Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of...

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Solar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations ChE-600 Kevin Sivula, Spring 2018

Transcript of Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of...

Page 1: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Solar Photovoltaics & Energy Systems

Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

ChE-600

Kevin Sivula, Spring 2018

Page 2: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Energy use today and in the future

Current global energy use 19 TW = 19·1012 W

Industry30%

Transport30%

Residential and service

40%

Page 3: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Energy use today and in the future

German Advisory Council on Global Change, ISBN 1-85383-882-9

100 EJ/year = 3.17 TW

Page 4: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Solar 86,000 TW

Wind14 TW

Hydro1.2 TWGeothermal

1.9 TW

Biomass7 TW

Energy use today and in the future

WEA 2000 and Lewis and Nocera PNAS 2006, 103, 15729.

Current global energy use 19 TW

Page 5: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Modern Si photovoltaic installations

7.7MW in Rion-des-Landes

Page 6: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Growth of photovoltaic installationsTW

6

New materials

Page 7: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Course ScheduleDate Lecturer Topic27-02-18 Kevin Sivula The nature of Solar irradiation, thermodynamics of solar energy

conversion, solar thermal approaches01-03-18 Kevin Sivula Thermodynamics of band-gap based solar energy conversion, the

SQ limit06-03-18 Wolfgang Tress pn Junctions, Silicon Solar Cells, and III-V materials08-03-18 Wolfgang Tress Heterojunction, Organic and Dye Sensitized Solar Cells13-03-18 Wolfgang Tress Perovskite and Third Generation Solar Cells15-03-18 Néstor Guijarro Solar-to-fuel conversion, motivation and approaches20-03-18 Néstor Guijarro Direct solar to fuel with photoelectrochemical and photocatalytic

systems22-03-18 Student presentations Advanced strategies and materials (Literature Project)27-03-18 Student presentations Advanced strategies and materials (Literature Project)29-03-18 Student presentations Advanced strategies and materials (Literature Project)

URL: https://go.epfl.ch/ChE_600_2018

Page 8: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Why does the sun shine?

Page 9: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Why do stars “shine”?

Page 10: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Energy production< 300 W m-3

Visible surface:5700 Kdensity 0.2 g m-3

Page 11: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Blackbody radiation

𝐵𝐵 𝑇𝑇, λ 𝑜𝑜𝑜𝑜 𝐵𝐵λ 𝑇𝑇spectral radiance

Page 12: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Classical theory vs. experimental results

Page 13: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Blackbody radiation

No of modesunit volume

Probability of occupying modes

Average energy per mode

Prediction

Classical 8𝜋𝜋𝑐𝑐𝜆𝜆2

Equal for all modes 12𝑘𝑘𝐵𝐵T 𝐵𝐵λ(𝑇𝑇) =

2𝑐𝑐𝑘𝑘𝐵𝐵𝑇𝑇λ4

Quantum 8𝜋𝜋𝑐𝑐𝜆𝜆2

Quantized modes: need hν to excite, higher modes less probable

ℎ𝑐𝑐𝜆𝜆

𝑒𝑒ℎ𝑐𝑐

𝜆𝜆𝑘𝑘𝐵𝐵𝑇𝑇 − 1

𝐵𝐵λ(𝑇𝑇) =2ℎ𝑐𝑐2

λ51

𝑒𝑒ℎ𝑐𝑐

𝜆𝜆𝑘𝑘𝐵𝐵𝑇𝑇 − 1

Page 14: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

0.0E+00

2.0E+21

4.0E+21

6.0E+21

8.0E+21

1.0E+22

1.2E+22

1.4E+22

1.6E+22

1.8E+22

2.0E+22

0 500 1000 1500 2000 2500 3000

Phot

on F

lux

(arb

.)

