Modeling and optimization of catalytic performance of SAPO-34 nanocatalysts synthesized sonochemically using a new hybrid of non-dominated sorting genetic algorithm-II based artificial neural networks (NSGA-II-ANNs)

Abstract

The effects of ultrasound-related variables on the catalytic properties of sonochemically prepared SAPO-34 nanocatalysts in methanol to olefins (MTO) reactions were investigated. Different catalytic behaviors are observed which can be explained by the differences in the catalysts’ physicochemical properties affected by ultrasonic (US) power intensity, sonication temperature, irradiation time and sonotrode size. This result confirms that the activity of SAPO-34 catalysts improves with the rise in US power, time and temperature. In order to find a catalyst with the maximum conversion of methanol, maximum light olefins content and maximum lifetime, a hybrid of non-dominated sorting genetic algorithm-II based artificial neural networks (NSGA-II-ANNs) was used. The multilayer feed forward neural networks with back-propagation structures were implemented using different training rules in the neural networks approach to relate the ultrasound-related variables and the catalytic performance of SAPO-34 catalysts. A comparison between experimental and artificial neural network (ANN) values indicates that the ANN model with a 3–10–3 structure using the Bayesian regulation training rule has the best fit and can be used as a fitness evaluation inside the non-dominated sorting genetic algorithm-II (NSGA-II). Also, multiple linear regression (MLR) was used to predict these objective functions. The results indicate a poor fit for the objective functions with a low coefficient of determination. This confirms that the ANN technique is more effective than the traditional statistical-based prediction models. Finally, this ANN model was linked to the NSGA-II and Pareto-optimal solutions were determined by the NSGA-II.

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1. Introduction

Because of the growing demand for light olefins and the shortage of petroleum resources in the future, the MTO technology, regarded as an alternative process for the production of light olefins from non-petroleum sources without using a large amount of energy, has received strong significant academic and industrial attention. A SAPO-34 catalyst shows exceptional selectivity for lower olefins and the complete conversion of methanol in the MTO reaction, although it is rapidly deactivated by coke, which completely blocks the internal channels of the SAPO-34 crystals.¹ Using a SAPO-34 catalyst with a small crystallite size enhances the accessibility of methanol into its cages, resulting in a better catalytic performance.^2–4 Since the diffusion of methanol in the SAPO-34 catalyst is limited by its small cages, only some cages near the external surface are active in the MTO reaction,⁵ therefore the effectiveness of the SAPO-34 catalyst is improved by reducing its crystallite size.⁴

The importance of nanoparticles, especially nanocatalysts and their uses in different industries, has attracted many researches. Materials at the nano-scale show different characteristics in comparison with their bulk state.^6,7 A great number of investigations about the effect of crystal size show that the best performance for SAPO-34 catalysts is with sizes of less than 500 nm.^2,8 Different techniques for improving the formation of SAPO-34 nanocrystals have been developed,^9–12 but recently there has been a rapid increase in the application of unconventional methods.^13,14 Sonochemical synthesis by US irradiation (20 kHz to 10 MHz¹⁵) is a new method for the synthesis of nanoparticles, especially nanocatalysts. Contrary to conventional methods (mainly hydrothermal synthesis), this method is very simple, fast, and does not need any complicated facilities. In this method, the size of particles can be easily controlled by changing the ultrasound-related variables. Chemical effects of US waves are due to acoustic cavitation phenomena in the solution. Due to the collapse of bubbles, a temperature and pressure of about 5000–25 000 K and 181.8 MPa (ref. 15) are produced. The collapsing of bubbles occurred in less than a nanosecond and so, a high rate of temperature decrease (10¹¹ K s⁻¹) takes place, which prevents the organization and agglomeration of particles.^6,7

A lot of parameters affect the sonochemical synthesis and its products. Some of these affecting parameters are: frequency and power of US waves, time of irradiation, solution temperature, type of solution, reaction vessel diameter, the kind of noble gas used in the reaction environment and even geometrical characteristics of the US device (e.g. horn type size).^16–19 The US irradiation typically used in common crystallization media (mainly aqueous media) falls in the low-frequency range (namely, 15, 20, 25 and 30 kHz).²⁰

