Production of Green Fuel Using a New Synthetic Magnetite Mesoporous Nano-Silica Composite Catalyst for Oxidative Desulfurization: Experiments and Process Modeling

Abstract

Producing an eco-friendly fuel with the least amount of sulfur compounds has been an ongoing issue for petroleum refineries. In this study, bentonite (which is a cheap material and is locally available in abundance) is employed to prepare a nano-silica catalyst with a high surface area to be used for the oxidative desulfurization of kerosene. Two composite catalysts of Fe/silica were supported on CAT-1 (0% HY-zeolite and 100% nano-silica) and CAT-2 (20% HY-zeolite and 80% nano-silica). The activity of the catalysts was evaluated in a batch ODS (oxidative desulfurization) process at temperatures of 30, 60, 90, and 120 °C, a pressure of 1 atm, and a reaction time of 30, 60, 90, and 120 min using 120 L/h of air as the oxidant. The results revealed that the highest total sulfur removal efficiency was 50% and 87.88% for 100% nano-silica (CAT-1) and 80% nano-silica (CAT-2), respectively. The experimental data were then used to construct and validate an accurate mathematical model of the process. The operational parameters for eliminating more than 99% of sulfur and producing eco-friendly fuel were then achieved by using the model. The testing methods for these characterizing materials included X-ray diffraction (XRD), thermal gravimetric examination (TGA), X-ray fluorescence (XRF), and surface area (BET). The outcomes indicated that the addition of HY-zeolite increased the activity of the catalyst (CAT-2 > CAT-1).

1. Introduction

Currently, the production of eco-friendly fuel to combat climate change is a major concern worldwide. Globally, 82% of energy is provided by fossil fuels, but it has been noticed that the use of fossil fuels has declined over the years [1]. The main source of energy used globally are petroleum fractions and their main usage is in the transportation sector, including jet fuel, diesel, and gasoline. The removal of toxic components such as sulfur [2] from the fuel that is widely used in transportation and in the petrochemical industry is absolutely essential from an environmental point of view. Fuels containing sulfur, like Benzothiophenes (BTs), Thiophenes (Ts), Dibenzothiophenes (DBTs), and Mercaptans (RSH), can be subjected to oxidative desulfurization, resulting in the production of sulfur-di-oxide, which can be easily removed from the fuel, thus producing clan fuel [3,4]. This process uses various types of oxidants, viz., air, H₂O₂, O₂, and O₃. Air is utilized due to its accessibility; it is inexpensive and causes no further pollution [5,6]. However, the choice of catalyst for sulfur compounds is considered the most critical issue in designing an efficient ODS process. Thus, researchers have continued to develop different types of catalysts for more rapid, economical, and efficient ODS processes in order to select the best type for the petroleum industry [7,8]. Magnetite mesoporous is an excellent catalyst, as it has a tremendous capacity to eliminate methyl mercaptan due to the robust molecular affinity of magnetite and the huge surface area and pore dimensions of mesoporous silica [9,10]. In this study, the authors have prepared a novel catalyst based on iron oxide magnetite supported on nano-silica and zeolite nanoparticles. The use of such catalysts has not yet entered the public domain. The primary objective of this study is to produce an eco-friendly fuel by eliminating sulfur compounds from kerosene in a batch reactor using the ODS technique. Another goal of this study is to create a mathematical model of the process that has excellent precision and can be optimized and simulated, providing in-depth understanding of the process. Experimental data and model predictions will be used within an optimization framework to estimate unknown model parameters (such as kinetic parameters).

2. Results and Discussion

2.1. Evaluation of Prepared Catalysts

2.1.1. Chemical Composition

X-Ray fluorescence (XRF) was used to determine the existence of metal oxides, the quantity of active metal (Fe₂O₃) and nano-silica support, and the structural composition.

Table 1. Results of the produced nano-silica’s XRF analysis.

Element	Symbol	Concentration %
Silicon	SiO2	71.14
Magnesium	MgO	0.0552
Aluminum	Al2O3	0.4179
Phosphorus	P2O5	0.5621
Sulfur	SO3	5.729
Chlorine	Cl	0.03876
Potassium	K2O	<0.0012
Calcium	CaO	0.1619
Titanium	TiO2	3.010
Chromium	Cr2O3	0.01824
Manganese	MnO	0.00367
Iron	Fe2O3	0.6977
Cobalt	CoO	<0.00039
Nickel	NiO	0.00648
Copper	CuO	0.00075
Zinc	ZnO	0.00274
Gallium	Ga	0.00045
Another element	-	0.11952
Sum of concentration	-	81.97

demonstrates the synthesized nano-silica’s purity, and

Table 2. The prepared catalysts’ XRF results.

Element	Symbol	Concentration %
Silicon	SiO2	51.56
Magnesium	MgO	0.0863
Aluminum	Al2O3	11.24
Phosphorus	P2O5	0.6316
Sulfur	SO3	3.652
Potassium	K2O	0.0927
Calcium	CaO	0.1326
Titanium	TiO2	2.995
Vanadium	V2O5	0.0314
Chromium	Cr2O3	0.00472
Manganese	MnO	0.01570
Iron	Fe2O3	10.88
Copper	CuO	0.00603
Zinc	ZnO	0.00510
Another component		0.091201
Sum of concentration		81.45

represents the XRF analysis of the prepared catalyst.

2.1.2. X-ray Diffraction

The oldest and most widely used technique for characterizing catalysts is thought to be X-ray diffraction. Lattice structural parameters are used to identify the crystalline phase within catalysts and to determine the size of the particles. The X-ray diffraction instrument used in this study was capable of generating Ni-filtered cuK radiation with a wavelength of 1.5418 Å. The instrument operated at 40 kV and 30 mA during the XRD measurements of the catalyst samples. The diffraction patterns were recorded over a 2θ angular range of 3° to 60°. X-ray diffraction is a well-established technique that can distinguish materials based on their characteristic diffraction patterns, with a scanning rate of 5°/min. It offers statistics on how far a specific structure deviates from the overall structure, in addition to section identification. High-resolution X-ray diffraction was used to record the crystal structure of the particles. Additionally, the presence of an amorphous silica component was confirmed by a large diffraction peak at 2 = 20°–24° [11]. The clearly defined nano-silica diffraction peak revealed the hypothesized existence of a highly structured hexagonal mesostructure. Figure 1 below illustrates the XRD for amorphous nano-silica.

2.1.3. Fourier Transform Infrared (FTIR)

A PerkinElmer FTIR spectrometer was utilized to obtain the Fourier transform infrared spectroscopy (FTIR) results depicted in Figure 2. The peaks observed at 1087.85 and 796.90 cm⁻¹ indicate the presence of asymmetric and symmetric stretching modes of the Si-O-Si bonds. Furthermore, a band centered at 462 cm⁻¹ was detected, which is attributed to the Si-O-bending vibration of the sample containing iron and no other elements aside from silicon and oxygen. Additionally, features at 92 cm⁻¹ are noted at DLA. The broad band around approximately 3481 cm⁻¹ is attributed to the increasing vibrational frequencies of the silanol group and the remaining absorbed water.

