Abstract

Spectral control of the emissivity of surfaces is essential in applications such as solar thermal and thermophotovoltaic energy conversion in order to achieve the highest conversion efficiencies possible. We investigated the spectral performance of planar aperiodic metal-dielectric multilayer coatings for these applications. The response of the coatings was optimized for a target operational temperature using needle-optimization based on a transfer matrix approach. Excellent spectral selectivity was achieved over a wide angular range. These aperiodic metal-dielectric stacks have the potential to significantly increase the efficiency of thermophotovoltaic and solar thermal conversion systems. Optimal coatings for concentrated solar thermal conversion were modeled to have a thermal emissivity <7% at 720K while absorbing >94% of the incident light. In addition, optimized coatings for solar thermophotovoltaic applications were modeled to have thermal emissivity <16% at 1750K while absorbing >85% of the concentrated solar radiation.

© 2009 OSA

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2009

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

J. T. K. Wan, “Tunable thermal emission at infrared frequencies via tungsten gratings,” Opt. Commun. 282(8), 1671–1675 (2009).
[CrossRef]

E. Rephaeli and S. Fan, “Absorber and emitter for solar thermo-photovoltaic systems to achieve efficiency exceeding the Shockley-Queisser limit,” Opt. Express 17(17), 15145–15159 (2009).
[CrossRef] [PubMed]

2008

E. Rephaeli and S. Fan, “Tungsten black absorber for solar light with wide angular operation range,” Appl. Phys. Lett. 92(21), 211107 (2008).
[CrossRef]

2007

2006

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in 2D-periodic metallic photonic crystal slabs,” Opt. Express 14(19), 8785–8796 (2006).
[CrossRef] [PubMed]

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016609 (2006).
[CrossRef] [PubMed]

2005

2004

I. Celanovic, F. O’Sullivan, M. Ilak, J. Kassakian, and D. Perreault, “Design and optimization of one-dimensional photonic crystals for thermophotovoltaic applications,” Opt. Lett. 29(8), 863–865 (2004).
[CrossRef] [PubMed]

H. Sai and H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85(16), 3399 (2004).
[CrossRef]

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

T. Trupke, P. Wurfel, and M. A. Green, “Comment on Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 84(11), 1997 (2004).
[CrossRef]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

2003

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[CrossRef]

N.-P. Harder and P. Wurfel, “Theoretical limits of thermophotovoltaic solar energy conversion,” Semicond. Sci. Technol. 18(5), S151–S157 (2003).
[CrossRef]

2001

1999

C. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59(6), 4736–4746 (1999).
[CrossRef]

1996

1992

Q.-C. Zhang and D. R. Mills, “Very low-emittance solar selective surfaces using new film structures,” J. Appl. Phys. 72(7), 3013 (1992).
[CrossRef]

1988

1965

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

R. N. Schmidt and K. C. Park, “High-Temperature Space-Stable Selective Solar Absorber Coatings,” Appl. Opt. 4(8), 917–925 (1965).
[CrossRef]

Akiyama, Y.

Alexander, R. W.

Bell, R. J.

Celanovic, I.

Chan, D. L. C.

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016609 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in 2D-periodic metallic photonic crystal slabs,” Opt. Express 14(19), 8785–8796 (2006).
[CrossRef] [PubMed]

Chang, Z. M.

Y.-B. Chen and Z. M. Chang, “Design of tungsten complex gratings for thermophotovoltaic radiators,” Opt. Commun. 269(2), 411–417 (2007).
[CrossRef]

Chao, C.-H.

Chen, G.

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

Chen, L.-Y.

Chen, Y.-B.

Y.-B. Chen and Z. M. Chang, “Design of tungsten complex gratings for thermophotovoltaic radiators,” Opt. Commun. 269(2), 411–417 (2007).
[CrossRef]

Chen, Y.-R.

Cheng, W.-C.

Cornelius, C.

C. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59(6), 4736–4746 (1999).
[CrossRef]

DeBell, G. W.

DeLuca, J. S.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Dobrowolski, J. A.

Dowling, J. P.

C. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59(6), 4736–4746 (1999).
[CrossRef]

Fan, S.

Fleming, J. G.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[CrossRef]

Green, M. A.

T. Trupke, P. Wurfel, and M. A. Green, “Comment on Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 84(11), 1997 (2004).
[CrossRef]

Hane, K.

Harder, N.-P.

N.-P. Harder and P. Wurfel, “Theoretical limits of thermophotovoltaic solar energy conversion,” Semicond. Sci. Technol. 18(5), S151–S157 (2003).
[CrossRef]

Ilak, M.

Joannopoulos, J. D.

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016609 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in 2D-periodic metallic photonic crystal slabs,” Opt. Express 14(19), 8785–8796 (2006).
[CrossRef] [PubMed]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

Kanamori, Y.

Karabacak, T.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Kassakian, J.

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Lee, Y. P.

Li, X.-F.

Lin, C.-F.

Lin, S. Y.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[CrossRef]

Lu, T.-M.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Luo, C.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Miao, J.

Mills, D. R.

Q.-C. Zhang and D. R. Mills, “Very low-emittance solar selective surfaces using new film structures,” J. Appl. Phys. 72(7), 3013 (1992).
[CrossRef]

Moreno, J.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[CrossRef]

Narayanaswamy, A.

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Newquist, L. A.

O’Sullivan, F.

Ordal, M. A.

Park, K. C.

Pereault, D.

I. Celanovic, D. Pereault, and J. Kassakian, “Resonant-cavity enhanced thermal emission,” Phys. Rev. B 72(7), 075127 (2005).
[CrossRef]

Perreault, D.

Querry, M. R.

Rephaeli, E.

Sai, H.

Schmidt, R. N.

Soljacic, M.

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in 2D-periodic metallic photonic crystal slabs,” Opt. Express 14(19), 8785–8796 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016609 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

Sullivan, B. T.

Ten Eyck, G. A.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Tikhonravov, A. V.

Trubetskov, M. K.

Trupke, T.

T. Trupke, P. Wurfel, and M. A. Green, “Comment on Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 84(11), 1997 (2004).
[CrossRef]

Wan, J. T. K.

J. T. K. Wan, “Tunable thermal emission at infrared frequencies via tungsten gratings,” Opt. Commun. 282(8), 1671–1675 (2009).
[CrossRef]

Wang, G.-C.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Wang, L. A.

Wang, P.-I.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Wurfel, P.

T. Trupke, P. Wurfel, and M. A. Green, “Comment on Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 84(11), 1997 (2004).
[CrossRef]

N.-P. Harder and P. Wurfel, “Theoretical limits of thermophotovoltaic solar energy conversion,” Semicond. Sci. Technol. 18(5), S151–S157 (2003).
[CrossRef]

Ye, D.

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

Yugami, H.

Zhang, Q.-C.

Q.-C. Zhang and D. R. Mills, “Very low-emittance solar selective surfaces using new film structures,” J. Appl. Phys. 72(7), 3013 (1992).
[CrossRef]

Zheng, Y.-X.

Zhou, P.

Appl. Opt.

Appl. Phys. Lett.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[CrossRef]

T. Trupke, P. Wurfel, and M. A. Green, “Comment on Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 84(11), 1997 (2004).
[CrossRef]

H. Sai and H. Yugami, “Thermophotovoltaic generation with selective radiators based on tungsten surface gratings,” Appl. Phys. Lett. 85(16), 3399 (2004).
[CrossRef]

E. Rephaeli and S. Fan, “Tungsten black absorber for solar light with wide angular operation range,” Appl. Phys. Lett. 92(21), 211107 (2008).
[CrossRef]

Comput. J.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

J. Appl. Phys.

Q.-C. Zhang and D. R. Mills, “Very low-emittance solar selective surfaces using new film structures,” J. Appl. Phys. 72(7), 3013 (1992).
[CrossRef]

