Nomenclature
cp,	specific heat;	u,	velocity component in the x-direction;
g,	gravitational acceleration;	v,	velocity component in the y-direction;
h,	heat transfer coefficient;	Sgen,	local entropy generation;
k,	thermal conductivity;	α,	thermal diffusivity;
Nu,	Nusselt number;	β,	thermal expansion coefficient;
p,	pressure;	μ,	dynamic viscosity;
Pr,	Prandtl number;	υ,	kinematic viscosity;
Ra,	Rayleigh number;	ϕ,	volume fraction;
T,	temperature;	ρ,	density.
Subscript
f,	base fluid;	s,	nanoparticle.
nf,	nanofluid;

1 Introduction

The natural convection heat transfer and fluid flow in a square cavity filled with traditional fluids have been investigated widely in the past years due to its wide engineering applications such as energy conservation, domestic refrigerator, electronic cooling system, solar energy collector, and heat exchanger. Traditional fluids such as water, oil, and ethylene glycol mixture has low thermal conductivity, resulting in a great limitation in improving the compactness and the effectiveness of the heat transfer. In order to overcome the problem mentioned above, Choi and Eastman^[1] proposed the fluid with nanoparticles suspended in the base liquid, called nanofluid. Since then, many studies about nanofluids have been carried out. Some researchers^[2-4] proposed that the heat transfer strength of nanofluids could be increased by 20% even with low nanoparticle concentration.

To have a better understanding about the features of the heat transfer and fluid flow for the natural convection in a cavity, various studies have been conducted numerically. Lee et al.^[2], Masuda et al.^[3], Xuan and Li^[4], and Santra et al.^[5] studied the heat transfer augmentation in a differentially heated square cavity, using the copper nanoparticle (100 nm). They found that the heat transfer strength decreased with an increase in the volume fraction of the nanoparticle at a certain Rayleigh number. However, when increasing the Rayleigh number at a fixed volume fraction, the heat transfer strength increased. Khanafer et al.^[6] investigated numerically the buoyancy-driven heat transfer in a two-dimensional nanofluid-filled enclosure. They showed that larger nanoparticle volume fraction led to larger average Nusselt number. Cianfrini et al.^[7] reported the natural convection of the nanofluid in an enclosure partially heated at one side for different nanoparticle diameters ranging from 25 nm to 100 nm. They showed the effects of the nanoparticle diameters on the heat transfer and fluid flow. Rashidi et al.^[8] studied the features about the heat transfer and fluid flow for the natural convection by use of the the Al₂O₃-water nanofluid in a square cavity. Ool and Popov^[9] used the radial basis integral equation (RBIE) method to study the natural convection of a nanofluid with nanoparticles of different shapes. They reported the advantages of the RBIM method and the effects of different nanoparticle shapes and sizes on the fluid flow and heat transfer. Garoosi et al.^[10] reported the effects of different design parameters on the heat transfer and flow fluid in a square cavity containing different pairs of heaters and coolers. They found that the optimal value of the volume fraction attributing to the strongest heat transfer existed. Bouhalleb and Abbassi^[11] studied the natural convection with the CuO-water nanofluid, and showed that when the aspect ratio was 0.5, the influence of the Rayleigh number was negligible. However, with a high Rayleigh number, the aspect ratio had a significant effect on the heat transfer and fluid flow. Abu-Nada et al.^[12] investigated the effects of the thermal conductivity and viscosity on the natural convection in enclosures. They reported the characteristics about the mean Nusselt number in a cavity by use of two different nanofluids (Al₂O₃-water and CuO-water nanofluids). Seyyedi et al.^[13] investigated the enhancement of the heat transfer for the natural convection by use of the Cu-water nanofluid in annulus enclosure. The control-volume finite element method (CVFEM) was adopted in their study and the concerned parameters such as the volume fraction, the aspect ratio, and the Rayleigh number were discussed to investigate the fluid flow and heat transfer. Hwang et al.^[14] investigated the buoyancy-driven heat transfer of the Al₂O₃-water nanofluid in a rectangular cavity. They theoretically concluded that the ratio of the heat transfer coefficient to the average Nusselt number decreased compared with the base fluid when the size of the nanoparticle increased. Mahmoodi and Sebdani^[15] studied the heat transfer and fluid flow of the Cu-water nanofluid in a square cavity with an adiabatic block. They found that the average Nusselt number increased with an increase in the volume fraction. In addition, when the size of the adiabatic block increased, the heat transfer rate decreased at Ra=10³ and Ra=10⁴ while increased at Ra=10⁵ and Ra=10⁶. Guo et al.^[16] investigated the thermal field and flow field of the Al₂O₃-water nanofluid in a square cavity by the lattice Boltzmann method (LBM), and showed that increasing the Rayleigh number could enhance the flow intensity and heat transfer and decrease the thermal boundary layer thickness near the vertical wall. They also put forward the relation between the Nusselt number and the values of Ra.