Notably, no peaks are observed in the 2800–3000 cm⁻¹ range in the spectra, indicating the absence of significant vibrational modes related to the water–silica matrix interactions. This actually implies that there were no actual natural mixes in the silica after controlled calcination. The top positions of the collected nano-silica showed no notable changes in the FTIR continuous flow. Silica with a large surface area is the best multipart for use as a catalyst backing. Additionally, it has been discovered that the surface area is related to the silica reactivity.

Spectroscopic investigation provided unequivocal evidence that the mesopore walls were made of a zeolite-like structure [12,13,14]. The scientific literature frequently cites bands in the range of 550 to 600 cm⁻¹ when investigating materials containing MFI-type and zeolite β seed structures. These bands are believed to correspond to the asymmetric stretching vibration of the 5-membered ring components within the material’s framework. The tetrahedral coordination of Al is a further intriguing characteristic of the aluminosilicate phase. This is significant because it emphasizes the existence of a potent Bronsted acid center of the zeolite variety, which is useful for research as a microporous material, particularly as a possible heterogeneous catalyst [13,14,15,16,17]. Figure 3 shows the results of the produced catalyst’s FTIR examination.

2.1.4. TGA (Thermal Gravimetric Analysis)

A TGA Model SDT 2960 device, owned by the Oil Research and Development Center located in Baghdad, Iraq, was utilized to examine the silica particles. The tests were conducted at a heating rate of 5 °C/min in a nitrogen environment from ambient to 900 °C. TGA profiles for the nano-silica and composite samples are shown in Figure 4A,B. It can evidently be seen from these figures that the mass losses of both samples were the same before and after the process of loading active metals. The curves display two areas of mass loss, the first of which (between 20 and 120 °C), with mass loss values of around 10%, is caused by the evaporation of water that has been physically adsorbed on the solid. In the TGA measurements, mass loss values of 16.74% for the support’s nano-silica and 16.79% for the catalysts were noted. Such favorable outcomes demonstrated the strong thermal stability of the supports and of the manufactured catalysts.

2.1.5. Particle Size Distribution

The connection between an assumed particle dimension and the frequency of that size or diameter is known as “particle size distribution”. Atomic Force Microscopy (AFM) is conducted by ethanol dispersion to measure the particle size distribution. The basis of this discussion rests on the perceived notion that the majority of particles exhibit remarkably similar shapes. It is crucial, when evaluating the catalytic properties of metal-backed catalysts, to account for the actual sizes and distributions of the particles. The physical dimensions of the metal particles determine their surface area, and consequently the potential number of active sites, as well as the electrical characteristics of the metal, which are dependent on particle size. Even though such catalysts may share similar average particle sizes, the distributions of particle extents can vary dramatically, and the characteristics of the compared catalysts may differ significantly as a result. Figure 5 explains the particle size distributions of nano-silica and the produced catalysts.

2.1.6. FESEM-EDX

The SiO₂ nanocatalyst’s surface shape and real particle size were visible in SEM, in large part due to the catalyst’s enormous surface area and mesoporous frequencies functioning as beds and supports. This technique was utilized using a Zeiss-EM10C-100KV FESEM device (Oberkochen, Germany). Figure 6 below displays a SEM picture of the nano-silica.

Figure 7 displays the SiO₂ nanocatalyst’s surface elemental composition as determined by EDX analysis. Si and O are present in every sample. The map displays the relative dispersion density and the EDX element map. The distribution and consistency of the dispersed particles can be examined through an EDX (energy-dispersive X-ray) image of the catalytic converter’s external surface. The EDX mapping analysis revealed the presence of the following elements: silicon (37.6% by weight), oxygen (32% by weight), carbon (8.7% by weight), sodium (5.7% by weight), iron (5.4% by weight), and aluminum (10.6% by weight). Figure 7, which depicts the EDX image for the nano-silica generated in this work, illustrates these details.

The bimetal nanocatalyst’s (Fe₃O₄/SiO₂) surface shape and particle size were demonstrated by FESEM. SiO₂ acted as a couch support for Fe₃O₄ and zeolite due to its variable surface and mesoporous frequencies. As seen in the image, several forms of iron were immobilized on a mesoporous silica surface. The SEM image of the catalyst (Fe/nano-silica) created in our work is given in Figure 8 below. Additionally, the micrographs (magnification of 1 m) demonstrate a uniform 100 distribution of the active metal (Fe) on the catalyst’s surface along with the presence of spherical particles and well-developed pores.

The scanning electron micrograph in Figure 9 depicts the Fe/SiO₂ nanocatalyst, and the corresponding energy-dispersive X-ray (EDX) analysis shows its surface component composition. The EDX results presented in Figure 9 characterize the sample compositions, which contain Si, O, and Fe components. As the metal oxide loading increased, the surface Fe concentration rose, while the Si concentration decreased, indicating a more uniform distribution of Fe on the SiO₂ nano-silica surface. The relative dispersion density and EDX elemental map are shown in the provided figure. When a sample was taken from the exterior surface of the catalytic converters and analyzed using EDX, the detected elements were found to be distributed uniformly across the converter’s outer surface. The EDX mapping reveals a composition of C (5% wt), O (34.2% wt), Na (4.7% wt), Si (34.1% wt), Fe (9.6% wt), and Al (12.4% wt).

2.2. Experimental Section

A lab-synthesized catalytic system with nanoparticles on zeolite was tested in a batch reactor. The influence of reaction temperature, reaction duration, and initial sulfur species concentration was investigated.

2.2.1. The Influence of Reaction Temperature

The reaction temperature investigated in the transformation of ODS reactions was 303 K, 333 K, 363 K, and 393 K. The results are illustrated in Figure 10.

The observed increase in reaction temperature appears to coincide with an enhanced conversion of sulfur-containing compounds. This observation can be attributed to the following factors:

• The higher temperature favorably influenced the reaction rate, as described by the Arrhenius equation. This leads to a notable rise in the conversion of sulfur-based compounds, as reported in the literature [18].

$$K=K_o{e}^{(-{\displaystyle \frac{EA}{RT}})}$$

(1)

In addition to the activation energy, the reactant temperature affects both the reaction rate constant and the Arrhenius equation in an inverse and direct proportion, respectively.

Higher temperatures increase the number of molecules with sufficient activation energy, facilitating the reaction between the oxidant and organic compound, thereby improving conversion. This behavior is consistent with the literature [19], suggesting that temperature can influence the oxidation reaction rate based on the activation energy.

