ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT AZOJETE June 2022. Vol. 18(2):307-318 Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria. Print ISSN: 1596-2644, Electronic ISSN: 2545-5818 www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 307 ORIGINAL RESEARCH ARTICLE KINETIC MODELING OF ETHYLBENZENE ISOMERIZATION USING BODENSTEIN APPROXIMATION TECHNIQUE A. S. Kolo*, A. S. Grema, and I. I. Maina Chemical Engineering Department, Faculty of Engineering, University of Maiduguri, P.M.B 1069, Maiduguri, Borno State, Nigeria *Corresponding author’s email address: askolo@unimaid.edu.ng 1.0 Introduction Ethylbenzene isomerization is an industrial reaction that is vital in industries that need petrochemical feedstocks. Products of this reaction are precursors to some very important polymer materials. A kinetic model of this reaction is handy when it comes to reactor design, optimization, operations and possible troubleshooting. Kinetic models developed for reactions of industrial importance are usually either empirical in nature or assume some rate-limiting steps (Kangas et al., 2008; Al-Khattaf et al., 2009; Waziri and Al-Khattaf, 2009; Farahani et al, 2020). The power rate-law type of equations usually used to study kinetic behaviors of complex reactions do not usually capture the multiple molecular events involved in such catalytic reactions and, therefore, cannot completely and reliably predict kinetic behavior of such systems. Consequently, models developed via these schemes are usually not reliable for direct scale-up purposes (Helfferich, 2004). Direct large scale-up of industrially significant reactions on the other hand need effective and reliable fundamental kinetic models reflecting the kinetics of the multi- step catalytic pathways of the reaction mechanism. An earlier study on isomerization of ethylbenzene on bifunctional Pt/Al2O3 and Pt-zeolite catalysts was investigated by Robschlager and Christoffel (1979). Their investigation showed that xylenes were produced over the Pt/Al2O3 catalyst via the route corresponding to prediction from a bifunctional mechanism in which skeletal rearrangements of tertiary carbocations only were considered to be the rate-determining steps. A systematic methodology to the elucidation of ARTICLE INFORMATION ABSTRACT Ethylbenzene isomerization reaction is a significant reaction employed in the production of xylene isomers which are used as petrochemical feedstocks. The reaction which proceeds over Pt/Al2O3 catalyst is multi-pathway and multi-cycle in topology. Kinetic model for the reaction was developed in this study using the general rate equation approach. Bodenstein approximation, cross-to-square, and Y-to-delta transformation techniques were used to reduce the complex reaction network to a single cycle network. In addition, the general rate equation for reduced single cycle networks was applied to derive the model for the reaction. The Nelder-Mead simplex optimization technique was used to estimate the kinetic parameters in the model. The structure of the model developed indicates that the model reasonably represents the mechanism of the reaction although few anomalies were observed in the values of the kinetic parameters estimated. The activation energy obtained for the rate constants follows the expected trend for multi-step reactions. © 2022 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved. Submitted 13 November, 2021 Revised 10 March, 2022 Accepted 16 March, 2022 Keywords: Isomerization Ethylbenzene kinetic model Bodenstein approximation reaction network http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 308 reaction networks with multiple numbers of steps and location of nodes was presented in Helfferich (1989). General equations for rate of reactions and yield ratio were developed with a view to elucidate complex reaction networks. Findings from the study indicate that the procedure used is handy in model development and discrimination applications. A kinetic modeling technique that simplifies complex reaction network by employing Bodenstein approximation for trace level