Abstract

A luminescent solar concentrator focuses solar light spectrally and spatially to convert it to electricity. Due to a large number of input parameters and their complex effect on the device efficiency, it is typically described by numerical simulations. Here, an analytical solution is presented, which significantly reduces the complexity of the optical efficiency evaluation. The equations were derived using probabilistic approach and validated by simulations. The implementation by a computer algebra program yields instant results for any set of input parameters. It allows a higher level of analysis, where an inverse task of finding parameters for a given efficiency can be readily solved. The obtained explicit expressions provide a clear standard for theoretical description of such devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

2018 (1)

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

2017 (5)

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

F. Meinardi, F. Bruni, and S. Brovelli, Nat. Rev. Mater. 2, 17072 (2017).
[Crossref]

2016 (3)

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

S. Nechayev, P. D. Reusswig, M. A. Baldo, and C. Rotschild, Sci. Rep. 6, 38576 (2016).
[Crossref]

2015 (2)

2014 (1)

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

2013 (1)

W. G. J. H. M. van Sark, Renew. Energy 49, 207 (2013).
[Crossref]

2012 (2)

F. Purcell-Milton and Y. K. Gun’ko, J. Mater. Chem. 22, 16687 (2012).
[Crossref]

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

1980 (1)

1979 (1)

1977 (1)

A. Goetzberger and W. Greubel, Appl. Phys. 14, 123 (1977).
[Crossref]

1976 (1)

1969 (1)

G. Keil, J. Appl. Phys. 40, 3544 (1969).
[Crossref]

1965 (1)

A. N. Kolmogorov, Probl. Inf. Transm. 1, 3 (1965).

Akkerrnan, Q. A.

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

Baldo, M. A.

S. Nechayev, P. D. Reusswig, M. A. Baldo, and C. Rotschild, Sci. Rep. 6, 38576 (2016).
[Crossref]

Batchelder, J. S.

Bergren, M. R.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

Bradshaw, L. R.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Brovelli, S.

F. Meinardi, F. Bruni, and S. Brovelli, Nat. Rev. Mater. 2, 17072 (2017).
[Crossref]

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Bruni, F.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

F. Meinardi, F. Bruni, and S. Brovelli, Nat. Rev. Mater. 2, 17072 (2017).
[Crossref]

Carlson, B.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

Carulli, F.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Chizhik, A. I.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Chizhik, A. M.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Cole, T.

Connell, R.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Dang, Z. Y.

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

Deng, L. G.

Dhamo, L.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Ehrenberg, S.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Eiselt, S.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

Erickson, C.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

Erickson, C. S.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Ferry, V. E.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Gamelin, D. R.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Gilbertson, J. D.

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Goetzberger, A.

A. Goetzberger and W. Greubel, Appl. Phys. 14, 123 (1977).
[Crossref]

Greubel, W.

A. Goetzberger and W. Greubel, Appl. Phys. 14, 123 (1977).
[Crossref]

Gughelmetti, R.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

Gun’ko, Y. K.

F. Purcell-Milton and Y. K. Gun’ko, J. Mater. Chem. 22, 16687 (2012).
[Crossref]

Haraldsson, T.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

Hill, S. K. E.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Hillmyer, M. A.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Hollinger, J.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Hu, X. M.

Huisken, F.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Jackson, A.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

Kang, R. D.

Keil, G.

G. Keil, J. Appl. Phys. 40, 3544 (1969).
[Crossref]

Kilburn, T. B.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

Klimov, V. I.

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

Kolmogorov, A. N.

A. N. Kolmogorov, Probl. Inf. Transm. 1, 3 (1965).

Kortshagen, U.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Lambe, J.

Li, H. B.

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

Lim, J.

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

Linnros, J.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

Luo, J. W.

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

Makarov, N. S.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

Manna, L.

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

Marinins, A.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

Mauri, M.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

McDaniel, H.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

McDowall, S.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Meinardi, F.

F. Meinardi, F. Bruni, and S. Brovelli, Nat. Rev. Mater. 2, 17072 (2017).
[Crossref]

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Meixner, A. J.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Nechayev, S.

S. Nechayev, P. D. Reusswig, M. A. Baldo, and C. Rotschild, Sci. Rep. 6, 38576 (2016).
[Crossref]

Papakonstantinou, I.

Park, S.

F. Meinardi, Q. A. Akkerrnan, F. Bruni, S. Park, M. Mauri, Z. Y. Dang, L. Manna, and S. Brovelli, ACS Energy Lett. 2, 2368 (2017).
[Crossref]

Patrick, D. L.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

C. S. Erickson, L. R. Bradshaw, S. McDowall, J. D. Gilbertson, D. R. Gamelin, and D. L. Patrick, ACS Nano 8, 3461 (2014).
[Crossref]

Peterson, C.

S. K. E. Hill, R. Connell, C. Peterson, J. Hollinger, M. A. Hillmyer, U. Kortshagen, and V. E. Ferry, ACS Photon. 6, 170 (2019).
[Crossref]

Pevere, F.

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

Popov, S.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

Potrick, K.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Purcell-Milton, F.

F. Purcell-Milton and Y. K. Gun’ko, J. Mater. Chem. 22, 16687 (2012).
[Crossref]

Ramasamy, K.

