Study on the Effect of Hole Size of Trombe Wall in the Presence of Phase Change Material for Different Times of a Day in Winter and Summer

: In this article, a numerical study is performed on a Trobme wall in a tropical city for two seasons, summer and winter. A 1 × 1.5 m Trobme wall with a thickness of 15 cm is designed and analyzed. A 1-inch-diameter tube ﬁlled with PCM is used to enhance efﬁciency. The wall is analyzed at different times of the day for the two cold and hot seasons for different sizes of wall holes in the range of 70 to 17.5 cm when the wall height is 20 cm. A ﬂuid simulation software is employed for the simulations. The problem variables include different hours of the day in the two cold and hot seasons, the presence or absence of PCM, as well as the size of the wall hole. The results of this simulation demonstrate that the maximum outlet temperature of the Trobme wall occurs at 2 P.M. Using PCM on the wall can allow the wall to operate for longer hours in the afternoon. However, the use of PCM reduces the outlet wall temperature in the morning. The smaller the size of the wall hole, the more air can be expelled from the wall. c ef during respectively. They also show that the c ef f (T) of PCM is directly proportional to the L-H of melting and inversely related to the melting TEM range, ∆ T k during the phase change period. T 1 K is the starting melting TEM of PCM, and T 2 K is the TEM at which PCM is completely melted while CHA.


Introduction
One of the most significant problems facing humanity in the future is energy challenges [1,2]. In the past, numerous researchers have looked into various options for reducing human energy usage [3][4][5][6][7]. With regard to the use of nanofluids, optimizations have been made, and some examples are given in references [8][9][10][11][12][13]. Some investigators analyzed the effect of using different fins to enhance the equipment efficiency. Many academics have regarded solar energy as one of the possible renewable energy sources [14][15][16][17][18]. One of the most important energy consumers is in buildings, where the researchers have aimed to reduce their energy consumption [19][20][21][22][23]. One approach to minimize energy usage in buildings is to employ solar energy. A Trobme wall (Tr-W) is a type of thermal storage wall that consists of a dark wall made of building materials and is covered with vertical glass. It is known as a storage and solar heating wall [24,25]. The Tr-W provides solar energy application, ventilation, and thermal comfort in buildings for different climatic regions [26]. The Tr-W's role is to absorb the solar beam and transform it into energy,

Problem Statement
The Tr-W studied in the present study is shown in Figure 1 for winter (left) and summer (right). The different parts of the Tr-W, including their thickness and dimensions, are presented in Table 1. There is an air passage hole at the top and bottom of the Tr-W, where its height is 20 cm (Figure 2). The width of the hole has three different sizes of 17.5, 35, and 70 cm ( Figure 3). Heat flux is applied to the back of the wall due to sunlight. Due to the natural convection, the air inside the wall moves. A tube with 1 inch in diameter filled with PCM is placed in the middle of the Tr-W horizontally. The wall of the tube is made of aluminum. Table 2 presents the properties of the PCM.  The different parts of the Tr-W, including their thickness and dimensions, are presented in Table 1. There is an air passage hole at the top and bottom of the Tr-W, where its height is 20 cm (Figure 2). The width of the hole has three different sizes of 17.5, 35, and 70 cm ( Figure 3). Heat flux is applied to the back of the wall due to sunlight. Due to the natural convection, the air inside the wall moves. A tube with 1 inch in diameter filled with PCM is placed in the middle of the Tr-W horizontally. The wall of the tube is made of aluminum. Table 2 presents the properties of the PCM.

Problem Statement
The Tr-W studied in the present study is shown in Figure 1 for winter (left) and summer (right). The different parts of the Tr-W, including their thickness and dimensions, are presented in Table 1. There is an air passage hole at the top and bottom of the Tr-W, where its height is 20 cm (Figure 2). The width of the hole has three different sizes of 17.5, 35, and 70 cm ( Figure 3). Heat flux is applied to the back of the wall due to sunlight. Due to the natural convection, the air inside the wall moves. A tube with 1 inch in diameter filled with PCM is placed in the middle of the Tr-W horizontally. The wall of the tube is made of aluminum. Table 2 presents the properties of the PCM.

