CHEMICAL ENGINEERING DIFFUSION-REACTION ANALYSIS VIA HELMHOLTZ EQUATION AND RADIAL BASIS FUNCTION METHODS
Keywords:
chemical engineering, Helmholtz equation, radial basis functions, meshless methods, diffusion-reactionDOI:
https://doi.org/10.17654/0972096025011Abstract
Analyzing steady-state concentration profiles in diffusion-reaction systems is crucial for designing and optimizing many chemical engineering processes, including catalytic reactors and separation units. These phenomena are frequently described by Helmholtz or modified Helmholtz equations. This study investigates the practical application of a meshless numerical technique, the radial basis function-method of approximate particular solutions (RBF-MAPS), for simulating such systems. The inherent meshless nature of RBF-MAPS offers significant advantages for handling the complex geometries often encountered in chemical engineering equipment. We apply the method to model two representative scenarios: reactant diffusion and first-order reaction within a square catalytic plate reactor incorporating an internal source term, and within a circular catalyst pellet with surface boundary conditions. The RBF-MAPS approach effectively discretizes the governing Helmholtz-type equations, and the resulting algebraic systems are robustly solved using singular value decomposition. To establish the reliability of the simulations for engineering analysis, the method’s implementation is rigorously verified using the method of manufactured solutions, demonstrating high-order convergence. Furthermore, convergence studies on specific application problems confirm the stability of the computed concentration profiles. This work validates RBF-MAPS as a flexible and accurate computational tool for chemical engineers needing to analyze diffusion-reaction behavior governed by Helmholtz equations in diverse reactor geometries.
Received: July 29, 2025
Revised: September 1, 2025
Accepted: September 5, 2025
References
[1] A. D. Polyanin and V. F. Zaitsev, Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd ed., CRC Press, 2015.
[2] S. Dasari and A. Parikh, Numerical solution of 2D inhomogeneous Helmholtz equation using the meshless radial basis function method, IJISET - International Journal of Innovative Science, Engineering and Technology 10(1) (2023), 133-139.
[3] W. W. Read, Analytical solutions for a Helmholtz equation with Dirichlet boundary conditions and arbitrary boundaries, Math. Comput. Modelling 24(2) (1996), 23-34.
[4] A. Van Hirtum, Quasi-analytical solution of two-dimensional Helmholtz equation, Applied Mathematical Modelling 47 (2017), 96-102.
[5] C. S. Chen, M. A. Goldberg and Y. C. Hon, The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, International Journal for Numerical Methods in Engineering 43(8) (1998), 1421-1435.
[6] A. Handlovičová and I. Riečanová, Numerical solution to the complex 2D Helmholtz equation based on finite volume method with impedance boundary conditions, Open Physics 14(1) (2016), 436-443.
[7] S. J. Kahlaf and A. A. Mhassin, Numerical solution of a two-dimensional Helmholtz equation with Dirichlet boundary conditions, Journal of Interdisciplinary Mathematics 24(4) (2021), 971-982.
[8] G. E. Fasshauer, Meshfree methods, Handbook of Theoretical and Computational Nanotechnology, American Scientific Publishers, 2005.
[9] A. Bouhamidi and K. Jbilou, Meshless thin plate spline methods for the modified Helmholtz equation, Computer Methods in Applied Mechanics and Engineering 1197(45-48) (2008), 3733-3741.
[10] M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, 2003.
[11] B. Fornberg and N. Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences, SIAM, Vol. 87, 2015.
[12] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics - I surface approximations and partial derivative estimates, Comput. Math. Appl. 19(8-9) (1990a), 127-145.
[13] E. J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics - II solutions to parabolic, hyperbolic, and elliptic partial differential equations, Comput. Math. Appl. 19(8-9) (1990b), 147-161.
[14] C. S. Chen, C. M. Fan and P. H. Wen, The method of approximate particular solutions for solving elliptic problems with variable coefficients, International Journal of Computational Methods 8(3) (2011), 545-559.
[15] L. Ponzellini Marinelli and E. Gaburro, A stabilized local integral method using RBFs for the Helmholtz equation with applications to wave chaos and dielectric microresonators, 2023. arXiv preprint.
[16] M. Xin, J. Li, W. Chen and Z. Fu, A new type of radial basis functions for problems governed by partial differential equations, PLoS ONE 18(11) (2023), e0294938.
[17] A. Fashamiha and D. Salac, Modified Hermite Radial Basis Functions, 2025. arXiv preprint arXiv:2503.05752.
[18] N. Egidi, J. Giacomini and P. Maponi, RBF approximation of the Lippmann-Schwinger equation, J. Math. Stat. 19(1) (2023), 28-36.
[19] P. J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, 1998.
[20] K. Salari and P. Knupp, Code verification by the method of manufactured solutions (SAND Report SAND2000-1444), Sandia National Laboratories, 2000.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
____________________________
Licensing Terms:
This work is published by Pushpa Publishing House and is subject to the following conditions:
Attribution: You must credit Pushpa Publishing House as the original publisher. Include the publication title and author(s) if applicable.
No Derivatives: Modifying the work or creating derivative works is not permitted without prior written permission.
For more information or permissions beyond the scope of this license, contact Pushpa Publishing House.
Publication count: 
Google h-index: 
Downloads: 