Analysis of Combined Natural Convection and Radiation Heat Transfer in a Partitioned Rectangular Enclosure with Semitransparent Walls

In this paper, we present a two-dimensional numerical analysis of the conjugate natural convection and radiation heat transfer in a double-space enclosure with two semitransparent walls. Two kinds of boundary conditions are considered, the first being the isothermal process of the opaque wall, and the other the incidence of a constant radiation flux in the left semitransparent wall. The renormalization group k–ε model is adopted to simulate the turbulent flow in the enclosure. To compute the radiation heat transfer in a semitransparent medium, the discrete ordinates model is used. We compare the behaviors of enclosures with single and double semitransparent walls and determine the difference in the results obtained for semitransparent and opaque partitions. The results indicate that a semitransparent partition facilitates a reduction in the heat loss or obtains a higher temperature distribution. The transmittance of a semitransparent wall has a great effect on the thermal and flow characteristics in an enclosure. The change of wall temperature is found to be significant when the thermal conductivity values range from 0.05 to 0.5 W/(m·K), and to be small when ranging from 0.5 to 10 W/(m·K). These conclusions are helpful for green design and energy saving in solar buildings.


Introduction
Due to its prime importance in various industrial and natural processes, determining the natural convection with radiation inside a square cavity filled with different fluids is a classical scientific problem. The model most often used to study this problem is a rectangular enclosure with two vertical isothermal walls and two horizontal adiabatic walls [1]. However, in terms of engineering applications, such as solar chimneys, double-skin facades, solar collectors, and Trombe walls, this simplified model cannot be correctly developed to investigate the flow and heat transfer characteristics [2], because the introduction of a semitransparent medium changes the enclosure characteristics. Compared to opaque materials, semitransparent materials allow a greater transmission of direct radiation. To use and understand the natural convection in buildings and systems, it is important to study the effect of a semitransparent medium on enclosures to provide references for practical applications.
Several theoretical and experimental studies have been conducted on enclosures with semitransparent walls. Moreno and Hernández [3] presented analytical solutions for a very simple room with glazing subjected to solar radiation, in which they identified the main physical mechanisms and the importance of the indoor temperature on the spectral and thermal parameters. Flores et al. [4] presented a study of the combined heat transfer in a cubic cavity containing a vertical semitransparent wall. From this parametric study, for a fixed absorbance of a solar-control coating, the contribution of the radiative heat transfer was found to be higher than that by convection, but both the convective and radiative heat transfers were determined to increase with an increase in the ambient temperature. Li et al. [5] investigated the effect of optical constants on the conjugate laminar natural convection and radiative heat transfer in a rectangular enclosure with one vertical semitransparent wall. It was found a marked effect of the optical constants of a semitransparent wall on the laminar natural convection in a rectangular enclosure. Wu and Lei [6] developed a transient heat balance model for predicting the thermal performance of a semitransparent water-wall system. A series of studies were conducted by Xamán et al. [7][8][9][10][11][12][13][14][15] regarding the heat transfer in a square cavity with a semitransparent wall. By performing numerical studies of the conjugate heat transfer in a square cavity with a semitransparent wall for laminar and turbulent flows, the authors obtained a set of correlations for the Nusselt number for both laminar and turbulent flows [7,8]. The unsteady Reynolds-averaged Navier-Stokes (URANS) model together with a low-Reynolds-number turbulence k-ω model was solved using the finite-volume method [9]. Using the glazing material available in the Mexican market, researchers have also investigated the thermal performance of a room with a double-glazed window [10,11]. A thermal evaluation of a room with a double-glazed window both with/without a solar-control film made in Mexico was also presented [12][13][14]. Based on the results, we know that a room with a double-pane window with a solar-control coating is associated with a smaller heat flux than one with a window without a solar-control coating. The effect of the roofing material on a room with a semitransparent wall has also been analyzed [15].
Natural convection in a differentially heated, partitioned enclosure with an opaque partition wall occurs in various situations both in natural settings and in engineering applications. Recently, several works have outlined the heat transfer characteristics when using partitioned enclosures. Rabhi et al. [16] studied the effects of surface radiation and the number of partitions on the heat transfer and flow structures in an inclined rectangular enclosure, and found that the total heat transfer in the enclosure increased in the presence of a thermal radiation heat flux, and reduced significantly with an increasing number of partitions. Selimefendigil and Öztop [17] used the finite element method to investigate conjugate natural convection and conduction heat transfer in an inclined partitioned cavity filled with different nanofluids on different sides of the partition. Sun et al. [18] presented extensive numerical studies on fluid flow and heat transfer in inclined and fully divided CO 2 enclosures with partitions on Mars. It was observed that three flow regimes formed in succession when the tilt angle increased, namely, Rayleigh-Bénard convection, transition convection, and single-cell convection. Sambou et al. [19] developed a onedimensional analytical model of the coupled heat transfer (conduction, convection, radiation) in enclosures divided by multiple vertical diffusive partitions, and found that the thermal resistance could be improved by decreasing the thermal conductivity of the walls, decreasing the emissivity of the partitions faces, or using very thin partitions. Oztop et al. [20] performed numerical simulations of the conductioncombined forced and natural convection heat transfer and fluid flow in a 2D lid-driven square enclosure divided by a partition with a finite thickness and finite conductivity. It was observed there to be higher heat transfer for a higher Richardson number in an upward moving wall for all thermal conductivity ratios. Khatamifar et al. [21] studied the effects of partition thickness and position over a wide range of Rayleigh numbers, and found that the average Nusselt number increases with the Rayleigh number, but decreases with partition thickness. They also found that the partition position had a negligible effect on the average Nusselt number. Most previous studies of natural convection in partitioned enclosures have been concerned with cases where the partition wall is opaque. There is insufficient knowledge regarding enclosures with a semitransparent wall partition, as this may be a better model for many real situations.
There are also two significant points that must be emphasized in the study of rectangular enclosures. First, for most flows, the Rayleigh number often exceeds the critical values and the fluid flow becomes turbulent in many studies. However, many studies consider only laminar flows. Second, the contribution of thermal radiation is an important topic with respect to the heat transfer in enclosures. Radiation heat transfer depends on several parameters, such as the wall temperature, surface emissivity, and thermophysical properties of the internal medium. In studies of turbulent natural convection, the radiative mode of heat transfer is occasionally neglected because of the overwhelming number of computational resources it demands. However, the radiation heat transfer has a significant effect on the system and cannot be ignored, because this phenomenon occurs in numerous engineering applications. Recently, a few studies have addressed the combination of natural convection and radiation heat transfer in rectangular enclosures. Miroshnichenko et al. [22][23][24][25] conducted a series of studies on transient turbulent natural convection combined with surface thermal radiation in a square cavity with a local heater. The Rayleigh number, thermal conductivity ratio, and internal surface emissivity 1 3 have a significant effect on the temperature and stream function contours within the enclosure [22]. In addition, they found that the presence of surface radiation leads to both an increase in the average total Nusselt number and intensive cooling of this type of system. A significant intensification of convective flow was also observed by an increase in the Rayleigh number [23,24]. The results show that an increase in the cavity inclination angle leads to a reduction in the radiative Nusselt number [25]. Turbulent natural convection in air-filled differentially heated cavities was numerically investigated using various RANS turbulence models and the discrete ordinates (DO) model [26]. The result of this study shows that the shear stress transport k-ω model performs best overall and the standard k-ε model has the worst overall performance. Sharma et al. [27] conducted a numerical investigation of turbulent natural convection in a transparent fluid medium and its interaction with surface thermal radiation in an inclined differentially heated enclosure. Mao and Zhang [28] conducted a numerical simulation of the turbulent natural convection of compressible air in a tall cavity, and found that the k-ε model had high accuracy in predicting the velocity distribution, whereas the large eddy simulation model had good performance in predicting temperature distributions.
In this study, our main aim was to investigate the effect of semitransparent walls on the natural convection flow field and temperature distribution in a double-spaced rectangular enclosure. Computation fluid dynamics can be used to simulate this problem. To compute the radiation heat transfer in a semitransparent medium, the DO model is used. The comparison of the enclosures with single and double semitransparent walls is conducted, and the difference of semitransparent partition as well as opaque partition is analyzed. The impact of different transmittance combinations on the semitransparent wall is examined. In addition, the influence of different thermal conductivities in a semitransparent medium on the performance of the proposed enclosure is also detailed.

Model Description
The study is focused on the effects of semitransparent walls on the natural convection flow field and temperature distribution in a rectangular enclosure. The physical model is shown in Fig. 1. The enclosure is formed by two horizontal adiabatic walls, and left and partition walls are semitransparent walls. The right wall is considered to be a blackbody diffuse surface. We assume the wall thicknesses, other than that of the semitransparent walls, to be negligible. The ambient temperature is set as 273 K and the external radiation temperature is assumed to be equal to T a . The heat transfer coefficient between the ambient and left semitransparent walls is 10 W/(m 2 K). The air inside the cavity is considered to be an incompressible Newtonian fluid that does not participate in the radiation process. The thickness of both semitransparent walls is 5 mm and their emissivity is 0.8. The height H of the enclosure is 500 mm and its length L is 300 mm. The aspect ratio of the cavity is computed as the length over its height, which in this case is L/H = 0.6. In order to have a comprehensive understanding, two different physical models have been studied. In Case 1, the blackbody diffuse wall is considered as an isothermal wall and its temperature is 313 K, as shown in Fig. 1a. In Case 2, the direction of the radiation beam is perpendicular to the left boundary and I is prescribed as 1000 W/m 2 , as shown in Fig. 1b.