T. Karabacak, J. S. DeLuca, P.-I. Wang, G. A. Ten Eyck, D. Ye, G.-C. Wang, and T.-M. Lu, “Low temperature melting of copper nanorod arrays,” J. Appl. Phys. 99(6), 064304 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

Y.-B. Chen and Z. M. Chang, “Design of tungsten complex gratings for thermophotovoltaic radiators,” Opt. Commun. 269(2), 411–417 (2007).
[CrossRef]

J. T. K. Wan, “Tunable thermal emission at infrared frequencies via tungsten gratings,” Opt. Commun. 282(8), 1671–1675 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

C. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59(6), 4736–4746 (1999).
[CrossRef]

Phys. Rev. B

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[CrossRef]

I. Celanovic, D. Pereault, and J. Kassakian, “Resonant-cavity enhanced thermal emission,” Phys. Rev. B 72(7), 075127 (2005).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

D. L. C. Chan, M. Soljacić, and J. D. Joannopoulos, “Thermal emission and design in one-dimensional periodic metallic photonic crystal slabs,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016609 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, “Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93(21), 213905 (2004).
[CrossRef] [PubMed]

Semicond. Sci. Technol.

N.-P. Harder and P. Wurfel, “Theoretical limits of thermophotovoltaic solar energy conversion,” Semicond. Sci. Technol. 18(5), S151–S157 (2003).
[CrossRef]

Other

C.E. Kennedy, Review of Mid- to High-Temperature Solar Selective Absorber Materials, NREL/TP-520–31267 (2002)

C. E. Kennedy, and H. Price, “Progress in development of high-temperature solar-selective coatings,” NREL/CP-520–36997 Proc. ISEC2005 2005 International Solar Energy Conference August 6–12, 2005, Orlando, Florida USA, ISEC2005–76039.

F. Burkholder and C. Kutscher, “Heat-Loss Testing of Solel’s UVAC3 Parabolic Trough Receiver,” NREL/TP-550–42394 (2008)

F. Burkholder and C. Kutscher, “Heat Loss Testing of Schott's 2008 PTR70 Parabolic Trough Receiver,” NREL/TP-550–45633 (2009)

A. Luque and V. M. Andreev, Concentrator Photovoltaics, Springer Series in Optical Sciences (Springer, New York), Chap. 9.

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Figures (12)

Fig. 1
Fig. 1

(a) Cutoff wavelength for an ideal absorber coating as a function of optimization temperature. (b) Spectral absorptivity of an ideal absorber for Tmerit = 720K or λc = 2.24 µm. Normalized spectral power densities for black body (BB) at 720K and solar spectrum (AM1.5) are shown for illustration.

Fig. 2
Fig. 2

Model of a heat collection element (HCE) of a CST system. An inner steel tube is coated with the selective absorber coating. The coating is isolated from air by the evacuated glass tube. This helps to protect the coating and reduces thermal losses. The heat transfer fluid (HTF) flows inside the steel tube and has an operation temperature of ~720K.

Fig. 3
Fig. 3

Design of the aperiodic metal-dielectric stacks optimized for planar geometry using layers of Mo, MgF2 and TiO2 . Optimized stacks with no of layers L = 5, 7, 9 and 11 are shown.

Fig. 4
Fig. 4

Design of aperiodic metal-dielectric stacks optimized for planar geometry using layers of W, MgF2 and TiO2. Optimized stacks with no of layers L = 5, 7, 9 and 11 are shown.

Fig. 5
Fig. 5

Spectral hemispherical absorptivity of aperiodic metal-dielectric stacks optimized using (a) Mo, TiO2 and MgF2 and (b) W, TiO2 and MgF2. The spectral absorptivity of an ideal absorber at 720K is also plotted for comparison.

