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 exergy
consumption with respect to main terrestrial processes. Based on Szargut’s definition of exergy for thermal radiation, a global
exergy balance of the thermodynamic system of the earth, driven by cosmic exergy flow originated from the temperature difference
between the sun and the cosmic background, is carried out to give the global cosmic exergy consumption as the multiplication
of the cosmic background microwave (CBM) radiation temperature and the global entropy generation due to irreversibility in
the earth system. Concrete formulae are derived for cosmic exergy, with emphasis on generalization of a simple blackbody
relationship between entropy and energy flux densities to the cases of gray body radiation with moderate or large emissivity
associated with the earth system. Global entropy generation is evaluated with a result compared very well with a widely accepted
datum based on satellite observations in earth science. A budget of the global entropy generation is made with respect to the
terrestrial radiation processes associated with the atmosphere and the earth’s surface and to the molecular transport phenomena
in the material earth. A mechanism of multiplication governing transformation between cosmic exergy and terrestrial exergy is
developed. An overall exergy budget of the earth system, based on the entropy budget by means of the Gouy-Stodola law, are
presented with essential implication to the problem of global sustainability.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Exergy; Entropy; Energy; Earth; Ecological modelling; Resource accounting; Environmental assessment; Sustainable development
1. Introduction
As a latest progress in ecological modeling, resource
accounting and environmental assessment, an escalating interest has been emerging (Jørgensen, 2001; Wall,
2002; Szargut, 2003; Svirezhev, 2001; Svirezhev et al.,
∗
Tel.: +86 10 62767167; fax: +86 10 62750416.
E-mail address: [email protected].
0304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2004.10.015
2003) for the adaptation and generalization of the concept of exergy originated in engineering thermodynamics (Szargut et al., 1988).
For a system given to have a direct bearing on
its local environment associated with the time and
length scales depending on the observer’s objectives
and knowledge (Woods, 1975, p. 5), exergy Ex is defined as the amount of work the system can perform
when it is brought into thermodynamic equilibrium
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
with the environment (Jørgensen, 2001), that is,
tot
Ex = T0 (Seq
− S tot )
(1)
tot
where T0 is the temperature of the environment, Seq
and Stot are the entropies in thermodynamic equilibrium and at the given deviation from equilibrium, respectively, of the total system as a combination of the
given system and the local environment.
With J standing for the exergy flux density (per unit
time and per unit area) vector through the boundaries
B of a general open system, n the normal vector to the
boundaries, σ the exergy consumption per unit time per
unit volume in the physical domain V of the system, an
exergy balance for an open system requires.
dEx
J · n da −
σ dv
(2)
=−
dt
B
V
that is, the time derivative of the exergy possessed by
the system equals to the rate of exergy reception by the
system from its external environment minus the rate
of exergy consumption in the system. For an arbitrary
domain, the second law of thermodynamics, often referred to as the law of entropy production, is alternatively expressed in terms of exergy consumption as
σ≥0
(3)
that is, exergy can never be created: it is always consumed in a real irreversible process, and the equality only holds for the idealized reversible or equilibrium process. Thus the fundamental resource to sustain a system is exergy, which is lost in driving the
irreversible process associated with the system as required by the second law of thermodynamics, rather
than energy, which is always in conservation and never
be consumed in any case according to the first law of
thermodynamics.
Exergy associated with a system may play distinctive roles as resource, buffering capacity and environmental impact, for the subject, system and environment, respectively. Firstly, for the subject originally as
an observer external to both the system and its local
environment, the exergy can stand as a unified measure of resource availability (Wall, 1977, 1990, 1997,
2000; Zaleta-Aguilar et al., 1998; Sciubba, 2001;
Dincer, 2002), in terms of the maximum amount of
work that can be extracted by the subject from the system in its process of reaching equilibrium with its local
environment. Secondly, for the system itself, the exergy
serves as a basic measure of the global buffering capacity (Mejer and Jørgensen, 1979; Jørgensen, 1981), in
terms of the self-organization and construction of the
system as contrast to the environment to represent the
aliveness and vitality of the system in a given environment. And finally, for the local environment, the exergy
of the system can act as an environmental impact (Wall
and Gong, 2001; Gong and Wall, 2001; Wall, 2002;
Sciubba, 2003), in terms of the potential to change the
state of the local environment in the natural process of
reaching equilibrium in the absence of external actions.
The planet earth has been driven by exergy flows
associated with the thermodynamic contrast between
the sun and the outer space. The exergy lost in earth
serves for the planet not only as the ultimate resource
to revitalize the meteorological system, feed the hydrological cycle, renovate the biosphere and make all
other natural and anthropogenic phenomena possible,
but at the same time also as a fundamental reference
for measuring the global environmental buffering capacity to sustain and resist environmental impact. A
budget of the global exergy loss with respect to main
phenomena within the planet such as solar and terrestrial radiation, atmosphere and ocean circulation, convective heat transfer, turbulence dissipation, transport
and precipitation of water, photosynthesis and human
activity is essential for evaluating the cost of a process or product in cosmic exergy equivalence. Then
the greatly concerned problem of global sustainability
might be examined with relation to exergy flows on the
earth.
Exergy of solar radiation has been studied by
Svirezhev and Steinborn (2001) and Svirezhev et al.