• Furthermore, temperature can affect various transport and physical properties of the gases and liquids involved, such as density, viscosity, gas diffusivity, and Henry’s constant. Specifically, an increase in temperature leads to an increase in both Henry’s constant and the diffusivity of reactants (sulfur and oxidant in the presence of catalyst) in the reaction media. Additionally, temperature impacts properties like viscosity, which decrease with increasing temperature. However, the oxidation reaction is favored with higher temperatures, as the absorption of molecular oxygen in the liquid, the dispersion of sulfur compounds, and the rate of dissolved oxygen reaching active sites inside the catalyst pores all increase with rising temperatures.
• The higher temperatures can be explained by the fact that faster-moving reactants have a higher probability of interacting. Moreover, for the catalyst used, the finest sulfones may have been readily desorbed from the surface at high temperatures [20].
• The catalysts’ oxidation temperature has a substantial effect on their activity, and the density of acidic centers, which determine the catalytic properties, depends on the temperature at which the catalyst has been calcined. This has led to enhanced removal of sulfur compounds from diesel fuel [21,22].

2.2.2. The Impact of Reaction Time

This study investigated an oxidation reaction at various time intervals, including 30 min, 60 min, 90 min, and 120 min. The results were presented graphically, and it was observed that the experimental findings for different temperatures were unique for each catalyst. In all cases, the desulfurization efficiency increased as the reaction time was extended, as a longer reaction time allowed for enhanced contact between the reactants. The interpretation of the figures can be summarized as follows: Figure 11A indicates that an increase in reaction time led to an increase in the conversion rate. With extended reaction time, the reactants had more opportunity to interact with the active sites of the catalyst, thereby enhancing the overall interaction process [23,24,25]. For instance, by simply increasing the reaction time from 30 min to 120 min at 303 K, the conversion of sulfur compounds rose from 33% to 79.83%. Images presenting the obtained results are provided in Figure 11 below.

2.3. Estimation of Kinetic Parameters

Kinetic parameter values are obtained when the model equations produce the closest correspondence to the experimental data [26,27,28]. In this study, the mathematical model was leveraged to estimate the kinetic parameters of the ODS process under various operational conditions and sulfur concentrations, where the maximum correlation with the measured data was achieved. Two methodologies were considered: linear growth and non-linear growth. The model takes into account the constant parameters detailed in

Table 3. Parameters of the ODS model and their values.

Parameter	Notation	Measurement Unit	Numerical Value
Initial amount	Ct	wt%	0.1966
Duration	time1, time2, time3, time4	min	time1 = 30, time2 = 60, time3 = 90, time4 = 120
Temperature	T1, T2, T3, T4	K	T1 = 303, T2 = 333, T3 = 363, T4 = 393
Kerosene density at 15.5 °C	${\mathsf{\rho }}_l$	gm/cm3	0.8205
Average boiling point	TmeABP	°R	787
Acceleration gravity	g	m/sec2	9.81
Gas constant	R	J/mole.°K	8.314
Pore volume per unit catalyst mass	Vg	cm3/gm	VgCAT-1 = 0.606 VgCAT-2 = 0.625
Particle surface area	Sg	cm2/gm	SgCAT-1 = 2,426,700 SgCAT-2 = 2,501,380
Catalyst particle volume	Vp	cm3	VpCAT-1 = 5.2158 × 10−19 VpCAT-2 = 5.2245 × 10−19
Particle’s external surface area	Sp	cm2	SpCAT-1 = 3.13297 × 10−12 SpCAT-2 = 3.1365 × 10−12
Density	ρB	gm/cm3	ρB.CAT-1 = 0.667 ρB.CAT-2 = 0.687
Molecular weight of kerosene	MwL	gm/mole	170
Molecular weight of sulfur	Mwi	gm/mole	32.06
Pore radius average	rg	nm	rg,CAT-1 = 4.9944 rg,CAT-2 = 4.9972

below.

2.3.1. Linear Approach

The optimal kinetic parameter values from the linear regression approach, presented in

Table 4. Parameters for CAT-1 obtained using linear regression.

Parameter	Value	Unit
n	2.000	_
k1	0.0106	${(\mathrm{w}\mathrm{t}}^{-1})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
k2	0.0233	${(\mathrm{w}\mathrm{t}}^{-1})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
k3	0.0544	${(\mathrm{w}\mathrm{t}}^{-1})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
k4	0.1296	${(\mathrm{w}\mathrm{t}}^{-1})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
SSE	2.9922 × 10−4	_

and

Table 5. CAT-2 model parameters obtained using linear regression.

Parameter	Value	Unit
n	1.9861	_
k1	0.0207	${(\mathrm{w}\mathrm{t}}^{-0.9861})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
k2	0.0538	${(\mathrm{w}\mathrm{t}}^{-0.9861})·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
k3	0.1165	${(\mathrm{wt}}^{-0.9861})·{\min }^{-1}$
k4	0.2278	${(\mathrm{wt}}^{-0.9861})·{\min }^{-1}$
SSE	2.1999 × 10−4	_

, enable absolute errors of less than 5%.

As can be seen in Table 4 and Table 5, the sequence of the reaction varies depending on the type of catalyst used. For CAT-1 and CAT-2, the ODS reaction order for sulfur compounds in kerosene was found to be 2 and 1.9860986, respectively.

Activation Energy (E)

Figure 12 and Figure 13 depict the Arrhenius equation, which was used to determine the activation energy for each catalyst by plotting (lnk) vs (1/T) and finding that the slope of the line is equal to (−E/R).

Table 6. Activation energy calculations using the linear regression technique.

Catalyst	E (kJ/mol)	Frequency Factor
CAT-1	25.8108	399.2151
CAT-2	26.2407	710.1669

displays the calculated activation energies.

2.3.2. Approach with Non-linear Regression Method

This study simultaneously assessed the reaction order, activation energy, and pre-exponential factor, with the results shown in

Table 7. Parameters for the kinetic model of CAT-1 obtained using non-linear regression.

Parameter	Value	Unit
n	1.9955	_
E	30.7646	KJ/mol.
ko	1455.7723	${\mathrm{w}\mathrm{t}}^{-0.9955}·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
SSE	1.4621 × 10−4	_

and

Table 8. Parameters for the kinetic model of CAT-2 obtained using non-linear regression.

Parameter	Value	Unit
n	1.9869	_
E	25.8635	KJ/mol.
ko	1681.5340	${\mathrm{w}\mathrm{t}}^{-0.9869}·{\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
SSE	1.5799 × 10−4	_

below.

These tables show that, under the same conditions, CAT-2 provides a higher conversion rate than the other catalysts. Under these conditions (reaction temperature = 393 K, reaction time = 120 min), CAT-1 converted sulfur compounds at a rate of 74.77%, while CAT-2 achieved a rate of 85.35%. The alternative catalysts demonstrate greater activation energy requirements for the reaction compared to CAT-2. This suggests that the use of CAT-2 leads to a reduction in the activation energy, which in turn correlates with an increase in the reaction rate. These parameters were estimated using both linear and non-linear methods, with the latter yielding lower sums of squared errors (SSEs). Given this, it follows that the second method yields more precise parameter estimation, at least to some extent, based on (SSE).

