exergy consumption of the earth

Ecological Modelling 184 (2005) 363–380

Exergy consumption of the earth

G.Q. Chen∗

National Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science,Peking University, Beijing 100871, China

Received 21 April 2004; received in revised form 15 October 2004; accepted 21 October 2004

Abstract

Presented in this paper are a systematic study on the global exergy consumption in the earth and a budget of the exergyconsumption with respect to main terrestrial processes. Based on Szargut’s definition of exergy for thermal radiation, a globalexergy balance of the thermodynamic system of the earth, driven by cosmic exergy flow originated from the temperature differencebetween the sun and the cosmic background, is carried out to give the global cosmic exergy consumption as the multiplicationof the cosmic background microwave (CBM) radiation temperature and the global entropy generation due to irreversibility inthe earth system. Concrete formulae are derived for cosmic exergy, with emphasis on generalization of a simple blackbodyrelationship between entropy and energy flux densities to the cases of gray body radiation with moderate or large emissivityassociated with the earth system. Global entropy generation is evaluated with a result compared very well with a widely accepteddatum based on satellite observations in earth science. A budget of the global entropy generation is made with respect to theterrestrial radiation processes associated with the atmosphere and the earth’s surface and to the molecular transport phenomenain the material earth. A mechanism of multiplication governing transformation between cosmic exergy and terrestrial exergy isd la law, arep©

K velopment

1

ai2

on-am-

onandtives

ormm

0

eveloped. An overall exergy budget of the earth system, based on the entropy budget by means of the Gouy-Stodoresented with essential implication to the problem of global sustainability.2004 Elsevier B.V. All rights reserved.

eywords:Exergy; Entropy; Energy; Earth; Ecological modelling; Resource accounting; Environmental assessment; Sustainable de

. Introduction

As a latest progress in ecological modeling, resourceccounting and environmental assessment, an escalat-

ng interest has been emerging (Jørgensen, 2001; Wall,002; Szargut, 2003; Svirezhev, 2001; Svirezhev et al.,

∗ Tel.: +86 10 62767167; fax: +86 10 62750416.E-mail address:[email protected].

2003) for the adaptation and generalization of the ccept of exergy originated in engineering thermodynics (Szargut et al., 1988).

For a system given to have a direct bearingits local environment associated with the timelength scales depending on the observer’s objecand knowledge (Woods, 1975, p. 5), exergyEx is de-fined as the amount of work the system can perfwhen it is brought into thermodynamic equilibriu

304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2004.10.015

364 G.Q. Chen / Ecological Modelling 184 (2005) 363–380

with the environment (Jørgensen, 2001), that is,

Ex = T0(Stoteq − Stot) (1)

whereT0 is the temperature of the environment,Stoteq

andStot are the entropies in thermodynamic equilib-rium and at the given deviation from equilibrium, re-spectively, of the total system as a combination of thegiven system and the local environment.

WithJ standing for the exergy flux density (per unittime and per unit area) vector through the boundariesB of a general open system,n the normal vector to theboundaries,σ the exergy consumption per unit time perunit volume in the physical domainV of the system, anexergy balance for an open system requires.

dEx

dt= −

∫BJ · nda −

∫Vσ dv (2)

that is, the time derivative of the exergy possessed bythe system equals to the rate of exergy reception by thesystem from its external environment minus the rateof exergy consumption in the system. For an arbitrarydomain, the second law of thermodynamics, often re-ferred to as the law of entropy production, is alterna-tively expressed in terms of exergy consumption as

σ ≥ 0 (3)

that is, exergy can never be created: it is always con-sumed in a real irreversible process, and the equal-ity only holds for the idealized reversible or equilib-r sus-t thei s re-q thert everb w oft

inc-t ron-m ron-m asa locale ea-s ,2 01;D ofw sys-t cal

environment. Secondly, for the system itself, the exergyserves as a basic measure of the global buffering capac-ity (Mejer and Jørgensen, 1979; Jørgensen, 1981), interms of the self-organization and construction of thesystem as contrast to the environment to represent thealiveness and vitality of the system in a given environ-ment. And finally, for the local environment, the exergyof the system can act as an environmental impact (Walland Gong, 2001; Gong and Wall, 2001; Wall, 2002;Sciubba, 2003), in terms of the potential to change thestate of the local environment in the natural process ofreaching equilibrium in the absence of external actions.

The planet earth has been driven by exergy flowsassociated with the thermodynamic contrast betweenthe sun and the outer space. The exergy lost in earthserves for the planet not only as the ultimate resourceto revitalize the meteorological system, feed the hy-drological cycle, renovate the biosphere and make allother natural and anthropogenic phenomena possible,but at the same time also as a fundamental referencefor measuring the global environmental buffering ca-pacity to sustain and resist environmental impact. Abudget of the global exergy loss with respect to mainphenomena within the planet such as solar and terres-trial radiation, atmosphere and ocean circulation, con-vective heat transfer, turbulence dissipation, transportand precipitation of water, photosynthesis and humanactivity is essential for evaluating the cost of a pro-cess or product in cosmic exergy equivalence. Thenthe greatly concerned problem of global sustainabilitym thee

byS l.( thec ptiona res-t theK en’sm teda endst t ofi di-a sonald aphi-c nds toc n of

ium process. Thus the fundamental resource toain a system is exergy, which is lost in drivingrreversible process associated with the system auired by the second law of thermodynamics, ra

han energy, which is always in conservation and ne consumed in any case according to the first la

hermodynamics.Exergy associated with a system may play dist

ive roles as resource, buffering capacity and enviental impact, for the subject, system and envient, respectively. Firstly, for the subject originallyn observer external to both the system and itsnvironment, the exergy can stand as a unified mure of resource availability (Wall, 1977, 1990, 1997000; Zaleta-Aguilar et al., 1998; Sciubba, 20incer, 2002), in terms of the maximum amountork that can be extracted by the subject from the

em in its process of reaching equilibrium with its lo

ight be examined with relation to exergy flows onarth.