(2003) with an information approach to measure the
change of energy, by the balance between absorption
and reflection of solar radiation and emission of terrestrial radiation, and the increment of information by the
Kullback measure. As a generalization of Jorgensen’s
maximal principle, a minimax principle was postulated
as that during the self-organization the vegetation tends
to maximize its exergy in respect to the increment of
information and to minimize it in respect to the radiation balance. This hypothesis was tested for seasonal
dynamics of several ecosystems located at geographically different sites, with radiation data for the long and
short spectral intervals. Using NASA satellite data to
calculate the global distribution of the annual mean of
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
exergy, they observed that the domains with maximal
values of exergy correspond to the main upwellings
of the ocean. With thermodynamic analogies applied
to the process of interaction of solar radiation with an
“active” planetary surface, in particular with vegetation cover, the difference between the radiation balance
and the exergy could be considered as the increment of
internal energy, of which the global pattern has been
shown very similar to that of vegetation.
In their general survey of exergy and sustainable
development, Wall and Gong (2001) made the first,
though brief, description of the global exergy losses in
the earth, and presented a chart for exergy flows on the
earth, not mentioning related evaluation procedures. It
is clearly pointed out that the exergy driving the flows
of energy and matter originates from the contrast between the sun and space, though the outer space seems
irrelevant in their quantitative calculation and a mean
temperature of the earth’s surface seems taken as the
reference environment temperature. It is illustrated that
the resource for the earth comes from the solar exergy
in value of 160,000 TW (T stands for trillion, i.e., 1012 ),
of which 30% is reflected back to the space associated
with the reflection of solar radiation and the remaining 70% is lost in driving the earth system, and the
society use of exergy of 12 TW amounts to be 13,000
times smaller than the total incipient associated with
solar radiation or 9100 times smaller than the total exergy consumption of the earth. With these small ratios
of order-of-magnitude, it seems hard to perceive the
threat of existing human impact to the global sustainability of the earth.
The work by Szargut (2003) on anthropogenic and
natural exergy losses made its appearance as the first
monograph on the global exergy consumption and its
budget with respect to main terrestrial processes in the
earth. The influence of the relict radiation of the cosmic
space is stressed in the exergy evaluation. While solar
exergy lost in the earth is estimated about 105, 100 TW,
close to the value given by Wall and Gong, an exergy of
relict radiation of the cosmic space is shown to be in the
value of 74,900 TW, an amount comparable to the consumed solar exergy. The exergy loses occurring near the
earth’s surface have been distinguished because they
are considered to represent the most accessible natural resources of exergy. The term of natural losses of
utilizable exergy has been proposed. These losses have
been compared with the anthropogenic ones caused by
365
the activity of humankind. The positive impact of the
natural exergy losses has been pointed out: they are the
main cause of the formation of the terrestrial natural
environment, of the non-renewable natural resources
of fuels, and of the generation of stable dissipative
structures in form of living beings. The sum of anthropogenic exergy losses was estimated at 6000 times
smaller than the natural losses of utilizable exergy.
With such a small ratio, it remains hard to illustrate the
problem of global sustainability with exergetics for the
earth.
This paper presents a systematic study on the global
exergy consumption of the earth and the budget of the
exergy consumption with respect to main terrestrial
processes. The thermodynamic system of the earth is
shown to be driven by cosmic exergy flow originated
from the temperature difference between the sun and
cosmic background, and a global balance of the cosmic exergy is carried out to give the global cosmic exergy consumption as the multiplication of the CBM
radiation temperature and the global entropy generation due to irreversibility of the earth system. Based
on the exergy definition for thermal radiation initially
proposed by Szargut, concrete formulae are derived
for cosmic exergy, with emphasis on generalization of
a simple blackbody relationship between entropy and
energy flux densities to the cases of gray body radiation with moderate or large emissivity associated with
the earth system. Global entropy generation is evaluated with a result compared very well with a widely
accepted datum based on satellite observations in earth
science. A detailed budget of the global entropy generation is made with respect to the radiation processes
associated with the atmosphere and the earth’s surface
and to the molecular transport phenomena in the material earth. The mechanism governing transformation
between cosmic exergy and terrestrial exergy is explored. An overall exergy budget of the earth system
are presented with essential implication to the problem
of global sustainability.
2. Cosmic exergy driving the earth system
Granted that thermodynamic effects connected
with the planetary motion braked by tides and heat
and material from inside the earth are negligible, the
concerned thermodynamic system of the earth has
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
a direct bearing on the cosmos thermodynamically
characterized of a singularity of the sun and a surrounding cosmic background. The earth system can be
appropriately considered as closed with relation to the
exchange of rest mass while open with relation to the
exchange of energy radiation with its cosmic environment. Commonly assumed of overriding importance,
the solar radiation as a whole may be more or less
considered as blackbody radiation, though the ultraviolet region (≤0.4 ␮m) of the solar spectrum deviates
remarkably from the visible and infrared regions in
terms of the equivalent blackbody temperature of
Ts = 5800 K (Liou, 1980, p.23). The thermodynamic
influence of the rest of the universe as an integral effect
is embodied in the cosmic background microwave
(CBM) radiation. Believed to be the relict remaining
after the Big Bang, the CBM radiation, corresponding
to an emission temperature of Tcbm = 2.73 K, has been
found with strikingly small anisotropies (Peacock,
1999, p. 597) and might be regarded as blackbody
radiation in the best sense of the term. The vastness of
the cosmos qualifies the ultimate thermal sink of the
blackbody associated with CBM radiation as a cosmic
cold reservoir. The fact of interception of some of the
solar radiation by the earth does not change the fundamental fate of the solar radiation to annihilate in and be
absorbed in the cosmic background: the solar radiation
intercepted by the earth, as a transformer of radiation,
is simply reflected or scattered back to the outer space,
or promptly transformed into terrestrial radiation
emitted to the outer space. Therefore related exergy
evaluations should be carried out with the blackbody
associated with CBM radiation as the reference environment. The exergy thus evaluated with Tcbm as the
environment temperature might be referred to as cosmic exergy. The thermodynamic difference between
the solar and CBM radiation results in a cosmic exergy
flow, of which a tiny branch is intercepted by the planet
earth and consumed in driving and sustaining the earth
system.
