Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 3, pp 1931–1942 | Cite as

Pore-scale modeling of rarefied gas flow in fractal micro-porous media, using lattice Boltzmann method (LBM)

  • H. Rostamzadeh
  • M. R. SalimiEmail author
  • M. Taeibi-Rahni


Due to the widespread use of rarefied gas flow in micro-porous media in industrial and engineering problems, a pore-scale modeling of rarefied gas flow through two micro-porous media with fractal geometries is presented, using lattice Boltzmann method. For this purpose, square- and circular-based Sierpinski carpets with fractal geometries are selected due to their inherent behavior for real porous media. Diffusive reflection slip model is used and developed for these porous media through this study. With this respect, the planar Poiseuille flow is selected as a benchmark and validated with the literature. The effect of Knudsen number (Kn) on the permeability is investigated and compared in each geometry. It is shown that as Knudsen number increases, the permeability will increase due to the gas slippage effect on the solid blocks. In addition, it is observed that the permeability is more sensitive to the gaseous flow behavior at the slip and beginning of transition flow regimes. At last, the permeability relationship with Knudsen number is presented with a higher coefficient of determination for both fractal geometries, showing that this relation is logarithmic.


Fractal micro-porous media Sierpinski carpet Rarefied gas flow Lattice Boltzmann method (LBM) Pore-scale simulation Diffusion reflection 

List of symbols



Total pores area (m2)


Total area (m2)


Discrete velocity (m s−1)


Sonic velocity (m s−1)


Fractal dimension


Distribution function

\( f_{\alpha }^{\text{eq}} \)

Equilibrium distribution function


Body force in x-direction (Pa m−1)


Height (m)


Permeability coefficient (m2)


Knudsen number


Length (m)


Characteristics length (m)


Lattice Boltzmann method


Cumulative number of capillary tubes


Number of nodes at y-direction


Pressure (Pa)


Velocity (m s−1)


Weight coefficient


Boundary node location


Fluid node location

Greek Symbols


Fractal coefficient factor


Mean free path (m)


Upper limit self-similar region


Lower limit self-similar region


Relaxation time (s)


Porosity coefficient


Density (kg m−3)


Dynamic viscosity (Pa s)


Kinematic viscosity (m2 s−1)


