Advertisement

Meccanica

, Volume 54, Issue 3, pp 505–524 | Cite as

Simulation of conjugate radiation–forced convection heat transfer in a porous medium using the lattice Boltzmann method

  • Yousef Kazemian
  • Saman Rashidi
  • Javad Abolfazli EsfahaniEmail author
  • Nader Karimi
Article
  • 177 Downloads

Abstract

In this paper, a lattice Boltzmann method is employed to simulate the conjugate radiation–forced convection heat transfer in a porous medium. The absorbing, emitting, and scattering phenomena are fully included in the model. The effects of different parameters of a silicon carbide porous medium including porosity, pore size, conduction–radiation ratio, extinction coefficient and kinematic viscosity ratio on the temperature and velocity distributions are investigated. The convergence times of modified and regular LBMs for this problem are 15 s and 94 s, respectively, indicating a considerable reduction in the solution time through using the modified LBM. Further, the thermal plume formed behind the porous cylinder elongates as the porosity and pore size increase. This result reveals that the thermal penetration of the porous cylinder increases with increasing the porosity and pore size. Finally, the mean temperature at the channel output increases by about 22% as the extinction coefficient of fluid increases in the range of 0–0.03.

Keywords

Lattice Boltzmann Conjugate radiation–forced convection Porous medium Porosity Conduction radiation ratio 

List of symbols

\( {\text{c}}_{\mathrm L} \)

Speed of light

c

Lattice streaming speed

c0

Parameter defined

c1

Parameter defined

cs

Speed of sound

ck

Lattice speed discreted in the main directions of the lattice

cp

Specific heat capacity

Da

Darcy number, \( = {\text{K}}/{\text{H}}^{2} \)

Data

Amount of each data per each mesh size

dp

Pore size diameter

ei

Discrete particle velocity in LBE model

\( {\text{e}}_{a}^{21} \)

Approximate relative error

\( e_{ext}^{21} \)

Extrapolated relative error

F

Total body force vector

fk

Density distribution function in LBE model

\( {\text{f}}_{{\mathrm{k}}}^{{{\mathrm{eq}}}} \)

Equilibrium density distribution function in LBE model

Fk

Discrete body force in LBE model

\( {\text{F}}_{\upvarepsilon} \)

Geometric function

gk

Temperature distribution function in LBE model

\( {\text{g}}_{{\mathrm{k}}}^{{{\mathrm{eq}}}} \)

Equilibrium temperature distribution function in LBE model

G

Incident radiation

\( GCI_{fine}^{21} \)

Fine-grid convergence index

H

Channel height

i

Direction index in LBE model (D2Q8 model)

I

Unit matrix

Ib

Radiation intensity of the black body

Ii

Radiation distribution function in LBE model

\( {\text{I}}_{{\mathrm{i}}}^{{{\mathrm{eq}}}} \)

Equilibrium radiation distribution function in LBE model

j

Kinematic viscosity ratio

k

Direction index in LBE model (D2Q9 model)

Ka

Absorbing coefficient

L

Channel length

M

Number of node

\( \dot{m}_{in} \)

Mass flow rate at the inlet of the channel

\( \dot{m}_{section} \)

Mass flow rate at each section of the channel

N

Conduction radiation ratio \( = \frac{{{\text{k}}_{\text{e}}\upbeta_{\text{e}} }}{{4\upsigma\left( {{\text{T}}_{\text{w}} - {\text{T}}_{\text{in}} } \right)^{3} }} \)

Nu

Nusselt number

p

Pressure

P

Apparent order

Pf

Scattering phase function

Pr

Prandtl number,\( = \frac{{\upupsilon_{\mathrm{e}} }}{{\upalpha_{\mathrm{e}} }} \)

\( \hbox{q}^{\mathrm{r}} \)

Dimensionless radiative heat flux

r

Ratio of the number of nodes

\( \vec{\hbox{r}}_{\mathrm{n}} \)

Position vector

RP

Radiation parameter \( = \frac{{\upbeta_{\rm f}^{2} \left( {1 - \upomega } \right)\hbox{H}^{2} }}{4\hbox{N}} \)

Re

Reynolds number \( = \frac{{\hbox{U}_{\rm in} \hbox{H}}}{{\upupsilon_{\rm f} }} \)

s

Constant parameter

\( \vec{s} \)