As illustrated in Fig. 1, there are exergy fluxes flowing through the outer boundary of the atmosphere, associated with the exchange of thermal radiation between the earth system and the cosmos. The essence
of this exchange involves an entering flux of solar radiation (short wave with wave length of λm ≈ 0.5 ␮m
corresponding to maximum energy), a leaving flux of
reflected (short wave with λm ≈ 0.5 ␮m) and backscat-
tered solar radiation (long wave with λm 0.5 ␮m),
a leaving flux of terrestrial radiation (long wave with
λm ≈ 10 ␮m) and an entering flux of CBM radiation
(very long wave with λm ≈ 1000 ␮m). According to
the Planck equation, the energy ε of a photon is given
by ε = hν (where h is the Planck constant and ν the frequency), thus the energy is inversely proportional to
the wave length of the radiation. The same amount of
energy associates with fewer photons in the form of
solar radiation than in the form of backscattered and
terrestrial radiation. In other words, the entering solar
radiation is more organized than the leaving backscatterd and terrestrial counterparts, and thus the amount
of exergy associated with the incoming solar radiation
is higher than that with the backscattered and terrestrial radiation. The CBM radiation emitted from the
reference environment is the most unorganized and
possesses no exergy at all. The earth system receives
high quality energy and returns low quality energy to
the cosmos. It is the contrast in energy quality rather
than in energy quantity provides the earth with an
exergy resource to drive and sustain the irreversible
system.
We consider the exergy balance of the earth with
a time scale of the climate cycles of years and a
length scale of the magnitude of the earth, that is,
exergy evaluation is time averaged over climate cycles
of years and area averaged over the earth’s surface.
With J standing for the exergy flux density vector
through the outer boundary A of the atmosphere, we
have
J · n da =
J · n da
(4)
B
V
A
σ dv =
A
φ da
(5)
where we have taken into account the fact that the
exergy flux vanishes on the lower boundary of the
system, and defined a rate of exergy consumption
corresponding to unit surface area as
+∞
φ≡
σ dz
(6)
0
with z standing for the altitude. Time and area average
of Eq. (2), into which Eqs. (4), (5) and (6) are
substituted, combined with the consideration of the
steadiness of the system with respect to the given
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
367
Fig. 1. Comic exergy driving the earth system.
time and length scales results in an overall balance
as
Φ = −Jn
(7)
where Φ standing for the mean rate of cosmic exergy
consumption per unit area, and Jn standing for the net
mean density of normal exergy flux. Thus the exergy
lost in the earth can be estimated by simply counting
the exergy flux at the top boundary of the atmosphere.
As first established by Szargut and elaborated by
Petela (Petela, 1964; Szargut et al., 1988), the exergy
of thermal radiation might be given as
J = (I − I0 ) − T0 (S − S0 )
(8)
where I and S are the flux densities of energy and
entropy radiation, I0 and S0 are the flux densities of energy and entropy radiation at the environment temperature of T0 , respectively. For the cosmic radiation under
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
consideration, corresponding cosmic exergy turns to be
Js = (Is − Icbm ) − Tcbm (Ss − Scbm )
(10)
to a hot thermal reservoir of solar radiation at an emission temperature of Ts and a cold thermal reservoir of
CBM radiation at an emission temperature of Tcbm , as
schematically illustrated in Fig. 2(a). The system receives energy of Iin , equal to Is , and entropy of Sin ,
equal to Ss , from the hot reservoir, and ejects energy of
Iout , equal to Isr + It − Icbm , and entropy of Sout , equal
to Ssr + St − Scbm , to the cold reservoir, and according
to Eqs. (14) and (16), we have
Jsr = (Isr − Icbm ) − Tcbm (Ssr − Scbm )
(11)
Iout = Iin
(18)
Jt = (It − Icbm ) − Tcbm (St − Scbm )
(12)
S ≡ Sout − Sin ≥ 0
(19)
Jc = (I − Icbm ) − Tcbm (S − Scbm )
(9)
Exergy flux densities for the incoming solar radiation,
Js , the leaving backscattered and reflected solar radiation, Jsr , and the leaving terrestrial radiation, Jt , can be
accordingly presented as
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 = −Tcbm S
(15)
where
S ≡ −Ss + Ssr + St − Scbm
(16)
is the total entropy increase in the radiation fluxes in
the earth system. Then from Eq. (7), we have
Φ = Tcbm S
(17)
thus the global exergy consumption amounts to the
multiplication of the CBM radiation temperature and
the total entropy increase in the radiation fluxes in the
earth, which can be simply deduced from the entropy
flux densities on the top of the atmosphere.
3. Cosmic exergy loss due to irreversibility of
the earth
The simple relationship of Eq. (17) between cosmic
exergy consumption and entropy increase associated
with the irreversibility of the earth system might be
illustrated in more basic contexts in thermodynamics
(Zemansky and Dittman, 1981, p. 199).
Consider the steady system of the earth as a thermodynamic machine of radiation transformation, subject
Now suppose that it is desired to produce exactly the
same change in the radiation fluxes through the radiation transformer of the earth with the concerned irreversible process, but by a reversible process only.
