# Conjugate heat transfer of Al_{2}O_{3}–water nanofluid in a square cavity heated by a triangular thick wall using Buongiorno’s two-phase model

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## 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

*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 0.5

*L*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:

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

*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:

*U*,

*V*), \(D_{\mathrm{B}0}=\frac{k_{\mathrm{b}} T_{\mathrm{c}}}{3 \pi \mu _{\mathrm{f}} d_{\mathrm{p}}}\) 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.

*s*is a segment along the interface between the solid wall and the nanofluid.

## Numerical method

*j*represents the internal nodes. The finite element analysis of a boundary-value problem should include the following basic steps:

- 1.
Discretization or subdivision of the domain into several small elements, connected with nodes.

- 2.
Weak (or weighted-integral) formulation of the governing dimensionless equations.

- 3.
Selection of the interpolation functions for providing an approximation of the unknown solution within an element.

- 4.
Development of the finite element model using the weak form.

- 5.
Assembly of finite elements to obtain the global system of algebraic equations where the Galerkin methods can be used.

- 6.
Imposition of the selected boundary conditions.

- 7.
Solving the system of equations.

*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

*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.

### Grid sensitivity study

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 |

| 6402 | \(-\)3.8072 | 6.2631 |

G7 | 25,722 | \(-\)3.8057 | 6.2631 |

## Results and discussion

*D*. The thermo-physical properties of the base fluid (water) and \(\text{Al}_2\text{O}_3\) solid phases are tabulated in Table 2.

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 |

| 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.

*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.

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 |

*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.

*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.

*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.

*n*values, the heat transfer decreases with the increase in nanoparticles, but in general, the heat transfer increases when the volume fraction is augmented.

*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.

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

- 1.
The nanoparticles migration is enhanced when thermophoresis and Brownian motion and also the velocity increase concurrent with the increase of

*Ra*number. - 2.
The local Nusselt number increases provided that the volume fraction of nanofluids is raised.

- 3.
The average Nusselt number is raised with the increase in the thermal conductivity ratio and the Rayleigh number.

- 4.
With the increase of \(\phi\), the heat transfer enhances for all values of

*D*. - 5.
The continuous reduction is seen for the mean Nusselt number via increasing

*D*for all*Ra*numbers. - 6.
The heat transfer enhances and the maximum heat transfer occurs at a higher thermal conductivity ratio with the increasing of \(\phi\).

- 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.

*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

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