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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.

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Figures

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.

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