This would require, in general, the service of Carnot
engines and refrigerators in the outer space, which, in
turn, would have to be operated in conjunction with
an auxiliary environment consisted of an auxiliary mechanical device and an auxiliary reservoir. For the auxiliary reservoir let us choose the one whose temperature
is the lowest at hand, i.e., Tcbm . With the aid of suitable
Carnot engines and refrigerators all operating in cycles, in conjunction with the auxiliary environment, it
is now possible to produce in the radiation flux through
the radiation transformer, by a reversible process only,
the same change that formerly brought about by the
irreversible process. If this is done, the entropy change
of the radiation fluxes is the same as before, since they
have gone from the same initial state to the same final
state. The auxiliary environment, however, must undergo an equal and opposite entropy change, because
the net entropy change of the total system consisted of
the radiation flux and the auxiliary environment during
the reversible process should be zero.
Since the entropy change of the radiation fluxes is
positive, the entropy change in the auxiliary system
should be negative. Therefore the auxiliary reservoir
at Tcbm must have ejected a certain amount of heat,
say, Φ. Since no extra energy has appeared in the earth
system as a mere radiation transformer and in the radiation flux, the heat Φ must have been transformed
into work in the auxiliary mechanical device. We have
the result therefore that, when the same change which
was formerly produced in the radiation flux by an irreversible process is brought about reversibly, an amount
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
369
Fig. 2. (a) Earth as a thermodynamic machine of radiation transformation operating between reservoirs at Ts and Tc bm . (b) Cosmic exergy loss
due to irreversibility of the earth.
of energy Φ leaves an auxiliary reservoir at Tcbm in the
form of heat and appears in the form of work in an
auxiliary mechanical device. In other words, energy Φ
is converted from a form free of exergy in which it
was completely unavailable for work into a form of exergy in which it is completely available for work. Since
the original process was not performed reversibly, the
energy Φ was not converted into work, and therefore
Φ is the energy that is rendered unavailable for work
because of the performance of the irreversible process.
It is then a simple matter to calculate the cosmic
exergy lost due to the irreversibility of the radiation
transformer of the earth system. If the same change is
brought about reversibly, the entropy change associated
with the radiation flux is the same as before, namely,
S. The entropy change of the auxiliary environment
is merely the entropy change of the auxiliary reservoir
due to rejection of Φ units of heat at temperature Tcbm ,
that is, −Φ/Tcbm . Since the sum of entropy changes of
the radiation flux and auxiliary environment is zero, we
have
S −
Φ
Tcbm
=0
(20)
equivalent to Eq. (17). Whence, the cosmic exergy loss
is Tcbm times the entropy increase in the radiation flux
due to the irreversibility of the earth.
4. Exergy of thermal radiation
The energy flux density of thermal radiation might
be written as
I=
iν (ξ)ξ · n dν dΩ(ξ)
(21)
ν
Ω
where iν (ξ) is the monochromatic energy flux density, ξ
the direction of the flow of radiation, Ω the solid angle.
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
For blackbody or gray body radiation,
I = εσT 4
where
(22)
where ε is the surface emissivity which is irrelevant to
frequency and for the perfect case of a blackbody equal
to 1, and σ the Stefan–Boltzmann constant, equal to
5.667 × 10−8 wm−2 K−4 .
The relationship between entropy flux density and
energy flux density was established by Planck (1959)
and derived in modern terms by Rosen (1954). The
monochromatic entropy flux density sν corresponds to
monochromatic energy flux density according to the
differential equation of
sν (ξ)
1
iν (ξ)
d
= ∗d
(23)
c
Tν
c
where c is the speed of light, and
−1
2hν3
hν
∗
ln 1 + 2
Tν =
k
c iν (ξ)
(24)
is referred to as the monochromatic brightness temperature, or emission temperature, associated with each
beam of radiation, with k standing for the Boltzmann’s
constant. Tν∗ is obtained by replacing by iν the Planck
function Bν (T) in the expression for complete thermodynamic equilibrium, and in general, might be dependent upon the direction of the flow of the radiation.
For the special case of blackbody radiation, Tν∗ has a
common value independent of both frequency and direction, and is the same as the thermal temperature of
the emitting surface under the conditions of thermodynamic equilibrium. Eq. (23) might be revised in explicit
form as
c 2 iν
c 2 iν
2kν2
Sv = 2
1+
ln 1 +
c
2hν3
2hν3
c 2 iν
c 2 iν
−
ln
(25)
2hν3
2hν3
Then entropy flux density can be obtained as
iν (ξ)ξ · n dν dΩ(ξ)
S=
ν
Ω
(26)
For gray body radiation (Landsberg and Tonge, 1979;
Stephens and O’Brien, 1993),
S = 43 σT 3 χ(ε)
(27)
∞ ε
ε
1+ x
ln 1 + x
e −1
e −1
0
ε
ε
(28)
ln x
x2 dx
− x
e −1
e −1
45
χ(ε) =
4π4
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 ln ε)
(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 of
ε ≥ 0.7.
For the case of blackbody or gray body radiation,
combination of Eqs. (22) and (27) gives
S=
4 χ(ε) I
3 ε T
(32)
which turns to be
S=
4I
3T
(33)
as an exact result for blackbody radiation, or an approximate one for gray body radiation with moderate or
large emissivity based on Eq. (31). The uniform emission temperature of
3I 1/4
T =
(34)
4εσ
might be replaced by an equivalent blackbody temperature of T* ,
1/4
3I
T ≈ T∗ ≡
(35)
4σ
for cases with moderate or large emmisivity, say,
ε ≥ 0.8, corresponding to a deviation less than 5%.