There are many references using Evolutionary Algorithm (EA) techniques in chemistry and catalysis including Genetic Algorithm (GA), Evolutionary Strategy (ES), Genetic Programming (GP), etc. Evolutionary Strategy (ES) is used for selection and optimization of heterogeneous catalytic materials.^21,22 Genetic programming has been employed very little. Baumes et al.^23,24 showed two examples of this very powerful technique. Genetic Algorithm (GA) has been done by various groups such as in the Pereira et al.²⁵ study. They reported a study of the effect of Genetic Algorithm (GA) configurations on the performance of heterogeneous catalyst optimization. Also, Gobin et al.^26,27 used multi-objective experimental design of experiments based on a genetic algorithm to optimize the combinations and concentrations of solid catalyst systems. Moreover, Genetic Algorithm merges with a knowledge based system²⁸ and has been boosted on a GPU hardware to solve a zeolite structure.^29,30 In addition, GA has been used for crystallography and XRD measurements^31,32 and as an Active Learning method for effective sampling.³³

2. Problem statement and objectives of the present study

Recently, we prepared SAPO-34 nanocrystals with a crystal size of about 50 nm by means of an efficient US procedure³⁴ and effects of ultrasound-related variables on the sonochemical synthesis of SAPO-34 nanocrystals were investigated.¹⁹ The physicochemical characteristics of SAPO-34 catalysts, i.e. crystallinity, BET surface area, crystal size and shape, are controlled by adjusting the US power intensity, sonotrode size, US irradiation time and sonication temperature. In this study, first the catalytic performance of the SAPO-34 nanocrystals synthesized sonochemically under various US conditions (US power intensity, sonication temperature, irradiation time and sonotrode size) in MTO reactions is studied. Afterwards, an artificial neural network and MLR model is applied for the prediction of catalytic performance. Finally, optimization of the catalytic performance is considered using the hybrid of NSGA-II based ANN. The ANN is used as a fitness evaluation inside the NSGA-II. The main objectives of optimization were to maximize conversion of methanol, maximize light olefins content and maximize lifetime.

3. Experimental

3.1. Catalyst preparation and characterization

The SAPO-34 catalysts were synthesized by the sonochemical method as described elsewhere^19,34 using a precursor gel with a molar ratio of 1.0Al₂O₃ : 1.0P₂O₅ : 0.6SiO₂ : 2.0TEAOH : 70H₂O. The sources for Al and P were aluminum isopropoxide [98% Al(OPrⁱ)₃, Merck] and H₃PO₄ [85 wt% aqueous solution, Merck], respectively. Tetraethylammonium hydroxide [TEAOH, 20 wt%, Aldrich] was used as a structure directing agent (SDA). Tetraethylorthosilicate [TEOS, Merck] was chosen as silica source for preparing primary gels. Aluminum isopropoxide was initially mixed with the template (TEAOH) and deionized water at room temperature and stirred for an hour. The silica source (TEOS) was then added and stirred. Finally, with continuous stirring, phosphoric acid was added dropwise to the above solution. The initial gel was irradiated with US waves at a frequency of 24 kHz. US irradiation was accomplished by means of an US Processor UP200H (Hielscher) using titanium sonotrodes with tip diameters of 3, 7 and 14 mm with different US power intensities. The amount of initial gel used for the sonication was 50 ml in each run. The sonication temperature was controlled at a temperature of 20, 30, 40 and 50 °C. The duration of irradiation was from 5 to 30 min. After sonication, the initial gel was placed in a 30 ml Teflon-lined stainless steel autoclave and heated in an oven at 200 °C for 1.5 h. The synthesis conditions of the samples are given in Table 1.

Table 1 Sonochemical synthesis conditions and properties of the catalysts

Sample	Ultrasonic power intensity (W cm^−2)	Ultrasonic irradiation time (min)	Sonication temperature (°C)	Sonotrode diameter (mm)	Relative crystallinity (%)	Distribution of acid sites (mmol NH3 g^−1)		Mean crystal diameter (nm)	BET surface area (m^2 g^−1)
						Weak (220–240 °C)	Strong (440–460 °C)
a Nd: Not determined.
S1	48	15	50	7	40	0.86	0.71	150	329
S2	108	15	50	7	48	0.8	0.94	100	369
S3	192	15	50	7	68	0.74	0.79	70	392
S4	300	15	50	7	77	1.01	0.78	50	493
S5	460	15	50	3	38	0.9	0.73	135	306
S6	105	15	50	14	100	—^Nd^a	—^Nd^a	66	444
S7	300	5	50	7	35	0.82	0.78	105	—^Nd^a
S8	300	30	50	7	88	—^Nd^a	—^Nd^a	58	429
S9	300	15	20	7	52	0.97	0.69	120	—^Nd^a
S10	300	15	30	7	47	0.94	0.60	115	—^Nd^a
S11	300	15	40	7	71	1.08	0.89	79	388