intermediates was reported in Chern and Helfferich (1990). The technique reduces complex networks in topology to networks with only pseudo-single steps between nodes; the simpler networks are then used to derive rate expressions for the reactants and products in the network. Fernendes et al. (1998) studied the hydroisomerization of ethylbenzene on Mordenite-based bifunctional catalysts with different platinum contents. The study concluded that hydroisomerization of ethylbenzene is affected by reactant diffusion inside zeolite pores in addition to zeolite acidity and that platinum content affected xylene selectivity due to the formation of hydrogenated intermediates. The concept of Bodenstein approximation and network reduction technique was extended to the modeling of single-cycle reaction network as reported in Chern (2000). Results presented in the study indicated that general rate equations derived is applicable to heterogeneous catalytic reactions involving first-order reactions with respect to intermediates formed in the reaction. Al-Zahrani et al (2002) studied the kinetics of transalkylation and isomerization of ortho diethylbenzene using trifluoromethane acid as a catalyst. The work tested power-law type model for transalkylation and disproportionation reaction and findings show that isomerization reaction follow first-order mechanism. Chen and Chern (2003) in their work titled the general rate equation and their application for catalytic reaction networks: pyramidal systems; systematically developed a general rate equation formula that can be used for fundamental kinetic modeling of complex reaction networks that have multi- pathways and multi-intersections. Their work seeks to cover reaction networks not captured by formulas and procedures presented by Chen and Chern (2002a) and Chen and Chern (2002b). The work by Chen and Chern (2009), discrimination of kinetic models for isomerization of n- butene to isobutene, presented the exploitation of analysis of mathematics of multi-step reactions to discriminate between rival mechanisms. The kinetic modeling of non-simple networks was reported in a work by Chern et al. (2014). Their study provided an extension to earlier techniques reported to include networks with branches from loops and non-simple networks. Oliveira et al. (2016) presented a review of kinetic modeling methodologies for complex processes and the paper analyzed in length advanced kinetic modeling strategies. Farahani et al. (2020) studied ethylbenzene/Xylene mixture isomerization over HZSM-5 zeolites catalyst. They developed a kinetic model that considers chemisorption, surface reaction, and diffusional processes for the reaction. Their findings provided an insight on the activation energy for the surface reaction of ethylbenzene into m-xylene. Considering the significance of general rate equation approach and the limitations of the nature of kinetic studies hitherto practiced, the algorithm of general rate equation was used to model the isomerization of ethylbenzene in this study. The study focused on developing a reliable and effective fundamental kinetic model for the isomerization of ethylbenzene over Pt/ Al2O3 catalyst using Bodenstein approximation technique on trace level intermediates. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng Kolo et al: Kinetic Modeling of Ethylbenzene Isomerization using Bodenstein Approximation Technique. AZOJETE, 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 309 2. Methodology The Bodenstein approximation of trace-level intermediates was the tool used in this work to reduce multi-step reaction networks to reactions with only pseudo-single steps between network nodes and end members. The net rate contribution of a multi-step network segment between nodes Xj and Xk represented by the reaction pathway below is given in equation 1 (Chern and Helfferich, 1990). 1 ... ...j j j i kX X X X        1j k jk j kj kr X X      The segment coefficients ( ) for the reaction pathway presented above are evaluated using equations 2 and 3. 