# Thermal Performance Analysis of Porous Foam-Assisted Flat-Plate Solar Collectors with Nanofluids

^{1}

^{2}

^{*}

## Abstract

**:**

^{−5}~10

^{−2}), types of nanoparticles, volume fraction (φ), and mixing ratio (φ

_{c}). The numerical findings indicate that the dominant factor in the channel is the global Nusselt number (Nu

_{g}). As the Darcy number rises, there is an improvement in the heat transfer performance within the channel. Simultaneously, for the case of Re = 234, φ = 3%, and φ

_{c}= 100%, the Nu

_{g}in the channel reaches a maximum value of 6.80, and the thermal efficiency can be increased to 70.5% with the insertion of rectangular porous blocks of Da = 10

^{−2}. Finally, the performance evaluation criteria (PEC) are employed for a comprehensive assessment of the thermal performance of FPSC. This analysis considers both the improved heat transfer and the pressure drop in the collector channel. The FPSC registered a maximum PEC value of 1.8 when rectangular porous blocks were inserted under conditions of Da = 10

^{−2}and Re = 234 and the nanofluid concentrations of φ = 3% and φ

_{c}= 100%. The findings can be provided to technically support the future commercial applications of FPSC. The findings may serve as a technical foundation for FPSC in upcoming porous media and support commercial applications.

## 1. Introduction

_{2}O

_{3}/water and CuO/water). The findings demonstrated that the introduction of nanoparticles and altering the flow direction in the FPSC channel induce an increase in the convective heat transfer coefficient. This phenomenon has been previously observed by Gupta et al. Gupta et al. [26] numerically tested the thermal performance of FPSCs with and without nanofluid (Al

_{2}O

_{3}/water). The outlet water temperature of FPSC without nanofluid was 5–10 °C lower than that the nanofluid was used. Moreover, Darbari and Rashidi [27] conducted a numerical simulation to explore the thermal efficiency of a flat plate thermosyphon solar water heater using various nanofluids. The study incorporated the examination of different parameters, including the volume fraction of nanoparticles, volumetric flow rate, solar radiation intensity, and more. The findings revealed that among the diverse nanoparticles tested, the inclusion of copper nanoparticles, followed by copper oxide, led to the most substantial enhancements in efficiency and useful energy. Esmaeili et al. [28] proposed an innovative solar collector design incorporating copper oxide (CuO) porous foam and nanoparticles with superior optical properties, aiming to enhance the thermal performance of the FPSC. Taking into account varying nanoparticle volume fractions, foam pore sizes, working fluid mass flow rates, as well as different thicknesses and positions of the porous layer within the FPSC channel, the numerical findings revealed that the efficiency of FPSC, whether partially or fully filled with metal foam, reached its peak value in the fully filled configuration. Additionally, compared with the water flow, the utilization of CuO nanofluid and metal foam led to an enhancement in collector efficiency by as much as 26.8% and 23.8%. Xiong et al. [29] conducted a numerical investigation exploring the impacts of hybrid nanofluid concentration (Ag-Al

_{2}O

_{3}/water), porosity, Darcy number, Reynolds number, and other factors that significantly influence the thermal performance of FPSC. Abolfazl et al. [30] propose solutions to improve the thermal performance of paraffin as a Phase-change Material (PCM) in solar flat-plate collector systems for domestic and industrial solar applications. Three methods are proposed: using 10 PPIs aluminum foams with 0.92 or 0.95 porosity, various types of 5%wt nanoparticles, and modifications to the geometry in three configurations: straight, wavy wall, and wavy wall Y-shaped fin combinations. The paper found that nano-powders reduce melting time by 18.15% and 40.70%, while metal foams or nanoparticles with foams improve cycling times.

_{2}O

_{3}), volume fraction, and mixing ratio of hybrid nanofluids are mainly taken into account. The global heat transfer (Nu

_{g}) in the FPSC channel is clarified at various Reynolds numbers. Meanwhile, the dimensionless pressure drop (friction coefficient (f

_{m})) is disclosed. Lastly, an analysis of the performance evaluation criteria (PEC) in the FPSC channel is conducted to identify an optimal configuration and operating parameters that yield maximum thermal efficiency.

