# Design and Optical Performance of Compound Parabolic Solar Concentrators with Evacuated Tube as Receivers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of Compound Parabolic Solar Concentrators with All-Glass Evacuated Solar Tube as Receiver

#### 2.1. Geometry of Compound Parabolic Solar Concentrator with a Tubular Absorber

_{a}) and upper reflector (0.5π + θ

_{a}< $\mathsf{\phi}$ ≤ 1.5π – θ

_{a}), and one can derive the expression of $\mathsf{\rho}$ from the string method as follows:

_{ap}) to the perimeter (P

_{abs,d}) of absorber based on which the reflectors of CPC are constructed, is uniquely determined by its acceptance half-angle (θ

_{a}) and is given by:

_{ap}= C

_{ideal}·P

_{abs,d}

_{t}into Equations (1) and (2) as follows:

_{t}is the edge-ray angle of CPCs after truncation (Figure 1), and θ

_{t}= θ

_{a}for full CPCs. The last term “0.5πr” in Equation (6) is the vertical depth of lowest point ($\mathsf{\phi}$ = 0.5π, a point on the involutes corresponding to dy/dx = 0) of reflectors relative to the x-axis. In turn, given C

_{t}and θ

_{a}, the edge-ray angle (θ

_{t}) can be obtained from Equation (5) by iterative calculations.

_{ap}) to the perimeter of the actual solar absorber (i.e., the inner tube of EST, P

_{abs,a}= 2πr) which might differ from P

_{abs,d}as seen in the next section.

#### 2.2. Design of Compound Parabolic Solar Concentrator with All-Glass Evacuated Tube as the Receiver

_{abs,d}is the perimeter of cover tube (2πR) instead of 2πr because CPC-1 is designed based on the cover tube of EST. In this case, the r in Equation (1) is set to be R, and $\mathsf{\rho}$ in this case is given by:

_{t}and θ

_{a}, the depth of CPC-1 is calculated by:

_{g,2}= 1/sinθ

_{a}due to P

_{abs,d}= P

_{abs,a}= 2πr, and the $\mathsf{\rho}$ in Equation (1) is given by Equation (2).

_{g,3}= 1/sinθ

_{a}, but the construction of involutes starts at point B of the cover tube (Figure 4 ) with φ = ϕ, thus the $\mathsf{\rho}$ in Equation (1) is given by:

_{ice}= (2π – 2ϕ)r + 2AB is the circumference of “ice-cream” shaped receiver, cos ϕ = r/R (due to OA = R), and AB = $\sqrt{{R}^{2}-{r}^{2}}$. The construction of involutes in this case starts at the lowest point (A) with φ = ϕ, and the $\mathsf{\rho}$ in Equation (1) for φ = ϕ is AB which is obviously larger than r·ϕ. Let AB = r(ϕ + γ), thus one has:

_{t}and θ

_{a}, the depth of CPC-4 is calculated by:

_{hat}= (2π – 4ϕ)r + 2BC is the circumference of “hat” shaped absorber, and cosϕ = r/R. As seen from Figure 6, the involute starts at the lowest point (A) of the “hat” with φ = 2ϕ, and the ρ in Equation (1) for φ = 2ϕ is BC (BC = $\sqrt{{R}^{2}-{r}^{2}}$) which is obviously less than 2rϕ. Let BC = r(2ϕ– ξ), thus one has:

_{a}, the geometric concentration factor (C

_{g}) and depth (H) of full CPCs differ for different CPC designs (Table 1 and Table 3). In turn, given θ

_{a}and C

_{t}, the edge-ray angle (θ

_{t}) of a truncated CPC differs for different CPC designs as shown in Table 2. Table 3 shows that the depth of full CPCs is very large and greatly reduced after truncation.

## 3. Gap Losses of Compound Parabolic Solar Concentrators

_{t}) and reflectivity of reflectors, therefore the optical loss through gaps of CPCs for radiation incident at any angle is hard to calculate analytically. In this exercise, the incident radiation is simply regarded as uniformly distributed over the CPC's acceptance angle. Realize, therefore, that the results for gap losses here do not pertain to a particular angle (θ), but rather are averaged over all incidence angles within the view field of full CPCs with perfect reflection on reflectors [16].

_{c-in}= r/R is the radiation transfer shape factor from cover tube to inner tube of EST.

_{A–C}is the radiation transfer shape factor from tubular absorber at A to the one at C.

_{AB-abs}= ϕ/tan ϕ is the radiative shape factor from AB to the tubular absorber (inner tube of EST).

_{1}= 0.1897, L

_{2}= 0.0747, L

_{3}= 0.031, L

_{4}= 0.03, L

_{5}= 0.0371 and L

_{6}= 0, respectively. Obviously, from the point of view of optical loss through gaps, CPC-6 is the optimal design, followed by CPC-4 and CPC-3, and CPC-1 is the worst design.