Math Theory
For two-dimensional steady flow and heat transfer of the presented physical model, the governing equations are listed as follows.
Continuity equation: where p is the pressure; T ′ is the fluctuating temperature; u i and u j are the mean velocity components in the i and j directions, respectively; u ′ i and u ′ j are the corresponding fluctuating velocity components in the i and j directions, respec- The choice of turbulence model is dictated not only by the type of flow but also by the availability of computational power and the desired accuracy. Buoyancy-driven air flow in the regular enclosure is the index for selecting a turbulence model. RANS models have been widely used in turbulent fluid flow simulations, and they demand a reasonable level of computational resources. In this paper, the two-equation renormalization group (RNG) k-ε model is selected to predict the turbulence in the cavity, as previous studies have reported this model to be computationally efficient and stable [29]. The turbulence kinetic energy, k, and the dissipation rate, ε, are obtained from the following transport equations: where G k represents the production of turbulence kinetic energy; G b represents the generation of ε; Y M represents the contribution of fluctuating dilatation in the compressible turbulence to the overall dissipation rate; and α k and α ε are the inverse effective Prandtl numbers for k and ε, respectively.
The turbulent viscosity μ t can be obtained as: [30].
In this study, the radiation problem is solved through using the DO radiation model. More importantly, only the DO model allows to solve semitransparent walls [31]. We used the DO model to solve the radiation transport equation (RTE) (given below) for a finite number of discrete solid angles, each associated with a vector direction s fixed in the global Cartesian system (x, y, z). The RTE in the direction s is written as follows: where r represents the position vector; s ′ represents the scattering direction vector. The conduction heat transfer equation of a semitransparent wall is given by the following equation: Once the intensity radiation field is known, the source term S T in the conduction heat transfer equation can be calculated as follows:

Grid Test
To numerically model turbulent natural convection, it is best to use many nodes near the boundaries of solid walls [32]. Therefore, we used a boundary layer with an initial size of 0.06 mm and a growth factor of 1.2 to capture the fluid flow that develops near the wall, as shown in Fig. 2. Grid independence in the solution scheme significantly affects the calculation results of the heat transfer and fluid flow. In order to verify the validity of the grid sensitivity, different mesh sizes are used to test grid independence. The heat flux (case 1) of wall-1 and average temperature of T w (case 2) for different mesh sizes are monitored and the results are shown in Table 1. It is found that almost the same temperature and heat flux for the two cases are obtained for different mesh sizes. Based on this result, we selected the mid-range mesh size of 180 × 160 for the simulation analysis.

Boundary Conditions
To solve governing equations, appropriate boundary conditions are necessary. The model is subjected to different types of boundary conditions and all walls are considered as nonslip wall boundary. The detailed boundary conditions can be obtained as follows: Topand bottom walls ∶ q rad + q conv = 0 where q 2,rad , q 3,rad , q 4,rad , and q w,rad are the radiative heat flux of wall-2, wall-3, wall-4, and the blackbody diffuse wall, respectively, W/m 2 ; q 2,conv , q 3,conv , q 4,conv , and q w,conv are the convective heat flux of wall-2, wall-3, wall-4, and the blackbody diffuse wall, respectively, W/m 2 .
In this study, we investigated the impact of a single factor on the enclosure. Thus, only the study object changes in the simulation and other factors are held constant. These constant values for the simulated parameters included the transmittance and thermal conductivity of the semitransparent wall, whose values were 0.8 and 0.5 W/(m·K), respectively.

Modeling Strategy
We performed all calculations reported hereafter using the general-purpose commercial code FLUENT, based on the finite-volume method. The governing equations were discretized by the finite-volume method on a non-uniform staggered grid system using the SIMPLEC algorithm. PRESTO! scheme was used for the pressure discretization. The secondorder upwind scheme was considered for the momentum and energy equations.