Exergy of solar radiation has been studiedvirezhev and Steinborn (2001)andSvirezhev et a

2003)with an information approach to measurehange of energy, by the balance between absornd reflection of solar radiation and emission of ter

rial radiation, and the increment of information byullback measure. As a generalization of Jorgensaximal principle, a minimax principle was postulas that during the self-organization the vegetation t

o maximize its exergy in respect to the incremennformation and to minimize it in respect to the ration balance. This hypothesis was tested for seaynamics of several ecosystems located at geogrally different sites, with radiation data for the long ahort spectral intervals. Using NASA satellite dataalculate the global distribution of the annual mea

G.Q. Chen / Ecological Modelling 184 (2005) 363–380 365

exergy, they observed that the domains with maximalvalues of exergy correspond to the main upwellingsof the ocean. With thermodynamic analogies appliedto the process of interaction of solar radiation with an“active” planetary surface, in particular with vegeta-tion cover, the difference between the radiation balanceand the exergy could be considered as the increment ofinternal energy, of which the global pattern has beenshown very similar to that of vegetation.

In their general survey of exergy and sustainabledevelopment,Wall and Gong (2001)made the first,though brief, description of the global exergy losses inthe earth, and presented a chart for exergy flows on theearth, not mentioning related evaluation procedures. Itis clearly pointed out that the exergy driving the flowsof energy and matter originates from the contrast be-tween the sun and space, though the outer space seemsirrelevant in their quantitative calculation and a meantemperature of the earth’s surface seems taken as thereference environment temperature. It is illustrated thatthe resource for the earth comes from the solar exergyin value of 160,000 TW (T stands for trillion, i.e., 1012),of which 30% is reflected back to the space associatedwith the reflection of solar radiation and the remain-ing 70% is lost in driving the earth system, and thesociety use of exergy of 12 TW amounts to be 13,000times smaller than the total incipient associated withsolar radiation or 9100 times smaller than the total ex-ergy consumption of the earth. With these small ratiosof order-of-magnitude, it seems hard to perceive thet ain-a

dn e firstm d itsb n thee mics solare TW,c y ofr thev con-s r thee theya natu-r s ofu haveb ed by

the activity of humankind. The positive impact of thenatural exergy losses has been pointed out: they are themain cause of the formation of the terrestrial naturalenvironment, of the non-renewable natural resourcesof fuels, and of the generation of stable dissipativestructures in form of living beings. The sum of an-thropogenic exergy losses was estimated at 6000 timessmaller than the natural losses of utilizable exergy.With such a small ratio, it remains hard to illustrate theproblem of global sustainability with exergetics for theearth.

This paper presents a systematic study on the globalexergy consumption of the earth and the budget of theexergy consumption with respect to main terrestrialprocesses. The thermodynamic system of the earth isshown to be driven by cosmic exergy flow originatedfrom the temperature difference between the sun andcosmic background, and a global balance of the cos-mic exergy is carried out to give the global cosmic ex-ergy consumption as the multiplication of the CBMradiation temperature and the global entropy genera-tion due to irreversibility of the earth system. Basedon the exergy definition for thermal radiation initiallyproposed by Szargut, concrete formulae are derivedfor cosmic exergy, with emphasis on generalization ofa simple blackbody relationship between entropy andenergy flux densities to the cases of gray body radia-tion with moderate or large emissivity associated withthe earth system. Global entropy generation is evalu-ated with a result compared very well with a widelya earths en-e ssesa rfacea ate-r tionb ex-p tema lemo

2

ctedw eata thec has

hreat of existing human impact to the global sustbility of the earth.

The work bySzargut (2003)on anthropogenic anatural exergy losses made its appearance as thonograph on the global exergy consumption an

udget with respect to main terrestrial processes iarth. The influence of the relict radiation of the cospace is stressed in the exergy evaluation. Whilexergy lost in the earth is estimated about 105, 100lose to the value given by Wall and Gong, an exergelict radiation of the cosmic space is shown to be inalue of 74,900 TW, an amount comparable to theumed solar exergy. The exergy loses occurring neaarth’s surface have been distinguished becausere considered to represent the most accessibleal resources of exergy. The term of natural lossetilizable exergy has been proposed. These losseseen compared with the anthropogenic ones caus

ccepted datum based on satellite observations incience. A detailed budget of the global entropy gration is made with respect to the radiation processociated with the atmosphere and the earth’s sund to the molecular transport phenomena in the mial earth. The mechanism governing transformaetween cosmic exergy and terrestrial exergy islored. An overall exergy budget of the earth sysre presented with essential implication to the probf global sustainability.

. Cosmic exergy driving the earth system

Granted that thermodynamic effects conneith the planetary motion braked by tides and hnd material from inside the earth are negligible,oncerned thermodynamic system of the earth


a direct bearing on the cosmos thermodynamicallycharacterized of a singularity of the sun and a sur-rounding cosmic background. The earth system can beappropriately considered as closed with relation to theexchange of rest mass while open with relation to theexchange of energy radiation with its cosmic environ-ment. Commonly assumed of overriding importance,the solar radiation as a whole may be more or lessconsidered as blackbody radiation, though the ultravi-olet region (≤0.4m) of the solar spectrum deviatesremarkably from the visible and infrared regions interms of the equivalent blackbody temperature ofTs = 5800 K (Liou, 1980, p.23). The thermodynamicinfluence of the rest of the universe as an integral effectis embodied in the cosmic background microwave(CBM) radiation. Believed to be the relict remainingafter the Big Bang, the CBM radiation, correspondingto an emission temperature ofTcbm= 2.73 K, has beenfound with strikingly small anisotropies (Peacock,1999, p. 597) and might be regarded as blackbodyradiation in the best sense of the term. The vastness ofthe cosmos qualifies the ultimate thermal sink of theblackbody associated with CBM radiation as a cosmiccold reservoir. The fact of interception of some of thesolar radiation by the earth does not change the funda-mental fate of the solar radiation to annihilate in and beabsorbed in the cosmic background: the solar radiationintercepted by the earth, as a transformer of radiation,is simply reflected or scattered back to the outer space,or promptly transformed into terrestrial radiatione ergye odya nvi-re cos-m eent rgyfl nete arths