## 2. Mathematical Model

#### 2.1. Mathematical Model and Assumptions

_{w}= 800 W/m

^{2}) [9,31]. The base fluid within the channel is water (Pr = 7), and for the purposes of this study, the different types of nanoparticles (Al

_{2}O

_{3}-H

_{2}O, Cu-H

_{2}O), nanoparticle volume fraction (φ = 1%, 2%, and 3%), and hybrid nanofluid mixing ratios (0%Cu/100%Al

_{2}O

_{3}, 25%Cu/75%Al

_{2}O

_{3}, 50%Cu/50%Al

_{2}O

_{3}, 75%Cu/25%Al

_{2}O

_{3}, and 100%Cu/0%Al

_{2}O

_{3}) have been designed to study the effects of the new type of heat exchanger. The influence of the Reynolds number on the thermal performance of the FPSC is examined as the incoming fluid enters the collector with uniform velocity (u

_{in}) and constant temperature (T

_{in}). It is presumed that the fluid flow inside the channel is laminar, stable, and incompressible. Also, the effects of its radiation and buoyancy are not taken into account. In this study, the geometric shapes of the porous media are constructed as rectangles (REC), trapezoids (TRA

_{1}, TRA

_{2}), and triangles (TRI), as shown in Figure 1. Inserting porous media into the FPSC channel is designed to investigate the impact of body shape and permeability (Da = 10

^{−5}~10

^{−2}) within the FPSC channel. In addition, the porous metal foam is constructed from isotropic alumina (Al

_{2}O

_{3}) material with a consistent distribution of pores. Its thermophysical properties are presumed to remain constant, as outlined in Table 1.

#### 2.2. Governing Equations

- Continuity Equation:$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$
- Momentum equations:
- Governing equations in the fluid domain.$$\rho \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\mu \left(\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}\right)$$$$\rho \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\mu \left(\frac{{\partial}^{2}v}{\partial {x}^{2}}+\frac{{\partial}^{2}v}{\partial {y}^{2}}\right)$$
- Governing equations in the porous medium domain.$$\frac{\rho}{{\epsilon}^{2}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\frac{{\mu}_{eff}}{\epsilon}\left(\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}\right)-\frac{\mu}{K}u-\frac{\rho F}{\sqrt{K}}\left|\overrightarrow{V}\right|u$$$$\frac{\rho}{{\epsilon}^{2}}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\frac{{\mu}_{eff}}{\epsilon}\left(\frac{{\partial}^{2}v}{\partial {x}^{2}}+\frac{{\partial}^{2}v}{\partial {y}^{2}}\right)-\frac{\mu}{K}v-\frac{\rho F}{\sqrt{K}}\left|\overrightarrow{V}\right|v$$
_{eff}for effective viscosity. $F=\frac{1.75}{\sqrt{150}}\xb7\frac{1}{{\epsilon}^{1.5}}$ is the inertial factor [33].

_{p}represents the pore diameter of the porous media matrix, and K is the permeability of a porous metal foam block, which is defined as follows:

_{h}denotes the equivalent diameter of the channel.

- Energy equation:$$\frac{1}{\epsilon}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\tau \left(\frac{{\partial}^{2}T}{\partial {x}^{2}}+\frac{{\partial}^{2}T}{\partial {y}^{2}}\right)$$

#### 2.3. Boundary Conditions

_{eff}is the effective thermal conductivity, which is defined as follows [34].

_{f}is the thermal conductivity of the fluid, and k

_{p}is the thermal conductivity of porous media.

#### 2.4. Nanofluid Parametric Definitions

_{f}denotes the nanofluid, ρ is the fluid density, and C

_{p}is the specific heat capacity of the nanofluid.

_{p}denotes the nanoparticle, and ρ

_{np}is the density of the nanoparticle.

_{p}) of a single nanoparticle can be calculated based on the calculations of [36]:

_{2}O

_{3}nanoparticles and Cu nanoparticles, respectively. The volume fraction of nanoparticles is defined as follows:

_{nf}is the mixed nanoparticle, while the density of the mixed nanoparticles is

_{p}) of the mixed nanoparticles can be calculated:

#### 2.5. Collector Control Parameters

_{m}is the friction coefficient, H represents the height of the collector channel, and L denotes the length of the collector channel.

