The exergy of thermal radiation and its relevance in solar energy conversion

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Relevance in Solar Energy Conversion

by

SEA N WRIGHT

B.Eng. (honours) McGill, 1996 M. A. Sc., Victoria, 1998

A Thesis Subm itted in Partial Fulfillment of the Requirements for the Degree Of

DOCTOR OF PHILOSOPHY

in the D epartm ent of Mechanical Engineering

We accept this thesis as conforming to the required standard

nD^St-SCott, Supervisor, Dept, of Mech. Eng., University of Victoria

Dr. M. A/Rosen, Co-Supervisor, Dept, of Mech. Eng., Rverson Polvtechnic University

DXj- B. H addow , Member, Dept, of Mech. E n g ., University of Victoria

Dr. M. FT WJialC Member, Dept, of Mech. E n g ., University of Victoria

D r . ^ . 1. Cooperstock, Member, Dept, of Physics, University of Victoria

_______________________________________________ Dr. D. Li, Ej^em al Examiner, Dept, of Mech. Eng., University of Toronto

© SEAN WRIGHT, 2000 University of Victoria

A ll rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author.

ABSTRACT

D riven by the im portance of optim izing energy system s and technologies, the field of exergy analysis was developed to better illum inate process inefficiencies and evaluate perform ance. Exerg}' analysis provides im portant inform ation and understanding that cannot be obtained from energy analysis. The field of exergy analysis is well form ulated and understood except for therm al radiation (TR) heat transfer. The exergy flux, or m axim um work obtainable, from TR has n o t been unam biguously determ ined. Moreover, m any therm odynam ic textbooks are m isleading by incorrectly im plying that the entropy and exergy transport w ith TR is calculated by using the same expressions that apply to heat conduction.

Research on the exergy of TR was carried out by Petela. However, m anv researchers have considered Petela's analysis of the exergy of TR to be irrelevant to the conversion of TR fluxes. Petela's therm odynam ic approach is considered irrelev an t because, others argue, that it neglects fundam ental issues that are specific to the conversion of fluxes, issues that are unusual in the context of exergy analvsis. The purpose of the research in this thesis is to determ ine, using fundam ental therm odynam ic principles, the exergy flux of TR w ith an arbitrary spectrum an d its relevance to solar radiation (SR) conversion.

In this thesis it is show n that Petela's result can be used for the exergy flux of blackbody radiation (BR) and represents the u p p er lim it to the conversion of SR approxim ated as BR. The thesis show s this by resolving a num ber of fundam ental issues:

1) Inherent Irreversibility

2) Definition of the Environm ent 3) Inherent Emission

4) Threshold Behaviour 5) Effect of Concentrating TR

This thesis also provides a new expression, based on inherent irreversibility, for the exergy flux of TR with an arbitrary spectrum. Previous analysis by K arlsson assum es that reversible conversion of non-blackbody radiation (NBR) is theoretically possible, w hereas this thesis presents evidence th at NBR conversion is inherently irreversible.

In ad d itio n the following conclusions and contributions are m ade in the thesis:

• Re-stated the general entropy and exergy balance equations for

therm odynam ic system s so that they correctly apply to TR heat transfer. • Provided second-law efficiencies for common solar energy conversion

processes such as single-cell Photovoltaics.

• Show ed that Om nicolor (infinite cell) conversion, the w idely held ideal conversion process for SR, is not ideal by explaining its non-ideal behaviour in terms of exergy destruction and exergy losses.

• Presented an ideal (reversible) infinite stage therm al conversion process for BR fluxes and presented two-stage thermal conversion as a practical alternative.

• Show ed that Prigogine's m inim um entropy production principle cannot be used as a governing principle in atm ospheric m odeling, and that in general, it m ay have little significance.

• Presented a graybody m odel of the planet that may prove useful in understanding the therm odynam ics of the Earth system.

• Show ed that the expression derived from the C lausius equality for

reversible processes is applicable, w hereas the statem ent for irreversible processes is not applicable, w hen there is significant h eat transfer bv TR. • Show ed that the 4 /3 coefficient in the BR entropv expression can be

obtained by sim ply using the concept of equilibrium an d the

Examiners:

PfT^P. S ^ gettrSupervisor, Dept, of Mech. Eng., University of Victoria

■n M. A. RpsénT Co-Supervisor, Dept, of Mech. Eng., -E v e rso n Polvtechnic Universitv

_______________________________________ p r J. B. H addow , Member, Dept, of Mech. E n g ., University of Victoria

______________________________ Dr. M. D. WhaWTMember, Dept, of Mech. E n g ., University of Victoria

_____________________________________ Dr. F. I. Cooperstock, Member, Dept, of Physics, University of Victoria

_________________________________________________ Dr. D. Li, Exftémal Examiner, Dept, of Mech. Eng., Universitv of Toronto

Table of C o n t e n t s --- v

List of Tables ---v iii Acknowledgem ents --- xiii

Part I

Background

1.1 M otivation --- 1

1.2 Objectives and S c o p e ---4

2 Principles of Exergy A n a ly s i s ---5

2.1 Exergy and the M ethod of Exergy A n aly sis--- 5

2.2 The E nvironm ent and S u r r o u n d i n g s --- 6

2.3 The Gouy-Stodola P r in c ip le ---7

2.4 Closed System Exergy and the Exergy Transfer w ith Mass F lo w 7 2.5 The Exergy Flux of Heat Conduction and C o n v e c tio n ---7

2.6 G eneral Balance Equations for a C ontrol Volume ( C V ) --- 8

2.7 Therm al Radiation (TR) and the Exergy C o n c e p t --- 9

3 Background on the Entropy of Thermal Radiation (TR) ---11

3.1 Expressions for the Entropy of Therm al R adiation--- 11

3.2 C om parison of the Entropy Flux of TR Emission to th at of H eat C o n d u c tio n --- 15

4 Background Research Related to the Exergy of T R --- 18

4.1 C onversion of Enclosed Blackbody Radiation (B R )--- 18

4.2 Approaches used to Determine the Exergy Flux of TR --- 22

4.2.1 Petela's approach for BR exergy: TR exchange betw een parallel blackbody p l a t e s --- 22

4.2.2 Karlsson's approach for the exergy of T R --- 25

4.3 M axim um C onversion Efficiencies in Solar Engineering --- 27

4.3.1 Single-stage thermal conversion --- 27

4.3.2 Single-cell quantum conversion --- 31

Part II

New Contributions

5 Exergy of Blackbody Radiation (B R )--- 38

5.1 Exergy of an Enclosed BR S y s t e m ---38

5.2 D efinition of the Environm ent for the Exergy of T R --- 39

5.3 Reversible Conversion of BR F l u x e s ---42

5.4 G eneral Balance Equations for a Control Volume C orrected tor TR heat t r a n s f e r ---52

6. The Effect o f Inherent Emission on TR Conversion --- 53

6.1 C om parison of Petela's A p p r o a c h e s ---53

6.2 An A dditive Inherent Emission Term for the Exergy Flux of TR — 56 6.3 N on-Zero Exergy Flux w hen there is no TR P re se n t--- 58