Governing Equations
Due to the high cost of experimental measurements, it is preferable to perform experiments on a model with a smaller scale than the original version. The most basic equations governing the fluid flow are conservation equations, including mass, momentum, and energy equations. The basis of the CFD technique is the use of numerical methods to solve conservation equations in the geometric domain of the flow system and to find the flow characteristics including velocity, pressure, TEM, concentration, and other flow properties.

Airflow and Wall Equations
In the present study, the Rayleigh number is 5.3 × 10 8 , indicating the turbulence flow regime. Thus, the k-ε realizable model is used. The continuity, momentum, and energy equations are as follows, respectively.

Governing Equations
Due to the high cost of experimental measurements, it is preferable to perform experiments on a model with a smaller scale than the original version. The most basic equations governing the fluid flow are conservation equations, including mass, momentum, and energy equations. The basis of the CFD technique is the use of numerical methods to solve conservation equations in the geometric domain of the flow system and to find the flow characteristics including velocity, pressure, TEM, concentration, and other flow properties.

Airflow and Wall Equations
In the present study, the Rayleigh number is 5.3 × 10 8 , indicating the turbulence flow regime. Thus, the k-ε realizable model is used. The continuity, momentum, and energy equations are as follows, respectively.

∂ρ ∂t
where k e f f (W/mK) is the effective thermal conductivity, τ ij e f f (N/m 2 ) is the effective stress tensor, and µ e f f (Pa.s) is the effective viscosity. The fluid motion is due to natural convection due to the body force applied on the fluid and the creation of a density gradient, resulting in the buoyancy force. The gradient of density is due to the gradient of TEM, and the body force is due to the gravitational acceleration. Because convective flow velocities are generally slower than forced convection, the convection heat transfer rate is also lower. The boundary layer of free convection is not limited to a laminar flow. Free convection flow is generally due to thermal instability, meaning that the warmer and lighter fluid moves upward and the colder and heavier fluid moves downward. However, hydrodynamic instability occurs in the flow similar to forced convection. Hence, small disturbances in the flow may be amplified, and the flow regime changes to a turbulent one. The conversion of the flow regime in the boundary layer of free convection depends on the ratio of the buoyancy force to the viscose force. The conversion of the flow regime is usually related to the Rayleigh number, which is defined as the product of the Grashof number and the Prandtl number [46].

