Research Papers: Conduction

Semi-Analytical Source Method for Reaction–Diffusion Problems

[+] Author and Article Information
K. D. Cole

Mechanical and Materials Engineering,
University of Nebraska–Lincoln,
W342 Nebraska Hall,
Lincoln, NE 65588-0656
e-mail: kcole1@unl.edu

B. Cetin

Mechanical Engineering,
Bilkent University,
Bilkent, Ankara 06800, Turkey

Y. Demirel

Chemical and Biomolecular Engineering,
University of Nebraska–Lincoln,
Lincoln, NE 68588-0643

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 25, 2017; final manuscript received December 15, 2017; published online April 11, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(6), 061301 (Apr 11, 2018) (10 pages) Paper No: HT-17-1499; doi: 10.1115/1.4038987 History: Received August 25, 2017; Revised December 15, 2017

Estimation of thermal properties, diffusion properties, or chemical–reaction rates from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Applications that motivate the present work include process control of microreactors, measurement of diffusion properties in microfuel cells, and measurement of reaction kinetics in biological systems. This study introduces a solution method for nonisothermal reaction–diffusion (RD) problems that provides numerical results at high precision and low computation time, especially for calculations of a repetitive nature. Here, the coupled heat and mass balance equations are solved by treating the coupling terms as source terms, so that the solution for concentration and temperature may be cast as integral equations using Green's functions (GF). This new method requires far fewer discretization elements in space and time than fully numeric methods at comparable accuracy. The method is validated by comparison with a benchmark heat transfer solution and a commercial code. Results are presented for a first-order chemical reaction that represents synthesis of vinyl chloride.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Froment, G. F. , and Bischoff, K. B. , 1979, Chemical Reactor Analysis and Design, Wiley, New York, pp. 202–204.
Heinrich, R. , and Schuster, S. , 1998, “The Modeling of Metabolic Systems. Structure, Control and Optimality,” BioSystems, 47(1–2), pp. 61–77. [CrossRef] [PubMed]
Burghardt, A. , and Berezowski, M. , 2003, “Periodic Solutions in a Porous Catalyst Pellet-Homoclinic Orbits,” Chem. Eng. Sci., 58(12), pp. 2657–2670. [CrossRef]
Gas, P. , Girardeaux, C. , Mangelinck, D. , and Portavoce, A. , 2003, “Reaction and Diffusion at Interfaces of Micro- and Nanostructured Materials,” Mater. Sci. Eng. B, 101(1–3), pp. 43–48. [CrossRef]
Demirel, Y. , and Sandler, S. I. , 2004, “Nonequilibrium Thermodynamics in Engineering and Science,” J. Phys. Chem. B, 108(1), pp. 31–43. [CrossRef]
Tevatia, R. , Demirel, Y. , and Blum, P. , 2014, “Influence of Subenvironmental Conditions and Thermodynamic Coupling on a Simple Reaction-Transport Process in Biochemical Systems,” Ind. Eng. Chem. Res., 53(18), pp. 7637–7647. [CrossRef]
Demirel, Y. , and Sandler, S. I. , 2001, “Linear Nonequilibrium Thermodynamics Theory for Coupled,” Int. J. Heat Mass Transfer, 44(13), pp. 2439–2451. [CrossRef]
Rodrigo, M. , and Mimura, M. , 2001, “Exact Solutions of Reaction-Diffusion Systems and Nonlinear Wave Equations,” Jpn. J. Ind. Appl. Math., 18, pp. 657–696. [CrossRef]
Turing, A. , 1952, “The Chemical Basis of Morphogenesis,” Philos. Trans. R. Soc. B, 237(641), pp. 37–72. [CrossRef]
Serna, H. , Munuzun, A. P. , and Barragan, D. , 2017, “Thermodynamic and Morphological Characterization of Turing Patterns in Non-Isothermal Reaction-Diffusion Systems,” Phys. Chem. Chem. Phys., 19(22), pp. 14401–14411. [CrossRef] [PubMed]
Lobanova, E. S. , Shnol, E. E. , and Ataullakhanov, F. I. , 2004, “Complex Dynamics of the Formation of Spatially Localized Standing Structures in the Vicinity of Saddle-Node Bifurcations of Waves in the Reaction-Diffusion Model of Blood Clotting,” Phys. Rev. E, 70(3), p. 032903. [CrossRef]
Berezhkovski, A. M. , Coppey, M. , and Shvartsman, S. Y. , 2009, “Signalling Gradients in Cascades of Two-State Reaction Diffusion Systems,” PNAS, 106(4), pp. 1087–1092. [CrossRef] [PubMed]
Soh, S. , Byrska, M. , Kandere-Grzybowska, K. , and Grzybowski, B. A. , 2010, “Reaction-Diffusion Systems in Intracellular Molecular Transport and Control,” Angew. Chem. Int. Ed. Engl., 49(25), pp. 4170–4198. [CrossRef] [PubMed]
Caplan, R. S. , and Essig, A. , 1999, Bioenergetics and Linear Nonequilibrium Thermodynamics: The Steady State, Harvard University Press, New York.
Demirel, Y. , and Sandler, S. I. , 2002, “Thermodynamics and Bioenergetics,” Biophys. Chem., 97(2–3), pp. 87–111. [CrossRef] [PubMed]
Anita, S. , and Capasso, V. , 2017, “Reaction-Diffusion Systems in Epidemiology,” eprint arXiv:1703.02760 https://arxiv.org/abs/1703.02760.