Wavelength [nm]

2500 K - coolest red dwarf star 5778 K - The Sun 40000 K - Hottest stars (Type O)

Blackbody radiation

x 2000

x 10000

Page 15: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Flux of photons from a blackbody

𝐵𝐵λ 𝑇𝑇 =2ℎ𝑐𝑐2

λ51

𝑒𝑒ℎ𝑐𝑐𝜆𝜆𝑘𝑘𝑇𝑇 − 1

[=]𝐸𝐸𝐸𝐸𝑒𝑒𝑜𝑜𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐𝑜𝑜𝑜𝑜𝑐𝑐𝑒𝑒𝑐𝑐 𝑏𝑏𝐸𝐸 𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝐸𝐸𝑜𝑜

𝑜𝑜𝑐𝑐𝑡𝑡𝑒𝑒 𝑝𝑝𝑜𝑜𝑜𝑜𝑝𝑝𝑒𝑒𝑐𝑐𝑜𝑜𝑒𝑒𝑐𝑐 𝑜𝑜𝑠𝑠𝑜𝑜𝑠𝑠𝑐𝑐𝑐𝑐𝑒𝑒 𝑐𝑐𝑜𝑜𝑒𝑒𝑐𝑐 𝑤𝑤𝑐𝑐𝑤𝑤𝑒𝑒𝑤𝑤𝑒𝑒𝐸𝐸𝐸𝐸𝑜𝑜ℎ 𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑐𝑐 𝑐𝑐𝐸𝐸𝐸𝐸𝑤𝑤𝑒𝑒

Sun Earth

dAsun dΩsun

SI unit of solid angle ?

Steradian (sr)4π sr in a sphere

Total solid angle of the sun from the viewpoint of the earth is Ω𝑠𝑠𝑠𝑠= 6.8 x 10-5 sr

Page 16: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Flux of photons from the sun

𝐵𝐵600nm 5778K Ω𝑠𝑠𝑠𝑠 =2ℎ𝑐𝑐2

λ51

𝑒𝑒ℎ𝑐𝑐𝜆𝜆𝑘𝑘𝑇𝑇 − 1

Ω𝑠𝑠𝑠𝑠 = 153169mW

cm2 sr nm1

𝑒𝑒4.15 − 1Ω𝑠𝑠𝑠𝑠

= 2453mW

cm2 sr nm6.2415 x 1015 eV

mW∙sphoton

2.067 eVΩ𝑠𝑠𝑠𝑠

= 7.405 x 1018photons

cm2 s sr nm (6.8 x 10−5 sr)

= 5.03 x 1014photonscm2 s nm measured value = 5.34 x 1014

photonscm2 s nm

measured value = 136.1 −136.2mWcm2∫𝜆𝜆=0

∞ 𝐵𝐵λ 5778𝐾𝐾 Ω𝑠𝑠𝑠𝑠𝑐𝑐𝜆𝜆 = 136.8 mWcm2

The solar constant:

136.8 mWcm2 (1.27 x 108 km2) = 174 PW

Page 17: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations
Page 18: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

0.0E+00

1.0E+14

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0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Phot

on F

lux

[pho

tons

/s/c

m2 /

nm]

Wavelength [nm]

Extraterrestrial spectra

Backbody 5778 K

Blackbody radiation

Page 19: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Solar energy conversion by the human eye

Page 20: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Spectral sensitivities of human cone cells

Page 21: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Trichromatic vision

iphone 4 display

Page 22: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Dichromatic vision

http://www.diycalculator.com/sp-cvision.shtml

Page 23: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Tetrachromatic vision

Page 24: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

0.0E+00

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0 500 1000 1500 2000 2500 3000

phot

on s

tera

nce

[ pho

tons

/ c

m2

/ s /

sr /

nan

omet

er ]

Photon wavelength [nm]

2500 K - coolest red dwarf star 5778 K - The Sun 40000 K - Hottest stars (Type O)

Another digression

x 2000

x 10000

Page 25: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Solar Spectra

0.0E+00

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Phot

on F

lux

[pho

tons

/s/c

m2 /

nm]

Wavelength [nm]

