH4 selectivity

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Fig. 6
figure 6

Contour (a) and 3D surface (b) plots for \(C_{5}^{ + }\) selectivity

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Figure 4 shows the surface plots and the contour of the CO conversion. Temperature and pressure exhibited an increasing effect on CO conversion; however, in high temperatures (543–573 K), this increasing effect of temperature changed. As the surface and contour plot has shown, by increasing both temperature and pressure the CO conversion raised. The surface plot and regression coefficients of the CO conversion equation in Table 3 indicated that pressure demonstrated a more significant effect than temperature.

Figure 5 shows the variation of CH4 selectivity with reaction conditions as contour and surface plots. CH4 selectivity generally demonstrates a decreasing trend with an increase in reactor pressure at any temperature. However, in low to moderate temperatures (523–543 K), CH4 selectivity maintained a constant value and in moderate to high temperatures (543–573 K) this value increased with increase in temperature. Generally by decreasing temperature and increasing pressure the minimum amount of CH4 selectivity occurs.

The change in \(C_{5}^{ + }\) selectivity with operating conditions (temperature and pressure) plotted as contour and surface curves are shown in Fig. 6. According to the figure, \(C_{5}^{ + }\) selectivity increased with the increase in pressure at any temperature. From low to moderate temperature, the selectivity remained approximately constant while an obvious increasing trend occurred at moderate to high temperatures. As can be seen, the maximum selectivity was occurred by increasing pressure and temperature.

4.3 Optimization

Optimization plays an important role in statistical modeling where in order to maximize desired products, detecting the best operating condition is a requirement. In this research, multi-component optimization was used and the optimum point which facilitated the obtaining of desired results was achieved. Temperature (around 558 K) and pressure (around 8 bar) resulted in maximized CO conversion and \(C_{5}^{ + }\) selectivity and minimized CH4 selectivity simultaneously.

5 Conclusions

As noted earlier, there have been few researchers who have investigated the Fischer–Tropsch products selectivity, quantitatively. The modeling of the CO conversion and hydrocarbon selectivity of the Fischer–Tropsch synthesis over Fe–Ni/Al2O3 catalyst was investigated by using coupled artificial neural networks and design of experiment. Pressure and temperature were selected as the variable parameters. ANN was employed to train networks related to original experimental data, and central composite design was used for designing of experiment in the exact range of temperature and pressure as the original operating condition. After training data by ANN and determination of DOE points by CCD, these results were combined together for producing simulated data to be used in response surface method. The approach of RSM provided the regression equations for CO conversion, CH4 selectivity, and \(C_{5}^{ + }\) selectivity according to the temperature and pressure. The analysis of variance was employed to test the significance of the parameters in the equations. The concluding equations were obtained (CO conversion, CH4 selectivity, and \(C_{5}^{ + }\) selectivity). The results illustrated that artificial neural network is a practical tool to model and estimate catalytic behavior in reactors. The results also indicated that CO conversion increased with an increase in both in temperature and pressure. The methane selectivity increased by increasing the temperature and decreasing pressure. The pressure possessed a positive effect on \(C_{5}^{ + }\) selectivity at any temperature. The \(C_{5}^{ + }\) selectivity also remained approximately constant at low to moderate temperatures while from moderate to high temperatures, it increased with temperature. These models show how the main parameters in the process can affect the products distribution. Finally, the optimization of the products distribution, in which maximization of \(C_{5}^{ + }\) and CO conversion and minimization of CH4 occurred simultaneously, was done. The optimum condition is around T = 558 K, P = 8 bar.