## 4. Optical Efficiency of Compound Parabolic Solar Concentrators (CPCs)

_{a}= 20°. It is shown that, except for CPC-1 and CPC-2, η as a function of θ increases with the increase of incidence angle as θ < 17°, then sharply decreases. This is because a considerable fraction of the incident radiation undergoes multiple reflections before arriving on the absorber when the incidence angle (θ) is small, and the radiation lost through gaps is high in the case of θ close to the acceptance angle. Whereas for CPC-1 and CPC-2, the situation is reversed, the gap losses are high when θ is small and low when θ is large, especially for CPC-2.

_{a}), the optical efficiency of CPC-6 is the highest and that of CPC-1 is the lowest (Figure 8 and Figure 9), and those of CPC-4 and CPC-5 are almost identical.

_{a}, the optical efficiency of CPC-5 and CPC-6 is lower than that of others due to the small edge-ray angles (Table 2). This means that, given θ

_{a}and C

_{t}, CPC-5 and CPC-6 are highly efficient to concentrate radiation on EST in the case of θ < θ

_{a}but less efficient as θ > θ

_{a}.

_{t}), $\overline{\mathsf{\eta}}$ of CPCs increases, a result of the fact that radiation incident on the upper portion of CPC reflectors would undergoes multiple reflections before arriving on the absorber [19], and the average reflection number of solar rays within the CPC cavity decreases with the decrease of C

_{t}. It must be noted that, for truncated CPCs, the $\overline{\mathsf{\eta}}$ merely represents the performance of CPCs for radiation over the acceptance angle (θ

_{a}) rather than the performance for radiation within θ

_{t}. In the case of radiation beyond its acceptance angle, the radiation on the absorber will be so low that the desired high temperature is not achievable. Therefore to provide high temperature heat, the sun must be kept within the acceptance angle of CPCs during the operation. As seen from Table 4, given θ

_{a}and C

_{t}, $\overline{\mathsf{\eta}}$ of CPC-6 is the highest, followed by CPC-4, thus, for high temperature applications, CPC-6 and CPC-4 are advisable due to high solar flux on the EST resulting from high η for radiation within the acceptance angle.

## 5. Annual Collectible Radiation on All-Glass Evacuated Solar Tube of CPC Collectors

_{b}is the instantaneous intensity of beam radiation; θ

_{in}is the real incident angle of solar rays on CPC collectors; g(θ

_{in}) is a control function, being 1 for cos θ

_{in}≥ 0, otherwise zero; I

_{abs,d}is the sky diffuse radiation received by EST of CPCs and estimated by:

_{x}= Min(0.5π – β, θ

_{t}); i is the directional intensity of sky diffuse radiation on the cross-section of EST, and i = 0.5I

_{d}for isotropic sky diffuse radiation [19,20]; I

_{d}is the sky diffuse radiation on the horizon. Thus, Equation (30) is rewritten as:

_{a}and C

_{t}, θ

_{t}of CPCs can be obtained based on the equation of CPC reflectors, then C

_{d1}and C

_{d2}can be obtained based the method aforementioned. At any moment of a day, θ

_{in}and θ in Equation (29) can be calculated from solar geometry [15]. Knowing the time variation of I

_{b}in a day, the daily radiation on EST is obtained by integrating Equation (29) over the daytime [17,19]:

_{a}) is estimated by summing H

_{day}in all days of a year. Given the monthly radiation on the horizon, the monthly average daily sky diffuse radiation, H

_{d}, time variation of I

_{b}in a day can be found [21].

_{day}is set to be 1 min, the η(θ) of CPCs at any moment is estimated based on θ at the moment and a linear extrapolation technique. The monthly horizontal radiation used in this work was taken from the book edited by Chen [22].

_{a}, two cases with β being yearly fixed (1T-CPC) and yearly adjusted four times at three tilts (3T-CPC), are considered. For 1T-CPCs, β = λ, θ

_{a}= 26° [23], whereas for 3T-CPCs, β = λ during periods of 23 days around both equinoxes, and adjusted to λ + 23 and λ – 23 in winters and summers, respectively [19,20]. Five sites with typical climatic conditions (Beijing: dry land with abundant solar resources; Shanghai, a site climatically characterized by rainy winters and sunny summers; Lhasa: a highland with extremely abundant solar resources; Chongqing: a site with poor solar resources; Kunming, a site climatically characterized by sunny winters and rainy summers) are selected as the representatives for the analysis.

#### 5.1. Annual Collectible Radiation on All-Glass Evacuated Solar Tube of 1T-CPCs

_{a}, the annual radiation collected by CPC-5 is the lowest; whereas for truncated CPCs with identical θ

_{a}and C

_{t}, the annual radiation by the CPC-1 is the lowest (Figure 10 and Figure 11).