M. R. Bergren, N. S. Makarov, K. Ramasamy, A. Jackson, R. Gughelmetti, and H. McDaniel, ACS Energy Lett. 3, 520 (2018).
[Crossref]

Reusswig, P. D.

S. Nechayev, P. D. Reusswig, M. A. Baldo, and C. Rotschild, Sci. Rep. 6, 38576 (2016).
[Crossref]

Rotschild, C.

S. Nechayev, P. D. Reusswig, M. A. Baldo, and C. Rotschild, Sci. Rep. 6, 38576 (2016).
[Crossref]

Schmidt, T.

T. Schmidt, A. I. Chizhik, A. M. Chizhik, K. Potrick, A. J. Meixner, and F. Huisken, Phys. Rev. B 86, 125302 (2012).
[Crossref]

Shafagh, R.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

Shi, L. J.

Simonutti, R.

F. Meinardi, S. Ehrenberg, L. Dhamo, F. Carulli, M. Mauri, F. Bruni, R. Simonutti, U. Kortshagen, and S. Brovelli, Nat. Photonics 11, 177 (2017).
[Crossref]

Song, H.

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

Sumner, R.

R. Sumner, S. Eiselt, T. B. Kilburn, C. Erickson, B. Carlson, D. R. Gamelin, S. McDowall, and D. L. Patrick, J. Phys. Chem. C 121, 3252 (2017).
[Crossref]

Sychugov, I.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

Tummeltshammer, C.

Van der Wijngaart, W.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

van Sark, W. G. J. H. M.

W. G. J. H. M. van Sark, Renew. Energy 49, 207 (2013).
[Crossref]

Veinot, J. G. C.

A. Marinins, R. Shafagh, W. Van der Wijngaart, T. Haraldsson, J. Linnros, J. G. C. Veinot, S. Popov, and I. Sychugov, ACS Appl. Mater. Interfaces 9, 30267 (2017).
[Crossref]

Weber, W. H.

Wu, K.

H. B. Li, K. Wu, J. Lim, H. Song, and V. I. Klimov, Nat. Energy 1, 16157 (2016).
[Crossref]

Yablonovitch, E.

Zewail, A. H.

Zhang, Y. Y.

Zhong, H. Z.

Zou, B. S.

Zunger, A.

I. Sychugov, F. Pevere, J. W. Luo, A. Zunger, and J. Linnros, Phys. Rev. B 93, 161413 (2016).
[Crossref]

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Other (1)

Maple file with formulas, https://doi.org/10.6084/m9.figshare.7868843 .

Supplementary Material (2)

NameDescription
» Dataset 1       Maple file with formulas
» Supplement 1       Detailed derivations

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Figures (4)

Fig. 1.
Fig. 1. Total optical path distribution from isotropic emitters in a rectangular slab (height h, width w, and thickness Δ) stems from in-plane p(r) and out-of-plane qr(l) distributions.
Fig. 2.
Fig. 2. Distribution p(r) of the optical path lengths for the photons from an isotropic emitter randomly placed in a 2D slab. Data points are simulated, and the lines are from Eq. (1) (h=6 and w=4 or two length units). The inset shows full 3D distribution q(l): bright lines are the exact solutions, and dark lines are the approximate solutions from Eq. (2).
Fig. 3.
Fig. 3. Comparison of analytical solutions [Eqs. (3)–(5)] for different loss mechanisms (lines) with numerical simulations (points) from [10,11].
Fig. 4.
Fig. 4. (top and left) Device power output (line) from the analytical solution of Eq. (7). The star and dots are experimental data and numerical simulations from [16]. (right) Critical aspect ratio βc, resulting in 7% efficiency, for different quantum yields and absorption coefficients.

Equations (18)

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p(r)={2w+2h2rπhw,0<r<w2r2r2w2πwr,w<r<h2r22hr2w22wr2h2πhwr,h<r<d.
qr(l)=12π·rll2r2,
q(l)=12πl23llp(r)rl2r2dr.
k1.14.
q(l)p(l/k).
q(l)={2w+2h2l/kπhwk,0<l<kw2l2l2(kw)2πlwk,kw<l<kh2l2/k2hl2(kw)22wl2(kh)2πlhwkkh<l<kd.
f(α)=0lmaxq(l)·exp(αl)dl,
f(α)=2α2hwπk2((h+w)αk(kdα+1)·ekdα+ekhα+ekwα1)I1I2,
I1=2wπkkwkdl2(kw)2lexp(αl)dl,
I2=2hπkkhkdl2(kh)2lexp(αl)dl.
χ0(αsc)=f(αsc).
χ1(αsc)=δ·(1f(αsc))·f(αsc).
χ(αsc)=i=0χi(αsc)=f(αsc)1δ·(1f(αsc)).
ξ(αre)=f(αre)1δ·QY·(1f(αre)),
ξ=λminλmaxI(λ)·ξ(αre(λ))dλ.
φ(αsc,αre)=f(αsc+αre)1δ·αsc+δ·QY·αreαsc+αre(1f(αsc+αre)).
g(αsc,αre,α)=f(αsc+αre+α)1δ·αsc+δ·QY·αreαsc+αre+α(1f(αsc+αre+α)).
γ=Φ·h·w·(1T)·δ·QY·η·g(αsc,αre,α),