PCM Equations
The three-dimensional thermal conductivity equation in PCM can be expressed as follows: where c p = c e f f (T) (J/kg·K). In this scheme, the L-H in the energy equation is modelled as an artificial behavior of specific heat within, where phase transition happens [47]. This amount of c e f f prevents the TEM from rising or lessening during the phase transition. Due to the nature of the artificial behavior, special attention should be paid to c e f f (T) to accurately predict the phase change kinetics and to accurately quantify the energy stored/released during charge/discharge (CHA/DCHA) cycles.
The c e f f (T) CHA/DCHA curves empirically obtained for the PCM-based product have been measured in the imitations [48].
Equations (5) and (6) express the EHC method, respectively. The c * m,K and c m,K represent the maximum amounts of c e f f (T) due to L-H during melting, respectively. They also show that the c e f f (T) of PCM is directly proportional to the L-H of melting and inversely related to the melting TEM range, ∆T k during the phase change period. T 1K is the starting melting TEM of PCM, and T 2K is the TEM at which PCM is completely melted while CHA.
According to Liu et al. [35], the EHC method does not work well in solving phasechange problems that have a very low phase change TEM range and cannot be used for the cases where phase change occurs at a constant TEM. Saadi and Zhai [19] also stated that to ensure accuracy and consider the L-H, fine-grid and small-time steps are needed. Time steps must be limited to ensure the accuracy of the simulation, because due to the low-TEM range of the phase change, the c e f f changes rapidly. Another important aspect of this method is its flexibility in defining the c e f f only by changing the values of c p,s , c p,l , T K , T 1K , T 2K , and L K during CHA and c p,s , c p,l , T s , T 1s , T 2s , and L s during DCHA. Finally, the TEM is the only dependent variable that is solved in the EHC method. This method allows using the same governing equations (Equation (4)) for liquid and solid phases apart from tracking the position of the melting zone. In fact, the melted PCM fraction is explicitly the average calculated for each time step as a function of the calculated TEM T as follows: Equation (9) is used for the CHA and DCHA processes of EHC using artificial profiles of c e f f (T) and AHS. At the time of determining f k and f s using Equation (10), the results are reported as a downward deviation of f(T) in the two-phase region. However, this two-part equation is retained for more general reasons. For example, in cases where the PCMs have different values of f K and f s , the model maker may want to specify such maximum values, especially of the melt fraction.
The value of c e f f is known for each TEM at any time in the simulations by using c e f f (T) obtained from the experimental measurements. Therefore, for a given TEM range of T 2 − T 1 that is small enough to consider a constant average value of c e f f in the molten PCM during CHA mode, the following relation can be obtained: When the PCM is in solid phase T ≤ T 1K that to ensure accuracy and consider the L-H, fine-grid and small-time steps are needed. Time steps must be limited to ensure the accuracy of the simulation, because due to the low-TEM range of the phase change, the changes rapidly. Another important aspect of this method is its flexibility in defining the only by changing the values of , , , , , , , and during CHA and , , , , , , , and during DCHA. Finally, the TEM is the only dependent variable that is solved in the EHC method. This method allows using the same governing equations (Equation (4)) for liquid and solid phases apart from tracking the position of the melting zone. In fact, the melted PCM fraction is explicitly the average calculated for each time step as a function of the calculated TEM T as follows: Equation (9) is used for the CHA and DCHA processes of EHC using artificial profiles of and AHS. At the time of determining and using Equation (10), the results are reported as a downward deviation of f(T) in the two-phase region. However, this twopart equation is retained for more general reasons. For example, in cases where the PCMs have different values of and , the model maker may want to specify such maximum values, especially of the melt fraction.
The value of is known for each TEM at any time in the simulations by using obtained from the experimental measurements. Therefore, for a given TEM range of − that is small enough to consider a constant average value of in the molten PCM during CHA mode, the following relation can be obtained: When the PCM is in solid phase T ≤ ، f(T) = 0.. When T ≥ (molten state), f(T) = 1. To simulate PCM CHA and DCHA processes at the TEM range of , , the term in Equation (11) must be replaced by the terms and f(T) = 0 (when T ≤ ) and f(T) = 1 (when T ≥ ). The values of heat flux applied on the Tr-W are given in Table 3 at different hours of the day in winter and summer. f(T) = 0. When T ≥ T 2K (molten state), f(T) = 1. To simulate PCM CHA and DCHA processes at the TEM range of T 1s , T 2s , the term L m in Equation (11) must be replaced by the terms L s and f(T) = 0 (when T ≤ T 1s ) and f(T) = 1 (when T ≥ T 2s ).
The values of heat flux applied on the Tr-W are given in Table 3 at different hours of the day in winter and summer.

Numerical Method, Boundary Conditions, Validation, and Grid Study
Since in any numerical analysis, grid independence is necessary, the simulations are first performed for different grid resolutions. Then, the grid with 1,675,460 nodes is selected for the solutions. Table 4 shows an example of studies performed on the number of grid points for the outlet TEM of the Tr-W to clarify the accuracy of the selected mesh in the presence of PCM at noon in the summer when the size of the hole is 17.5 cm. After generating the grid and applying the boundary conditions on the geometric surfaces using the software, the grid is imported by industry-leading fluid simulation software. Then, the problem is simulated by solving the governing equations (Equations (1)-(3)), applying the boundary conditions, and employing the Boussinesq approximation. The control volume method is used to solve the equations. There is also a separate solution on the geometry of the PCM-filled tube for a more accurate solution. To solve the problem for each different hour of the day, the amount of heat flux and other boundary conditions are imposed on the wall, and the results are obtained. One of the boundary conditions used for the wall is a constant thermal flux. Additionally, the TEM of the air entering the Tr-W is always considered to be 23.5 • C. The boundary condition of the exit from the Tr-W hole is also considered to be a constant pressure equal to the amount of room air pressure, i.e., 1 atm. The tube that is filled with PCM is placed in the wall, and the process is simulated separately using a constant TEM boundary condition.
In this paper, the volume control method is used to algebraize the differential equations. In this method, different terms of the differential equation are written algebraically at different points of the solution domain. For this purpose, the solution field is first divided into a number of control volumes in such a way that each node is surrounded by a control volume, and at the same time, the control volumes do not have common volumes with each other. The differential equation is then integrated into each volume control. To calculate integrals, it is sometimes necessary to assume functions for dependent variables. Thus, algebraic equations created satisfy the conservation of mass, energy, and momentum and are consistent with the physics of the problem. The advantage of using the control volume method is that the conservation law is established for quantities such as mass, momentum, and energy in each control volume. This is valid for all nodes and even when the number of nodes decreases. Therefore, even the answer for the large grid leads to the exact balance. Thus, the solution domain is of particular importance. The algebraic steps in the control volume method are as follows:

1.
Selection of the appropriate control volume; 2.
Integration of the equation on the control volume; 3.
Selection of the appropriate profile on the control volume; 4.
Extraction of an algebraic equation.
The simulations are verified by comparing the present results with the experimental results of Attalla et al. [49] who studied a Tr-W. To compare two works, the amount of wall TEM in the middle of the wall is compared at different times of the day (Table 5). It can be seen that the wall TEM results are close to each other under constant conditions.

Results and Discussion
Initially, PCM melting contours in the tube are presented. Figure 4 shows the PCM melting contours at different times from 0 to 5000 s. It can be seen that the PCM inside the tube begins to melt over time. Melting starts from the outer surface of the tube and moves inwards. First, the upper parts of the pipe are melted, and then, parts of solid PCM are melted at the bottom of the tube due to free convection in the PCM. It can be seen that the entire PCM is not melted after 5000 s from the beginning of the process, which is due to the large diameter and volume of the PCM inside the tube. This can cause the freezing time to be longer. Hence, the heat is transferred from the tube to the air inside the wall for a longer time.

Results and Discussion
Initially, PCM melting contours in the tube are presented. Figure 4 shows the PCM melting contours at different times from 0 to 5000 s. It can be seen that the PCM inside the tube begins to melt over time. Melting starts from the outer surface of the tube and moves inwards. First, the upper parts of the pipe are melted, and then, parts of solid PCM are melted at the bottom of the tube due to free convection in the PCM. It can be seen that the entire PCM is not melted after 5000 s from the beginning of the process, which is due to the large diameter and volume of the PCM inside the tube. This can cause the freezing time to be longer. Hence, the heat is transferred from the tube to the air inside the wall for a longer time.   Figure 5 demonstrates the TEM contours of a PCM-filled tube from 0 to 5000 s with a time interval of 1000 s. It is seen that the tube is full of cold PCM, and its TEM is uniformly cold. In this case, all PCM inside the tube is solid. It can be seen that the outer wall is heated due to solar radiation, leading to that the PCM is melted. The hot TEM created in the wall of the tube enters the tube over time and heats all the PCM inside it. Then,  Figure 5 demonstrates the TEM contours of a PCM-filled tube from 0 to 5000 s with a time interval of 1000 s. It is seen that the tube is full of cold PCM, and its TEM is uniformly cold. In this case, all PCM inside the tube is solid. It can be seen that the outer wall is heated due to solar radiation, leading to that the PCM is melted. The hot TEM created in the wall of the tube enters the tube over time and heats all the PCM inside it. Then, more of the tube space is marked by the red-color contour. Additionally, the middle parts of the PCM inside the tube remain solid and not melted over time. In this area, the tube is still cold and the PCM is solid. t = 5000 s t = 4000 s Figure 4. PCM melting contours in a tube filled with PCM from 0 to 5000 s with a time interval of 1000 s. Figure 5 demonstrates the TEM contours of a PCM-filled tube from 0 to 5000 s with a time interval of 1000 s. It is seen that the tube is full of cold PCM, and its TEM is uniformly cold. In this case, all PCM inside the tube is solid. It can be seen that the outer wall is heated due to solar radiation, leading to that the PCM is melted. The hot TEM created in the wall of the tube enters the tube over time and heats all the PCM inside it. Then, more of the tube space is marked by the red-color contour. Additionally, the middle parts of the PCM inside the tube remain solid and not melted over time. In this area, the tube is still cold and the PCM is solid.   