Research Papers: Heat and Mass Transfer

Analytical Solution for 2D Non-Fickian Transient Mass Transfer With Arbitrary Initial and Periodic Boundary Conditions

[+] Author and Article Information
Yaohong Suo

State Key Laboratory for Strength and
Vibration of Mechanical Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China;
School of Science,
Xi'an University of Science and Technology,
Xi'an 710054, China
e-mail: Yaohongsuo@126.com

Shengping Shen

State Key Laboratory for Strength and
Vibration of Mechanical Structures,
School of Aerospace,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: sshen@mail.xjtu.edu.cn

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received August 23, 2012; final manuscript received January 29, 2013; published online July 18, 2013. Assoc. Editor: Andrey Kuznetsov.

J. Heat Transfer 135(8), 082001 (Jul 18, 2013) (9 pages) Paper No: HT-12-1453; doi: 10.1115/1.4024352 History: Received August 23, 2012; Revised January 29, 2013

Two-dimensional non-Fickian diffusion equation is solved analytically under arbitrary initial condition and two kinds of periodic boundary conditions. The concentration field distributions are analytically obtained with a form of double Fourier series, and the damped diffusion wave transport is discussed. At the same time, the numerical simulation is carried out for the problem with homogeneous boundary condition and arbitrary initial condition, which shows that the concentration field gradually changes from the initial distribution to the steady distribution and it changes faster for the smaller Vernotte number. The numerical results agree well with the experimental results.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Sherief, H. H., Hamza, F. A., and Saleh, H. A., 2004, “The Theory of Generalized Thermoelastic Diffusion,” Int. J. Eng. Sci., 42, pp. 591–608. [CrossRef]
Crank, J.1979, The Mathematics of Diffusion, Oxford University Press, Oxford, UK, Chap. 2.
Bowen, R. N., 1976, Theory of Mixture, Continuum Physics Vol. III, A. C. Eringen, Academic Press, New York.
Vrentas, J. S., Vrentas, C. M., and Huang, W. J., 1997, “Anticipation of Anomalous Effects in Differential Sorption Experiments,” J. Appl. Polym. Sci., 64, pp. 2007–2013. [CrossRef]
Neogi, P., 1983, “Anomalous Diffusion of Vapors Through Solid Polymers. Part I: Irresversible Thermodynamics of Diffusion and Solution Processes,” AICHE J., 29, pp. 829–833. [CrossRef]
Afif, A. E., and Grmela, M., 2002, “Non-Fickian Mass Transport in Polymer,” J. Rheol., 46, pp. 591–628. [CrossRef]
Huai, X. L., Wang, G. X., Jiang, R. Q., and Li, B., 2004, “Non-Classical Diffusion Model for Heat and Mass Transfer in Laser Drying,” J. Univ. Sci. Technol. Beijing, 11(5), pp. 455–461.
Dong, H. X., and Jiang, R. Q., 1996, “Non-Fick Effect of Transient Diffusion Mass Transfer With First Order Homogeneous Chemical Reaction,” J. Harbin Eng. Univ., 18, pp. 101–105 (in Chinese).
Dong, H. X., Li, M. S., and Jiang, R. Q., 2002, “The Non-Fick Analytics in Transient Diffusion Mass Transfer With the Second Order Homogeneous Chemical Reaction,” J. Basic Sci. Eng., 10, pp. 347–351 (in Chinese). [CrossRef]
Long, F. A., and Richman, D., 1960, “Concentration Gradients for Diffusion of Vapors in Glassy Polymers and Their Relations to Time-Dependent Diffusion Phenomena,” J. Am. Chem. Soc., 82, pp. 513–519. [CrossRef]
Richman, D., and Long, F. A., 1960, “Measurement of Concentration Gradients for Diffusion of Vapors in Polymers,” J. Am. Chem. Soc., 82, pp. 509–513. [CrossRef]
Huai, X. L., Jiang, R. Q., Liu, D. Y., and Meng, Q., 2000, “Experimental and Theoretical Investigation on the Non-Fick Effects During the Rapid Transit Mass Diffusion,” J. Eng. Thermophys., 21, pp. 595–599 (in Chinese).
Maxwell, J. C., 1845, “On the Dynamic Theory of Gases,” Philos. Trans. R. Soc., 157, pp. 49–88. [CrossRef]
