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LBM study of natural convection heat transfer from a porous cylinder in an enclosure

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Abstract

Natural convection heat transfer from a porous cylinder put at various positions in a square, cooled enclosure, with air as the working fluid, is investigated in this work. The following setups are taken into account: The hot cylinder is placed in the middle of the enclosure, near the bottom, top, right sides, along diagonal as top-diagonal and bottom-diagonal. The cylinder and the enclosure walls are kept hot and cold, respectively. The lattice Boltzmann method is used to perform a numerical analysis for Rayleigh number \(10^{4}\le \) Ra \(\le 10^{6}\) and Darcy number \(10^{-6}\le \) Da \(\le 10^{-2}\). The results are plotted as streamlines, isotherms, and local and mean Nusselt number values. The amount of heat transported from the heated porous cylinder is determined by varying Ra, Da, and the cylinder location. Even at a lower Rayleigh number (\(10^{4}\)), the average Nusselt number grows by nearly 70 % as the cylinder moves from the centre to the bottom and 105% as it moves to bottom-diagonal location when \({Da}=10^{-2}\). At Ra \(=10^{6}\) and Da \(=10^{-2}\), the heat transfer rate of the cylinder located near the corner of the enclosure at the bottom wall increases by approximately 33% when compared to the case of the cylinder in the centre. Convective effects are more noticeable when the cylinder is positioned towards the enclosure’s bottom wall. This research is applicable to electronic cooling applications in which a collection of electronic components is arranged in a circular pattern inside a cabinet.

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Abbreviations

\({C}_{1}\), \({C}_{2}\) :

Binary constants

D :

Diameter of cylinder (m)

\(c_\mathrm{F}\) :

Non-dimensional Forchheimer term

\(\delta \) :

Distance from centre (\(\delta {x}\) or \(\delta y\) in x and y direction)

\(c_{s}\) :

Speed of sound (\(\hbox {ms}^{-1}\))

G:

Body force due to gravity (N)

Da :

Darcy number \(\frac{K}{D^{2}}\)

N :

Number of lattices on the cylinder

\(d_{p}\) :

Particle diameter (m)

\({Nu}_\mathrm{L}\) :

Local Nusselt number \(\frac{\partial \theta }{\partial n}\)

\(e_{i}\) :

Discrete lattice velocity in direction i, \(\frac{\Delta x_{i}}{\Delta t}\)

Nu :

Nusselt number \(\frac{hL}{K}\)

F :

Body force due to presence of the porous medium (N)

p :

Dimensionless pressure \(\frac{p*}{\rho u_{\infty }^{2}}\)

\(F_{i}\) :

Total force term due to porous medium (N)

Pr :

Prandtl number \(\frac{\nu }{\alpha }\)

\(F_{b}\) :

Boussinesq force term (N)

Ra :

Rayleigh number \(\frac{g\beta \Delta TL^{3}}{\alpha \nu }\)

g:

Gravitational acceleration (ms\(^{-2}\))

u :

Non-dimensional x-component velocity (ms\(^{-1}\))

\(f_{i}\) :

Particle distribution function along ith link direction

v :

Non-dimensional y-component velocity (ms\(^{-1}\))

\(f_{i}^{eq}\) :

Equilibrium distribution function along ith link direction

U:

Actual velocity (ms\(^{-1}\))

V:

Auxiliary velocity (ms\(^{-1}\))

\(g_{i}\) :

Temperature distribution function along ith link direction

\(w_{i}\) :

Weighing factor in direction i

\(g_{i}^{eq}\) :

Equilibrium distribution function of temperature ith link direction

x,y :

Non-dimensional horizontal and vertical coordinate

L :

Length of enclosure (m)

\(x^{*}\) ,y \(^{*}\) :

Dimensional horizontal and vertical coordinate

\(\rho \) :

Fluid density (kg m\(^{-3}\))

\(\epsilon \) :

Porosity

\(\tau \) :

Dimensionless relaxation time for density

\(\nu \) :

Fluid kinematic viscosity (m\(^{2}\)s\(^{-1}\))

\(\tau '\) :

Dimensionless relaxation time for temperature

\(\mu \) :

Fluid dynamic viscosity (N  s  m\(^{-2}\))

t :

Non-dimensional time \(\frac{t{*u}_{\infty }}{H}\)

\(\varLambda \) :

Viscosity ratio \(\frac{\mu _{e}}{\mu }\)

\(\Delta t\) :

Time step (s)

\(\alpha \) :

Thermal diffusivity (m\(^{2}\)s\(^{-1}\))

\(\Delta x\) :

Lattice space

\(\beta \) :

Thermal expansion coefficient \(\frac{Ra\, \alpha \, \nu }{\, g\Delta TL^{3}}\) (K\(^{-1}\))

\(\theta \) :

Dimensionless temperature \(\frac{T-T_{\infty }}{T_{w}-\, T_{\infty }}\)

\(\sigma \) :

Thermal conductivity ratio

avg:

Average

w:

Wall

\(\infty \) :

Far field value

e:

Effective

\(\circ \) :

Inlet value

f:

Fluid

M:

Mean value

l:

Local value of variable

i :

Lattice link direction

\(*\) :

Dimensional form of variables

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Acknowledgements

One of the authors (S.D) acknowledges the funds received from the Science and Engineering Research Board (SERB), a statutory body of Department of Science & Technology (DST), Government of India, through a Project Grant (File no. MTR/2019/001440).

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Authors and Affiliations

  1. The Centre for Fluid Dynamics, Department of Mechanical Engineering, Indian Institute of Technology Indore, Simrol, Indore, 453 552, India

    B. Shruti & S. Dhinakaran

  2. Center for Turbulence Control, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China

    Md. Mahbub Alam

  3. Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, 453 552, India

    A. Parkash

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  1. B. Shruti
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Correspondence to S. Dhinakaran.

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Shruti, B., Alam, M.M., Parkash, A. et al. LBM study of natural convection heat transfer from a porous cylinder in an enclosure. Theor. Comput. Fluid Dyn. 36, 943–967 (2022). https://doi.org/10.1007/s00162-022-00632-z

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