In comparison to the concurrent determination of kinetic parameters through non-linear regression, the magnitude of error introduced during the computation of activation energy and pre-exponential factor via the linearization of the Arrhenius equation is notably higher. The final sulfur concentration and reaction rate models obtained by non-linear methods for CAT-1 and CAT-2 are as follows:

For CAT-1

• Reaction rate:

$$={-r}_{sulfur}=578.64{{\mathrm{ƞ}}_0}_{}{e}^{(-{\displaystyle \frac{4126.2184}{T}})}{C_{sulfur}}^2$$

(2)

• Concentration profile:

$$C_{sulfur}={[1.33155+578.64{\mathrm{ƞ}}_0{e}^{(-{\displaystyle \frac{4126.2184}{T}})}.t]}^{-1}\dots$$

(3)

For CAT-2

• Reaction rate:

$${-r}_{sulfur}=1681.534{{\mathrm{ƞ}}_0}_{}{e}^{(-{\displaystyle \frac{4182.68994}{T}})}{C_{sulfur}}^{1.975313}$$

(4)

• Concentration profile:

$$C_{sulfur}={[1.32217+1640.021971{\mathrm{ƞ}}_0{e}^{(-{\displaystyle \frac{4182.68994}{T}})}.t]}^{-1.025311}$$

(5)

2.4. Experimental Findings and Simulation Outcomes

The oxidation process was simulated after the optimal kinetic parameters had been estimated.

Table 9. Experimental and simulation results for CAT-1 obtained using linear regression.

Temperature (K)	Time (min)	Sulfur Content (wt. %)		Conversion		Error %
		Experimental	Predicted	Experimental	Predicted
303	30	0.1902	0.1851	0.0325	0.0590	2.6940
303	60	0.1815	0.1748	0.0768	0.1108	3.6760
303	90	0.1709	0.1656	0.1307	0.1576	3.0688
303	120	0.1641	0.1574	0.1653	0.1998	4.0846
333	30	0.1749	0.1729	0.0874	0.1210	1.1594
333	60	0.1578	0.1542	0.1973	0.2156	2.2465
333	90	0.1457	0.1392	0.2589	0.2919	4.4216
333	120	0.1334	0.1269	0.3214	0.3545	4.8582
363	30	0.1534	0.1488	0.2197	0.2431	2.9650
363	60	0.1227	0.1197	0.3758	0.3911	2.3924
363	90	0.0991	0.1002	0.4959	0.4908	1.0974
363	120	0.0894	0.0861	0.5452	0.5625	3.6793
393	30	0.1099	0.1114	0.4409	0.4333	1.3928
393	60	0.0799	0.0777	0.5953	0.6047	2.6920
393	90	0.0622	0.0597	0.6836	0.6964	4.0148
393	120	0.0496	0.0484	0.7477	0.7536	2.3081

,

Table 10. Experimental and simulation results for CAT-2 obtained using linear regression.

Temperature (K)	Time (min)	Sulfur Content (wt. %)		Conversion		Error %
		Experimental	Predicted	Experimental	Predicted
303	30	0.1694	0.1747	0.1383	0.1113	3.1557
303	60	0.1499	0.1572	0.2375	0.2004	4.8965
303	90	0.1398	0.1429	0.2889	0.2731	2.2202
303	120	0.1295	0.1309	0.3413	0.3341	1.1193
333	30	0.1416	0.1483	0.2797	0.2456	4.7459
333	60	0.1152	0.1189	0.4140	0.3952	3.2956
333	90	0.0988	0.0993	0.4975	0.4949	0.5145
333	120	0.0876	0.0852	0.5544	0.5667	2.7597
363	30	0.1099	0.1153	0.4409	0.4140	4.8970
363	60	0.0784	0.0814	0.6012	0.5859	3.8378
363	90	0.0624	0.0629	0.6826	0.6805	0.7395
363	120	0.0497	0.0512	0.7472	0.7400	2.9511
393	30	0.0795	0.0825	0.5956	0.5804	3.7622
393	60	0.0496	0.0520	0.7614	0.7354	4.8996
393	90	0.0399	0.0379	0.7970	0.8070	4.9001
393	120	0.0288	0.0298	0.8535	0.8482	3.5993

,

Table 11. Second-approach experimental and simulation results for CAT-1.

Temperature (K)	Time (min)	Sulfur Content (wt. %)		Conversion		Error %
		Experimental	Predicted	Experimental	Predicted
303	30	0.1902	0.1885	0.0325	0.0417	0.8947
303	60	0.1815	0.1810	0.0768	0.0793	0.2555
303	90	0.1709	0.1741	0.1307	0.1144	1.8965
303	120	0.1641	0.1678	0.1653	0.1469	2.2248
333	30	0.1749	0.1741	0.0874	0.1144	0.4521
333	60	0.1578	0.1562	0.1973	0.2054	0.9967
333	90	0.1457	0.1417	0.2589	0.2794	2.7661
333	120	0.1334	0.1296	0.3214	0.3413	2.8569
363	30	0.1534	0.1485	0.2197	0.2446	3.1823
363	60	0.1227	0.1193	0.3758	0.3931	2.7662
363	90	0.0991	0.0997	0.4959	0.4933	0.5862
363	120	0.0894	0.0856	0.5452	0.5651	4.2606
393	30	0.1099	0.1153	0.4409	0.4140	4.8952
393	60	0.0799	0.0815	0.5953	0.5854	2.0034
393	90	0.0622	0.0630	0.6836	0.6795	1.3054
393	120	0.0496	0.0514	0.7477	0.7388	3.5296

and

Table 12. Second-approach experimental and simulation results for CAT-2.

Temperature (K)	Time (min)	Sulfur Content (wt. %)		Conversion		Error %
		Experimental	Predicted	Experimental	Predicted
303	30	0.1694	0.1738	0.1383	0.1158	2.6066
303	60	0.1499	0.1557	0.2375	0.2078	3.8948
303	90	0.1398	0.1410	0.2889	0.2825	0.8931
303	120	0.1295	0.1289	0.3413	0.3444	0.4810
333	30	0.1416	0.1477	0.2797	0.2486	4.3185
333	60	0.1152	0.1182	0.4140	0.3986	2.6218
333	90	0.0988	0.0985	0.4975	0.4989	0.3038
333	120	0.0876	0.0844	0.5544	0.5707	3.6631
363	30	0.1099	0.1145	0.4409	0.4178	4.1502
363	60	0.0784	0.0806	0.6012	0.5900	2.7996
363	90	0.0624	0.0621	0.6826	0.6839	0.4212
363	120	0.0497	0.0505	0.7472	0.7429	1.6689
393	30	0.0795	0.0825	0.5956	0.5805	3.7277
393	60	0.0496	0.0520	0.7614	0.7354	4.8722
393	90	0.0399	0.0379	0.7970	0.8070	4.9152
393	120	0.0288	0.0298	0.8535	0.8482	3.6126

below show the predicted results compared to the experimental findings under a variety of operational scenarios.