7 Inherent Irreversibility and Non-Blackbody Radiation (NBR) Exergy ---- 59

7.1 Petela's Approach Extended for NBR F lu x es--- 59

7.2 Review of Karlsson's Approach for NBR Exergy and Com parison to P etela's Approach Extended for NBR Fluxes---60

7.3 NBR Exergy Flux based on the Analysis of an Enclosed NBR System 62 7.4 Inherent Irreversibility and the Exergy of N B R --- 66

7.4.1 Non-Equilibrium NBR e n t r o p y --- 69

7.5 C om parison of the Result of this Study to Karlsson's R esult for NBR E x e r g y --- 70

8 Significance of TR Exergy in Solar Energy Conversion --- 73

8.1 TR Exergy - the U p p er Limit to SR Conversion --- 73

8.2 Second-Law Efficiencies of Common SR Conversion Processes - 75 8.3 T he Non-Ideal C haracter of Omnicolor C o n v e r s io n --- 77

8.4 The Effect of Concentration on Conversion Efficiency --- 80

9 On the Entropy of TR in Engineering Thermodynamics --- 82

9.1 M otivation for R e s e a r c h --- 82

9.2 A pplicability of the Clausius Statem ents for Reversible and Irreversible Processes w hen TR Transfer is Involved --- 83 9.3 A lternative Straightforw ard D erivation of the 4 /3 Coefficient for

10 Planetary Entropy Production and its Relevance

in Atmospheric M o d e li n g --- 90

10.1 M otivation for R e s e a r c h ---90

10.2 Simple G raybody Planetary M o d e l--- 91

10.2.1 Discussion of planetary model r e s u l t s ---98

10.2.2 Relevancy of the sim ple graybody radiative m odel to the therm odynam ics of the E a r t h --- 101

10.2.3 Exergy analysis of the Earth and b io s p h e r e --- 103

10.3 Stephens and Obrien's Support of a M axim um Dissipation Conjecture --- 104

10.4 Prigogine's Minimum Entropy Production P rin c ip le --- 107

10.4.1 Prigogine's result applied to one-dim ensional heat conduction in a p l a t e --- 110

10.5 Irreversible Therm odynam ics of the A tm o s p h e r e --- 112

11 Summary --- 114

11.1 Conclusions and Encapsulation of New R e su lts---112

11.2 R eco m m en d atio n s--- 121

List of Tables

N -l Thermal radiation (TR) energy, entropy an d exergy.

3-1 Accuracy of the approxim ation for the entropy of GR for two cases: m=ci and

m=C2-C3S-4-1 Maximum first-law energy conversion efficiencies (in %) for SR. 5-1 Maximum first-law energy conversion efficiencies for BR Conversion. 7-1 Exergy quantities for isotropic TR.

8-1 Maximum second-law efficiencies (in %) for SR (approxim ated as BR) conversion.

10-1 M ean planetary tem perature and entropy production rates.

List of Figures

3-1 Percent difference in entropy between GR an d BR with the sam e energy, as a function of emissivity.

3-2 The coefficient n for GR.

3-3 Illustration of several TR spectrums.

4-1 Piston-Cylinder device and the associated PV diagram for P V-*/3=const. 4-2 Qualitative plot of Petela's efficiency rip versus tem perature T a n d the

analogous plot for heat conduction.

4-3 BR system connected to a reversible C am o t Heat engine. 4-4 TR exchange betw een two parallel blackbody surfaces. 4-5 Exergy balance for the control volume.

4-6 Monochromatic TR incident on a blackbody surface at To. 4-7 Single-Stage therm al (SST) conversion device.

4-8 Single-Cell q u antum (SCQ) conversion.

4-9 Qualitative single-cell quantum emission a n d incident BR. 4-10 Infinite tandem of q uantum cells.

4-11 Conversion efficiency versus system tem perature for a single collector cascaded quantum -therm al converter.

5-1 BR system connected to a reversible C am o t heat engine. 5-2 Temperature of the environment; the slope of (i=ii(S).

5-3 Energy and entropy flows for the reversible C am ot h eat engine. 5-4 Infinitesimal pencil of rays of isotropic radiation.

5-5 Blackbody thermal conversion device.

5-6 Entropy production rate versus work production rate. 5-7 M ultistage thermal conversion device.

5-8 Two-stage thermal (TST) conversion device.

5-9 Energy spectrums for two-stage therm al conversion of BR.

5-10 M aximum first-law energy conversion efficiencies for BR; Petela's ideal efficiency, single-stage (SST) and two-stage (TST) therm al processes.

6-1 Petela's parallel-plate approach.

6-2 Black-box model for ideal BR conversion.

7-1 TR exchange between tw o parallel surfaces. 7-2 C onversion of an enclosed NBR system.

7-3 Energy spectrums for the conversion of an enclosed NBR system. 7-4 Black-box model for ideal NBR conversion.

7-5 NPD versus T/To = .r ', for T > To (.r < 1). 7-6 NPD versus T/To = .r^ for 0.5To < T < 2.5To.

8-1 Energy spectrums for TR exchange betw een two adjacent cells in a set for om nicolor thermal conversion.

8-2 Source an d emission energy spectrum s for omnicolor therm al conversion. 8-3 Energy flow versus tem perature.

9-1 Solid BB sphere system.

9-2 N et TR transfer for both h o t and cold cases.

9-3 A BR system contained in an evacuated cavity inside a n isotherm al solid.

10-1 SR incident on Earth.

10-2 An illustration of diffusely reflected solar radiation. 10-3 One-dimensional heat conduction in a solid.

Table N -l: Thermal Radiation (TR) Energy, Entropy and Exergy

Energy

Entropy

Exergy*

Symbol Units Symbol Units Sym bol Units

Internal U ₁ 5 l/K 1 Specific ** It 1 / m® s l/m®K r_*3 1/m® Flow Rate k W 5 W /K W Irradiance or Flux H W /m - _/ W /K m - M W/m® Spectral Irradiance _K 1 /m - _/v 1/Km- M . 1/m® Radiance K W / m ’sr L W /K m ’sr N W / m®sr Spectral Radiance _K 1/m -sr _L l/K m ’sr N.. l/m®sr D im en sio n less

Spectral Radiance ]fy none - V none

not

defined

---H and ç are the G reek letters corresponding to the English X and .r, respectively. T he symbols M and N were ch osen follow ing the pattern H.K and f,L for energy and entropy quantities. The specific energy and entropy for TR are per unit volum e rather than per u nit m ass as they are for material related quantities, for exam ple see Equ. (2.3).

a Alternative sy m b o l for the specific

exergy o f a m aterial system (1/kg);

the sym bol % is u sed in this thesis

a BR constant = 4 a / c =(7.6l)l0 '^l/m^K^

a Planetary a lb ed o (no units)

A Alternative sy m b o l for the exergy

o f a material s y ste m (J); the sy m b o l 5 is used in this thesis

A Surface area o f collector (m-)