_{w}is the wall temperature, and T

_{m}is the mean volume temperature of the fluid. The latter can be calculated using the following formula [21]:

_{i}signifies the inlet spatial location of each porous block.

_{g}denotes the global Nusselt number, while N signifies the count of porous blocks within the channel. Through the measurement of the incoming and outgoing temperatures of the working fluid, the net rate of effective heat absorption by the working fluid can be determined using the following formula [37]:

_{out}and T

_{in}denoting the temperatures of the working fluid as it exits and enters the solar collector, respectively.

_{c}is the collector unit lighting area.

## 3. Numerical Details

#### 3.1. Numerical Method

^{−8}is recognized as convergence criteria are established, and for the steady-state flow issue in this study, the laminar flow model is also adopted.

#### 3.2. Grid Independence Validation

^{−4}, Re = 234, s = 0.6 H, the porous block was rectangular, water served as the heat transfer fluid within the channel, and the parameter s was set to 0.6 H to validate the precision of the numerical solution. Figure 2 illustrates the changes in the total Nusselt number and friction coefficient with various grid positions. With more grid points, we observe a drop in Nu

_{g}and f

_{m}. The channel Nu

_{g}and f

_{m}curves tend to stabilize when the grid points rise to 150 × 1000. In this instance, the global Nusselt number and channel friction coefficient are not significantly impacted by changes in the grid points. Therefore, it may be said that 150 × 1000 is the ideal mesh size. In addition, Table 2 conducts a comparison between the aforementioned model and the empty channel while taking the effect of various mesh sizes on channel outlet temperature (T

_{out}) into account. The results show that the percentage difference between the porous block and non-porous block in the NO.3 channel is 0.5% and 0.4%, respectively, while that of the NO.4 channel is 0.08% and 0.09%, respectively; this is computed using the expression, i.e., the percentage error $=({Nu}_{g(No.n+1)}-{Nu}_{g(No.n)})/{Nu}_{g(No.n+1)}$. Therefore, with the above two verification methods, it can be confirmed that the grid point selection of 150 × 1000 can obtain sufficient accuracy.

^{−4}, Re = 234, and s = 0.6 H.

#### 3.3. Numerical Verification

_{g}and f

_{m}are 2.23% and 1.35%, respectively, which can attest to the validity of this study’s numerical method. It can be demonstrated that the results of this study’s numerical calculations are trustworthy enough by the double verification of mesh size and numerical approach. As a result, the following numerical investigations employ this numerical approach.

## 4. Results and Discussions

#### 4.1. Heat Transfer Performance and Resistance Loss of the Collector

_{p}

^{2}) was calculated using the empirical Equation (6). To examine the impact of various Da values of porous blocks on the flow within FPSC channels, Figure 4 depicts the changes in the global Nusselt number (Nu

_{g}) in the FPSC channel under the base fluid condition (water) with different permeability shapes of porous blocks. Clearly, it is noticeable that, for a specific shape of the porous block, Nu

_{g}rises with an increase in Da, and vice versa. This is due to the fact that at higher permeability (Da = 10

^{−2}and Da = 10

^{−3}), more fluid passes through the interior of the porous block, enhancing heat exchange. Furthermore, for porous blocks with the same Da values, the configuration of the porous block also exerts a noteworthy influence on Nu

_{g}, exhibiting an almost monotonic increase as the shape transitions from a triangle (TRI) to a rectangle (REC). It is worth noting that the shape changes of TRI and TRA

_{2}porous blocks have little effect under low Da (Da = 10

^{−4}and Da = 10

^{−5}) conditions. This is because the rectangular shape of the porous block provides a larger solid–liquid heat exchange area compared to the triangular and trapezoidal shapes. Additionally, a close examination of Figure 4a,b reveals that the Nu

_{g}of the channel at Re = 468 exhibits a higher value in comparison to the base fluid at Re = 234. According to Equation (20), as Re increases, fluid velocity accelerates, leading to an increase in Nu

_{g}. In a comprehensive comparison and analysis of the results in Figure 4, it is evident that under the conditions of base fluid Re = 468 and a Da = 10

^{−2}for REC porous blocks, the heat transfer efficiency within the FPSC channel is at its peak, attaining a maximum Nu

_{g}value of 7.20.

