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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 161–176 | Cite as

Conjugate heat transfer of Al2O3–water nanofluid in a square cavity heated by a triangular thick wall using Buongiorno’s two-phase model

  • A. I. AlsaberyEmail author
  • T. Armaghani
  • A. J. Chamkha
  • I. Hashim
Article

Abstract

The present study investigates the conjugate heat transfer in a square cavity heated by a triangular solid and saturated with \(\text{Al}_2\text{O}_3\)–water nanofluid. Two-phase Buongiorno’s model is used for modeling the nanofluid heat transfer. The finite element method is used for numerical solution of the dimensionless governing equations subject to the boundary conditions. Comparisons of the proposed method with previously published experimental and numerical works show a good agreement. The effects of some parameters such as the Rayleigh number, thermal conductivity ratio, dimensionless triangular wall thickness and nanofluid volume fraction on heat transfer and nanoparticle distributions are completely studied and discussed. The results show clockwise rotations for streamlines and nanoparticle migration. Also the Nusselt number increases with the nanofluid volume fraction. A continuous reduction is seen for the mean Nusselt number by increasing the dimensionless triangular wall thickness for all the considered values of the Rayleigh number.

Keywords

Nanofluid Heat transfer Brownian motion Thermophoresis effect Nanoparticle distribution Triangular solid 

List of symbols

\(C_{\mathrm{p}}\)

Specific heat capacity (J kg\(^{1}\) K\(^{1}\))

d

Width and height of triangular solid wall (m)

\(d_{\mathrm{f}}\)

Diameter of the base fluid molecule (nm)

\(d_{\mathrm{p}}\)

Diameter of the nanoparticle (nm)

D

Dimensionless triangular wall thickness, \(D=d/L\)

\(D_{\mathrm{B}}\)

Brownian diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))

\(D_{\mathrm{B}0}\)

Reference Brownian diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))

\(D_{\mathrm{T}}\)

Thermophoretic diffusivity coefficient (kg m\(^{-1}\) s\(^{-1}\))

\(D_{\mathrm{T}0}\)

Reference thermophoretic diffusion coefficient (kg m\(^{-1}\) s\(^{-1}\))

g

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

k

Thermal conductivity (Wm\(^{-1}\) K\(^{-1}\))

\(K_{\mathrm{r}}\)

Triangular wall to nanofluid thermal conductivity ratio, \(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\)

L

Width and height of enclosure (m)

Le

Lewis number

\(N_{\mathrm{BT}}\)

Ratio of Brownian to thermophoretic diffusivity

\(\overline{Nu}\)

Average Nusselt number

Pr

Prandtl number

Ra

Rayleigh number

\(Re_{\mathrm{B}}\)

Brownian motion Reynolds number

T

Temperature (K)

\(T_0\)

Reference temperature (310 K)

\(T_{\mathrm{fr}}\)

Freezing point of the base fluid (273.15 K)

\(\mathbf v\), \(\mathbf V\)

Velocity and dimensionless velocity vector (ms\(^{-1}\))

\(u_{\mathrm{B}}\)

Brownian velocity of the nanoparticle (ms\(^{-1}\))

x, y and X, Y

Space coordinates and dimensionless space coordinates

Greek symbols

\(\alpha\)

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

\(\beta\)

Thermal expansion coefficient (K\(^{-1}\))

\(\theta\)

Dimensionless temperature

\(\mu\)

Dynamic viscosity (kg m\(^-1\) s\(^-1\))

\(\nu\)

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

\(\rho\)

Density (kg m\(^{-3}\))

\(\varphi\)

Solid volume fraction

\(\varphi ^*\)

Normalized solid volume fraction

\(\phi\)

Average solid volume fraction

Subscript

c

Cold

f

Base fluid

h

Hot

nf

Nanofluid

p

Solid nanoparticles

w

Solid wall

Introduction

Enhancing the natural convection heat transfer in a cavity is one of the very important challenges in recent years because of its many engineering applications, such as solar cells, cooling devices, food drying, chemical processes, power generation, conventional heat transfer fluids such as air, water, ethylene glycol (EG), etc [1]. The use of nanofluids as a new technique for reaching to a higher heat transfer was introduced by Chol and Eastman [2]. Nanofluids which are composed of a base fluid and nano-sized particles have been extensively used in various heat transfer applications. A recent study reporting experimentally and analytically the effects of particle types on thermal conductivity of nanofluid showed the improvement of the heat transfer characteristics with a little pressure drop as compared to base fluids [3]. Ilyas et al. [4] experimentally studied the natural convection heat transfer in a rectangular cavity filled with an alumina-thermal oil-based nanofluids. They concluded that the heat transfer coefficient for nanofluids was higher than that of a pure oil. Using control volume based finite element method (CVFEM), Sheikholeslami and Seyednezhad [5] investigated the effect of an electric field on natural convection in a 2D porous cavity filled with Fe\(_3\)O\(_4\)–Ethylene glycol nanofluid. Their results showed that the heat transfer was an increasing function of the Darcy number.

Two approaches based on conservation equations have been introduced in the literature to investigate the numerical heat transfer of nanofluids: single-phase model and two-phase model [6]. In a single-phase model, a homogenous volume fraction distribution is assumed for nanofluids and the thermo-physical properties of the pure fluid are replaced by the nanofluid properties. Then, the conservation equations are solved for nanofluids. The followings are some works focusing on nanofluid natural convection in various shapes of cavity using the single-phase model: Chamkha and Ismael [7] investigated the nanofluid conjugate heat transfer in a porous cavity which is heated by a triangular thick wall; Ismael et al. [8] investigated the conjugate heat transfer in a cavity saturated with a porous medium and heated by a triangular solid. They used the nanofluid as a working fluid. Chamkha et al. [9] examined the nanofluid MHD natural convection heat transfer in a C-shaped cavity. The results showed an enhancement of heat transfer by the addition of the nanoparticles in a pure fluid. Ghalambaz et al. [10] examined nanofluid natural convection in a porous cavity considering viscous dissipation and radiation effects. Jbeili et al. [11] conducted a numerical study on the evaluation of thermal and power performances of nanofluid flows through square in-line cylinder arrays. They considered the effects of nanoparticle size, concentration and the Reynolds number and found that the thermal conductivity of the nanoparticle has a negligible effect on the nanofluid performance. Recently, Alsabery et al. [12] considered the unsteady natural convective heat transfer in a 2D cavity filled with a nanofluid and heated by sinusoidal boundary conditions. They found that heat transfer is enhanced by the addition of the nanofluid. Akar et al. [13] numerically considered the second law of thermodynamic analysis of the nanofluid turbulent flow around a rotating cylinder.

