Abstract

A detailed characterization is performed to calibrate pyrgeometers, using a newly developed apparatus that contains a blackbody radiation source and the means to vary the temperatures of the pyrgeometer under testing. Calibration measurements cover the parameter space of radiation and instrument temperatures that prevail during field measurements. Dome-temperature measurements, normally provided on pyrgeometers, are inadequate for accurate corrections of the dome emission. A new temperature measurement with three sensors inside the dome at 45° elevation is proposed and has been implemented on several test instruments. This modification and the detailed characterization measurements permit an improved evaluation, based on thorough analysis of the thermal balance of the instrument, leading to a sensitivity factor C and three correction factors, k 1,2,3. Test measurements demonstrate the substantial improvement achieved on the accuracy of atmospheric and terrestrial long-wave radiation measurements, down to ±2 W m−2.

© 1995 Optical Society of America

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References

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  1. J. T. Suttles, G. Ohring, eds., “Workshop on surface radiation budget for climate applications,” WMO/TD-No. 109 (World Meteorological Organization, Geneva, 1985).
  2. World Climate Research Programme, “Workshop on implementation of the baseline surface radiation network,” WMO/TD-No. 406 (World Meteorological Organization, Geneva, 1990); “Second workshop on implementation of the baseline surface radiation network Davos,” WMO/TD-No. 453 (World Meteorological Organization, Geneva, 1991).
  3. A. Weiss, “On the performance of pyrgeometers with silicon domes,” J. Appl. Meteorol. 20, 962–965 (1981).
    [CrossRef]
  4. L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
    [CrossRef]
  5. R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
    [CrossRef]
  6. J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.
  7. K. Dehne, U. Bergholter, F. Kasten, “IEA comparison of long-wave radiometers in Hamburg 1989/90,” Rep. IEA-SHCP-9F-3 (International Energy Agency, Paris, 1993).
  8. A. J. Drummond, W. J. Scholes, J. H. Brown, “A new approach to the measurement of terrestrial long-wave radiation,” WMO Tech. Note 104 (World Meteorological Organization, Geneva, 1970), pp. 383–387.
  9. F. Miskolczi, R. Guzzi, “Effect of nonuniform spectral dome transmittance on the accuracy of infrared radiation measurements using shielded pyrradiometers and pyrgeometers,” Appl. Opt. 32, 3257–3265 (1993).
    [CrossRef]
  10. B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
    [CrossRef]
  11. J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
    [CrossRef]
  12. B. Albrecht, S. K. Cox, “Procedures for improving pyrgeometer performance,” J. Appl. Meteorol. 16, 188–197 (1977).
    [CrossRef]
  13. G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).
  14. Ch. Betz, “Entwicklung eines Eichstrahlers für Pyrgeometer,” Diplomarbeit (Universität Stuttgart, Stuttgart, Germany, 1993).
  15. E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
    [CrossRef]
  16. J. Olivieri, “Pyrgeometry: spectral considerations,” Centre Radiometrique, F-84200 Carpentras, France (personal communication, 1994).
  17. M. Shiobara, S. Asano, “The dome emission effect on the performance of pyrgeometers with silicon domes,” Pap. Meteorol. Geophys. 43, 17–31 (1992).
    [CrossRef]
  18. A. Heimo, Swiss Meteorological Institute, Aerological Station, CH-1530 Payerne, Switzerland (personal communication, 1993).

1993

1992

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

M. Shiobara, S. Asano, “The dome emission effect on the performance of pyrgeometers with silicon domes,” Pap. Meteorol. Geophys. 43, 17–31 (1992).
[CrossRef]

1988

L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
[CrossRef]

1986

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

1981

A. Weiss, “On the performance of pyrgeometers with silicon domes,” J. Appl. Meteorol. 20, 962–965 (1981).
[CrossRef]

1977

B. Albrecht, S. K. Cox, “Procedures for improving pyrgeometer performance,” J. Appl. Meteorol. 16, 188–197 (1977).
[CrossRef]

1975

J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
[CrossRef]

1974

B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
[CrossRef]

1962

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
[CrossRef]

Alados-Arboledas, L.