_{m}) for the same operating conditions as depicted in Figure 4. It showcases the non-linear behavior of the in-channel friction coefficient (f

_{m}) concerning different porous block (Al

_{2}O

_{3}) shapes, Darcy numbers, and Reynolds numbers (Re). It is evident that, for a specified configuration of a porous block, when its Da number diminishes while maintaining a constant pore diameter, the porosity and pore density (PPI) also decrease, resulting in elevated permeability resistance loss and an inevitable increase in the friction coefficient as per Equation (6). Additionally, the friction coefficient (f

_{m}) demonstrates an almost linear escalation as the configuration of the inserted porous block transitions from TRI to REC. Specifically, the REC porous block, maintaining an equivalent base width dimension, occupies the most extensive channel space compared to the TRA- and TRI-shaped porous blocks. As a result, fluid flow through the porous media matrix experiences greater viscous and inertial resistance, leading to the largest in-channel pressure drop and the highest f

_{m}value. Moreover, a comprehensive comparative analysis of Figure 5a,b reveals that, under low Reynolds number (Re = 234) conditions, the f

_{m}value is consistently higher than that observed under high Reynolds number conditions (Re = 468). The variation can be ascribed to the reduced flow rate within the FPSC channel under the condition of low Reynolds number, resulting in lower inlet kinetic energy and, thus, a relatively higher calculated value of the f

_{m}. In summary, it is apparent that the f

_{m}maximum value of 7.91 is attained in the case of the REC porous block with Da = 10

^{−5}and Re = 234.

_{g}) and friction coefficient (f

_{m}) within the channel, Figure 6 delves deeper into the analysis, building upon the findings from Figure 4 and Figure 5. Figure 6 presents the impact of inserting REC porous blocks with Da values ranging from 10

^{−7}to 10

^{−1}on the channel Nu

_{g}and f

_{m}under the Re = 234 operating condition. It is evident that as Da increases from 10

^{−7}to 10

^{−2}, the channel Nu

_{g}also increases. However, it is worth noting that as Da further increases to 10

^{−1}, the Nu

_{g}of the channel exhibits a decreasing trend.

_{g}. This indicates that for Re = 234, the REC porous block with Da = 10

^{−2}yields the most favorable conditions for achieving the maximum value of FPSC channel Nu

_{g}, which reaches a maximum value of 5.91. Furthermore, by examining Figure 6, it becomes apparent that the channel’s f

_{m}consistently decreases with an increase in the porous block Da. As a result, a minimum value of 1.01 is observed at Da = 10

^{−1}for the channel’s f

_{m}.

_{g}) and pressure drop (ΔP) within the channel. This analysis is conducted under the operational scenario of an inserted porous block with a rectangular shape and Da = 10

^{−2}. It is evident that the Nu

_{g}and ΔP of the channel increase with rising fluid Re in the laminar flow regime. At a high Re value (Re = 2000), the channel achieves its maximum Nu

_{g}and ΔP, which are 11.58 and 3.48.

^{−2}, and fluid Re = 234. Figure 8a presents the variation of channel Nu

_{g}with different volume fractions of Al

_{2}O

_{3}-H

_{2}O nanofluids flowing through the FPSC channel. It is evident that higher Nu

_{g}values are observed with increased volume fractions, and the maximum value is achieved with the REC porous block. This improvement can be ascribed to the augmentation resulting from the addition of Al

_{2}O

_{3}nanoparticles, which contributes to an increase in the thermal conductivity, density, and viscosity of the fluid; a decrease in specific heat capacity; and a notable enhancement in heat transfer within the channel is observed in comparison to the base fluid, thereby amplifying the overall heat transfer efficiency of the channel. In Figure 8b, the nanofluid is Cu-H

_{2}O, and it is apparent that the inclusion of Cu nanoparticles also improves the heat transfer efficiency of the channel. A comparison between Figure 8a,b reveals that, under the same volume fraction condition, Cu-H

_{2}O has a significantly higher impact on the channel’s heat transfer performance compared to Al

_{2}O

_{3}-H

_{2}O. In summary, under the conditions of an inserted rectangular-shaped porous block and an increased volume fraction of Cu-H

_{2}O to 3%, the Nu

_{g}within the channel attains its peak at a value of 6.80, signifying a notable improvement of 20.7% in comparison to the base fluid.