In Buongiorno’s two-phase model, the volume fraction distribution equation is added to the conservation equations [6, 14]. Buongiorno [15] studied the effects of seven slip mechanisms on nanofluid heat transfer. He showed that the Brownian motion and thermophoresis effects are the dominant terms for studying the nanofluid heat transfer. Many researchers used the Buongiorno two-phase model for numerical analysis of nanofluid heat transfer [16, 17, 18, 19]. Free convection due to an inclined porous fin embedded in a cavity was examined by Zargartalebi et al. [19]. They used Buongiorno’s two-phase model with local thermal non-equilibrium (LTNE) condition for porous media. Garoosi et al. [20] studied mixed convection heat transfer where a two-phase mixture model was used to simulate the nanofluid in a two-sided lid-driven cavity with several pairs of heaters and coolers (HACs). Extending the works in [22, 23, 24], Sheremet et al. [21] studied natural convection flow in a porous cavity filled with Buongiorno’s nanofluid. The bottom and top walls are wavy, and the vertical walls are subjected to sinusoidal temperature distributions. Natural convection of alumina–water nanofluid in an inclined cavity using Buongiorno’s two-phase model has been studied recently by Motlagh and Soltanipour [25]. Sheremet and Pop [26] preformed a numerical study on the effect of Buongiorno’s nanofluid model on natural convection in a differentially heated cubical cavity under the Marangoni effect from the upper free surface.

Based on the previously mentioned papers and to the authors’ best knowledge, there have been no studies on conjugate heat transfer of \(\text{Al}_2\text{O}_3\)–water nanofluid in a square cavity heated by a triangular thick wall using Buongiorno’s two-phase model. We shall study the effects of nanoparticle migration on conjugate conduction convection heat transfer. The triangular solid wall occupies one corner of the square cavity making an inclined interface between the solid and the nanofluid.

Mathematical formulation

Consider the steady laminar two-dimensional natural convection problem in a square cavity of length L with a solid triangular wall of length d at the lower left corner of the cavity. Here, d is varied in such a way to keep the overall domain as a square [7] as illustrated in Fig. 1a. An isothermal heater is placed on the lower left corner of the cavity (triangular wall), while the top right corner with length 0.5L is maintained at a constant cold temperature \(T_{\mathrm{c}}\). The remainder of these walls are kept adiabatic. The boundaries of the square domain are assumed to be impermeable; the fluid within the cavity is a water-based nanofluid having \(\text{Al}_2\text{O}_3\) nanoparticles. The Boussinesq approximation is applicable. Considering these assumptions, the continuity, momentum and energy equations for the laminar and steady-state natural convection can be written as follows:
$$\begin{aligned}&\nabla \cdot \mathbf v = 0, \end{aligned}$$
(1)
$$\begin{aligned}&\rho _{\mathrm{nf}}{} \mathbf v \cdot \nabla \mathbf v =-\nabla p + \nabla \cdot \mu _{\mathrm{nf}}\nabla \mathbf v +(\rho \beta )_{\mathrm{nf}}(T-T_{\mathrm{c}}) \mathbf{g}, \end{aligned}$$
(2)
$$\begin{aligned}&(\rho C_{\mathrm{p}})_{\mathrm{nf}}{} \mathbf v \cdot \nabla T_{\mathrm{nf}}=-\nabla \cdot k_{\mathrm{nf}}\nabla T_{\mathrm{nf}} - C_{\mathrm{p, p}} J_{\mathrm{p}}\cdot \nabla T_{\mathrm{nf}}, \end{aligned}$$
(3)
$$\begin{aligned}&\mathbf v \cdot \nabla \varphi =- \frac{1}{\rho _{\mathrm{p}}}\nabla \cdot J_{\mathrm{p}}, \end{aligned}$$
(4)
where \(\mathbf v\) is the velocity vector, \(\mathbf{g}\) is the gravitational acceleration vector, \(\varphi\) is the local volume fraction of nanoparticles, and \(J_{\mathrm{p}}\) is the nanoparticles mass flux.
The energy equation of the solid wall is
$$\begin{aligned} \nabla T_{\mathrm{w}} = 0. \end{aligned}$$
(5)
Based on Buongiorno’s model nanoparticles mass flux can be written as:
$$\begin{aligned}&J_{\mathrm{p}}=J_{\mathrm{p,B}}+J_{\mathrm{p,T}}, \end{aligned}$$
(6)
$$\begin{aligned}&J_{\mathrm{p,B}}=-\rho _{\mathrm{p}} D_{\mathrm{B}} \nabla \varphi ,\quad D_{\mathrm{B}}=\frac{k_{\mathrm{b}} T}{3 \pi \mu _{\mathrm{f}} d_{\mathrm{p}}}, \end{aligned}$$
(7)
$$\begin{aligned}&J_{{\mathrm{p,T}}}=-\rho _{\mathrm{p}} D_{\mathrm{T}} \nabla T,\quad D_{\mathrm{T}}=0.26\frac{k_{\mathrm{f}}}{2k_{\mathrm{f}}+k_{\mathrm{p}}} \frac{\mu _{\mathrm{f}}}{\rho _{\mathrm{f}} T}\varphi \end{aligned}$$
(8)
As reported by Buongiorno, the effects of Brownian motion and thermophoresis are most significant and have higher orders of magnitude compared to the other drift-flux effects such as diffusiophoresis, Magnus effect, shear-induced and gravity. Brownian motion represents the random motion of small colloidal particles suspended in a liquid or gas medium and is caused by the collision of the medium’s molecules with the particles. Thermophoresis accounts for the diffusion of particles under the effect of a temperature gradient. This thermophoresis phenomenon is the ‘particle’ equivalence of the well-known Soret effect for gaseous or liquid mixtures.
Fig. 1

a Physical model of convection in a square cavity, and b grid-points distribution for grid size of G6