The results in

Table 13. Comparison of mean squared errors.

Parameters	The Summation of Squared Errors (SSE)
	Linear Approach	Non-Linear Approach
CAT-1	299.2197 × 10−6	146.213 × 10−6
CAT-2	219.9992 × 10−6	157.995 × 10−6

show that for the same number of training data points and depending on the target function, the estimations of the parameters provided by the second approach were definitely more accurate than the ones given by the first approach. It was because of this characteristic that, for the same number of training data points, the second approach actually gave rise to a smaller sum of squared errors (SSEs). The experimental errors obtained through a linearization (first method) of the Arrhenius equation to find activation energy (E) and pre-exponential factor were far above those for the non-linear fit of data (second method).

The predicted sulfur conversion outcomes are depicted in Figure 14 and Figure 15, alongside the experimental results. The figure displays experimental values (x-axis) versus observed values (y-axis) under the same conditions, demonstrating reasonable agreement between the experimental data and the simulations. As can be observed from the figures, the predicted and experimental data exhibit a high degree of conformity, with the ratio between them being close to a linear relationship with a slope value near one.

2.5. Optimal Conversion to Eco-Friendly Fuel

Using best-fit kinetic parameters,

Table 14. Optimal conditions for applying the ODS process with CAT-1.

Parameter	Value	Unit
Csulfur,t	0.1911	wt.%
T	546	K
Time	199	min
Conversion	99.0432	%

and

Table 15. Optimal conditions for applying the ODS process with CAT-2.

Parameter	Value	Unit
Csulfur,t	0.2443	wt.%
T	550	K
Time	193.6332	min
Conversion	99.1042	%

show the best operating conditions for each catalyst.

The utilization of CAT-2 under particular conditions, which are 550 K reaction temperature, 193.63318 min contact time, and an initial sulfur compound concentration of 0.2443 (wt%), meet environmental standards and attain a low sulfur content, resulting in high-quality fuel oil, as shown in Table 12 and Table 13.

Maximum conversion (almost 99%) was achieved using the optimal operating conditions, as demonstrated in this section. The following is a brief synopsis of the prerequisites that need to be met in order to have a product with a low level of sulfur dioxide, which is better for the environment.

Temperature: As temperature rises, the sulfur compound is more likely to be converted. This is because the reaction rate constant increases with temperature, as described by the Arrhenius equation. Elevating the temperature also enhances other critical factors like diffusivity and viscosity. Consequently, the sulfur compound’s diffusion rate, air molecule penetration through catalyst pores, and air absorption into kerosene all increase with higher temperature.

Time: As the reaction goes on for longer, there will be more time for the reactants to make contact with one another on the active sites of the catalyst. As a direct consequence of the prolonged residence time, a significant amount of reaction product is produced.

Table 16. Comparison of results.

Sulfur Compound	Oxidant/Catalyst	Sulfur Removal	Reference
TH	n-heptane, H2O2/TiO2/ZSM-12	60%	[29]
DBT	Diesel, H2O2/Fe2O3/ AC	71%	[30]
S-compounds	Kerosene, H2O2/formic acid	87%	[31]
S-compounds	Kerosene, air/20% HY-zeolite and 80% nano-silica	87.88%	Present study

presents a comparison between the present study and some previous studies.

3. Experimental Methodology

This study investigated the oxidation of sulfur in kerosene using a novel composite nanocatalyst and atmospheric oxygen as the oxidizing agent. Four distinct catalysts were prepared following a series of steps.

(a)

Nano-silica was made using bentonite.

(b)

Using the precipitation process, two composite supports (nano-silica + HY-zeolite) were prepared with varying percentages of HY-zeolite: support-1: nanoparticles of silica only; support-2: nanoparticle composed of 80% nano-silica and 20% HY-zeolite.

(c)

Using the impregnation process, two types of nanocatalysts were created:

• Fe₂O₃/nano-silica nanoparticle (CAT-1).
• Fe₂O₃/(80% nano-silica + 20% HY-zeolite) nanoparticle (CAT-2).

The efficiency of the two prepared catalysts was evaluated in a batch reactor undergoing oxidative desulfurization at different operational conditions, shown in

Table 17. Variables and operating conditions for this work.

Variables	Conditions
Catalyst	CAT-1, CAT-2
Temperature (°C)	30, 60, 90, 120
Reaction time (min)	30, 60, 90, 120

.

3.1. Materials and Equipment

3.1.1. Kerosene Feedstock (Fuel Feed)

Kerosene was utilized as the fuel feedstock here. The consumption of this type of fuel is significantly high in daily life, and in aircraft, its consumption is rapidly increasing. Commercial kerosene, with a total sulfur content equal to 0.1966 wt. %, was provided by the Baiji refinery, North Refineries Company. The detailed specification of the feedstock is shown in

Table 18. Specification of kerosene.

Property	Value
Flash point, °C	40.5
API	47.5
Density at 15.6 °C, g/cm3	0.7904
Color	+20
IBP, °C	147
EP, °C	265
Sulfur, ppm	1966

.

3.1.2. Oxidant

Utilizing a compressor, air was delivered primarily as an oxidizing mediator. Sulfones were created by oxidizing the sulfur compound with oxygen found in the air.

3.1.3. Active Components for Catalyst Preparation

The chemical compounds that were utilized to provide catalysts for the active components in the present research study are listed in

Table 19. Materials and supplies needed to prepare catalysts.

Materials	Purity %	Company	Role
Deionized water	-	SDI	Solvent of active materials
Fe(NO3)3·9H2O	98	Himedia	Active metal

.

3.1.4. HY-Zeolite

HY-zeolite was synthesized and characterized by the Iraqi Ministry of Oil’s Laboratory. The particle properties and XRF data are shown in

Table 20. HY-zeolite nanoparticle specifications.

Characteristic	Value
Crystallinity percentage	92.3
Pore volume (cm3/g)	0.388
Surface area (m2/g)	499.4
Particle size (nm)	94.31
Bulk density (g/cm3)	0.648
Pore size (nm)	31.14

and

Table 21. XRF results for HY-zeolite nanoparticles.

HY-Zeolite Component	Wt. %
SiO2	57.96
Al2O3	19.01
Na2O	1.954
SiO2/Al2O3 molar ratio	5.183

, respectively.

3.1.5. Equipment

Table 22. The main equipment used.