O3

O2

H2O

H2O

H2O, CO2

Page 26: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Atmospheric effects

Page 27: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Solar Spectra

Mecherikunnel et al., Applied Optics 1983, 22, 1534

Page 28: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Variations in solar irradiance

http://www.newport.com/Introduction-to-Solar-Radiation

Page 29: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Averaged Solar Radiation 1990-2004

Assuming 8% energy conversion efficiency

Page 30: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Energy conversion thermodynamics

T1

T2

Q1

Q2

W

= heat flow = energytime

• How can we understand the limits of solar energy conversion?– Let’s start with a simple thermodynamic

model (Carnot Engine)

– Add linear (conduction) heat transfer (Curzon-Ahlborn Engine)

– Add radiative heat transfer (The Stefan-Boltzmann Engine)

– Include a Sun-planet model (The MüserEngine)

– Include a “band gap” to derive the Shockley–Queisser limit

Reference: Thermodynamics of Solar Energy Conversion by Alexis De Vos ISBN: 978-3-527-40841-2

Page 31: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Thermodynamics: The Carnot engine• The Carnot Engine

Normally T1>T2

• Axiom 1 (first law) ΣQ + ΣW = dU = 0

• Axiom 2 (second law) ΣS = 0 (for reversible processes)

• Postulate: heat flow is accompanied by an entropy flow Q/T, and work flow is accompanied by an entropy flow of 0, then Σ(Q/T) = 0

• It follows: Q1 = W + Q2 and Q1/T1 = Q2/T2

• Introduce “efficiency” W = ηC Q1

• ηC = 1 T2/T1

T1

T2

Q1

Q2

W

= heat flow = energytime

If W>0: positive work flow, “heat engine” If W<0: negative work flow, two subcases:1) T1>T2 this is a “heat pump” with ηC > 02) T2>T1 this is a “refrigerator” with ηC < 0

Page 32: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Carnot engine with linear conduction

• The Curzon-Ahlborn EngineNormally T1>T2 as in CarnotTwo irreversible components Reversible Carnot engine between T3 and T4

• Axiom 1 (first law)ΣQ + ΣW = 0 ⟹ Q1 = W + Q2

• Axiom 2 (second law)Σ(Q/T) = 0 ⟹ Q1/T3 = Q2/T4

• ηCA = 1 T4/T3• Fourier’s law: • Q = -λth𝛻𝛻T (assume linear conduction, λth thermal

conductivity)Q = g ∆T (g = λth /L)

• Q1 = g1(T1 T3)• Q2 = g2(T4 T2)

T1

T2

Q1

Q2

W

T3

T4

g1

g2

Page 33: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Curzon-Ahlborn Engine

• Q1 = g𝑡𝑡T1−T2−T1ηCA

1−ηCA, g𝑡𝑡 = g1g2

g1+g2

• W = g𝑡𝑡ηCA(T1−T2−T1ηCA)

1−ηCA

• ΣS = g𝑡𝑡(T1−T2−T1ηCA)2

T1T2(1−ηCA)

T1

T2

Q1

Q2

W

T3

T4

g1

g2

Page 34: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Curzon-Ahlborn Engine

T1

T2

Q1

Q2

W

T3

T4

g1

g2

CA

CA

CA

𝜂𝜂𝐶𝐶𝐶𝐶 = 1 −𝑇𝑇2𝑇𝑇1

𝑄𝑄1,𝑠𝑠𝑐𝑐 = g𝑡𝑡 (𝑇𝑇1 − 𝑇𝑇2)

𝜂𝜂𝐶𝐶𝐶𝐶,𝑜𝑜𝑐𝑐 = 1 −𝑇𝑇1𝑇𝑇2

For ηCA > ηC T3>T1>T2>T4 (heat pump)For -∞<ηCA <0 , T1>T3, T4>T3, T4>T2 (refrigerator)

Page 35: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Curzon-Ahlborn Engine

