Modeling and simulation of ray tracing for compound parabolic thermal solar collector

International Communications in Heat and Mass Transfer 87 (2017) 169–174

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Modeling and simulation of ray tracing for compound parabolic thermal solar collector

MARK

Zhongyuan Sua,b, Shengyan Guc, Kambiz Vafaib,⁎ a b c

Jiangsu Provincial Key Laboratory of Solar Energy Science and Technology, School of Energy & Environment, Southeast University, Nanjing 210096, China Department of Mechanical Engineering, University of California, Riverside, CA 92521, USA Geology and Mineral Resources Research Institute of Jiangsu Province, Nanjing 210008, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Compound parabolic solar collector Ray tracing Optical efficiency

A ray tracing model for the compound parabolic collector (CPC) is presented in this work. The pertinent parameters for the compound parabolic thermal solar collector are analyzed and calculated, and the ray tracing model is further investigated. The ray tracing model is validated by comparing our ray tracing model results with a commercial optical software. Each ray is traced by the CPC model, so the incident angle is calculated when solar ray enters the absorption tube. The ray tracing model was applied to the thermal efficiency analysis of the CPC, and the thermal performance results obtained by the model and test results were compared.

1. Introduction Due to the rapid consumption of fossil fuels in the world, it is necessary to find alternative energy sources to meet current daily needs. At the same time, people are increasingly interested in concentrating solar thermal systems as an alternative and/or supplement to existing fossil fuels or nuclear fuel generation systems [1–2]. Solar concentrating technology aims to concentrate the direct normal insolation (DNI) in solar energy, to convert it directly and efficiently into thermal energy. Low-cost concentrator technologies are important research topics [3]. Compound parabolic concentrator (CPC) is a non-imaging lowfocus concentrator, designed according to the principle of optics. The receiver can collect the sunlight in a certain range of the incident angle of the incident ray so that it can receive both DNI and diffuse radiation. Aiming at the tubular CPC, the ray tracing software TracePro was used to analyze the CPC and the optimization scheme was proposed according to the simulation results. The CPC collectors with the concentration ratio of three and six were tested, and the heat loss and thermal efficiency of the thermal properties, used in such as industrial processes and solar refrigeration were investigated. The CPC collectors are desirable and reachable within the temperature range of 80 to 250 °C, as they do partly collect diffuse radiation and can avoid the high cost of accurate tracking system [4]. Buttinger et al. [5] demonstrated a new, flat, non-tracking V-type collector with a concentration ratio of 1.8. The prototype of this collector shows an efficiency of about 50% under the irradiation intensity of 1000 W/m2 and 150 °C, collector temperature. Harmim et al. [6] developed a novel box-shaped CPC solar

⁎

Corresponding author. E-mail address: [email protected] (K. Vafai).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.06.021

0735-1933/ © 2017 Elsevier Ltd. All rights reserved.

cooker with asymmetry and analyzed its thermal properties using a detailed theoretical model. Baig et al. [7] worked out a CPC with a geometric concentration ratio of 7.5 and their experiment showed that the peak efficiency could exceed 45% when using a vacuum tube receiver. The general solar water heater system can meet the demand of low temperature hot water, but it is less efficient when it is more than 55°. Pei et al. [8] built a CPC hot water system test setup and investigated its high water temperature performance. Their experiments show that winter thermal efficiency is more than 43%. Using U-tube solar energy set for the water heater system, the higher water temperature results in a much better thermal performance. Tang et al. [9] analyzed the amount of annual radiation that CPC, which is mounted on the stand oriented E-W in China can receive. The solar absorption ratio of absorber tube depends on the incident angle when solar ray enters the absorption tube, and the commercial optical software cannot account for the absorption ratio with the incident angle changes. As such the commercial software can not accurately carry out the simulation. In this work, the ray tracing model is used to simulate the process of concentrating sunlight, the number of reflections of the sunlight and the incident angle of the sunlight when it enters the absorber tube. The CPC concentrating sunlight process is programmed according to the established model while addressing applications.

