# Performance Comparison of Different Flat Plate Solar Collectors by Means of the Entropy Generation Rate Using Computational Fluid Dynamics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}). The velocities, pressures, and temperature contours of the three geometries were compared under the average climatic conditions that prevail in the state of Guanajuato, Mexico. Finally, the global and local entropy generation rates due to fluid viscosity, heat losses, and heat transfer were discussed.

## 2. Geometries and Mathematical Model

#### 2.1. Configurations of the Geometries of the Solar Collectors

#### 2.2. Equations Used in the Model

#### 2.3. Computing the Irreversibilities Using the Entropy Generation Rate

#### 2.4. Boundary Conditions of the Model

^{2}∙day, for one year in the state of Guanajuato, Mexico, was used. A pressure outlet boundary condition was applied at the exit of the FPC, and a heat flux (${\dot{Q}}_{useful}$) in the walls of the FPC was defined. The useful heat was obtained analytically by subtracting the optical and thermal losses in the tubes and the headers from the total amount of heat received due to solar radiation, as presented by Li et al. [29] and Budihardjo et al. [30]. Experimentally, the useful heat is the heat required for increasing the temperature of the water between the entrance and the exit (Equation (14)). All these boundary conditions were considered in the CFD numerical model in order to simulate the effect of solar radiation and predict the thermal and hydraulic performance of the flat plate solar collectors.

#### 2.5. Numerical Approach

^{®}v.18.1 software to solve the governing equations described in Section 2.2 and Section 2.3 (see Figure 2). The SIMPLE algorithm was used to solve Equations (1)–(7) and to obtain the velocity and temperature fields inside the FPC. The convergence criterion was considered once the residuals were lower than 10

^{−6}. Finally, the entropy generation analysis was computed through user-defined functions (UDFs) (Equations (8)–(15)).

## 3. Results

#### 3.1. Thermal Performance Comparison of the FPCs: Variable Volumetric Flow Rates

^{2}∙day. As depicted, the outlet temperatures obtained in the zigzag type B (case 3) geometry were higher than those in the conventional (case 1) and zigzag type A (case 2) geometries. The highest temperature of the water at the outlet of the solar collector, approximately 324.5 K, was achieved in the zigzag type B geometry at a volumetric flow rate of 1.0 L/min, while the lowest temperature of the water at the outlet of the solar collector, approximately 303 K, was achieved in the conventional case (case 1) at a volumetric flow rate of 9.0 L/min. Moreover, the maximum temperature differences were reached at 1.0 L/min, where the temperature difference between case 3 and case 1 was approximately 7.4 K, and the temperature difference between case 3 and case 2 was approximately 3.1 K. The minimum temperature differences were reached at 9.0 L/min, where the temperature difference between case 3 and case 1 was approximately 1.5 K, and between case 3 and case 2, the temperature difference was approximately 0.6 K (Figure 4a). It is important to highlight that this behavior was mainly due to the differences in the area of the walls of the parallel tubes. Case 3 had a larger wall area than conventional case 1, which was 62% greater than case 1 and 31% greater than case 2. Despite this, the three geometries occupied the same area of 1.75 m

^{2}, as shown in Figure 1.

#### 3.2. Performance Comparison of the FPCs at 3.0 L/min

#### 3.3. Entropy Generation Analysis

_{h}), fluid viscosity (S

_{µ}), and heat loss (S

_{q}), and the global entropy generation rates (S

_{total}) at different volumetric flow rates ranging from 1.0 to 9.0 L/min for the three cases. In general, for all three cases, S

_{h}decreased as the volumetric flow rate increased. For example, at volumetric flow rates of 1.0, 5.0, and 9.0 L/min, the conventional geometry had S

_{h}values of approximately 0.0955, 0.0639, and 0.0384 W/K, respectively, while the zigzag type A had S

_{h}values of approximately 0.0954, 0.0755 and 0.0457 W/K, respectively, and the zigzag type B had S

_{h}values of approximately 0.1276, 0.1070 and 0.0545 W/K, respectively. These findings indicate that the temperature gradients inside the FPC tubes decreased when the volumetric flow rates increased. As can be observed, the S

_{h}values between cases 1 and 2 were almost the same, 0.095 W/K, for volumetric flow rates lower than 2.5 L/min, while case 3 had an increase of approximately 40% at the same volumetric flow rates. Moreover, as the volumetric flow rate increased from 2.5 to 9.0 L/min, the entropy generation due to heat transfer was higher in case 2 compared to the case 1, up to 25%. Although the trends in the entropy generation decreased in all cases as the volumetric flow rate increased, the maximum difference in the entropy generation between case 1 and case 3 was approximately 70%, and between case 3 and 2 was approximately 42%. Furthermore, the entropy generation rate due to heat transfer for cases 1, 2, and 3 diminished to 60%, 52%, 57%, respectively, for the lowest volumetric flow rate (1.0 L/min) and the highest volumetric flow rate (9.0 L/min).