Equipment	Specification	Supplier
Tubular Furnace	Heating length 300 mm, Max. temperature 1200 °C	SAFTherm, Luoyang, China
Magnetic heater stirrer	Stirring speed 100–1500 rpm, capacity 100–3000 mL, Max. temperature 380 °C	JISICO, Seoul, Republic of Korea
Vacuum pump	1/4 hp, model VE125N	Manufacturer is Value, DEIRA DUBAI, Dubai, UAE
X-ray diffraction instrument	XRD-6000 model	Shimadzu company, Tokyo, Japan
Atomic Force Microscope device	SPM-AT 3000, Scanning Probe Microscope, AA 3000	Angstrom-Advance Inc., contact model Stoughton, MA, USA
FESEM device	Zeiss-EM10C-100KV	ZEISS Sigma company, Oberkochen, Germany

shows the required equipment used in the present study.

3.2. Nano-Silica Preparation

Mesoporous silicate/aluminosilicate materials (M41S phase) were synthesized in 1992 with a pore size range of 2 to 30 nm. These materials belong to the mesoporous silica molecular sieves (MCM) series—MCM-41, featuring a hexagonal distribution of mesopores; MCM-48, cubic; and MCM-50, layered [32]. The complicated steric arrangements of these materials, which are well-organized mesoporous solids, have encouraged scientists to develop new methods for synthesizing new types that have unconventional physical and chemical characteristics. Therefore, the synthesis of zeolite from these groups maintains the mesopores’ construction integrity.

In other words, the optimal porous material comprises small, clean pores surrounded with crystalline walls, in such a way that the mass transfer is carried out with the maximum possible efficiency, and thermal and hydrothermal stability are coupled with better control of its active properties. The compounds and materials required for the precipitation process—which yields nano-silica—are listed in

Table 23. Nano-silica production materials.

Materials and Components	Purity%	Supplier
Sodium hydroxide (NaOH)	98%	Sigma Aldrich (Burlington, MA, USA)
Ammonium hydroxide (NH4OH)	28%	GCC Ltd. Company (London, UK)
Sulfuric acid (H2SO4)	98%	GCC Ltd. company
Bentonite clay	97%	GCC Ltd. company
Deionized water	-	CHD Chemicals Ltd. Company (Chandigarh, India)
Ethanol (C2H5OH)	98%	Analytical laboratory (Telford, PA, USA)

.

The sand was mixed with sodium hydroxide, which led to the creation of very small (around 45 nanometer) particles. This mixture was then stirred together with bentonite sand and heated in a programmable electric furnace at 550 degrees Celsius for 30 min. Water was added to the heated mixture, forming a consistent, transparent liquid. Concentrated sulfuric acid (98%) was slowly introduced into this liquid until the pH reached 3. The gel that formed was separated by filtering, and then it was rinsed once with filter paper. The filtered gel was then washed with deionized water (pH 7) to remove sulfate ions. Finally, the gel was dried overnight at 110 degrees Celsius, producing the material. The overall process of preparing this nano-silica support material is summarized in Figure 16.

3.3. Preparation of Iron-Supported Nano-Silica Catalyst

Incorporating different elements into microporous and mesoporous materials has been an active area of research. It is driven by the need to develop more stable and effective materials for applications like catalysis, separation, coatings, and sensing [33,34]. The discovery of the useful catalytic properties of titanium silicate molecular sieves sparked interest in incorporating other metals into similar porous materials [34]. Iron-containing zeolites and related microporous substances were found to have unique catalytic abilities applicable to gas-phase reactions like hydrocarbon oxidation, ammonia decomposition, nitrous oxide reduction, and selective catalytic reduction of nitric oxide and nitrous oxide [35,36]. In this study, the researchers thoroughly mixed 3 g of ferric nitrate nonahydrate with 75 milliliters of water for 30 min using an ultrasonic mixer. Then, 22 g of nano-silica was slowly added to the mixture, which was further stirred in an ultrasonic water bath for 60 min to ensure thorough mixing. The resulting gel was filtered using a Buchner funnel and a vacuum pump, and the dry sample was dried overnight at 110 °C. To prepare the catalyst, a calcination process was carried out at 550 °C for 3 h. The catalyst preparation process and the loading of the active metal are illustrated in Figure 17.

The catalyst was tested at the Petroleum Research Center in Baghdad, and its physical properties were examined and are shown in

Table 24. The surface area, pore volume, and bulk density of the catalysts.

Catalyst Type	Surface Area, m2/g	Pore Volume, cm3/g	Bulk Density, g/cm3
SiO2 (support)	341.31	0.853	0.932
CAT-1 (0% HY-zeolite and 100% nano-silica)	242.67	0.606	0.667
CAT-2 (20% HY-zeolite and 80% nano-silica)	250.138	0.625	0.687

.

3.4. Oxidative Desulfurization Reactor

3.4.1. Batch Reactor

Oxidative desulfurization was carried out in a 250 mL three-necked round-bottom flask. One neck connects to a condenser that separates kerosene vapor from air. Another has an air inlet connected to a compressor to force air into the flask. The third neck allows for temperature monitoring and sampling. The contents are heated and stirred using a magnetic stirrer. A diagram is shown in Figure 18.

3.4.2. The ODS Reaction

The following section presents the likely mechanism behind the oxidation of sulfur compounds during the heterogeneous desulfurization of fuels using a catalyst. The complete reaction of converting ethyl mercaptan to a disulfide is achieved with the aid of 5% iron core/(silica + zeolite) nanoparticles. The reactions start with the metals reacting with oxygen in the air to form peroxide species [37,38].

The peroxide compounds incorporate the sulfur atom and react with the sulfur atoms in ethyl mercaptan. This causes the sulfur atoms to be oxidized, forming a disulfide compound. At the same time, the primary metal oxide is produced as a byproduct of this reaction. The resulting disulfide can then be easily extracted using a polar solvent, which is a suitable method for driving the reaction [37,39,40]. The overall reaction can be summarized as follows:

2RSH + O₂ → 2RSSR + H₂O

3.5. Experimental Testing

The experimental runs for the ODS process of kerosene fuel were conducted under different operating conditions, as shown in Table 17.

A controlled airflow of 120 L/h was adjusted to obtain a pressure of 1 atm. The points taken in the experiment were carefully selected so that all maxima and minima were considered while also taking into account safe and moderate conditions in between.

Experimental Procedure for ODS Reactions

To prepare for the experiments, sulfur-supplying feedstock oil, specifically kerosene, was used. This included the preparation of the necessary catalysts and other equipment required. During each experimental run, the following steps were carried out:

• A round-bottom flask was charged with 80 mL of the feedstock before proceeding with other steps.
• The flask was placed in a heating mantle stirrer, and an air hose was connected to a condenser as well.
• Measures were taken to ensure that heat was not transferred to the condenser through the kerosene vapor, and the condenser was supplied with cooled water.
• A thermometer measured the temperature at which the overall reaction occurred.

The catalyst was added to the reactor once the required temperature was reached, and the time was recorded. The sulfur content of the final product was then analyzed to determine the reaction rate.

