# Comparative Study of a Fixed-Focus Fresnel Lens Solar Concentrator/Conical Cavity Receiver System with and without Glass Cover Installed in a Solar Cooker

^{1}

^{2}

## Abstract

**:**

^{®}7.0 software. An experimental setup was then constructed to test the thermal performance of the system. The results show that the optical efficiency of a system without a glass cover is much higher than that with a glass cover. The difference between them remains unchanged for incidence angle at a range of 0–20°. The time constant of the system with a glass cover is much less than that without a glass cover, in the ranges of 29–33 s and 48–59 s, respectively. The system with a glass cover for a wide range of higher temperature differences also has better thermal efficiency.

## 1. Introduction

^{®}7.0 software to obtain the optical efficiency of the system and flux distribution of the conical cavity receiver. An experimental setup of the system was built to test and analyze the system’s thermal performance under various concentrated sunlight incidence angles.

## 2. System Description

#### 2.1. FSC-C System

#### 2.2. Testing System

## 3. Optical Characteristics

#### 3.1. Definition of Optical Efficiency, Incidence Angle Modifier and Uniformity Factor

^{®}was employed to simulate ray-tracking for designing, analyzing, and optimizing optical and illumination systems. The angular subtense of the sun at any point on earth (ξ = 32′) was considered during ray tracing to improve the model’s accuracy. The variation of θ on the optical efficiency was evaluated, defined as the fraction of direct solar radiation intercepted by the Fresnel lens absorbed by the conical cavity receiver. The mathematical expression for the optical efficiency is given as follows:

_{ab}(W), I

_{d}(W/m

^{2}) represents the direct solar radiation on the Fresnel lens surface, and A

_{c}(m

^{2}) is the area of the Fresnel lens. To analyze the impact of the θ on the system’s optical performance, a global parameter K(θ) [25] is used, which considers the incidence angle modifier and the variation in the intercept factor of the conical cavity receiver. Therefore, the optical efficiency of the proposed system can be expressed as follows:

#### 3.2. The Radiation Flux Profile of the Absorbing Surface and the Optical Efficiency of the System

_{0}(θ) mainly decreases with the increase in θ. The η

_{0}(θ) changes slowly before θ = 25°and then falls quickly. For the case with a glass cover, the f is 980 mm, 985 mm, 990 mm, 995 mm, 1000 mm, 1005 mm, 1010 mm, 1015 mm, and 1020 mm. As θ = 0°, the η

_{0}(θ) are 0.6790, 0.6804, 0.6815, 0.6825, 0.6834, 0.6844, 0.6854, 0.6863, and 0.6871, respectively. As θ = 25°, the η

_{0}(θ) are 0.6769, 0.6782, 0.6792, 0.6798, 0.6802, 0.6813, 0.6786, 0.6782, and 0.6752, respectively. As θ = 60°, the η

_{0}(θ) are 0.4418, 0.4840, 0.5104, 0.5246, 0.5264, 0.5131, 0.4870, 0.4429, and 0.3839, respectively. For the case without a glass cover, the f is 980 mm, 985 mm, 990 mm, 995 mm, 1000 mm, 1005 mm, 1010 mm, 1015 mm, and 1020 mm. As θ = 0°, the η

_{0}(θ) are 0.7527, 0.7543, 0.7555, 0.7566, 0.7575, 0.7587, 0.7597, 0.7607, and 0.7614, respectively. As θ = 25°, the η

_{0}(θ) are 0.7571, 0.7583, 0.7506, 0.7598, 0.7601, 0.7530, 0.7554, 0.7564, and 0.7526, respectively. As θ = 60°, the η

_{0}(θ) are 0.5777, 0.6407, 0.6826, 0.7015, 0.6944, 0.6585, 0.5896, 0.5033, and 0.4156, respectively. It means that the η

_{0}(θ) of the FSC-C system is comparable to that of the general Fresnel lens solar collector system (the optical axis of the Fresnel lens coincides with the recenter axis of the cavity receiver) when the θ < 25°. To show the effect of the glass cover on the η

_{0}(θ) of the FSC-C system objectively, the optical efficiency difference between the groups with and without glass cover under various θ is shown in Figure 7. The η