A simple relationship between entropy and energy
flux densities might be obtained by the combination of
Eqs. (33) and (35) as
S = S(I) ≡
4 1/4 3/4
σ I
3
(36)
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
Derived under a wide range of conditions and not involving such details as emissivity and emission temperature, the idealized relation of Eq. (36), corresponding
to Eqs. (33) and (35), may be conjectured as approximately correct for non-equilibrium radiation fields of
more complicated implications. Though in the simplest form corresponding to blackbody radiation, the
idealized formula has been found representing the actual long wave entropy flux quite accurately, by Lesins
(1990) and Goody and Abdou (1996) in their entropy
flux calculations at the top of the atmosphere using
energy flux over the entire spectrum based on the
earth radiation budget experiment (ERBE) data, and
by Goody and Abdou (1996) in numerical simulation
of a variety of atmospheres with different solar fluxes
and cloud amounts, and is employed in the present
work in the estimation of associated global entropy
flux densities associated with long wave radiation processes, including the terrestrial radiation leaving for the
outer space, that leaving the atmosphere for the earth’s
surface and that leaving the earth’s surface for the
atmosphere.
For the case of blackbody or gray body radiation,
the cosmic exergy as defined in Eq. (9) can be obtained
in a certain expression as
4 χ(ε) Tcbm
4 χ(ε)
Jc = 1−
− 1 Icbm
I+
3 ε T
3 ε
(37)
As Tcbm is much smaller than T associated with solar
radiation, backscattered and reflected radiation, and terrestrial radiation, Icbm is several orders smaller in magnitude than I under concern, then the second term in
the right hand side of the above equation is neglected,
and we get
4 χ(ε) Tcbm
Jc = 1 −
I
3 ε T
(38)
For cases of blackbody radiation or gray body radiation
with moderate or large emissivity associated with the
concerned terrestrial radiation processes, the result can
be further simplified into
4 Tcbm
Jc = 1 −
I
3 T
(39)
371
5. Cosmic exergy consumption of the earth
As the sun as a whole can be regarded as a blackbody, we readily have
Is = 41 Cs = 340 W m−2
1 Cs
3 Ts
= 0.078 W m−2 K−1
4 Tcbm
1
Js = Cs 1 −
= 339.8 W m−2
4
3 Ts
Ss =
(40)
(41)
(42)
where Cs 1360 W m−2 is the solar constant, defined
as the amount of solar radiation incident per unit area
and per unit time on a surface normal to the direction of
propagation and situated at the earth’s mean distance
from the sun (Peixoto and Oort, 1992). An uncertainty
of 21 wm−2 , corresponding to 1.6% of Cs , as issued
by NASA (Thekaekara, 1976) brings about the same
percentage of uncertainty in evaluating the solar exergy.
The factor of one fourth in Eq. (40) is due to the fact
that the global surface area of the earth is four times
the area of its cross section.
The mean broad-brand albedo α for the backscattering and reflection of the solar radiation has been
typically found in the range of about 0.29–0.31 (Liou,
1980, p. 189), for which a mean value of 0.30 (Peixoto
and Oort, 1992) is taken in the present study, with a
uncertainty of about 3.3%. But detailed records with
spectral resolution high enough for a reliable entropy
evaluation for the back-scattering and reflection of the
solar radiation has yet to be achieved. As a first approximation, we assume
α = αsc + αrf
(43)
where αsc is the albedo for perfect back-scattering in the
manner of diffuse reflection, equivalent to gray body
radiation, and αrf is the albedo for perfect reflection,
i.e., specular or mirror-like reflection, corresponding
to blackbody radiation, then corresponding monochromatic energy flux densities for the backscattered and
reflected solar radiation are given as
isc = εsc Bν (Ts )
(44)
irf = εrf Bν (Ts )
(45)
where εsc = αsc is the equivalent emissivity for the
back-scattered solar radiation and εrf = αrf is that for
the reflected solar radiation, θ is the zenith angle of the
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
sun, Ω0 the solid angle of the sun for the earth, equal
to 67.7 × 10−6 . With αsc = 0.06, αrf = 0.24 (Peixoto
and Oort, 1992, p. 94), and averaged cos θ = 0.25,
we have mean emissivity ε̄sc = 1.61 × 10−6 , corresponding to ␹(ε̄sc ) = 7.52 × 10−6 according to Eq.
(30). With formulae given in the previous section, we
have
Isc = αsc Is
(46)
Irf = αrf Is
(47)
and
Ssc = αsc
χ(ε̄sc )
Ss
ε̄sc
Srf = αrf Ss
(48)
(49)
Then
Isr = Isc + Irf = αIs = 102 W m−2
(50)
χ(ε̄sc )
Ssr = Ssc + Srf = αsc
+ αrf Ss
ε̄sc
= 0.547Ss = 0.043 W m−2 K−1
(51)
Thus a net value of 1.209 wm−2 K−l is obtained for the
global density of entropy increase of S. The net solar entropy as the contribution from the first two terms
of Ss and Ssr , equal to 0.035 wm−2 K−1 , amounts to
only 2.89% of the global entropy increase, and can
be neglected for simplicity, considering the total uncertainty near 5% resulted from that of 1.6% with the
solar constant and that of 3.3% with the albedo. Then
we might choose to simply take the global density of
entropy increase as the terrestrial entropy density of
1.244 wm−2 K−l . This value of S is very close to the
widely accepted value in the earth science of 1.25 from
the earth radiation budget experiment (ERBE) observations, with a deviation of merely 0.48%, well within
the claimed uncertainty of about 3% associated with
the latter.
The global value of Φ for cosmic exergy consumption of the earth, according to Eqs. (13) or (17), equals
to 3.30 wm−2 , based on the value of 1.209 wm−2 K−l
for the global density of entropy increase of S. This
is only about 1% of the mean solar energy density
entering the earth. With the global surface area of
510Tm2 , the global cosmic exergy loss is calculated
as about 1700 TW, equal to 1% of the amount of
the total energy flow of solar radiation meeting the
earth.