A number of experiments have been conducted to study the characteristics about the heat transfer and fluid flow of nanofluids. Wen and Ding^[17] reported the heat transfer of the TiO₂-water nanofluid by experiments. They claimed that the heat transfer coefficient decreased when the volume fraction of the nanoparticle increased. Ho et al.^[18] studied the heat transfer of the Al₂O₃-water nanofluid in a square by experiments. The sizes of the square enclosure, the volume fraction, and the Rayleigh number were changed. They found that, when Ra≥ 10⁷ and the cavity size was the largest, the heat transfer efficiency enhanced significantly when the volume fraction of the nanoparticle was 0.001. Heris et al.^[19] conducted experiments to study the effects of the inclination angle on the natural convection heat transfer in a square cavity by use of the Al₂O₃, TiO₂, and CuO nanofluids dispersed in turbine oil. They showed that the mean Nusselt number was the highest for turbine oil among all the studied fluids. Li et al.^[20] investigated the natural convection of the mixtures consisting of ethylene glycol (EG), water, and ZnO nanoparticles with different mass fractions, and indicated that more EG contained in the mixtures led to worse heat transfer efficiency. Moradi et al.^[21] investigated the natural convection heat transfer of the Al₂O₃ and TiO₂-water nanofluids in a cylindrical enclosure. They found that the volume fraction, the aspect ratio, and the inclination angle at a low Rayleigh number had more significant effects on the heat transfer than those at a high Rayleigh number.

Meanwhile, some researchers investigated the entropy generation of the natural convection filled with nanofluids. Mahian et al.^[22] reviewed the entropy generation caused by the flow of nanofluids in microchannels, square conduits, circular tubes, coiled tubes, and heat exchangers. In their study, some suggestions were given about the study of entropy generation in the future. It mainly aimed to make researchers aware of the importance of the entropy generation in heat transfer and fluid flow. Cho^[23] studied the heat transfer and entropy generation of the natural convection in a cavity with a partially heated wave surface. The Rayleigh number, volume fraction, and shapes of the wave surface were discussed to reveal the effects they had on the average Nusselt number, total entropy generation, and Bejan number. It was found that increasing the volume fraction resulted in a decrease in the average Nusselt number and an increase in the total entropy generation. Shahi et al.^[24] investigated the entropy generation of the natural convection in a square cavity with the heat source placed at different positions. It revealed that the minimum entropy generation and the maximum average Nusselt number were obtained when the heat source was placed at the bottom of the cavity.

Although many publications have investigated the characteristics about the natural convection heat transfer and fluid flow in a square cavity filled with nanofluids, no investigators have reported the natural convection heat transfer, fluid flow, and entropy generation in a different heated square cavity filled with nanofluids with a thermal square column. In many engineering fields, the column at the center of the square cavity can be considered to be a temperature or flow rate controller, which can affect the fluid flow and heat transfer such as the baffles in a heat exchanger and the devices controlling the heat transfer from the hot parts in electronic chips or shells in nuclear reactors. Therefore, it is significant for engineering innovations to conduct the study on the natural convection in a square cavity with a thermal column. In the present study, the effects of the relevant parameters (the Rayleigh number, the temperature of the thermal column, and the volume fraction) on the fluid flow, heat transfer, and entropy generation are investigated, and then relevant phenomena and conclusions are obtained.