T1

T2

Q1

Q2

W

T3

T4

g1

g2

CA

𝜂𝜂𝐶𝐶𝐶𝐶 = 1 −𝑇𝑇2𝑇𝑇1

Q1 = g T1 T1 − T2 , g =g1g2

g1 + g2

W = g T1 − T22

ΣS = gT1 − T2

2

T1T2

Page 36: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

𝑄𝑄𝐴𝐴

= 𝑞𝑞 = 𝜆𝜆=0

∞𝐵𝐵λ 𝑇𝑇 𝑐𝑐𝜆𝜆

𝜃𝜃=0

2𝜋𝜋𝑐𝑐𝜃𝜃

𝜑𝜑=0

𝜋𝜋/2cos𝜑𝜑 sin𝜑𝜑 𝑐𝑐𝜑𝜑

𝜎𝜎 =2𝜋𝜋5𝑘𝑘𝐵𝐵4

15𝑐𝑐2ℎ3

𝜎𝜎 = 5.670 × 10−8 𝑊𝑊 m−2K−4

Stefan-Boltzmann law:

𝑄𝑄/𝐴𝐴 = ε𝜎𝜎𝑇𝑇4

For two (infinite) parallel plates of the same area and same material :

𝑄𝑄 = ε𝜎𝜎 𝑇𝑇14 − 𝑇𝑇24

T2

T1

36

Heat transfer by RadiationThe maximum hear transfer from the surface of a blackbody can be calculated from the spectral distribution:

= 𝜋𝜋𝜆𝜆=0

∞𝐵𝐵λ 𝑇𝑇 𝑐𝑐𝜆𝜆 =

𝜆𝜆=0

∞ 2𝜋𝜋ℎ𝑐𝑐2

λ51

𝑒𝑒ℎ𝑐𝑐

𝜆𝜆𝑘𝑘𝐵𝐵𝑇𝑇 − 1𝑐𝑐𝜆𝜆 = 𝜎𝜎𝑇𝑇4

(grey body, 𝛼𝛼(𝜆𝜆) = ε(𝜆𝜆))Ε = emissivity of the material (ε=1 for a blackbody)

Page 37: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

5.1 Heat transfer by Radiation

37

Material Average Emissivity

Aluminum foil 0.03

Aluminum, anodized 0.9

Asphalt 0.88

Brick 0.90

Concrete, rough 0.91

Copper, polished 0.04

Copper, oxidized 0.87

Glass, smooth (uncoated) 0.95

Ice 0.97

Limestone 0.92

Marble (polished) 0.89 to 0.92

Paint (including white) 0.9

Paper, roofing or white 0.88 to 0.86

Plaster, rough 0.89

Silver, polished 0.02

Silver, oxidized 0.04

Snow 0.8 to 0.9

Water, pure 0.96

Page 38: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Stefan-Boltzmann Engine

• Q = σεAT4 (assume radiative heat transfer)– Q1 = g1 T14 − T34

– Q2 = g2 T44 − T24

– ηSB = 1 T4/T3

• Q1 = g1g2(1−ηSB)4T14−T24

g1(1−ηSB)+g2(1−ηSB)4

• W = g1g2ηSB(1−ηSB)4T14−T24

g1(1−ηSB)+g2(1−ηSB)4

• ΣS = g1g2T1T2

( (1−ηSB)𝑇𝑇1−𝑇𝑇2)((1−ηSB)4T14−T24)g1(1−ηSB)+g2(1−ηSB)4

T1

T2

Q1

Q2

W

T3

T4

g1

g2

Page 39: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Stefan-Boltzmann Engine

T1

T2

Q1

Q2

W

T3

T4

g1

g2

For 𝑑𝑑𝑑𝑑𝑑𝑑ηSB

= 0Order 8 in ηSBDepends on g1/g2

SB

SB

SB

1

Page 40: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Stefan-Boltzmann Engine

