*3.1. Hydraulic Behavior*

Figures 4 and 5 show the velocity streamlines with the two analyzed air mass flow rates (flow 1: 0.01 kg/s and flow 2: 0.02 kg/s, respectively). In Figure 4 (flow 1: 0.01 kg/s), the configurations with an inlet from below (a and b), a sudden expansion occurs near the inlet of the cavity, forming an eddy in the upper part of the collector. While in the configurations with the entrance at the top (c and d), the eddy forms at the bottom. Further, in all configurations except (b), it is observed that after the air enters the cavity, several families of eddies form until the end of the collector (z = 2.0 m). On the other hand, in configuration (b), a large eddy is observed in the first half of the cavity, and then the formation of some smaller eddies. Finally, it is essential to note that the highest magnitude velocities are generated at the elbow and at the beginning of the duct (second pass).

**Figure 4.** Streamlines at the collector cross-section with air mass flow 1: 0.01 kg/s. (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

**Figure 5.** Streamlines at the collector cross-section with air mass flow 2: 0.02 kg/s. (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

In Figure 5, a similar behavior to the one described for flow 1 is shown for flow 2 (0.02 kg/s), where a sudden expansion of air near the inlets occurs. For configurations (a) and (b) (inlet from the bottom), a visible jet can be observed at the bottom of the cavity; here, an eddy at the upper region can also be observed. Contrarily, in the configurations with an inlet from above, the jet is formed at the top, whereas the eddy occurs at the bottom region. Additionally, in configurations (a) and (c) (either both inlet and outlet from above or below), there is a primary air current with high speed. In configuration (a), the current goes up and down the cavity, generating several eddies at the upper and bottom sides opposite to the main flow. Contrarily, in configurations (b) and (d), which have inlet and outlet in opposite positions on the *y*-axis, a large eddy is formed near the cavity inlet, observing a larger eddy in configuration (b) that moves towards the exit in a disorderly manner. Moreover, the largest speed occurs at the elbow of the collector, having higher speeds in the configurations with a bottom outlet.

One reason that explains a disordered and asymmetric flow is because of the collector tilt (see Figure 1). In addition, the pressure increase justifies the phenomenon of the sudden air contraction caused by the elbow area reduction. Furthermore, a higher mass flow influences the amplitude and turbulence of the eddies found in the first section of the cavity.

#### *3.2. Thermal Behavior*

Figures 6 and 7 show the temperature fields of the four analyzed configurations. In Figure 6, for flow 1, it is observed that there is an extended region at low temperatures for all configurations in the cavity inlet, which is related to the air inlet in the form of a jet described in Figure 4. In all configurations, a region with low temperature is generated related to the jet of cold air that enters the cavity. For the configurations with an inlet from the bottom, the zone is located in the upper left corner. This is due to the presence of the primary eddies observed in the hydrodynamic behavior of the fluid (Figures 4 and 5). In configurations with a top inlet, the area is also extended towards the middle of the cavity, which is related to less turbulence in the movement of the fluid.

**Figure 6.** Air temperature fields at the collector cross-section with air mass flow 1: 0.01 kg/s. (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

**Figure 7.** Air temperature fields at the collector cross-section with air mass flow 2: 0.02 kg/s. (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

When the air enters from above, a zone of hot air is generated in the region near the entrance and another near the exit, while configurations with a bottom inlet have a heat recovery since the air enters the cavity.

For all configurations, the region with the highest temperature is located in the proximity of the receiver, and it is related to the heat generation on the receiver plate. Consequently, in the final part of the duct, there is another region with high temperatures. High temperatures are highly desirable since heat is extracted from the plate to the working fluid.

In Figure 7 (flow 2: 0.02 kg/s), the temperatures are lower than the ones observed in flow 1 (0.01 kg/s). In the four configurations, the temperature fields are highly dependent on the movement of the fluid in the cavity. The phenomena that drive the low temperatures in the section near the cavity inlet are the presence of a cold air jet from the inlet air and the consequent formation of eddies throughout the cavity. In the configurations with an inlet from below (a and b), there is another circumstance that causes the low temperatures, and it is due to the formation of the main eddy in the upper part of the cavity that promotes the stagnation of cold air in this section. Similar to the behavior described for temperatures of flow 1, for flow 2, the highest temperatures are always found in the regions neighboring the receiving duct.

Figures 8 and 9 show the four configurations of the flat plate receiver temperature contours at flow 1 (0.01 kg/s) and flow 2 (0.02 kg/s), respectively.