1 , 1 1 1, 2 3 i k i i i j jk jk i k i i i j kj jk D D                 Where  represents the pseudo-first order rate coefficients of quasi-single molecular steps and are also the products of the actual rate coefficients and the concentration of any co-reactants of the respective steps (Chern and Helfferich, 1990). The numerators of the segment coefficient expressions given in equation 2 and 3 are the products of the forward rate coefficients and reverse rate coefficients in the pathway respectively. The operator jkD is evaluated using equation 4. 1 1 , 1 , 1 1 1 1 4 i kk jk m m m m i j m j m D                      A loop coefficient is used to reduce parallel pathways in a network into a single lumped pathway. The expressions for forward and reverse loop coefficients are shown in equation 5 and 6 (Chen and Chern, 2009). ( ) 1 ( ) 1 5 6 m k ij ij k m k ji ji k L L         Collective coefficients ( ij ) are used to group segment rate coefficients in a network segment so as to reduce network portions with higher number of coefficients to a pseudo-single step. 2.1 Reaction Mechanism and Network The mechanism of the reaction is presented below: EB + S ⇌ 𝐸𝐵𝑆 EBS ⇌ 𝑂𝑋𝑆 http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 310 𝑂𝑋S ⇌ 𝑃𝑋𝑆 𝑃𝑋S ⇌ 𝑀𝑋𝑆 𝑀𝑋S ⇌ 𝐸𝐵𝑆 𝑂𝑋S ⇌ 𝑂𝑋 + 𝑆 𝑃𝑋S ⇌ 𝑃𝑋 + 𝑆 𝑀𝑋S ⇌ 𝑀𝑋 + 𝑆 The reaction network and steps involved in reduction of the complex network into a single-cycle network for modeling is presented in steps 1-8 below: 1. The network depicting the mechanism for the reaction showing multi-pathways and multi- intersections is shown in Figure 1. Figure 1: Reaction network for the four-node pyramidal isomerization reaction. 2. The network in Figure 1 can be viewed as pyramidal network with four nodes in the base triangle. The four nodes being X1, X0, X3, and X4. The pyramidal network with the four nodes in the base triangle is shown in Figure 2. Figure 2: A pyramidal reaction network with four-nodes in the base triangle. 3. Cross-to-square transformation technique was used to transform the cross-network represented by the pathways X2 to X4 and X1 to X3 to a square network represented by the pathways X4 to X1, X1 to X2, X2 to X3, X3 to X4. Figure 3 shows the transformed inner cross- network (dash lines). X0 was considered as an intersection node in the transformation. Figure 3: Transformed cross-network into square network. A B C D X1 X2 X3 X4 X0 X2 X1 X3 X4 X0 X4 X2 X3 X1 D C B A file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng Kolo et al: Kinetic Modeling of Ethylbenzene Isomerization using Bodenstein Approximation Technique. AZOJETE, 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 311 4. The parallel pathways between X3 and X2, X2 and X1, X1 and X4, and X4 and X3 are lumped into loop coefficients (Lij) where (i,j) stand for (3,2), (2,3), (2,1), (1,2), (1,4), (4,1), (4,3), (3,4). The system is now reduced to a pyramidal network with three nodes in the base triangle. The network with three nodes in the base triangle is presented in Figure 4. Figure 4: Pyramidal network with three nodes in the base triangle A cyclic representation of the pyramidal network with three nodes in the base triangle showing inner Y-type network is presented in Figure 5. Figure 5: Cyclic representation of pyramidal network 5. Applying Y-to-delta transformation to the Y-network in Figure 5, an equivalent delta-type network is obtained. Transferring the delta-type network into the main cycle, the complete network of the reaction is now reduced to two cycles with two pathways between each node. Figure 6 shows the network containing two cycles and pathways between the three nodes. The dash lines show the delta-type network. Figure 6: Reduced network containing two cycles and three pathways between nodes. X4 X3 X2 X1 X4 X2 X1 X3 X1 X4 X3 http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 312 6. The three parallel pathways between the three nodes in Figure 6 are lumped into loop coefficients to yield a single-cycle reaction network. The single-cycle network is shown in Figure 7. Figure 7: Reduced single cycle reaction network for ethylbenzene isomerization reaction. The general rate equation for reduced single-cycle developed in Chern and Chen (2003) is then applied to the reduced single-cycle network in figure 6 to get the rate equation through the cycle. Equation 12 shows the general equation for the rate through the cycle.  