International Communications in Heat and Mass Transfer 87 (2017) 169–174

Z. Su et al.

θt θl ηo ηob ηod ξb ξd ρb ρd αb αd

Nomenclature As AAperture cr r Ib Id k

overflowing area (m2) CPC aperture area (m2) concentration ratio absorber diameter (m) solar beam radiation intensity (W/m2) diffuse radiation intensity (W/m2) slope

Greek symbols α θ0

absorption ratio the angle between ray and the surface of the absorbing coating (°) half-acceptance angle (°)

θ1

transversal projection angle (°) longitudinal projection angle (°) thermal efficiency of the absorber tube DNI optical efficiency diffuse optical efficiency DNI geometric optical efficiency diffuse geometric optical efficiency reflectivity of DNI reflectivity of diffuse radiation DNI absorption ratio diffuse absorption ratio

Abbreviations CPC DNI

dyT

2. Model description

dθR 2.1. Physical model

compound parabolic collector direct normal insolation

=

r 1 − cos θR )[sin θ1 sin θR − (π + 2θa ) sin(θR − θ1)] 2 {( (1 − cos θR ) − sin θR[sin θ1(1 − cos θR ) + (π + 2θ1) cos(θR − θ1)]}

(7)

The solar collector section is composed of parabolas and involutes, which are shown in Fig. 1. This CPC has a half-acceptance angle of θ1. The right parabola has an entrance aperture of DC and a focal point F1. The Ray tracing process for CPC in collecting solar energy will be analyzed based on the assumption that the direction of sunlight passing through the glass envelope does not change.

The parametric equation of the parabola's normal slope k2n is obtained as:

2.2. Mathematical model

k 2n = − 1 k 2

dx T r 1 − cos θR )[cos θa sin θR + (π + 2θa ) cos(θR − θa )] = 2 {( dθR (1 − cos θR ) − sin θR[cos θa (1 − cos θR ) + (π + 2θa ) sin(θR − θa )]}

2.3. Model validation In order to verify the accuracy of the ray tracing model, the tracing result was compared with the results obtained by the commercial optical software TracePro. The characteristics of the CPC are listed in Table 1. The black curves in Fig. 3 represent the CPC, the circle in the middle of Fig. 3 is the collector, and blue lines are the traced rays. Transversal projection angle θt is defined as the angle between the collector normal vector (nc) and the sun position vector (SOL) projected

(1)

The coordinates of the point T(x,y) on the parabolas are expressed by the following parametric equations:

xT =

r [cos θ1(1 − cos θR ) + (π + 2θ1) sin(θR − θ1)] 1 − cos θR

(2)

yT =

r [sin θ1(1 − cos θR ) + (π + 2θ1) cos(θR − θ1)] 1 − cos θR

(3)

(9)

Based on Eqs. (1)–(9), the reflected light can be obtained from the incident light using the reflection law. Fig. 2 shows the programming flowchart for CPC ray tracing.

For the CPC curve with Cartesian coordinates, expressions can be deduced from analytical geometry, accounting for the fact that the symmetry axis of the parabolas is tangent to the absorber. T(x,y) is a point on the parabolas and point O is the Cartesian plane origin. The coordinates of a point on the involutes are given by the following parametric equations:

⎧ x = r (sin θ − θ cos θ) ⎨ ⎩ y = − r (cos θ + θ sin θ)

(8)

b Y

where θR is the angle between the line from point T(x,y) to the parabolic focus F1 and the right parabolic symmetry axis. According to the involute equation, the parametric equation of the involute tangent slope in Cartesian coordinate system is obtained as: 1

dy dy dθ −rθ cos θ 1 k1 = = = =− dx dx dθ rθ sin θ tan θ

(4)

The parametric equation of the involute normal slope k1n in the Cartesian coordinate system is calculated as:

k1n = − 1 k1 = tan θ

T( x,y)

1

R

F

(5)

2

F

1

X

The parametric equation of the parabola's tangent slope k2 is represented as:

k2 =

dyT dx T

=

>

dyT dθR dx T dθR

(6)

where the numerator and denominator are derived as follows:

Fig. 1. Basic schematic of CPC.