Elias, J. , and Clairambault, J. , 2014, “Reaction-Diffusion Systems for Spatio-Temporal Intracellular Protein Networks: A Beginners Guide With Two Examples,” Comp. Struct. Biotechnol. J., 10(16), pp. 12–22. [CrossRef]
Fahmy, E. S. , and Abdusalam, H. A. , 2009, “Exact Solutions for Some Reaction Diffusion Systems With Nonlinear Reaction Polynomial Terms,” Appl. Math. Sci., 3(11), pp. 533–540. http://m-hikari.com/ams/ams-password-2009/ams-password9-12-2009/fahmyAMS9-12-2009.pdf
Tuncer, N. , Madzvamuse, A. , and Meir, A. J. , 2015, “Projected Finite Elements for Reaction-Diffusion Systems on Stationary Closed Surfaces,” Appl. Numer. Math., 96, pp. 45–71. [CrossRef]
Stakgold, I. , 1979, Green's Functions and Boundary Value Problems, 1st ed., Wiley, New York, Chap. 9.
Taigbenu, A. E. , and Onyejekwe, O. O. , 1999, “Green's Function-Based Integral Approaches to Nonlinear Transient Boundary-Value Problems (II),” Appl. Math. Model., 23(3), pp. 241–253. [CrossRef]
Jones, M. R. , and Solovjov, V. P. , 2010, “Green's Function Approach to Nonlinear Conduction and Surface Radiation Problems,” ASME J. Heat Transfer, 132(2), p. 024502. [CrossRef]
Flint, T. F. , Francis, J. A. , Smith, M. C. , and Vasileiou, A. N. , 2018, “Semi-Analytical Solutions for the Transient Temperature Fields Induced by a Moving Heat Source in an Orthogonal Domain,” Int. J. Therm. Sci., 123, pp. 140–150. [CrossRef]
Johansson, B. T. , and Lesnic, D. , 2008, “A Method of Fundamental Solutions for Transient Heat Conduction,” Eng. Anal. Boundary Elem., 32(9), pp. 697–703. [CrossRef]
Dong, C. F. , 2009, “An Extended Method of Time-Dependent Fundamental Solutions for Inhomogeneous Heat Equation,” Eng. Anal. Boundary Elem., 33(5), pp. 717–725. [CrossRef]
Yan, L. , Yang, F. , and Fu, C. , 2009, “A Meshless Method for Solving an Inverse Spanwise-Dependent Heat Source Problem,” J. Comput. Phys., 228(1), pp. 123–136. [CrossRef]
Axelsson, O. , Glushdov, E. , and Glushkova, N. , 2009, “The Local Greens Function Method in Singularly Perturbed Convection-Diffusion Problems,” Math. Comp., 78, pp. 153–170. [CrossRef]
Mandaliya, D. D. , Moharir, A. S. , and Gudi, R. D. , 2013, “An Improved Greens Function Method for Isothermal Effectiveness Factor Determination in One- and Two-Dimensional Catalyst Geometries,” Chem. Eng. Sci., 91, pp. 197–211. [CrossRef]
Lugo-Mendez, H. D. , Valdes-Parada, F. J. , Porter, M. L. , Wood, B. D. , and Ochoa-Tapia, J. A. , 2015, “Upscaling Diffusion and Nonlinear Reactive Mass Transport in Homogeneous Porous Media,” Transp. Porous Med., 107(3), pp. 683–716. [CrossRef]
Cole, K. D. , Beck, J. V. , Woodbury, K. A. , and de Monte, F. , 2014, “Intrinsic Verification and a Heat Conduction Database,” Int. J. Therm. Sci., 78, pp. 36–47. [CrossRef]
Demirel, Y. , 2006, “Non-Isothermal Reaction-Diffusion System With Thermodynamically Coupled Heat and Mass Transfer,” Chem. Eng. Sci., 61(10), pp. 3379–3385. [CrossRef]
Cole, K. D. , de Monte, F. , McMasters, R. L. , Woodbury, K. A. , Haji-Sheikh, A. , and Beck, J. V. , 2016, “Steady Heat Conduction in Slab Bodies With Generalized Boundary Conditions,” International Mechanical Engineering Congress and Exposition, Phoenix, AZ, Nov. 13–16, Paper No. IMECE2016-66605.
Cole, K. D. , Beck, J. V. , Haji-Shiekh, A. , and Litkouhi, B. , 2011, Heat Conduction Using Green's Functions, CRC Press, Boca Rotan, FL.
Patankar, S. V. , 1980, Numerical Heat and Fluid Flow, McGraw-Hill, New York, Chap. 4.
University of Nebraska–Lincoln, 2017, “EXACT Analytical Conduction Toolbox,” University of Nebraska, Lincoln, NE, accessed July 28, 2017, http://exact.unl.edu
University of Nebraska–Lincoln, 2017, “Green's Function Library,” University of Nebraska, Lincoln, NE, accessed July 28, 2017, http://greensfunction.unl.edu
Haji-Sheikh, A. , and Beck, J. V. , 1990, “Greens Function Partitioning in Galerkin-Based Integral Solution of the Diffusion Equation,” ASME J. Heat Transfer, 112(1), pp. 28–34. [CrossRef]


Grahic Jump Location
Fig. 1

(a) Temperature versus time in transient fin at three locations and (b) error between exact solution (Eq. (35)) and SAS method at three locations, for conditions M=N=40 and m2=0.1

Grahic Jump Location
Fig. 2

Results from SAS method at dimensionless times t=0.0,0.25,0.5,1.0: (a) concentration C/Cs, (b) temperature T/Ts, (c) source g1, and (d) source g2

Grahic Jump Location
Fig. 3

Results from SAS method at dimensionless times t=0.0,0.75,1.5,3.0: (a) concentration C/Cs, (b) temperature T/Ts, (c) source g1, and (d) source g2

Grahic Jump Location
Fig. 4

Relative error in concentration (θ) at t = 0.5 as discretization parameters N and M are varied. A very coarse mesh (18 elements, 6 timesteps) is adequate for 0.4% precision.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In