# A Realistic Approach of the Maximum Work Extraction from Solar Thermal Collectors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review up Today

#### 2.1. Maximum Work Extraction from Solar Energy

_{ex}) is given. The ambient (reference) temperature (T

_{amb}) can be taken at 298 K (for instance) and the sun temperature (T

_{sun}) at 5770 K.

#### 2.2. Maximum Work Extraction Using Solar Collectors

_{r}) of the solar collector has been suggested to be calculated by solving the Equation (6) [13,16].

_{L}). The optimum temperature, in this case, was calculated according to Equation (7):

## 3. Suggested Model

#### 3.1. General Description

- -
- The optical efficiency of the collector (η
_{opt}), which is the product of the receiver absorbance, the cover transmittance, the concentrator reflectance; it also includes the impact of the incident angle. - -
- The radiation thermal losses of the receiver are taken into account using the emittance of the receiver (ε
_{r}). This parameter usually takes low values (close to 0.05~0.10) for high-quality collectors. It has been assumed the collector exchanges irradiation with the ambient and not with the sky. - -
- The solar irradiation is considered as heat and not irradiation in the model, something that is not performed in similar studies [13].
- -
- The convection losses are neglected because by using vacuum between glass cover and absorber (in parabolic trough collectors, for instance), these thermal losses become negligible. This assumption is reasonable because the present work aims to determine the maximum work extraction using a real concentrating solar collector. So, this assumption corresponds to a high quality concentrating collector. In any case, an increase in the emittance (about 0.1) can be performed in order to take into account the convection thermal losses.

_{opt}).

#### 3.2. Mathematical Description

#### 3.2.1. Energy Balance in the Receiver

#### 3.2.2. Energy Balance in the Thermal Engine

#### 3.2.3. Total System Energy Balance

#### 3.2.4. Optimum Operation Temperature

_{r,opt}) is calculated. By substituting the value (X

_{opt}) in Equation (16), the maximum possible system efficiency is calculated. By knowing this parameter, the maximum work (W

_{max}) is calculated easily, as the product of maximum system efficiency to the available solar energy (Q

_{solar}).

## 4. Results—Discussion

#### 4.1. Solution of the Optimization Polynomial

_{opt}) and (q), for a great range of (q) values. This great range practically covers all the possible solar collectors.

#### 4.2. Impact of Design Parameters in the System Performance

_{amb}) was selected to be 298 K, and the solar irradiation (G) was kept constant at 1000 W/m

^{2}, which are typical values for the evaluation of solar collectors. This solar irradiation corresponds to the possible irradiation utilized from every collector in every case. More specifically, in flat technologies, the examined irradiation includes both beam and diffuse parts, while in concentrating only the beam part. At this point, it is important to state that the solar irradiation value of 100 W/m

^{2}is a high value that can be obtained close to solar noon in the summer period (for instance June or July). However, this is a typical value for the theoretical investigation of the solar thermal systems.

_{opt}), and the absorber emittance (ε

_{r}). Figure 4a–c illustrates the system efficiency, while Figure 4d–f gives the optimum receiver temperature. It is obvious that higher concentration ratio increases the system efficiency and the optimum operating temperature. Moreover, lower emittance leads to higher system efficiency and simultaneously to greater optimum temperature. The optical efficiency influences the system efficiency in a direct way, while it has the lower impact on the optimum temperature level. More specifically, lower optical efficiency causes the optimum temperature to decrease slightly, while the decrease in system efficiency is extremely great.

^{2}. It is important to state that the emittance in the flat plate case is increased in order to include the heat convection losses.

#### 4.3. The Application of the Model in a Real Solar Collector

_{L}. In this case, this parameter was calculated from Equation (24), as Equation (26) shows:

#### 4.4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | Area, m^{2} |

C | Concentration ratio, - |

f | Geometric factor, - |

G | Solar irradiation, W/m^{2} |

p | Efficiency parameter, - |

q | Polynomial parameter, - |

Q | Heat or radiant energy, W |

T | Temperature, K |

U_{L} | Thermal loss coefficient, W/m^{2} K |

W | Work, W |

X | Ratio of receiver temperature to ambient temperature, - |

Z | Collector quality parameter, - |

Greek symbols | |

α | Absorbance, - |

γ | Intercept factor, - |

δ | Cone half-angle, ° |

ε | Emittance, - |

η | Efficiency, - |

θ | Incident angle, ° |

ρ | Reflectance, - |

σ | Stefan–Boltzmann constant, [=5.67 × 10^{−8} W/m^{2} K^{4}] |

τ | Transmittance, - |

Subscripts and Superscripts | |

a | Aperture |

amb | Ambient |

carnot | Carnot cycle |

coll | Collector |

ex | exergy |

loss | Thermal losses |

max | Maximum |

opt | Optimum |

out | Output heat |

r | Receiver |

r,opt | Optimum receiver |

solar | Solar energy |

sun | Sun |

sys | System |

th | Thermal |

u | Useful |

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**Figure 3.**Maximum system efficiency as a function of parameter (p) for various optical efficiency values.

**Figure 4.**Maximum system efficiency (

**a**,

**b**,

**c**) and optimum receiver temperature (

**d**,

**e**,

**f**) for various values of the collector parameters.

Collector | C | η_{opt} | ε_{r} | T_{r,opt} (K) | η_{sys} |
---|---|---|---|---|---|

Flat plate collector | 1 | 0.8 | 0.25 | 401 | 0.1449 |

Evacuated tube collector | 1 | 0.7 | 0.10 | 451 | 0.1777 |

Compound parabolic collector | 3 | 0.6 | 0.10 | 527 | 0.2079 |

Parabolic trough collector | 30 | 0.75 | 0.10 | 884 | 0.4400 |

Solar dish collector | 100 | 0.75 | 0.10 | 1110 | 0.5011 |

C | η_{opt} | ε_{r} | T_{amb} (K) | G (W/m^{2}) |
---|---|---|---|---|

14.36 | 0.79 | 0.3 | 298 | 1000 |

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Bellos, E.; Tzivanidis, C. A Realistic Approach of the Maximum Work Extraction from Solar Thermal Collectors. *Appl. Syst. Innov.* **2018**, *1*, 6.
https://doi.org/10.3390/asi1010006

**AMA Style**

Bellos E, Tzivanidis C. A Realistic Approach of the Maximum Work Extraction from Solar Thermal Collectors. *Applied System Innovation*. 2018; 1(1):6.
https://doi.org/10.3390/asi1010006

**Chicago/Turabian Style**

Bellos, Evangelos, and Christos Tzivanidis. 2018. "A Realistic Approach of the Maximum Work Extraction from Solar Thermal Collectors" *Applied System Innovation* 1, no. 1: 6.
https://doi.org/10.3390/asi1010006