3.6. Analysis of Liquid Samples

The ASTM-D4294 method combined with X-ray fluorescence was used to analyze the overall content of sulfur in the oil feedstock and the final products [40]. All the available analytical methods were used to characterize the properties of the feedstock. In every operational condition, the analysis of products was duplicated for each sample so that the consistency and validation of the results can be demonstrated. The results obtained from each run were taken as average and a maximum permissible variance of 0.5 percent was assumed for all runs.

4. Mathematical Modeling of ODS Process

A mathematical model was developed for the ODS process based on the following assumptions:

• The reactor is isothermal at a constant pressure.
• A reactant that is in excess is in the gaseous state.
• Reactor design produces a homogeneous environment throughout.

4.1. Mass Balance Equation

The general mass balance equation for the sulfur component of the catalytic batch reactor at an increment of time (dt) is as follows:

[input] = [output] + [consumption by reaction] + [accumulation]

(6)

• Input of sulfur, mass/time = 0.
• Output of sulfur, mass/time = 0.
• Consumption of sulfur by reaction, mass = −(r_RS)V dt.

Accumulation of sulfur, mass = V ${\displaystyle \frac{d{C}_{RS}}{dt}}$ dt.

Introducing these terms into Equation (6) will give:

$$0=0+(-{\mathrm{r}}_{\mathrm{RS}})\mathrm{Vdt}+\mathrm{V}{\displaystyle \frac{d{C}_{RS}}{dt}}\mathrm{dt}$$

(7)

Then,

$$-(-{\mathrm{r}}_{\mathrm{RS}})={\displaystyle \frac{d{C}_{RS}}{dt}}$$

(8)

So:

$$\mathrm{dt}={\displaystyle \frac{-d{C}_{RS}}{{(-r}_{RS})}}$$

(9)

where:

• at t = 0, C_RS = C_RSO
• t = t, C_RS = C_RS

Thus:

$$\underset{0}{\overset{t}{\int }}dt=\underset{\mathrm{C}\mathrm{R}\mathrm{S}\mathrm{O}}{\overset{\mathrm{C}\mathrm{R}\mathrm{S}}{\int }}{\displaystyle \frac{-d{C}_{RS}}{{(-r}_{RS})}}$$

(10)

Finally, the design correlation for the ideal batch reactor is written in terms of batch time to be

$$\mathrm{t}=\underset{\mathrm{C}\mathrm{R}\mathrm{S}\mathrm{O}}{\overset{\mathrm{C}\mathrm{R}\mathrm{S}}{\int }}{\displaystyle \frac{-d{C}_{RS}}{{(-r}_{RS})}}$$

(11)

Equation (9) can be integrated to obtain the concentration profile of sulfur compounds in the catalytic reactor, although the term (−rRS) is primarily dependent on the sulfur concentration.

4.2. Reaction Rate

The oxidation of sulfur by air is a complex multi-step process. The reaction kinetics can be described using an nth-order rate equation to capture the complexity.

(−r_RS) = k_app C_RSⁿ

(12)

The apparent kinetic constant k_app is controlled by the intrinsic kinetic constant k_in, modified by the catalyst effectiveness factor η_o [41]:

k_app = ƞ_o k_in

(13)

Substituting this expression into the main Equation (6) and integrating it yields the final expression for the catalytic oxidation reaction of sulfur with nth-order kinetics.

$$C_{RS}={[{C_{RSO}}^{\left(1-n\right)}+\left(n-1\right).t.k_{app}]}^{\left({\displaystyle \frac{1}{\left(1-n\right)}}\right)}$$

(14)

4.3. Efficiency of Reactors

Equation (6) is rewritten as follows (using ${\mathrm{ƞ}}_o$, k_in instead of $k_{app}$) [41] to account for more physical factors:

$$C_{RS}={[{C_{RSO}}^{\left(1-n\right)}+\left(n-1\right).t.k_{in}.{\mathrm{ƞ}}_o]}^{\left({\displaystyle \frac{1}{\left(1-n\right)}}\right)}$$

(15)

The Arrhenius equation may be used to calculate the reaction rate constant (K_in) for each ODS reaction:

$$k_{in}=K_0{e}^{-{\displaystyle \frac{EA}{RT}}}$$

(16)

K₀ represents the frequency or pre-exponential factor, while (E) denotes the activation energy of the reaction. This expression fits experiments conducted over extensive temperature ranges and is widely advocated from various perspectives as a highly accurate approximation of the actual temperature dependence. The chemical reaction rate can thus be formulated as follows:

$$C_{RS}={[{C_{RSO}}^{\left(1-n\right)}+\left(n-1\right).t.K_0{e}^{-{\displaystyle \frac{EA}{RT}}}.{\mathrm{ƞ}}_o]}^{\left({\displaystyle \frac{1}{\left(1-n\right)}}\right)}$$

(17)

Many factors, including diffusivities, oil viscosity, and efficacy factors, are involved in the oxidation process of sulfur compounds. These parameters are obtained using the following formulae.

4.3.1. Effectiveness Ratio (${\mathrm{ƞ}}_{\mathrm{o}}$)

The effectiveness factor is dependent on the Thiele modulus (Φ) and is determined by the following correlation, which is applicable for spherical particles [41,42]:

$${\mathrm{ƞ}}_{\mathrm{o}}={\displaystyle \frac{3(\mathrm{\Phi }\coth \left(\mathrm{\Phi }\right)-1)}{{\mathrm{\Phi }}^2}}$$

(18)

The Thiele modulus for an nth-order irreversible reaction is calculated using the following generalized equation [43,44]:

$$\mathrm{\Phi }={\displaystyle \frac{V_p}{S_p}}\sqrt{{\displaystyle \frac{(n+1)}{2}}{\displaystyle \frac{K_{in}{C_{RS}}^{(n-1)}{\rho }_p}{D_{ei}}}}$$

(19)

4.3.2. Catalyst Surface (S_p) and External Volume (V_p)

For a spherical particle, the external surface area and volume of the catalyst can be determined as follows:

$${\mathrm{V}}_{\mathrm{p}}={\displaystyle \frac{4}{3}}\pi ·{\left({\mathrm{r}}_{\mathrm{p}}\right)}^3$$

(20)

S_p = 4π (r_p)²

(21)

4.3.3. Effective Diffusivity (Dei)

Effective diffusivity characterizes the catalyst structure (porosity and tortuosity) by considering the pore network within the particle, as described below [41,43]:

$$D_{ei}={\displaystyle \frac{{ϵ}_B}{\tau }}{\displaystyle \frac{1}{\frac{1}{D_{mi}}+\frac{1}{D_{ki}}}}$$

(22)

The catalyst porosity (ϵ_B) is determined using the following two equations based on experimental data:

$${ϵ}_B={\rho }_p{V}_g$$

(23)

$${\rho }_p={\displaystyle \frac{{\rho }_B}{1-{ϵ}_B}}$$

(24)

The values of the tortuosity factor (τ) of the pore network are from 2 to 7 [44]. τ values in studies in the literature have been reported to be 4 [41,44].

