Numerical Simulation by the Lattice Boltzmann Method of the Natural Convective Flow in a Square Cavity Containing a Nanofluid

The importance of heat and mass transfers appears in many applications. Natural convection is a precious study in the diverse cavity with different gradient temperatures [1-4]. The enhancement of heat transfer is a big challenge for researchers, which let them focus to use the nanoparticles in pure fluid, and they showed the necessity of nanofluids for increasing the thermal flux.

Choi and Eastman [5] studied the enhancement of the thermal conductivity of fluid with nanoparticles. They showed that the use of nanofluids enhances heat transfer against conventional fluids. Oztop and Abu-Nada [6] made a numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. They indicated that heat transfer increase with the increase of heater size, the value of Rayleigh number, and the volume fraction. Boualit et al. [7] investigated the natural convection in a square cavity filled with nanofluids using a dispersion model. They have concluded that the using the nanoparticles with augmentation of their size enhance the heat transfer in different value of Rayleigh number. Parametric study on natural convection of nanofluid in a heated chamber presented by Bara et al. [8] Their results showed that heat transfer augments when volume fractions augment especially in a low value of Ra the conduction was dominance; the Copper give more efficacies on flux thermal than Al₂O₃ and TiO₂.

Lattice Boltzmann method (LBM) is one of the best methods that give researchers an advantage for studying the challenging geometries which were hard to solve with old techniques. The fundamental bases of this method are outlined in the book of Mohamad [9]. Yao et al. [10] made an analysis of nanofluids phase transition in a pipe using the lattice Boltzmann method. As result, they concluded that heat transfer is enhanced in nanofluids more than in pure fluid, but it becomes weak with augmenting the size of nanoparticles. Mahmoudi et al. [11] simulated the conduction radiation heat transfer in a planar medium with lattice Boltzmann. The effects of the scattering albedo, the wall emissivity, and conduction-radiation parameters on temperature distribution in the medium were tested and they noticed a good agreement result between CDM (Collapsed dimension method) and LBM. Mliki et al. [12] used the lattice Boltzmann to simulate the magnet-hydro-dynamic (MHD) natural convection in an L-shaped enclosure. They went on to remark that the parameters of aspect ratio and Hartmann number reduced the free convection, otherwise, the heat transfer increased with augmentation of volume fraction. Khakrah et al. [13] studied a thermal lattice Boltzmann simulation of natural convection in a multi-pipe sinusoidal-wall cavity filled with Al₂O₃-Ethylene Glycol nanofluid. They concluded that heat transfer improved by adding nanoparticles, on the other hand the total entropy generation was reduced, and that was acceptable in the thermal component. Naseri et al. [14] reviewed a lattice Boltzmann simulation of natural convection heat transfer of a nanofluid in an L-shape enclosure with a baffle. As a consequence, at low Rayleigh values, the natural convection increased when they added baffles, otherwise, at high Rayleigh values, the natural convection was more enhanced just with a longer baffle (L=0.3) especially in case C configuration. Zhang et al. [15] studied numerically the mixed convection of nanofluid inside an inlet/outlet inclined cavity under the effect of Brownian motion using LBM. They tried to study the influence of cavity angle, volume fraction and location of hot obstacle. The thermal conductivity is augmented when the volume fraction increases, also the increasing of the Richardson number reduced the effect of the volume fraction. Besides that the change of angle from 0° to 60° with the change of Ri number influenced the heat transfer, also heat transfer is enhanced when the hot obstacle is put in the flow path.

The lattice Boltzmann method (LBM) gives more advantages for studying the complex flows and geometries with easy implementation and less calculation time, which was hard with other conventional methods. For this reason, the main objective of this study is to analyze the flow and heat transfer by convection in a differentially heated square cavity using the lattice Boltzmann method as a computational tool. We plan to determine the influence of several control parameters such as the Rayleigh number and the volume fraction of the nanoparticles on the flow structure and average Nusselt number.

Natural convection flow is assumed to be steady and laminar, so the fluid is incompressible and Newtonian. Except for the density, which is calculated using the Boussinesq approximation, the physical properties are constants.