Tr X D   12 Where; , 1 1, 0 0 n n l l l l i i L L             , and           , , 0 , , 0 , ,0 ho hh i jk i h i j k ii jj kk kl kk il ii il ii jl jj i l ppath pj k p i j k pl jl jj kl kk ho hh pl PP i p l h i j k D D D D C D C D C D C D S D C C D C D D C D S                                           The notation  in the numerator of equation 12 represents the difference between the product of the forward loop coefficients of the reduced single-cycle network and the product of the reverse rate coefficients. The forward loop coefficients are 1 1 1 13 34 41, ,L L L and the reverse loop coefficients are 1 1 1 31, 43 14,L L L . Hence the notation  can be expressed in the form shown in equation 14. 1 1 1 1 1 1 13 34 41 31 43 14 14L L L L L L   The expression for D is evaluated by considering each term in the expression: 1. , ,ii jj kkD D and D (i, j, and k represent the indices of the node intermediates of the final reduced cycle; that is 1, 3, and 4) are obtained from the loop coefficients of the final reduced X3 X1 X4 file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng Kolo et al: Kinetic Modeling of Ethylbenzene Isomerization using Bodenstein Approximation Technique. AZOJETE, 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 313 cycle shown in figure 6. The jkD operator formula shown in equation 4 is used in evaluating the D terms. Equation 15 shows the expression of , ,ii jj kkD D and D 1 1 1 1 1 1 11 34 41 31 14 31 43 1 1 1 1 1 1 33 41 13 43 31 43 14 1 1 1 1 1 1 44 13 34 14 43 14 31 15 D L L L L L L D L L L L L L D L L L L L L          2. , , 0 ho hh h i j k D S   This term denotes the expression for concentration of missed center nodes X2 used in Y-to-delta transformation. Indices i, j, and k represent indices of the node intermediates of the final reduced cycle; that is 1, 3, and 4. And S0 is the sum of the segment coefficients (  ) of the segments from the missed center node intermediate to the adjacent node intermediates (that is; X1, X3, and X4). Consequently, after relevant substitutions, the term is presented in equation 16. , , 10 11 30 33 40 44 0 01 02 03 04 16 ho hh h i j k D D D D S            3.   k i kl kk il ii l C D C D   This term denotes the expression for summation of the concentration of missed intermediates (represented by l ) in the pathway k-to-i (k stands for 1 and i stands for 3) due to network reduction. This is expression is 0 (zero) because there are no intermediates in the pathway from X1 to X3. 4.   i j il ii jl jj l C D C D   This term denotes the expression for summation of the concentration of missed intermediates (represented by l ) in the pathway i-to-j (i stands for 3 and j stands for 4) due to network reduction. This is expression is 0 (zero) because there are no intermediates in the pathway from X3 to X4. 5.   j k jl jj kl kk l C D C D   . This term denotes the expression for summation of the concentration of missed intermediates (represented by l ) in the pathway j-to-k (j stands for 4 and k stands for 1) due to network reduction. This is expression is also 0 (zero) because there are no intermediates in the pathway from X4 to X1. http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 314 6.     , , 0 , ,0 ppath pp i j k pl ho hh pl PP p l h i j k C D C D S              This generally represents the concentration of missed intermediates as a result of network reduction between center-node and i, center-node and j, and center node and k (i, j, and k represent the indices of the nodes that make the final reduced cycle; that is 1, 3, and 4). This expression is also 0 (zero) because there are no intermediates between the nodes that make up the final reduced cycle and the centre-node. Following relevant substitutions, the D term is presented in equation 17.       