Lattice Boltzmann simulation of natural convection in a square enclosure with discrete heating

doi:10.1016/j.matcom.2020.07.025

Mathematics and Computers in Simulation

Volume 179, January 2021, Pages 265-278

https://doi.org/10.1016/j.matcom.2020.07.025 Get rights and content

Abstract

In the present work, natural convection heat transfer in a differentially heated cavity is considered, in which, source–sink pairs are located on vertical walls with a constant temperature while other wall parts of the enclosure are insulated. The effects of different arrangements, sizes, and number of heat source–sink pairs on the flow and thermal fields are investigated by using the Lattice Boltzmann Method to solve the flow and thermal field equations. In this paper, streamlines, isotherms, the average Nusselt number, and the entropy generation are presented to get a better insight into the nature of the problem. Results of this study showed that the arrangement of the heat source–sink pairs could strongly affect the flow and thermal fields in the cavity due to the formation of the vortices. It was also found that the highest amount of heat transfer with the lowest entropy generation was achieved by splitting the discrete heat source–sink pairs into the smaller segments and putting them alternately on one sidewall.

Introduction

Natural convection can be seen in a wide range of applications such as an electronic component cooling system since it can be regarded as a cheap, safe and noise-free mechanism. In this mechanism, a temperature gradient in a flow field causes density differences which can induce the natural convection heat transfer. Many research projects have been conducted to optimize this heat transfer mechanism [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Among these studies, a rectangular enclosure with discrete heat source–sink pairs seems to be an interesting topic among scientists because it has many industrial applications in different fields. In these rectangular enclosures, natural convection is proved dominant heat transfer mechanism and flow field structure is strongly affected by the arrangements and locations of the discrete heat source–sink pairs. Some of the studies in this field are as follows.

Sezai and Mohamad [9] studied the effects of different length and width ratios of heat sources in a horizontal enclosure. Their results showed that the maximum and minimum heat transfer rates were achieved at the edges of the discrete heat sources and at the center of the enclosure, respectively. Mezrhab et al. [5] conducted research on the inclined differentially heated square cavity by using numerical simulation. Their results indicated that inclined cavities had higher average Nu numbers than vertical ones for a low Rayleigh (Ra) numbers, while opposite trends were seen for high Ra numbers. Abu-Nada et al. [1] studied the effects of inclination angle and volume fraction of nanoparticle in a square enclosure. They found that the addition of nanoparticles remarkably enhanced the heat transfer in comparison with pure fluid. This trend was more significant for low Ra numbers. Furthermore, their results proved that heat transfer could be controlled by an inclination angle. Mirabedin et al. [6] investigated the effects of the Ra number and a central angle on natural convection heat transfer in circular enclosures. They found that the heat transfer enhanced by increasing the Ra number and decreasing the central angle. It was also seen that there was a linear trend in heat transfer rate after achieving critical Ra numbers. Baghsaz et al. [2] conducted research on natural convection heat transfer in a porous cavity filled with nanofluid by considering entropy generation. Their results indicated that irreversibility of the process and natural convection heat transfer increased by increasing the values of porosity and Ra numbers. Zhan et al. [11] studied the natural convective heat transfer in a closed cavity by using an experimental approach and reported that, by increasing the distance between heat sources, the heat transfer was improved and then decreased. Their studies also proved that heat transfer could be enhanced by increasing the temperature differences between heat sources and sinks.

Reviewing the previous literature has shown that no attempts have been made in order to consider the effects of different number and arrangements of discrete heat source–sink pairs on the natural convection heat transfer and entropy generation in a square cavity using the Lattice Boltzmann Method (LBM). Therefore, in the present work, two and three heat source–sink pairs at different $Ra$ numbers are considered to find the best case.

Section snippets

Mathematical formulation

In this paper, the problem of interest is to simulate natural convection heat transfer in a two-dimensional square enclosure with discrete heating on the vertical sidewalls and filled with air. The flow inside the cavity is considered steady and laminar ( $Ra = 1 0^{2} - 1 0^{7}$ ). The governing equations involve the continuity, conservation of momentum, and energy equations.

Numerical method and its validation

In this study, the natural convection heat transfer in a square cavity is studied using LBM implemented by an in-house computer code written in FORTRAN. In order to consider the mesh independence study and validation of the model, the results of the work done by Davis [10] are selected. In [10] natural convection in a square cavity is studied where $Pr = 0.707$ . In addition, the temperatures of the vertical walls are assumed to be $T_{h} = 1$ and $T_{c} = 0$ while the horizontal walls are insulated (Fig. 1).

Hydrodynamic and thermal analysis

As mentioned above and illustrated in Fig. 2, two and three heat source–sink pairs with three different arrangements for each of them are studied in this paper by considering their effects on the flow field, temperature field and rate of heat transfer. In this work, the $Pr$ number is kept constant at 0.707 and the $Ra$ number is varied from $1 0^{3}$ to $1 0^{7}$ .

Streamlines and isotherms for a given Ra number are depicted in Fig. 3, Fig. 4 for different cases. These figures show that the flow moves upwards

Conclusions

Natural convection heat transfer in a square enclosure with discrete heat source–sink pairs is simulated in this work by using LBM. The effects of the different arrangements, and number of heat source–sink pairs and Ra numbers are considered and listed as follows:

•
The flow is not affected much by thermal boundary layers in the central region of the cavity and the temperature gradient is not steep.
•
Temperature gradients are severe in the vicinity of the heat source–sink pairs due to the contracted

Nomenclature

$c$	Lattice speed, m/s
$c_{s}$	Speed of sound, m/s
$f$	Density distribution function
$f^{eq}$	Equilibrium density distribution function
$g$	Energy distribution function
$g^{eq}$	Equilibrium energy distribution function
g	Gravitational acceleration
$H$	Height of enclosure
$Nu$	Nusselt number
P	Pressure
$Pr$	Prandtl number
$Ra$	Rayleigh number
$S^{‴}$	Volumetric rate of entropy generation, $W ∕ (m^{3} K)$
$S^{*}$	Dimensionless volumetric entropy generation rate
$T$	Temperature
$U$	Dimensionless velocity component in X direction
$V$	Dimensionless velocity component in Y direction
$X$

References (11)

Abu-NadaE. et al.
Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid
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(2009)
BaghsazS. et al.
Numerical investigation of transient natural convection and entropy generation analysis in a porous cavity filled with nanofluid considering nanoparticles sedimentation
J. Mol. Liq.
(2019)
MezrhabA. et al.
Lattice-Boltzmann modelling of natural convection in an inclined square enclosure with partitions attached to its cold wall
Int. J. Heat Fluid Flow
(2006)
MirabedinSeyed Milad et al.
Natural convection in circular enclosures heated from below for various central angles
Case Stud. Therm. Eng
(2016)
SezaiI. et al.
Natural convection from a discrete heat source on the bottom of a horizontal enclosure
Int. J. Heat Mass Transfer
(2000)

There are more references available in the full text version of this article.

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Citation Excerpt :
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Original articlesLattice Boltzmann simulation of natural convection in a square enclosure with discrete heating

Abstract

Introduction

Section snippets

Mathematical formulation

Numerical method and its validation

Hydrodynamic and thermal analysis

Conclusions

Nomenclature

Int. J. Heat Fluid Flow

J. Mol. Liq.

Int. J. Heat Fluid Flow

Case Stud. Therm. Eng

Int. J. Heat Mass Transfer

Original articles
Lattice Boltzmann simulation of natural convection in a square enclosure with discrete heating