_{0}(θ) of the group without a glass cover is higher than that with a glass cover. The optical efficiency difference changes obviously for the θ of 20–60° but remains unchanged for the θ of 0–20°. The closer the focal length is to 995 mm, the faster the difference increases with the increase in θ. Otherwise, the slower the difference increases and even decreases. For f, it is 980 mm, 985 mm, 990 mm, 995 mm, 1000 mm, 1005 mm, 1010 mm, 1015 mm, and 1020 mm. As θ = 0°, the optical efficiency difference is 0.0738, 0.0739, 0.0740, 0.0741, 0.0741, 0.0743, 0.0744, 0.0744, and 0.0743, respectively. As θ = 25°, the optical efficiency difference is 0.0802, 0.0801, 0.0713, 0.0800, 0.0799, 0.0717, 0.0768, 0.0782, and 0.0774, respectively. As θ = 60°, the optical efficiency difference is 0.1359, 0.1568, 0.1722, 0.1769, 0.1681, 0.1454, 0.1026, 0.0604, and 0.0317, respectively. These observations can be explained by the nature of natural light, which consists of entirely polarized light. When light travels through a transparent medium, the incident and emergent light are coplanar, with the normal plane being the plane of incidence. Additionally, the complete polarization of natural light can be decomposed into p-polarized and s-polarized components. As indicated by Ma et al. [27], the relationships of reflectivities of p-polarized lights R

_{p}and reflectivities of s-polarized lights R

_{s}to the incident angles are shown in Figure 8. The relative index of refraction of glass cover to air (n

_{21}) is 1.5. It is noted that the R

_{p}and R

_{s}change slightly before the θ = 20°, but the changing trend subsequently increases with the increasing of θ. Thus, the optical efficiency difference remains unchanged before θ = 20° but changes obviously for the θ of 20°–60°.

^{3}+ 0.0002θ

^{2}− 0.0031θ + 1.0055; R² = 0.9967

^{3}+ 0.0002θ

^{2}− 0.0031θ + 1.0059; R² = 0.9782

## 4. Thermal Performance

#### 4.1. Time Constant of the FSC-C System

_{o}(τ) represents the collector outlet fluid temperature (°C) at time τ (s), T

_{o,ss}represents the steady-state working fluid outlet temperature (°C), T

_{i}represents the collector inlet fluid temperature (°C).

#### 4.2. Thermal Efficiency of the FSC-C System

_{t}) of the FSC-C system can be calculated using the following equation [30,31]:

_{p}(kJ/kg/°C) is the heat capacity of the working fluid. The heat capacity of synthetic heat transfer oil can be found in Appendix A to evaluate the thermal efficiency of the FSC-C system under different θ. The instantaneous thermal efficiency equation of the FSC-C system, similar to other energy conversion devices, can be expressed as follows [32,33]:

_{r}(°C) represents the average inner surface temperature of the cavity receiver, while T

_{amb}(°C) is the ambient temperature, U

_{a}(W/m

^{2}/°C) is the overall heat loss coefficient based on T

_{r}, and A

_{r}(m

^{2}) is the area of the conical cavity receiver. Determining the exact value of T

_{r}can be challenging. However, the inlet and outlet temperatures of the conical cavity receiver can be easily measured. Therefore, the average temperature of the conical cavity receiver T

_{m}= (T

_{i}+ T

_{o})/2 can approximate T

_{r}in the efficiency equation [34]. The equation can be expressed as follows:

_{L}(W/m

^{2}/°C) is related to T

_{m}, the mean fluid temperature of the collector, and F′, the collector efficiency factor. F′ is the ratio of the actual useful energy gain to the gain if the collector absorbing surface was at the local fluid temperature. The value of F′ depends on the collector heat exchange structure. Using the above equations, the thermal efficiency of the FSC-C system can be calculated and plotted against the temperature difference (T

_{m}− T

_{amb})/I

_{d}.

## 5. Results and Discussion

#### 5.1. Time Constant of Different Incidence Angle

_{d}, wind speed V

_{w}, and ambient temperature T

_{amb}, with a heat transfer fluid discharge rate of 0.008 kg/s. The direct solar radiation exhibited an increase from 708 W/m

^{2}at solar time 9:00 to 781 W/m

^{2}at solar time 11:46. The wind speed varied between 0–3.2 m/s, while the ambient temperature fluctuated within the range of 20.84–22.24 °C throughout the day. The air quality was also assessed as “good”, considering a relative humidity of 72%, a PM2.5 concentration of 35, and a PM10 concentration of 55.

#### 5.2. Thermal Efficiencies of Different Incidence Angles

_{m}− T

_{amb})/I

_{d}, is shown in Figure 14a–d. Linear equations were used to fit the experimental data and provide the characteristic parameters of the FSC-C system. The values of F′η

_{0}(θ) and F′U

_{L}/C for the FSC-C system with and without a glass cover at each incidence angle are presented in Table 2.