_{µ}, is shown in Figure 9. As was expected, the entropy due to fluid viscosity increased as the volumetric flow rate increased. Figure 9 shows that the lowest and highest entropy generation rates for the FPC in all the cases were reached at 1.0 L/min and 9.0 L/min, respectively. The maximum differences in the entropy generation rate due to the fluid viscosity between case 3 and case 1 were more than twice the value of case 1 at 9.0 L/min. This variation between the conventional and the zigzag type B geometries was due to the zigzag effect in fluid flow. In other words, the fluid flow through the parallel tubes from the lower header to the upper header mostly flowed in one direction, whereas the fluid flow through the zigzag geometry experienced several deviations to arrive at the upper header from the lower header. Hence, the difference in S

_{µ}between case 3 and case 2 at 9.0 L/min was smaller, approximately 23%, as both geometries had the same zigzag effect. Case 3 had the highest S

_{µ}value because its geometry had longer tubes compared to case 2.

_{q}, for the conventional (case 1), zigzag type A (case 2), and zigzag type B (case 3) configurations of the FPCs, considering volumetric flow rates ranging from 1.0 L/min to 9.0 L/min, is illustrated in Figure 10. As can be observed, there was a decrease in the values of S

_{q}for all three cases at volumetric flow rates of 1.0 L/min to 9 L/min. For example, in case 1, the S

_{q}values for volumetric flow rates of 1, 2, 3, and 5 L/min were 3.1334, 1.9730, 1.9634 and 1.9588 W/K, respectively. In case 2, the S

_{q}values for volumetric flow rates of 1, 2, 3, and 5 L/min were 3.9952, 2.3223, 2.3130 and 2.3127 W/K, respectively. Similarly, in case 3, the S

_{q}values for volumetric flow rates of 1, 2, 3, and 5 L/min were 5.6719, 3.04851, 3.0375 and 3.0334 W/K, respectively. It is worth mentioning that the entropy generation rate due to heat loss was mainly related to the absorptivity and transmittivity of the material used in the collectors, along with the weather conditions such as ambient temperature, wind velocity, direct radiation, diffuse radiation, and total radiation. For all the volumetric flow rates, the highest S

_{q}values were related to case 3, and the lowest values were related to case 1. The maximum difference between case 3 and case 1 was approximately 81%, and for case 3 and 2 it was approximately 42%, and both were observed at 1.0 L/min. These results were due to differences in the area of the walls of the parallel tubes, as was discussed in Section 3.1.

_{total}, for the conventional (case 1), zigzag type A (case 2), and zigzag type B (case 3) configurations of the FPCs considering volumetric flow rates of 1.0 L/min to 9.0 L/min. As expected, the S

_{total}exhibited the same trend as the S

_{q}(Figure 10). This behavior was due to the significant contribution of the entropy generation rate due to the heat loss associated with the construction materials (pipes, headers, type of cover) of the FPCs and the related phenomena such as optical and heat transfer losses by solar radiation.

_{µ}were smaller in comparison to the entropy generation due to heat transfer and heat loss. Therefore, it can be established that for the operating conditions considered for these three geometries, S

_{µ}was negligible. It can also be observed that the maximum contribution of the S

_{h}to the S

_{total}was approximately 4.2% for the zigzag type B geometry with a volumetric flow rate of 3.0 L/min.

#### 3.4. Maps of the Local Entropy Generation Rates inside the Tubes of the FPCs at 3.0 L/min

_{h}and S

_{µ}in the interior of the pipes for the conventional (case 1), zigzag type A (case 2), and zigzag type B (case 3) configurations of the FPCs, considering a volumetric flow rate of 3.0 L/min, are illustrated in Figure 12 and Figure 13. It can be seen that the local S

_{h}throughout the lower and upper headers and the pipes of the FPCs (Figure 12) was related to the high temperature difference between the water and the surface of the pipe (Figure 5). Therefore, the zones with high entropy generation rates are related to the zones where the temperature gradients are high (Figure 5). For example, the conventional geometry exhibited the highest temperature gradients in the lower and upper headers, opposite the inlet and the outlet of the fluid, respectively. Consequently, these zones showed the highest values in the entropy generation rate due to heat transfer (Figure 5 and Figure 12).