B Alternative sy m b o l for exergy flux;

the sym bol H is u sed in this th esis

c Speed of l i g h t = (2.9979)10® m /s

C C oncentrating factor

k Energy flo w rate (VV)

h Planck's co n sta n t = (6.626)10 " J s

H Energy irradiance or flux (W /m -)

H^. Spectral energy irradiance (J/m-)

l(z) Function for GR en tro p y in Eq. (3.7)

/ Rate o f irreversibility (w )

/ Entropy irradiance or flux (W /K m -)

/,. Spectral entropy irradiance (1/Km-)

/i G en eralized th erm od ynam ic flux

(i.e. Jm, Jvt)

k B oltzm ann's constan t = (i.38)i0'^ l/ K

K E nergy radiance (VV/m’sr)

K^, Spectral energy radian ce (1/m -sr)

L E ntropy radiance (W/BCm-sr)

Spectral entropy rad ian ce (l/BCm-sr)

Lij P h en o m en o lo g ica l coefficien ts

m Function for GR entropy in Eq. (3.9)

M TR exergy irradiance (W /m -) Spectral exergy irradiance (J/m -)

n(e) Coefficient for GR entropy, Eq. (3.14)

N TR exergy radiance (w/m-sr)

N^. Spectral exergy radiance (j/m-sr)

Ni Mol num ber of species i in the en v iro n m en t

NPD Percent difference in exergy

radiance N, see Equ. (7.17)

p State of polarization, p=l for plane polarized, p=2 for unpolarized P Pressure (N /m -)

q H eat flux (W /m -) Integrand of Eq. (7.7)

dO Infinitesimal heat transfer (J)

Q H eat transfer rate (W)

r Mean radius of planetary orbit (m)

R Radius of the planet (m)

s Specific entropv-per unit volume (j/m^K)

S Entropy of the system (J/K) S Entropy flow rate (W /K )

Integrand of Eq. (7.8)

T M aterial em ission tem perature (K) Ty Spectral tem perature (K)

u Specific internal energy-per unit volum e (J/m^)

U Internal energy of the system (J)

V Volume (m^)

IT Work transfer rate (W)

X T em perature ratio T</T

.r Non-dim ensional group, hv/kT

X, G eneralized therm odynam ic force (i.e. Xm, Xtii)

X(e) Function for GR entropy in Eq. (3.7)

dY Differential unit of area (Karlsson's notation, norm ally dA)

a Unit step function in Eq. (4.35)

E Emissivity for gray radiation (GR) (p, (j) Spherical C oordinates

X] Efficiency

|i Q uantum excitation energy V Frequency (s-^)

v,dv Frequency interval dv about frequency v

Vo Cutoff (threshold) frequency Physical constant, 3.14159...

T Entropy production rate per unit volume (W /Km^)

n Entropy production (J/K ) n Entropy production rate (W /K )

0 Tem perature ratio T /T o (9 = l/.r) a Stefan-Boltzmann constant =

(5.67)10^ W/m-K^

(J Entropy production rate per unit surface area

n Solid angle (sr)

H Internal exergy of the system (J) H Exergy Flux (W), note that Karlsson

[21] uses the sym bol B

Note th at tw o symbols, a and .t, are conventionally used for different quantities so it is assum ed th at their svm bolism will be determ ined from context.

SUBSCRIPTS/SUPERSCRIPTS

A Stage A o Environm ent

Abs Absorbed out O utgoing

B Stage B _p Petela

BR at To- Blackbody Radiation with p Planet

em ission tem perature To Q Q uantum system

c C onduction Q fT h Q u an tu m /T h erm al hybrid

C C onverter r Radiation

Dest Destruction Ref Reflected

Emi Emission RM R adiation-m atter

f Final s Source

fm Fundam ental m inim um S Sun

i Initial Sf Surface of the Sun

Inc Incident T at tem perature T

m M aterial system V Frequency

N N et

ABBREVIATIONS

BB Blackbody

BR Blackbody Radiation CV Control Volume

DBR Diluted Blackbody Radiation

ERBE National Aeronautics and Space A dm inistration (NASA) Earth Radiation Budget Experiment GR G raybody Radiation

GS Guoy-Stodola Theorem HE Heat Engine

NBR Non-Blackbody Radiation

SCQ Single-Cell Q uantum C onversion SR Solar R adiation

SST Single-Stage Thermal C onversion 1ER Therm al Energy Resevoir

TOA Top of the Atmosphere TR Therm al Radiation

ACKNOWLEDGEMENTS

Forem ost I w ould like to thank Dr. David S. Scott for his su p p o rt over the four years that w e have worked together. I greatly benefited from his style and perspective of engineering which is at times unique and very effective. I respect his im partiality and his com m itm ent to science and hum anity related issues. Dr. Scott sparked my interest in energy systems developm ent and exergy analysis an d his enthusiasm has been very encouraging.

I w ould also like to acknowledge and thank Dr. Marc Rosen m y co-supervisor. His detailed and prom pt reviews of my w ork have been both insightful and beneficial. I appreciate the time th at he devoted to this project considering his heavy career responsibilities. He gave much encouragem ent and su p p o rt both academ ically and financially for this project.

A special thanks to Dr. James H addow for our m any productive technical discussions in all areas of therm odynam ics. I benefited from his m anv years of experience and his ability to sharpen my reasoning abilities through posing interesting problem s. I respect his integrity as a teacher and his abilitv in conducting his classes.

A dditional thanks to Dr. Fred Cooperstock and Dr. M. W hale for the their tim e and efforts sp en t on the committee.

Lastly, I w ish to recognize and thank Dr. Scott for financial su p p o rt d raw n from his research funds. Dr. Rosen for financial support through NSERC funding, the U niversity of Victoria for funding through UVic graduate fellow ships (1998/1999 an d 1999/2000), and the departm ent of Mechanical E ngineering for financial su p p o rt through the teaching assistant program .

1.1 M otivation

Engineering texts are often m isleading because they state that heat transfer has three forms, conduction, convection, and radiative transfer, and proceed to evaluate the entropy and exergy flux in a wav that does not apply to TR heat transfer. For example, see Moran and Shapiro's [1] text on the fundam entals of engineering therm odynam ics. The entropy and exergy flux of heat conduction are given by the energy flux divided by the local tem perature (q/T) and £y(l-

To/T), respectively, w here To is the environm ental tem perature. The en tro p y

flux of BR is not given by the q /T relation and recently it was show n that the entropy flux of non-blackbody radiation (NBR) is even farther rem oved from

q / T [2], see section 3.2. However, a general exergy flux expression for TR w ith

an arbitrary spectrum has not been clearly determ ined. Also, a discordance exists betw een researchers who have used an exergy analysis approach and researchers w ho have investigated the upper limit to the conversion efficiency or m axim um w ork o u tp u t from solar radiation (SR) conversion.