1 1 1 1 1 1 34 41 31 14 31 43 10 11 30 33 40 44 1 1 1 1 1 1 1 1 1 1 1 1 01 02 03 0441 13 43 31 43 14 13 34 14 43 14 31 17 L L L L L L D D D D L L L L L L L L L L L L                          Substituting equations 14 and 17 into equation 12, the expression for the rate through the reduced single cycle is obtained.         1 1 1 1 1 1 13 34 41 31 43 14 1 1 1 1 1 1 34 41 31 14 31 43 10 11 30 33 40 44 1 1 1 1 1 1 1 1 1 1 1 1 01 02 03 0441 13 43 31 43 14 13 34 14 43 14 31 18T L L L L L L r X L L L L L L D D D L L L L L L L L L L L L                            Equation 18 shows the rate equation through the reduced single cycle network presented in figure 6. 2.2 Parameter Estimation A well-known task in the concise mathematical representation of reaction network and mechanism is the estimation of the reaction rate constants. Parameter estimation is essentially an optimization problem whereby the unknown parameters are obtained by minimizing a suitable objective function (Englezos and Kalogerakis, 2001). The objective function represents a measure of disparity between observed reaction rates from experimental data to rate from developed model. The Nelder- Mead simplex optimization technique was used in this study to minimize the objective function. The Nelder-Mead simplex algorithm is a direct search method for multidimensional unconstrained minimization. The algorithm seeks to minimize a scalar valued nonlinear function of n real variables using only scalar function values without any derivative. The optimization routine was carried out using MATLAB computer software. The MATLAB solver that executes optimization routine based on Nelder-Mead technique is the nonlinear programming solver fminsearch. The solver uses the Nelder-Mead optimization algorithm as elucidated by Lagarias et al (1998). 3. Results and Discussion The model developed is presented in equation 18. Table 1 shows the rate constants for each reaction step. The calculated values of activation energy and frequency factor for the model is presented in Table 2. file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng Kolo et al: Kinetic Modeling of Ethylbenzene Isomerization using Bodenstein Approximation Technique. AZOJETE, 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 315 Table 1: Estimated Rate Constants Rate constants (mol h-1 gcat -1) T = 593 K T = 613 K T = 633K T = 673 K K01 5.1845 -10.2268 10.404 1.4783 K02 2.4713 -964 -1.4413 -0.3023 K03 13.7876 -15.6994 4.129 -0.2102 K04 5.3871 0.9333 -12.9715 -0.5389 K10 -26.2378 -55.2825 -31.5937 -0.8001 K20 -1.6024 -2.9293 1.2157 2.7953 K30 6.3507 6.5799 6.7756 2.1114 K40 3.646 16.5405 15.957 1.4358 K12 -10.3324 12.9313 -14.6375 1.3043 K21 3.3084 6.7447 23.2739 1.3722 K23 -0.2239 -7.7967 7.8819 0.5419 K32 2.7019 0.1671 0.7595 1.906 K34 7.7078 -4.0816 0.4403 -1.1509 K43 6.9018 4.6199 12.9419 0.0532 K14 5.9099 13.9132 12.0624 5.4539 The model developed in this study clearly depicts the multi-step nature of the reaction. The embedded partial pressure terms in the denominator of the rate expression indicate the adsorption of the reacting species on the surface of the catalyst as the reaction mechanism suggests (Fogler, 2004). The loop coefficients in the models expressively show that the reaction consist of several pathways as represented by the reaction network. The model also shows the contribution of each elementary step in the reaction network to the kinetics of the overall reaction. This is as indicated by the appearance of the rate constants of each step in the model developed. The L1 ij terms appearing in the model equations constitute the rate constants of the all steps. As it is customary to observe the effect of temperature variation on the estimated rate constants, a general pattern is not observable in the results obtained. Some of the values obtained for the rate constants deviate from expected trend. Comparing the values for the constants with that reported by Farahani et al (2020) further indicates a departure although with a justification. In their work, the rate constants for the reaction step producing m-xylene at temperatures of 653K, 663K, and 673K were reported to be 48.74 mol h-1 gcat -1, 58.47 mol h-1 gcat -1, and 72.29 mol h-1 gcat -1 