Thermohydraulic Performance and Entropy Generation of a Triple-Pass Solar Air Heater with Three Inlets

In this paper, a triple-pass solar air heater with three inlets is analytically investigated. The effects of airflow ratios of the second and third passes (ranging from 0 to 0.4), and the Reynolds number of the third pass (ranging from 8000 to 18,000) on the thermohydraulic efficiency and entropy generation are assessed. An absorber plate equipped with rectangular fins on both sides is used to enhance heat transfer. The air temperature change in the passes is represented by ordinary differential equations and solved by numerical integration. The results demonstrate that the effect of the third pass airflow ratio on the thermohydraulic efficiency and entropy generation is more significant than that of the second pass airflow ratio. The difference in air temperature through the collector shows an insignificant reduction, but the air pressure loss is only 50% compared with that of a traditional triple-pass solar air heater. Increasing the air flow ratios dramatically reduces entropy generation. Multi-objective optimization found a Reynolds number of 11,156 for both the airflow ratio of the second pass of 0.258 and airflow ratio of the third pass of 0.036 to be the an optimal value to achieve maximum thermohydraulic efficiency and minimum entropy generation.


Introduction
Hot air is an essential source of thermal energy for industrial fields such as regarding drying, pre-heating of bottles, textile production processes, painting processes, and conditioning tobacco. For moderate-temperature heating, a solar air heater (SAH) can be employed, which reduces fossil fuel energy consumption [1]. SAHs are the simplest energy conversion devices, as they can be made from locally available materials. However, SAHs reveal some limitations that arise from the heat transfer fluid of air, which cannot be stored as hot water can, and the low convection heat transfer coefficient of air. Heat transfer improvement between the airflow and absorption plate of a SAH is a topic of constant interest to researchers. Inserts including rib roughness, fins, porous media, or baffles in the SAH duct have led to increases in the heat transfer rate due to the elimination of the laminar sub-layer close to the absorber plate and airflow reconfiguration [2,3]. The small convection heat transfer coefficient of the air leads to a high absorber plate temperature, resulting in a large top heat loss of the SAH. To reduce this heat loss, airflow with multiple passes is structured such that the air receives heat from both sides of the absorber plate, glass, and back plate. [4,5]. Tuncer et al. [6] used triple and quadruple-pass SAHs to dry food. They reported that pressure losses of 3.8 and 4.5 Pa were obtained for triple and quadruple-pass SAHs, respectively. In addition, air temperature increases of 18.2 and 20.1 • C were observed in the triple and quadruple-pass SAHs, respectively.
Khanlari et al. [7] dried municipal sewage sludge using triple and quadruple-pass SAHs. A V-groove absorber plate was designed to enhance heat transfer. Their experimental results proved that the collector efficiency can be up to 81.7%. Potato drying using a triple-pass SAH has been experimentally examined by Kesavan et al. [8]. A wire mesh in the second pass and sand as a thermal storage medium were adopted in order to increase the heat transfer rate and extend the drying time. The highest thermal and exergy efficiencies of 66% and 87%, respectively, were deduced. Sopian et al. [9] discussed a double-pass solar collector with porous media in the second pass. They confirmed that the collector had a higher thermal efficiency compared to that of a single-pass SAH.
Another modification to multi-pass SAHs is the adjustment of the air flow in each channel by recycling flow or adding air inlets. This correction can improve collector performance, as the multi-pass air collector has a large pressure loss and the temperature difference between the fluid and the surface in each channel differs. Table 1 summarizes studies on the flow patterns changed in the multi-pass SAHs. Ho et al. [10] explored a double-pass SAH with recycled flow. Wire mesh, as a heat transfer enhancement media, was inserted in the channel between the absorber plate and back plate. They reported an optimal recycle ratio of 1.0 to obtain the highest collector efficiency improvement. Yeh and Ho [11] divided the space between the absorber plate and back plate into two air channels. Longitudinal fins were attached to the absorber plate in order to intensify the heat exchange. They concluded that the collector efficiency can be improved through an increase in the reflux ratio. Singh and Dhiman [12] added wire mesh packing in the main air channel between a glass cover and absorber plate. The recycle ratio of 1.8 yielded the largest thermohydraulic efficiency. Ho et al. [13] utilized corrugated absorber and back plates to improve heat transfer in a double-pass SAH with recycled flow. It was shown that the thermal performance improved with an increasing recycle ratio and decreasing air flow rate. The optimum recycle ratio of 0.5 was found considering the trade-off of heat transfer improvement and pressure loss penalty. Abo-Elfadl et al. [14] recently added an air inlet port in the second pass of a double-pass SAH. The absorber plate was equipped with pin fins for heat transfer augmentation. They confirmed that 66.7% of the air supply at the port led to the highest energy and exergy efficiencies. More recently, Ahmadkhani et al. [15] investigated two flow configurations, including recycled flow to the back plate or upper glass cover. A packed bed matrix was utilized in the main air channel to perform the parametric study. It was identified that the recycled flow to the back plate led to a higher air temperature rise. Table 1. Summary of multi-pass SAH studies on flow pattern change.

