Multi-Step DTM simulation of Transesterification reactions model

Document Type : Research Paper

Authors

1 Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran.

2 Faculty of Mechanical, Process and Energy, TU Bergakademie Freiberg, Freiberg, Germany.

Abstract

The multi-step differential transform method (DTM) adopted from the standard DTM is employed in this case study to solve a model of the transesterification reaction. The DTM is considered in a sequence of time intervals. The accuracy of the proposed method is confirmed by comparing its results with those of the fourth-order RungeKutta (RK4) method. In addition, the experimental results are investigated with the Multi-step DTM to demonstrate the efficiency and effectiveness of these chemical reactions obtained in the laboratory. The present findings confirmed the effectiveness of using the multi-step DTM in validating the chemical models obtained in laboratories.

Keywords

Main Subjects

References

  • [1] H. Ahmad, M. N. Alam, and M. Omri, New computational results for a prototype of an excitable system, Results in Physics, 28 (2021), 104666.
  • [2] M. N. Alam and C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predatorprey system, Alexandria Engineering Journal, 55(2) (2016), 1855-1865.
  • [3] M. N. Alam and C. Tunc, New solitary wave structures to the (2+ 1)-dimensional KD and KP equations with spatio-temporal dispersion, Journal of King Saud University-Science, 32(8) (2020), 3400-3409.
  • [4] M. N. Alam and C. Tun, The new solitary wave structures for the (2+ 1)-dimensional time-fractional Schrodinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations, Alexandria Engineering Journal, 59(4) (2020), 2221-2232.
  • [5] M. N. Alam, A. R. Seadawy, and D. Baleanu, Closed-form wave structures of the space-time fractional HirotaSatsuma coupled KdV equation with nonlinear physical phenomena, Open Physics, 18(1) (2020), 555-565.
  • [6] M. Alam, I. Talib, O. Bazighifan, D. N. Chalishajar, and B. Almarri, An Analytical Technique Implemented in the Fractional Clannish Random Walkers Parabolic Equation with Nonlinear Physical Phenomena, Mathematics, 9(8) (2021), 801.
  • [7] M. N. Alam, E. Bonyah, M. Fayz-Al-Asad, M. Osman, and K. M. Abualnaja, Stable and functional solutions of the Klein-Fock-Gordon equation with nonlinear physical phenomena, Physica Scripta, 96(5) (2021), 055207.
  • [8] M. N. Alam and M. Osman, New structures for closed-form wave solutions for the dynamical equations model related to the ion sound and Langmuir waves, Communications in Theoretical Physics, 73(3) (2021), 035001.
  • [9] M. B. Almatrafi, A. R. Alharbi, and C. Tun, Constructions of the soliton solutions to the good Boussinesq equation, Advances in Difference Equations, 2020(1) (2020), 1-14.
  • [10] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, Exact solutions for the fractional differential equations by using the first integral method, Nonlinear engineering, 4(1) (2015), 15-22.
  • [11] H. Aminikhah, A. H. R. Sheikhani, T. Houlari, and H. Rezazadeh, Numerical solution of the distributed-order fractional Bagley-Torvik equation, IEEE/CAA Journal of Automatica Sinica, 6(3) (2017), 760-765.
  • [12] F. Ayaz, Solutions of the system of differential equations by differential transform method, Applied Mathematics Computation, 147(2) (2004), 547-567.
  • [13] E. Abuteen, S. Momani, and A. Alawneh, Solving the fractional nonlinear Bloch system using the multi-step generalized differential transform method, Computers Mathematics with Applications, 68(12) (2014), 2124-2132.
  • [14] J. Biazar and M. Eslami, Differential transform method for quadratic Riccati differential equation, International Journal of Nonlinear Science, 9(4) (2010), 444-447.
  • [15] M. E. Borges and L. Daz, Recent developments on heterogeneous catalysts for biodiesel production by oil esterification and transesterification reactions: A review, Renewable Sustainable Energy Reviews, 16(5) (2012), 2839-2849.
  • [16] S. S. Chen, Application of the differential transformation method to the free vibrations of strongly non-linear oscillators, Nonlinear analysis: Real world applications, 10(2) (2009), 881-888.
  • [17] K. M. Dharmalingam, M. Veeramuni, H. Rezazadeh, and C. Tun, Variation Iteration Method for Solving Ethanol and Acetaldehyde Concentrations in a Fixed Bed Laboratory Reactor, Applications and Applied Mathematics: An International Journal (AAM), 16(1) (2021), 20.
  • [18] M. Diasakou, A. Louloudi, and N. J. F. Papayannakos, Kinetics of the non-catalytic transesterification of soybean oil, Fuel, 77(12) (1998), 1297-1302.
  • [19] M. Eslami and H. Rezazadeh, Multi-step conformable fractional differential transform method for solving and stability of the conformable fractional differential systems, Caspian Journal of Mathematical Sciences, 9(2) (2020), 340-348.