#### 5.2. Annual Collectible Radiation on All-Glass Evacuated Solar Tube of 3T-CPCs

_{a}, the CPC-1 yearly concentrates the most solar radiation, followed by CPC-4 and CPC-3, and the CPC-5 concentrates the least radiation. The effect of the reflector’s reflectivity on the S

_{a}of full CPCs is presented in Figure 12 and the same situation as seen in Table 5 was found. This is because, given θ

_{a}, the geometric concentration factor of full CPC-1 is the largest and that of full CPC-5/6 is the smallest (Table 1). It is also seen that, for truncated CPCs with identical θ

_{a}and C

_{t}, the annual radiation collected by CPC-4 is the highest, followed by CPC-3, and the CPC-1 annually collected least radiation for the case of $\mathsf{\rho}$ > 0.85 otherwise CPC-5 annually collects the least radiation (Figure 13). Effect of geometric concentration factor on S

_{a}of truncated 3T-CPCs is shown in Figure 14, and it is seen that the S

_{a}linearly increases with the increase of C

_{t}, the CPC-4 annually concentrates most radiation and CPC-1 annually collects the least radiation.

## 6. Conclusions

_{a}) but the least efficient when θ > θ

_{a}as compared to other designs. This means that, for high temperature applications, CPC-6 and CPC-4 are advisable due to the high solar flux on the EST resulting from the high optical efficiency for radiation within the acceptance angle.

_{a}, CPC-1 concentrates the most radiation due to its largest geometric concentration, and CPC-5 collects the least radiation; whereas for truncated CPCs with identical θ

_{a}and C

_{t}, CPC-4 is the best solution, and CPC-1 is the inferior solution. In practical applications, CPCs are usually truncated to save reflector materials and reduce the depth of CPCs due to the lesser contribution of upper reflectors to radiation concentration, therefore, it is concluded that CPC-4 is the optimal design, and CPC-1 is the worst solution in terms of annual collectible radiation on the EST.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclatures

A_{ap} | area of CPCs’ aperture, m^{2} |

C_{g} | geometric concentration of full CPCs, dimensionless |

C_{ideal} | geometric concentration of ideal CPCs (1/sin ${\mathsf{\theta}}_{a}$), dimensionless |

C_{t} | geometric concentration of truncated CPCs, dimensionless |

F_{a-b} | radiative shape factor from surface a to surface b, dimensionless |

H | depth of CPC, mm |

H_{day} | daily radiation on unit length of solar tubes, MJ/m |

H_{d} | daily sky diffuse radiation on the horizon, J/m^{2} |

h | depth of V-groove, mm |

I | instantaneous radiation intensity, W/m^{2} |

i | directional intensity of sky diffuse radiation, W/m^{2}·rad |

P_{abs,d} | perimeter of absorber based on which CPC is designed, mm |

P_{abs,d} | perimeter of actual absorber of CPC with EST (2$\mathsf{\pi}r$), mm |

R | radius of the cover tube, mm |

r | radius of the inner tube, mm |

S_{a} | annual collectible radiation on solar tubes, MJ/m |

t | solar time, s |

## Greek Letters

$\mathsf{\beta}$ | tilt-angle of the aperture of CPCs from the horizon, degree |

$\mathsf{\lambda}$ | site latitude, degree |

$\mathsf{\varphi}$ | angle given by cos$\mathsf{\varphi}=\frac{r}{R}$, radian |

$\mathsf{\phi}$ | the angle used to describe the coordinate of any point on reflectors of CPCs, radian |

$\mathsf{\eta}\left(\mathsf{\theta}\right)$ | optical efficiency factor, dimensionless |

$\mathsf{\theta}$ | projection incident angle of solar rays on the cross-section of CPC-trough, radian |

${\mathsf{\theta}}_{a}$ | acceptance half-angle of CPCs, degree |

${\mathsf{\theta}}_{in}$ | real incidence angle of solar rays on the aperture of CPCs, radian |

${\mathsf{\theta}}_{t}$ | edge-ray angle of truncated CPCs, degree |

$\mathsf{\rho}$ | reflectivity of reflectors, dimensionless; a parameter to describe coordinates of points on reflectors |

ψ | Opening angle of “V” groove, radian |

## Subscripts

abs | absorber |

ap | aperture of CPCs |

b | beam radiation |

d | sky diffuse radiation |

day | daily solar gain |

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**Figure 4.**CPC-3, designed based on inner tube of EST with the involutes near the inner tube being truncated.