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The solid product was recovered and washed three times by centrifuging with distilled water, and then dried at 110 °C. The as-synthesized crystals were calcined at 560 °C in air for 5 h to remove the organic template molecules. Powder X-ray diffraction (XRD) patterns were recorded in step scanning on a Philips PW3050 X-ray diffractometer using CuKα radiation (λ = 1.54 Å) operating at 40 kV and 40 mA. The phase purity and the overall crystallinity were appointed by XRD. Crystallinity characterization of the samples was calculated using the following equation:

where I is the line intensity of the sample and I_r is the line intensity of the reference sample, using the product with the highest crystallinity, as identified by XRD. The line intensities of the XRD pattern at 2 h equal to 9.5 and 20.5 were employed for these calculations. The crystal morphology was analyzed using scanning electron microscopy (SEM, Philips XL30). The mean crystal diameters of the SAPO-34 samples were estimated by measuring the particle size of 100 particles in SEM images using Microstructure Measurement software. From the results, the mean crystal size was calculated as d, using the formula mentioned below:¹⁹d is a measure of volume/outer surface (d_i = size of particles). The BET surface areas of the calcined samples were determined from isotherm data of nitrogen adsorption data in the relative pressure (P/P₀) range of 0.05–0.30 obtained at 77.35 K using a Quantachrome Autosorb-1 analyzer. The surface acidity of the catalysts was measured using temperature programmed desorption of ammonia (NH₃-TPD) using a Micrometrics TPD/TPR 2900 analyzer with a TCD. The amount of ammonia desorbed from the catalyst was measured by comparing the TPD areas with that of the standard sample.

3.2. Catalytic performance

The conversion reactions of MTO over SAPO-34 catalysts were carried out in a fixed bed reactor made of quartz glass (i.d. 15 mm) with a continuous-flow system containing a preheater and a catalyst bed under atmospheric pressure. The catalyst (0.6 g) charged in the catalyst bed at the center of the quartz reactor was activated at 500 °C in a nitrogen flow of 100 ml min⁻¹ for 1 h before starting each reaction run and then cooled to the reaction temperature of 450 °C. A liquid mixture of methanol in water (20 vol%) was fed into the reactor. The feed rate was adjusted to 0.29 ml min⁻¹. The weight hourly space velocity (WHSV) was 4.5 h⁻¹. In the preheating zone (upper part of the reactor), the temperature increased to 450 °C at atmospheric pressure and the feed evaporated. In the catalyst bed at the center of the reactor, methanol vapor was converted to light olefins.

The analysis of the reaction products was performed using an on-line gas chromatograph Agilent GC (6890 N), equipped with a flame ionization detector (FID) and Plot-Q column. Methanol conversion (X_CH3OH) and light olefins content (C₂H₄–C₃H₆) are defined as below:

3.3. ANN modeling

ANNs are widely accepted as an information processing methodology inspired by the working process of the human brain. ANNs are efficient in handling the non-linear relationship of data.³⁵ The empirical models and correlations developed by conventional methods such as different types of multiple regression are complex in nature, difficult for predicting non-linear relationships, less accurate, and require long computing time. The ANN has numerous advantages, including accurate approximations of complex problems, greater efficiency than traditional statistical-based prediction models such as regression even for multiple response computations, and greater effectiveness even with incomplete and noisy input data.³⁶

ANNs constitute a branch of artificial intelligence which has recently undergone rapid evolution and progress.³⁷ ANNs act as a black box model, which is composed of interconnected processing units called artificial neurons or nodes.³⁵ The ANN approach has the ability to learn highly non-linear relationships and processes information by its dynamic system response to external inputs.³⁶

There are several types of ANNs such as the multilayer feed forward neural network (FFNN), recurrent, radial basis, function networks and self-organizing maps. The universal approximation theorem for neural networks states that every continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multilayer perceptron with just one hidden layer. This result holds for a wide range of activation functions, e.g. for the sigmoidal functions. So in this study, a multilayer feed forward neural network is used. The multilayer feed forward neural network consists of an input layer, one or more hidden layers and an output layer.^35,38 In this study, the input layer is composed of 3 nodes, which are the US intensity, US irradiation time and sonication temperature. The output layer has three nodes, which are the maximum conversion of methanol, maximum light olefins content and nanocatalyst lifetime which is the time on stream where the methanol conversion is more than 73%. There is no general rule for the determination of the optimum number of nodes in the hidden layer and usually it is determined through trial and error.^35,38–40

The nodes between each layer are connected with adaptable weights. The general working principle of the artificial neuron or node can be demonstrated as:³⁵

where, x_j is the input from the previous node j, w_ij is the weight that connects node i and node j, n is the total number of previous nodes connecting with node i, b_i is the bias of node i, and f is the activation (transfer) function. Feed forward neural networks with one hidden layer can approximate virtually any linear or non-linear function to an acceptable accuracy, if sufficient hidden layer nodes are provided with the sigmoid function as the hidden layer activation function and the linear function as the output layer activation function.³⁵ In this study, the tansig function and the purelin function are used as the activation functions in the hidden layer and the output layer, respectively. Therefore, a feed forward neural network with one hidden layer is applied to predict the experimental data in this study.