  1. 1.
    Yu B, Li J, Li Z, Zou M. Permeabilities of unsaturated fractal porous media. Int J Multiph Flow. 2003;29:1625–42.CrossRefGoogle Scholar
  2. 2.
    Soric A, Ferrasse J-H, Roche N. Microcalorimetric qualitative analysis of biofilm development in porous media used in wastewater treatment by constructed wetland. J Therm Anal Calorim. 2011;104:113–8.CrossRefGoogle Scholar
  3. 3.
    Alizadeh R, Karimi N, Arjmandzadeh R, Mehdizadeh A. Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media. J Therm Anal Calorim. 2018; 1–18. Google Scholar
  4. 4.
    Zheng Q, Yu B, Duan Y, Fang Q. A fractal model for gas slippage factor in porous media in the slip flow regime. Chem Eng Sci. 2013;87:209–15.CrossRefGoogle Scholar
  5. 5.
    Zheng Q, Yu B. A fractal permeability model for gas flow through dual-porosity media. J Appl Phys. 2012;111:024316.CrossRefGoogle Scholar
  6. 6.
    Nasiri H, Jamalabadi MYA, Sadeghi R, Safaei MR, Nguyen TK, Shadloo MS. A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows. J Therm Anal Calorim. 1–9.Google Scholar
  7. 7.
    Heydari A, Akbari OA, Safaei MR, Derakhshani M, Alrashed AA, Mashayekhi R, et al. The effect of attack angle of triangular ribs on heat transfer of nanofluids in a microchannel. J Therm Anal Calorim. 2018;131:2893–912.CrossRefGoogle Scholar
  8. 8.
    Klinkenberg L. The permeability of porous media to liquids and gases. In: Klinkenberg L, editor. Drilling and production practice. Washington: American Petroleum Institute; 1941.Google Scholar
  9. 9.
    Civan F. Effective correlation of apparent gas permeability in tight porous media. Transp Porous Media. 2010;82:375–84.CrossRefGoogle Scholar
  10. 10.
    Hooman K, Tamayol A, Dahari M, Safaei M, Togun H, Sadri R. A theoretical model to predict gas permeability for slip flow through a porous medium. Appl Therm Eng. 2014;70:71–6.CrossRefGoogle Scholar
  11. 11.
    Song W, Yu W. Heat transfer through fibrous assemblies by fractal method. J Therm Anal Calorim. 2012;110:897–905.CrossRefGoogle Scholar
  12. 12.
    Chądzyński G, Staszczuk P, Sternik D, Błachnio M. Studies of physico-chemical properties and fractal dimensions of selected high-temperature superconductor surfaces. J Therm Anal Calorim. 2008;94:623–6.CrossRefGoogle Scholar
  13. 13.
    Zheng J, Zhang W, Zhang G, Yu Y, Wang S. Effect of porous structure on rarefied gas flow in porous medium constructed by fractal geometry. J Nat Gas Sci Eng. 2016;34:1446–52.CrossRefGoogle Scholar
  14. 14.
    Khabbazi AE, Hinebaugh J, Bazylak A. Analytical tortuosity–porosity correlations for Sierpinski carpet fractal geometries. Chaos, Solitons Fractals. 2015;78:124–33.CrossRefGoogle Scholar
  15. 15.
    Othman M, Helwani Z. Simulated fractal permeability for porous membranes. Appl Math Model. 2010;34:2452–64.CrossRefGoogle Scholar
  16. 16.
    Cui H, Wang Y, Zhang M, Wang W, Zhao C. A fractal method to calculate the permeability for compressible gas flow through a porous restrictor in aerostatic bearings. Int J Heat Mass Transf. 2018;121:437–52.CrossRefGoogle Scholar
  17. 17.
    Miao T, Long Z, Chen A, Yu B. Analysis of permeabilities for slug flow in fractal porous media. Int Commun Heat Mass Transfer. 2017;88:194–202.CrossRefGoogle Scholar
  18. 18.
    Xu P, Yu B, Qiu S, Cai J. An analysis of the radial flow in the heterogeneous porous media based on fractal and constructal tree networks. Phys A. 2008;387:6471–83.CrossRefGoogle Scholar
  19. 19.
    Song W, Yao J, Li Y, Sun H, Yang Y. Fractal models for gas slippage factor in porous media considering second-order slip and surface adsorption. Int J Heat Mass Transf. 2018;118:948–60.CrossRefGoogle Scholar
  20. 20.
    Li C, Xu P, Qiu S, Zhou Y. The gas effective permeability of porous media with Klinkenberg effect. J Nat Gas Sci Eng. 2016;34:534–40.CrossRefGoogle Scholar
  21. 21.
    Sadeghi R, Shadloo M. Three-dimensional numerical investigation of film boiling by the lattice Boltzmann method. Numer Heat Transf A Appl. 2017;71:560–74.CrossRefGoogle Scholar
  22. 22.
    Sadeghi R, Shadloo MS, Jamalabadi MYA, Karimipour A. A three-dimensional lattice Boltzmann model for numerical investigation of bubble growth in pool boiling. Int Commun Heat Mass Transfer. 2016;79:58–66.CrossRefGoogle Scholar
  23. 23.
    Sadeghi R, Shadloo M, Hopp-Hirschler M, Hadjadj A, Nieken U. Three-dimensional lattice Boltzmann simulations of high density ratio two-phase flows in porous media. Comput Math Appl. 2018;75:2445–65.Google Scholar
  24. 24.
    Goodarzi M, Safaei M, Karimipour A, Hooman K, Dahari M, Kazi S, et al. Comparison of the finite volume and lattice Boltzmann methods for solving natural convection heat transfer problems inside cavities and enclosures. Abstr Appl Anal. 2014;2014:1–15.Google Scholar