_{µ}for the three FPCs occurred near the walls of the tubes, in the sections where the fluid flow of the lower header was divided and distributed in the seven tubes of the FPCs, and in the section where the seven tubes fed the upper header of the FPCs (Figure 13). Finally, a higher entropy generation was observed in the zigzag geometries, specifically in cases 2 and 3, in the areas where the zigzag was formed. These behaviors were related to the velocity gradients that were inside the tubes of the FPCs (Figure 7). However, as discussed previously, this contribution of the S

_{µ}was insignificant.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ahmadia, A.; Ehyaei, M.A.; Doustgani, A.; El, A.M.; Hmida, A.; Jamalif, D.H.; Kumarg, R.; Li, Z.X.; Razmjoo, A. Recent residential applications of low-temperature solar collector. J. Clean. Prod.
**2021**, 279, 123549. [Google Scholar] [CrossRef] - Chow, T.T.; Dong, Z.; Chan, L.S.; Fong, K.F.; Bai, Y. Performance evaluation of evacuated tube solar domestic hot water systems in Hong Kong. Energy Build.
**2011**, 43, 3467–3474. [Google Scholar] [CrossRef] - Bhatia, S.C. Solar thermal energy, chapter 4. In Advanced Renewable Energy Systems, 1st ed.; Taylor & Francis Group: New York, NY, USA, 2014; pp. 94–143. [Google Scholar] [CrossRef]
- Tang, R.; Cheng, Y.; Wu, M.; Li, Z.; Yu, Y. Experimental and modeling studies on thermosiphon domestic solar water heaters with flat-plate collectors at clear nights. Energy Convers. Manag.
**2010**, 51, 2548–2556. [Google Scholar] [CrossRef] - Elsheikh, A.H.; Sharshir, S.W.; Mostafa, M.E.; Essa, F.A.; Ali, M.K. Applications of nanofluids in solar energy: A review of recent advances. Renew. Sustain. Energy Rev.
**2018**, 82, 3483–3502. [Google Scholar] [CrossRef] - Olfian, H.; Ajarostaghi, S.S.; Ebrahimnataj, M. Development on evacuated tube solar collectors: A review of the last decade results of using nanofluids. Sol. Energy
**2020**, 211, 265–282. [Google Scholar] [CrossRef] - Ashour, A.F.; Ahmed, E.A.; Mohsen, A.T. Numerical investigation on the thermal performance of a flat plate solar collector using ZnO & CuO water nanofluids under Egyptian weathering conditions. Energy
**2022**, 240, 122743. [Google Scholar] [CrossRef] - Cao, Y.; Ayed, H.; Hashemian, M.; Issakhov, A.; Jarad, F.; Makatar, W. Inducing swirl flow inside the pipes of flat-plate solar collector by using multiple nozzles for enhancing thermal performance. Renew. Energy
**2021**, 180, 1344–1357. [Google Scholar] [CrossRef] - Vengadesan, E.; Senthil, R. Experimental study on the thermal performance of a flat plate solar water collector with a bifunctional flow insert. Sustain. Energy Technol. Assess.
**2022**, 50, 101829. [Google Scholar] [CrossRef] - Wang, D.; Mo, Z.; Liu, Y.; Ren, Y.; Fan, J. Thermal performance analysis of large-scale flat plate solar collectors and regional applicability in China. Energy
**2022**, 238, 121931. [Google Scholar] [CrossRef] - Azad, E. Experimental analysis of thermal performance of solar collectors with different numbers of heat pipes versus a flow-through solar collector. Renew. Sustain. Energy Rev.
**2018**, 82, 4320–4325. [Google Scholar] [CrossRef] - Alwan, N.T.; Shcheklein, S.E.; Ali, O.M. Experimental analysis of thermal performance for flat plate solar water collector in the climate conditions of Yekaterinburg, Russia. Mater. Today Proc.
**2021**, 42, 2076–2083. [Google Scholar] [CrossRef] - Hashim, W.M.; Shomran, A.T.; Jurmut, H.A.; Gaaz, T.S.; Kadhum, A.A.H.; Al-Amiery, A.A. Case study on solar water heating for flat plate collector. Case Stud. Therm. Eng.