Investigators (Year) Flow Pattern Heat Transfer Enhancement Main Finding
Ho et al., 2013 [10] Energies 2021, 14, 6399 2 of 20 °C were observed in the triple and quadruple-pass SAHs, respectively. Khanlari et al. [7] dried municipal sewage sludge using triple and quadruple-pass SAHs. A V-groove absorber plate was designed to enhance heat transfer. Their experimental results proved that the collector efficiency can be up to 81.7%. Potato drying using a triple-pass SAH has been experimentally examined by Kesavan et al. [8]. A wire mesh in the second pass and sand as a thermal storage medium were adopted in order to increase the heat transfer rate and extend the drying time. The highest thermal and exergy efficiencies of 66% and 87%, respectively, were deduced. Sopian et al. [9] discussed a double-pass solar collector with porous media in the second pass. They confirmed that the collector had a higher thermal efficiency compared to that of a single-pass SAH. Another modification to multi-pass SAHs is the adjustment of the air flow in each channel by recycling flow or adding air inlets. This correction can improve collector performance, as the multi-pass air collector has a large pressure loss and the temperature difference between the fluid and the surface in each channel differs. Table 1 summarizes studies on the flow patterns changed in the multi-pass SAHs. Ho et al. [10] explored a double-pass SAH with recycled flow. Wire mesh, as a heat transfer enhancement media, was inserted in the channel between the absorber plate and back plate. They reported an optimal recycle ratio of 1.0 to obtain the highest collector efficiency improvement. Yeh and Ho [11] divided the space between the absorber plate and back plate into two air channels. Longitudinal fins were attached to the absorber plate in order to intensify the heat exchange. They concluded that the collector efficiency can be improved through an increase in the reflux ratio. Singh and Dhiman [12] added wire mesh packing in the main air channel between a glass cover and absorber plate. The recycle ratio of 1.8 yielded the largest thermohydraulic efficiency. Ho et al. [13] utilized corrugated absorber and back plates to improve heat transfer in a double-pass SAH with recycled flow. It was shown that the thermal performance improved with an increasing recycle ratio and decreasing air flow rate. The optimum recycle ratio of 0.5 was found considering the trade-off of heat transfer improvement and pressure loss penalty. Abo-Elfadl et al. [14] recently added an air inlet port in the second pass of a double-pass SAH. The absorber plate was equipped with pin fins for heat transfer augmentation. They confirmed that 66.7% of the air supply at the port led to the highest energy and exergy efficiencies. More recently, Ahmadkhani et al. [15] investigated two flow configurations, including recycled flow to the back plate or upper glass cover. A packed bed matrix was utilized in the main air channel to perform the parametric study. It was identified that the recycled flow to the back plate led to a higher air temperature rise. °C were observed in the triple and quadruple-pass SAHs, respectively. Khanlari et al. [7] dried municipal sewage sludge using triple and quadruple-pass SAHs. A V-groove absorber plate was designed to enhance heat transfer. Their experimental results proved that the collector efficiency can be up to 81.7%. Potato drying using a triple-pass SAH has been experimentally examined by Kesavan et al. [8]. A wire mesh in the second pass and sand as a thermal storage medium were adopted in order to increase the heat transfer rate and extend the drying time. The highest thermal and exergy efficiencies of 66% and 87%, respectively, were deduced. Sopian et al. [9] discussed a double-pass solar collector with porous media in the second pass. They confirmed that the collector had a higher thermal efficiency compared to that of a single-pass SAH. Another modification to multi-pass SAHs is the adjustment of the air flow in each channel by recycling flow or adding air inlets. This correction can improve collector performance, as the multi-pass air collector has a large pressure loss and the temperature difference between the fluid and the surface in each channel differs. Table 1 summarizes studies on the flow patterns changed in the multi-pass SAHs. Ho et al. [10] explored a double-pass SAH with recycled flow. Wire mesh, as a heat transfer enhancement media, was inserted in the channel between the absorber plate and back plate. They reported an optimal recycle ratio of 1.0 to obtain the highest collector efficiency improvement. Yeh and Ho [11] divided the space between the absorber plate and back plate into two air channels. Longitudinal fins were attached to the absorber plate in order to intensify the heat exchange. They concluded that the collector efficiency can be improved through an increase in the reflux ratio. Singh and Dhiman [12] added wire mesh packing in the main air channel between a glass cover and absorber plate. The recycle ratio of 1.8 yielded the largest thermohydraulic efficiency. Ho et al. [13] utilized corrugated absorber and back plates to improve heat transfer in a double-pass SAH with recycled flow. It was shown that the thermal performance improved with an increasing recycle ratio and decreasing air flow rate. The optimum recycle ratio of 0.5 was found considering the trade-off of heat transfer improvement and pressure loss penalty. Abo-Elfadl et al. [14] recently added an air inlet port in the second pass of a double-pass SAH. The absorber plate was equipped with pin fins for heat transfer augmentation. They confirmed that 66.7% of the air supply at the port led to the highest energy and exergy efficiencies. More recently, Ahmadkhani et al. [15] investigated two flow configurations, including recycled flow to the back plate or upper glass cover. A packed bed matrix was utilized in the main air channel to perform the parametric study. It was identified that the recycled flow to the back plate led to a higher air temperature rise.  