_{0}(θ) value of the FSC-C system is obtained at an incident angle of 16°, while the lowest F′U

_{L}/C value is obtained at this angle. Hence, according to Equation (12), the thermal efficiency of the FSC-C system is the highest at this incident angle. Comparing the thermal efficiencies of the FSC-C system with and without glass cover under different incident angles, it is observed that the thermal efficiency with glass cover is higher than that without glass cover for a wide range of temperature differences. Based on the fitting curves, it can be observed that at θ = 16° and (T

_{m}− T

_{amb})/I

_{d}= 0.03, the η

_{t}for the cases with and without a glass cover are 0.5006 and 0.5304, respectively. At (T

_{m}− T

_{amb})/I

_{d}= 0.21, the η

_{t}for the cases with and without a glass cover are 0.2215 and 0.1688, respectively. At θ = 24° and (T

_{m}− T

_{amb})/I

_{d}= 0.03, the η

_{t}for the cases with and without a glass cover are 0.4887 and 0.5212, respectively. At (T

_{m}− T

_{amb})/I

_{d}= 0.21, the η

_{t}for the cases with and without a glass cover are 0.194959 and 0.139112, respectively. Similarly, at θ = 32° and (T

_{m}− T

_{amb})/I

_{d}= 0.03, the η

_{t}for the cases with and without a glass cover are 0.4765 and 0.5143, respectively. At (T

_{m}− T

_{amb})/I

_{d}= 0.21, the η

_{t}for the cases with and without a glass cover are 0.1612 and 0.0978, respectively. Lastly, at θ = 40° and (T

_{m}− T

_{amb})/I

_{d}= 0.03, the η

_{t}for the cases with and without a glass cover are 0.4515 and 0.4890, respectively. At (T

_{m}− T

_{amb})/I

_{d}= 0.21, the η

_{t}for the cases with and without a glass cover are 0.1119 and 0.0584, respectively. The value of F′η

_{0}(θ) is the dominant parameter for small temperature differences, i.e., (T

_{m}− T

_{amb})/I

_{d}< 0.0951, (T

_{m}− T

_{amb})/I

_{d}< 0.0962, (T

_{m}− T

_{amb})/I

_{d}< 0.0973 and (T

_{m}− T

_{amb})/I

_{d}< 0.1043 for θ of 16°, 24°, 32° and 40°, respectively, whereas F′U

_{L}/C is the dominant parameter for higher temperature differences. The value of F′η

_{0}(θ) without a glass cover is greater than that with a glass cover in the lower temperature differences range. Therefore, according to Equation (12), the thermal efficiency without a glass cover is greater than that with a glass cover in this range. However, in higher temperature differences range, i.e., (T

_{m}− T

_{amb})/I

_{d}> 0.0951, (T

_{m}− T

_{amb})/I

_{d}> 0.0962, (T

_{m}− T

_{amb})/I

_{d}> 0.0973 and (T

_{m}− T

_{amb})/I

_{d}> 0.1043 for θ of 16°, 24°, 32° and 40°, respectively, the value of F′U

_{L}/C with glass cover is smaller than that without glass cover. Thus, the thermal efficiency with a glass cover in this range is higher than that without a glass cover.

## 6. Conclusions

- (1)
- The optical efficiency of the FSC-C system without a glass cover is higher than that with a glass cover. The difference between them is more significant at incidence angles of 20–60°. The increase in differences is faster with the decrease in the focal length, while it is slower or even decreasing when increasing the focal length;
- (2)
- The incidence angle has a significant influence on optical efficiency. Two models were proposed to predict the effect of incidence angle on the optical performance of conical cavity receiver coupled fixed-focus Fresnel lens solar concentrator;
- (3)
- The time constant of the FSC-C system with a glass cover is less than that without a glass cover. The time constant for incidence angles 25°, 30°, 35°, and 40° with and without glass cover was determined as 31 s, 29 s, 31 s, 33 s and 49 s, 59 s, 57 s, 48 s, respectively. The incident angle variations during the experiment were small, with a maximum of 1.064° over a 5-min interval;
- (4)
- Comparing the thermal efficiencies of the FSC-C system with and without glass cover under different incidence angles, it is found that the thermal efficiency with glass cover is higher for a wide range of higher temperature differences. The parameters F′η
_{0}(θ) and F′U_{L}/C dominate at lower and higher temperature differences, respectively.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