-i , as-s be-t senceo ra-dc x ofr t-

tered solar radiation (long wave withλm 0.5m),a leaving flux of terrestrial radiation (long wave withλm ≈ 10m) and an entering flux of CBM radiation(very long wave withλm ≈ 1000m). According tothe Planck equation, the energyε of a photon is givenby ε=hν (whereh is the Planck constant andν the fre-quency), thus the energy is inversely proportional tothe wave length of the radiation. The same amount ofenergy associates with fewer photons in the form ofsolar radiation than in the form of backscattered andterrestrial radiation. In other words, the entering solarradiation is more organized than the leaving backscat-terd and terrestrial counterparts, and thus the amountof exergy associated with the incoming solar radiationis higher than that with the backscattered and terres-trial radiation. The CBM radiation emitted from thereference environment is the most unorganized andpossesses no exergy at all. The earth system receiveshigh quality energy and returns low quality energy tothe cosmos. It is the contrast in energy quality ratherthan in energy quantity provides the earth with anexergy resource to drive and sustain the irreversiblesystem.

We consider the exergy balance of the earth witha time scale of the climate cycles of years and alength scale of the magnitude of the earth, that is,exergy evaluation is time averaged over climate cyclesof years and area averaged over the earth’s surface.With J standing for the exergy flux density vectorthrough the outer boundaryA of the atmosphere, weh∫

∫

w thee thes tionc

φ

w ageos thes iven

mitted to the outer space. Therefore related exvaluations should be carried out with the blackbssociated with CBM radiation as the reference eonment. The exergy thus evaluated withTcbm as thenvironment temperature might be referred to asic exergy. The thermodynamic difference betw

he solar and CBM radiation results in a cosmic exeow, of which a tiny branch is intercepted by the plaarth and consumed in driving and sustaining the eystem.

As illustrated inFig. 1, there are exergy fluxes flowng through the outer boundary of the atmosphereociated with the exchange of thermal radiationween the earth system and the cosmos. The esf this exchange involves an entering flux of solariation (short wave with wave length ofλm ≈ 0.5morresponding to maximum energy), a leaving flueflected (short wave withλm ≈ 0.5m) and backsca

ave

BJ · nda =

∫A

J · nda (4)

Vσ dv =

∫A

φ da (5)

here we have taken into account the fact thatxergy flux vanishes on the lower boundary ofystem, and defined a rate of exergy consumporresponding to unit surface area as

≡∫ +∞

0σ dz (6)

ith zstanding for the altitude. Time and area averf Eq. (2), into which Eqs. (4), (5) and (6) areubstituted, combined with the consideration ofteadiness of the system with respect to the g


Fig. 1. Comic exergy driving the earth system.

time and length scales results in an overall balanceas

Φ = −Jn (7)

whereΦ standing for the mean rate of cosmic exergyconsumption per unit area, andJn standing for the netmean density of normal exergy flux. Thus the exergylost in the earth can be estimated by simply countingthe exergy flux at the top boundary of the atmosphere.

As first established by Szargut and elaborated byPetela (Petela, 1964; Szargut et al., 1988), the exergyof thermal radiation might be given as

J = (I − I0) − T0(S − S0) (8)

where I and S are the flux densities of energy andentropy radiation,I0 andS0 are the flux densities of en-ergy and entropy radiation at the environment tempera-ture ofT0, respectively. For the cosmic radiation under


consideration, corresponding cosmic exergy turns to be

Jc = (I − Icbm) − Tcbm(S − Scbm) (9)

Exergy flux densities for the incoming solar radiation,Js, the leaving backscattered and reflected solar radia-tion,Jsr, and the leaving terrestrial radiation,Jt, can beaccordingly presented as

Js = (Is − Icbm) − Tcbm(Ss − Scbm) (10)

Jsr = (Isr − Icbm) − Tcbm(Ssr − Scbm) (11)

Jt = (It − Icbm) − Tcbm(St − Scbm) (12)

The total flux density can be readily obtained as

Jn = −Js + Jsr + Jt = (−Is + Isr + It − Icbm)

− Tcbm(−Ss + Ssr + St − Scbm) (13)

As thermal balance of the earth system requires

−Is + Isr + It − Icbm = 0 (14)

Eq.(13) turns into

Jn = −TcbmS (15)

where

S ≡ −Ss + Ssr + St − Scbm (16)

is the total entropy increase in the radiation fluxes inthe earth system. Then from Eq.(7), we have

Φ

t them ndt thee opyfl

3t

ce iatedw bei ics(

rmo-d ject

to a hot thermal reservoir of solar radiation at an emis-sion temperature ofTs and a cold thermal reservoir ofCBM radiation at an emission temperature ofTcbm, asschematically illustrated inFig. 2(a). The system re-ceives energy ofI in, equal toIs, and entropy ofSin,equal toSs, from the hot reservoir, and ejects energy ofIout, equal toIsr + It − Icbm, and entropy ofSout, equalto Ssr +St −Scbm, to the cold reservoir, and accordingto Eqs.(14)and(16), we have

Iout = Iin (18)

S ≡ Sout − Sin ≥ 0 (19)

Now suppose that it is desired to produce exactly thesame change in the radiation fluxes through the radi-ation transformer of the earth with the concerned ir-reversible process, but by a reversible process only.This would require, in general, the service of Carnotengines and refrigerators in the outer space, which, inturn, would have to be operated in conjunction withan auxiliary environment consisted of an auxiliary me-chanical device and an auxiliary reservoir. For the aux-iliary reservoir let us choose the one whose temperatureis the lowest at hand, i.e.,Tcbm. With the aid of suitableCarnot engines and refrigerators all operating in cy-cles, in conjunction with the auxiliary environment, itis now possible to produce in the radiation flux throughthe radiation transformer, by a reversible process only,the same change that formerly brought about by theirreversible process. If this is done, the entropy changeo theyh finals un-d causet ed oft ingt

s isp tems voira eat,s earths e ra-d edi avet hichw re-v ount

= TcbmS (17)

hus the global exergy consumption amounts toultiplication of the CBM radiation temperature a

he total entropy increase in the radiation fluxes inarth, which can be simply deduced from the entrux densities on the top of the atmosphere.