O ur m otivation for considering the exergy flux of TR stems from the fact that exergy analysis is an effective and illum inating form of second law analvsis

that:

a) identifies entropy production or exergy destruction as the true indicator of non-ideal behaviour,

b) pinpoints com ponents or sub-processes contributing m ost to the overall non-ideal behaviour of a system,

c) correctly evaluates the significance of emissions to the surroundings such as exhaust gases from a coal-fired pow er plant, and

d) provides second-law efficiencies that give a true evaluation of perform ance by com paring perform ance to ideal operation, in contrast to first-law energy efficiencies w hich are usually m isleading. Exergy analysis gives a m uch better indication than energy analysis of w hether it is beneficial and cost effective to m odify or re-design equipm ent. Bv

efforts on im proving performance.

T urning to the exergy of TR, all m atter emits TR continuously as a result of its tem perature so TR is an inherent p art of our environm ent and it is an im portant energy transfer m echanism in the therm odynam ic analysis of m any systems in addition to solar conversion such as (1) industrial boilers and furnaces, (2) spacecraft cooling and solar pow er systems, (3) heating and lighting system s th at act by means of radiation, (4) biochemical (photosynthesis) processes th at occur in plants, (5) cryogenic devices, and (6) circulation of the atm osphere. Petela carried out research to determ ine the exergy of blackbody radiation (BR). How ever, om nicolor conversion is considered by m any researchers to be the ideal theoretical process for the conversion of solar radiation (SR) approxim ated as BR. For example, H aught [3] states regarding om nicolor conversion that the "results obtained are independent of the specific form of the therm al and quantum radiation conversion device and serve as an upper bound on the efficiency w ith which radiant energy can be converted to useful work in any actual device." In agreem ent with H aught, De Vos and Pauwels [4] also state that an infinite series of optim ized onm icolor collectors is "the therm odynam ically optim al device for converting solar energy into work." Petela's [5] blackbody radiation (BR) exergy result is thought to neglect fundam ental theoretical issues that are specific to the conversion of TR fluxes. For exam ple, H aught [3] states that "Therm odynam ic treatm ents of the radiation field w hich derive the conversion efficiency from the available w o rk content of the radiant flux neglect the lim itations (re-radiation, threshold absorption, etc.) inherent in the conversion process." G iving cause for this view point, some therm odynam icists have stated th at the issue of inherent irreversibilities an d inherent em ission (re-radiation) can be ignored in order to determ ine the m axim um w ork obtainable from TR conversion.

depends on the system and its environm ent, so how can an environm ent be defined for TR? Bejan [6] states that "there is no such thing as an "environm ent" of isotropic blackbody radiation (and pressure), as is assum ed m ost visibly in the availability type derivation."

Second, how is it appropriate to assum e that the conversion of BR fluxes can be reversible even though it appears th at the conversion of TR fluxes is inherently irreversible? De Vos and Pauwels [4] state that "the conversion of radiation into w ork cannot be perform ed...w ithout entropy creation." It is true that entropv is created by any conversion device in practice but this does not m ean that the conversion process is irreversible even in the theoretical case as is implied. Third, how does the inherent em ission of TR affect the m axim um w ork obtainable? Any device that absorbs TR for conversion m u st also em it TR. De Vos and Pauwels [7] state that the "pow er flow from the solar cell is rightly considered lost." Also Landsberg [8] comments on the effect of inherent em ission when he notes that Petela's efficiency is "pulled dow n below the C am o t efficiency because of the black-body emission from the converter which does n o t contribute to the useful w ork output."

As m entioned above exergy analysis is an effective form of second-law analysis. Thus, it is im portant that these issues reg ard in g TR exergy are resolved. This will allow second-law efficiencies to be determ ined for com m on solar energy conversion processes an d provide insight th at m ay lead to practical device im provem ents.

Note th at the m otivation for the research presented in chapter 9 (On the E ntropy of Therm al Radiation in Engineering Therm odynam ics) and chapter 10 (Planetary Entropy Production and its Relevance in A tm ospheric Modeling) is discussed in sections 9.1 and 10.1, respectively.

The difficulties th at arise in determ ining the exergy of TR and its relevance in solar energy conversion may be sum m arized and addressed as follows:

1) Exergy is a quantity that depends on the system and its environm ent, so how is the environm ent defined for TR? 2) The conversion of TR appears to be inherently

irreversible. Is this the case and if so how does it affect tire m axim um w ork obtainable from TR conversion?

3) Any device that absorbs TR m ust also em it TR. W hat is the effect of this inherent em ission on TR exergy?

4) W hat is the relevance of TR exergy to solar energy conversion?

It is the objective of the present w ork to resolve these four m ain issues and to im prove the effectiveness of exergy analysis of engineering system s w hen TR transfer is significant. This research can help us to better model, analyze, design, and optim ize systems w here TR transfer is im portant.

Furtherm ore, we have two m ore sets of objectives relating to atm ospheric m odeling and to the applicability of fundam ental therm odynam ic equations w hen TR is involved. Regarding the therm odynam ics of the Earth system o u r purpose is to:

1) Determ ine the relevance of Prigogine's m inim um entropy production principle as a governing principle in atm ospheric m odeling.

2) Present a graybody m odel of the planet to im prove o u r therm odynam ic understanding of the Earth system.

3) Examine the significance of Stephens and O brien's conclusion that the Earth is near a state of m axim um entropy production.

R egarding the entropy of therm al radiation in engineering therm odynam ics our objectives are to;

1) Determ ine the applicability of the Clausius statem ents for reversible and irreversible processes w hen TR transfer is involved.

2) Present an alternative straightforw ard derivation of BR entropy in an attem p t to find a physical reason for the 4 /3 coefficient (see section 3.1).

2.1

Exergy and the Method of Exergy Analysis

Exergyi analysis is an effective and illum inating form of second law analysis. Exergy is defined as the m axim um am ount of w ork that can be produced by a stream of m aterial or a system as it comes into equilibrium w ith its environm ent. Exergy may be loosely interpreted as a universal m easure of the w ork potential or quality of different forms of energy in relation to a given environm ent. The first law states that energy is never 'used up' it is sim ply degraded or converted from one form to another. For example, a hot beverage will cool to the tem perature of its environm ent, a balloon will burst if it's punctured, and a fuel will b u m if it's ignited. Although energy is not destroyed, the exerg}' or potential for a system to do w ork is lost.

To illustrate the benefits of second law analysis consider the operation of a typical therm al power-plant. For example, a pow er plant may have roughly 1 /5 of the energy of the incoming fuel leaving by the smoke stack, 2 /5 rejected by waste heat, and 2 /5 leaving as the desired w ork output usually in the form of electricity. The first-law perspective is that 3 /5 of the incom ing energv source is em itted or lost to the surroundings. How ever, second law analvsis show s that even under ideal operation w ork cannot be obtained w ith o u t sending a certain m inim um am ount of energy and products to the environm ent. The corresponding m axim um ideal level of w ork o u tp u t th at can be obtained is the exergy of the fuel in a particular environm ent. In the exam ple pow er-plant considered above 3 /5 of the fuel energy m av be directed to the environm ent but it is of relatively low quality and thus m av onlv represent a sm all fraction of the exergy of the fuel, in the order of 1/10.

indicators of perform ance. Exergetic (second-law) efficiencies com pare the

exergetic value of the desired outpu t to the ideal lim it, the exergy of the energy

source. First-law energy efficiencies are m isleading for therm al or chem ical processes because they com pare the desired energy output to the energy input. For example, first-law efficiencies for heat engines com pare the w ork o u tp u t to a theoretically unachievable u p p er limit, the energy of the heat source.