_{m}). This occurrence is a result of the elevated density and dynamic viscosity of the nanofluid when contrasted with the base fluid, increasing the channel’s frictional losses. Additionally, an increase in nanoparticle volume fraction is accompanied by a corresponding rise in f

_{m}. It is noteworthy that the rate of f

_{m}increment gradually diminishes. This behavior is attributed to an increased presence of nanoparticles, which promotes improved mixing of the boundary layer, consequently reducing frictional resistance. Comparative analysis of Figure 9a,b reveals that, under the same volume fraction condition, the f

_{m}for Cu-H

_{2}O is slightly higher than that for Al

_{2}O

_{3}-H

_{2}O. This difference can be explained by the significantly higher density of Cu-H

_{2}O nanofluid in comparison to Al

_{2}O

_{3}-H

_{2}O nanofluid. In summary, when the porous block takes a rectangular shape and the concentration of Cu-H

_{2}O is increased to 3%, the channel’s f

_{m}reaches a maximum value of 3.96, representing a 33.8% increase in drag loss compared to an empty channel.

_{2}O

_{3}and Cu nanoparticles (φ

_{c}is the proportion of Cu nanoparticles) on the heat transfer efficiency of the FPSC channel is depicted. This analysis considers diverse nanoparticle volume fractions and porous block shapes, with the porous block (Al

_{2}O

_{3}) characterized by Da = 10

^{−2}and fluid Re = 234. The corresponding overall Nu

_{g}number in the FPSC channel is big for the condition of a high Cu nanoparticle proportion (φ

_{c}= 100%), regardless of the nanoparticle volume fraction and the configuration of the porous block, exhibiting enhanced thermal performance. The Nu

_{g}experiences a substantial increase when the configuration of the porous block transitions from TRI to REC. Additionally, the shape of the porous block will influence the mixing ratio of blended nanoparticles. It can be seen from the curve variation that the influence of the nanofluid mixing ratio on Nu

_{g}increases with the change in the shape of the porous block: TRI-TRA-REC. Because more nanofluids flow through the porous block as it transforms from a triangle to a rectangle, the effect of nanofluid mixing becomes more substantial than that of Nu

_{g}. This is the cause of the occurrence. Additionally, it is observed that, as the nanoparticle volume fraction increases from 1% to 3%, the shape of the porous block and the ratio of nanofluid mixing do not impact the extent to which the Nusselt number (Nu

_{g}) of the channel increases. FPSC exhibits the best thermal performance when the mixing ratio of nanofluids is φ

_{c}= 100% (Cu), and upon a comprehensive examination of the Nu

_{g}depicted in Figure 10, it is noted that the Nu

_{g}can attain a value of 6.80 within the channel when the shape of the porous block is rectangular and the nanoparticle volume fraction is 3%. The heat transfer performance is increased by roughly 13.3% when compared to φ

_{c}= 0% (Al

_{2}O

_{3}).

#### 4.2. Outlet Temperature of the Collector

^{−2}, the channel’s outlet temperature gradient is significantly higher than that observed in cases with lower Darcy numbers (10

^{−5}~10

^{−3}). Furthermore, the channel’s mean outlet temperature consistently ranks highest, followed by a gradual decrease in the mean outlet temperature as the Darcy number decreases. This observation indicates that the insertion of porous blocks with high Darcy numbers is advantageous for enhancing FPSC thermal performance. This effect is attributed to the high-resistance porous block, which restricts fluid flow through the porous block, resulting in limited heat exchange between the fluid and the collector plate, particularly near the absorber plate.

^{−2}. By applying both the median and interquartile range as measures of location, it is evident that, similarly to the operational conditions depicted in Figure 11, the outlet temperature gradient increases gradually in most locations at the channel’s exit and significantly increases in a small area close to the collector. With an increase in volume fraction, the peak and valley values rise. In Figure 12a,b, the maximum observed outlet temperature peaks are 306.8 °C and 305.9 °C, respectively, when the porous block shape is REC. The average outlet temperature of the channel reaches its highest value when the porous block shape is REC in comparison to TRI and TRA, and the outlet temperature experiences a continuous increase as the shape of the porous block transitions from TRI to TRA to REC. Furthermore, upon comparing Figure 12a,b, it becomes evident that the average outlet temperature of the channel is higher when the Reynolds number is low (Re = 234) and vice versa. As a result, when the porous block is rectangular, Da = 10

^{−2}and Re = 234, and the fluid is the base fluid, the channel’s average outlet temperature can achieve a peak value of 302.1 °C.