**2018**, 12, 666–671. [Google Scholar] [CrossRef] - Wei, L.; Yuan, D.; Tang, D.; Wu, B. A study on a flat-plate type of solar heat collector with an integrated heat pipe. Sol. Energy
**2013**, 97, 19–25. [Google Scholar] [CrossRef] - Kargarsharifabad, H.; Behshad, S.M.; Taeibi, R.M.; Abbaspour, M. Exergy Analysis of a Flat Plate Solar Collector in Combination with Heat Pipe. Int. J. Environ. Res.
**2014**, 8, 39–48. [Google Scholar] [CrossRef] - Mansour, M.K. Thermal analysis of novel minichannel-based solar flat-plate collector. Energy
**2013**, 60, 333–343. [Google Scholar] [CrossRef] - Robles, A.; Duong, V.; Martin, A.J.; Guadarrama, J.L.; Diaz, G. Aluminum minichannel solar water heater performance under year-round weather conditions. Sol. Energy
**2014**, 110, 356–364. [Google Scholar] [CrossRef] - Deng, Y.; Zhao, Y.; Wang, W.; Quan, Z.; Wang, L.; Yu, D. Experimental investigation of performance for the novelflat platesolar collector with micro-channel heat pipe array (MHPA-FPC). Appl. Therm. Eng.
**2013**, 54, 440–449. [Google Scholar] [CrossRef] - Azad, E. Assessment of three types of heat pipe solar collectors. Renew. Sustain. Energy Rev.
**2012**, 16, 2833–2838. [Google Scholar] [CrossRef] - Dovic, D.; Andrassy, M. Numerically assisted analysis of flat and corrugated plate solar collectors thermal performances. Sol. Energy
**2012**, 86, 2416–2431. [Google Scholar] [CrossRef] - García, A.; Herrero-Martin, R.; Solano, J.P.; Pérez-García, J. The role of insert devices on enhancing heat transfer in a flat-plate solar water collector. Appl. Therm. Eng.
**2018**, 132, 479–489. [Google Scholar] [CrossRef] - Gunjo, D.G.; Mahanta, P.; Robi, P.S. Exergy and energy analysis of a novel type solar collector under steady state condition: Experimental and CFD analysis. Renew. Energy
**2017**, 114, 655–669. [Google Scholar] [CrossRef] - Alim, M.A.; Abdin, Z.; Saidur, R.; Hepbasli, A.; Khairul, M.A.; Rahim, N.A. Analyses of entropy generation and pressure drop for a conventional flat plate solar collector using different types of metal oxide nanofluids. Energy Build.
**2013**, 66, 289–296. [Google Scholar] [CrossRef] - Jilani, J.; Thomas, C. Effect of thermo-geometric parameters on entropy generation in absorber plate fin of a solar flat plate collector. Energy
**2014**, 70, 35–42. [Google Scholar] [CrossRef] - Jouybari, H.J.; Saedodin, S.; Zamzamian, A.; Nimvari, M.E. Experimental investigation of thermal performance and entropy generation of a flat-plate solar collector filled with porous media. Appl. Therm. Eng.
**2017**, 127, 1506–1517. [Google Scholar] [CrossRef] - Alklaibi, A.M.; Sundar, L.S.; Sousa, A.C.M. Experimental analysis of exergy efficiency and entropy generation of diamond/water nanofluids flow in a thermosyphon flat plate solar collector. Int. Commun. Heat Mass Transf.
**2019**, 120, 105057. [Google Scholar] [CrossRef] - Launder, B.; Spalding, D. Lectures in Mathematical Models of Turbulence, 1st ed.; Academic Press: London, UK, 1972. [Google Scholar]
- Takashima, T.; Tanaka, T.; Doi, T.; Kamoshida, J.; Tani, T.; Horigome, T. New proposal for photovoltaic-thermal solar energy utilization method. Sol. Energy
**1994**, 52, 241–245. [Google Scholar] [CrossRef] - Li, Z.; Chen, C.; Luo, H.; Zhang, Y.; Xue, Y. All-glass vacuum tube collector heat transfer model used in forced-circulation solar water heating system. Sol. Energy
**2010**, 84, 1413–1421. [Google Scholar] [CrossRef] - Budihardjo, I.; Morrison, G.L. Performance of a water-in-glass evacuated tube solar water heater. Sol. Energy
**2009**, 83, 49–56. [Google Scholar] [CrossRef] - Zambolin, E.; Del, C.D. Experimental analysis of thermal performance of flat plate and evacuated tube solar collectors in stationary standard and daily conditions. Sol. Energy
**2010**, 84, 1382–1396. [Google Scholar] [CrossRef]

**Figure 3.**Comparison of the average temperature at the outlet of the flat plate solar collector: model developed and experimental data [31].