**Figure 11.**As in Figure 10 but for truncated 1T-CPCs.

**Figure 12.**As in Figure 10 but for full 3T-CPCs.

**Figure 13.**As in Figure 10 but for truncated 3T-CPCs.

θ_{a} | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 |
---|---|---|---|---|---|---|

20° | 3.608 | 2.924 | 2.924 | 3.014 | 2.431 | 2.431 |

26° | 2.815 | 2.281 | 2.281 | 2.351 | 1.897 | 1.897 |

C_{t} | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 |
---|---|---|---|---|---|---|

C_{t} = 2.0 | 79.3° | 61.4° | 61.4° | 64.1° | 44.8° | 44.8° |

C_{t} = 2.4 | 63.8° | 45.3° | 45.3° | 48.1° | 24.7° | 24.7° |

Size of CPC | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 |
---|---|---|---|---|---|---|

C_{t} = 2.0 | 103.1 | 144.2 | 144.2 | 137.0 | 206.7 | 218.0 * |

C_{t} = 2.4 | 165.1 | 245.4 | 245.4 | 229.7 | 466.3 | 477.6 * |

Full CPCs | 862.2 | 698.7 | 698.7 | 701.0 | 686.3 | 697.6 * |

**Table 4.**Average optical efficiency $\overline{\mathsf{\eta}}$ of CPCs with ${\mathsf{\theta}}_{a}={20}^{\mathrm{o}}$.

Size of CPCs | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 |
---|---|---|---|---|---|---|

C_{t} = 2.0 | 0.74326 | 0.86212 | 0.87657 | 0.87967 | 0.86477 | 0.89542 |

C_{t} = 2.1 | 0.74180 | 0.85869 | 0.87307 | 0.87592 | 0.86146 | 0.89326 |

C_{t} = 2.2 | 0.74005 | 0.85574 | 0.86892 | 0.87173 | 0.85883 | 0.89061 |

C_{t} = 2.3 | 0.73940 | 0.85288 | 0.86444 | 0.86657 | 0.85660 | 0.88750 |

C_{t} = 2.4 | 0.73841 | 0.84933 | 0.86125 | 0.86179 | 0.85263 | 0.88292 |

Full CPC | 0.71110 | 0.83424 | 0.84379 | 0.83205 | 0.84675 | 0.87611 |

Site | Full 1T-CPCs | Truncated 1T-CPCs (C_{t} = 1.8) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 | |

Beijing | 1322 | 1252 | 1289 | 1327 | 1075 | 1106 | 1013 | 1161 | 1151 | 1166 | 1069 | 1101 |

Shanghai | 1004 | 949 | 977 | 1007 | 816 | 838 | 800 | 899 | 888 | 902 | 813 | 836 |

Lhasa | 2051 | 1943 | 1999 | 2060 | 1669 | 1715 | 1558 | 1789 | 1773 | 1797 | 1656 | 1704 |

Chongqing | 728 | 683 | 705 | 726 | 589 | 606 | 597 | 659 | 649 | 659 | 588 | 605 |

Kunming | 1293 | 1219 | 1256 | 1294 | 1048 | 1078 | 1017 | 1143 | 1130 | 1145 | 1042 | 1073 |

Site | Full 3T-CPCs | Truncated 3T-CPCs (C_{t} = 2) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 | CPC-1 | CPC-2 | CPC-3 | CPC-4 | CPC-5 | CPC-6 | |

Beijing | 1981 | 1726 | 1818 | 1871 | 1519 | 1597 | 1328 | 1384 | 1426 | 1442 | 1341 | 1398 |

Shanghai | 1446 | 1265 | 1331 | 1369 | 1112 | 1167 | 1008 | 1046 | 1071 | 1085 | 994 | 1036 |

Lhasa | 3158 | 2743 | 2894 | 2979 | 2417 | 2544 | 2076 | 2168 | 2243 | 2266 | 2123 | 2214 |

Chongqing | 996 | 875 | 920 | 946 | 768 | 805 | 723 | 746 | 759 | 770 | 696 | 724 |

Kunming | 1891 | 1650 | 1739 | 1789 | 1452 | 1526 | 1295 | 1345 | 1383 | 1399 | 1291 | 1346 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Wang, J.; Tang, R.
Design and Optical Performance of Compound Parabolic Solar Concentrators with Evacuated Tube as Receivers. *Energies* **2016**, *9*, 795.
https://doi.org/10.3390/en9100795

**AMA Style**

Wang Q, Wang J, Tang R.
Design and Optical Performance of Compound Parabolic Solar Concentrators with Evacuated Tube as Receivers. *Energies*. 2016; 9(10):795.
https://doi.org/10.3390/en9100795

**Chicago/Turabian Style**

Wang, Qiang, Jinfu Wang, and Runsheng Tang.
2016. "Design and Optical Performance of Compound Parabolic Solar Concentrators with Evacuated Tube as Receivers" *Energies* 9, no. 10: 795.
https://doi.org/10.3390/en9100795