4
χ(ε̄sc )
Tcbm
Jsr ≡Isc +Irf = α −
αsc
Is
+ αrf
3
Ts
ε̄sc
= 101.9 W m−2
(52)
The net entropy flux density of solar radiation
(Stephens and O’Brien, 1993) is then given as
Sns ≡ Ss − Ssr = 0.453Ss = 0.035 W m−2 K−1
(53)
For the terrestrial radiation, the energy flux density is
It = (1 − α)Is = 238 W m−2
(54)
corresponding to an equivalent blackbody temperature
of Tt∗ = 255 K. With the idealized blackbody formula
of Eq. (33), we have
4 It
= 1.244 W m−2 K−1
3 Tt∗
4 Tcbm
It = 234.6 W m−2
Jt = 1 −
3 Tt∗
St =
(55)
(56)
6. Global entropy budget
Since the appearance of the seminal papers by
Paltridge (1975, 1978) to explore the possibility of
a thermodynamic description of the climate system
based on a variational principle for maximization of
the rate of entropy production, emphasis on the application of the second law of thermodynamics has been
placed in the earth science with the result that many papers on entropy budget of the earth have been published
(e.g., Peixoto and Oort, 1992; Stephens and O’Brien,
1993; O’Brien and Stephens, 1995; Goody and Abdou,
1996; Goody, 2000; Pauluis and Held, 2002a, 2002b).
Within the physical domain of the earth system there
is a two phase continuum, composed of a real material
phase with rest mass and a radiation phase of photons
with no rest mass. We may treat the material and radiation phases in the earth as two separate but interacting subsystems. The internal entropy production is the
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
sum of the entropy production in the subsystem of material phase due to irreversible molecular process and
that in the radiation phase due to irreversibility associated with interactions between radiation and material,
namely,
mat
rad
Ṡirr = Ṡirr
+ Ṡirr
(57)
The material is considered in a state of local thermodynamic equilibrium. External interactions, whether they
are heat, work or the transfer of species, take place
through a series of local equilibrium and are locally reversible. However, transport phenomena due to interactions between material molecules, such as momentum
transfer associated with velocity gradient due to viscosity, heat transfer associated with temperature gradient
due to conductivity and mass transfer associated with
concentration gradient due to diffusivity, are internal
to the material; they all lead to irreversible increases
of entropy, tending to bring the material phase close to
complete equilibrium.
The radiation term involves interaction between radiation and material. A first consequence of the interaction is a locally reversible heat exchange between
the material and radiation, corresponding to entropy
exchange between the subsystems with zero net entropy production for the earth system as a whole. However, for a reversible heat exchange with a material
373
in thermodynamic equilibrium, collisional rearrangement of energy levels, associated with a very small
proportion of molecules in a local region of negligible extent, would be required, as absorbed or emitted photons generally have an emission temperature
different from the local kinetic temperature. This second consequence of the interaction involves no energy
exchange, but increases entropy in the radiation field,
rather than in the material phase. A third consequence
is scattering, with no heat exchange takes place, but the
direction and perhaps the polarizations of photons are
changed. Scattered radiation is less organized and has
higher entropy than incident radiation. The irreversible
production of entropy due to thermalization and scattering has been generally studied by Essex (1984) on
the violation of the bilinearity in the thermodynamics of irreversible process and by Goody and Abdou
(1996) in a special assessment of reversible and irreversible sources of radiation entropy in the climate
system.
rad ,
To estimate the irreversible radiation term of Ṡirr
we refer to the radiation balance of the total system
modeled as consisted of three boxes of the earth’s
surface, atmosphere and outer space, as illustrated
in Figs. 3 and 4, which is proposed on two typical
schematic diagrams of global radiation balance of Fig.
6.3 in Peixoto and Oort (1992, p. 94) (which was
adopted from “Understanding Climatic Change, U.S.
Fig. 3. Schematic diagram of the global budget of solar radiation.
374
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
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 shown
on the left-hand side of Fig. 3, 46 units are intercepted by the atmosphere, from which six units are
scattered and 20 units are reflected back to the outer
space; the remaining 54 units reach the earth’s surface, over there 50 units are absorbed and 4 units
are reflected back to the outer space. With the 50
units absorbed by the earth’s surface and 20 units absorbed by the atmosphere from short-wave solar radiation, the long-wave terrestrial radiation has a budget as shown in Fig. 4: 121 units of terrestrial radiation leave the earth’s surface for the atmosphere,
and in return 101 units leave the atmosphere for the
earth’s surface, and 70 units leave the atmosphere for
the outer space. The data of Ss , Ssc and Srf obtained
in the above section for the primary, backscattered
and reflected solar radiation, and the simple blackbody relation of Eq. (36) for long wave terrestrial radiation processes associated with the earth’s surface
and atmosphere are employed. Then, for the earth’s
surface,
rad
Ṡirr
|es
=S
100
54
121
4
Is −S
Is −
Ss + Srf
100
100
100
24
= 0.203 W m−2 K−1
(58)
and for the atmosphere,
70
101
121
rad
|atm = S
Is + S
Is − S
Is
Ṡirr
100
100
100
20
6
Ss + Ssc −
−
Ss
100
100
= 1.011 W m−2 K−1
(59)
Thus the total entropy production in the radiation field,
due to irreversible radiation interaction with the material phase, is obtained as
rad
rad
rad
Ṡirr
= Ṡirr
|es + Ṡirr
|atm = 1.213 W m−2 K−1
(60)
This value is very close to the net entropy increase of
1.209 wm−2 K−1 for the planetary albedo of 0.3, with a
difference of only 0.3%. Should this estimation be reliable, the value of entropy production due to irreversible
molecular effects, Ṡmat , must be within the uncertainty
of about 5% in estimating the net entropy increase.
