Heat and Mass Transfer

, Volume 53, Issue 1, pp 151–160 | Cite as

Discrete and continuous modelling of convective heat transport in a thin porous layer of mono sized spheres

  • Per E. C. BurströmEmail author
  • Vilnis Frishfelds
  • Anna-Lena Ljung
  • T. Staffan Lundström
  • B. Daniel Marjavaara


Convective heat transport in a relatively thin porous layer of monosized particles is here modeled. The size of the particles is only one order of magnitude smaller than the thickness of the layer. Both a discrete three-dimensional system of particles and a continuous one-dimensional model are considered. The methodology applied for the discrete system is Voronoi discretization with minimization of dissipation rate of energy. The discrete and continuous model compares well for low velocities for the studied uniform inlet boundary conditions. When increasing the speed or for a thin porous layer however, the continuous model diverge from the discrete approach if a constant dispersion is used in the continuous approach. The new result is thus that a special correlation must be used when using a continuous model for flow perpendicular to a thin porous media in order to predict the dispersion in proper manner, especially in combination with higher velocities.


Porous Medium Continuous Model Dissipation Rate Heat Transport Discrete Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Specific surface area (m−1)

A, Bo, Bn



Fluid specific heat capacity (J/kg K)


Diameter of sphere (m)


Dispersion (m2/s)


Solid enthalpy (J/kg)


Heat transfer coefficient (W/mK)


Height of bed (m)


Fluid enthalpy (J/kg)


Order of rotational symmetry


Fluid thermal conductivity (W/mK)


Solid thermal conductivity (W/mK)


Pressure (Pa)


Radius of the particle in given cross section (m)


Radius of pellet (m)


Area of cross section (m2)

Sf, Ss

Energy sources (W/m2)


Time (s)


Temperature (K)


Energy transfer (W)


Superficial velocity (m/s)


Velocity (m)


Volumetric flow rate (m3/s)


Height (m)







Density (kg/m3)


Angle (rad)


Stream function (m2/s)


Vorticity (s−1)


Dynamic viscosity (kg/ms)