. Cosmic exergy loss due to irreversibility ofhe earth

The simple relationship of Eq.(17)between cosmixergy consumption and entropy increase associth the irreversibility of the earth system might

llustrated in more basic contexts in thermodynamZemansky and Dittman, 1981, p. 199).

Consider the steady system of the earth as a theynamic machine of radiation transformation, sub

f the radiation fluxes is the same as before, sinceave gone from the same initial state to the sametate. The auxiliary environment, however, mustergo an equal and opposite entropy change, be

he net entropy change of the total system consisthe radiation flux and the auxiliary environment durhe reversible process should be zero.

Since the entropy change of the radiation fluxeositive, the entropy change in the auxiliary syshould be negative. Therefore the auxiliary resert Tcbm must have ejected a certain amount of hay,Φ. Since no extra energy has appeared in theystem as a mere radiation transformer and in thiation flux, the heatΦ must have been transform

nto work in the auxiliary mechanical device. We hhe result therefore that, when the same change was formerly produced in the radiation flux by an irersible process is brought about reversibly, an am


Fig. 2. (a) Earth as a thermodynamic machine of radiation transformation operating between reservoirs atTs andTcbm. (b) Cosmic exergy lossdue to irreversibility of the earth.

of energyΦ leaves an auxiliary reservoir atTcbm in theform of heat and appears in the form of work in anauxiliary mechanical device. In other words, energyΦ

is converted from a form free of exergy in which itwas completely unavailable for work into a form of ex-ergy in which it is completely available for work. Sincethe original process was not performed reversibly, theenergyΦ was not converted into work, and thereforeΦ is the energy that is rendered unavailable for workbecause of the performance of the irreversible process.

It is then a simple matter to calculate the cosmicexergy lost due to the irreversibility of the radiationtransformer of the earth system. If the same change isbrought about reversibly, the entropy change associatedwith the radiation flux is the same as before, namely,S. The entropy change of the auxiliary environmentis merely the entropy change of the auxiliary reservoirdue to rejection ofΦ units of heat at temperatureTcbm,that is,−Φ/Tcbm. Since the sum of entropy changes of

the radiation flux and auxiliary environment is zero, wehave

S − Φ

Tcbm= 0 (20)

equivalent to Eq.(17). Whence, the cosmic exergy lossis Tcbm times the entropy increase in the radiation fluxdue to the irreversibility of the earth.

4. Exergy of thermal radiation

The energy flux density of thermal radiation mightbe written as

I =∫

ν

∫Ω

iν(ξ)ξ · ndν dΩ(ξ) (21)

whereiν(ξ) is the monochromatic energy flux density,ξ

the direction of the flow of radiation,Ω the solid angle.


For blackbody or gray body radiation,

I = εσT 4 (22)

whereε is the surface emissivity which is irrelevant tofrequency and for the perfect case of a blackbody equalto 1, andσ the Stefan–Boltzmann constant, equal to5.667× 10−8 wm−2 K−4.

The relationship between entropy flux density andenergy flux density was established byPlanck (1959)and derived in modern terms byRosen (1954). Themonochromatic entropy flux densitysν corresponds tomonochromatic energy flux density according to thedifferential equation of

d

[sν(ξ)

c

]= 1

T ∗ν

d

[iν(ξ)

c

](23)

wherec is the speed of light, and

T ∗ν = hν

k

[ln

(1 + 2hν3

c2iν(ξ)

)]−1

(24)

is referred to as the monochromatic brightness temper-ature, or emission temperature, associated with eachbeam of radiation, withk standing for the Boltzmann’sconstant.T ∗

ν is obtained by replacing byiν the PlanckfunctionBν(T) in the expression for complete thermo-dynamic equilibrium, and in general, might be depen-dent upon the direction of the flow of the radiation.For the special case of blackbody radiation,T ∗

ν has acommon value independent of both frequency and di-r re oft dy-n itf

S

T

S

F 9;S

S

where

χ(ε) = 45

4π4

∫ ∞

0

[(1 + ε

ex − 1

)ln

(1 + ε

ex − 1

)

− ε

ex − 1ln

(ε

ex − 1

)]x2 dx (28)

with x=hν/kT. It is easy to establish that,

χ(0) = 0, χ(1) = 1 (29)

For the case of very small emissivity, say,ε≤ 0.1,

χ(ε) ≈ ε(0.9652− 0.2777 lnε) (30)

while for the case of moderate or large emissivity,

χ(ε) ≈ σ (31)

which is with a deviation less than 5% for the case ofε≥ 0.7.