The above literature review indicates that multiple passes and various air flows within the passes can improve the thermohydraulic performance of an SAH. However, a study for triple-pass SAHs with air flow modification was not found. Furthermore, studies focused on the parameters related to the second law of thermodynamics are lacking. In this study, a triple-pass SAH with three inlets each at an air pass is investigated. The main aim of the present study was to determine the flow rate of each pass such that the collector efficiency can be augmented and the entropy generation lowered.

Model Description
The schematic diagram of a three-inlet triple-pass air collector is shown in Figure 1. In this study, the collector dimensions are fixed, including the collector length L, width W, and depth of an air channel D. The collector includes two glass covers, one absorber plate, and one back plate. The air travels through the surfaces in turn, forming three passes. Each pass has an air inlet with ambient temperature Ta, as shown in the figure. The air flow through the third pass is the total flow, with the mass flow rate . Let y and z be the additional air fractions supplied to passes 2 and 3, respectively. The air fraction through the first pass is (1 − y − z). The two sides of the absorber plate was equipped with longitudinal fins in order to enhance the heat transfer for the plate. The mathematical model is established with the following assumptions: - The flow in a channel is considered one-dimensional in the x-direction.

-
The flow is steady and the air fluid is incompressible. - The thermophysical properties of the air and the SAH are temperature-independent; and - The bottom plate and the edges of the SAH are perfectly insulated from the surrounding environment.
The heat balance equation for the glass, plates, and air in the three channels is as follows: The above literature review indicates that multiple passes and various air flows within the passes can improve the thermohydraulic performance of an SAH. However, a study for triple-pass SAHs with air flow modification was not found. Furthermore, studies focused on the parameters related to the second law of thermodynamics are lacking. In this study, a triple-pass SAH with three inlets each at an air pass is investigated. The main aim of the present study was to determine the flow rate of each pass such that the collector efficiency can be augmented and the entropy generation lowered.

Model Description
The schematic diagram of a three-inlet triple-pass air collector is shown in Figure 1. In this study, the collector dimensions are fixed, including the collector length L, width W, and depth of an air channel D. The collector includes two glass covers, one absorber plate, and one back plate. The air travels through the surfaces in turn, forming three passes. Each pass has an air inlet with ambient temperature Ta, as shown in the figure. The air flow through the third pass is the total flow, with the mass flow rate . Let y and z be the additional air fractions supplied to passes 2 and 3, respectively. The air fraction through the first pass is (1 − y − z). The two sides of the absorber plate was equipped with longitudinal fins in order to enhance the heat transfer for the plate. The mathematical model is established with the following assumptions: - The flow in a channel is considered one-dimensional in the x-direction. - The flow is steady and the air fluid is incompressible. - The thermophysical properties of the air and the SAH are temperature-independent; and - The bottom plate and the edges of the SAH are perfectly insulated from the surrounding environment.
The heat balance equation for the glass, plates, and air in the three channels is as follows:

Corrugated absorber and back plates
Optimal recycle ratio of 0.5 Abo-Elfadl et al., 2020 [14] Energies 2021, 14, 6399 3 of 20 Singh and Dhiman, 2018 [12] Wire mesh packing Optimal recycle ratio of 1. The above literature review indicates that multiple passes and various air flows within the passes can improve the thermohydraulic performance of an SAH. However, a study for triple-pass SAHs with air flow modification was not found. Furthermore, studies focused on the parameters related to the second law of thermodynamics are lacking. In this study, a triple-pass SAH with three inlets each at an air pass is investigated. The main aim of the present study was to determine the flow rate of each pass such that the collector efficiency can be augmented and the entropy generation lowered.