Figure 6 shows the TEM in three parts of the wall, including the outlet, the absorber plate, and the glass at different times of the day with and without PCM in the summer. Since radiation is directly related to the TEM of different points, the amount of TEM during the day is enhanced with the radiation. As time passes and the amount of solar radiation is reduced, the amount of wall TEM decreases. Another important point is the strong dependence of the outlet TEM on the absorber plate TEM. Therefore, as the TEM of the adsorbent plate is enhanced, the outlet TEM increases. Additionally, since the PCM is  Figure 6 shows the TEM in three parts of the wall, including the outlet, the absorber plate, and the glass at different times of the day with and without PCM in the summer. Since radiation is directly related to the TEM of different points, the amount of TEM during the day is enhanced with the radiation. As time passes and the amount of solar radiation is reduced, the amount of wall TEM decreases. Another important point is the strong dependence of the outlet TEM on the absorber plate TEM. Therefore, as the TEM of the adsorbent plate is enhanced, the outlet TEM increases. Additionally, since the PCM is used on the back of the absorber plate, the TEM of the absorber plate has the highest value for all cases. Therefore, the output TEM is in the best condition. It can be seen that the use of PCM causes the TEM to reduce in the morning, but on the other hand, the outlet TEM as well as the TEM of the absorber plate and glass increase in the afternoon. Especially, the increment in the outlet TEM in summer makes the airflow in the Tr-W faster, and as a result, more airflow exits from the house. used on the back of the absorber plate, the TEM of the absorber plate has the highest value for all cases. Therefore, the output TEM is in the best condition. It can be seen that the use of PCM causes the TEM to reduce in the morning, but on the other hand, the outlet TEM as well as the TEM of the absorber plate and glass increase in the afternoon. Especially, the increment in the outlet TEM in summer makes the airflow in the Tr-W faster, and as a result, more airflow exits from the house.  Figure 7 illustrates the TEM in three parts of the wall, including the outlet, the absorber plate, and the glass at different times of the day with and without PCM in the winter. It can be seen that all TEMs are reduced due to the reduction in solar radiation. The presence of PCM causes the wall TEMs, especially the outlet TEM in the afternoon, to rise, and the wall is able to heat the house for more hours. Since the PCM is located in the tube inside the Tr-W and the solar heat flux is reduced in winter, the PCM reaches the threshold of a melting point later. The operation of the Tr-W with PCM is a function of the radiation intensity of the region. Thus, the PCM used in the wall should be selected based on the radiation intensity and TEM to employ the maximum L-H energy of the PCM.  Figure 7 illustrates the TEM in three parts of the wall, including the outlet, the absorber plate, and the glass at different times of the day with and without PCM in the winter. It can be seen that all TEMs are reduced due to the reduction in solar radiation. The presence of PCM causes the wall TEMs, especially the outlet TEM in the afternoon, to rise, and the wall is able to heat the house for more hours. Since the PCM is located in the tube inside the Tr-W and the solar heat flux is reduced in winter, the PCM reaches the threshold of a melting point later. The operation of the Tr-W with PCM is a function of the radiation intensity of the region. Thus, the PCM used in the wall should be selected based on the radiation intensity and TEM to employ the maximum L-H energy of the PCM. Figure 8 shows the outlet TEM variations of the Tr-W for different dimensions of the wall hole without PCM at different times of the day in summer. The size of the hole is such that both valves become equally smaller in both cases. In the first case, the length of the inlet and outlet valves is 70 cm. In the next two cases, the length is 35 cm and 17.5 cm. For all cases, the width is kept constant as 20 cm. In each case, the size of the hole is halved compared to the previous case. As can be seen, the smaller the outlet valve, the higher the TEM. At 2 P.M., the rate of TEM enhancement from a 70 cm hole to a 35 cm hole is 3.7% and from a 35 cm hole to a 17.5 cm hole is 3.2%. The reason for this is the reduction in pressure at the outlet section. Additionally, the enhancement in the velocity accelerates the vortices, which improves the heat transfer.