For the case of blackbody or gray body radiation,combination of Eqs.(22)and(27)gives

S = 4

3

χ(ε)

ε

I

T(32)

which turns to be

S = 4I

3T(33)

as an exact result for blackbody radiation, or an ap-proximate one for gray body radiation with moderate orl -s

T

m per-a

T

f ay,ε .

rgyfl n ofE

S

ection, and is the same as the thermal temperatuhe emitting surface under the conditions of thermoamic equilibrium. Eq.(23)might be revised in explic

orm as

v = 2kν2

c2

[(1 + c2iν

2hν3

)ln

(1 + c2iν

2hν3

)

− c2iν

2hν3ln

(c2iν

2hν3

)](25)

hen entropy flux density can be obtained as

=∫

ν

∫Ω

iν(ξ)ξ · ndν dΩ(ξ) (26)

or gray body radiation (Landsberg and Tonge, 197tephens and O’Brien, 1993),

= 43σT

3χ(ε) (27)

arge emissivity based on Eq.(31). The uniform emision temperature of

=(

3I

4εσ

)1/4

(34)

ight be replaced by an equivalent blackbody temture ofT* ,

≈ T ∗ ≡(

3I

4σ

)1/4

(35)

or cases with moderate or large emmisivity, s≥ 0.8, corresponding to a deviation less than 5%

A simple relationship between entropy and eneux densities might be obtained by the combinatioqs.(33)and(35)as

= S(I) ≡ 4

3σ1/4I3/4 (36)


Derived under a wide range of conditions and not in-volving such details as emissivity and emission temper-ature, the idealized relation of Eq.(36), correspondingto Eqs.(33) and(35), may be conjectured as approx-imately correct for non-equilibrium radiation fields ofmore complicated implications. Though in the sim-plest form corresponding to blackbody radiation, theidealized formula has been found representing the ac-tual long wave entropy flux quite accurately, byLesins(1990)andGoody and Abdou (1996)in their entropyflux calculations at the top of the atmosphere usingenergy flux over the entire spectrum based on theearth radiation budget experiment (ERBE) data, andby Goody and Abdou (1996)in numerical simulationof a variety of atmospheres with different solar fluxesand cloud amounts, and is employed in the presentwork in the estimation of associated global entropyflux densities associated with long wave radiation pro-cesses, including the terrestrial radiation leaving for theouter space, that leaving the atmosphere for the earth’ssurface and that leaving the earth’s surface for theatmosphere.

For the case of blackbody or gray body radiation,the cosmic exergy as defined in Eq.(9) can be obtainedin a certain expression as

Jc =[1−4

3

χ(ε)

ε

Tcbm

T

]I +

[4

3

χ(ε)

ε− 1

]Icbm (37)

A rr d ter-r g-n int cted,a

J

F tionw thec lt canb

J

5. Cosmic exergy consumption of the earth

As the sun as a whole can be regarded as a black-body, we readily have

Is = 14Cs = 340 W m−2 (40)

Ss = 13CsTs

= 0.078 W m−2 K−1 (41)

Js = 1

4Cs

(1 − 4

3

Tcbm

Ts

)= 339.8 W m−2 (42)

whereCs 1360 W m−2 is the solar constant, definedas the amount of solar radiation incident per unit areaand per unit time on a surface normal to the direction ofpropagation and situated at the earth’s mean distancefrom the sun (Peixoto and Oort, 1992). An uncertaintyof 21 wm−2, corresponding to 1.6% ofCs, as issuedby NASA (Thekaekara, 1976) brings about the samepercentage of uncertainty in evaluating the solar exergy.The factor of one fourth in Eq.(40) is due to the factthat the global surface area of the earth is four timesthe area of its cross section.

The mean broad-brand albedoα for the back-scattering and reflection of the solar radiation has beentypically found in the range of about 0.29–0.31 (Liou,1980, p. 189), for which a mean value of 0.30 (Peixotoand Oort, 1992) is taken in the present study, with auncertainty of about 3.3%. But detailed records withspectral resolution high enough for a reliable entropyevaluation for the back-scattering and reflection of thes rox-i

α

w them odyr n,i ingt hro-m andr

i

i

w ebt e

s Tcbm is much smaller thanT associated with solaadiation, backscattered and reflected radiation, anestrial radiation,Icbm is several orders smaller in maitude thanI under concern, then the second term

he right hand side of the above equation is neglend we get

c =[1 − 4

3

χ(ε)

ε

Tcbm

T

]I (38)

or cases of blackbody radiation or gray body radiaith moderate or large emissivity associated withoncerned terrestrial radiation processes, the resue further simplified into

c =(

1 − 4

3

Tcbm

T

)I (39)

olar radiation has yet to be achieved. As a first appmation, we assume

= αsc + αrf (43)

hereαscis the albedo for perfect back-scattering inanner of diffuse reflection, equivalent to gray b

adiation, andαrf is the albedo for perfect reflectio.e., specular or mirror-like reflection, correspondo blackbody radiation, then corresponding monocatic energy flux densities for the backscattered

eflected solar radiation are given as

sc = εscBν(Ts) (44)

rf = εrfBν(Ts) (45)

here εsc = αsc is the equivalent emissivity for thack-scattered solar radiation andεrf = αrf is that for

he reflected solar radiation,θ is the zenith angle of th


sun,Ω0 the solid angle of the sun for the earth, equalto 67.7× 10−6. With αsc= 0.06, αrf = 0.24 (Peixotoand Oort, 1992, p. 94), and averaged cosθ = 0.25,we have mean emissivityεsc= 1.61× 10−6, corre-sponding to (εsc) = 7.52× 10−6 according to Eq.(30). With formulae given in the previous section, wehave

Isc = αscIs (46)

Irf = αrfIs (47)

and

Ssc = αscχ(εsc)

εscSs (48)

Srf = αrfSs (49)

Then

Isr = Isc + Irf = αIs = 102 W m−2 (50)

Ssr = Ssc + Srf =αsc

χ(εsc)

εsc+ αrf

Ss

= 0.547Ss = 0.043 W m−2 K−1 (51)

Jsr≡Isc+Irf =α − 4

3

αsc

χ(εsc)

εsc+ αrf

Tcbm

Ts

Is

T ion(

S

F is

I

c tureo lao

S

J

Thus a net value of 1.209 wm−2 K−l is obtained for theglobal density of entropy increase ofS. The net so-lar entropy as the contribution from the first two termsof Ss andSsr, equal to 0.035 wm−2 K−1, amounts toonly 2.89% of the global entropy increase, and canbe neglected for simplicity, considering the total un-certainty near 5% resulted from that of 1.6% with thesolar constant and that of 3.3% with the albedo. Thenwe might choose to simply take the global density ofentropy increase as the terrestrial entropy density of1.244 wm−2 K−l . This value ofS is very close to thewidely accepted value in the earth science of 1.25 fromthe earth radiation budget experiment (ERBE) obser-vations, with a deviation of merely 0.48%, well withinthe claimed uncertainty of about 3% associated withthe latter.