3.1 Navier-Stokes method

The dimensionless partial differential equations (PDEs) that describe the heat transfer and nanofluid flow used in this study are:

-Dimensionless continuity equation:

$\frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\frac{\partial \mathrm{V}}{\partial \mathrm{Y}}=0$       (1)

- Dimensionless momentum equations:

$\frac{\partial \mathrm{U}}{\partial \tau}+\mathrm{U} \frac{\partial \mathrm{U}}{\partial \mathrm{X}}+\mathrm{V} \frac{\partial \mathrm{U}}{\partial \mathrm{Y}}=-\frac{\partial \mathrm{P}}{\partial \mathrm{X}}$$+\frac{1}{(1-\phi)^{2.5}} \sqrt{\frac{\operatorname{Pr}}{\mathrm{Ra}}}\left(\frac{\partial^2 U}{\partial \mathrm{X}^2}+\frac{\partial^2 U}{\partial \mathrm{Y}^2}\right)$       (2)

$\frac{\partial \mathrm{V}}{\partial \tau}+\mathrm{U} \frac{\partial \mathrm{V}}{\partial \mathrm{X}}+\mathrm{V} \frac{\partial \mathrm{V}}{\partial \mathrm{Y}}$$=-\frac{\partial \mathrm{P}}{\partial \mathrm{Y}}+\left(1-\phi+\phi \mathrm{R}_\beta\right) \theta$$+\frac{1}{(1-\phi)^{2.5}} \sqrt{\frac{\operatorname{Pr}}{\mathrm{Ra}}}\left(\frac{\partial^2 \mathrm{~V}}{\partial \mathrm{X}^2}+\frac{\partial^2 \mathrm{~V}}{\partial \mathrm{Y}^2}\right)$       (3)

- Dimensionless energy equation:

$\frac{\partial \theta}{\partial \tau}+U \frac{\partial \theta}{\partial X}+V \frac{\partial \theta}{\partial Y}$$=\frac{\left(2+\mathrm{R}_{\mathrm{k}} \frac{1+2 \phi}{1-\phi}\right)}{\left(\mathrm{R}_{\mathrm{k}}+\frac{2+\phi}{1-\phi}\right)\left(1-\phi+\phi \mathrm{R}_{\rho \mathrm{C}}\right)} \frac{1}{\sqrt{\mathrm{RaPr}}} \nabla^2 \theta$       (4)

With the following reference variables:

$\mathrm{X}=\frac{\mathrm{x}}{\mathrm{H}}$; $\mathrm{Y}=\frac{\mathrm{y}}{\mathrm{H}}$; $\mathrm{U}=\frac{\mathrm{uH}}{\sqrt{\mathrm{g} \beta \mathrm{H} \Delta \mathrm{T}}}$; $\mathrm{V}=\frac{\mathrm{vH}}{\sqrt{\mathrm{g} \beta \mathrm{H} \Delta \mathrm{T}}}$; $\theta=\frac{\mathrm{T}-\mathrm{Tc}}{\mathrm{Th}-\mathrm{Tc}}$; $P=\frac{p}{\rho_{n_f}(\sqrt{g \beta H \Delta T})^2}$; $\tau=\frac{\mathrm{t} \sqrt{g \beta H \Delta T}}{H}$.

3.2 Nanofluid formulations

The following are the density, heat capacity, and thermal expansion equations for nanofluids, respectively:

$\frac{\rho_{\mathrm{nf}}}{\rho_{\mathrm{f}}}=1-\phi+\phi \mathrm{R}_\rho$       (5)

$\frac{(\rho \mathrm{Cp})_{\mathrm{nf}}}{(\rho \mathrm{Cp})_{\mathrm{f}}}=1-\phi+\phi \mathrm{R}_{\rho \mathrm{Cp}}$       (6)

$\frac{\beta_{\mathrm{nf}}}{\beta_{\mathrm{f}}}=1-\phi+\phi \mathrm{R}_\beta$       (7)

The dynamic viscosity and thermal conductivity are calculated respectively using the Brinkman [16] and Maxwell [17] models, as follows:

$\frac{\mu_{\mathrm{nf}}}{\mu_{\mathrm{f}}}=\frac{1}{(1-\phi)^{2.5}}$       (8)

$\frac{\mathrm{k}_{\mathrm{nf}}}{\mathrm{k}_{\mathrm{f}}}=\frac{2+\mathrm{R}_{\mathrm{k}} \frac{1+2 \phi}{1-\phi}}{\mathrm{R}_{\mathrm{k}}+\frac{2+\phi}{1-\phi}}$        (9)

With $\mathrm{R}_{\mathrm{\rho}}=\frac{\rho_{\mathrm{p}}}{\rho_{\mathrm{f}}}$; $\mathrm{R}_{\rho \mathrm{Cp}}=\frac{(\rho \mathrm{Cp})_{\mathrm{p}}}{(\rho \mathrm{p})_{\mathrm{f}}}$; $\mathrm{R}_\beta=\frac{\beta_{\mathrm{p}}}{\beta_{\mathrm{f}}}$; $\mathrm{R}_{\mathrm{k}}=\frac{\mathrm{k}_{\mathrm{p}}}{\mathrm{k}_{\mathrm{f}}}$.