  • [20] M. El-Shahed, Application of differential transform method to non-linear oscillatory systems, Communications in Nonlinear Science Numerical Simulation, 13(8) (2008), 1714-1720.
  • [21] V. S. Erturk, S. Momani, and Z. Odibat, Application of generalized differential transform method to multi-order fractional differential equations, Communications in Nonlinear Science Numerical Simulation, 13(8) (2008), 16421654.
  • [22] H. Fukuda, A. Kondo, and H. Noda, Biodiesel fuel production by transesterification of oils, Journal of bioscience bioengineering, 92(5) (2001), 405-416.
  • [23] S. Islam, M. Alam, M. Al-Asad, and C. Tun, An analytical Technique for Solving New Computational Solutions of the Modified Zakharov-Kuznetsov Equation Arising in Electrical Engineering, Journal of Applied Computational Mechanics, 7(2) (2021), 715-726.
  • [24] O. Kolebaje, M. Ojo, O. Ojo, and A. Omoliki, On the Application of the Multistage Differential Transform Method to the Rabinovich-Fabrikant System, The African Review of Physics, 9 (2014), 169-176.
  • [25] A. M. Liaquat, H. H. Masjuki, M. Kalam, I. R. Fattah, M. Hazrat, M. Varman, M. Mofijur, and M. J. P. E. Shahabuddin, Effect of coconut biodiesel blended fuels on engine performance and emission characteristics, Procedia Engineering, 56 (2013), 583-590.
  • [26] J. Liu, Biodiesel Synthesis via Transesterification Reaction in Supercritical Methanol: a) A Kinetic Study, b) Biodiesel Synthesis Using Microalgae Oil, 2013.
  • [27] F. Ma and M. A. Hanna, Biodiesel production: a review, Bioresource technology 70(1) (1999), 1-15.
  • [28] M. Merdan, Numerical solution of the fractional-order Vallis systems using multi-step differential transformation method, Applied Mathematical Modelling,37(8) (2013), 6025-6036.
  • [29] S. Momani and Z. Odibat, A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula, Journal of Computational Applied Mathematics Computation, 220(1-2) (2008), 85-95.
  • [30] S. Murillo, J. Miguez, J. Porteiro, E. Granada, and J. Moran, Performance and exhaust emissions in the use of biodiesel in outboard diesel engines, J Fuel Processing Technology, 86(12-13) (2007), 1765-1771.
  • [31] M. Nourifar, A. A. Sani, and A. Keyhani, Efficient multi-step differential transform method: Theory and its application to nonlinear oscillators, Communications in Nonlinear Science Numerical Simulation, 53 (2017), 154183.
  • [32] B. Nas and A. Berktay, Energy potential of biodiesel generated from waste cooking oil: an environmental approach, Energy Sources, Part B: Economics, Planning, Policy, 2(1) (2007), 63-71.
  • [33] Z. M. Odibat, C. Bertelle, M. Aziz-Alaoui, and G. H. Duchamp, A multi-step differential transform method and application to non-chaotic or chaotic systems, Computers Mathematics with Applications, 59(4) (2010), 14621472.
  • [34] H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation, Modern Physics Letters B, 33(17) (2019), 1950196.
  • [35] M. S. Shehata, H. Rezazadeh, E. H. Zahran, E. Tala-Tebue, and A. Bekir, New optical soliton solutions of the perturbed Fokas-Lenells equation, Communications in Theoretical Physics, 71(11) (2019), 1275.
  • [36] M. Shi, Z. Wang, and M. Du, A modified multi-step differential transform method for solving fractional dynamic systems, Journal of Computational Nonlinear Dynamics, 8(1) (2013).
  • [37] Q. Shu, Q. Zhang, G. Xu, Z. Nawaz, D. Wang, and J. Wang, Synthesis of biodiesel from cottonseed oil and methanol using a carbon-based solid acid catalyst, J Fuel Processing Technology, 90(7-8) (2009), 1002-1008.
  • [38] B. Thangaraj, P. R. Solomon, B. Muniyandi, S. Ranganathan, and L. Lin, Catalysis in biodiesel productiona review, Clean Energy, 3(1) (2019), 2-23.
  • [39] H. Wang, M. N. Alam, O. A. Ilhan, G. Singh, and J. Manafian, New complex wave structures to the complex Ginzburg-Landau model, AIMS Mathematics, 6(8) (2021), 8883-8894.
  • [40] Z. Wen, X. Yu, S. T. Tu, J. Yan, and E. Dahlquist, Synthesis of biodiesel from vegetable oil with methanol catalyzed by Li-doped magnesium oxide catalysts, J Applied Energy, 87(3) (2010), 743-748.
  • [41] W. Xie and H. Li, Alumina-supported potassium iodide as a heterogeneous catalyst for biodiesel production from soybean oil, Journal of Molecular Catalysis A: Chemical, 255(1-2) (2006), 1-9.
  • [42] A. Yildirim, A. Gökdogan, and M. Merdan, Chaotic systems via multi-step differential transformation method, Canadian Journal of Physics, 90(4) (2012), 391-406.
  • [43] J. Zhou, Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan, China, 1986.

 

Computational Methods for Differential Equations
Volume 13, Issue 2
March 2025
Pages 634-645

Files

History

  • Receive Date: 07 August 2021
  • Revise Date: 27 April 2024
  • Accept Date: 26 May 2024

Share

How to cite

Statistics