The global value ofΦ for cosmic exergy consump-tion of the earth, according to Eqs.(13)or (17), equalsto 3.30 wm−2, based on the value of 1.209 wm−2 K−l

for the global density of entropy increase ofS. Thisis only about 1% of the mean solar energy densityentering the earth. With the global surface area of510Tm2, the global cosmic exergy loss is calculatedas about 1700 TW, equal to 1% of the amount ofthe total energy flow of solar radiation meeting theearth.

6. Global entropy budget

s byP fa temb oft ppli-c eenp y pa-p ished( ien,1 dou,1 2b

erei erialp otonsw radi-a ract-i the

= 101.9 W m−2 (52)

he net entropy flux density of solar radiatStephens and O’Brien, 1993) is then given as

ns ≡ Ss − Ssr = 0.453Ss = 0.035 W m−2 K−1 (53)

or the terrestrial radiation, the energy flux density

t = (1 − α)Is = 238 W m−2 (54)

orresponding to an equivalent blackbody temperaf T ∗

t = 255 K. With the idealized blackbody formuf Eq.(33), we have

t = 4

3

It

T ∗t

= 1.244 W m−2 K−1 (55)

t =(

1 − 4

3

Tcbm

T ∗t

)It = 234.6 W m−2 (56)

Since the appearance of the seminal paperaltridge (1975, 1978)to explore the possibility othermodynamic description of the climate sys

ased on a variational principle for maximizationhe rate of entropy production, emphasis on the aation of the second law of thermodynamics has blaced in the earth science with the result that maners on entropy budget of the earth have been puble.g.,Peixoto and Oort, 1992; Stephens and O’Br993; O’Brien and Stephens, 1995; Goody and Ab996; Goody, 2000; Pauluis and Held, 2002a, 200).

Within the physical domain of the earth system ths a two phase continuum, composed of a real mathase with rest mass and a radiation phase of phith no rest mass. We may treat the material andtion phases in the earth as two separate but inte

ng subsystems. The internal entropy production is


sum of the entropy production in the subsystem of ma-terial phase due to irreversible molecular process andthat in the radiation phase due to irreversibility associ-ated with interactions between radiation and material,namely,

Sirr = Smatirr + Srad

irr (57)

The material is considered in a state of local thermody-namic equilibrium. External interactions, whether theyare heat, work or the transfer of species, take placethrough a series of local equilibrium and are locally re-versible. However, transport phenomena due to interac-tions between material molecules, such as momentumtransfer associated with velocity gradient due to viscos-ity, heat transfer associated with temperature gradientdue to conductivity and mass transfer associated withconcentration gradient due to diffusivity, are internalto the material; they all lead to irreversible increasesof entropy, tending to bring the material phase close tocomplete equilibrium.

The radiation term involves interaction between ra-diation and material. A first consequence of the inter-action is a locally reversible heat exchange betweenthe material and radiation, corresponding to entropyexchange between the subsystems with zero net en-tropy production for the earth system as a whole. How-ever, for a reversible heat exchange with a material

in thermodynamic equilibrium, collisional rearrange-ment of energy levels, associated with a very smallproportion of molecules in a local region of negligi-ble extent, would be required, as absorbed or emit-ted photons generally have an emission temperaturedifferent from the local kinetic temperature. This sec-ond consequence of the interaction involves no energyexchange, but increases entropy in the radiation field,rather than in the material phase. A third consequenceis scattering, with no heat exchange takes place, but thedirection and perhaps the polarizations of photons arechanged. Scattered radiation is less organized and hashigher entropy than incident radiation. The irreversibleproduction of entropy due to thermalization and scat-tering has been generally studied byEssex (1984)onthe violation of the bilinearity in the thermodynam-ics of irreversible process and byGoody and Abdou(1996) in a special assessment of reversible and ir-reversible sources of radiation entropy in the climatesystem.

To estimate the irreversible radiation term ofSradirr ,

we refer to the radiation balance of the total systemmodeled as consisted of three boxes of the earth’ssurface, atmosphere and outer space, as illustratedin Figs. 3 and 4, which is proposed on two typicalschematic diagrams of global radiation balance of Fig.6.3 in Peixoto and Oort (1992, p. 94)(which wasadopted from “Understanding Climatic Change, U.S.

of the
Fig. 3. Schematic diagram global budget of solar radiation.


Fig. 4. Schematic diagram of the global budget of terrestrial radiation.

National Academy of Sciences, Washington, D.C.,1975, p. 14) and Fig. 8.19 in Liou (1981, p. 328).Of the 100 units of incoming solar radiation shownon the left-hand side ofFig. 3, 46 units are inter-cepted by the atmosphere, from which six units arescattered and 20 units are reflected back to the outerspace; the remaining 54 units reach the earth’s sur-face, over there 50 units are absorbed and 4 unitsare reflected back to the outer space. With the 50units absorbed by the earth’s surface and 20 units ab-sorbed by the atmosphere from short-wave solar ra-diation, the long-wave terrestrial radiation has a bud-get as shown inFig. 4: 121 units of terrestrial ra-diation leave the earth’s surface for the atmosphere,and in return 101 units leave the atmosphere for theearth’s surface, and 70 units leave the atmosphere forthe outer space. The data ofSs, Ssc andSrf obtainedin the above section for the primary, backscatteredand reflected solar radiation, and the simple black-body relation of Eq.(36) for long wave terrestrial ra-diation processes associated with the earth’s surfaceand atmosphere are employed. Then, for the earth’ssurface,