2 Investigated domain

The simulation domain is shown in Fig. 1. It consists of a two-dimensional square cavity and a thermal square column. The lengths of the square cavity and the thermal column are H and L, respectively. The left and right walls of the square cavity are considered to be the hot wall (T_h) and the cold wall (T_c), respectively. The top and bottom horizontal walls of the cavity are kept adiabatic. The thermal column is adjusted to different temperatures. The square cavity is filled with the SiO₂-water nanofluid, which is assumed to be in thermal equilibrium, and has the same flow velocity. In addition, the nanofluid is considered to be Newtonian, laminar, and incompressible. The thermo-physical properties of the base water and nanoparticle are listed in Table 1, all of which are considered to be constant except for the density given by the Boussinesq approximation^[25].

3 Method

The steady state equations of the continuity, momentum, energy, and local entropy generation for buoyancy-driven laminar natural convection are expressed as follows.

The continuity equation is

(1)

The momentum equations are

(2)

(3)

The energy equation is

(4)

The local entropy generation equation is

(5)

The thermo-physical properties of the nanofluid about the density, heat capacity, thermal expansion coefficient, thermal conductivity, and dynamic viscosity are obtained from the following equations^[27].

The density of the nanofluid can be given by

(6)

where φ is the volume fraction of the nanoparticle, and the subscripts nf, f, and s mean the nanofluid, the pure water, and the nanoparticle, respectively.

The heat capacitance of the nanofluid can be given by

(7)

The thermal expansion coefficient of the nanofluid can be described by

(8)

The thermal conductivity of the nanofluid can be determined by^[27]

(9)

The dynamic viscosity of the nanofluid can be obtained by^[28]

(10)

The dimensionless variables are set as follows:

(11)

The governing equations can be changed into the following dimensionless forms:

(12)

(13)

(14)

(15)

where is the Rayleigh number.

The dimensionless form of the local generation entropy is defined by

(16)

where the Prandtl number Pr and the Eckert number Ec are defined by

(17)

The dimensionless form of the total entropy generation is

(18)

The Bejan number Be indicates the distribution of the heat transfer irreversibility in the total entropy generation, and it is defined by

(19)

The boundary conditions in the dimensionless form for Eqs. (12) -(15) are considered to be as follows:

(i) On the left wall of the cavity, U=V=0, and θ =1.

(ii) On the right wall of the cavity, U=V=0, and θ =0.

(iii) On the adiabatic walls of the cavity, U=V=0, and , where n is perpendicular to the cavity walls.

(iv) On the thermal square column, U=V=0, and θ =0.0, 0.5, 1.0, 1.5.

The average Nusselt number is defined by

(20)

The mass, momentum, and energy governing equations (12) -(15) are discretized by a finite volume method. The SIMPLE algorithm is used to solve the coupling of the pressure and the velocity. Meanwhile, in this study, a second-order central scheme is used to discretize the diffusion terms in the governing equations. The convective terms are discretized by the second-order upwind scheme. The convergence criterion is set to be 10^-8. In order to validate the numerical scheme, two cases are investigated and compared. One is the natural convection in a square cavity filled with air (Pr=0.71). Table 2 shows the values of the average Nusselt number obtained in this study and previous works. The other is the natural convection in a square cavity filled with the SiO₂-water nanofluid (Pr=6.2). The values of the average Nusselt number compared with Kefayati et al.^[26] are shown in Fig. 2. It can be obviously found that the results obtained in the two cases agree well with the previous works.