The physical properties of the base fluid (water) and nanoparticles (Al₂O₃) used in this study are summarized in Table 1.

Table 1. Thermo-physical proprieties of water and Al₂O₃

3.3 Lattice Boltzmann method

Lattice Boltzmann Model used a double distribution function for the flow field (f) and temperature (g). These distributions are used to compute the density, velocity, and temperature of the macroscopic field. To write the general version of the lattice Boltzmann equations, the BGK (Bhatnagar-Gross-Krook) approach can be used.

$f_i(\boldsymbol{x}+\boldsymbol{c} \Delta t, t+\Delta t)=f_i(\boldsymbol{x}, t)$      (10)

$\mathrm{g}_{\mathrm{i}}(\mathrm{x}+\operatorname{ci} \Delta \mathrm{t}, \mathrm{t}+\Delta \mathrm{t})$$=g_i(\boldsymbol{x}, t)-\frac{1}{\tau_T}\left(g_i(\boldsymbol{x}, t)-g_i^{e q}(\boldsymbol{x}, t)\right)$$-\frac{1}{\tau_f}\left(f_i(\boldsymbol{x}, t)-f_i^{e q}(\boldsymbol{x}, t)\right)+\Delta t \boldsymbol{c}_{\boldsymbol{i}} \boldsymbol{F}_{\boldsymbol{i}}$      (11)

where, $f_i(\mathbf{x}, t), g_i(\mathbf{x}, t)$ are the density and temperature distribution function at lattice $\mathbf{x}$ and time $t$ in the i direction; $\Delta t$ is the time step and $\tau_f, \tau_T$ are correspondent the relaxation times; $F_i$ is the external force term. $f_i^{e q}(x, t)$ and $g_i^{e q}(x, t)$ are the equilibrium distribution for density and temperature, respectively and can be calculated as follow:

$f_i^{e q}=\omega_i \rho\left[1+3 \frac{\left(\boldsymbol{c}_{\boldsymbol{i}} \cdot \boldsymbol{u}\right)}{c_s^2}+\frac{9}{2} \frac{\left(\boldsymbol{c}_{\boldsymbol{i}} \cdot \boldsymbol{u}\right)^2}{2 c_s^4}-\frac{3}{2} \frac{\boldsymbol{u}^2}{c_s^2}\right]$      (12)

$g_i^{e q}=\omega_i T\left[1+\frac{3\left(\boldsymbol{c}_{\boldsymbol{i}} \cdot \boldsymbol{u}\right)}{c_s^2}\right]$         (13)

The buoyancy force $F_i$ appearing in Eq. (10) can be determined by:

$\boldsymbol{F}_i=3 \boldsymbol{\omega}_{\boldsymbol{i}} \rho g \beta(T-T c)$        (14)

where, $\boldsymbol{c}_{\boldsymbol{i}}$ and $\boldsymbol{\omega}_{\boldsymbol{i}}$, are the lattice velocity vectors, weighting factor, respectively. $c_s$ is lattice velocity sound and is equal to $1 / \sqrt{3}$.

For this work, we used D₂Q₉ as a two-dimensional, nine-velocity model, as shown in Figure 2.

2.png

Figure 2. Discrete velocity for D₂Q₉

The discrete velocity as bellow:

The weighing factor is calculated as:

The macrospic variables $T, \rho, u$, can finally be determined as follows:

Flow density : $\rho=\sum_{i=0}^8 f_i$       (15)

Velocity : $\boldsymbol{u}=\frac{1}{\rho} \sum_{i=0}^8 \boldsymbol{c}_i f_i$         (16)

Temperature $: T=\sum_{i=0}^8 g_i$        (17)

In this paper, the lattice Boltzmann method is used to study the natural convection in a differentially heated square cavity containing Al₂O₃-water nanofluid. The D₂Q₉ scheme is used to determine, at the mesoscopic scale, the two distribution functions (f for the flow and g for the temperature) for obtain the macroscopic variables (U, V, q). A Matlab code is developed to solve the finite difference discretized Boltzmann equation. The effect of the Rayleigh number as well as the volume fraction of the nanoparticles on the structure of the flow is determined. The increases the intensity of the flow due to augmentation of Ra, while the increase in the volume fraction decreases the intensity of the flow thus favoring the transfer of heat by conduction which appears clearly in average Nusselt number values. The phenomena found are illustrated in two correlations of the maximum vertical velocity and the average Nusselt number with the control parameters used in this study (Rayleigh number and nanofluid volume fraction).