Geometric distance

t

Time

T

Temperature

Tin

Inlet temperature

Tw

Wall temperature

u

Fluid velocity vector

u

Fluid velocity in x direction

uo

Velocity at outlet

U

Non-dimensional fluid velocity, = u/Uin

Uin

Inlet fluid velocity

\( u_{section} \)

Local velocity in each section

\( URF \)

Under relaxation factor

\( \varDelta u^{ + } \)

Velocity difference with inlet section of the channel

v

Temporal velocity vector defined in Eq. (20)

v

Fluid velocity in y direction

wgi

Weighting coefficient for radiation

wk

Weighting coefficient for energy and momentum

x

Axial coordinate

X

Non-dimensional axial coordinat, = x/H

y

Transversal coordinate

Y

Non-dimensional transversal coordinat, = y/H

Greek symbols

\( \upalpha_{\mathrm f} \)

Thermal diffusivity coefficient for fluid

\( \upalpha_{\mathrm s} \)

Thermal diffusivity coefficient for solid

\( \upalpha_{\mathrm e} \)

Effective Thermal diffusivity coefficient

\( \Delta \hbox{t} \)

Time step

\( \Delta \hbox{x} \)

Space step

\( \upvarepsilon \)

Porosity

\( \upepsilon \)

Difference of data

\( \upbeta_{\mathrm e} \)

Effective extinction coefficient

\( \upbeta_{\mathrm s} \)

Extinction coefficient for solid

\( \upbeta_{\mathrm f} \)

Extinction coefficient for fluid

\( \upupsilon_{\mathrm e} \)

Effective kinematical viscosity

\( \uptau_{{\upalpha_{\mathrm{e}} }} \)

Effective relaxation time for temperature distribution

\( \uptau_{{\upupsilon_{\mathrm e} }} \)

Effective relaxation time for velocity distribution

\( \upupsilon_{\mathrm{f}} \)

Kinematical viscosity for fluid

\( \upgamma \)

Thermal diffusivity ratio

\( \upsigma \)

Stefan–Boltzmann constant

\( \upsigma_{\mathrm s} \)

Scattering coefficient

\( \upvarepsilon_{\mathrm{e}} \)

Emissivity coefficient

\( \upomega \)

Scattering albedo

\( \uptheta \)

Dimensionless temperature, \( = \frac{{\hbox{T} - \hbox{T}_{\mathrm{in}} }}{{\hbox{T}_{\mathrm{w}} - \hbox{T}_{\mathrm{in}} }} \)

\( \theta_{\mathrm m} \)

Non-dimensional mean temperature

\( \theta_{\mathrm w} \)

Non-dimensional temperature on the channel wall

\( \hbox{K}_{\mathrm a} \)

Absorbing coefficient

\( \uprho_{\mathrm{in}} \)

Density at inlet

\( \uprho_{\mathrm o} \)

Density at outlet

\( \Omega \)

Space angle

\( \varvec{\upgamma} \)

Polar angle

\( \updelta \)

Horizon angle

\( {{\upvarphi }}_{ext}^{21} \)

Extrapolated values

Notes

Compliance with ethical standards

Conflict of interest

All authors declare that they have neither conflict of interest, nor external funding.