T1

T2

Q1

Q2

W

T3

T4

g1

g2

For 𝑑𝑑𝑑𝑑𝑑𝑑ηSB

= 0Order 8 in ηSBDepends on g1/g2

SB

g1 ≪ g2g1 = g2g1 ≫ g2

Page 41: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine

T1

T2

Q1

Q2

W

T3

k1

T =T2=TP T =T3

Ts

Ωse

TP

Page 42: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine

T1

T2

Q1

Q2

W

T3

g1

T =T2=TP T =T3

Ts

Ωse TP

Q1 = 𝑠𝑠𝜎𝜎𝑇𝑇𝑠𝑠4 + 1 − 𝑠𝑠 𝜎𝜎𝑇𝑇𝑝𝑝4 − 𝜎𝜎T34

𝑠𝑠 =𝑐𝑐𝑝𝑝𝑝𝑝𝑐𝑐𝑜𝑜𝑒𝑒𝐸𝐸𝑜𝑜 𝑐𝑐𝑜𝑜𝑒𝑒𝑐𝑐 𝑜𝑜𝑠𝑠𝑏𝑏𝑜𝑜𝑒𝑒𝐸𝐸𝑐𝑐𝑒𝑒𝑐𝑐 𝑏𝑏𝐸𝐸 𝑜𝑜𝑠𝑠𝐸𝐸𝑐𝑐𝑝𝑝𝑝𝑝𝑐𝑐𝑜𝑜𝑒𝑒𝐸𝐸𝑜𝑜 𝑐𝑐𝑜𝑜𝑒𝑒𝑐𝑐 𝑜𝑜𝑠𝑠 𝑜𝑜ℎ𝑒𝑒 ℎ𝑒𝑒𝑡𝑡𝑐𝑐𝑜𝑜𝑝𝑝ℎ𝑒𝑒𝑜𝑜𝑒𝑒

=6.8 × 10−5

𝜋𝜋= 2.16 × 10−5

=∬Ω cos𝜃𝜃 𝑐𝑐Ω∬2𝜋𝜋 cos𝜃𝜃 𝑐𝑐Ω

≈Ω𝑠𝑠𝑠𝑠𝜋𝜋

= 𝜎𝜎 𝑠𝑠𝑇𝑇𝑠𝑠4 + 1 − 𝑠𝑠 𝑇𝑇𝑝𝑝4 − T34T1 = 𝑠𝑠𝑇𝑇𝑠𝑠4 + 1 − 𝑠𝑠 𝑇𝑇𝑝𝑝4

1/4

g1= 𝜎𝜎

The Müser engine efficiency:W = 𝜂𝜂𝑀𝑀Q1

Solar energy efficiency:W = 𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜𝑠𝑠𝜎𝜎T𝑠𝑠4

Qsun-earth = 𝑠𝑠𝜎𝜎𝑇𝑇𝑠𝑠4

Q1 = g1 T14 − T34

Page 43: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine

T1

T2

Q1

Q2

W

T3

g1

T =T2=TP T =T3

Ts

Ωse TP

W = 𝜂𝜂𝑀𝑀 𝑠𝑠𝜎𝜎𝑇𝑇𝑠𝑠4 + 1 − 𝑠𝑠 −1

1 − 𝜂𝜂𝑀𝑀 4 𝜎𝜎𝑇𝑇𝑝𝑝4

T3 =𝑇𝑇𝑝𝑝

1 − 𝜂𝜂𝑀𝑀

𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜 = 𝜂𝜂𝑀𝑀 1 +1 − 𝑠𝑠 1 − 𝜂𝜂𝑀𝑀 4 − 1

1 − 𝜂𝜂𝑀𝑀 4𝑇𝑇𝑝𝑝4

𝑠𝑠𝑇𝑇𝑠𝑠4

-0.020.000.020.040.060.080.100.120.14

0 0.2 0.4 0.6 0.8 1

Sola

r ene

rgy

conv

ersi

on

effic

ienc

y

Müser engine efficiency

Page 44: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine on other planets

Planet Avg. distance

(Gm)

𝑠𝑠 S (W m-2) Tp (K) Max. 𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜(%)