Model Description
The schematic diagram of a three-inlet triple-pass air collector is shown in Figure 1. In this study, the collector dimensions are fixed, including the collector length L, width W, and depth of an air channel D. The collector includes two glass covers, one absorber plate, and one back plate. The air travels through the surfaces in turn, forming three passes. Each pass has an air inlet with ambient temperature Ta, as shown in the figure. The air flow through the third pass is the total flow, with the mass flow rate . Let y and z be the additional air fractions supplied to passes 2 and 3, respectively. The air fraction through the first pass is (1 − y − z). The two sides of the absorber plate was equipped with longitudinal fins in order to enhance the heat transfer for the plate. The mathematical model is established with the following assumptions: - The flow in a channel is considered one-dimensional in the x-direction. - The flow is steady and the air fluid is incompressible. - The thermophysical properties of the air and the SAH are temperature-independent; and - The bottom plate and the edges of the SAH are perfectly insulated from the surrounding environment.
The heat balance equation for the glass, plates, and air in the three channels is as follows: The above literature review indicates that multiple passes and various air flows within the passes can improve the thermohydraulic performance of an SAH. However, a study for triple-pass SAHs with air flow modification was not found. Furthermore, studies focused on the parameters related to the second law of thermodynamics are lacking. In this study, a triple-pass SAH with three inlets each at an air pass is investigated. The main aim of the present study was to determine the flow rate of each pass such that the collector efficiency can be augmented and the entropy generation lowered.

Model Description
The schematic diagram of a three-inlet triple-pass air collector is shown in Figure 1. In this study, the collector dimensions are fixed, including the collector length L, width W, and depth of an air channel D. The collector includes two glass covers, one absorber plate, and one back plate. The air travels through the surfaces in turn, forming three passes. Each pass has an air inlet with ambient temperature Ta, as shown in the figure. The air flow through the third pass is the total flow, with the mass flow rate . Let y and z be the additional air fractions supplied to passes 2 and 3, respectively. The air fraction through the first pass is (1 − y − z). The two sides of the absorber plate was equipped with longitudinal fins in order to enhance the heat transfer for the plate. The mathematical model is established with the following assumptions: - The flow in a channel is considered one-dimensional in the x-direction. - The flow is steady and the air fluid is incompressible. - The thermophysical properties of the air and the SAH are temperature-independent; and - The bottom plate and the edges of the SAH are perfectly insulated from the surrounding environment.
The heat balance equation for the glass, plates, and air in the three channels is as follows:

Packed bed matrix
Recycle flow to back plate (upper figure) led to higher air temperature rise The above literature review indicates that multiple passes and various air flows within the passes can improve the thermohydraulic performance of an SAH. However, a study for triple-pass SAHs with air flow modification was not found. Furthermore, studies focused on the parameters related to the second law of thermodynamics are lacking. In this study, a triple-pass SAH with three inlets each at an air pass is investigated. The main aim of the present study was to determine the flow rate of each pass such that the collector efficiency can be augmented and the entropy generation lowered.

Model Description
The schematic diagram of a three-inlet triple-pass air collector is shown in Figure 1. In this study, the collector dimensions are fixed, including the collector length L, width W, and depth of an air channel D. The collector includes two glass covers, one absorber plate, and one back plate. The air travels through the surfaces in turn, forming three passes. Each pass has an air inlet with ambient temperature T a , as shown in the figure. The air flow through the third pass is the total flow, with the mass flow rate . m. Let y and z be the additional air fractions supplied to passes 2 and 3, respectively. The air fraction through the first pass is (1 − y − z). The two sides of the absorber plate was equipped with longitudinal fins in order to enhance the heat transfer for the plate. The mathematical model is established with the following assumptions: - The flow in a channel is considered one-dimensional in the x-direction.

-
The flow is steady and the air fluid is incompressible. - The thermophysical properties of the air and the SAH are temperature-independent; and - The bottom plate and the edges of the SAH are perfectly insulated from the surrounding environment.  Boundary conditions for the ordinary differential Equations (2), (4), and (6) are as follows: The temperature of the air entering the first pass (Tf1(x = 0)) is equal to the ambient temperature (Ta): The air entering the second pass with a flow fraction (1 − z) is a mixture of the air exiting the first pass with the flow fraction (1 − y − z) and the air from the second inlet with the flow fraction (y). The mixing temperature is calculated from the massweighted average as follows: -Similarly, the temperature boundary condition for the air entering the third pass can also be expressed as: The radiative heat transfer coefficient in the above system of equations can be evaluated through the following: -Radiation from glass 1 to the surroundings: -Radiation of glasses: ( ) -Radiation of glass 1 and the absorber plate: -Radiation of the absorber plate and back plate: Figure 1. Triple-pass solar air heater with three inlets and notation.