Reciprocal tortuosity



The authors express their gratitude to HLRC for supporting and financially backing this work.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Degroot CT, Straatman AG (2011) Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media. Int J Heat Mass Tranf 54:5039–5048CrossRefzbMATHGoogle Scholar
  2. 2.
    Degroot CT, Straatman AG (2012) Thermal dispersion in high-conductivity porous media. In: Delgado JMPQ et al (eds) Numerical analysis of heat and mass transfer in porous media. Springer, HeidelbergGoogle Scholar
  3. 3.
    Jiang P-X, Fan M-H, Si G-S, Ren Z-P (2001) Thermal-hydraulic performance of small scale micro-channel and porous-media heat-exchangers. Int J Heat Mass Transf 44:1039–1051CrossRefGoogle Scholar
  4. 4.
    Burström PEC, Lundström TS, Marjavaara BD, Töyrä S (2010) CFD-modelling of selective non-catalytic reduction of NOx in grate-kiln plants. Prog Comput Fluid Dyn 10:284–291CrossRefzbMATHGoogle Scholar
  5. 5.
    Ljung A-L, Lundström TS, Marjavaara BD, Tano K (2011) Convective drying of an individual iron ore pellet—analysis with CFD. Int J Heat Mass Transf 54:3882–3890CrossRefzbMATHGoogle Scholar
  6. 6.
    Burström PEC, Antos D, Lundström TS, Marjavaara BD (2015) A CFD-based evaluation of selective non-catalytic reduction of nitric oxide in iron ore grate-kiln plants. Prog Comput Fluid Dyn 15:32–46CrossRefGoogle Scholar
  7. 7.
    Larsson IAS, Lundström TS, Marjavaara BD (2015) Calculation of kiln aerodynamics with two RANS turbulence models and by DDES. Flow Turbul Combust 94:859–878CrossRefGoogle Scholar
  8. 8.
    Gronli MG, Melaaen MC (2000) Mathematical model for wood pyrolysis—comparison of experimental measurements with model predictions. Energy Fuels 14:791–800CrossRefGoogle Scholar
  9. 9.
    Guo B-Y, Maldonado D, Zulli P, Yu A-B (2008) CFD modelling of liquid metal flow and heat transfer in blast furnace hearth. ISIJ Int 48:1676–1685CrossRefGoogle Scholar
  10. 10.
    Qing GL, Ma L, Zhang XS, Zhou JL, Kuwabara M (2010) Numerical investigation of gas flow through blast furnace shaft with designed layered structure of ore and coke burdens. Ironmak Steelmak 37:546–552CrossRefGoogle Scholar
  11. 11.
    Chen W, Qu M (2014) Analysis of the heat transfer and airflow in solar chimney drying system with porous absorber. Renew Energy 63:511–518CrossRefGoogle Scholar
  12. 12.
    Yamoah S, Akaho EHK, Ayensu NGA, Asamoah M (2012) Analysis of fluid flow and heat transfer model for the pebble bed high temperature gas cooled reactor. Res J Appl Sci Eng Technol 4:1659–1666Google Scholar
  13. 13.
    Zhang J, Datta AK, Mukherjee S (2005) Transport processes and large deformation during baking of bread. AIChE J 51:2569–2580CrossRefGoogle Scholar
  14. 14.
    Datta AK (2007) Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: problem formulations. J Food Eng 80:80–95CrossRefGoogle Scholar
  15. 15.
    Dotto GL, Pinto LAA, Moreira MFP (2015) Determination of the effective thermal diffusivity in a porous bed containing rice grains: effects of moisture content and temperature. Heat Mass Transf. doi: 10.1007/s00231-015-1604-5 Google Scholar
  16. 16.
    Kluge NEJ, Lundström TS, Westerberg LG, Olofsson K (2015) Compression moulding of SMC: modelling with CFD and validation. J Reinf Plast Comp 34:479–492CrossRefGoogle Scholar
  17. 17.
    Whitaker S (1977) Simultaneous heat and momentum transfer in porous media: a theory of drying. Adv Heat Transf 13:119–203CrossRefGoogle Scholar
  18. 18.
    Whitaker S, Chou W (1983) Drying granular porous media-theory and experiment. Dry Technol 1:3–33CrossRefGoogle Scholar
  19. 19.
    Nasrallah SB, Perre P (1988) Detailed study of a model of heat and mass transfer during convective drying of porous media. Int J Heat Mass Transf 31:957–967CrossRefzbMATHGoogle Scholar
  20. 20.
    Liu W, Peng SW, Mizukami K (1995) A general mathematical modelling for heat and mass transfer in unsaturated porous media: an application to free evaporative cooling. Heat Mass Transf 31:49–55CrossRefGoogle Scholar
  21. 21.
    Baggio P, Bonacina C, Schrefler BA (1997) Considerations on modeling heat and mass transfer in porous media. Transp Porous Med 28:233–251CrossRefGoogle Scholar
  22. 22.
    Jourak A, Frishfelds V, Lundström TS, Herrmann I, Hedström A (2013) The calculations of dispersion coefficients inside two-dimensional randomly packed beds of circular particles. AIChE J 59:1002–1011CrossRefGoogle Scholar
  23. 23.
    Jourak A, Hellström JGI, Lundström TS, Frishfelds V (2014) Numerical derivation of dispersion coefficients for flow through three-dimensional randomly packed beds of monodisperse spheres. AIChE J 60:749–761CrossRefGoogle Scholar
  24. 24.
    Nijemeisland M, Dixon AG (2004) CFD study of fluid flow and wall heat transfer in a fixed bed of spheres. AIChE J 50:906–921CrossRefGoogle Scholar
  25. 25.
    Magnico P (2003) Hydrodynamic and transport properties of packed beds in small tube-to-sphere diameter ratio: pore scale simulation using an eulerian and a lagrangian approach. Chem Eng Sci 58:5005–5024CrossRefGoogle Scholar
  26. 26.
    Dixon AG, Nijemeisland M (2001) CFD as a design tool for fixed-bed reactors. Ind Eng Chem Res 40:5246–5254CrossRefGoogle Scholar
  27. 27.
    Succi S (2001) The lattice Boltzmann equation: for fluid dynamics and beyond. Clarendon Press, OxfordzbMATHGoogle Scholar
  28. 28.
