Performance Analysis of Euler and Fourth-Order Runge-Kutta Methods in Modeling the Transient Response of a Continuous Stirred Tank Reactor

Nurul Nadiya Abu Hassan; Nurmarni Athirah Abdul Wahid

doi:https://doi.org/10.14445/22315373/IJMTT-V72I9P102

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 9 | Year 2026 | Article Id. IJMTT-V72I9P102 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I9P102

Performance Analysis of Euler and Fourth-Order Runge-Kutta Methods in Modeling the Transient Response of a Continuous Stirred Tank Reactor

Nurul Nadiya Abu Hassan, Nurmarni Athirah Abdul Wahid

Received	Revised	Accepted	Published
12 Jul 2026	18 Aug 2026	11 Sep 2026	28 Sep 2026

Citation :

Nurul Nadiya Abu Hassan, Nurmarni Athirah Abdul Wahid, "Performance Analysis of Euler and Fourth-Order Runge-Kutta Methods in Modeling the Transient Response of a Continuous Stirred Tank Reactor," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 9, pp. 34-44, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I9P102

Abstract

This paper presents a comparison study of the explicit Euler and fourth-order Runge-Kutta (RK4) methods for predicting the transient reactant concentration of the Continuous Stirred Tank Reactor (CSTR). The numerical simulations were performed in MATLAB for the stepsize of ℎ=0.01,0.05,0.1,0.1 and 0.5. The results obtained from this study were compared with the existing analytical solution and the stability criteria derived from the governing system eigenvalue is analyzed. The RK4 method achieved a better result, maintaining the maximum relative errors below 4.43×10⁻⁴% across all step sizes. Meanwhile, for Euler’s method, the maximum error obtained is 7.687429725% for ℎ=0.5. Although both methods satisfied linear stability bounds where |1+ℎ𝜆|<1 and −2.78<ℎ𝜆<0, RK4 gives a better balance between step-size tolerance and computational accuracy.

Keywords

CSTR, Numerical Methods, Euler Method, Runge-Kutta Method, Tank Reactor.

References

[1] Dale E. Seborg et al., Process Dynamics and Control, 4^th ed., John Wiley and Sons, 2017.
[Google Scholar] [Publisher Link]

[2] H. Scott Fogler, Elements of Chemical Reaction Engineering, 6^th ed., Pearson, 2022.
[Publisher Link]

[3] Tianzhou Ye et al., “Particle Movement Characteristics in a Gas–Solid Vertical Single Helical Ribbon Agitated Reactor,” Chemical Engineering Journal, vol. 429, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[4] Jie Yang et al., “Influence of Reflux Ratio on the Anaerobic Digestion of Pig Manure in Leach Beds Coupled with Continuous Stirred Tank Reactors,” Waste Management, vol. 97, pp. 115-122, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[5] Amit K. Thakur et al., “Multi-Objective Optimization of an Industrial Slurry Phase Ethylene Polymerization Reactor,” International Journal of Chemical Reactor Engineering, vol. 20, no. 6, pp. 649-659, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Steven C. Chapra, ‎and Raymond P. Canale, Numerical Methods for Engineers, 8^th ed., McGraw-Hill Education, 2021.
[Google Scholar] [Publisher Link]

[7] J. Douglas Faires, and Richard L. Burden, Numerical Analysis, 11^th ed., Cengage Learning, 2021.
[Publisher Link]

[8] Teoh Ai Kee, and Rahifa Ranom, “Comparison of Numerical Techniques in Solving Transient Analysis of Electrical Circuits,” ARPN Journal of Engineering and Applied Sciences, vol. 13, no, 1, pp. 314-320, 2018.
[Google Scholar] [Publisher Link]

[9] Chijioke Godswill Igwe, P.C. Jackreece, and Isobeye George, “Comparative Convergence Analysis of Runge-Kutta Fourth Order and Runge-Kutta-Fehlberge Methods Implementation in Matlab and Python Applied to a Series RLC Circuit,” International Journal of Innovative Science and Research Technology, vol. 10, no. 3, pp. 2957-2975, 2025.
[CrossRef] [Publisher Link]

[10] Mario D. Fiore, and Ciro Natale, “Discrete-Time Closed-Loop Inverse Kinematics: A Comparison between Euler and RK4 Methods,” 2021 29^th Mediterranean Conference on Control and Automation, Puglia, Italy, pp. 584-589, 2021.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Muhirwa Jean Pierre, “Mathematical Modelling and Analysis of Deformable Perturbed Continuously Stirred Tank Reactors (CSTRs),” Doctoral Dissertation, Nelson Mandela African Institution of Science and Technology, pp. 1-158, 2022.
[Google Scholar]

[12] F.J. Hurtado, Antonio S. Káiser, and Blas Zamora, “Fluid Dynamic Analysis of a Continuous Stirred Tank Reactor for Technical Optimization of Wastewater Digestion,” Water Research, vol. 71, pp. 282-293, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Octave Levenspiel, “Chemical Reaction Engineering,” Industrial & Engineering Chemistry Research, vol. 38, no. 11, pp. 4140-4143, 1999.
[CrossRef] [Google Scholar] [Publisher Link]

[14] John Charles Butcher, Numerical Methods for Ordinary Differential Equations, 3^rd ed., John Wiley & Sons, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[15] Liangcheng Suo et al., “Study on the Nonlinear Dynamics of the Continuous Stirred Tank Reactors,” Processes, vol. 8, no. 11, pp. 1-18, 2020.
[CrossRef] [Google Scholar] [Publisher Link]

[16] Tapio Salmi et al., “Determination of Kinetic Constants by using Transient Temperature Data from Continuous Stirred Tank Reactors,” Chemical Engineering Science, vol. 248, pp. 1-8, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[17] Jun Yin et al., “Generalized Reactor Neural ODE for Dynamic Reaction Process Modeling with Physical Interpretability,” Chemical Engineering Journal, vol. 452, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Changrui Xie et al., “Hybrid Dynamic Modeling of an Industrial Reactor Network with First-Principles and Data-Driven Approaches,” Chemical Engineering Science, vol. 289, 2024.
[CrossRef] [Google Scholar] [Publisher Link]