Model Verification
To evaluate the validity of the numerical model, we conducted a simulation according to the work in Ref. [5]. A comparison of the temperature distribution at mid-height of enclosure is given in Fig. 3. The R-square method is used to compare the simulation results with the experimental measurements, and the equation is Eq. (20). The value of   R-square is 0.8783, which means that there is good agreement between the reference's results and the computational fluid dynamics (CFD) predicted data.
where y sim,i is the simulation data, y exp,i is the experimental data from Ref. [5], and ȳ exp,i is the mean value of the experimental data.

Case 1
The temperature fields of the enclosure with and without a semitransparent partition are shown in Fig. 4. For the nonpartition enclosure, the temperature is distributed evenly from bottom to top and the isothermals are substantially parallel, except for the highest one. For the partitioned enclosure, it is observed that the temperature field is divided into two parts, with the temperature of the right part (E 2 ) much higher than that of the left part (E 1 ). The average temperature of the left semitransparent wall (S left ) decreased by about 3.02 K, as compared to that of the enclosure without a partition. The heat flux on wall-1 is 22.4% lower in the partitioned enclosure, which means that the heat transfer in a partitioned rectangular cavity is considerably lower than that in a non-partitioned cavity. This numerical result indicates that a semitransparent partition can decrease heat loss effectively. The flow patterns of the enclosure with and without a (20) semitransparent partition are shown in Fig. 5. At the center of the cavity, the streamlines are nearly horizontal, which indicates fluid stratification, but at the boundaries adjacent to the adiabatic walls there are significant differences, as shown in Fig. 5a. In contrast, in Fig. 5b, the streamlines are not parallel. The maximum stream function value of E 2 is greater than that of E 1 .

Case 2
The temperature distributions of an enclosure with and without a semitransparent partition under a constant radiation flux are shown in Fig. 6. A uniform temperature gradient is observed in Fig. 6a. The temperature field of the enclosure with double semitransparent walls can be considered to have two parts, and the average temperature of E 2 is 21.9 K higher than that of E 1 . This result indicates that T w of the enclosure with double semitransparent walls is 360.8 K, as compared to 359.7 K for that with a single wall. This means that an additional semitransparent partition is beneficial for improving the absorbed temperature. Figure 7 shows the stream functions for single and double semitransparent walls, respectively. In Fig. 7a, the flow pattern is very similar to that in Fig. 5a. For double semitransparent walls, the stream function of E 1 is higher than that of E 2 , and the streamlines tend to more turbulence than that occurring with a single wall. The comparison of T w at different transmittances (τ) in an enclosure with and without a semitransparent partition wall is shown in Fig. 8. There appears to be a positive correlation between τ and T w for both research objects. When τ drops to 0.5, the T w value of the enclosure without a semitransparent partition is slightly greater than that with such a wall. This indicates that with a decrease in the transmittance, a semitransparent partition would fail to maintain a higher temperature distribution. Figure 9 displays the temperature fields of enclosures with semitransparent and opaque partition walls, respectively, in which it is obvious that the temperature of an enclosure with a semitransparent partition is higher than that with an opaque partition. The average temperature of E 1 for a semitransparent partition is 4.2 K higher than that for an opaque partition. This is because radiation can be transmitted by the semitransparent partition from wall-4 to wall-3. However, with an opaque partition, the heat transfer process to wall-3 can only occur by conduction. The simulation results show that the heat fluxes from the left semitransparent wall are 212.4 W/m 2 and 142.1 W/m 2 for semitransparent and opaque partitions, respectively. As shown in Fig. 10, the stream function value in E 1 with a semitransparent partition is higher than that in one with an opaque wall, and lower than that in the right part.

Effect of Transmittance
The transmittance of a semitransparent wall has a significant effect on the thermal and flow characteristics in an enclosure. In this section, different combinations of transmittance in an enclosure with two semitransparent walls have been studied. The transmittance combination is expressed as (τ left , τ mid ). Figure 11 shows the temperature contours of four transmittance combinations, in which we can see that the isothermals in E 1 are similar in all cases. In addition, when the transmittance of S mid is varied from 0.6 to 0.9, the higher temperature region in E 2 is expanded, and the lower temperature region is constricted. The comparison among the temperatures of E 1 , E 2 , S left and S mid is shown in Fig. 12, in which we can also observe that the temperatures of E 1 and S left increase with the greater transmittance of the semitransparent partition. This is because the ability of the semitransparent material to transfer and distribute radiative energy intensifies the temperature distribution in the enclosure which becomes more homogeneous [5]. The results also show that the heat flux from wall-1 of (0.6, 0.6) is 34.5 W/m 2 greater than that of (0.9, 0.9). For the combinations of (0.6, 0.9) and (0.9, 0.6), the heat fluxes are almost the same. In Fig. 13, the stream function is shown for the four cases in which we find that the flow patterns are basically similar. Figure 14 shows the temperature contours of four transmittance combinations under a constant radiation flux. The comparison among the temperatures of E 1 , E 2 , S left , and S mid is shown in Fig. 15. For a normal constant radiation flux on wall-1, the high-temperature region in E 2 expands with an increase in the transmittance. The temperature gradient is reduced in E 1 as τ left decreases from 0.9 to 0.6. For the combinations (0.6, 0.9) and (0.9, 0.6), it could be found that the temperature of E 2 for (0.9, 0.6) is much higher than that for (0.6, 0.9), as shown in Fig. 14b, c. However, the T w of (0.9, 0.6) is 1.5 K lower than that of (0.9, 0.9), with the average temperature of E 2 for (0.9, 0.6) being 0.7 K greater than the latter. Figure 16 shows the stream functions for the four cases, in which we can see that the flow pattern in E 1 is similar to all four cases. The stream function values in E 2 are less than half of those in E 1 , which means the flow is weak.