The heat balance equation for the glass, plates, and air in the three channels is as follows: -Glass 1: Solar irradiation absorbed by the glass balances the heat exchanged by convection and radiation from the top and bottom sides of the glass [4,16]: where the sky temperature is calculated by T sky = 0.0552T a 1.5 . - The heat gain of first pass air in an infinitesimal change (dx) is equal to the heat exchange by convection with glasses 1 and 2 as: Therefore, the variation in the first pass air temperature along the x-direction due to the convection heat transfer of the air with the two glasses is expressed as follows: -Similarly, for glass 2: -Similar to the change of the first pass air temperature, the air in the second pass exchanges heat with glass 2 and receives thermal energy by convection with the finned absorber plate. The convection heat transfer of the air with the fins and their prime surface is calculated through the temperature difference of the absorber plate (T p ) and the air (T f2 ) with the fin efficiency. Hence, the change in the second pass air temperature can be written as: where φ is the area-weighted fin efficiency. - The absorber plate can be expressed as: Energies 2021, 14, 6399 The change in the third pass air temperature can be expressed as: -The back plate-received radiation heat transfer from the absorber plate and transfer heat to the third pass air by convection is expressed as: Boundary conditions for the ordinary differential Equations (2), (4) and (6) are as follows: The temperature of the air entering the first pass (T f1(x = 0) ) is equal to the ambient temperature (T a ): - The air entering the second pass with a flow fraction (1 − z) is a mixture of the air exiting the first pass with the flow fraction (1 − y − z) and the air from the second inlet with the flow fraction (y). The mixing temperature is calculated from the massweighted average as follows: -Similarly, the temperature boundary condition for the air entering the third pass can also be expressed as: The radiative heat transfer coefficient in the above system of equations can be evaluated through the following: -Radiation from glass 1 to the surroundings: h r,a = σε g1 T g1 2 + T sky 2 T g1 + T sky ; -Radiation of glasses: -Radiation of glass 1 and the absorber plate: -Radiation of the absorber plate and back plate: The convective heat transfer coefficient of the air in the ducts can be estimated from the empirical correlation. The correlation was developed for the rectangular solar heater duct in the thermally developing flow [17] and has been adopted by many studies on solar air heater ducts [18][19][20]. where Pr and D e are the Prandtl number and hydraulic diameter of the air duct, respectively, and Re 1 , Re 2 , and Re 3 are the respective Reynolds numbers in passes 1, 2, and 3. These quantities are defined by: .
The wind loss heat transfer coefficient for glass 1 can be determined using the McAdams formula [21]: The area-weighted fin efficiency can be estimated as [11,18,19]: where A f is the total surface area of the fins and A c is the collector surface area with A f = 2nW f L and A c = LW, in which n and W f are the number of fins and fin height, respectively. The fin efficiency is defined as: where M = 2 The useful heat gain is determined from the air temperature difference of the collector as follows: The air pumping power can be calculated by: where ∆P is the air pressure difference through the collector, which sums the pressure drops of passes, as expressed by: The friction factor of each pass is given as [17,18]: (25c) Therefore, the thermohydraulic efficiency is defined as: where C o is the conversion factor of the mechanical work to heat (C o = 0.2) [4,22]. Entropy generation of a SAH exists as a result of irreversibility. The irreversibility may consist of the absorption of radiation by the absorber plate, heat transfer to the working air, heat loss to the environment, and frictional loss of the working air. The entropy generation can be determined using the entropy balance for a steady control volume without work transfer [23,24]: where T s is the solar temperature (T s = 5777 K) and Q s is the solar energy absorbed by the absorber plate (Q s = I(τα)LW).