^{−2}and Re = 234. By examining the median and interquartile range of the data in Figure 13, it becomes evident that irrespective of the nanoparticle volume fraction, temperature valleys are consistently observed near the insulating plate, while temperature peaks occur in proximity to the upper collector plate. Furthermore, both the peaks and valleys exhibit an increase with higher nanoparticle volume fractions. In addition, the growth trend of the outlet temperature gradient in the vicinity of the upper collector plate is particularly pronounced, in contrast to the behavior of the outlet temperature gradient near the insulating plate. Additionally, it is evident that the average outlet temperature of the channel increases with a rise in nanoparticle volume fraction. Consequently, higher average exit temperatures are observed with higher volume fractions (φ = 3%). Upon comparing Figure 13a,b, it can be observed that Cu-H

_{2}O nanofluid, when flowing through the channel, results in a higher average exit temperature compared to Al

_{2}O

_{3}-H

_{2}O. In summary, it is noteworthy that the conditions of REC porous block (Da = 10

^{−2}, Re = 234) and Cu-H

_{2}O nanofluid lead to a maximum average exit temperature of 303.1 °C for the channel.

#### 4.3. Thermal Efficiency of Solar Collector

^{−2}, demonstrates a peak thermal efficiency of 67%. Moreover, upon examination of Figure 14b, it is evident that the addition of nanoparticles significantly enhances the thermal efficiency of the collector, and this enhancement is positively correlated with the increase in the volume fraction of nanoparticles. When the added nanoparticles contain both Cu and Al

_{2}O

_{3}, a noteworthy observation is that a higher percentage of Cu nanoparticles results in a higher thermal efficiency for the collector. This indicates that at a nanoparticle volume fraction of 3% and φ

_{c}= 100%, the collector’s thermal efficiency reaches a maximum value of 70.5%, representing a 2.5% increase compared to Al

_{2}O

_{3}nanoparticles.

#### 4.4. Performance Evaluation Criteria (PEC)

^{−2}, 10

^{−3}) conditions. This is due to the observation that under conditions of elevated permeability, more fluid passes through the porous block, leading to a significantly enhanced effect of heat transfer area variation on the channel’s heat transfer performance. Overall, it is evident that the peak thermal performance is observed at Da = 10

^{−2}for REC shapes, establishing the FPSC channel as consistently demonstrating the highest PEC value. Specifically, as depicted in Figure 15a for the base fluid Re = 234, under conditions of low Darcy numbers (Da = 10

^{−4}, 10

^{−5}), the PEC value can achieve 1.68, the impact of porous block shape on the PEC value is not significant, and the PEC value persists below that of high Darcy number conditions (Da = 10

^{−2}, 10

^{−3}). This implies that irrespective of the shape of the porous block, the impact of pressure loss prevails in the FPSC channel at low Darcy number conditions, posing a hindrance to its objective of improving thermal performance. Additionally, under high Darcy number conditions (Da = 10

^{−2}, 10

^{−3}), for Reynolds numbers Re = 234 and 468, the PEC values consistently surpass 1.0. This signifies that the enhanced thermal performance of the FPSC channel outweighs the adverse impact of pressure loss. In conclusion, based on the synthesis of FPSC thermal performance evaluation, the configuration of the FPSC channel featuring an incorporated REC porous block, with a Darcy number of Da = 10

^{−2}and Re = 234, exhibits the most effective heat transfer enhancement.