4 Results and discussion

In this paper, the heat transfer, fluid flow, and entropy generation of the natural convection about the SiO₂-water nanofluid in a square cavity containing a thermal column are investigated numerically. The temperatures T=0.0, 0.5, 1.0, 1.5 are attached to the thermal column, respectively. Meanwhile, the temperature of the hot wall is considered to be T=1.0, and the cold wall is considered to be T=0.0. The other parameters are as follows: the Rayleigh number ranges from 10³ to 10⁶, the temperature of the thermal column ranges from T=0.0 to T=1.5, and the volume fraction ranges from 0.01 to 0.04. The results are expressed by the effects of the Rayleigh number, the temperature of the thermal square column, the volume fraction on the heat transfer, the fluid flow, and the entropy generation.

4.1 Effects of temperature and Ra values on streamlines and isotherms

Figure 3 displays the streamlines and isotherms of the SiO₂-water nanofluid in a square cavity containing a thermal column at a particular volume fraction φ =0.02. From Fig. 3, it can be noted that, at all Rayleigh numbers, when the thermal column is attached to different temperatures, the flow patterns show the difference. It is the reason that four kinds of temperature differences causing natural convection exist between the thermal column and the square cavity when T=0.0, 0.5, 1.0, 1.5. A variety of flow patterns turn up at the same time when the Rayleigh number increases due to different buoyant forces. It can be seen from Fig. 3 that at a low Rayleigh number, e.g., Ra=10³, for all the temperatures, the streamlines circle around the thermal column, owing to the reason that the low buoyant forces lead to the steady flow. The streamlines gradually move towards the left side of the thermal column when the temperature of the thermal column increases. Especially, when T=1.5, the density of the streamlines near the top-left diminishes. However, near the top-right, it increases. When Ra increases, vortexes start to form in the left upper corner of the cavity. At Ra=10⁴, when T=0.0, a vortex on the left side of the thermal column establishes, and the streamlines take up the whole cavity. When T=0.5, streamlines circle around the thermal column due to the equal temperature difference between the thermal column and the two-side walls of the cavity. Contrasting to T=0.0, when T=1.0, a vortex stands on the right side of the thermal column. When T=1.5, the streamlines are full of all the cavity with the exception of the top-left. When Ra is increased to 10⁶, the flow intensity enhances significantly. It can be seen that more streamlines have occupied the cavity. When T=0.0, small and big vortexes establish at the top-right and bottom-left of the cavity, respectively. When T=0.5, compared with Ra=10⁵,

the center vortex breaks up and two vortexes establish at the top-left and bottom-right of the thermal column. As far as T=1.0 is concerned, an interesting thing can be found that the feature about the streamlines is contrasting to T=0.0 due to the contrasting temperature difference between the vertical walls of the cavity and the thermal column. When T=1.5, more vortexes form in the upside of the thermal column.

In regard to the isotherms in Fig. 3, it can be seen that, at all Rayleigh numbers, different temperatures of the thermal column contribute to different structures and variations of the isotherms. For all temperatures of the thermal column shown in Fig. 3, when Ra=10³, the isotherms are distributed vertical to the two insulted walls and parallel to the hot and cold walls, implying that heat conduction dominates this area. When the Rayleigh number increases, it can be easily observed that more compacter strips generate at the bottom-left and top-right of the cavity. At the same time, the isotherms are distributed uniformly and parallel to the two insulted walls, which indicates that the heat transfer mechanism is changing from heat conduction to heat convection. In addition, at a low Rayleigh number, an evident thermal boundary layer forms along the hot wall and the cold wall and around the thermal column. When the Rayleigh number increases, the thickness of the thermal boundary layers becomes thinner when the temperature gradient increases due to the effects of the natural convection strength. Moreover, as far as a certain temperature is concerned, when T=0, at Ra=10³, it can be observed that the isotherms occupy the whole cavity with the exception of the lower right region of the cavity because that no temperature differences exist between the thermal column and the cold wall. When the Rayleigh number increases, the isotherms take up more portions of the cavity due to the stronger heat convection. When T=0.5, the square cavity is filled with the isotherms for all Rayleigh numbers due to the uniform heat transfer between the thermal column and the two-side walls of the cavity. When T=1.0, 1.5, the isotherms in the top side of the thermal column get denser than those in the bottom side.