Fig. 6
Fig. 6

Merit function evaluation at 720K for optimized aperiodic stacks as a function of the number of layers L in the stack. The merit was evaluated for stacks composed of layers of Mo, TiO2 and MgF2 (squares) and W, TiO2 and MgF2 (circles), respectively. When determining the number of layers L in a stack, the metal substrate also counts as a layer, thus the uncoated stack has L = 1.

Fig. 7
Fig. 7

Spectral directional absorptivity of (a) Mo TiO2 MgF2 and (b) W TiO2 MgF2 coatings with L = 11, optimized for operation at 720K. A skewed colorbar was used in order to provide more color contrast in spectral range of high absorption.

Fig. 8
Fig. 8

Schematic model of a STPV system. Incoming concentrated sunlight is absorbed by a selective absorber coating. The absorber is in thermal contact with a cylindrical emitter. The thermal emission from the selective emitter is tuned to the bandgap of the photovoltaic cells which are mounted at the inside of a hollow cylinder.

Fig. 9
Fig. 9

Design of aperiodic metal-dielectric stacks optimized for planar geometry using (a) Mo, MgO and (b) W, MgO. The stacks are optimized for operation at 1750K.

Fig. 10
Fig. 10

Spectral hemispherical absorptivity of aperiodic metal-dielectric stacks optimized for planar geometry using (a) Mo and MgO and (b) W and MgO. The spectral absorptivity of an ideal absorber at 1750K is also plotted for comparison.

Fig. 11
Fig. 11

Merit function evaluation at 720K for optimized aperiodic stacks as a function of the number of layers L in the stack. The merit was evaluated for stacks composed of layers of Mo, TiO2 and MgF2 (squares) and W, TiO2 and MgF2 (circles), respectively. When determining the number of layers L in a stack, the metal substrate also counts as a layer, thus the uncoated stack has L = 1.

Fig. 12
Fig. 12

Spectral directional absorptivity of (a) Mo MgO and (b) W MgO coatings with L = 4, optimized for operation at 1750K. A skewed colorbar was used in order to provide more color contrast in spectral range of high absorption.

Tables (4)

Tables Icon

Table 1 Details on selected aperiodic metal-dielectric coatings with layers of Mo, MgF2 and TiO2. L is the number of layers in the stack, Topt is the temperature of operation, α s o l a r is the absorbed solar fraction, ε t h e r m a l is the thermal emissivity and F is the merit evaluation. When determining the number of layers L in a stack, the metal substrate also counts as a layer.

Tables Icon

Table 2 Details on selected aperiodic metal-dielectric coatings with layers of W, MgF2 and TiO2. When determining the number of layers L in a stack, the metal substrate also counts as a layer. Values are also given for a commercial coating (PTR70) and a complex optimized multilayer coating (NREL 6A) for comparison.

Tables Icon

Table 3 Spectral selectivity of aperiodic metal-dielectric coatings with Mo and MgO optimized for T = 1750K. L is the number of layers in the stack, Topt is the temperature of operation, α s o l a r is the absorbed solar fraction, ε t h e r m a l is the thermal emissivity and F is the merit evaluation. When determining the number of layers L in a stack, the metal substrate also counts as a layer.

Tables Icon

Table 4 Spectral selectivity of aperiodic metal-dielectric coatings with W and MgO optimized for T = 1750K. When determining the number of layers L in a stack, the metal substrate also counts as a layer.

Equations (3)

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F ( T m e r i t ) = α s o l a r × [ 1 ε t h e r m a l ( T m e r i t ) ]
α s o l a r = 0 α ( θ , λ ) L s u n ( λ ) d λ 0 L s u n ( λ ) d λ     with    θ = 0   (normal)
ε t h e r m a l ( T m e r i t ) = M c o a t i n g M B B = 2 π 0 π 2 0 ε ( θ , λ ) L B B ( λ , T m e r i t ) sin ( θ ) cos ( θ ) d λ d θ π 0 L B B ( λ , T m e r i t ) d λ  

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