respectively. That of the reaction step producing p-xylene at the same temperatures were reported to be 9.72E 06 mol h-1 gcat -1, 1.11E 06 mol h-1 gcat -1, and 1.28E 06 mol h-1 gcat -1 respectively. The rate constants for the reaction step producing m-xylene in this study are 7.7078 mol h-1 gcat - 1 and 0.4403 mol h-1 gcat -1 at 593K and 633K. While the rate constants for the reaction step producing p-xylene at the same temperatures are 7.8819 mol h-1 gcat -1 and 0.5419 mol h-1 gcat -1 respectively. A significant discrepancy can be observed in the reported values when compared. This can be attributed to the different modelling techniques adopted in the two works, slight difference in the starting material, and the choice of catalyst. It is already established that the nature of catalyst used for the reaction of interest affects the reaction rate constants (Farahani, 2020). An anomaly was also observed in the values of pre-exponential factor calculated. The values as shown in table 2 appear to be constant all through. This did not compare favorably well http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 316 with expected trend although literature values are scanty as much work is not reported on the methodology adopted in this study. The expected anomaly in the value of activation energy for multi-step reactions is confirmed in the results presented in table 2. This is evident in some negative values of the activation energy calculated. Table 2: Calculated Arrhenius Parameters in the Model Rate Constants Activation Energy (kJ/mol) Frequency Factor K01 2.49E-05 2.722635 K02 8.31E-04 2.722635 K03 2.49E-04 2.722635 K04 4.16E-04 2.722635 K10 0.00E+00 2.722635 K20 6.65E-04 2.722635 K30 4.16E-04 2.722635 K40 2.49E-05 2.722635 K12 4.16E-05 2.722635 K21 -4.16E-05 2.722635 K23 -8.31E-05 2.722635 K32 4.16E-05 2.722635 K34 3.32E-04 2.722635 K43 8.31E-05 2.722635 K14 -8.31E-05 2.722635 K41 -5.82E-04 2.722635 For multi-step reactions, the activation energy may be negative and the rate decreases with increasing temperature (Helfferich, 2004). This indicates that the rates of reverse steps in the reaction network increases more sharply with temperature than those of forward steps for cases in which the activation energy is negative. 4. Conclusions The Bodenstein approximation of trace-level intermediates was exploited in this study to effectively model ethylbenzene isomerization reaction over Pt/Al2O3. Reasonable assumptions were considered in the model development alongside network transformation techniques that helped reduce the complex reaction network with multi pathways and intersection into a single cycle network. Few deviations were seen in the results presented in the study even though the model development procedure compares well with the available literature. Discrepancies were seen in the rate constant values reported in an earlier study but the non-alignment can be file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng Kolo et al: Kinetic Modeling of Ethylbenzene Isomerization using Bodenstein Approximation Technique. AZOJETE, 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818, www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 317 attributed to the choice of modeling technique and catalyst used. Findings from the study re-echo the reliability of fundamental approach in up-scale applications; this is evident in the consideration of the individual molecular steps of the reaction in the model development. General rate equation approach to kinetic study is recommended for elucidating the kinetics of complex reactions and Bodenstein approximation alongside network transformation techniques are also recommended for effective modeling of multi-step reactions without loss in vital information. Nomenclature EB, A Ethylbenzene OX, B o-xylene PX, C p-xylene MX, D m-xylene EBS, X1 Ethylbenzene complex OXS, X2 o-xylene complex PXS, X3 p-xylene complex MXS, X4 m-xylene complex S, X0 free catalyst site i, j indices indicating reaction step ij  Pseudo-first order rate coefficient D Denominator of segment coefficient equation  Segment coefficient Xj intermediate at jth position Xi intermediate at ith position j kr  Rate of reaction from j-to-k XT Total catalyst concentration References Al-Khattaf, S., Tukur, NM. and Rabiu S. 2009. Ethylbenzene Transformation over ZSM-5-Based Catalyst in a Riser Simulator. Industrial and Engineering Chemistry Research, 48: 2836-2843. Al- Zahrani, SM., Al-kinany, MC., Al-humaizi, KI. and Al-khowaitu, SH. 2002. Transalkylation and Isomerization of ortho-diethylbenzene with Benzene using Trifluromethanesulphonic acid Catalysts: Kinetic analysis. Chemical Engineering and Processing. 