4.2 Effects of temperature and Ra on mean Nusselt number

Figure 4 exhibits the variations of the average Nusselt number along the hot wall for different temperatures of the thermal column (T=0.0, 0.5, 1.0, 1.5) at a fixed volume fraction φ =0.02. It can be observed from the left figure in Fig. 4 that, when T=0.0, 0.5, 1.0, 1.5, the average Nusselt number of the hot wall generally increases with the increase in the Rayleigh number. It is the reason that a larger Rayleigh number leads to stronger heat transfer efficacy. It can also be found that the average Nusselt number decreases with the increase in the temperature of the thermal column. In addition, at all Rayleigh numbers, the average Nusselt number along the hot wall is the maximum for T=0.0 and Ra=10⁶, while is the minimum for T=1.5 and Ra=10³.

It can be seen from the right figure in Fig. 4 that the value of the Nusselt number increases almost linearly with the decrease in the temperature of the thermal column at all the Rayleigh numbers. It is mainly due to the different temperature existing between the thermal column and the hot wall. Meanwhile, it can also be seen that at Ra=10³, when T=1.5, the average Nusselt number of the hot wall is less than zero. At this moment, the heat transfer mechanism is mainly heat conduction, and the temperature of the thermal column is higher than the left side wall of the cavity. Therefore, the hot wall may absorb the heat from the outside. However, when Ra≥10⁴, the heat transfer mechanism is changing from heat conduction to heat convection. Especially, at Ra=10⁶, the increasing rate of the Nusselt number is the highest when the temperature of the thermal column decreases. It indicates that the heat transfer in the square cavity is enhanced when the buoyant forces increase.

4.3 Effects of volume fraction on streamlines and isotherms

Figures 5 and 6 illustrate the effects of the volume fraction of the nanoparticle on the streamlines and isotherms with different volume fractions (φ =0.01, 0.02, 0.03, 0.04) at Ra=10³ and Ra=10⁶, respectively. As can be seen in Fig. 6 that as far as all temperatures of the thermal column are concerned, no matter at a low or high Rayleigh number, the streamlines and the isotherms show no significant differences when the volume fraction at the fixed temperature of the thermal column increases. In general, increasing the volume fraction leads to the increases in the viscosity and the thermal conductivity of the nanofluid, which affects the flow patterns and heat transfer. From Fig. 6, however, no evident change turns up, which implies that when the thermal column is given to different temperatures, the volume fraction of the nanoparticle has a negligible effect on the flow field and the temperature field in the cavity at low or high Ra. This result can also be illustrated in the next section.

4.4 Effects of volume fraction on Nusselt number

Table 3 shows the values of the average Nusselt number along the hot wall for different volume fractions and Rayleigh numbers. It can be found in this table that no matter at Ra=10³ or Ra=10⁶, the value of the average Nusselt number has a small increment with the increase in the volume fraction. This can be explained that the volume fraction has a slight effect on the heat transfer when the thermal conductivity and the volume fraction increase. This phenomenon has also been stated in the above section. The volume fraction has an insignificant effect on the heat transfer and fluid flow. Meanwhile, it can be found that the increasing increment of the Nusselt number is higher at Ra=10⁶ than at Ra=10³. This is mainly due to that, at a high Rayleigh number, the heat transfer form in the square cavity is dominated by the heat convection. However, it can be found that, when T=1.5, at Ra=10³, the value of the average Nusselt number is less than zero. The reason has been mentioned above. At this time, the heat transfer mechanism is mainly heat conduction, and the temperature of the thermal column is higher than the left-side wall of the cavity. Therefore, the hot wall may absorb heat from the outside.