_{2}O

_{3}-H

_{2}O mixing ratio is displayed in Figure 16. Calculations were conducted for the REC porous block with Da = 10

^{−2}, revealing analytical results that indicate a noteworthy elevation in the PEC value of the channel upon the addition of nanoparticles in comparison to the base fluid. This holds true irrespective of the nanoparticle type, and the PEC value is positively correlated with the nanoparticle volume fraction, i.e., the larger the volume fraction, the larger the PEC, implying a better performance of the FPSC. In addition, we can find that at any nanoparticle volume fraction, the PEC increases as the percentage of Cu(φ

_{c}) nanoparticles in Cu-Al

_{2}O

_{3}-H

_{2}O increases. It is also worth mentioning that in condition φ

_{c}= 0%, nanoparticles consist solely of Al

_{2}O

_{3}, and when the volume fraction is increased to φ

_{c}= 100%, nanoparticles consist solely of Cu. Consequently, it is evident that Cu-H

_{2}O can yield higher PEC values compared to Al

_{2}O

_{3}-H

_{2}O. A comparison between Figure 16a and b distinctly indicates that the PEC value is higher at low Reynolds numbers (Re = 234) and decreases with increasing Re. Based on the analysis and comparison, it is concluded that under the conditions of REC porous block with Da = 10

^{−2}and Re = 234, when the nanoparticle volume fraction reaches 3%, along with the nanoparticle mixing ratio φ

_{c}= 100%, the channel reaches its maximum PEC value of 1.9. This represents an enhancement in performance of approximately 90% compared to the empty channel.

## 5. Conclusions

- (1)
- Regarding the impact of the flow state within the channel, observations indicate that fluids with high Reynolds numbers (within the laminar range) can markedly improve heat transfer performance within the FPSC channel. However, this improvement is counterbalanced by increased pumping power requirements. In scenarios with high Darcy numbers, the shape and permeability of the porous block demonstrate a substantial influence on heat transfer performance.
- (2)
- The introduction of nanoparticles has a pronounced effect on the heat transfer performance within FPSC channels. It is apparent that a higher volume fraction of nanoparticles results in enhanced heat transfer performance. Additionally, the type of nanoparticles plays a critical role. Cu nanoparticles exhibit enhanced heat transfer performance compared to Al
_{2}O_{3}nanoparticles. Consequently, when both Al_{2}O_{3}and Cu nanoparticles are introduced, a higher percentage of Cu nanoparticles results in improved heat transfer performance. Notably, the highest Nusselt number (Nu_{g}) achieved for the channel is 6.80, achieved with a nanoparticle volume fraction of 3% and φ_{c}= 100%. - (3)
- Additional analysis of the thermal efficiency in the FPSC channel reveals that at Re = 234, the thermal efficiency is high and is notably affected by the shape of the porous block, with superior performance observed in the REC porous block configuration and the large Da values. For instance, when the volume percentage of nanoparticles is 3% and φ
_{c}= 100%, the thermal efficiency can reach a maximum of 70.5%, which is roughly 2.5% greater than that of φ_{c}= 0% (the nanoparticle is Al_{2}O_{3}). - (4)
- A comprehensive performance evaluation criteria (PEC) analysis for FPSC indicates that the rectangular REC porous block configuration performs optimally for high Darcy numbers (Da = 10
^{−2}). Taking into account the influence of the flow regime in the FPSC channel, the PEC value reaches its peak at 1.68 under conditions of low Reynolds numbers (Re = 234), representing a 68% enhancement compared to an empty channel. Furthermore, it is crucial to emphasize that the PEC is affected by the characteristics of the heat transfer fluid, with a maximum value of 1.90 attained for the channel PEC under the condition of a nanoparticle volume fraction of 3% and φ_{c}= 100%.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Verification of grid independence through the utilization of the Nusselt number and friction.

**Figure 3.**Comparison of FPSC channel average Nusselt number (Nu

_{g}) and friction coefficient (f

_{m}) with the experimental data [21].

**Figure 4.**Variation of the global Nusselt number (Nu

_{g}) in the FPSC channel with the shape change of inserted porous blocks at various Da and Reynolds numbers (Re): (

**a**) Re = 234; (

**b**) Re = 468.

**Figure 5.**Variation of the friction coefficient (f

_{m}) in the FPSC channel with the shape change of inserted porous blocks at various Da and Reynolds numbers (Re): (

**a**) Re = 234; (

**b**) Re = 468.

**Figure 6.**Variation of the Nusselt number (Nu

_{g}) and friction coefficient (f

_{m}) in the FPSC channel with the Da change of inserted porous blocks.

**Figure 7.**Variation of the Nusselt number (Nu

_{g}) and pressure drop (ΔP) in the FPSC channel with the Reynolds number change.