The parameters used as input for the mathematical model are reported in Table 2. Most of the parameters were obtained from the study of Ramani et al. [16] for the sake of validating numerical computation in the present study. The thermophysical parameters of air were estimated at ambient temperature. Temperature gradient equations-that is, Equations (3), (5) and (7)-can be solved by numerical integration in the form of Equation (28) [25]. The code for the solution of the governing equations was implemented in the software EES [26] using the built-in integral function. The details of the solution procedure are presented in the textbook of Nellis and Klein [27]. Figure 2 shows a comparison of the air temperature distribution in a double-pass solar air collector. There is good agreement between the results obtained in this study and the published results. Thus, the formulation of the governing equations and solution strategy can ensure accuracy of the results.  Table 3 presents the range of key parameters that were used to investigate the thermohydraulic performance and entropy generation in this study. It should be noted that the case with y = z = 0 is the traditional triple-pass collector. This case is considered as the base case for comparison with cases where y or z is greater than zero. For the multi-objective optimization of maximum thermohydraulic efficiency and minimum entropy generation, a genetic algorithm (GA) was adopted. The settings of the GA are provided in Table  4, which were partially adapted to our previous study [29]. The optimum curve can be   Table 3 presents the range of key parameters that were used to investigate the thermohydraulic performance and entropy generation in this study. It should be noted that the case with y = z = 0 is the traditional triple-pass collector. This case is considered as the base case for comparison with cases where y or z is greater than zero. For the multi-objective optimization of maximum thermohydraulic efficiency and minimum entropy generation, a genetic algorithm (GA) was adopted. The settings of the GA are provided in Table 4, which were partially adapted to our previous study [29]. The optimum curve can be displayed by means of a Pareto front [30]. To specify the ultimate optimal solution in terms of Pareto optimality, the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) decision-making process was adopted. Another common decision-making technique is LINMAP (Linear Programming Technique for Multi-dimensional Analysis of Preference). The LINMAP method chooses the best solution by finding the shortest distance to the ideal point. The TOPSIS is attributed to the better method because the idea of this technique is to seek out the point that is nearest to the ideal point and the furthermost from the non-ideal point [31]. The lengths from a point to the ideal point (l i+ ) and non-ideal point (l i− ) are estimated as [32]:

Key Parameter Range
Reynolds number in the third pass Re 3 = 8000-18,000 Airflow ratio in the second pass y = 0-0.4 Airflow ratio in the third pass z = 0-0.4  Table 3 Selection type Tournament Enhances chance of selection for the fittest individual [33] Crossover To meet the condition of TOPSIS, the ultimate optimal solution has the biggest value of the ratio, as given by:

Results and Discussion
The influence of key parameters on the thermohydraulic efficiency, entropy generation, and the determination of optimal parameters is presented in this section. Figures 3-8 show the influence of air flow ratios on the thermo-hydraulic parameters and entropy generation at Re 3 = 10,000. The effect of air flow ratios on the temperature difference through the collector is presented in Figure 3. A general trend can be seen that as the ratio increases, the temperature difference decreases. In the base case (y = z = 0), the air temperature difference was the largest. This is because the reduced air flow in passes 1 or 2 decreased the convection heat exchange coefficient, thereby reducing the heat transfer capacity. It can be clearly seen, however, that the reduction of the temperature difference with the flow ratios was not significant. In the case where y = z = 0.4, the air temperature difference was about 0.15 K lower than that of the base case. This was due to the fact that radiant heat transfer existed in the multi-pass air collector. Figure 4 shows the average radiant heat transfer coefficients which were calculated from the average temperature of the surfaces. It is clear that as the Reynolds number increased, the radiant heat transfer coefficient decreased due to a decrease in the surface temperatures. However, at certain Reynolds numbers, all radiant heat transfer coefficients of the three-inlet SAH with airflow ratios of 0.4 were greater than those of the base case. This resulted in a negligible decrease in the air temperature difference with increasing airflow ratios. As the air flow decreased, the temperature of the heat exchanger surfaces increased, thereby increasing the temperature difference between the fluid and the heat transfer surface. The heat transfer rate was proportional to the temperature difference. The temperature distributions of the air flows and heat transfer surfaces are depicted in Figure 5 with respect to the extremes of the airflow ratios (i.e., 0 and 0.4). decreased, the temperature of the heat exchanger surfaces increased, thereby increasing the temperature difference between the fluid and the heat transfer surface. The heat transfer rate was proportional to the temperature difference. The temperature distributions of the air flows and heat transfer surfaces are depicted in Figure 5 with respect to the extremes of the airflow ratios (i.e., 0 and 0.4).   h ra h ra h r,b,p h r,b,p h r,g1,g2 h r,g1,g2 h r,g2,p h r,g2,p h ra h ra h r,b,p h r,b,p h r,g1,g2 h r,g1,g2 h r,g2,p h r,g2,p y = z = 0 y = z = 0.4 The air temperature increased gradually through the passes. In the case of three inlets (Figure 5b), air temperatures entering the second and third passes were reduced compared to the air temperatures exiting the first and second passes, respectively. The temperatures in the first and second passes were reduced due to the fact that cool ambient air entered the collector from the additional inlets with the airflows of and , respectively. It is clear that when decreasing the air flow rate of passes 1 and 2, the air temperature difference at these passes increases. The third pass had the same flow rate for both cases in Figure 5. However, the air temperature difference of the third pass in the threeinlet collector was much larger than that of the traditional triple-pass SAH: about 6 K vs. 4 K. This is because the average temperature of the absorber plate of the base case was about 41 °C, while that in the case where y = z = 0.4 was about 43 °C. In the base case (Figure 5a), the airflow in the first pass received heat mainly from glass 2. The difference in air temperature through the second pass was the largest, as this was the main pass (i.e., the air exchanged heat with the absorber plate). Hence, the temperature cross occurred between the second pass air temperature (Tf2) and temperature of glass 2 (Tg2). The absorber plate temperature profile was in line with the second pass air temperature. In the case y = z = 0.4 (Figure 5b), the first pass air received heat from the upper glass until x = 0.5 m, as the lower air flow led to higher glass temperatures. The absorber plate temperature (Tp) reached its maximum in the middle of the collector (x = 1.1 m), as the heat exchange occurring at the second and third passes was almost the same. However, the direction of the temperature rise of these two passes was opposite, such that the absorber plate temperature reached its maximum in the middle of the collector.