41: 321-327. Chern, J.-M. 2000. General Rate Equations and Their Applications for Cyclic Reaction Network: Single Cycle Systems. Industrial and Engineering Chemistry Research, 39: 4100-4105. Chen, T.-M and Chern, J. 2002a. General Rate Equations and Their Applications for Cyclic Reaction Networks: Multi-Cycle systems. Chemical Engineering Science, 57: 457-467. Chen, T.-M and Chern, J. 2002b. General Rate Equations and Their Applications for Cyclic Reaction Networks: Multi-Pathway Systems. Chemical Engineering Science, 57: 5011-5020. Chen, T. and Chern, J. 2003. General Rate Equations and their Applications for Cyclic Reaction Networks: Pyramidal Systems. Chemical Engineering Science, 58: 1407-1415. http://www.azojete.com.ng/ mailto:askolo@unimaid.edu.ng Arid Zone Journal of Engineering, Technology and Environment, June, 2022; Vol. 18(2):307-318. ISSN 1596-2644; e-ISSN 2545-5818; www.azojete.com.ng Corresponding author’s e-mail address: askolo@unimaid.edu.ng 318 Chen, T. and Chern, J. 2009. Discrimination of the Kinetic Models for Isomerization of n-Butene to Isobutene. Journal of Chemical Engineering of Japan, 42(1): 79-84. Chern, J. and Helfferich, FG. 1990. Effective Kinetic Modeling of Multistep Homogeneous Reactions. American Institute of Chemical Engineers Journal, 36: 1200-1208. Chern, F., Chen T. and Chern, J. 2014. Effective Kinetic Modeling for Homogeneous Reaction Networks with Branches from Loops. Industrial and Engineering Chemistry Research, 53: 12983- 12992. Englezos, P., and Kalogerakis N. 2001. Applied Parameter Estimation for Chemical Engineers. Marcel Dekker, Inc. New York. Farahani SH., Falamaki C. and Alavi, SM. 2020. Kinetic Modeling of Ethylbenzene/Xylene Isomerization Reaction over HZSM-5 Zeolites Revisited. International Journal of Chemical Kinetics, 52(6): 368-377. Fogler HS. 2004. Elements of Chemical Reaction Engineering, 3rd Edition, Prentice-Hall, New Delhi. Fernendes, LD., Corma, A., Martinez, A., Aguiar EF. and Monteiro, JC. 1998. Hydroisomerization of Ethylbenzene on Mordenite-Based Bifunctional Catalysts with Different Platinum Contents. Brazilian Journal of Chemical Engineering, 15(2): https://doi.org/10.1590/S0104- 66321998000200009. Helfferich, FG. 1989. Systematic Approach to Elucidation of Multistep Reaction Networks. Journal of Physical Chemistry, 93: 6676. Helfferich, FG. 2004. Kinetics of Multi-step Reactions. Amsterdam: Elsevier Science. Kangas, M., Salmi, T. and Murzin, DU. 2008. Skeletal Isomerization of Butene in Fixed bed. Part 2. Kinetic and Flow Modeling. Industrial and Engineering Chemistry Research, 47: 5413-5426. Lagarias, JC., Reeds, JA., Wright, MH. and Wright, PE. 1998. Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM Journal of Optimization, 9 (1): 112–147. Oliveira, LP., Hudebine, D., Guillaume, D. and Verstraete, JJ. 2016. A Review of Kinetic Modeling Methodologies for Complex Processes. Oil & Gas Science and Technology-Rev. IFP Energies nouvelles, 71. 45. Robschlager, KH. and Christoffel, EG. 1979. Reaction Mechanism of Ethylbenzene Isomerization. Industrial and Engineering Chemistry Product Research and Development, 18 (4). 347-352. doi: 10.1021/13600729. Waziri, SM. and Al-Khattaf, S. 2009. Kinetics of Ethylbenzene Ethylation with Ethanol over a ZSM- 5-Based Catalyst in a Riser Simulator. Industrial and Engineering Chemistry Research, 48: 8341- 8348 file:///C:/user/Downloads/azojete143/www.azojete.com.ng mailto:askolo@unimaid.edu.ng https://doi.org/10.1590/S0104-66321998000200009 https://doi.org/10.1590/S0104-66321998000200009