    Chen S, Doolen GD (2003) Lattice boltzmann method for fluid flows. Annu Rev Fluid Mech 30:329–364MathSciNetCrossRefGoogle Scholar
  29. 29.
    Manz B, Gladden LF, Warren PB (1999) Flow and dispersion in porous media: lattice-boltzmann and NMR studies. AIChE J 45:1845–1854CrossRefGoogle Scholar
  30. 30.
    Freund H, Bauer J, Zeiser T, Emig G (2005) Detailed simulation of transport processes in fixed-beds. Ind Eng Chem Res 44:6423–6434CrossRefGoogle Scholar
  31. 31.
    Shakhawath Hossain Md, Chen XB, Bergstrom DJ (2015) Fluid flow and mass transfer over circular strands using the lattice Boltzmann method. Heat Mass Transf 51:1493–1504CrossRefGoogle Scholar
  32. 32.
    Brady JF, Sierou A (2001) Accelerated stokesian dynamics simulations. J Fluid Mech 448:115–146zbMATHGoogle Scholar
  33. 33.
    Gunjal PR, Ranade VV, Chaudhari RV (2005) Computational study of a single-phase flow in packed beds of spheres. AIChE J 51:365–378CrossRefGoogle Scholar
  34. 34.
    McKenna TF, Spitz R, Cokljat D (1999) Heat transfer from catalysts with computational fluid dynamics. AIChE J 45:2392–2410CrossRefGoogle Scholar
  35. 35.
    Hellström JGI, Frishfelds V, Lundström TS (2010) Mechanisms of flow-induced deformation of porous media. J Fluid Mech 664:220–237MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Van Der Westhuizen J, Plessis JPD (1996) An attempt to quantify fibre bed permeability utilizing the phase average Navier-Stokes equation. Compos A 27:263–269Google Scholar
  37. 37.
    Barry SI, Aldis GK (1993) Radial flow through deformable porous shells. J Aust Math Soc B 34:333–354MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ljung A-L, Frishfelds V, Lundström TS, Marjavaara BD (2012) Discrete and continuous modeling of heat and mass transport in drying of a bed of iron ore pellets. Dry Technol 30:760–773CrossRefGoogle Scholar
  39. 39.
    Jourak A, Frishfelds V, Hellström JGI, Lundström TS, Herrmann I, Hedström A (2013) Longitudinal dispersion coefficient: effects of particle-size distribution. Transp Porous Med 99:1–16CrossRefGoogle Scholar
  40. 40.
    Gunn DJ, Pryce C (1969) Dispersion in packed beds. Trans Inst Chem Eng 47:341–350Google Scholar
  41. 41.
    Frishfelds V, Hellström JGI, Lundström TS (2014) Flow-induced deformations within random packed beds of spheres. Transp Porous Med 104:43–56MathSciNetCrossRefGoogle Scholar
  42. 42.
    Chen X, Papathanasiou TD (2008) The transverse permeability of disordered fiber arrays: a statistical correlation in terms of the mean nearest interfiber spacing. Transp Porous Med 71:233–251CrossRefGoogle Scholar
  43. 43.
    Anikeenko AV, Gavrilova ML, Medvedev NN (2009) Shapes of delaunay simplexes and structural analysis of hard sphere packings. In: Gavrilova ML (ed)Generalized voronoi diagram: a geometry-based approach to computational intelligence. Studies in computational intelligence 158, Springer, Berlin, Heidelberg, pp. 13–45Google Scholar
  44. 44.
    Thompson KE (2002) Fast and robust Delaunay tessellation in periodic domains. Int J Numer Methods Eng 55:1345–1366CrossRefzbMATHGoogle Scholar
  45. 45.
    Berlyand L, Panchenko A (2007) Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions. J Fluid Mech 578:1–34MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Batchelor GK (2000) An introduction to fluid dynamics. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  47. 47.
    Akbari M, Sinton D, Bahrami M (2011) Viscous flow in variable cross-section microchannels of arbitrary shapes. Int J Heat Mass Transf 54:3970–3978CrossRefGoogle Scholar
  48. 48.
    Kress J, Yun TS, Narsilio GA, Evans TM, Lee D-S (2012) Evaluation of hydraulic conductivity in 3D random and heterogeneous particulate materials using network model. Comput Geotech 40:45–52CrossRefGoogle Scholar
  49. 49.
    Drummond JE, Tahir MI (1984) Laminar viscous flow through regular arrays of parallel solid cylinders. Int J Multiph Flow 10:515–540CrossRefzbMATHGoogle Scholar
  50. 50.
    Frishfelds V, Lundström TS, Jakovics A (2003) Permeability of clustered fiber networks: modeling of the unit cell. Mech Compos Mater 39:265–272CrossRefGoogle Scholar
  51. 51.
    Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
  52. 52.
    Lundström TS, Gebart BR (1995) Effect of perturbation of fibre architecture on permeability inside fibre tows. J Compos Mater 29:424–443CrossRefGoogle Scholar
  53. 53.
    Wakao N, Kaguei S, Funazkri T (1979) Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds. Chem Eng Sci 34:325–336CrossRefGoogle Scholar
  54. 54.
    Wakao N, Kaguei S (1982) Heat and mass transfer in packed beds. Gordon and Breach Science Publishers Inc, New YorkGoogle Scholar
  55. 55.
    Incropera FP, Dewitt DP, Bergman TL, Lavine AS (2007) Fundamentals of heat and mass transfer, 6th edn. Wiley, New YorkGoogle Scholar
  56. 56.
    Sutherland W (1893) The viscosity of gases and molecular force. Philos Mag 36(5):507–531CrossRefzbMATHGoogle Scholar
  57. 57.
    Sonntag RE, Borgnakke C, Wylen GJV (2003) Fundamentals of thermodynamics, 6th edn. Wiley, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Per E. C. Burström
    • 1
    Email author
  • Vilnis Frishfelds
    • 2
  • Anna-Lena Ljung
    • 1
  • T. Staffan Lundström
    • 1
  • B. Daniel Marjavaara
    • 3
  1. 1.Division of Fluid and Experimental MechanicsLuleå University of TechnologyLuleåSweden
  2. 2.Liepaja UniversityLiepajaLatvia
  3. 3.LKABKirunaSweden

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