References

  1. 1.
    Hosseini R, Rashidi S, Esfahani JA (2017) A lattice Boltzmann method to simulate combined radiation–force convection heat transfer mode. J Br Soc Mech Sci Eng 39:3695–3706CrossRefGoogle Scholar
  2. 2.
    Nazari M, Ramzani S (2014) Cooling of an electronic board situated in various configurations inside an enclosure: lattice Boltzmann method. Meccanica 49(3):645–658MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amiri Rad E, Salimi M (2017) Investigating the effects of shear rate on the collapse time in a gas–liquid system by lattice Boltzmann. Meccanica 52(4–5):915–924CrossRefGoogle Scholar
  4. 4.
    Muley A, Kiser C, Sundén B, Shah RK (2012) Foam heat exchangers: a technology assessment. Heat Transf Eng 33:42–51ADSCrossRefGoogle Scholar
  5. 5.
    Rashidi S, Esfahani JA, Karimi N (2018) Porous materials in building energy technologies—a review of the applications, modelling and experiments. Renew Sustain Energy Rev 91:229–247CrossRefGoogle Scholar
  6. 6.
    Rashidi S, Esfahani JA, Rashidi A (2017) A review on the applications of porous materials in solar energy systems. Renew Sustain Energy Rev 73:1198–1210CrossRefGoogle Scholar
  7. 7.
    Qin J, Zhou X, Zhao CY, Xu ZG (2018) Numerical investigation on boiling mechanism in porous metals by LBM at pore scale level. Int J Therm Sci 130:298–312CrossRefGoogle Scholar
  8. 8.
    Lei H, Dong L, Ruan C, Ren L (2017) Study of migration and deposition of micro particles in porous media by lattice-Boltzmann method. Energy Procedia 142:4004–4009CrossRefGoogle Scholar
  9. 9.
    Dai Q, Yang L (2013) LBM numerical study on oscillating flow and heat transfer in porous media. Appl Therm Eng 54:16–25CrossRefGoogle Scholar
  10. 10.
    Kefayati GHR, Tang H, Chan A, Wang XA (2018) Lattice Boltzmann model for thermal non-Newtonian fluid flows through porous media. Comput Fluids 176:226–244MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ma Q, Chen Z (2014) Numerical study on gas diffusion in isotropic and anisotropic fractal porous media (gas diffusion in fractal porous media). Int J Heat Mass Transf 79:925–929CrossRefGoogle Scholar
  12. 12.
    Yun H, Fangfang MA, Guo X, Chen B (2017) Mesoscopic pore-scale simulations of natural convection of porous media in closed square cavity by using LBM. Procedia Eng 205:4009–4016CrossRefGoogle Scholar
  13. 13.
    Jiang PX, Si GS, Li M, Ren ZP (2004) Experimental and numerical investigation of forced convection heat transfer of air in non-sintered porous media. Exp Therm Fluid Sci 28(6):545–555CrossRefGoogle Scholar
  14. 14.
    Thakur NS, Saini JS, Solanki SC (2003) Heat transfer and friction factor correlations for packed bed solar air heater for a low porosity system. Sol Energy 74(4):319–329ADSCrossRefGoogle Scholar
  15. 15.
    Dehghan M, Rahmani Y, Domiri Ganji D, Saedodin S, Valipour MS, Rashidi S (2015) Convection-radiation heat transfer in solar heat exchangers filled with a porous medium: homotopy perturbation method versus numerical analysis. Renew Energy 74:448–455CrossRefGoogle Scholar
  16. 16.
    Bovand M, Rashidi S, Esfahani JA (2016) Heat transfer enhancement and pressure drop penalty in porous solar heaters: numerical simulations. Sol Energy 123:145–159ADSCrossRefGoogle Scholar
  17. 17.
    Parmananda M, Dalal A, Natarajan G (2018) The influence of partitions on predicting heat transfer due to the combined effects of convection and thermal radiation in cubical enclosures. Int J Heat Mass Transf 121:1179–1200CrossRefGoogle Scholar
  18. 18.
    Gao D, Chen Z, Chen L, Zhang D (2017) A modified lattice Boltzmann model for conjugate heat transfer in porous media. Int J Heat Mass Transf 105:673–683CrossRefGoogle Scholar
  19. 19.
    Vijaybabu TR, Anirudh K, Dhinakaran S (2018) LBM simulation of unsteady flow and heat transfer from a diamond-shaped porous cylinder. Int J Heat Mass Transf 120:267–283CrossRefGoogle Scholar
  20. 20.
    Chen S, Doolen GD (1998) Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 30(1):329–364ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    He X, Luo L-S (1997) Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys Rev E 56(6):6811–6817ADSCrossRefGoogle Scholar
  22. 22.
    Nithiarasu P, Seetharamu KN, Sundararajan T (1997) Natural convective heat transfer in a fluid saturated variable porosity medium. Int J Heat Mass Transf 40:3955–3967CrossRefzbMATHGoogle Scholar
  23. 23.