**Figure 8.**Variation of the global Nusselt number (Nu

_{g}) in the FPSC channel with the shape change of inserted porous blocks in different kinds of nanofluids: (

**a**) Al

_{2}O

_{3}-H

_{2}O; (

**b**) Cu-H

_{2}O.

**Figure 9.**Variation of the friction coefficient (f

_{m}) in the FPSC channel with the shape change of inserted porous blocks in different kinds of nanofluids: (

**a**) Al

_{2}O

_{3}-H

_{2}O; (

**b**) Cu-H

_{2}O.

**Figure 10.**Variation of the global Nusselt number (Nu

_{g}) in the FPSC channel with the shape change of inserted porous blocks in different volume fractions and mixing ratios of nanofluids.

**Figure 11.**Comparisons of the outlet temperature of FPSC for the effects of various Darcy numbers at Re = 468.

**Figure 12.**Comparisons of the outlet temperature of FPSC for the effects of different porous block shapes at various Reynolds numbers: (

**a**) Re = 234; (

**b**) Re = 468.

**Figure 13.**Comparisons of the outlet temperature of FPSC for the effects of different volume fraction nanofluids at various Reynolds numbers: (

**a**) Re = 234; (

**b**) Re = 468.

**Figure 14.**Thermal efficiency of FPSC channel at Re = 234: (

**a**) the base fluid is water; (

**b**) REC porous block Da = 10

^{−2}.

**Figure 15.**Variation of the performance evaluation criteria (PEC) in the FPSC channel with the shape change of inserted porous blocks at various Da and Reynolds numbers (Re): (

**a**) Re = 234; (

**b**) Re = 468.

**Figure 16.**Variation of the performance evaluation criteria (PEC) versus the different volume fractions and mixing ratios of nanofluids for the cases of Reynolds numbers: (

**a**) Re = 234; (

**b**) Re = 468.

Parameters | Value | Unit |
---|---|---|

Total length | 0.36 | m |

Height | 0.0078 | m |

Density | 3.5 × 10^{3} | kg/m^{3} |

Thermal conductivity | 29 | W/(m∙K) |

Specific heat capacity | 750 | J/(kg∙K) |

Darcy number | Da = 10^{−5}, 10^{−4}, 10^{−3}, 10^{−2} | |

Material type | Al_{2}O_{3} |

No. | Grid Size | T_{out} | Percentage Difference (%) | s = 0.6 H, Re = 234, Da = 10^{−4} | Percentage Difference (%) |
---|---|---|---|---|---|

s = 0 | |||||

1 | 25 × 100 | 292.9 | 295.7 | ||

2 | 50 × 250 | 296.4 | 1.20 | 300.4 | 1.60 |

3 | 100 × 500 | 297.9 | 0.50 | 301.6 | 0.40 |

4 | 150 × 1000 | 298.2 | 0.08 | 301.9 | 0.09 |

5 | 200 × 1300 | 298.4 | 0.06 | 302.2 | 0.08 |

6 | 250 × 1500 | 298.5 | 0.05 | 302.4 | 0.07 |

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## Share and Cite

**MDPI and ACS Style**

Lin, X.; Xia, Y.; Cheng, Z.; Liu, X.; Fu, Y.; Li, L.; Zhou, W.
Thermal Performance Analysis of Porous Foam-Assisted Flat-Plate Solar Collectors with Nanofluids. *Sustainability* **2024**, *16*, 693.
https://doi.org/10.3390/su16020693

**AMA Style**

Lin X, Xia Y, Cheng Z, Liu X, Fu Y, Li L, Zhou W.
Thermal Performance Analysis of Porous Foam-Assisted Flat-Plate Solar Collectors with Nanofluids. *Sustainability*. 2024; 16(2):693.
https://doi.org/10.3390/su16020693

**Chicago/Turabian Style**

Lin, Xinwei, Yongfang Xia, Zude Cheng, Xianshuang Liu, Yingmei Fu, Lingyun Li, and Wenqin Zhou.
2024. "Thermal Performance Analysis of Porous Foam-Assisted Flat-Plate Solar Collectors with Nanofluids" *Sustainability* 16, no. 2: 693.
https://doi.org/10.3390/su16020693