Also, exergy analysis gives a true indication of w hether it is beneficial and cost effective to m odify or re-design the equipm ent. Exergy analysis identifies exergy destruction (entropy production) as the true indicator of non-ideal perform ance. Analysis of a m ulti-com ponent system pinpoints the com ponents contributing m ost to the overall irreversibility of the system. Also, the decision to utilize any exergy 'losses' w ith emissions from the system (such as 'w aste' heat) or by-products of the process is based on their exergetic value.

2.2

The Environment and. Surroundings

M oran [9, p. 45] describes the terms environm ent and surroundings as: "The surroundings com prom ise everything not included in the system. O ne p art of the surroundings is some portion of the Earth an d its atm osphere, the intensive properties of w hich do not change significantly as a result of any of the processes u n d e r consideration. It is to this that the term environm ent applies."

We m ust develop a m odel for the environm ent because the physical w orld is com plicated an d cannot be described in every detail. The characteristics of the environm ent m odel are:

• The environm ent is assum ed to be large w ith respect to the system and to have hom ogeneous and tim e-invariant intensive properties. The changes in the extensive properties are so sm all relative to the size o f the environm ent that the intensive properties rem ain unchanged.

• The environm ent experiences only internally reversible processes in which the sole w ork mode is associated w ith volum e change (PciV). • The environm ent experiences heat transfer at a uniform tem perature To.

2.3 The Gouy-Stodola Principle

The Gouy-Stodola principle [10, p. 24] states that for any fixed environm ent the exergy destruction rate, term ed the rate of irreversibility ( / ) , is directly proportional to the entropy production rate (IT). The entropy production rate is a n absolute quantity w hereas the exergy destruction rate is a relative quantity th at depends on the choice of reference heat reservoir, norm ally taken as the tem perature of the environm ent To:

/ = To n (2.1)

2.4 Closed System Exergy and the Exergy Transfer with

Mass Flow

Exergy is a m easure of the departure of a closed system from that of its environm ent. The exergy- (H) of a system closed to mass flow is the theoretical m axim um am ount of w ork that can be produced as the system reaches the dead state (equilibrium w ith its environm ent)

H = ( C /- f /> P ,( F - T ,) - 7 ;,( 5 - ^ ,) (2.2) w here U is internal energy, V is volum e, S is entropy, Po and To are the environm ent pressure and tem perature, and Uo, Vo and So are the dead state properties of the system (see for exam ple Bejan [11]). The system is in the dead state w hen m utual equilibrium is reached betw een the environm ent and the system , w hen the system has pressure Po and tem perature To. Note th a t in all

- The Greek sym bols H and c correspond to the upper and low er case English letter .r for erergy. The sym bols A , B and a are som etim es used in place o f H, 2 and c and originate from the alternative nam e for exergy, availability.

the equations introduced in this section kinetic energy, g ravitational potential energy, and chemical exergy have not been included. O n a per mass basis equation (2.2) becomes

Ç = (w - w„) + - %,) - (2.3) The rate of exergy transfer when mass flows across a system boundary is given

by

E = (2.4)

w here h is the enthalpy {h = ii+Pi').

2.5 The Exergy Flux of Heat Conduction and Convection

C onsider heat transfer Q (heat conduction and convection) betw een a control mass and some other system where initially the control m ass is at the dead state (equilibrium w ith its environm ent), and the process experienced by the control mass is internally reversible and constant volume. It can be show n that the exergy of the control mass in its final state is

H = (2.5)

oV ‘i J

w here To is the environm ental tem perature, o denotes the dead state, / denotes the final state, and clQ denotes an infinitesim al heat transfer at the boundary of the control m ass w here the tem perature is Is. Note that for this particular process the exergy of the system is strictly therm al exergy (no mechanical o r chemical exergy). The integrand of equation (2.5) times dQ gives the exergy transfer with the heat transfer dQ. The exergy of heat transfer is the m axim um w ork that can be obtained using the environm ent as a reservoir of zero-grade therm al energy. The C am ot heat engine (HE) provides a theoretical means of producing this m axim um ideal w ork o u tp u t (reversibly).

The exergy balance equation for a system takes into account the exergy transfer rates across the system boundary by heat and w ork transfer, and the exergy destruction rate that occurs w ithin the system due to irreversibilities:

d=^y_ f f. r.

j dA -\W (.y-P,,—^ \ ^ Y Z : - i r v (2.6)

C l' himndarv \ J ^ Ü I J ^

w here the subscript CV denotes control volum e, the subscript h denotes the b oundary of the CV, q is the heat flux across the system boundary w here the tem perature is Tb, and dA is a unit of area on the system boundary. The second last term accounts for exergy transfer w ith mass flow and is a sum m ation term because m ultiple mass flows may cross the boundary of a system. Note that the subscript i is used to indicate mass flows into the system so any outgoing m ass flows w ould have a negative sign.

The therm al-m echanical exergy balance equation is a sim ple com bination

of the energy _ dt d z . ç y _ d E ç y d S f ~ y ( ' J T \ dt dt " dt ' jq dA (2.8) C r b<)unJdr\'

an d the entropy balance equations

= J ^ d . A (2.9)

2.7

Thermal Radiation (TR) and the Exergy Concept

TR is generated by the acceleration of electric charges in atoms or m olecules of a m aterial as a result of its tem perature (therm al vibrations). TR can travel through a vacuum w hereas heat conduction and convection are forms of energy transfer by direct m aterial interaction and movement. TR does not interact w ith itself so one w ould expect that there is no flow w ork as w ith mass flows.

In trying to define the exergy radiance of TR one expects that the exergy depends on: (1) the energy and entropy carried by the radiation, (2) the polarization of the TR, and (3) the environm ental character. How ever, the entropy of TR is com pletely determ ined by the energy spectrum so entropy m ay not be an independent param eter. Secondly, polarization is a geom etric param eter so it does not play a role in determ ining a fundam ental form of the expression for the exergy flux of TR (the polarization p is simply a coefficient in the entropy expression (3.2)). For simplicity, in this thesis it is assum ed that the TR is unpolarized (p=2). Once the fundam ental form of the exergy flux expression for TR is determ ined it may be altered to take into account polarization.

Chapter 3 Background on the Entropy of

Thermal Radiation (TR)

3.1

Expressions for the Entropy of Thermal Radiation

The correct evaluation of the entropy flux of TR can be im portant w h e n determ ining the second-law performance of energy conversion devices. M any therm odynam ic texts incorrectly imply that the entropy flux of TR heat tran sfer has the sam e form as that of conductive heat transfer, that is the heat flux divided by the local tem perature (q/T). Wright [2] has show n that m isuse of the

q/T relation for TR transfer from an energy conversion device can cause v e ry

significant erro rs and always causes the irreversibility of the device to be underestim ated w hether it is h o t or cold relative to its surroundings.