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(13)

where As is the overflowing area and AAperture is the CPC aperture area. Eq. (11) can then be amended as follows:

Incident Ray

ηob = ξ b⋅ρb ⋅αb⋅ξ

No No Ray intersect the absorber?

As tan θl AAperture

ξ=1−

Begin

Reflected ray by CPC

3.2. Transmittance and reflection

Ray is overflowing?

Optical surface properties of the utilized materials for Ray tracing are listed in Table 2. The direct reflection ratio ρb of CPC selects the calculation method proposed by Rabl [12,13]:

Yes

Yes

(14)

ρ b = ρ nb

(15)

where nb is the number of reflections for direct normal insolation for the CPC.

End Fig. 2. Programming flowchart for CPC ray tracing.

3.3. Absorption Table 1 The characteristics of the CPCs. Parameters

Symbol

Units

Value

Half-acceptance angle Concentrator ratio Absorber diameter

θ1 cr r

degree Dimensionless m

45 1.29 0.028

Each absorbed ray passes through the transmission and reflection. According to experimental investigation of selective absorbing coating by magnetron sputtering, the fitting equation for the relationship between the absorption ratio and the angle between the ray and the surface of the absorbing coating for designed CPC is as follows [14,15]:

α = − 8.43 × 10−7θ03 + 4.56 × 10−5θ02 + 2.22 × 10−5θ0 + 0.9185

in a plane perpendicular to the trough axis of the collector (Transversal axis). θl is longitudinal projection angle, which is the angle between nc and SOL projected in a plane parallel with the trough axis (Longitudinal axis) of the collector. The values of θt and θl during an entire day are calculated [10]. The simulation results of this model at the incident angle θt equal to 0°, 15°, 30° and 45° are shown in Fig. 3 (a), (c), (e) and (g). The simulation of the TracePro software at the incident angle θt at 0°, 15°, 30°, and 45° are shown in Fig. 3 (b), (d), (f), (h). A very good agreement can be observed between our results and the commercial optical software TracePro.

(16)

3.4. Efficiency of diffuse radiation The efficiency of the diffuse radiation ζd is greatly influenced by the weather and the environment. As such it is difficult to find a general and accurate model for it. It can be presented as: π

ξd =

∫0 2 ξbdθ π

∫0 2 dθ

(17)

3. Absorbing heat from solar irradiance

4. Results and discussions

3.1. Optical efficiency

The truncated CPC solar collector in Reference [4], which possesses concentration ratios of about 3.06 (3 ×) with half-acceptance angle of 10°, is selected for our simulations. Ray tracing of the selected CPC is shown in Fig. 4 with θt equal to 5° and the red circle is represented as the glass envelope. The thermal efficiency can be obtained utilizing the reflection, the transmittance and the absorption at different incident angle during each ray tracing. As such the DNI efficiency can be computed. Fig. 5 presents the DNI efficiency at different transversal projection angles θt, when longitudinal projection angle θl is equal to 0. The efficiency of the diffuse radiation can be obtained using Eq. (17). The thermal performance results during the entire day are shown in Figs. 6 and 7, where the thermal efficiency is measured in a clear day. As shown in Fig. 7, the relative error for the thermal efficiencies computed by our model and the experimental results for the CPC is roughly within the − 18% to 3% range, while the relative error between the calculated and test results is roughly within the − 5% to 28% range in Reference [4] using a commercial optical software. Relative error is greater, when the proportion of diffuse radiation in the effective irradiation is larger. The efficiency obtained by our model is lower than the measured values for the optical surface properties of the utilized materials. Considering the uncertainty associated in the physical processes, the measured and theoretical values are in quite a good agreement.