Before using an artificial neural network, it is necessary to train the network. There are different kinds of training methods, of which back-propagation is a common method.³⁷ In the current study, the back-propagation structure is implemented using different learning rules in the neural network approach.

Before using any method for training, an ANN has to normalize input and output. So input and output data are normalized between −1 and 1 using the following equation:

The prediction performance of the network is assessed by using statistical coefficients, i.e. root mean square error (RMSE), mean square error (MSE), correlation coefficient (R²) and mean relative error (MRE) values.^37,40 In this study, the mean square error (MSE) and correlation coefficient (R²), which are calculated using the following expressions, are used as the correlation performance indicators of the network:

where ȳ_actual and ȳ_calculated are arithmetic means of the actual and calculated values, respectively.

3.4. NSGA-II main process

The genetic algorithm is an optimization tool based on Darwinian evolution.^41–43 The principles, on which the NSGA-II relies, are the same as those of the single-objective optimization. The strongest individuals (or chromosomes) are combined to create the offspring by crossover and mutation and this scheme is repeated over many generations. However, the multi-objective optimization algorithm must consider the fact that there are many “best solutions”, which modify the selection process. NSGA-II sorts individuals based on the non-domination rank and the crowding distance to ensure a high level of performance as well as good dispersion of results.^44–48 In this study, the ANN was used as a fitness evaluation inside the NSGA-II. The flowchart of the hybrid of the non-dominated sorting genetic algorithm-II based artificial neural network (NSGA-II-ANN) program is shown in Fig. 1.

Fig. 1 Flowchart of the NSGA-II-ANN program.

4. Results and discussion

4.1. Catalytic performance of SAPO-34 nanocatalysts

The methanol conversion and light olefins content (wt%) (C2=C3=) over SAPO-34 catalysts synthesized by using the sonotrode with a tip diameter of 7 mm with different US power intensities are presented in Fig. 2. As indicated in Fig. 2, all SAPO-34 catalysts suffer activation in the first minutes of the process. The incomplete conversion of methanol in the early time on stream is due to the existence of an induction period⁴⁹ which can be explained by the hydrocarbon pool (HP) mechanism. Haw suggested that the MTO reaction proceeds by a HP mechanism with cyclic organic species such as hexamethylbenzene (HMB) as reaction centers for light olefins production.^49,50 The time for the formation of these cyclic organic species causes the induction period, i.e. an increase in activity before maximum conversion. Except for sample S1, all catalysts show a high conversion for a long time especially catalyst S4 which maintains its activity for 540 min. The S1 catalyst just shows a maximum conversion of 81% after 60 min. A rapid fall in conversion is subsequently observed and it reaches 12% after 420 min.

Fig. 2 Methanol conversion and light olefins content (C₂H₄–C₃H₆) over SAPO-34 catalysts synthesized with different ultrasound power intensities; S1: 48 W cm⁻², S2: 108 W cm⁻², S3: 192 W cm⁻², S4: 300 W cm⁻².

The nano-sized catalyst S4 shows a high light olefins (ethylene and propylene) content (wt%) for 180 min (Fig. 2) and the light olefins content decreases afterwards. Even after the reaction time of 540 min the light olefins content can remain at 10% over S4. The decrease in light olefins formation corresponds to an increase in the formation of DME which can be explained by the catalyst deactivation.⁵¹ At this reaction time, DME is the main product (79 wt%) and other products are butene and the alkanes (C1–C5 alkanes), mainly methane.

The catalysts S2 and S3 also show the same trend for light olefins content but with lower amounts compared to that of the catalyst S4. In comparison, the content of light olefins over the SAPO-34 catalyst S1 is much lower than that of the others and it decreases rapidly (ESI, Table I†).