A recent and typical budget for entropy production
due to irreversible molecular effects has been presented
by Goody (2000) as follows
Ṡturb = 0.002 W m−2 K−1
(61)
Ṡdiss = 0.011 W m−2 K−1
(62)
Ṡwater = 0.019 W m−2 K−1
(63)
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
where Ṡturb is the inventory for convective heat transport in atmospheric turbulence, which is dominant in
the category of heat transfer, Ṡdiss is the inventory for
dissipation in the atmosphere, dominant in the category
of momentum transfer, and Ṡwater is the inventory for
water-vapor, dominant in the category of mass transfer.
The total value for entropy production due to molecular
effects is
irr
Ṡmat
= Ṡturb + Ṡdiss + Ṡwater = 0.032 W m−2 K−1
(64)
rad and
The total entropy production Ṡirr as sum of Ṡirr
rad
−2
−1
Ṡmat , equal to 1.245 wm K , is close to the value
of 1.209 wm−2 K−1 for the net entropy increase with
the planetary albeto of 0.3, with a deviation of 2.98%,
within the uncertainty of about 5% in estimating the net
entropy increase, and agrees very well with the value
of 1.25 wm−2 K−l obtained by Stephen and O’Brien
(1993) based on the ERBE data. The entropy generation in the material phase amounts to only 2.5% of
the global entropy generation. According to the GouyStodola law (Szargut et al., 1988), exergy loss is proportional to the entropy generation, it is then concluded
375
that only a very small proportion of 2.5% of the cosmic
exergy availability is lost in driving the material earth.
7. Transformation of exergy
It seems trivial to recall that exergy for a system is
defined as the maximal amount of work that can be
extracted from the system in the process of reaching
equilibrium with its local environment. But subtlety
arises when pondering a little bit over the choosing of
the system and its local environment, which has a direct
bearing on the behavior of the system, with respect to
the time and length scales, depending on the observer’s
objectives and knowledge (Woods, 1975, p. 5). As exergy is defined in terms of the thermodynamic contrast
between system and its local environment defined with
relation to the observer’s time and length scales, the
same system as commonly referred to as the exergy
carrier may possesses different amounts of exergy corresponding to different environments associated with
the change of the involved time and length scales.
In the category of overlapping multi-component media involving no electromagnetic effects, Wall (1977)
Fig. 5. A subsystem in the terrestrial environment which in its turn is located in the cosmic environment.
376
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
considered the case of a subsystem in a local environment which in its turn is included in a global environment, and for exergy transformation it is found as a
mechanism of additivity that the exergy of the subsystem with respect to the global environment is the sum
of the original exergy with respect to the local environment and the contribution of exergy due to the deviation
of the local environment from the global environment,
as given by the original exergy expression associated
with the local environment with the intensive parameters of the global environment replacing those of the
subsystem. In the present study with a particular electromagnetic effect of thermal radiation playing an essential role, we need to consider a subsystem in the
terrestrial environment, which in its turn is included in
the cosmic environment, as illustrated in Fig. 5.
Consider an exergy impact external to the existing
‘equilibrium’ of the earth system, that is, external to
the existing characteristic state and process of the earth
as it has been, is released in some subsystem in the
earth. External impacts could be enforced in a variety
of forms, such as incidental cosmic matter or energy received from the out space other than the solar and CBM
radiation, which have been already accounted for associated with the existing ‘equilibrium’, or anthropogenic
utilization of heat and material from inside the earth.
Assume the impact is small enough as not to effect the
global ‘equilibrium’ of the earth system, then associated exergy impact of δetx can be calculated with the
existing earth system as local reference environment,
and it may be referred to as terrestrial exergy loss. The
exergy of δetx might be carried in the form of heat or
would be eventually turned into heat, which would then
join in the terrestrial radiation leaving the earth for the
extended environment of the cosmic background, resulting in an increment of terrestrial radiation flux density of
δIt = δetx
(65)
According to Eq. (55), associated increment of the terrestrial entropy associated with the increment of terrestrial radiation leaving the earth is
δSt
δSt δTt∗
δet
δSt =
δIt = ∗x
+ ∗
(66)
δIt
δTt δIt
Tt
corresponding to an increment of cosmic exergy flux
density associated with the incremental terrestrial ra-
diation of
Tcbm
δJt = δIt − Tcbm δSt = 1 − ∗
Tt
δetx
(67)
Then according to Eq. (17) combined with Eq. (16), the
net increment of the global cosmic exergy consumption
is obtained as
Tcbm
δΦ = Tcbm δSt = ∗ δetx = 0.0107δetx
(68)
Tt
Conversely, terrestrial exergy impact is related to an
increment of global cosmic exergy consumption as
δetx =
Tt∗
δΦ = 93.4δΦ
Tcbm
(69)
that is, about 93 units of terrestrial exergy use results
in one unit of increment in cosmic exergy loss. This
might be referred to as a mechanism of multiplication
for the exergy transformation for the case of thermal
radiation.
The cosmic exergy δΦ lost in material process of
mat in value of
the earth system, equal to Tcbm Ṡirr
0.087 wm−2 , transforms into a terrestrial exergy loss
of δetx ≈ 8.13 wm−2 , corresponding to a global terrestrial exergy of δExt ≈ 4100 TW.
The simple transformation relation of Eq. (66) is
proposed for small exergy change denoted with the
symbol of δ. For exergy change large enough to effect the global equilibrium, nonlinear response of the
system out of the description of above differential operations would be essential in the transformation process.
8. Exergy budget of the earth
The budget of cosmic exergy associated with the
earth system is promptly presented in part as the results
of the global exergy balance and in part based on the
entropy budget according to the Gouy-Stodola law.