The energy and entropy of unpolarized^ TR is correctly calculated using the spectral energy and entropy expressions derived by Max Planck [12] u sin g equilibrium statistical mechanics:

u c- — (3.1) - I and L = pku' ( , c'K^ 11+— ^ Inr, c'K1+— %- — ]In I 2Au-' J 2Au' 2hv‘ J (3.2)

w here p=2 for unpolarized TR considered in this thesis. Equation (3.1) expresses the spectral energy radiance of BR, i.e., the energy flow rate per unit frequencv, area, and solid angle. A plot of versus frequency v for various values of tem perature T gives a family of BR energy spectra. After substituting (3.1) in to (3.2) we can obtain a family of BR entropy spectra.

For an arbitrary TR, the entropy spectrum is found by substituting data, rather th an (3.1) for BR, into (3.2). The entropy radiance L of any TR spectrum can be calculated by integrating over frequency. That is, the entropy radiance

L is the area u n d er the spectrum.

Landsberg and Tonge [13] used a non-equilibrium statistical mechanics approach to obtain the same result as Planck. They concluded- "This result, usually obtained from equilibrium statistical mechanics, is therefore of w ider significance and represents a non-equilibrium entropy."

The energy spectrum of TR em itted from a solid depends on the nature of the em itting m aterial and its tem perature. Some m aterials can be adequately approxim ated as blackbody (BE) or graybody w hile others may have a unique spectra that caimot be adequately modeled as either BB or gray body.

For BR, equations (3.1) and (3.2) can be integrated over frequency and solid angle to obtain the energy and entropy irradiances (fluxes):

Hbr = kKbr = gT-^ /gR = TcLgR=j oT^ (3.3)

Irradiances (H and f) are the integration of the radiances (K and L) over solid angle an d have units of energy or entropy flow rate per unit area. To com pare the entropy flux of BR emission to that of heat conduction we can express the entropy irradiance as

' 3 - ^ (3.4)

N ote th at TR transfer is generally a net transfer betw een incident, reflected, and em itted TR but in (3.4) we are considering TR em ission alone.

- Two assum ptions were specified for this result to be exact: (1) the probability of finding N, bosons in quantum state / is independent of the occupation numbers o f the other quantum states, and (2) the probability o f an additional particle occup ying a state; is ind ep en d en t o f the num ber already in that state.

By definition the spectral energy irradiance for isotropic gray radiation (GR) em ission is

^GR = (3.5)

H ow ever, the entropy is not as easily calculated because the spectral en tro p y is not a linear function of the spectral energy. For GR the entropy irradiance is

2;r/:’* - U g \ / s \ f s ^ f g Al

w here x = fiv/kT. The entropy of GR (3.6) is a simple cubic function of the m aterial emission tem perature. This is also true of any TR em ission th at has an energy spectrum w ith a fixed shape independent of emission tem perature [2]. The definite integral in (3.6) is a function of e only and is called f(e) here, b u t was first recognized by Landsberg and Tonge [14] as eX(e)4;ry45. Thus, we have:

^ = (3.7)

The integral 1(e) has not been solved in closed form. Stephens and O brien [15] presented an infinite series solution and Landsberg and Tonge [14] presented the approxim ate lim iting solution for s < 0.10:

X { s ) % 0.9652 - 0.2777 In £ + 0.0511 £ (3 .8 )

Note that Landsberg and Tonge [14] refer to gray radiation (GR) as diluted blackbody radiation (DBR).

W right [2] presented an approxim ation for the entropy of GR that is accurate over a large range of emissivities:

/(£ ) s: £ { ^ - m l n £ I (3.9) w here m is optim ized for a specific emissivity range. The approxim ation (3.9) is such th at the blackbody result is obtained for e = 1 and the w hitebody result for 8 = 0. U sing (3.9) the entropy of GR becomes:

(3.10)

If m is approxim ated as a constant the resulting expressions are w ithin 0.8% of the num erical integration data over the em issivity range 0.005<s<l. The accuracy of the approxim ation can be increased by replacing the constant for m w ith a linear function of emissivity. Table 3-1 show s the m axim um percent error in the entropy calculation for various em issivity ranges, for tw o cases m=Ci and m=C2-CiS. Note that the accuracy of the approxim ation is strongly

dep en d en t on the low er lim it of the emissivity range.

T able 3-1: Accuracy of the A pproxim ation for the Entropy of GR for two cases: m = Ci and m=C2- Cje.

Emissivity Range

Case 1 Case 2

c,

0.005 to 1.0 2.317 0.77 2.336 0.260 0.33 0.010 to 1.0 2.310 0.71 2.328 0.200 0.33 0.050 to 1.0 2.285 0.49 2.311 0.175 0.16 0.200 to 1.0 2.319 0.23 2.292 0.150 0.03

Alternatively, the entropy radiance of any NBR can be approxim ated as that of BR w ith the same energy radiance

^ (3.11)

and for GR

(3.12)

The error is relatively low in using the approxim ation (3.12) for GR over a w ide range of emissivities (e.g., less than 1% for e>0.5). Likewise, this en tro p y approxim ation is relatively accurate for any NBR as long as the spectral em issivities are n o t too low. Note that Equations (3.11) and (3.12) alw avs overestim ate the entropy of NBR because BR represents the state of m axim um

entropy for a given energy. Figure 3-1 shows the percentage th at the entropy of GR is overestim ated by using (3.12).

S l 8

I

Emissivity (e)

Figure 3-1: Percent difference in entropy betw een GR and BR w ith the same energy, as a function o f emissivity.

For em issivity values greater th a n 0.50 the percent difference in en tro p y is very small, less than 1%. At e = 0.20 the percent difference is ab o u t 5% and at s = 0.10 it is 10.5%.

3.2

Comparison of the Entropy Flux of TR Emission to that of

Heat Conduction

For heat conduction the entropy flux is the ratio of the heat flux an d the local tem perature (q/T). For TR this relationship does n o t hold. The e n tro p y flux of isotropic^ TR emission can be compared to th at of heat conduction by expressing the entropy flux as

(H.firm

(3.13)

^ Isotropy is specified so that w e can consider fluxes (irradiances), rather than radiances, in a sim p le manner.

w here n is dependent o n the character* of the TR and the subscript Emi denotes emission. For BR n = 4 /3 as shown in equation (3.4). For GR the coefficient n is given by [2]

^ = (3.M)

Figure 3-2 illustrates the function n for GR using the approxim ation m = 2.311- 0.175e listed in Table 3-1.

n(s) evaluated w ith equation (3.14)

2.5

1.5 —

0 0.4 0.6 0.8 1

E m issivity (e)

Figure 3-2: The coefficient n for GR.

For GR the coefficient n increases as emissivity decreases. For exam ple, n reaches 2.5 for e =0.05. So the entropy flux of NBR emission can be m uch g reater th an th at given by u sin g the relation for heat conduction (cf/T). It is su rp risin g th at n for NBR is g reater than 4/3. BR is know n as equilibrium TR and as m axim um entropy. H ow ever, BR has a m inim um ratio of entropy to en erg y for all TR w ith the sam e em ission tem perature. This observation m eans th at the entropy flux of NBR em ission is farther rem oved from q / T than th a t of BR em ission [2].