The hydraulic loss with flow ratios is shown in Figure 6. It can be clearly observed that the air pressure difference decreased notably with increasing flow ratios: 25.5 Pa in the base case and 13 Pa at y = z = 0.4 (i.e., pressure drop by a half). Figure 7 illustrates the variations of useful heat gain and pumping power, as assessed by thermohydraulic x-coordinate (m) In other words, the thermohydraulic parameters were affected more strongly by the change of feed air into the third pass. Figure 8 shows the entropy generation with flow ratios. We observed that the entropy generation decreases with the increase of airflow ratios. This behavior results from a change in the airflow ratio, which had little effect on the outlet temperature but greatly reduced the pressure loss. The effect of the irreversibility due to the frictional loss can be expressed by the last term of Equation (27). Entropy generation decreased from 2.442 W/K to 2.384 W/K when increasing flow ratios from 0 to 0.4.   The air temperature increased gradually through the passes. In the case of three inlets (Figure 5b), air temperatures entering the second and third passes were reduced compared to the air temperatures exiting the first and second passes, respectively. The temperatures in the first and second passes were reduced due to the fact that cool ambient air entered the collector from the additional inlets with the airflows of y . m and z . m, respectively. It is clear that when decreasing the air flow rate of passes 1 and 2, the air temperature difference at these passes increases. The third pass had the same flow rate for both cases in Figure 5. However, the air temperature difference of the third pass in the three-inlet collector was much larger than that of the traditional triple-pass SAH: about 6 K vs. 4 K. This is because the average temperature of the absorber plate of the base case was about 41 • C, while that in the case where y = z = 0.4 was about 43 • C. In the base case (Figure 5a), the airflow in the first pass received heat mainly from glass 2. The difference in air temperature through the second pass was the largest, as this was the main pass (i.e., the air exchanged heat with the absorber plate). Hence, the temperature cross occurred between the second pass air temperature (T f2 ) and temperature of glass 2 (T g2 ). The absorber plate temperature profile was in line with the second pass air temperature. In the case y = z = 0.4 (Figure 5b), the first pass air received heat from the upper glass until x = 0.5 m, as the lower air flow led to higher glass temperatures. The absorber plate temperature (T p ) reached its maximum in the middle of the collector (x = 1.1 m), as the heat exchange occurring at the second and third passes was almost the same. However, the direction of the temperature rise of these two passes was opposite, such that the absorber plate temperature reached its maximum in the middle of the collector.
The hydraulic loss with flow ratios is shown in Figure 6. It can be clearly observed that the air pressure difference decreased notably with increasing flow ratios: 25.5 Pa in the base case and 13 Pa at y = z = 0.4 (i.e., pressure drop by a half). Figure 7 illustrates the variations of useful heat gain and pumping power, as assessed by thermohydraulic efficiency, with air flow ratios. Taking the aforementioned discussions into account, we determined that the efficiency can reach a maximum at a certain airflow ratio. Greater efficiencies were achieved at flow ratios less than 0.1. This is because the performance of an SAH is dominated by heat transfer compared to pumping power. The thermohydraulic efficiency reached a maximum of 64.65% at z = 0 and y = 0.2. At z = 0.4, the efficiency dropped from 64.42 to 64.08% when increasing y from 0 to 0.4. At y = 0.4, the efficiency decreased from 64.62 to 64.08% as z increased from 0 to 0.4. In other words, the thermohydraulic parameters were affected more strongly by the change of feed air into the third pass. Figure 8 shows the entropy generation with flow ratios. We observed that the entropy generation decreases with the increase of airflow ratios. This behavior results from a change in the airflow ratio, which had little effect on the outlet temperature but greatly reduced the pressure loss. The effect of the irreversibility due to the frictional loss can be expressed by the last term of Equation (27). Entropy generation decreased from 2.442 W/K to 2.384 W/K when increasing flow ratios from 0 to 0.4.