As shown in Fig. 6 the solar radiation in the cosmic background evokes an exergy flux entering the atmosphere with an intensity of 173,300 TW, of which
52,000 TW amounting to 30% leaves the atmosphere
associated with the backscattering and reflection of the
solar radiation by the earth, 119,600 TW amounting
to 69% leaves the atmosphere associated with the terrestrial radiation. The remaining intensity of 1700 TW,
amounting to about 1% of the entering intensity, is lost
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
377
Fig. 6. Budget of cosmic exergy associated with earth system.
in the domain of the earth system. As the most of the
remaining 1%, an intensity of 1690 TW is consumed
in the radiation field due to its irreversible interaction
with the material earth, of which 1410 TW is occurred
in the radiation process associated with the atmosphere,
280 TW is occurred in that with the earth’s surface. The
ultimate part lost in the material earth is down to 45 TW,
amounting to 2.5% of the remaining 1%, that is, 4000
times smaller than the original entering exergy intensity associated with solar radiation.
As shown in Fig. 7, as the fundamental driving force
to sustain the material earth, the cosmic exergy flux
intensity of about 45 TW lost in the material earth
corresponds to a terrestrial exergy intensity of about
4100 TW, of which 260 TW is lost in heat transfer associated with thermal diffusivity, 1450 TW lost in momentum transfer associated with turbulent dissipation,
and 2380 TW lost in mass transfer associated with the
water cycle as water transport and precipitation. Some
other details have been given by Szargut (2003). Atmospheric and oceanic circulations possess an exergy
intensity of 370 TW. A prevailing part of the potential
exergy of the clouds is destroyed by precipitation, and
only a very small part of about 5 TW is transformed
into the potential exergy of river flows. The droplets
of liquid or solid water contained in clouds represent
a renewable resource of fresh water associated with a
chemical exergy of 22 TW, of which 6 TW is available
on the land. The active part of the vegetation receives
an exergy of about 37 TW and transforms about 2.9 TW
into chemical exergy of plants, of which about 1 TW
is consumed by the human society. Exergy loss connected with the planetary motion braked by the tides
is about 3 TW, that connected with the transfer of heat
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G.Q. Chen / Ecological Modelling 184 (2005) 363–380
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, mainly
fossil fuels, is about 12 TW.
For the human society, the main source of exergy is
fossil and nuclear fuels, and the products of photosynthesis comprising food, fuels and building timber. The
sum of the anthropogenic exergy losses has been evaluated by Szargut (2003) at 13 TW, or by Wall and Gong
(2001) at 12 TW. This intensity is already in the orderof-magnitude of 1% of the global terrestrial exergy consumption in the material earth. Though about 340 times
smaller than the global exergy loss, this amount, as is
lost near the earth’s surface, might be compared with
the losses associated with certain processes occurred
near the earth’s surface, such as the amount of 37 TW
for the photosynthesis associated with the ecosphere
and 31 TW for the geothermal effect essential for the
landscape. While the global intensities are comparable,
local or instant density of anthropogenic exergy impact
can be greater than that of natural exergy consumption
for some local regions, and anthropogenic impact may
dominant some ecosystems over there. That fact is of
essential implication to the highly concerned problem
with resources, environment and sustainability.
9. Conclusions
Exergy as a triad of resource availability, environmental impact and buffering capacity associated with
a system in its local environment, in the latest development in the fields of ecological modeling, resource
accounting, and environmental impact assessment, is
reviewed and assessed.
G.Q. Chen / Ecological Modelling 184 (2005) 363–380
The thermodynamic system of the earth is illustrated
to be driven by cosmic exergy flow originated from
the temperature difference between the sun and cosmic background, and a global balance of the cosmic
exergy, based on the exergy definition for thermal radiation initially proposed by Szargut, is carried out to
give the global cosmic exergy consumption as the multiplication of the CBM radiation temperature and the
global entropy generation due to thermodynamic irreversibility in the earth system.
Concrete formulae are derived for cosmic exergy,
with emphasis on generalization of a simple blackbody
relationship between entropy and energy flux densities to the cases of gray body radiation with moderate
or large emissivity associated with the earth system.
Global entropy generation is evaluated with a result
compared very well with the widely accepted datum
based on satellite observations in earth science.
A self consistent budget of the global entropy generation is made with respect to the radiation processes
associated with the atmosphere and the earth’s surface
and to the molecular transport phenomena in the material earth.
A mechanism of multiplication governing transformation between cosmic exergy and terrestrial exergy is
developed. As exergy of a system is defined in terms of
thermodynamic difference between the system and its
local environment corresponding to the time and length
scale of the observer, the system carries different values
of exergy as the environment is changed with varying
time and length scales. One unit of cosmic exergy corresponds to about 93 units of terrestrial exergy for the
case of small exergy impact to the earth system.
An overall exergy budget of the earth system are presented with implication to the problem of global sustainability. Of the cosmic exergy associated with short
wave solar radiation intercepted by the earth, 30% is
scattered or reflected back to and 69% is carried by
the long wave terrestrial radiation leaving for the outer
space. What lost in the domain of the earth system is
merely the remaining 1%, of which most is lost in the
radiation field due to its irreversible interaction with
the material earth, only a tiny fraction of about 2.5% is
consumed in driving the material earth, which is about
4000 times smaller than the intercepted solar exergy.
Anthropogenic exergy use is already in the order-ofmagnitude of 1% of the global exergy consumption
of the material earth and can be dominant for some
379
terrestrial processes, which provides an essential evidence for illustrating the seriousness of the globally
concerned problem of resources, environment and sustainability.
Acknowledgement
Project supported by the National Natural Science
Foundation of China (Grant No. 10372006).
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