* T he coefficient n is o n ly a constant independent o f em ission temperature if the sh a p e of the

NBR en ergy spectrum is invariant within the temperature range o f interest This condition is by definition satisfied for BR and GR.

Figure 3-3 illustrates the BR and NBR spectrum s at the same em ission tem perature, Tnigh, and the BR spectrum at tem perature Timv th at has the sam e energy as the NBR spectrum at Tnigh. Note th at the area under the spectrum gives the energy radiance K. The ratios of entropy to energy irradiances (fluxes) for the three spectrum s in Figure 3-3 m ay be expressed as a continued inequality: - I > - ] > 'rtf* (3.15) 'r t f *

NBR at Tnigh

Frequency (

)

Chapter 4 Background Research Related to

the Exergy of TR

Research on the ideal conversion of therm al radiation (TR) usually follows one of three general approaches. First, some researchers consider the w ork o u tp u t from an enclosed blackbody rad iatio n (BR) system as discussed in section 4.1. A second ap p ro ach (section 4.2) is based on the G ouy-Stodoia theorem an d the entropy production rate w hen a TR source flux is absorbed by a surface at the environm ent tem perature To. Thirdly, as discussed in section 4.3, researchers in solar en g in eerin g have calculated the m axim um conversion efficiencies of various solar conversion processes.

4.1 Conversion of Enclosed Blackbody Radiation (BR)

A num ber of researchers have considered the ideal conversion of an enclosed BR system an d Bejan [6] provides an excellent review and unifying interpretation of the different approaches used.

In 1964 Petela [5] used a piston-cyUnder approach to determ ine the m axim um w ork o u tp u t obtainable from an enclosed equilibrium BR svstem . Figure 4-1 show s the evacuated enclosure w ith perfectly reflecting w alls and a m oveable piston. T he enclosed ra d ia tio n sy stem is th erm ally iso lated from its su rro u n d in g s. BR at tem perature To occupies the volum e of space o n the backside of the piston.

process:

pr/*3 = constant

Perfectly reflecting (isolating) enclosure Fncnonless oiston

BR at r. BR svstem

at'r

The entropy a n d internal energy of the BR system are

U = a f V (4.1)

and

S = ^ a f V (4.2)

So the fundam ental equation of state for the BR system is

C/ = (flF )'H è-^)' (4.3) The system d eliv ers m axim um w ork as it settles into the d e ad state by an isentropic (reversible and adiabatic) expansion process. Using the relation for BR p r e s s u r e , P = L //3 F , it can be straig h tfo rw ard ly sh o w n from (4.3) th at

5 = constant for a n isentropic process m eans that = constant as illustrated in Figure 4-1. The w o rk produced during this process is given by

I',

= 1 (P -P .) dV (4.4) r

where P = constant After carrying out the integration the w ork o u tp u t can be expressed as

K = 4 ^ - R r + K } (4.5)

Petela proceeds to define the m axim um conversion efficiency as the ratio of the maximum w ork o u tp u t to the initial energy of the system:

£

J j (4.6)

Figure 4-2 show s the qualitative plot of ;] p versus tem perature T and also the analogous plot for h eat conduction in materials, l-T^/T.

Petela's efficiency can also be expressed as

= + (4.7)

where x is the ratio of the environm ent tem perature to the tem perature of the BR source, To/T. This sam e result was obtained by other researchers using different

approaches. For exam ple Edgerton [16] uses the straightforw ard application of the concept of non-flow exergy:

A-4U-U,)*PiV-K)-l{S-S,] (4.8)

to arrive at Petela's efficiency.

300 Petela's BR Conversion Efficiency c-ioo Carnot's Heat Engine Efficiency -200 0 1 2 3 4 5

Tem perature Ratio, x’* = T / To

Figure 4-2: Q ualitative p lot of Petela's efficiency r\ p versus tem perature T and the analogous plot for heat conduction.

C o n sid er the therm al c o u n te rp a rt of P etela's m echanical p isto n -cy lin d er approach for the conversion of an enclosed BR system as depicted in Figure 4-3. The BR system a t tem perature T is in therm al contact w ith the environm ent through a reversible C am ot heat engine. The therm ally conductive section of the enclosure is very small so that changes in its internal energy an d entropy m ay be neglected. The therm ally conductive section is also thin and has a high therm al conductivity such th at en tro p y p ro d u ctio n d u e to heat con d u ctio n m ay be neglected. The rem aining wall of the enclosure is perfectly insulating so that its internal energy and entropy m ay be neglected as well. The w o rk o u tp u t d u rin g the constant volum e cooling process is:

The first law applied to the BR system show s that dQ = -dll, and since U=aT*V w e have dQ = -4aV'PdT, so by substituting this (4.9) becomes

=

-wi ( \ - TJT) f d T

After carrying o u t the integration we arrive at Petela's result:

(4.11) If the initial tem perature T, is less than To then the HE depicted in Figure 4-3 will o p erate w ith the environm ent at To, an d the BR system , as the high and low tem perature reservoirs, respectively, and (4.11) will give the correct w ork output.

The BR system is placed in thermal

contact with the environm ent through a reversible

Cam ot heat engine.

= ( i - T k e BR system at temperature Tin a vacuum with V=cst. ciQiHit Perfectly isolating enclosure. Thermally conductive section with an infinite thermal conductivity Environment at temperature T,

Figure 4-3: BR system connected to a reversible Camot heat engine.

H o w ev er, at least tw o other m axim um conversion efficiencies have been presented in literature. The m ost com m on are Sparm er's efficiency [17]:

^Spemnar ^ 3

su p p o rted by G ribik and Osterle [18,19], and Jeter's efficiency [20]:

(4.12)

Bejan [6] show s that the differences between these efficiencies is sim ply based on their definitions. Bejan considers three steps;

(1) reversible filling of the evacuated enclosure w ith BR at tem perature T, (2) reversible cooling of the BR system from T to To, and

(3) reversible em ptying of the BR system at To from the enclosure.

All three efficiencies are the ratio of w ork output to the initial internal energy of the BR system at tem perature T, but the work o u tp u t is defined differently for each efficiency. In Petela's efficiency the work o u tp u t is extracted in step 2 only. In Spanner's efficiency the net w ork output is from steps 2 and 3 com bined and in Jeter's efficiency it is from steps 1, 2, and 3.

4.2 Approaches used to Determine the Exergy Flux of TR

4.2.1 Petela's approach for BR exergy: TR exchange between parallel blackbody plates

In Petela's approach two parallel regions of m aterial are separated by a vacuum , as show n in the Figure 4-4. Surface A is m aintained at tem p eratu re T bv a therm al energy reservoir (TER) w hereas surface E is in therm al contact w ith the Environm ent at tem perature To. W hen T To there is a net transfer of radian t

energy an d entropy betw een the surfaces. Both surfaces are of blackbodies so TR em itted by one surface is completely absorbed by the other surface.