L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
[CrossRef]

Albers, L. U.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
[CrossRef]

Albrecht, B.

B. Albrecht, S. K. Cox, “Procedures for improving pyrgeometer performance,” J. Appl. Meteorol. 16, 188–197 (1977).
[CrossRef]

B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
[CrossRef]

Asano, S.

M. Shiobara, S. Asano, “The dome emission effect on the performance of pyrgeometers with silicon domes,” Pap. Meteorol. Geophys. 43, 17–31 (1992).
[CrossRef]

Baker, D. G.

J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
[CrossRef]

Bergholter, U.

K. Dehne, U. Bergholter, F. Kasten, “IEA comparison of long-wave radiometers in Hamburg 1989/90,” Rep. IEA-SHCP-9F-3 (International Energy Agency, Paris, 1993).

Betz, Ch.

Ch. Betz, “Entwicklung eines Eichstrahlers für Pyrgeometer,” Diplomarbeit (Universität Stuttgart, Stuttgart, Germany, 1993).

Brogniez, G.

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

Brown, J. H.

A. J. Drummond, W. J. Scholes, J. H. Brown, “A new approach to the measurement of terrestrial long-wave radiation,” WMO Tech. Note 104 (World Meteorological Organization, Geneva, 1970), pp. 383–387.

Buriez, J-C.

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

Cox, S. K.

B. Albrecht, S. K. Cox, “Procedures for improving pyrgeometer performance,” J. Appl. Meteorol. 16, 188–197 (1977).
[CrossRef]

B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
[CrossRef]

Dehne, K.

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

K. Dehne, U. Bergholter, F. Kasten, “IEA comparison of long-wave radiometers in Hamburg 1989/90,” Rep. IEA-SHCP-9F-3 (International Energy Agency, Paris, 1993).

DeLuisi, J.

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

Drummond, A. J.

A. J. Drummond, W. J. Scholes, J. H. Brown, “A new approach to the measurement of terrestrial long-wave radiation,” WMO Tech. Note 104 (World Meteorological Organization, Geneva, 1970), pp. 383–387.

Eckert, E. R. G.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
[CrossRef]

Enz, J. W.

J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
[CrossRef]

Field, R. T.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Fouquart, Y.

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

Fritschen, L. J.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Guzzi, R.

Heimo, A.

A. Heimo, Swiss Meteorological Institute, Aerological Station, CH-1530 Payerne, Switzerland (personal communication, 1993).

Jimenez, J. I.

L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
[CrossRef]

Kanemasu, E. T.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Kasten, F.

K. Dehne, U. Bergholter, F. Kasten, “IEA comparison of long-wave radiometers in Hamburg 1989/90,” Rep. IEA-SHCP-9F-3 (International Energy Agency, Paris, 1993).

Klink, J. C.

J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
[CrossRef]

Konzelmann, K.

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

Kustas, W. P.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Miskolczi, F.

Ohmura, A.

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

Olivieri, J.

J. Olivieri, “Pyrgeometry: spectral considerations,” Centre Radiometrique, F-84200 Carpentras, France (personal communication, 1994).

Peollet, M.

B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
[CrossRef]

Scholes, W. J.

A. J. Drummond, W. J. Scholes, J. H. Brown, “A new approach to the measurement of terrestrial long-wave radiation,” WMO Tech. Note 104 (World Meteorological Organization, Geneva, 1970), pp. 383–387.

Shiobara, M.

M. Shiobara, S. Asano, “The dome emission effect on the performance of pyrgeometers with silicon domes,” Pap. Meteorol. Geophys. 43, 17–31 (1992).
[CrossRef]

Smith, E. A.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Sparrow, E. M.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
[CrossRef]

Stewart, J. B.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Vanhoutte, J-C.

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

Verma, S. B.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

Vida, J.

L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
[CrossRef]

Vogt, R.

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

Weiss, A.

A. Weiss, “On the performance of pyrgeometers with silicon domes,” J. Appl. Meteorol. 20, 962–965 (1981).
[CrossRef]

Appl. Opt.