The thermo-physical properties of the nanofluid can be determined as follows:

The heat capacitance of the nanofluids \((\rho C_{\mathrm{p}})_{\mathrm{nf}}\) is
$$\begin{aligned} (\rho C_{\mathrm{p}})_{\mathrm{nf}}=(1-\varphi )(\rho C_{\mathrm{p}})_{\mathrm{f}}+\varphi (\rho C_{\mathrm{p}})_{\mathrm{p}}. \end{aligned}$$
(9)
The effective thermal diffusivity of the nanofluids \(\alpha _{\mathrm{nf}}\) is given as
$$\begin{aligned} \alpha _{\mathrm{nf}}=\frac{k_{\mathrm{nf}}}{(\rho C_{\mathrm{p}})_{\mathrm{nf}}}. \end{aligned}$$
(10)
The effective density of the nanofluids \(\rho _{\mathrm{nf}}\) is
$$\begin{aligned} \rho _{\mathrm{nf}}=(1-\varphi )\rho _{\mathrm{f}}+\varphi \rho _{\mathrm{p}}. \end{aligned}$$
(11)
The thermal expansion coefficient of the nanofluids \(\beta _{\mathrm{nf}}\) can be determined by
$$\begin{aligned} {(\rho \beta )_{\mathrm{nf}}} = (1 - \varphi ){(\rho \beta )_{\mathrm{f}}} + \varphi {(\rho \beta )_{\mathrm{p}}}. \end{aligned}$$
(12)
The dynamic viscosity ratio of water–\(\text{Al}_2\text{O}_3\) nanofluids for 33nm particle-size in the ambient condition was derived in reference [27] as follows:
$$\begin{aligned} \frac{\mu _{\mathrm{nf}}}{\mu _{\mathrm{f}}} = 1/\left( 1-34.87\left( d_{\mathrm{p}}/d_{\mathrm{f}}\right) ^{-0.3}\varphi ^{1.03}\right) . \end{aligned}$$
(13)
The thermal conductivity ratio of water–\(\text{Al}_2\text{O}_3\) nanofluids is calculated by the Corcione model [27]
$$\begin{aligned} \frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}}= 1 + 4.4 \textit{Re}_{\mathrm{B}}^{0.4} \textit{Pr}^{0.66} \left( \frac{T}{T_{\mathrm{fr}}}\right) ^{10} \left( \frac{k_{\mathrm{p}}}{k_{\mathrm{f}}}\right) ^{0.03} \varphi ^{0.66}. \end{aligned}$$
(14)
where \(\textit{Re}_{\mathrm{B}}\) is
$$\begin{aligned} \textit{Re}_{\mathrm{B}}= & {} \frac{\rho _{\mathrm{f}} u_{\mathrm{B}} d_{\mathrm{p}}}{\mu _{\mathrm{f}}}, \end{aligned}$$
(15)
$$\begin{aligned} u_{\mathrm{B}}= & {} \frac{2k_{\mathrm{b}} T}{\pi \mu _{\mathrm{f}} d_{\mathrm{p}}^2}. \end{aligned}$$
(16)
Here, \(k_{\mathrm{b}}=1.380648 \times 10^{-23}\) (J K\(^{-1}\)) is the Boltzmann constant, \(l_{\mathrm{f}}=0.17\) nm is the mean path of fluid particles, and \(d_{\mathrm{f}}\) is the molecular diameter of water given by Corcione [27]
$$\begin{aligned} d_{\mathrm{f}} = \frac{6 M}{N \pi \rho _{\mathrm{f}}}, \end{aligned}$$
(17)
where M is the molecular weight of the base fluid, N is the Avogadro number, and \(\rho _{\mathrm{f}}\) is the density of the base fluid at room temperature (310 K). Accordingly, and basing on water as a base fluid, the value of \(d_{\mathrm{f}}\) is obtained:
$$\begin{aligned} d_{\mathrm{f}} = \left( \frac{6 \times 0.01801528}{6.022 \times 10^{23} \times \pi \times 998.26}\right) ^{1/3}=3.85 \times 10^{-10} \text{m}. \end{aligned}$$
(18)
According to the study of Corcione [27] and their comparisons with experimental results, the two-phase mixture method is more accurate than the single-phase model. However, this model is valid for temperatures ranging from 294 to 324 K, particle diameters ranging from 10 to 150 nm, and the range of the particle volume fraction is from 0.002 to 0.09 for the thermal conductivity data as in Eq. (14). In addition, it is also valid for temperatures ranging from 293 to 323 K, particle diameters ranging from 25 to 200 nm, and the range of the particle volume fraction is from 0.0001 to 0.071 for the dynamic viscosity data as in Eq. (13).
Now we introduce the following non-dimensional variables:
$$\begin{aligned}&X=\frac{x}{L},\,\, Y=\frac{y}{L},\,\, {\mathbf{V}} = \frac{{\mathbf{v}} L}{\nu _{\mathrm{f}}}, \,\, P = \frac{p L^2}{\rho _{\mathrm{nf}} \nu _{\mathrm{f}}^2}, \nonumber \\&\varphi ^*=\frac{\varphi }{\phi }, \,\, D^*_{\mathrm{B}}=\frac{D_{\mathrm{B}}}{D_{\mathrm{B}0}}, \,\, D^*_{\mathrm{T}}=\frac{D_{\mathrm{T}}}{D_{\mathrm{T}0}}, \nonumber \\&\delta = \frac{T_{\mathrm{c}}}{{{T_{\mathrm{h}}} - {T_{\mathrm{c}}}}}, \quad \theta _{\mathrm{nf}}=\frac{T_{\mathrm{nf}}-T_{\mathrm{c}}}{T_{\mathrm{h}}-T_{\mathrm{c}}},\quad {\theta _{\mathrm{w}}} = \frac{{{T_{\mathrm{w}}} - {T_{\mathrm{c}}}}}{{{T_{\mathrm{h}}} - {T_{\mathrm{c}}}}}. \end{aligned}$$
(19)
This then yields the dimensionless governing equations:
$$\begin{aligned}&\nabla \cdot \mathbf V =0, \end{aligned}$$
(20)
$$\begin{aligned}&\mathbf V \cdot \nabla \mathbf V = - \nabla P + \frac{\rho _{\mathrm{f}}}{\rho _{\mathrm{nf}}} \frac{\mu _{\mathrm{nf}}}{\mu _{\mathrm{f}}} \nabla ^2 \mathbf V + \frac{(\rho \beta )_{\mathrm{nf}}}{\rho _{\mathrm{nf}} \beta _{\mathrm{f}}} \frac{1}{Pr} Ra \cdot \theta _{\mathrm{nf}}, \end{aligned}$$