**Figure 4.**Performance comparisons at different volumetric flows: (

**a**) temperature outlet, and (

**b**) pressure drop.

**Figure 5.**Temperature distribution inside the FPCs at 3.0 L/min: (

**a**) conventional, (

**b**) zigzag type A, and (

**c**) zigzag type B.

**Figure 6.**Pressure distribution inside the FPCs at 3.0 L/min: (

**a**) conventional, (

**b**) zigzag type A, and (

**c**) zigzag type B.

**Figure 7.**Velocity distribution inside the FPCs at 3.0 L/min: (

**a**) conventional, (

**b**) zigzag type A, and (

**c**) zigzag type B.

**Figure 12.**Local entropy generation rate due to heat transfer inside the FPCs at 3.0 L/min: (

**a**) conventional, (

**b**) zigzag type A, and (

**c**) zigzag type B.

**Figure 13.**Local entropy generation rate due to fluid viscosity inside the FPCs at 3.0 L/min: (

**a**) conventional, (

**b**) zigzag type A, and (

**c**) zigzag type B.

Description | Specification |
---|---|

Number of tubes, [-] | 7 |

External diameter, [m] | 0.0254 |

Internal diameter, [m] | 0.025 |

Width of the FPC, [m] | 1 |

Length of the FPC, [m] | 1.75 |

Area of the FPC, [m^{2}] | 1.75 |

Horizontal inclination of the FPC, [°] | 21 |

Density, [kg∙m^{−3}] | Boussinesq, Equation (7) |

Thermal expansion coefficient, [K^{−1}] | 0.000206 |

Specific heat, [J∙kg^{−1}∙K^{−1}] | 4182 |

Thermal conductivity, [W/m∙K] | 0.6 |

Viscosity, kg∙m^{−1}∙s^{−1} | 001003 |

Number of Elements | Average Temperature [K] | Variation between the Previous Value of Average Temperature [K] | Convergence Time [Hour] |
---|---|---|---|

30,253 | 306.50 | - | 0.58 |

82,341 | 307.22 | 0.72 | 1.5 |

278,864 | 307.71 | 0.49 | 5.5 |

649,887 | 308.07 | 0.36 | 8.2 |

1,734,153 | 308.06 | 0.01 | 17.5 |

Number of Elements | Average Temperature [K] | Variation between the Previous Value of Average Temperature [K] | Convergence Time [Hour] |
---|---|---|---|

516,833 | 308.90 | - | 7.5 |

1,260,857 | 309.85 | 0.95 | 13.8 |

2,501,458 | 309.81 | 0.04 | 22.6 |

Number of Elements | Average Temperature [K] | Variation between the Previous Value of Average Temperature [K] | Convergence Time [Hour] |
---|---|---|---|

405,458 | 310.50 | - | 6.7 |

906,544 | 311.84 | 1.34 | 11.0 |

1,923,992 | 312.63 | 0.79 | 21.2 |

4,589,410 | 312.52 | 0.11 | 39.8 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramírez-Minguela, J.J.; Rangel-Hernández, V.H.; Alfaro-Ayala, J.A.; Elizalde-Blancas, F.; Ruiz-Camacho, B.; López-Núñez, O.A.; Alvarado-Rodríguez, C.E.
Performance Comparison of Different Flat Plate Solar Collectors by Means of the Entropy Generation Rate Using Computational Fluid Dynamics. *Entropy* **2023**, *25*, 621.
https://doi.org/10.3390/e25040621

**AMA Style**

Ramírez-Minguela JJ, Rangel-Hernández VH, Alfaro-Ayala JA, Elizalde-Blancas F, Ruiz-Camacho B, López-Núñez OA, Alvarado-Rodríguez CE.
Performance Comparison of Different Flat Plate Solar Collectors by Means of the Entropy Generation Rate Using Computational Fluid Dynamics. *Entropy*. 2023; 25(4):621.
https://doi.org/10.3390/e25040621

**Chicago/Turabian Style**

Ramírez-Minguela, J. J., V. H. Rangel-Hernández, J. A. Alfaro-Ayala, F. Elizalde-Blancas, B. Ruiz-Camacho, O. A. López-Núñez, and C. E. Alvarado-Rodríguez.
2023. "Performance Comparison of Different Flat Plate Solar Collectors by Means of the Entropy Generation Rate Using Computational Fluid Dynamics" *Entropy* 25, no. 4: 621.
https://doi.org/10.3390/e25040621