Contrib. Atmos. Phys.

G. Brogniez, J-C. Buriez, J-C. Vanhoutte, Y. Fouquart, “An improvement of the calibration of the Eppley pyrgeometer for the case of airborne measurements,” Contrib. Atmos. Phys. 59, 538–551 (1986).

J. Appl. Meteorol.

A. Weiss, “On the performance of pyrgeometers with silicon domes,” J. Appl. Meteorol. 20, 962–965 (1981).
[CrossRef]

J. W. Enz, J. C. Klink, D. G. Baker, “Solar radiation effects of pyrgeometer performance,” J. Appl. Meteorol. 14, 1297–1302 (1975).
[CrossRef]

B. Albrecht, S. K. Cox, “Procedures for improving pyrgeometer performance,” J. Appl. Meteorol. 16, 188–197 (1977).
[CrossRef]

J. Atmos. Oceanic Technol.

L. Alados-Arboledas, J. Vida, J. I. Jimenez, “Effects of solar radiation on the performance of pyrgeometers with silicon domes,” J. Atmos. Oceanic Technol. 5, 666–670 (1988).
[CrossRef]

J. Geophys. Res.

R. T. Field, L. J. Fritschen, E. T. Kanemasu, E. A. Smith, J. B. Stewart, S. B. Verma, W. P. Kustas, “Calibration, comparison, and correction of net radiation instruments used during FIFE,” J. Geophys. Res. 97, 18681–18695 (1992).
[CrossRef]

J. Heat Transfer

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer, 73–81 (1962).
[CrossRef]

Pap. Meteorol. Geophys.

M. Shiobara, S. Asano, “The dome emission effect on the performance of pyrgeometers with silicon domes,” Pap. Meteorol. Geophys. 43, 17–31 (1992).
[CrossRef]

Rev. Sci. Instrum.

B. Albrecht, M. Peollet, S. K. Cox, “Pyrgeometer measurements from aircraft,” Rev. Sci. Instrum. 45, 33–38 (1974).
[CrossRef]

Other

A. Heimo, Swiss Meteorological Institute, Aerological Station, CH-1530 Payerne, Switzerland (personal communication, 1993).

J. Olivieri, “Pyrgeometry: spectral considerations,” Centre Radiometrique, F-84200 Carpentras, France (personal communication, 1994).

Ch. Betz, “Entwicklung eines Eichstrahlers für Pyrgeometer,” Diplomarbeit (Universität Stuttgart, Stuttgart, Germany, 1993).

J. DeLuisi, K. Dehne, R. Vogt, K. Konzelmann, A. Ohmura, “First results of the baseline surface radiation network (BSRN) broadband infrared radiometer intercomparison at FIRE II,” in International Radiation Symposium ’92, S. Keevallik, O. Kärner, eds. (Deepak, Hampton, Va., 1993), pp. 559–564.

K. Dehne, U. Bergholter, F. Kasten, “IEA comparison of long-wave radiometers in Hamburg 1989/90,” Rep. IEA-SHCP-9F-3 (International Energy Agency, Paris, 1993).

A. J. Drummond, W. J. Scholes, J. H. Brown, “A new approach to the measurement of terrestrial long-wave radiation,” WMO Tech. Note 104 (World Meteorological Organization, Geneva, 1970), pp. 383–387.

J. T. Suttles, G. Ohring, eds., “Workshop on surface radiation budget for climate applications,” WMO/TD-No. 109 (World Meteorological Organization, Geneva, 1985).

World Climate Research Programme, “Workshop on implementation of the baseline surface radiation network,” WMO/TD-No. 406 (World Meteorological Organization, Geneva, 1990); “Second workshop on implementation of the baseline surface radiation network Davos,” WMO/TD-No. 453 (World Meteorological Organization, Geneva, 1991).

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Figures (8)

Fig. 1
Fig. 1

Upper part of the pyrgeometer, with the thermopile mounted on the cold junctions (body of the instrument) and the protecting silicon dome. The new dome-temperature measurement consists of three thermistors glued at 45° to the interference filter at the inside wall of the dome.