    Pepona M, Favier J (2016) A coupled immersed boundary-lattice Boltzmann method for incompressible flows through moving porous media. J Comput Phys 321:1170–1184ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li Q, He YL, Tang GH, Tao WQ (2010) Improved axisymmetric lattice Boltzmann scheme. Phys Rev E 81:056707ADSCrossRefGoogle Scholar
  25. 25.
    Guo Z, Zhao TS (2002) Lattice Boltzmann model for incompressible flows through porous media. Phys Rev E 66:036304ADSCrossRefGoogle Scholar
  26. 26.
    Guo Z, Zheng C, Shi B (2002) Discrete lattice effects on the forcing term in the Lattice Boltzmann method. Phys Rev E 65:046308ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Hsu CT, Cheng P (1990) Thermal dispersion in a porous medium. Int J Heat Mass Transf 33(8):1587–1597CrossRefzbMATHGoogle Scholar
  28. 28.
    Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
  29. 29.
    Vafai K (1984) Convective flow and heat transfer in variable-porosity media. J Fluid Mech 147:233–259ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Yan YY, Zu YQ (2008) Numerical simulation of heat transfer and fluid flow past a rotating isothermal cylinder—a LBM approach. Int J Heat Mass Transf 51:2519–2536CrossRefzbMATHGoogle Scholar
  31. 31.
    Mishra SC, Lankadasu A, Beronov K (2005) Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction radiation problem. Int J Heat Mass Transf 48:3648–3659CrossRefzbMATHGoogle Scholar
  32. 32.
    Raj R, Prasad A, Parida PR, Mishra SC (2006) Analysis of solidification of a semi-transparent planar layer using the lattice Boltzmann method and the discrete transfer method. Numer Heat Transf, Part A: Appl 49:279–299ADSCrossRefGoogle Scholar
  33. 33.
    Mishra SC, Roy HK (2007) Solving transient conduction–radiation problems using the lattice Boltzmann method and the finite volume method. J Comput Phys 233:89–107ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Howell JR, Siegel R, Menguc MP (2011) Thermal radiation heat transfer, 5th edn. CRC Press, FloridaGoogle Scholar
  35. 35.
    Modest MF (2003) Radiative heat transfer, 3rd edn. Academic Press, New YorkzbMATHGoogle Scholar
  36. 36.
    Mishra SC, Asinari P, Borchiellini R (2010) A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transf, Part B: Fundam 57:126–146ADSCrossRefGoogle Scholar
  37. 37.
    Asinari P, Mishra SC, Borchiellini R (2010) A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transf, Part B: Fundam 57(2):126–146ADSCrossRefGoogle Scholar
  38. 38.
    Succi S (2001) The lattice Boltzmann method for fluid dynamics and beyond. Oxford University Press, New YorkzbMATHGoogle Scholar
  39. 39.
    Modest MF (2013) Radiative heat transfer, 3rd edn. Academic Press, New YorkGoogle Scholar
  40. 40.
    Mishra SC, Roy HK, Mishra N (2006) Discrete ordinate method with a new and a simple quadrature scheme. J Quant Spectrosc Radiat Transf 101:249–262ADSCrossRefGoogle Scholar
  41. 41.
    Sukop MC, Thorne DT Jr (2006) Lattice Boltzmann modeling lattice Boltzmann modeling. Springer, BerlinGoogle Scholar
  42. 42.
    Guo Z, Zhao TS (2005) A lattice Boltzmann model for convection heat transfer in porous media. Numer Heat Transf, Part B 47(2):157–177ADSCrossRefGoogle Scholar
  43. 43.
    Roache PJ (1998) Verification and validation in computational science and engineering. Hermosa, AlbuquerqueGoogle Scholar
  44. 44.
    Celik IB, Ghia U, Roache PJ, Freita ChJS, Coleman H, Raad PE (2008) Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. ASME J Fluids Eng 130:078001CrossRefGoogle Scholar
  45. 45.
    Talukdar P, Mishra SC, Trimis D, Durst F (2004) Combined radiation and convection heat transfer in a porous channel bounded by isothermal parallel plates. Int J Heat Mass Transf 47(5):1001–1013CrossRefzbMATHGoogle Scholar
  46. 46.
    Mahmud S, Fraser RA (2005) Flow, thermal, and entropy generation characteristics inside a porous channel with viscous dissipation. Int J Therm Sci 44(1):21–32CrossRefGoogle Scholar
  47. 47.
    McCulloch R (2015) Advances in radiation transport modeling using Lattice Boltzmann methods. Doctoral dissertation, Kansas State UniversityGoogle Scholar
  48. 48.
    Talukdar P, Mishra SC, Trimis D, Durst F (2004) Combined radiation and convection heat transfer in a porous channel bounded by isothermal parallel plates. Int J Heat Mass Transf 47:1001–1013CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.School of EngineeringUniversity of GlasgowGlasgowUK

Personalised recommendations