Sradirr |es = S

(121

100Is

)−S

(100

100Is

)− 54

100Ss + 4

24Srf

= 0.203 W m−2 K−1 (58)

and for the atmosphere,

Sradirr |atm = S

(70

100Is

)+ S

(101

100Is

)− S

(121

100Is

)

− 20

100Ss +

(Ssc − 6

100Ss

)

= 1.011 W m−2 K−1 (59)

Thus the total entropy production in the radiation field,due to irreversible radiation interaction with the mate-rial phase, is obtained as

Sradirr = Srad

irr |es+ Sradirr |atm = 1.213 W m−2 K−1 (60)

This value is very close to the net entropy increase of1.209 wm−2 K−1 for the planetary albedo of 0.3, with adifference of only 0.3%. Should this estimation be reli-able, the value of entropy production due to irreversiblemolecular effects,Smat, must be within the uncertaintyof about 5% in estimating the net entropy increase.

A recent and typical budget for entropy productiondue to irreversible molecular effects has been presentedby Goody (2000)as follows

Sturb = 0.002 W m−2 K−1 (61)

Sdiss = 0.011 W m−2 K−1 (62)

Swater = 0.019 W m−2 K−1 (63)


whereSturb is the inventory for convective heat trans-port in atmospheric turbulence, which is dominant inthe category of heat transfer,Sdiss is the inventory fordissipation in the atmosphere, dominant in the categoryof momentum transfer, andSwater is the inventory forwater-vapor, dominant in the category of mass transfer.The total value for entropy production due to moleculareffects is

S irrmat = Sturb + Sdiss+ Swater = 0.032 W m−2 K−1

(64)

The total entropy productionSirr as sum ofSradirr and

Sradmat, equal to 1.245 wm−2 K−1, is close to the value

of 1.209 wm−2 K−1 for the net entropy increase withthe planetary albeto of 0.3, with a deviation of 2.98%,within the uncertainty of about 5% in estimating the netentropy increase, and agrees very well with the valueof 1.25 wm−2 K−l obtained by Stephen and O’Brien(1993) based on the ERBE data. The entropy gener-ation in the material phase amounts to only 2.5% ofthe global entropy generation. According to the Gouy-Stodola law (Szargut et al., 1988), exergy loss is pro-portional to the entropy generation, it is then concluded

that only a very small proportion of 2.5% of the cosmicexergy availability is lost in driving the material earth.

7. Transformation of exergy

It seems trivial to recall that exergy for a system isdefined as the maximal amount of work that can beextracted from the system in the process of reachingequilibrium with its local environment. But subtletyarises when pondering a little bit over the choosing ofthe system and its local environment, which has a directbearing on the behavior of the system, with respect tothe time and length scales, depending on the observer’sobjectives and knowledge (Woods, 1975, p. 5). As ex-ergy is defined in terms of the thermodynamic contrastbetween system and its local environment defined withrelation to the observer’s time and length scales, thesame system as commonly referred to as the exergycarrier may possesses different amounts of exergy cor-responding to different environments associated withthe change of the involved time and length scales.

In the category of overlapping multi-component me-dia involving no electromagnetic effects,Wall (1977)

ent w
Fig. 5. A subsystem in the terrestrial environm hich in its turn is located in the cosmic environment.


considered the case of a subsystem in a local environ-ment which in its turn is included in a global environ-ment, and for exergy transformation it is found as amechanism of additivity that the exergy of the subsys-tem with respect to the global environment is the sumof the original exergy with respect to the local environ-ment and the contribution of exergy due to the deviationof the local environment from the global environment,as given by the original exergy expression associatedwith the local environment with the intensive param-eters of the global environment replacing those of thesubsystem. In the present study with a particular elec-tromagnetic effect of thermal radiation playing an es-sential role, we need to consider a subsystem in theterrestrial environment, which in its turn is included inthe cosmic environment, as illustrated inFig. 5.

Consider an exergy impact external to the existing‘equilibrium’ of the earth system, that is, external tothe existing characteristic state and process of the earthas it has been, is released in some subsystem in theearth. External impacts could be enforced in a varietyof forms, such as incidental cosmic matter or energy re-ceived from the out space other than the solar and CBMradiation, which have been already accounted for asso-ciated with the existing ‘equilibrium’, or anthropogenicutilization of heat and material from inside the earth.Assume the impact is small enough as not to effect theglobal ‘equilibrium’ of the earth system, then associ-ated exergy impact ofδet

x can be calculated with theexisting earth system as local reference environment,a Thee orw henj thee , re-s en-s

δ

A er-r res-t

δ

c fluxd l ra-

diation of

δJt = δIt − TcbmδSt =(

1 − Tcbm

T ∗t

)δet

x (67)

Then according to Eq.(17)combined with Eq.(16), thenet increment of the global cosmic exergy consumptionis obtained as

δΦ = TcbmδSt = Tcbm

T ∗t

δetx = 0.0107δet

x (68)

Conversely, terrestrial exergy impact is related to anincrement of global cosmic exergy consumption as

δetx = T ∗

t

TcbmδΦ = 93.4δΦ (69)

that is, about 93 units of terrestrial exergy use resultsin one unit of increment in cosmic exergy loss. Thismight be referred to as a mechanism of multiplicationfor the exergy transformation for the case of thermalradiation.

The cosmic exergyδΦ lost in material process ofthe earth system, equal toTcbmSmat

irr in value of0.087 wm−2, transforms into a terrestrial exergy lossof δet

x ≈ 8.13 wm−2, corresponding to a global terres-trial exergy ofδEt

x ≈ 4100 TW.The simple transformation relation of Eq.(66) is

proposed for small exergy change denoted with thesymbol of δ. For exergy change large enough to ef-fect the global equilibrium, nonlinear response of thesystem out of the description of above differential oper-a ess.