Frisch, H. L., 1966, “Irreversible Thermodynamics of Internally Relaxing Systems in the Vicinity of the Glass Transition, in Non-Equilibrium Thermodynamics,” Variational Techniques and Stability, R. J.Dennelly, R.Herman, and I.Prigogine, eds., University of Chicago Press, Chicago., pp. 277–280.
Crank, J., and Park, G. S., 1968, Diffusion in Polymers, Academic Press, London.
Jiang, R. Q., Kong, X. Q., Dong, H. X., and Liu, S. L.1996, “Non-Fick's Effect Study of Mass Transfer Law for Transient Mass Transfer Process,” J. Eng. Thermophys., 17, pp. 139–142 (in Chinese).
Sobolev, S. L., 1995, “Local Non-Equilibrium Model for Rapid Solidification of Undercooled Melts,” Phys. Lett. A., 199, pp. 383–386. [CrossRef]
Kuang, Z. B., 2010, “Variational Principles for Generalized Thermodiffusion Theory in Pyroelectricity,” Acta Mech., 214, pp. 275–289. [CrossRef]
Atefi, G., and Talaee, M. R., 2010, “Non-Fourier Temperature Field in a Solid Homogeneous Finite Hollow Cylinder,” Arch. Appl. Mech., 81, pp. 569–583. [CrossRef]
Moosaie, A., 2007, “Non-Fourier Heat Conduction in a Finite Medium Subjected to Arbitrary Periodic Surface Disturbance,” Int. Commun. Heat Mass Transfer, 34, pp. 996–1002. [CrossRef]
Moosaie, A., 2009, “Axisymmetric Non-Fourier Temperature Field in a Hollow Sphere,” Arch. Appl. Mech., 79, pp. 679–694. [CrossRef]
Tang, D. W., and Araki, N., 1996, “Non-Fourier Heat Conduction in a Finite Medium Under Periodic Surface Thermal Disturbance,” Int. J. Heat Mass Transfer, 39, pp. 1585–1590. [CrossRef]
Sharma, K. R., 2005, Damped Wave Transportand Relaxation, Elsevier, Amsterdan, The Netherlands, Chap. 2.
Kapila, D., and Plawsky, J. L., 1995, “Diffusion Processes for Integrated Waveguide Fabrication in Glasses: A Solid State Electrochemical Approach,” Chem. Eng. Sci., 50, pp. 2589–2600. [CrossRef]
Krishna, R., and van den Broeke, L. J. P., 1995, “The Maxwell-Stefan Description of Mass Transport Across Zeolite Membranes,” Chem. Eng. J., 57, pp. 155–162.
Hassanizadeh, S. M., and Leijnse, A., 1995, “A Non-Linear Theory of High-Concentration-Gradient Dispersion in Porous Media,” Adv. Water Resour., 18, pp. 203–215. [CrossRef]
Hassanizadeh, S. M., 1996, “On the Transient Non-Fickian Dispersion Theory,” Transport Porous Media, 23, pp. 107–124. [CrossRef]
Vadasz, J. J., Govender, S., and Vadasz, P., 2005, “Heat Transfer Enhancement in Nano-Fluids Suspensions: Possible Mechanisms and Explanations,” Int. J. Heat and Mass Transfer, 48, pp. 2673–2683. [CrossRef]
Vadasz, J. J., and Govender, S., 2010, “Thermal Wave Effects on Heat Transfer Enhancement in Nanofluids Suspensions,” Int. J. Therm. Sci., 49, pp. 235–242. [CrossRef]
Vadasz, P., 2006, “Heat Conduction in Nanofluid Suspensions,” J. Heat Transfer, 128, pp. 465–477. [CrossRef]
Vadasz, P., 2005, “Lack of Oscillations in Dual-Phase-Lagging Heat Conduction for a Porous Slab Subject to Imposed Heat Flux and Temperature,” Int. J. Heat Mass Transfer, 48, pp. 2822–2828. [CrossRef]
Chen, X., Zhao, S. F., and Zhai, L., “Moisture Absorption and Diffusion Characterization of Molding Compound,” ASME Trans. J. Electron. Packag., 127, pp. 460–465. [CrossRef]
Gutkin, B. S., Laing, C. R., and Colby, C. L., 2001, “Turning On and Off With Excitation: The Role of Spike-Timing Asynchrony and Synchrony in Sustained Neural Activity,” J. Comput. Neurosci., 11, pp. 121–134. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Sketch of 2D non-Fickian diffusion

Grahic Jump Location
Fig. 2

The evolution of concentration field for Ve2 = 1. (a) T = 0, (b) T = 0.1, (c) T = 0.2, and (d) T = 10.

Grahic Jump Location
Fig. 3

The evolution of concentration field for Ve2 = 5. (a) T = 0, (b) T = 0.1, (c) T = 0.2, and (d) T = 13.

Grahic Jump Location
Fig. 4

The comparison between Fickian and non-Fickian diffusion

Grahic Jump Location
Fig. 5

The comparison between non-Fickian diffusion and experimental result in Ref. [12]




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