Mercury 57 1.48 × 10-4 9228 440 13.5

Venus 108 4.14 × 10-5 2586 733 0.95

Earth 150 2.16 × 10-5 1353 288 11.8

Mars 227 9.37 × 10-6 586 220 13.6

Jupiter 778 8.01 × 10-7 50 124 12.3

Saturn 1426 2.38 × 10-7 15 95 11.3

Uranus 2868 5.89 × 10-8 4 59 15.2

Neptune 4497 2.40 × 10-8 2 59 8.75

𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜 = 𝜂𝜂𝑀𝑀 1 +1 − 𝑠𝑠 1 − 𝜂𝜂𝑀𝑀 4 − 1

1 − 𝜂𝜂𝑀𝑀 4𝑇𝑇𝑝𝑝4

𝑠𝑠𝑇𝑇𝑠𝑠4

A first approximation: 𝑇𝑇𝑝𝑝4 = 𝑓𝑓

4𝑇𝑇𝑠𝑠4

Page 45: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Concentrating solar irradiation

Solar energy conversion device

T =T2=TP T =T3

Ts

Ω

TP

Concentration factor:C > 1

Qsun = C𝑠𝑠𝜎𝜎T𝑠𝑠4

Maximum concentration factor:

𝜎𝜎T𝑠𝑠4 = C𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝜎𝜎T𝑠𝑠4

C𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑠𝑠−1 = 46300

Page 46: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine with a concentrator

T1

T2

Q1

Q2

W

T3

g1

T =T2=TP T =T3

Ts

Ω

TP

Q1 = 𝜎𝜎 C𝑠𝑠T𝑠𝑠4 + 1 − C𝑠𝑠 T𝑝𝑝4 − T34

Solar energy efficiency:

W = 𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜C𝑠𝑠𝜎𝜎T𝑠𝑠4

-0.10

0.10

0.30

0.50

0.70

0.90

0 0.2 0.4 0.6 0.8 1

Sola

r ene

rgy

conv

ersi

on e

ffic

ienc

y

Müser engine efficiency

C = 1C = 10

C = 100

C = Cmax

Page 47: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine with a concentrator

-0.10

0.10

0.30

0.50

0.70

0.90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sola

r ene

rgy

conv

ersi

on e

ffic

ienc

y

Müser engine efficiency

C = 1 C = 10 C = 100 C = Cmax

85.4% maximum

Page 48: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

ηsolar(T3) for different Cso

lar

For Cmax = Toptimum = 2310 K

Page 49: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

The Müser Engine with concentrator on other planets

Planet Avg. distance

(Gm)

f S (W m-2) Tp (K) Max. 𝜂𝜂𝑜𝑜𝑜𝑜𝑤𝑤𝑐𝑐𝑜𝑜(%)

With conc.

Mercury 57 1.48 × 10-4 9228 440 79.7

Venus 108 4.14 × 10-5 2586 733 75.0

Earth 150 2.16 × 10-5 1353 288 85.4

Mars 227 9.37 × 10-6 586 220 88.1

Jupiter 778 8.01 × 10-7 50 124 92.5

Saturn 1426 2.38 × 10-7 15 95 93.9

Uranus 2868 5.89 × 10-8 4 59 95.6

Neptune 4497 2.40 × 10-8 2 59 95.7

Pluto 5874 1.18 × 10-8 2 44 96.6

Page 50: Solar Photovoltaics & Energy SystemsSolar Photovoltaics & Energy Systems Lecture 1 – The nature of Solar irradiation, thermodynamics of solar energy conversion, solar thermal limitations

Summary

• The Curzon-Ahlborn engine provides the framework for energy conversion with heat transfer

• The Stefan-Boltzmann engine uses radiative heat transfer and shows that the highest efficiency can be gained if the thermal transfer to the cold sink is infinitely good

• The Müser engine approximates this and reveals that only about 12% solar energy conversion efficiency is possible even with a Carnot cycle.

• Concentration allows the maximum efficiency of 85.4%