8

thee sultso thee .

s-m e at-m ich5 herea f thes tingt e ter-r W,a lost

nd it may be referred to as terrestrial exergy loss.xergy ofδet

x might be carried in the form of heatould be eventually turned into heat, which would t

oin in the terrestrial radiation leaving the earth forxtended environment of the cosmic backgroundulting in an increment of terrestrial radiation flux dity of

It = δetx (65)

ccording to Eq.(55), associated increment of the testrial entropy associated with the increment of terrial radiation leaving the earth is

St =(δSt

δIt+ δSt

δT ∗t

δT ∗t

δIt

)δIt = δet

x

T ∗t

(66)

orresponding to an increment of cosmic exergyensity associated with the incremental terrestria

tions would be essential in the transformation proc

. Exergy budget of the earth

The budget of cosmic exergy associated witharth system is promptly presented in part as the ref the global exergy balance and in part based onntropy budget according to the Gouy-Stodola law

As shown inFig. 6 the solar radiation in the coic background evokes an exergy flux entering thosphere with an intensity of 173,300 TW, of wh2,000 TW amounting to 30% leaves the atmospssociated with the backscattering and reflection oolar radiation by the earth, 119,600 TW amouno 69% leaves the atmosphere associated with thestrial radiation. The remaining intensity of 1700 Tmounting to about 1% of the entering intensity, is


Fig. 6. Budget of cosmic exergy associated with earth system.

in the domain of the earth system. As the most of theremaining 1%, an intensity of 1690 TW is consumedin the radiation field due to its irreversible interactionwith the material earth, of which 1410 TW is occurredin the radiation process associated with the atmosphere,280 TW is occurred in that with the earth’s surface. Theultimate part lost in the material earth is down to 45 TW,amounting to 2.5% of the remaining 1%, that is, 4000times smaller than the original entering exergy inten-sity associated with solar radiation.

As shown inFig. 7, as the fundamental driving forceto sustain the material earth, the cosmic exergy fluxintensity of about 45 TW lost in the material earthcorresponds to a terrestrial exergy intensity of about4100 TW, of which 260 TW is lost in heat transfer as-sociated with thermal diffusivity, 1450 TW lost in mo-mentum transfer associated with turbulent dissipation,

and 2380 TW lost in mass transfer associated with thewater cycle as water transport and precipitation. Someother details have been given bySzargut (2003). At-mospheric and oceanic circulations possess an exergyintensity of 370 TW. A prevailing part of the potentialexergy of the clouds is destroyed by precipitation, andonly a very small part of about 5 TW is transformedinto the potential exergy of river flows. The dropletsof liquid or solid water contained in clouds representa renewable resource of fresh water associated with achemical exergy of 22 TW, of which 6 TW is availableon the land. The active part of the vegetation receivesan exergy of about 37 TW and transforms about 2.9 TWinto chemical exergy of plants, of which about 1 TWis consumed by the human society. Exergy loss con-nected with the planetary motion braked by the tidesis about 3 TW, that connected with the transfer of heat


Fig. 7. Budget of terrestrial exergy in the material earth.

and hot material from inside the earth is about 31 TW,and that connected with depletion of mineral, mainlyfossil fuels, is about 12 TW.

For the human society, the main source of exergy isfossil and nuclear fuels, and the products of photosyn-thesis comprising food, fuels and building timber. Thesum of the anthropogenic exergy losses has been evalu-ated bySzargut (2003)at 13 TW, or byWall and Gong(2001)at 12 TW. This intensity is already in the order-of-magnitude of 1% of the global terrestrial exergy con-sumption in the material earth. Though about 340 timessmaller than the global exergy loss, this amount, as islost near the earth’s surface, might be compared withthe losses associated with certain processes occurrednear the earth’s surface, such as the amount of 37 TWfor the photosynthesis associated with the ecosphereand 31 TW for the geothermal effect essential for the

landscape. While the global intensities are comparable,local or instant density of anthropogenic exergy impactcan be greater than that of natural exergy consumptionfor some local regions, and anthropogenic impact maydominant some ecosystems over there. That fact is ofessential implication to the highly concerned problemwith resources, environment and sustainability.

9. Conclusions

Exergy as a triad of resource availability, environ-mental impact and buffering capacity associated witha system in its local environment, in the latest devel-opment in the fields of ecological modeling, resourceaccounting, and environmental impact assessment, isreviewed and assessed.


The thermodynamic system of the earth is illustratedto be driven by cosmic exergy flow originated fromthe temperature difference between the sun and cos-mic background, and a global balance of the cosmicexergy, based on the exergy definition for thermal ra-diation initially proposed by Szargut, is carried out togive the global cosmic exergy consumption as the mul-tiplication of the CBM radiation temperature and theglobal entropy generation due to thermodynamic irre-versibility in the earth system.

Concrete formulae are derived for cosmic exergy,with emphasis on generalization of a simple blackbodyrelationship between entropy and energy flux densi-ties to the cases of gray body radiation with moderateor large emissivity associated with the earth system.Global entropy generation is evaluated with a resultcompared very well with the widely accepted datumbased on satellite observations in earth science.

A self consistent budget of the global entropy gen-eration is made with respect to the radiation processesassociated with the atmosphere and the earth’s surfaceand to the molecular transport phenomena in the mate-rial earth.

A mechanism of multiplication governing transfor-mation between cosmic exergy and terrestrial exergy isdeveloped. As exergy of a system is defined in terms ofthermodynamic difference between the system and itslocal environment corresponding to the time and lengthscale of the observer, the system carries different valuesof exergy as the environment is changed with varyingt cor-r thec

pre-s us-t hortw % iss d byt ters m ism ther itht isc out4 rgy.A -of-m tiono ome

terrestrial processes, which provides an essential ev-idence for illustrating the seriousness of the globallyconcerned problem of resources, environment and sus-tainability.

Acknowledgement

Project supported by the National Natural ScienceFoundation of China (Grant No. 10372006).

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