These catalytic behaviors can be explained by the differences in the catalysts’ physicochemical properties which are affected by the US power intensity. According to the results obtained previously,¹⁹ the crystallinity changes markedly with the US power intensity and a higher crystallinity is obtained with increasing US intensity. Therefore, the catalyst S1 synthesized at the lowest power intensity has the lowest crystallinity as shown in Table 1. Increasing the crystallinity results in the formation of more structural pores and it may enhance the diffusion rate and quantity of reactants into structure channels in crystallites.¹⁹ Also, different US power intensities applied in the synthesis of SAPO-34 crystals alter the morphology and agglomeration of the products. It is clearly seen in Table 1 that with an increase in the US intensity the average crystal size of SAPO-34 crystals gradually decreases, and the formation of uniform spherical nanocrystals (ESI, Fig. S1†) is preferred over the spherical aggregates of cube type SAPO-34 crystals seen previously.¹⁹ By comparing the acidity of the SAPO-34 catalysts represented in Table 1, the S2 and S1 samples have the highest and lowest concentration of strong acid sites, respectively, which could be active for MTO conversion, though the strength of their acid sites are similar. Therefore, the short lifetime of the catalyst S1 cannot be related to the strength and number of acid sites but can be correlated with the crystallinity and crystal sizes of the catalyst. For larger crystals of the catalyst S1, the residence time in a crystal for the hydrocarbons is long because of the longer diffusion length. Saturated hydrocarbons and/or aromatic compounds cannot escape from the pores of SAPO-34, and successive polymerizations readily occur because of the long reaction time.² Therefore, the catalyst S4 possessing the higher crystallinity, smaller size and better dispersion¹⁹ has a lower diffusion limitation and a longer lifetime.⁵¹ Additionally, the formation of coke is inhibited over the smaller crystals of catalyst S4 as a result of the reduced resistance to diffusion, resulting in a much greater production of light olefins for longer reaction times.

Effects of the sonotrode size on the catalytic properties of SAPO-34 catalysts S4–S6 are shown in Fig. 3. The catalysts S4 and S6 synthesized using the sonotrodes with tip diameters of 7 and 14 mm show a much higher conversion of methanol and maintain it for a longer time as well as a higher light olefins content compared to that of the catalyst synthesized with the sonotrode with a tip diameter of 3 mm (S5). This can be explained by an increase in US power with the increasing sonotrode size from 3 to 14 mm. Although the smallest sonotrode possesses the highest US intensity, i.e. 460 W cm⁻² (related to 100% amplitude setting) compared to 300 and 105 W cm⁻² (max. intensities of sonotrodes with tip diameters of 7 and 14 mm, respectively), it has the smallest horn tip area, resulting in a lower emitted US power. The increase in the US power results in the higher turbulence of the mixture and satisfactory mixing can be achieved much more rapidly. The difference in the US power results in different catalyst activities which can be attributed to the difference in the crystallinity and thus the BET surface area of the catalysts. It is clear from Table 1 that by increasing the US power the crystallinity of the samples increases (ESI, Fig. II†), resulting in the rise of BET surface areas.

Fig. 3 Effects of the sonotrode size on the catalytic properties of SAPO-34 catalysts; S5: 3 mm, S4: 7 mm, S6: 14 mm.

Fig. 4 shows the role of US irradiation time on the methanol conversion and light olefins production. The catalyst irradiated for 5 min (S7) shows the lowest methanol conversion and the lowest olefins content on account of the absence of the crystal phase. In short times, the US wave fails to blend the solution uniformly and only a few nuclei are formed. Some small crystal particles cannot grow further due to the shorter US irradiation time. Applying US irradiation for longer times produces more apparent crystals and no large crystals can grow, because more nuclei will occur continuously until the level of super-saturation becomes very low.^19,20 With the extension of the US irradiation time from 5 to 15 min, the methanol conversion as well as the olefins content increases sharply, and the samples irradiated for 15 and 30 min show the highest olefin selectivity. This result confirms that the activity of the SAPO-34 samples improves with the rise in crystallinity.

Fig. 4 Effects of ultrasonic irradiation time on the methanol conversion and light olefins production over SAPO-34 catalysts; S7: 5 min, S4: 15 min, S8: 30 min.