(21)
$$\begin{aligned}&\mathbf V \cdot \nabla \theta _{\mathrm{nf}} = \frac{(\rho C_{\mathrm{p}})_{\mathrm{f}}}{(\rho C_{\mathrm{p}})_{\mathrm{nf}}} \frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}} \frac{1}{Pr} \nabla ^2 \theta _{\mathrm{nf}} + \frac{(\rho C_{\mathrm{p}})_{\mathrm{f}}}{(\rho C_{\mathrm{p}})_{\mathrm{nf}}} \frac{D^*_{\mathrm{B}}}{Pr \cdot Le} \nabla \varphi ^*\cdot \nabla \theta _{\mathrm{nf}} \nonumber \\&\quad \quad \quad +\, \frac{(\rho C_{\mathrm{p}})_{\mathrm{f}}}{(\rho C_{\mathrm{p}})_{\mathrm{nf}}} \frac{D^*_{\mathrm{T}}}{Pr \cdot Le \cdot N_{\mathrm{BT}}} \frac{\nabla \theta _{\mathrm{nf}} \cdot \nabla \theta _{\mathrm{nf}}}{1 + \delta \theta _{\mathrm{nf}}}, \end{aligned}$$
(22)
$$\begin{aligned}&\mathbf V \cdot \nabla \varphi ^*= \frac{D^*_{\mathrm{B}}}{Sc} \nabla ^2 \varphi ^*+ \frac{D^*_{\mathrm{T}}}{Sc \cdot N_{\mathrm{BT}}} \frac{\nabla ^2 \theta _{\mathrm{nf}}}{1 + \delta \theta _{\mathrm{nf}}}, \end{aligned}$$
(23)
$$\begin{aligned}&\nabla \theta _{\mathrm{w}} = 0, \end{aligned}$$
(24)
where \(\mathbf V\) is the dimensionless velocity vector (UV), \(D_{\mathrm{B}0}=\frac{k_{\mathrm{b}} T_{\mathrm{c}}}{3 \pi \mu _{\mathrm{f}} d_{\mathrm{p}}}\) is the reference Brownian diffusion coefficient, \(D_{\mathrm{T}0}=0.26\frac{k_{\mathrm{f}}}{2k_{\mathrm{f}}+k_{\mathrm{p}}} \frac{\mu _{\mathrm{f}}}{\rho _{\mathrm{f}} \theta }\phi\) is the reference thermophoretic diffusion coefficient, \(Sc=\nu _{\mathrm{f}}/D_{\mathrm{B}0}\) is the Schmidt number, \(N_{\mathrm{BT}}=\phi D_{\mathrm{B}0}T_{\mathrm{c}}/D_{\mathrm{T}0}(T_{\mathrm{h}}-T_{\mathrm{c}})\) is the diffusivity ratio parameter (Brownian diffusivity/thermophoretic diffusivity), \(Le=k_{\mathrm{f}}/(\rho C_{\mathrm{p}})_{\mathrm{f}} \phi D_{\mathrm{B}0}\) is the Lewis number, \(Ra=g \rho _{\mathrm{f}} \beta _{\mathrm{f}}(T_{\mathrm{h}} - T_{\mathrm{c}})L^{3}/(\mu _{\mathrm{f}} \alpha _{\mathrm{f}}\)) is the Rayleigh number for the base fluid, and \(Pr=\nu _{\mathrm{f}}/\alpha _{\mathrm{f}}\) is the Prandtl number for the base fluid.
The dimensionless boundary conditions of Eqs. (20) and (24) are:
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0, \end{aligned}$$
(25)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_{\mathrm{T}}}{D^*_{\mathrm{B}}} \frac{1}{N_{\mathrm{BT}}} \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\, \theta _{\mathrm{w}}=1 \,\, \text{on the horizontal bottom wall},\nonumber \\ &\quad 0\le X \le D,\, Y=0 \,\, \text{and on the vertical left wall},\, 0\le Y \le D,\, X=0, \end{aligned}$$
(26)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0,\,\, \text{(for the adiabatic parts of the remainder walls),} \end{aligned}$$
(27)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_{\mathrm{T}}}{D^*_{\mathrm{B}}} \frac{1}{N_{\mathrm{BT}}} \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\, \theta _{\mathrm{nf}}=0 \,\, \text{on the horizontal top wall},\nonumber \\&\quad 0\le X \le 0.5,\, Y=1 \,\, \text{and on the vertical right wall},\, 0\le Y \le 0.5,\, X=1, \end{aligned}$$
(28)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}=0,\, \frac{\partial \theta _{\mathrm{nf}}}{\partial n}=0,\,\, \text{(for the adiabatic parts of the remainder walls),} \end{aligned}$$
(29)
$$\begin{aligned}&\theta _{\mathrm{nf}} = {\theta _{\mathrm{w}}},\,\, \text{at the outer solid wall surface}, \end{aligned}$$
(30)
$$\begin{aligned}&U = V = 0,\,\, \frac{\partial \varphi ^*}{\partial n}= - \frac{D^*_{\mathrm{T}}}{D^*_{\mathrm{B}}} \frac{1}{N_{\mathrm{BT}}} \frac{1}{1 + \delta \theta _{\mathrm{nf}}} \frac{\partial \theta _{\mathrm{nf}}}{\partial n},\quad \frac{\partial \theta _{\mathrm{nf}}}{\partial n} = K_{\mathrm{r}}\frac{\partial \theta _{\mathrm{w}}}{\partial n}, \end{aligned}$$
(31)
where \(K_{\mathrm{r}} = {k_{\mathrm{w}}}/{k_{\mathrm{nf}}}\) is the thermal conductivity ratio and \(D = {d}/{L}\) is the dimensionless triangular wall thickness.
The local Nusselt number evaluated at the interface is as defined in [7],
$$\begin{aligned} Nu_{i} = -\frac{k_{\mathrm{nf}}}{k_{\mathrm{f}}} \left( \frac{\partial \theta _{\mathrm{nf}}}{\partial n}\right) _{\text {i}}. \end{aligned}$$
(32)
Finally, the average Nusselt number can be calculated on the interface using the following formula:
$$\begin{aligned} \overline{Nu}_{\mathrm{nf}} = \frac{1}{\sqrt{2}D} \int _0^{\sqrt{2}D} Nu_{\text {i}} {} \mathrm{{d}}s, \end{aligned}$$
(33)
where s is a segment along the interface between the solid wall and the nanofluid.