**Figure 8.** Temperature contours in the flat plate receiver, first pass (Flow 1: 0.01 kg/s).

**Figure 9.** Temperature contours in the flat plate receiver, first pass (Flow 2: 0.02 kg/s).

Figure 8 (flow 1: 0.01 kg/s) shows that when the air enters from below (configurations a and b), it removes heat from the first part of the receiver since it enters the cavity. In contrast, when the air enters from above (configurations c and d), the zones with lower temperatures are displaced from around positions z = 0.30 m to z = 0.65 m. The displacement of the presence of the lower temperature zones is a direct consequence of the formation of the main eddy, which helps to remove heat from the receiver. On the other hand, in configuration (a) Down–Down, a higher temperature of around 370 K is observed, indicating that the heat would not be uniformly removed in the first pass. This high-temperature zone is explained by contrasting with the hydrodynamic behavior observed in Figure 4, since when the air enters from the bottom, it heats up when it comes into contact with the receiver and rises, then continues its movement mainly through the upper part and then descend when looking for the exit that is in the lower part of the cavity. In contrast, configurations (b), (c), and (d) have a medium–high temperature zone in the center of the receiver. When the air leaves the collector cavity (z = 2.0 m), low-temperature zones are generated in the configurations with an outlet from below (configurations (a) and (d)).

In contrast, in configurations (c) and (d), the existence of two zones of low and medium–high temperature indicates that although heat removal is heterogeneous, energy is recovered in the zone between z = 0.30 m and z = 0.8 m, which is related to the presence of a large eddy in the inlet zone surroundings. In configuration (b), two low-temperature zones are observed at the inlet and outlet of the collector and a medium–high temperature zone in the center.

In Figure 9, similar temperatures but lesser magnitude can be observed, corresponding to the prevalence of high velocities due to the application of a large mass flow rate (0.02 kg/s). Low-temperature regions are observed near the inlets in the receiver plates of configurations (a) and (b), where the inlet is from below. Configurations (a) and (d), with outlets from below, have low-temperature regions near the outlets meaning that heat removal mainly occurs in those sections. On the other hand, configurations (a) and (b) have a sizeable high-temperature area in their central zone, noting that this area comprises most of the receiver plate extension. For configuration (c), a zone from z = 0.2 m to z = 1.0 m with lower temperatures is observed, and even though its outlet is from above; it also has a small region with low temperatures near the exit. Contrastingly, configuration (d) has most of its receiver flat-plate with high temperatures, except for its final part (around z = 1.7 m to z = 2.0 m). Still, it is essential to note that the high temperatures observed for the receiver plates of Figure 9 (around 345 K) are of lesser magnitude than those observed in Figure 8 (around 370 K), and thus higher heat removal was accomplished with the application of flow 2 (0.02 kg/s).

Table 5 shows the average temperatures in the flat plate receiver (*Tp,avg*) for both airflow rates. The highest temperature, as expected, is obtained with flow 1, 362.2 K with configuration (a); the lowest is obtained with flow 2, with configuration (c), of 338.5 K. The flat-plate temperature differences between flows 1 and 2 are greater than 19.7 K.


**Table 5.** Mean temperatures in the flat plate receiver of the U-shape double-pass CPC.

Figure 10 displays the first and second pass air temperature profiles for the four analyzed configurations. The blue line represents the air temperature profile in the first pass, and the red line is the profile of the second pass. In addition, the scale of the horizontal axis of the graphs (z-position) is inverted to facilitate the interpretation of the results since it allows visualizing the air outlet of the collector on the far right.

**Figure 10.** Air temperature profiles. (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

In the first pass for the configurations with the inlet from the bottom, (a) Down–Down and (b) Down–Up, the temperature profile has a slight increase near the entrance (z = 0.1 m), then it continues to increase until near the end of the cavity (z = 1.9 m) where it has another sharp increase. In the configurations that have an entrance to the cavity from the top, (c) Up–Up and (d) Up–Down, it is observed that in the first pass, there is a sudden increase in temperature in the section close to the entrance of the cavity (z = 0.1 m). In addition, all the temperature profiles are smooth for both flow rates; nevertheless, the temperature profiles have a less pronounced slope for configurations (a), (c), and (d).

Furthermore, in the second pass of all configurations, the temperature profile is smooth, with a sustained increase in temperature, reaching similar outlet temperatures of 327 K for flow 1 and 313 K for flow 2.

In the second pass for all the studied configurations, the temperature profile steadily increases from z = 2.0 m to z = 0.0 m, where it is observed that the temperature profiles are smooth for both flow rates. The differences in the behavior among the configurations in the second pass are only slight differences in the temperatures at the inlet and outlet, where the lowest inlet temperature is found in configuration (c) Up–Up, and the highest in (a) Down–Down.

On the other hand, the configurations that have an outlet from above, (b) Down–Up and (c) Up–Up, present a higher temperature at the outlet end of the cavity (end of the first pass) than at the entrance to the duct (beginning of the second pass). The phenomenon is caused by stratification in the cavity, so the air enters the connecting elbow at a temperature lower than that shown in Figure 10 (See Table 6). This occurs with both flows but is more significant with 0.01 kg/s due to the greater air stagnation in the cavity.