Fig. 2
Fig. 2

Calibration apparatus composed of a thermally well-insulated blackbody radiation source and an instrument housing. The temperature of the inside wall of the cylindrical cavity can be set from −30 to +60 °C. Cooling and heating systems in the instrument housing allow the user to adjust and control the body and dome temperatures of the instrument to be calibrated. A closed-cycle ventilation system simulates field measurement conditions during calibration.

Fig. 3
Fig. 3

Large temperature differences (gradients) observed between zenith and rim of the silicon dome. The temperature difference between the dome and the body is plotted versus the temperature difference between the blackbody radiator and the body of the instrument.

Fig. 4
Fig. 4

Temperature difference between old and new dome-temperature measurements plotted versus the temperature difference between the dome and the body.

Fig. 5
Fig. 5

Dome- and body-temperature evolution during field measurements of an unshaded pyrgeometer on a clear day at the end of October in Davos: (a) Temperature difference between the dome and the body measured with the old and the new measurements and body-temperature evolution during the day. (b) Evolution of the dome-temperature difference between south and north compared with the global radiation.

Fig. 6
Fig. 6

Pyrgeometer mount for remote Alpine stations that provides heating and dome ventilation. The fixed shadow band, which casts a shadow on the instrument at solar noon, provides information on the correct functioning of the instrument.

Fig. 7
Fig. 7

Comparison of three modified pyrgeometers, of which only PMOD2 is shaded: (a) Total downwelling long-wave irradiance E L of the three instruments, measured in Davos on the 12 September. (b) Difference of PMOD2 and PMOD3 and the reference instrument PMOD1 in watts per square meter. (c) Evolution of the temperature difference between the dome and the body of the three instruments.

Fig. 8
Fig. 8

Comparison of three modified unshaded pyrgeometers: (a) Total downwelling long-wave irradiance E L of the three instruments, measured in Davos during the week of 14 to 20 February. (b) Difference of PMOD2 and PMOD3 and the reference instrument PMOD1 in watts per square meter.

Tables (1)

Tables Icon

Table 1 Sensitivity C and Correction Factors k 1,2,3 with Standard Deviation of Five Modified Pyrgeometers, Calibrated According to the New Methoda

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

F net = F in - F out .
F in = α S τ D E L + α S ɛ D σ T D 4 + α S ɛ S ρ D σ T S 4 ,
F out = ɛ S σ T S 4 ,
F net = α S τ D E L + α S ɛ D σ T D 4 + α S ɛ S ρ D σ T S 4 - ɛ S σ T S 4 .
F net = T S - T B c ,             U emf = T S - T B γ .
T S = T B + γ U emf ,             T S 4 = T B 4 + 4 T B 3 γ U emf + .
E L = γ U emf c ɛ S τ D + ( 1 - ɛ S ρ D ) T D σ T B 4 + ( 1 - ɛ S ρ D ) τ D σ 4 T B 3 γ U emf - ɛ D τ D σ T D 4 .
E L = γ U emf c ɛ S τ D + ( 1 - ɛ S ρ D ) τ D σ 4 T B 3 γ U emf + ( 1 - ɛ S ρ D - ɛ D ) τ D σ T B 4 - ɛ D τ D σ ( T D 4 - T B 4 ) .
C = c ɛ S τ D γ ,
k 1 = 4 γ C ( 1 - ɛ S ρ D ) τ D ,             k 2 = ( 1 - ɛ S ρ D - ɛ D ) τ D , k 3 = ɛ D τ D ,
E L = U emf c ( 1 + k 1 σ T B 3 ) + k 2 σ T B 4 - k 3 σ ( T D 4 - T B 4 ) .
E L A = ɛ A σ T A 4 + ɛ D + ɛ D ( 1 - ɛ A ) σ T D 4 + ɛ S τ D ( 1 - ɛ A ) σ ( T B + γ U emf ) 4 ,
E L = U emf C ( 1 + k 1 σ T B 3 ) + k 2 σ T B 4 - k 3 σ ( T D 4 - T B 4 ) - f Δ T S - N + g U emf .

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