Numerical method

The governing dimensionless equations (20)–(24) subject to the boundary conditions (25)–(31) are solved by the Galerkin weighted residual finite element method. The computational domain was divided into sub-domains (finite elements), and each of the velocity distribution, pressure, temperature and the nanoparticle distribution is approximated by using the basis set \(\{\varPhi _{\text j}\}_{{\text j}=1}^{\text N}\) as,
$$\begin{aligned}&\mathbf V \approx \sum _{j=1}^N \mathbf V _{\text j} \varPhi _{\text j} (X,Y),\quad P\approx \sum _{j=1}^N P_{\text j} \varPhi _{\text j} (X,Y),\quad \theta \approx \sum _{j=1}^N \theta _{\text j} \varPhi _{\text j} (X,Y), \nonumber \\&\phi \approx \sum _{j=1}^N \phi _{\text j} \varPhi _{\text j} (X,Y), \end{aligned}$$
(34)
where \(\varPhi\) represents the basis functions and j represents the internal nodes. The finite element analysis of a boundary-value problem should include the following basic steps:
  1. 1.

    Discretization or subdivision of the domain into several small elements, connected with nodes.

     
  2. 2.

    Weak (or weighted-integral) formulation of the governing dimensionless equations.

     
  3. 3.

    Selection of the interpolation functions for providing an approximation of the unknown solution within an element.

     
  4. 4.

    Development of the finite element model using the weak form.

     
  5. 5.

    Assembly of finite elements to obtain the global system of algebraic equations where the Galerkin methods can be used.

     
  6. 6.

    Imposition of the selected boundary conditions.

     
  7. 7.

    Solving the system of equations.

     
The details of such a procedure are clearly described in Reddy [28]. Following the above-mentioned steps, the computational domain is discretized into triangular elements as shown in Fig. 1b. Triangular Lagrange finite elements of different orders are used for each of the flow variables within the computational domain. Residuals for each conservation equation are obtained by substituting the approximations into the governing equations. To simplify the nonlinear terms in the momentum equations, a Newton–Raphson iteration algorithm was used. The convergence of the solution is assumed when the relative error for each of the variables satisfies the following convergence criterium:
$$\begin{aligned} \left| \frac{\varGamma ^{{\text i}+1}-\varGamma ^{\text i}}{\varGamma ^{{\text i}+1}}\right| \le \eta , \end{aligned}$$
where i represents the iteration number and \(\eta\) is the convergence criterion. In this study, the convergence criterion was set at \(\eta =10^{-6}\).

Validation

Comparisons with other published works

The numerical solution is achieved by computing the Nusselt number for the cavity by using in-house computer code. For the validation of the present work, the result is compared with the earlier published works of Chamkha and Ismael [7] for the case of conjugate heat transfer in a porous cavity heated by a triangular thick wall and filled with nanofluid, as shown in Fig. 2. In addition, the case of natural convection in a square cavity heated horizontally and filled with \(\text{Al}_2\text{O}_3\)–water nanofluid for different Ra, \(d_{\mathrm{p}}=33\) nm, \(2\le \varDelta T\le 10\), \(\phi =0.03\), \(Pr=4.623\) and \(D=0\) using Buongiorno’s model is shown in Fig. 3, where the average Nusselt number was compared with the experimental work of Ho et al. [29], numerical results of Sheikhzadeh et al. [30] and numerical results of Motlagh and Soltanipour [25]. Generally, it is found that the present results are in very good agreement with the numerical results of Sheikhzadeh et al. [30] and Motlagh and Soltanipour [25]. We note that a deviation happens between the numerical result and the experimental data, essentially at low Ra numbers. This is a fact that developing a numerical model which can completely describe all hydro-thermal behaviors of nanofluids is hard, particularly at high particle volume fractions. Buongiorno’s model takes into consideration some important slip mechanisms, but it lacks in the formulation some complex phenomena such as particle–particle and particle–wall collisions. Besides, the correlations of nanofluid viscosity and thermal conductivity have their own approximation [25]. Thereby, the difference between numerical results and experimental data is somewhat acceptable. Based on these validations, it is found that the present result is in a very good agreement with the results of the previously published works.
Fig. 2

Streamlines (a), Chamkha and Ismael [7] (left), present study (right), isotherms (b), Chamkha and Ismael [7] (left), present study (right) for \(D=0.1\) (top) and \(D=1\) (bottom) at \(Ra=500\), \(\phi =0.1\) and \(K_{\mathrm{r}}=1\)