Case 1
The temperature profiles of different walls are shown in Fig. 17a and b, in which we can see that T 1 and T 3 increase quickly when λ increases from 0.05 to 0.5 W/(m·K). However, as λ continues to increase, the rate of increase in T 1 and T 3 slows and then becomes stable. In contrast, T 2 and T 4 decrease rapidly as λ increases from 0.05 to 0.5 W/ (m·K), then begins to stabilize. Figure 17c shows the variation in the heat flux in wall-1 with the thermal conductivity of the semitransparent wall. In the figure, we can see that the heat flux increases rapidly when λ ranges from 0.05 to 0.5 W/(m·K), then tends to remain constant as λ continues to increase. The vertical velocity distributions at H/2 for different thermal conductivities are shown in Fig. 17d. If we compare the vertical velocities in E 1 , the lowest vertical velocity occurs at 0.05 W/(m·K) and the others are almost the same. The vertical velocity in E 2 is the highest.

Case 2
As shown in Fig. 18a and b, T 1 increases quickly when λ increases from 0.05 to 0.5 W/(m·K). However, as λ continues to increase, the change in T 1 slows and tends to become stable. In contrast, T 2 , T 3 , and T 4 decrease rapidly as λ increases from 0.05 to 0.5 W/(m·K), and then begin to stabilize. In Fig. 18c, when λ increases from 0.05 to 0.5 W/ (m·K), T w decreases quickly. As λ continues to increase, the change in T 1 slows and tends to become stable. T 2 , T 3 , T 4 , and T w demonstrate the same pattern, which indicates that a higher temperature distribution would be achieved when λ decreases below a critical value. When the thermal conductivity of a semitransparent wall is greater than 0.5 W/(m·K), the change is weak. Figure 18d shows the vertical velocity distributions at H/2 for different thermal conductivities, in which we can see that the velocity is low for λ = 0.05 W/(m·K) in the two parts, and little difference is observed in the others.

Conclusion
In this paper, we presented our study of the natural convection flow field and temperature distribution in a rectangular enclosure with semitransparent walls. Two different cases were considered based on their boundary conditions, the first was the isothermal for an opaque wall, and the other was the radiation flux on the left semitransparent wall. RNG k-ε model was adopted to simulate the turbulent flow in the enclosure. To compute the radiation heat transfer in a semitransparent medium, the discrete ordinates model was used. Then, we compared enclosures with single and double semitransparent walls and analyzed the different results obtained using semitransparent and opaque partitions. In addition, the effects of the transmittance and thermal conductivity of a semitransparent wall on the fluid flow and temperature distribution in an enclosure have been investigated. From our simulation results, we can draw the following major conclusions.
(1) The presence of a semitransparent partition has a significant impact on the flow field and temperature distribution, as compared to an enclosure without any partition. This result indicates that a semitransparent partition is beneficial for reducing heat loss, as shown in Case 1, and for obtaining a higher temperature distribution, as shown in Case 2. (2) From our comparison of an enclosure with either semitransparent or opaque walls, the results indicate that semitransparency allows radiation to pass through the material, thus enhancing the air temperature distribution in the enclosure. (3) The transmittance of a semitransparent wall has a great effect on the thermal and flow characteristics in an enclosure. (4) The change in wall temperature is significant, with thermal conductivity values ranging from 0.05 to 0.5 W/ (m·K), and is slight with values ranging from 0.5 to 10 W/(m·K) for both cases. For Case 1, the heat flux also increases rapidly for λ values ranging from 0.05 to 0.5 W/(m·K) and tends to become constant as λ continues to increase.