The total efficiency model of CPC can be obtained by the following analysis:

ηo =

Ib⋅ηob + Id⋅ηod Ib + Id

(10)

ηob = ξ b⋅ρb ⋅αb

(11)

ηod = ξd⋅ρd ⋅αd

(12)

where ηo is the thermal efficiency of the absorber tube. ηob and ηod are the direct normal insolation (DNI) and diffuse optical efficiencies respectively. Ib and Id are the solar beam and the diffuse radiation intensity (W/m2) respectively on the CPC aperture plane; ξb and ξd are CPC DNI and diffuse geometric optical efficiency; ρb, ρd are the reflectivity of DNI and diffuse radiation respectively and αb, αd are CPC DNI and diffuse absorption ratio respectively. The influence of longitudinal projection angle for optical efficiency is based on the end of the collector, since part of the rays is not absorbed at the end of the CPC tube. The loss is equal to the projected overflowing area in the aperture plane, according to the geometric relationship of the designed CPC. 171

International Communications in Heat and Mass Transfer 87 (2017) 169–174

120

120

100

100

80

80 y / mm

y / mm

Z. Su et al.

60 40

60 40

20

20

0

0

-20

-20 -40

-40 -100

-50

0 x / mm

50

-100

100

-50

50

100

(e)

(a)

(b)

(f)

120

120

100

100

80

80 y / mm

y / mm

0 x / mm

60 40

60 40

20

20

0

0

-20

-20 -40

-40 -100

-50

0 x / mm

50

-100

100

-50

0 x / mm

(c)

(g)

(d)

(h)

50

100

Fig. 3. Comparison of the CPC Ray tracing obtained by our model and TracePro. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

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0.7

Materials

Instantaneous thermal efficiency

Table 2 Transmittance and reflection. Incident angles

Transmittance of the glass envelope [11] Reflection of mirror film [4]

0°

20°

40°

60°

80°

0.88 0.94

0.87 0.94

0.84 0.94

0.75 0.94

0.36 0.94

400 350 300

Measured thermal efficiency Thermal efficiency by the model

0.65 0.6 0.55 0.5 0.45 0.4 0.35

y / mm

250 8

9

10

200

11

12

13

14

15

16

Solar time / hh:mm

150

Fig. 6. Comparison of our model's thermal efficiency computations with the experimental data.

100 30

50 0

20 0

50

100

150

200

250

Relative error / %

-250 -200 -150 -100 -50

x / mm Fig. 4. Selected CPC ray tracing.

Efficiency of DNI at different transversal projection angle

0.8 0.7 0.6

10

0

-10

-20

0.5

-30 0.4

8

9

10

11

12

13

14

15

16

Solar time / hh:mm

0.3

Fig. 7. Relative error between our model's thermal efficiency and the experiments.

0.2

Acknowledgements

0.1

This work was supported by Natural Science Foundation of China (Grant No. 51476099) and National High Technology Research Program (Grant No. 2013GH710501) as well as through a grant by the Chinese Scholarship Council.

0 0

10

20

30

40

50

transversal projection angle

60

70

80

90

/

Fig. 5. DNI efficiency at different transversal project angles.

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5. Conclusions A theoretical ray tracing model based on the CPC collector is investigated in this work. A detailed theoretical analysis was performed. The ray tracing model was applied to thermal efficiency analysis of the CPC. The following conclusions can be drawn: 1. The ray tracing model is validated by comparing the results between our ray tracing model and the commercial optical software. 2. Each ray can be traced by the CPC model, so the incident angle can be calculated when the solar ray enters the absorber tube. The solar absorption of absorber tube depends on the angle of incidence. 3. The thermal performance results obtained by our model and those obtained through the test results show that our model is effective in analyzing the CPC efficiency. 173

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