presence of PCM causes the wall TEMs, especially the outlet TEM in the afternoon, to rise, and the wall is able to heat the house for more hours. Since the PCM is located in the tube inside the Tr-W and the solar heat flux is reduced in winter, the PCM reaches the threshold of a melting point later. The operation of the Tr-W with PCM is a function of the radiation intensity of the region. Thus, the PCM used in the wall should be selected based on the radiation intensity and TEM to employ the maximum L-H energy of the PCM.  Figure 8 shows the outlet TEM variations of the Tr-W for different dimensions of the wall hole without PCM at different times of the day in summer. The size of the hole is such that both valves become equally smaller in both cases. In the first case, the length of the inlet and outlet valves is 70 cm. In the next two cases, the length is 35 cm and 17.5 cm. For all cases, the width is kept constant as 20 cm. In each case, the size of the hole is halved compared to the previous case. As can be seen, the smaller the outlet valve, the higher the TEM. At 2 PM, the rate of TEM enhancement from a 70 cm hole to a 35 cm hole is 3.7% and from a 35 cm hole to a 17.5 cm hole is 3.2%. The reason for this is the reduction in pressure at the outlet section. Additionally, the enhancement in the velocity accelerates the vortices, which improves the heat transfer.     Figure 10 illustrates the outlet TEM variations from the Tr-W for different dimensions of the wall hole in the presence of PCM at different times of the day in winter. The TEM at the wall outlet is reduced due to the reduction in solar radiation in winter. In the middle hours of the day, the amount of radiation is higher, and therefore, the output TEM is higher. It is seen that the smaller the hole in the wall, the higher the outlet TEM of the air from the wall. The effect of the size of the outlet and inlet valve on the outlet TEM of the wall is presented in Figure 11 for different times of the day in the presence of PCM. Adding PCM to the wall reduces the amount of output TEM in the morning but increases the amount of output TEM in the afternoon. It can be seen that the shrinkage of the air outlet hole also enhances the amount of TEM in the outlet of the wall. At 2 PM, the increment in the TEM from a 70 cm hole to a 35 cm hole is 4.5% and from a 35 cm hole to a 17.5 cm hole is 3.7%.  Figure 10 illustrates the outlet TEM variations from the Tr-W for different dimensions of the wall hole in the presence of PCM at different times of the day in winter. The TEM at the wall outlet is reduced due to the reduction in solar radiation in winter. In the middle hours of the day, the amount of radiation is higher, and therefore, the output TEM is higher. It is seen that the smaller the hole in the wall, the higher the outlet TEM of the air from the wall.  Figure 10 illustrates the outlet TEM variations from the Tr-W for different dimensions of the wall hole in the presence of PCM at different times of the day in winter. The TEM at the wall outlet is reduced due to the reduction in solar radiation in winter. In the middle hours of the day, the amount of radiation is higher, and therefore, the output TEM is higher. It is seen that the smaller the hole in the wall, the higher the outlet TEM of the air from the wall. The effect of the size of the outlet and inlet valve on the outlet TEM of the wall is presented in Figure 11 for different times of the day in the presence of PCM. Adding PCM to the wall reduces the amount of output TEM in the morning but increases the amount of output TEM in the afternoon. It can be seen that the shrinkage of the air outlet hole also enhances the amount of TEM in the outlet of the wall. At 2 PM, the increment in the TEM from a 70 cm hole to a 35 cm hole is 4.5% and from a 35 cm hole to a 17.5 cm hole is 3.7%. The effect of the size of the outlet and inlet valve on the outlet TEM of the wall is presented in Figure 11 for different times of the day in the presence of PCM. Adding PCM to the wall reduces the amount of output TEM in the morning but increases the amount of output TEM in the afternoon. It can be seen that the shrinkage of the air outlet hole also enhances the amount of TEM in the outlet of the wall. At 2 P.M., the increment in the TEM from a 70 cm hole to a 35 cm hole is 4.5% and from a 35 cm hole to a 17.5 cm hole is 3.7%.