Fig. 3

Comparison of the mean Nusselt number obtained from present numerical simulation with the experimental results of Ho et al. [29], numerical results of Sheikhzadeh et al. [30] and numerical results of Motlagh and Soltanipour [25] for different values of Rayleigh numbers

Grid sensitivity study

For code verification, the grid study is done for the current work. We have used different grid sizes to calculate the minimum strength of the flow circulation (\(\varPsi _{\min }\)) and the average Nusselt number for the case \(Ra=10^{5}\), \(\phi =0.02\), \(K_{\mathrm{r}}=1\) and \(D=0.4\). The results shown in Table 1 and Fig. 4 indicate insignificant differences for the G6 grids and above. Therefore, for all the computations in this paper for similar problems to this subsection, the G6 uniform grid is employed.
Table 1

Grid testing for \(\varPsi _{\min }\) and \(\overline{Nu}_{\mathrm{nf}}\) at different grid sizes for \(Ra=10^{5}\), \(\phi =0.02\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Grid size

Number of elements

\(\varPsi _{\min }\)

\(\overline{Nu}_{\mathrm{nf}}\)

G1

186

\(-\)4.1866

5.5045

G2

286

\(-\)4.0434

5.7236

G3

648

\(-\)3.8902

6.0122

G4

1008

\(-\)3.8507

6.1379

G5

1980

\(-\)3.8148

6.2629

G6

6402

\(-\)3.8072

6.2631

G7

25,722

\(-\)3.8057

6.2631

Fig. 4

Variation of the average Nusselt number (\(\overline{Nu}_{\mathrm{nf}}\)) with grid sizes for \(Ra=10^{5}\), \(\phi =0.02\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Results and discussion

In this section, we present numerical results for the streamlines, isotherms and nanoparticle distributions for various values of the nanoparticle volume fraction (\(0\le \phi \le 0.04\)), the Rayleigh number \((10^{2}\le Ra \le 10^{6})\), thermal conductivity ratio (\(K_{\mathrm{r}}=0.44\), 1, 2.40, 9.90 and 23.8) (epoxy–water: 0.44, brickwork–water: 1, glass–water: 2.40, epoxy–air: 9.90, stainless steel–water: 23.8), dimensionless triangular wall thickness (\(0.1 \le D \le 1\)), where the values of Prandtl number, Lewis number, Schmidt number, ratio of Brownian to thermophoretic diffusivity and normalized temperature parameter are fixed at \(Pr=4.623\), \(Le=3.5\times 10^{5}\), \(Sc=3.55\times 10^{4}\), \(N_{\mathrm{BT}}=1.1\) and \(\delta =155\). respectively. The values of the average Nusselt number are calculated for various values of \(\phi\) and D. The thermo-physical properties of the base fluid (water) and \(\text{Al}_2\text{O}_3\) solid phases are tabulated in Table 2.
Table 2

Thermo-physical properties of water with Al\(_{2}\)O\(_{3}\) nanoparticles at \(T=310\) K [25, 31]

Physical properties

Fluid phase (water)

Al\(_{2}\)O\(_{3}\)

\(C_{\mathrm{p}}\)/J kg\(^{-1}\) K\(^{-1}\)

4178

765

\(\rho\)/kg m\(^{-3}\)

993

3970

k/W m\(^{-1}\) K\(^{-1}\)

0.628

40

\(\beta \times 10^{5}\)/1/K

36.2

0.85

\(\mu \times 10^{6}\)/kg ms\(^{-1}\)

695

\(d_{\mathrm{p}}\)/nm

0.385

33

The direction of the fluid flow and the strength of the flow are defined by the labels of the contour level (clockwise or anticlockwise direction). The positive \(\varPsi\) denotes anticlockwise fluid flow, whereas the negative \(\varPsi\) designates the clockwise fluid flow. The extreme values of the stream function are defined by \(\varPsi _{{\min }}\). These values are essential values for presenting the minimum change of the flow. The nanoparticles tend to take negative values following the same direction of the fluid flow, which is in the clockwise direction.

Before we move on to the discussion of the results, it is worthy to check the Brownian and thermophoretic mechanisms effects on the flow, temperature and nanoparticle distributions. Figure 5 displays the streamlines, isotherms and the nanoparticle distributions with the presence of the Brownian diffusion coefficient \(D_{\mathrm{B}}\) and thermophoretic diffusivity coefficient \(D_{\mathrm{T}}\) in the energy Eq. (22). With the presence of both the Brownian diffusion coefficient (\(D_{\mathrm{B}}\)) and the thermophoretic diffusivity coefficient (\(D_{\mathrm{T}}\)), the strength of the flow circulation tends to decrease from 3.82 to 3.79 due to the effect of \(D_{\mathrm{B}}\) and \(D_{\mathrm{T}}\) which slows down the viscosity of the nanofluid. Also, we can observe from this figure that the thermophoresis and Brownian motion have an equal effect on particle migrations of a nanofluid. The nanoparticles move from the right wall and the upper right corner middle of the cavity to the left and upper left corner of the cavity due to the effects of Brownian motion and thermophoresis. Also with both of them, the same scenario intensifies.
Fig. 5