**Table 6.** Air temperatures in the elbow inlet and outlet of the U-shape double-pass CPC.

Table 6 shows the elbow inlet and outlet temperatures for both air flow rates. The presented elbow inlet temperature refers to the average temperature at the outlet of the cavity (z = 2.0 m) in positions y = 0.1255 m to y = 0.1645 m, where the height of the elbow is 0.036 m. As expected, the temperatures corresponding to flow 1 (0.01 kg/s) are higher than flow 2 (0.02 kg/s). Therefore, all the air temperatures at the elbow inlet are lower than the air temperatures at the outlet of the cavity z = 2.0 m. On the other hand, the elbow outlet air temperatures and the duct inlet air temperatures are the same.

Figure 11 shows the heat transfer coefficients (HTCp-cav) in the cavity for both air flow rates. As expected, the coefficients are lower when the airflow is lower and higher when the air flow rate is higher. Configurations with an inlet from below ((a) and (b)) have a high coefficient at the entry, which decreases to subsequently increase until it reaches a maximum near the outlet for configuration (a) and at the outlet (z = 2.0 m) for configuration (b). It is also noted that configuration (a) has a sharp increase and then a decrease from z = 0.2 m to z = 0.8 m, which is caused by the rise of the jet until it reaches the cover (see Figures 4a and 5a). In contrast, configurations with an inlet from the top (c and d) have an increase in the HTCp-cav in the region near the entry; then, it sharply declines to later gradually increase towards the region near the outlet where it reaches its maximum value. For example, the maximum HTCp-cav values with flow 1 are 15.0 W/(m<sup>2</sup> K), 15.8 W/(m<sup>2</sup> K), 16.0 W/(m<sup>2</sup> K), and 15.9 W/(m<sup>2</sup> K), while for flow 2 of 24.4 W/(m2 K), 22.3 W/(m<sup>2</sup> K), 21.9 W/(m2 K), and 25.0 W/(m2 K), for configurations (a), (b), (c) and (d).

**Figure 11.** Flat-plate receiver heat transfer coefficients (HTCp-cav). (**a**) Down–Down; (**b**) Down–Up; (**c**) Up–Up; (**d**) Up–Down.

In configurations where the air enters from the top ((c) and (d)), there is a high HTCp-cav at z = 0.1 m; nevertheless, all configurations have the maximum HTCp-cav in the region near the outlet. The above observations indicate that the HTC p-cav maximums correspond to the presence of the eddies produced by the sudden expansion at the inlet. While in the region near the outlet, there are also high HTCp-cav at z = 2.0 m in configuration (b), but in configurations (a), (c), and (d), it occurs around z = 1.8 m to z = 1.9 m. For configurations (c) and (d), this occurs because the fluid becomes turbulent in the final region of the cavity as air is forced out of the manifold elbow.

Table 7 shows the pressure drop in each configuration for the two mass flow rates. First, the highest pressure drop is seen in the bottom outlet configurations (a and d). Moreover, the pressure drop increases three to four times with the highest air flow rate.



#### *3.3. Efficiency*

Table 8 shows the temperature increments in the first and second passes and the outlet temperature for each configuration. The temperature increase resulting from the first pass (Δ*Tps*1) and the second pass (Δ*Tps*2) are more significant with flow 1. Moreover, for each flow, the U-shape double-pass CPC collector outlet temperature (*Tps*2,*out*) of all the configurations is very similar, with differences between 0.7 K and less.

**Table 8.** U-shape double-pass CPC temperature increments and outlet temperature (*Tf*1,*in* = 298.15 K).


Table 9 shows the contribution of the first pass (cavity) and second pass (receiver duct) to the useful energy gain inside the U-shape double-pass CPC collector and the efficiency for each configuration applying Equations (16) and (17). For instance, in configurations (a) and (b), between 40% and 45% of the total heat is extracted in the first pass with both flows. While in configurations (c) and (d), the heat recovery depends on the operating flow and is between 36% and 42% in the first pass.

Moreover, the efficiencies are higher with flow 2 due to the better heat transfer and lower heat losses to the ambient. In addition, it is observed that configuration (a) Down– Down is the most efficient, while configuration (c) Up–Up provides the lower efficiency of the cases analyzed at flow 2.

Configuration (a) has the highest efficiency because the air flows predominantly through the bottom section of the cavity, which is closest to the receiver plate. This surface has the highest temperature, thus is where heat recovery is desired. In addition, as the air descends, the final part of the cavity recovers heat since the fluid is forced to exit from the bottom.


**Table 9.** U-shape double-pass CPC useful energy gain percentage and thermal efficiency (*Tf*1,*in* = 298.15 K).