Within the catalyst particle, D_ei comprises two diffusivity types: molecular diffusivity (D_mi) and Knudsen diffusivity (D_ki).

Molecular diffusivity is computed using the Tyn–Calus equation [45,46]:

$$D_{mi}=8.93\ast {10}^{-8}{\displaystyle \frac{{\mathrm{\upsilon }}_L^{0.267}T}{{\mathrm{\upsilon }}_{RS}^{0.433}{\mu }_L}}$$

(25)

Knudsen diffusivity is determined according to the following procedure [42,47]:

$$D_{ki}=9700r_g{\left({\displaystyle \frac{T}{M_i}}\right)}^{0.5}$$

(26)

4.3.4. Viscosity

The viscosity of light naphtha can be calculated using the equation below [48]:

$${\mu }_L=3.141\ast {10}^{10}{\left(T-460\right)}^{-3.444}{\left[{log}_{10}API\right]}^a$$

(27)

$$a=10.313\left[{log}_{10}\left(T-460\right)\right]-36.447$$

(28)

This equation is used to estimate the API.

$$\mathrm{API}={\displaystyle \frac{141.5}{{sp.gr}_{15.6}}}-131.5$$

(29)

4.3.5. Molar Volume

The molar volume of the model sulfur compound is estimated according to the following correlation [49]:

$${\mathrm{\upsilon }}_{RS}=0.285{{(\mathrm{\upsilon }}_{cRS})}^{1.048}$$

(30)

The critical volume of liquid (light naphtha) is obtained using a Riazi–Daubert equation [49]:

$${\mathrm{\upsilon }}_L=0.285{\left({\mathrm{\upsilon }}_{cL}\right)}^{1.048}$$

(31)

$${\mathrm{\upsilon }}_{cL}=(7.5214\ast {10}^{-3}{{(T}_{meABP})}^{0.2896}{{(\rho }_{L,15.6})}^{-0.7666})M_L$$

(32)

The mathematical model equations for the ODS reaction outlined in this section were implemented and solved concurrently using the gPROMS (general Process Modeling System—Version 5.6) package.

5. Technique Used to Estimate Kinetic Parameters

Model predictions can be guaranteed only if the unknown parameters are estimated correctly. The design and control strategies of the process can be compromised if an accurate process model is not used.

In this work, two methodologies were applied to find the optimum kinetic parameters that depend on the amount of sulfur in the ODS process under diverse conditions of operation. The methods are given below according to references [49,50,51]:

In general, the reaction order (n) and the reaction rate constant (k) are first determined using linear regression. Subsequently, the activation energy (E) and the pre-exponential factor (K_o) are determined by linear regression using the Arrhenius equation.

The simultaneous evaluation of the reaction order (n), activation energy (E), and pre-exponential factor (K_o) was performed using non-linear regression.

The following function was minimized in order to find the best value for the kinetic parameter:

$$OBJ={\sum }_{n=1}^{Nt}{(C_{RS}^{exp}-C_{RS}^{pred})}^2$$

(33)

In Equation (33), Nt is the number of runs, $C_{RS}^{exp}$ is the concentration of sulfur obtained experimentally, and $C_{RS}^{pred}$ is the value predicted by the model.

Optimization Problem Formulation for Parameter Estimation

The optimization problem formulation to estimate the parameters is as follows:

Given	The reactor configuration, the type of catalyst used, and the process conditions of a specific process
Determine	The order of the oxidation reaction (n1) and the reaction rate constant (k) at different temperatures (303, 333, 363, 393 K) for each catalyst and calculate the activation energy (E) and pre-exponential factor using linear regression of the Arrhenius equation.

So as to minimize The sum of squared error (SSE).

Subjected to

All optimization variables are subject to conversion rate limits and linear limitations.

The optimization problem may be expressed mathematically using linear regression as follows:

Min	SSE (OBJ)
$n^j,k_i^j,(i=1-4$, j = CAT-1, CAT-2, CAT-3 and CAT-4),	S.t. f(z, x(z),ẋ(z), u(z), v) = 0 (batch reactor)

The problem is formulated as follows when employing the non-linear regression technique. Here, reaction order (n), activation energy (E), and the pre-exponential factor (ko) are estimated simultaneously for each catalyst.

Min	SSE
$n^j,{EA}^j,k_i^j$, j = CAT-1, CAT-2, CAT-3 and CAT-4),	S.t.f (z, x(z),ẋ(z), u(z), v) = 0 (batch reactor)

The equation f(z, x(z), x’(z), u(z), v) = 0 represents a model where u(z) is the decision variable, z is the independent variable, x(z) are the variables, and x’(z) are their derivatives. Variable v is the design variable, and C, CL, CU, L, U are related to concentration and bounds.

The optimization technique used is a two-step method described in [52]. Step 1 simulates the process for a decision variable and calculates the objective and constraints. Step 2 updates the optimization variables.

6. Conclusions

The synthesis of new efficient catalysts through oxidative desulfurization will change oil pollutants into clean fuel. In view of the environmental concerns of high sulfur content and the difficulties in designing potential catalysts to develop better-quality fuel, achieving this synthesis is one of the most essential tasks. Oxidation reactions are very promising when catalyzed by nanoparticles. Silica can possibly be safely, economically, and environmentally extracted from bentonite. Therefore, when compared to findings obtained in the literature, this study’s preparation of nano-silica from the cheapest raw material (bentonite) produced an excellent nano material with a high surface area. Here, two composite catalysts (CAT-1 (0% HY-zeolite and 100% nano-silica) and CAT-2 (20% HY-zeolite and 80% nano-silica)) of Fe/silica were experimentally designed. The results revealed that the highest total sulfur removal efficiency was 50% and 87.88% for CAT-1 and 80% CAT-2, respectively. CAT-2 showed remarkable enhancement in the performance of the ODS process due to the positive impact of HY-zeolite on sulfur removal efficiency and the improved activity and selectivity of the catalyst in sulfur compound removal. Compared to previous studies, the obtained conversion provides a promising technology for efficient ODS processes.

The novel nanocatalyst designated as CAT-2 exhibited a markedly enhanced capacity to convert sulfur-containing compounds when air was utilized as the oxidizing agent under the predetermined conditions of a temperature of 393 Kelvin and a duration of 120 min. Both linear and non-linear optimization methods were employed in order to obtain the optimal fit of kinetic parameters, with the non-linear approach proving to be more accurate. The final mean absolute error of the model remained below 5% across all conditions tested. The efficiencies achieved with this catalyst reached over 99% of the process at a temperature of 550 Kelvin and a batch time of 193.63318 min. Ultimately, the new nanocatalyst enabled a reduction in fuel sulfur content with high quality while also lowering environmental pollution.