Variations of the streamlines (left), isotherms (middle) and nanoparticle distributions (right) by the presence of the Brownian diffusion coefficient \(D_{\mathrm{B}}\) and thermophoretic diffusivity coefficient \(D_{\mathrm{T}}\) in the energy Eq. (22), where a in the absence of \(D_{\mathrm{B}}\) and \(D_{\mathrm{T}}\), b in the presence of \(D_{\mathrm{T}}\) only, c in the presence of \(D_{\mathrm{B}}\) only d in the presence of \(D_{\mathrm{B}}\) and \(D_{\mathrm{T}}\); for \(Ra=10^{5}\), \(\phi =0.01\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Figure 6 shows the streamlines, isotherms and the nanoparticle distributions when \(D, K_{\mathrm{r}}\) and Ra are constant, while \(\phi\) has varying quantities. As seen in this figure, the strength of cells diminishes with the increase of \(\phi\). The viscosity of the nanofluid is aggrandized as \(\phi\) increases, and as a result, the velocity and strength of cells are lowered and the streamlines will have a lower velocity as well. Isothermal lines are observed in Fig. 6b, in which the curved lines adjacent to the cold wall are converted to lines parallel to borderline of nanofluid and the triangular solid wall. The temperature gradient and the Nusselt number are expected to be increased with the increase of \(\phi\). The distribution of nanoparticles is indicated in Fig. 6c. As obvious in the shape, considering the low temperature gradient close to the cold wall and also the low speed in this region, thermophoresis and Brownian effects and nanoparticle migration are slight. While beside the warm wall and also near the left wall, thermophoresis effect and nanoparticle migration are high, as the temperature gradient is great. Nanoparticles have CW rotation and their migration abates when \(\phi\) increases and velocity decreases.
Fig. 6

Variations of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by the solid volume fraction (\(\phi\)) for \(Ra=5\times 10^{4}\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Table 3 presents the thermo-physical properties of Al\(_{2}\)O\(_{3}\)–water nanofluid for different \(\phi\) at \(Ra=10^{5}\), \(K_{\mathrm{r}}=1\) and \(D=0.4\). It can be seen from this table that the presence of \(\text{Al}_2\text{O}_3\) nanoparticles with an average solid volume fraction of \(\phi =0.02\) clearly enhances the properties of the base fluid for which the density of the base fluid (water) increases from 993 to 1052.5, thermal conductivity increases from 0.628 to 0.755, and the dynamic viscosity increases from 695 to 830.5. However, the specific heat capacity of the base fluid decreases from 4178 to 4109.7 and the thermal expansion reduces form \(36.2\times 10^{5}\) to \(35.4\times 10^{5}\).
Table 3

Thermo-physical properties of nanofluid (Al\(_{2}\)O\(_{3}\)–water) for different \(\phi\) at \(Ra=10^{5}\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

\(\phi\)

\((C_{\mathrm{p}})_{\mathrm{nf}}\)

\(\rho _{\mathrm{nf}}\)

\(k_{\mathrm{nf}}\)

\(\beta _{\mathrm{nf}} \times 10^{5}\)

\(\mu _{\mathrm{nf}} \times 10^{6}\)

0.01

4143.9

1022.8

0.70845

35.8

755.4

0.02

4109.7

1052.5

0.755

35.4

830.5

0.03

4075.6

1082.3

0.79411

35.1

923.9

0.04

4041.5

1112.1

0.82885

34.8

1004.2

Figure 7 shows the mentioned parameters at fixed \(D, K_{\mathrm{r}}\) and \(\phi\) and varying with Ra number. Given the direct impact of Ra number on momentum equation, the increase in this number leads to the increase in streamline’s velocity and strength. Since convection is the dominant term, with the increase of Ra, the circular shape of streamlines extends horizontally and is transformed into two cells at \(Ra=10^6\). The augmentation of convection and velocity in isothermal line is clearly observable, so that the slant and nearly-parallel isothermal lines exhibit some curves inclined to the right of rigid wall. As indicated in Fig. 7c, nanoparticles migration is enhanced when thermophoresis and Brownian motion and also velocity increases, concurrent with the increase of Ra number.
Fig. 7

Variations of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by the Rayleigh number (Ra) for \(\phi =0.02\), \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Figure 8 represents the influences of \(K_{\mathrm{r}}\) variation on the streamlines, isotherms and the nanoparticle distribution. As evident in Fig. 8a, the velocity is raised with the increase of \(K_{\mathrm{r}}\) and accordingly the increase in heat transfer by the triangular solid from the warm walls and the increase in the temperature of the triangular solid and the nanofluid. In the isothermal lines adjacent to the rigid triangular wall, the temperature gradient increases with the increase of \(K_{\mathrm{r}}\). As the temperature rises and therefore, the thermophoresis effects are improved, and the nanoparticles migrate from the border line of the triangular solid wall and the left wall to the right, while they have a clockwise rotation and this migration is augmented with the increase of \(K_{\mathrm{r}}\).
Fig. 8

Variations of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by the thermal conductivity ratio (\(K_{\mathrm{r}}\)) for \(Ra=5\times 10^{5}\), \(\phi =0.02\) and \(D=0.4\)

Figure 9 shows the alterations streamlines, isotherms and the nanoparticle distribution as a result of D variations. The streamlines are improved with the increase of D. The extreme value of the stream function (\(\mid \varPsi _{{\min }}\mid =6.08\)) occurs at \(D=0.3\). As D rises from 0.3 to 1, the velocity reduces and the streamlines form a single cell. Given the fact that the temperature of the nanofluid changes by conduction from the triangular solid and its borderline with the nanofluid, get away from each other and this trend accounts for the reduction in the Nusselt number along the increase of D. Taking the density of the isotherms near the triangular solid and the left wall into consideration, for moderate values of D, the migration of nanoparticles is significant, considering the increase in the thermophoresis effects. As the density of lines close to the triangular solid wall decreases, the thermophoresis effects and accordingly, the migration of nanoparticles diminish. This is obviously indicated in Fig. 9c.
Fig. 9

Variations of the streamlines (left), isotherms (middle) and nanoparticle distribution (right) evolution by the dimensionless triangular wall thickness (D) for \(Ra=5\times 10^{5}\), \(\phi =0.02\) and \(K_{\mathrm{r}}=1\)

Figure 10 represents the variations of the local Nusselt number for different Ra numbers and at \(D=0.1\) and \(D=1\). With the increase of Ra and the convection term, the heat transfer is supposed to rise as well. The variations are more tangible for \(Ra=10^5\) and \(10^6\). At \(Ra=10^6\) in Fig. 10b, the occurrence of two peaks is attributed to the arrangement of the isothermal lines and accordingly, the resulting temperature gradient.
Fig. 10