4.5 Effects of temperature and Ra on local entropy generation and total entropy generation

The fluid friction irreversibility and heat transfer irreversibility caused by the temperature gradients and viscosity of the nanofluid contribute to the local entropy generation. Figures 7 and 8 describe the local entropy generation for different Rayleigh numbers and different volume fractions. It can be seen that at a low Rayleigh number, the volume fraction has an inappreciable effect on the local entropy generation. But at a high Rayleigh number, with the augment in the volume fraction, the local entropy generation starts to show the difference. Whether this difference is caused by the increase in the volume fraction will be explained in the following section. Meanwhile, at a low Rayleigh number, when T=0.0, the local entropy generation in the left side of the thermal column is more significant than the rest area due to the significant temperature gradients. In terms of T=0.5, the local entropy generation in the top-left side and the bottom-right side of the thermal column shows no obvious changes due to the same temperature difference between the thermal column and the two cavity sides. While when T=1.0 and T=1.5, the remarkable local entropy generation occurs in the right-side of the thermal column. At a high Rayleigh number, due to the larger temperature gradients and stronger fluid friction irreversibility caused by the heat convection, the local entropy generation in the lower left and upper right corner of the cavity is more significant, and its values are larger than those at Ra=10³. Meanwhile, when T=0.0 and T=1.5, the local entropy generation near the thermal column is severer than that when T=0.5 and T=1.0.

Figure 9 shows the total entropy generation of the square cavity filled with the SiO₂-water nanofluid. It can be seen in this figure that, when T≤0.5, the total entropy generation decreases with the increase in the temperature of the thermal column. However, when T≥ 1.0, the total entropy generation of the cavity increases with the increase in the temperature. In addition, it can also be found that, when the volume fraction increases, at a low Rayleigh number, the total entropy increases slightly, indicating that the volume fraction shows an insignificant effect on the local entropy generation at a low Rayleigh number. This phenomenon has also been stated in Subsection 4.5. But at a high Rayleigh number, the growth of the values of the total entropy generation becomes more significant. It is the reason that, with the increases in the volume fraction and Ra, the thermal conductivity and buoyant forces increase, attributing to stronger irreversibility of the heat transfer and fluid friction. The Bejan number in the present study for all the conditions is kept at a constant value 0.8. This demonstrates that the heat transfer irreversibility has a more pronounced effect than the fluid friction on the local entropy generation. Moreover, increasing the thermal conductivity via increasing the volume fraction results in the decrease in the temperature gradient and thus the decrease in the entropy generation. Therefore, at a high Rayleigh number, the heat convection condition is the main parameter bringing about an obvious increase in the total entropy generation rather than the volume fraction. Moreover, when T=0.5, the total entropy generation is the minimum, while when T=1.5, it is the maximum.

5 Conclusions

In the present study, the fluid flow, heat transfer, and entropy generation for the natural convection in a square cavity containing a thermal column filled with the SiO₂-water nanofluid are investigated numerically. The effects of the Rayleigh number, the volume fraction, the temperature of the thermal column on the fluid flow, the heat transfer, and the entropy generation are revealed. Based on the results mentioned above, we can obtain the general conclusions as follows:

(i) The volume fraction of the nanoparticle shows no evident effects on the flow field and temperature field, no matter at a low or high Rayleigh number. When the volume fraction increases, for all the temperatures of the thermal column, the Nusselt number increases slightly. The Nusselt number at a high Rayleigh number increases more significantly than that at a low Rayleigh number.

(ii) With an increase in the temperature of the thermal column, the value of the average Nusselt number decreases nearly linearly for all the Rayleigh numbers. In addition, the decreasing rate of the Nusselt number is higher at a high Rayleigh number than at a low Rayleigh number. Moreover, when Ra=10⁶ and T=0.0, the value is the maximum, while when Ra=10³ and T=1.5, it is the minimum.

(iii) When T≤ 0.5 and the temperature of the thermal column increases, there is a reduction in the total entropy generation. But, when T≥1.0, a growth occurs in the total generation when the temperature increases. When T=0.5, the value of the total entropy is the minimum. Meanwhile, at a low Rayleigh number, the volume fraction shows no significant effects on the total entropy generation, while at a high Rayleigh number, the heat convection mechanism is the main factor, which has a significant effect on the growth of the total entropy generation rather than the volume fraction.