  25. 25.
    Yuan Y, Rahman S. Extended application of lattice Boltzmann method to rarefied gas flow in micro-channels. Phys A. 2016;463:25–36.CrossRefGoogle Scholar
  26. 26.
    In-Won Park M-SS, Sung-Joon Byun J-Y. Simulation of gas flow in a microchannel by lattice Boltzmann method. In: Fluid machinery and fluid mechanics: 4th international symposium (4th ISFMFE); 2010.Google Scholar
  27. 27.
    Nojoomizadeh M, D’Orazio A, Karimipour A, Afrand M, Goodarzi M. Investigation of permeability effect on slip velocity and temperature jump boundary conditions for FMWNT/Water nanofluid flow and heat transfer inside a microchannel filled by a porous media. Physica E. 2018;97:226–38.CrossRefGoogle Scholar
  28. 28.
    Boltzmann L. Lectures on gas theory. North Chelmsford: Courier Corporation; 2012.Google Scholar
  29. 29.
    Foroughi S, Masihi M, Jamshidi S, Pishvaie MR. Investigating the permeability–porosity relation of percolation-based porous media using the lattice Boltzmann method. J Porous Media. 2017; 20:899–919.Google Scholar
  30. 30.
    Abbaszadeh M, Salehi A, Abbassi A. Lattice Boltzmann simulation of heat transfer enhancement in an asymmetrically heated channel filled with random porous media. J Porous Media. 2017; 20:175–91.Google Scholar
  31. 31.
    Tang G, Tao W, He Y. Gas slippage effect on microscale porous flow using the lattice Boltzmann method. Phys Rev E. 2005;72:056301.CrossRefGoogle Scholar
  32. 32.
    Bird G. Molecular gas dynamics and the direct simulation monte carlo of gas flows, vol. 508. Oxford: Clarendon; 1994. p. 128.Google Scholar
  33. 33.
    Alder BJ, Wainwright TE. Studies in molecular dynamics. I. General method. J Chem Phys. 1959;31:459–66.CrossRefGoogle Scholar
  34. 34.
    Zhang Y-H, Gu X-J, Barber RW, Emerson DR. Capturing Knudsen layer phenomena using a lattice Boltzmann model. Phys Rev E. 2006;74:046704.CrossRefGoogle Scholar
  35. 35.
    Shan X, Yuan X-F, Chen H. Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J Fluid Mech. 2006;550:413–41.CrossRefGoogle Scholar
  36. 36.
    Kim SH, Pitsch H, Boyd ID. Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers. J Comput Phys. 2008;227:8655–71.CrossRefGoogle Scholar
  37. 37.
    Chen Z-L, Wang N-T, Sun L, Tan X-H, Deng S. Prediction method for permeability of porous media with tortuosity effect based on an intermingled fractal units model. Int J Eng Sci. 2017;121:83–90.CrossRefGoogle Scholar
  38. 38.
    Yu B, Li J. Some fractal characters of porous media. Fractals. 2001;9:365–72.CrossRefGoogle Scholar
  39. 39.
    Kaviany M. Fluid mechanics. In: Kaviany M, editor. Principles of heat transfer in porous media. Berlin: Springer; 1991. p. 15–113.CrossRefGoogle Scholar
  40. 40.
    Verhaeghe F, Luo L-S, Blanpain B. Lattice Boltzmann modeling of microchannel flow in slip flow regime. J Comput Phys. 2009;228:147–57.CrossRefGoogle Scholar
  41. 41.
    Park I-W, Shin M-S, Byun S-J, Yoon J-Y (2009) Simulation of gas flow in a microchannel by lattice Boltzmann method. In: Fluid machinery and fluid mechanics. Springer, p. 195–200.Google Scholar
  42. 42.
    Guo Z, Shu C. Lattice Boltzmann method and its applications in engineering. Singapore: World Scientific; 2013.CrossRefGoogle Scholar
  43. 43.
    Sofonea V, Sekerka RF. Boundary conditions for the upwind finite difference Lattice Boltzmann model: evidence of slip velocity in micro-channel flow. J Comput Phys. 2005;207:639–59.CrossRefGoogle Scholar
  44. 44.
    Mohamad AA. Lattice Boltzmann method: fundamentals and engineering applications with computer codes. Berlin: Springer; 2011.CrossRefGoogle Scholar
  45. 45.
    Ohwada T, Sone Y, Aoki K. Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard-sphere molecules. Phys Fluids A. 1989;1:2042–9.CrossRefGoogle Scholar
  46. 46.
    Cercignani C, Lampis M, Lorenzani S. Variational approach to gas flows in microchannels. Phys Fluids. 2004;16:3426–37.CrossRefGoogle Scholar
  47. 47.
    Dongari N, Sharma A, Durst F. Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes. Microfluid Nanofluid. 2009;6:679–92.CrossRefGoogle Scholar
  48. 48.
    Deng Z, Chen Y, Shao C. Gas flow through rough microchannels in the transition flow regime. Phys Rev E. 2016;93:013128.CrossRefPubMedPubMedCentralGoogle Scholar
  49. 49.
    Knudsen M. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann Phys. 1909;333:75–130.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • H. Rostamzadeh
    • 1
  • M. R. Salimi
    • 2
    Email author
  • M. Taeibi-Rahni
    • 1
  1. 1.Department of Aerospace EngineeringSharif University of TechnologyTehranIran
  2. 2.Aerospace Research Institute, Ministry of Science, Research and TechnologyTehranIran

Personalised recommendations