Fig. 5 shows the catalytic performance of the products synthesized under different sonication temperatures of 20, 30, 40 and 50 °C. With an increase in the sonication temperature, the crystallinity of the SAPO-34 products increases, smaller crystals with a uniform size distribution are formed and the morphology of the product alters from cubic crystals to uniform nanocrystals since the temperature can affect the cavitation threshold. Generally, the cavitation threshold limit has been found to decrease with an increase in temperature. This means that cavitation bubbles are more easily produced as the temperature is raised. Therefore, a higher sonication temperature leads to fast nucleation, which results in smaller particles in comparison to those from the lower sonication temperature.¹⁹ According to Table 1, the catalysts S9 and S10 have the lowest strong acid site concentration at 440–460 °C and it falls in a narrow interval for samples S9 and S10, i.e., 0.69 and 0.60, respectively. Also, the acid strength of all the samples is almost the same. Therefore, it can be concluded that the differences between the catalytic behaviors of samples could be more related to the crystal size and crystallinity of the catalyst.

Fig. 5 Catalytic performance of the SAPO-34 products synthesized under different sonication temperatures; S9: 20 °C, S10: 30 °C, S11: 40 °C, S4: 50 °C.

4.2. ANN results

In this study, the multilayer feed forward neural networks with back-propagation structure are implemented using different learning rules in the neural network approach such as Bayesian regulation (BR), Levenberg–Marquardt (LM), scaled conjugated gradient (SCG) and RPROP back-propagation (RP). In the Bayesian regulation rule, the input data are divided into two parts; 70% and 30% of the data are used for training and the test, respectively, but in other learning rules the input data are divided into three parts; 70%, 15% and 15% of the data are used for training, validation and the test, respectively. The values of the training and test data are normalized between −1 and 1 using eqn (5). The number of data values used in the network is 54. In this data set the sonotrode size is 7 mm. The range of all experimental conditions used for modeling with ANNs is shown in Table 2.

Table 2 Range of all experimental conditions used for modeling with ANNs

Inputs
Sonication temperature (°C)	20–50
Ultrasound intensity (W cm^−2)	12, 27, 48, 75, 108, 147, 192, 243, 300
Ultrasonic irradiation time (min)	5–30

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Outputs
Methanol conversion (%)	70–100
Light olefins content (wt%)	36–84
Lifetime (min)	50–540

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The results of the network models for different neuron numbers in the hidden layer using different training algorithms are presented in Table 3. The correlation performance of the network is assessed using the mean square error (MSE) and correlation coefficient (R²) values. It can be seen that the Bayesian regulation back-propagation algorithm with 10 neurons in the hidden layer is the best training procedure that achieved the highest R² and lowest MSE values. Thus, the optimum number of neurons is used to create the network topologies which were 3–10–3. Here, the numbers in the expressions of the network topologies represent the neuron numbers in the input layer, the hidden layer and the output layer, respectively.

Table 3 Performance of different ANN methods for the prediction of catalytic performance of the SAPO-34 products synthesized in MTO reactions

Learning rule	No. of neurons	R^2				MSE
		Methanol conversion	Light olefins content	Lifetime	Mean	Methanol conversion	Light olefins content	Lifetime	Mean
LM	5	0.9808	0.9712	0.8308	0.9276	0.0045	0.0058	0.0204	0.0102
	10	0.9932	0.9759	0.9839	0.9843	0.0017	0.0049	0.0020	0.0029
	15	0.9957	0.9760	0.9929	0.9882	0.0010	0.0059	0.0009	0.0026
BR	5	0.9907	0.9756	0.9917	0.9860	0.0022	0.0049	0.0009	0.0027
	10	0.9957	0.9796	0.9922	0.9892	0.0010	0.0042	0.0009	0.0020
	15	0.9949	0.9538	0.9523	0.9670	0.0013	0.0094	0.0080	0.0062
SCG	5	0.9846	0.9673	0.9037	0.9519	0.0036	0.0065	0.0109	0.0070
	10	0.9904	0.9751	0.9397	0.9684	0.0022	0.0049	0.0069	0.0047
	15	0.9782	0.9606	0.8143	0.9177	0.0052	0.0080	0.0239	0.0123
RP	5	0.7583	0.8336	0.7886	0.7935	0.0580	0.0332	0.0260	0.0391
	10	0.8849	0.8121	0.7781	0.8250	0.0267	0.0377	0.0250	0.0298
	15	0.8146	0.8671	0.713	0.7982	0.0450	0.0274	0.0327	0.0350

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Fig. 6 explains the comparison plots between the network output and the corresponding experimental data of the catalytic performance of SAPO-34 nanocatalysts synthesized sonochemically, for training and test data using the BR training rule and a 3–10–3 topology. This figure shows that there is a very good agreement between this ANN model and experimental values. The comparison between network-predicted and experimental data can be seen in Table II of the ESI.† Fig. 7 shows the performance graph. It is observed that the desired goal has been reached in 1000 epochs, and the ANN with ten hidden neurons could achieve convergence. The errors in data attained by the optimum ANN model are plotted versus the frequency of data in Fig. 8. A normal distribution of variation brings about a specific bell-shaped curve (Gaussian curve), with the highest point in the middle and smoothly curving symmetrical slopes on both sides of the center. These figures illustrate an approximately normal distribution of errors produced by the model. The Gaussian curve reveals that our results are symmetrical.³⁸

Fig. 6 Comparison between experimental and ANN values for (a) methanol conversion (%), (b) light olefins content (wt%) and (c) lifetime.