Blackbody Material Connected to a TER at Temperature T Surface A Heat .

Conduction---Vacuum Blackbody Material

at Temperature To Surface £ Heat Conduction BR at T Emitted by surface A: H E m, } E m BR at To Incident on surface A : H i n c, f i n e

et Transfer of Energy and Entrop

H n et ~ C j, j n e t _q,Cf^a

In this approach the Guoy-Stodola principle is used as a basis for determ ining the exergy of the TR fluxes. This principle im plies that the exergy destruction rate, term ed the rate of irreversibility, is equal to the product of the total entropy production rate and the environm ental tem perature, that is / = 7^FI. The exergy of the radiation is then calculated w ith an exergy balance at either of the m aterial surfaces.

The following limitations are imposed in Petela's analysis for simplicité': 1) A vacuum exists betw een the two surfaces.

2) The exchange of radiation is steady state.

3) All TR is isotropic and the geometry is such that the problem is ID . Note that all fluxes and entropy production rates in this analysis are per m- of surface area.

C onsider the case w hen the net energy flux is tow ards the environm ent, that is surface A is hot (T>To) relative to surface E, as show n in Figure 4-5. H eat conduction entering the control volume has an energy flux q, an entropy flux t/T, and an exergy flux of q(l-To/T).

Blackbody with Surface at Temperature T Surface A Heat Conduction: energy flux q entropy flux ^ exergy flux q ( l - T / T )

Control Volume

_(o7r

Vacuum - Exergy destruction occurs in the CV Net transfer of energy Hv,,, entropy t Jsetf and exergy Mv„ by TR Blackbody at Temperature To Surface E

Figure 4-5: Exergy balance for the control volum e.

W henever there is h eat transfer (q^O) betw een the plates en tro p y p ro d u ctio n occurs due to radiation-m atter interaction as can be determ in ed by an entropy balance usin g Planck's spectral en tro p y radiance (3.2). This en tro p y

production occurs in layer of m aterial at each surface. This layer, referred to here as the interaction region, is defined as the region of material th at contains TR that has or w ill travel in to /fro m the region exterior to the solid m aterial. This distinction betw een exterior an d interior is required because all atom s including those w ithin a solid continuously em it therm al radiation. For sim plicity of discussion, the therm al conductivity is suficiently large so th at the temperature^ in the interaction region is approxim ately uniform. The control volum e is defined such that it includes the radiation-m atter interaction region.

The net exergy flux leaving the CV by TR transfer m ust be equal to the exergy flux entering by heat conduction at T minus the exergy destruction rate in the CV:

^^Se, = -

^)

j

-The net energy flux of the TR is equal to that of the heat conduction (H.vef = q) and the net entropy flux by TR transfer is:

A . = (415)

Thus, based on the Guoy-Stodola theorem the net exergy flux by TR is:

Set Exergy Flux tovards S e t Energy Flux towards Set Entropy Flux towards the Environment the Environment the Environment

= ( ^ S e t ) - I ( 4 r ) (4.16)

The net energy an d entropy fluxes by TR are

H ^ - o ( r - r , ) (4.17)

I _{Strictly speaking the temperature in the interaction region is not uniform. So it may be argued}

that the emission temperature and the temperature at the inner boundary of the interaction region are not strictly equal However, the temperature variation is very small because (1) the interaction region is usually very thin, e.g. on the order o f a few micrometers for metals, and (2) the temperature gradient decreases to zero at the surface of die solid because heat conduction decreases to zero at the surface. Furthermore, in a theoretical sense the temperature difference can be made arbitrarily small by specifying a sufficiently large thermal conductivify.

respectively. Petela assum es that the exergy flux of BR at To is zero. So the exergy flux of BR at T is equal to the net exergy flux. Using (4.17) in (4.16) the exergy flux of BR at T becomes :

= f j 3 r - 4 r / > + 7r) (4.18)

or alternatively

(4.19)

w here x is the ratio of tem perature T o/T . This is the sam e as the result Petela obtained for the exergy of a n enclosed BR system.

4.2.2 Karlsson's approach for the exergy of TR

Karlsson's [21] approach to determ ine the exergy of NBR is based on calculating the e n tro p y p ro d u ctio n th at occurs as a TR source flux is absorbed bv a blackbody surface a t To. The environm ent reference state selected is BR at tem perature To. The geom etry of Karlsson's approach is depicted in Figure 4-6. M onochromatic TR in the frequency interval v,v+dv is incident on a small area

d Y of a blackbody surface area (Y) at the environm ent tem p eratu re To. The

m onochrom atic TR is contained in a solid angle d fl w ith direction perpendicular to the blackbodv surface.

BR at To incident in d n except at one frequency w here the

spectral energy and entropy radiances are

K L Blackbody Surface at To • • % Flow BR a t BR at T,

BR at To is em itted by the plate in every direction (over the solid angle 2 k ) , including w ithin the solid angle dÇï, and BR at To is incident from every direction (over the solid angle 2 k ) except in the solid angle dQ. By energy conservation the heat flow into the plate is given as

Q = Kd lÆ ïd Y - K ^ d vdOdY (4.20)

w here K and Ko are the spectral energy radiances of the incident m onochrom atic TR and the BR at To, respectively.

The entropy production is given as

AS = Q / l + LJvdOdY - LdudQdY (4.21)

Karlsson states th at the spectral exergy radiance of the incom ing NBR is the entropy production rate times the environmental tem perature:

B = T,AS (4.22)

After substituting for Q in Equation (4.21) using Equation (4.20), Karlsson then claims that the exergy of the monochromatic beam is

B = [ { K - K , ) - l ( L - L ^ ) y v d n d Y (4.23)

4.3 Maximum Conversion Efficiencies in Solar Engineering

In the field of solar engineering a variety of conversion processes have been considered: therm al, quantum or hybrid therm al-quantum processes. In section 4.3.1 we consider pure therm al single-stage absorption systems w ith blackbody absorptivity. Inherently q u an tu m system s operate w ith a threshold o r cutoff frequency. By choosing special coatings therm al system s m ay operate w ith a threshold frequency as well, this is discussed a t the end of section 4.3.1. In section 4.3.2 w e consider pure q u an tu m single-cell system s. Finally w e will examine omnicolor multiple threshold systems (section 4.3.3).

4.3.1 Single-stage thermal conversion

In thermal conversion the TR source is absorbed by a receiver or collector as heat. The heat transfer rate from the collector is the difference betw een the absorbed and emitted TR energy flow rates. Figure 4-7 depicts a basic therm al conversion device for isotropic TR fluxes. The blackbody collector is thermally connected to the environm ent through a reversible C am ot h eat engine (HE). The absorbing m aterial has blackbodv characteristics w ithout threshold behaviour.

O u tgoin g em itted BR at Tc

Radiation-matter interaction produces entropy at the rate II_'■R.W

Incom ing source TR: energy flux H entropy flux /

Infinite thermal conductivity ensures that the tem perature is uniform and that

there is no entropy production d u e to heat conduction. System Boundary 0 = E — ctATt versibie Qout Environm ent (T„)