Variations of local Nusselt number interfaces with n by the different Ra for a\(D=0.1\) and b\(D=1\) at \(\phi =0.02\) and \(K_{\mathrm{r}}=1\)

The impact of increasing the nanoparticles volume fraction on the heat transfer is illustrated in Fig. 11. As observed in the figure, for \(D=0.1\) and \(D=1\), the heat transfer and the local Nusselt number increase provided that the volume fraction of nanofluids is raised. However, considering the reduction in the velocity due to the increase in the viscosity, and since the variation of the viscosity is a determining factor in natural convection, for a number of n values, the heat transfer decreases with the increase in nanoparticles, but in general, the heat transfer increases when the volume fraction is augmented.
Fig. 11

Variations of local Nusselt number interfaces with n by the different \(\phi\) for a\(D=0.1\) and b\(D=1\) at \(Ra=5\times 10^{5}\) and \(K_{\mathrm{r}}=1\)

The variations of the mean Nusselt number for different Rayleigh numbers and based on the variations of \(\phi\), are shown in Fig. 12. The average Nusselt number increases continuously with the increase of \(\phi\) at \(Ra=10^3\). However for Ra numbers of \(10^4\)\(10^6\), based on the variation of \(\phi\), an optimum situation for the heat transfer is taken into account. At \(Ra=10^4\) when \(0.01<\phi <0.015\), at \(Ra=10^5\) when \(0.02<\phi <0.025\) and at \(Ra=10^6\) when \(0.025<\phi <0.03\) the maximum Nusselt number, and therefore, the optimum heat transfer can be observed.
Fig. 12

Variations of average Nusselt number with \(\phi\) for different Ra for \(K_{\mathrm{r}}=1\) and \(D=0.4\)

Based on the elaborations of the isothermal lines related to \(K_{\mathrm{r}}\) variations, when \(K_{\mathrm{r}}\), and therefore, the temperature gradient near the triangular solid wall increases, the heat transfer and the Nusselt number increase accordingly, as shown in Fig. S1. At the corner of the left wall and the triangular solid wall, the density of isotherms decreases with the increase of \(K_{\mathrm{r}}\). Therefore, at this region, the Nusselt number decreases with the increase of \(K_{\mathrm{r}}\). The average Nusselt number is also supposed to rise with the increase of \(K_{\mathrm{r}}\), and this can be observed in Fig. S2(a), in which the average Nusselt number is raised with the increase of \(K_{\mathrm{r}}\) and Ra.

Figure S3(a) represents the variations of the average Nusselt number with \(\phi\) and for different \(K_{\mathrm{r}}\). The heat transfer increases with \(K_{\mathrm{r}}\), and the heat transfer from the triangular solid to the fluid increases. The reduction heat resistant is experienced when \(K_{\mathrm{r}}\) increases; therefore, the heat transfer enhances, and the maximum heat transfer occurs at \(K_{\mathrm{r}}=23.8\). With the increase of \(\phi\), the increase in the mean Nusselt number is observed for all values of \(K_{\mathrm{r}}\). This increase is shown in Fig. S3(b) for a variety of D values. With the increase in the triangular solid area and accordingly, the decrease in the heat transfer to the fluid, the heat transfer diminishes by the increase of D from 0.1 to 1. Figure S4 indicates the variations of the average Nusselt number for different amounts of the volume fraction and D, and for Ra numbers of \(10^2\) and \(10^6\). As expected, with the increase of \(\phi\), the heat transfer enhances for all values of D. However at a certain \(\phi\), the heat transfer decreases with the increase of D. Figure S5(a) shows the continuous reduction for the mean Nusselt number via increasing D for all Ra numbers. This reduction is more sensible for high Ra numbers. Figure S5(b) exhibits the reduction scenario for the average Nusselt numbers for all \(K_{\mathrm{r}}\).

Conclusions

Conjugate heat transfer of Al\(_2\)O\(_3\)–water nanofluid in a square cavity heated by a triangular thick wall is investigated numerically using Buongiorno’s two-phase model. The results have led to the following concluding remarks:
  1. 1.

    The nanoparticles migration is enhanced when thermophoresis and Brownian motion and also the velocity increase concurrent with the increase of Ra number.

     
  2. 2.

    The local Nusselt number increases provided that the volume fraction of nanofluids is raised.

     
  3. 3.

    The average Nusselt number is raised with the increase in the thermal conductivity ratio and the Rayleigh number.

     
  4. 4.

    With the increase of \(\phi\), the heat transfer enhances for all values of D.

     
  5. 5.

    The continuous reduction is seen for the mean Nusselt number via increasing D for all Ra numbers.

     
  6. 6.

    The heat transfer enhances and the maximum heat transfer occurs at a higher thermal conductivity ratio with the increasing of \(\phi\).

     
  7. 7.

    At \(Ra=10^4\) numbers when \(0.01<\phi <0.015\), at \(Ra=10^5\) when \(0.02<\phi <0.025\) and at \(Ra=10^6\) when \(0.025<\phi <0.03\) the maximum Nusselt number, and therefore, the optimum heat transfer can be observed.

     
Finally, the results show that at low Ra number, adding nanoparticles leads to increase the Nu number. Also, at high Ra number, the increment of heat transfer is seen in low nanofluid volume fractions.

Notes

Acknowledgements

The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010. We thank the respected reviewers for their constructive comments which clearly enhanced the quality of the manuscript.

Supplementary material

10973_2018_7473_MOESM1_ESM.pdf (102 kb)
Supplementary material 1 (pdf 102 KB)

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Refrigeration & Air-conditioning Technical Engineering Department, College of Technical EngineeringThe Islamic UniversityNajafIraq
  2. 2.School of Mathematical Sciences, Faculty of Science & TechnologyUniversiti Kebangsaan MalaysiaBangiMalaysia
  3. 3.Department of Engineering, Mahdishahr BranchIslamic Azad UniversityMahdishahrIran
  4. 4.Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the EnvironmentPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  5. 5.RAK Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates

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