Fig. 7 Performance graph.

Fig. 8 Error histogram.

The methanol conversion, light olefins content and catalyst lifetime are illustrated versus US power intensity, sonication temperature and irradiation time in the ESI (Fig. III–V in the ESI†). Also a Matlab user-friendly code is given as supplementary data for verification of the ANN results (‘ANN.m’, ‘deNormalize FCN.m’ and ‘Normalize FCN.m’ in the ESI†).

3-Fold cross-validation (3-CV) of the model is used in this section because of the low number of points. The original sample is randomly partitioned into three equally sized subsamples. Of the three subsamples, a single subsample is retained as the validation data for testing the model, and the remaining two subsamples are used as training data. The cross-validation process is then repeated three times (the folds), with each of the three subsamples used exactly once as the validation data. The three results from the folds are averaged to produce a single estimation. The results of the 3-CV models are presented in Table 4. This shows that this model has achieved a very high R² and low MSE. The advantage of this method is that all observations are used for both training and validation, and each observation is used for validation exactly once.

Table 4 The results of the 3-CV models

No. of run	R^2				MSE
	Methanol conversion	Light olefins content	Lifetime	Mean	Methanol conversion	Light olefins content	Lifetime	Mean
1	0.9999	1.0000	1.0000	1.0000	0.0002	0.0009	0.0008	0.0006
2	0.9998	0.9171	0.9564	0.9717	0.0001	0.0031	0.0010	0.0014
3	0.9994	0.9983	0.9998	0.9992	0.0001	0.0008	0.0007	0.0005
Mean	0.9997	0.9718	0.9854	0.9856	0.0001	0.0016	0.0008	0.0008

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4.3. MLR model results

In this section, the traditional models of objection functions including methanol conversion, light olefins content and catalyst lifetime, obtained through linear regression analysis, are presented. These models used US power intensity, sonication temperature and irradiation time as input variables. These models are shown in eqn (8)–(10) and comparison plots between the model output and its corresponding experimental data are given in Fig. 9.

X_CH3OH = 0.1082 × P + 0.7721 × t + 1.1298 × T (8)

LO = 0.0946 × P + 0.5296 × t + 0.9030 × T − 0.9122 (9)

τ = 0.4919 × P + 0.1942 × t + 4.4912 × T (10)

where X_CH3OH, LO, τ, P, t and T are methanol conversion (%), light olefins content (wt%), lifetime (min), US power intensity (W cm⁻²), sonication temperature (°C) and irradiation time (min), respectively.

Fig. 9 Comparison between experimental and MLR model values for (a) methanol conversion (%), (b) light olefins content (wt%) and (c) lifetime.

The results indicate a poor fit between experimental and predicted data with a low coefficient of determination. Thus, the objective functions prediction of using these traditional models may not be reliable. Table 5 shows a comparison between the ANN and MLR model results. This confirms that the ANN technique is more effective at predicting the objective functions tested in this study than the traditional statistical-based prediction models.

Table 5 Comparison between the optimum ANN and MLR model results

Method	R^2				MSE
	Methanol conversion	Light olefins content	Lifetime	Mean	Methanol conversion	Light olefins content	Lifetime	Mean
MLR	0.7005	0.7411	0.5121	0.6512	60.7065	68.1923	9207.9348	3112.2779
ANN	0.9957	0.9796	0.9922	0.9892	0.0010	0.0042	0.0009	0.0020

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4.4. Optimization results using the NSGA-II

In this section, the hybrid NSGA-II based ANN is used to optimize catalytic performance of SAPO-34 nanocatalysts synthesized sonochemically in the MTO reaction and obtains Pareto-optimal solutions. In other words, the ANN is used as a fitness evaluation inside the NSGA-II. A Pareto-optimal set is a series of solutions that are non-dominated with respect to each other. While moving from one Pareto solution to another, there is always a certain amount of sacrifice in at least one objective to achieve a certain amount of gain in another.