δ | Sun declination angle (degree) |

η_{0} | Optical efficiency |

η_{t} | Instantaneous thermal efficiency |

θ | Concentrated sunlight incidence angle (degree) |

τ | Time constant (s) |

Φ | Local latitude angle (degree) |

φ | Tilt angle of cavity receiver with the horizontal (degree) |

ω | Solar hour angle (degree) |

A_{c} | Area of Fresnel lens (m^{2}) |

A_{r} | Area of conical cavity receiver (m^{2}) |

C | Geometric concentrating ratio |

c_{p} | Heat capacity of working fluid (kJ/kg/°C) |

D | Aperture diameter of Fresnel lens (mm) |

f | Focal length of Fresnel lens (mm) |

F′ | Collector efficiency factor |

I_{d} | Direct solar radiation (W/m^{2}) |

K | Global incidence angle modifier |

$\stackrel{\xb7}{m}$ | Mass flow rate of fluid flow (kg/s) |

Q_{ab} | Radiation energy received by the cavity receiver surface (W) |

T_{amb} | Ambient temperature (°C) |

T_{in} | Collector inlet fluid temperature (°C) |

T_{m} | Average temperature of conical cavity receiver (°C) |

T_{o} | Collector outlet fluid temperature (°C) |

T_{o,ss} | Steady-state working fluid outlet temperature (°C) |

T_{r} | Average inner surface temperature of cavity receiver (°C) |

U_{a} | Overall heat loss coefficient base on Tr (W/m^{2}/°C) |

U_{L} | Overall heat loss coefficient base on Tm (W/m^{2}/°C) |

Acronyms | |

FSC-C | Fixed-focus Fresnel lens solar concentrator/conical cavity receiver |

UF | Uniformity factor |

## Appendix A

^{3}at 25 °C

Temperature (°C) | 20 | 50 | 80 | 110 | 140 | 170 | 200 |

Heat capacity (kJ/kg/°C) | 1.625 | 1.730 | 1.850 | 1.945 | 2.045 | 2.150 | 2.255 |

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**Figure 5.**Rays and the flux distribution for the conical cavity receiver with five representative incidence angles (f = 1000 mm, the green rays represent the incident sunlight, while the blue and red rays represent the sunlight after reflection or re-reflection).

**Figure 8.**Characteristics of reflectivities of p-polarized lights and s-polarized lights change with incidence angles.

**Figure 12.**(

**a**)The time constant for FSC-C system with glass cover; and (

**b**) the time constant for FSC-C system without glass cover.

**Figure 13.**The changes of incident angle variations with solar time for different time-intervals (

**a**) and solar declination angles (

**b**).

**Figure 14.**Thermal efficiency of the FSC-C system with and without glass cover at various incidence angles (

**a**) 16°, (

**b**) 24°, (

**c**) 32°, and (

**d**) 40°.

No. | Measurements | Type | Uncertainty |
---|---|---|---|

1 | Temperature | PT-100 sensor | ±0.1 °C |

2 | Mass flow rate | Flow meter | 0.5 lpm |

3 | Wind speed | Hot wire anemometer | ±0.01 m/s |

4 | Direct solar irradiance | TBS-2-2 | 2% |

5 | Tracking accuracy | Polar-axis tracking mechanism | ±0.1° |

System Type | Incident Angle | 16° | 24° | 32° | 40° |
---|---|---|---|---|---|

With glass cover | F′η_{0}(θ) | 0.5471 | 0.5377 | 0.529 | 0.5081 |

Without glass cover | F′U_{L}/C | 1.5503 | 1.6321 | 1.7514 | 1.8869 |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.
Comparative Study of a Fixed-Focus Fresnel Lens Solar Concentrator/Conical Cavity Receiver System with and without Glass Cover Installed in a Solar Cooker. *Sustainability* **2023**, *15*, 9450.
https://doi.org/10.3390/su15129450

**AMA Style**

Wang H.
Comparative Study of a Fixed-Focus Fresnel Lens Solar Concentrator/Conical Cavity Receiver System with and without Glass Cover Installed in a Solar Cooker. *Sustainability*. 2023; 15(12):9450.
https://doi.org/10.3390/su15129450

**Chicago/Turabian Style**

Wang, Hai.
2023. "Comparative Study of a Fixed-Focus Fresnel Lens Solar Concentrator/Conical Cavity Receiver System with and without Glass Cover Installed in a Solar Cooker" *Sustainability* 15, no. 12: 9450.
https://doi.org/10.3390/su15129450