The impacts of the Reynolds number in the third pass (Re 3 ) and the airflow ratios are shown in Figures 9-12. As the Reynolds number increased, both heat transfer and pressure loss increased. The convection heat transfer coefficient was proportional to the 0.8 power of the Reynolds number (Equation (14)) and the air pumping power was proportional to the 2.75 power of the Reynolds number (Equations (23)- (25)). Hence, the efficiency peaked at some Reynolds number when the ratios were fixed, as seen in Figures 9 and 11. It can be inferred from Figure 9 that the maximum efficiency at each z-value was roughly the same. However, when z was larger, the optimum Re 3 was also greater. At z = 0, the optimal Re number was around 10,000, while the optimal Re number was around 13,000 at z = 0.4. This is a promising outcome in terms of increasing the air flow through the triple-pass collector with additional inlets. In addition, when Re 3 > 11,000, the base case performance was minimal. Entropy generation increased with an increasing Reynolds number and with decreasing flow ratios, as can be observed from Figures 10 and 12. The increase of the entropy generation with the Reynolds number was mainly due to increased pressure loss penalty (∆P). It can be noticed that the effect of z on the efficiency and entropy generation was more pronounced than that of y, as the third pass was the main heat exchange channel and the pressure loss was the largest for the triple-pass SAH with three inlets. From Figures 9 and 11, it can be concluded that a three-inlet air collector should be used when the Reynolds number is greater than 11,000 in terms of the thermohydraulic performance.
pressure loss penalty (ΔP). It can be noticed that the effect of z on the efficiency and entropy generation was more pronounced than that of y, as the third pass was the main heat exchange channel and the pressure loss was the largest for the triple-pass SAH with three inlets. From Figures 9 and 11, it can be concluded that a three-inlet air collector should be used when the Reynolds number is greater than 11,000 in terms of the thermohydraulic performance.       From the above parametric study, it can be seen that the Reynolds number and airflow ratios had opposite effects on collector efficiency and entropy generation. The maximum efficiency was achieved at a certain Reynolds number and at certain airflow ratios in the surveyed range. Meanwhile, entropy generation increased with the increase of Re and decrease of flow ratios. Therefore, to assign optimal values of independent parameters, a Pareto front was constructed, as shown in Figure 13. The Pareto front curve was established using the genetic algorithm with the settings shown in Table 4. The curve was a series of optimal solutions that yielded a large efficiency and small entropy generation. Ideal and non-ideal points located at the vertices of the rectangle formed by the Pareto curve were found. To determine the final solution, the TOPSIS decision-making technique was applied, which found Sgen = 2.4691 W/K and ηeff = 65.38%, as shown in the graph. This is the point that was closest to the ideal point (bottom-right corner) and furthest from the non-ideal point (top-left corner). The optimal parameters at the selected point were found to be Re3 = 11,156, y = 0.258, and z = 0.036. In addition, Figure 13 also shows the maximum efficiency value corresponding to the maximum entropy generation and vice versa. From the above parametric study, it can be seen that the Reynolds number and airflow ratios had opposite effects on collector efficiency and entropy generation. The maximum efficiency was achieved at a certain Reynolds number and at certain airflow ratios in the surveyed range. Meanwhile, entropy generation increased with the increase of Re and decrease of flow ratios. Therefore, to assign optimal values of independent parameters, a Pareto front was constructed, as shown in Figure 13. The Pareto front curve was established using the genetic algorithm with the settings shown in Table 4. The curve was a series of optimal solutions that yielded a large efficiency and small entropy generation. Ideal and non-ideal points located at the vertices of the rectangle formed by the Pareto curve were found. To determine the final solution, the TOPSIS decision-making technique was applied, which found S gen = 2.4691 W/K and η eff = 65.38%, as shown in the graph. This is the point that was closest to the ideal point (bottom-right corner) and furthest from the non-ideal point (top-left corner). The optimal parameters at the selected point were found to be Re 3 = 11,156, y = 0.258, and z = 0.036. In addition, Figure 13 also shows the maximum efficiency value corresponding to the maximum entropy generation and vice versa.