The objectives of optimization were to minimize unreacted methanol, maximize light olefins content and maximize lifetime. Thus, the objectives included:

Objective 1: maximizing X_CH3OH (11)

Objective 2: maximizing light olefins content (wt%) (12)

Objective 3: maximizing lifetime (min) (13)

The constraints were:

20 °C ≤ sonication temperature ≤ 50 °C (14)

US intensity = 12, 27, 48, 75, 108, 147, 192, 243, 300 W cm⁻² (15)

5 min ≤ US irradiation time ≤ 30 min (16)

The objective functions are optimized fulfilling the constraints given in eqn (14)–(16). The sonotrode size is 7 mm. A population size of 100 is chosen with a crossover probability of 0.7 and mutation probability of 0.1. Arithmetic crossover and the gauss method were used as crossover and mutation methods, respectively.

The optimum solutions are listed in Table 6. This table shows that each of the solutions is better than the other in at least one of the objective functions. Thus, the user has to decide on the ultrasound-related variables based on the ease of operation, experience, the cost involved and also the quality of the product (Fig. VI in the ESI†). Three samples of optimum solutions are validated experimentally. Table 7 shows a comparison between experimental and predicted values. The results indicate that there is an acceptable agreement between the experimental and predicted results by the NSGA-II-ANN. The methanol conversion, light olefins content and catalyst lifetime of the optimum solutions are illustrated versus US power intensity, sonication temperature and irradiation time in the ESI (Fig. VII–IX in the ESI†).

Table 6 Pareto-optimal solution set after 200 generations

No.	Ultrasonic intensity (W cm^−2)	Ultrasonic irradiation time (min)	Sonication temperature (°C)	Methanol conversion (%)	Light olefins content (wt%)	Lifetime (min)
1	300	25	29	100	92	486
2	300	22	29	100	87	515
3	300	22	30	100	90	496
4	243	23	31	95	72	583
5	243	23	30	93	68	668
6	243	22	31	95	71	615
7	243	21	30	90	63	706
8	243	20	33	95	73	531
9	243	14	48	97	70	521
10	192	13	48	96	65	546

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Table 7 Comparison between experimental and predicted values of the optimum solutions

Test no.	Ultrasonic intensity (W cm^−2)	Ultrasonic irradiation time (min)	Sonication temperature (°C)	Methanol conversion (%)		Light olefins content (wt%)		Lifetime (min)
				Experimental	Predicted	Experimental	Predicted	Experimental	Predicted
1	300	25	29	100	100	86	92	495	486
2	243	23	30	90	93	74	68	655	668
3	243	14	48	98	97	76	70	535	521

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Conclusions

The MTO reaction was investigated over SAPO-34 nanocatalysts synthesized sonochemically under various US conditions. The sonochemically prepared catalysts were used to elucidate the effects of US power intensity, sonication temperature, irradiation time and sonotrode size on their catalytic performance, especially in terms of methanol conversion, light olefins content and catalyst lifetime. The catalysts synthesized with a higher US power (adjusted by either the size of the sonotrode or US power intensity) show a much higher conversion of methanol and maintain it for a long time as well as a high light olefins content. By increasing the sonication temperature and irradiation time, the methanol conversion as well as the light olefins content increases. Afterwards, the multilayer feed forward neural network with a back-propagation structure was implemented using different learning rules in the neural network approach. The ANN with a 3–10–3 architecture and a Bayesian regulation training rule has the best fit and was selected as the optimum ANN model for prediction. The tansig and purelin functions were used as the activation functions in the hidden and output layers, respectively. In the optimum model, the input data were divided into two parts; 70% and 30% of the data were used for training and the test, respectively. The results indicate that there is a very good agreement between the experimental and predicted results by the ANN. Also, multiple linear regression (MLR) was used to predict these objective functions. The results indicate a poor fit for the objective functions with a low coefficient of determination. This confirms that the ANN technique is more effective than the traditional statistical-based prediction models. Finally, the hybrid NSGA-II based ANN was used and introduced the best catalytic performance of SAPO-34 nanocatalysts synthesized sonochemically under various US conditions in the MTO reaction and the ANN was used as a fitness evaluation inside the NSGA-II. Such a methodology (hybrid of the ANN and NSGA-II) allowed the researchers to find a near optimum solution to their problem where multiple input and output variables are interacting. Despite the relative simplicity of the search space, human understanding and capture of the variables’ relationships still remain complicated. Therefore such an approach shows how the use of both a modeling and an optimization strategy allow this challenge to be tackled.

Acknowledgements

The financial supports from the Iran National Science Foundation are gratefully acknowledged.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra03764f

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