**Overall Availability Analysis of
Earth, Part **2

Table of Contents to Part 2

Computing the Lost Work or Maximum Reversible Work for Earth’s Control Volume (continued from Part 1)

Miscellaneous Fluxes and Other Things Calculated and Not Calculated

Average Grey-Body Emissivity of Earth

Difficulties and Uncertainties Due to Photons Not in Equilibrium and Irreversibilities

** **

[**Note by Dave Bowman (8-4-97).**
Due to the
albedo, the higher frequency radiation does not enter our computation,
therefore we may treat the Earth as more or less a black body with
respect to
the preponderant far-infrared radiation leaving the Earth in all
directions as
‘junk heat’ – except that most authors
would not treat this radiation as heat
in Second Law analysis because of the large irreversibilities due to
finite
temperature differences. Nevertheless, we use the heat-out
term in the First
and Second Laws Combined to pump power out of the CV from _{} to _{} later
on in this development.]

The
second part of _{} is _{}. Since
the Sun’s emissivity _{} is taken
to be 1.0, the
portion of _{} considered
in this study is (1
- a)_{}
, the nature of which is well
enough understood because the albedo got rid of the more mysterious
higher frequencies.

Since
the far-infrared radiation emitted from the Earth as junk heat
behaves essentially as blackbody radiation, essentially because the
troublesome
radiation was rejected by the albedo, the emissivity = 1.0
also. And, despite
global warming, if it occurs, nature keeps the Earth sufficiently well
in
energy balance that we may claim that _{}.

_{}

_{}

_{}_{}

If
we subtract _{} from
both sides to account for
the albedo, set _{} let _{}_{} replace
_{} by
_{},
use _{} =
_{} ,
and note
that _{} =
_{} since
the Sun ‘sees’
the Earth (plus its atmosphere) as a flat disk of area _{}, whereas
the Earth emits far-infrared radiation from its entire spherical
area 4 _{}_{} .
The two areas differ by a factor of four; so, after we have canceled _{},
we get _{} _{} .

_{}

_{} ,

which implies

_{} ,

Then,
_{}.

Thus, _{} =
0.0441142 _{} =
0.0441142 _{}5760
K = 254 K .

T_{cv}
= 254 K .

This temperature may seem too low; however, we must not forget that the surface of the control volume is 100 miles above the surface of the Earth. The U-2 pilots tell me that it’s cold up there.

_{}_{}
.

Let’s
see how well _{} matches the output _{}.

_{}_{}_{}

_{}

which matches the input admirably. Thus,

_{} _{}

This concludes our discussion of the First Law as it applies to photons. We should begin our analysis of the Second Law by writing an entropy balance over the entire Universe:

_{} ,

where
_{} and
are
the rates at which entropy leaves and enters the control
volume. We have
agreed already that the CV shall be in steady state with respect to
entropy, so
_{}.
Moreover, the Second Law tells us that _{}must be
positive. Therefore,
the rate at which entropy leaves the CV may not be less than the rate
at which
entropy enters the CV, otherwise the rate at which the entropy of the
Universe
changes would be negative, which we know cannot happen. For
convenience, let’s
denote the minimum value of the rate at which entropy leaves our
control volume
as _{},
which, of course, is just
equal to _{}.

We may
now introduce the appropriate First Law variables into our
Second Law discussion. According to Yourgrau et al. [5], the
entropy of a
photon gas can be written _{} ,
where _{} is the
rate
at which internal energy enters the CV. Therefore,

_{}_{}_{} ,

since _{} .
Then,

_{} = _{} _{} = _{}_{} = _{}_{} .

Thus,

_{} _{}_{}_{} .
Eq. (I-10)

Equation
I-10 can be rewritten in terms of internal energies or
enthalpies, since H =_{}:

_{}

But,
since H_{i}
= H_{e},

_{}

and

_{} .

Finally,
setting _{}for
this calculation, we get
the thermodynamic lost power in two steps.

_{}

Normally, in engineering problems, T_{o}
is the temperature
of the coldest thermal reservoir (heat sink capable of absorbing
unlimited
thermal energy without changing temperature and, strictly speaking,
differing
in temperature from the CV only infinitesimally). Clearly,
the heat transfer
in the normal statement of the Second Law is reversible, but one can
take some
liberties in most practical applications. Engineers set T_{o}
equal to the temperature of the
air or the nearest body of water, whichever they are polluting in the
case
under consideration. The meaning of T_{o}
is the temperature at which a heat engine rejects
thermal energy.

The
first step was the simplified First Law calculation of _{}involving
the difference in two enthalpies, the enthalpy of the photons entering
our CV
and the least enthalpy the photons might carry out of the CV without
violating
the Second Law. The second step was the work that would have
to be done by a
Carnot heat pump to raise the power entering the control volume from
254 K to
the environmental temperature 288 K; so that the world would be
restored to
what it would have been without irreversible activity. The
first step is like
replacing a bottle of champagne you have broken; the second step is
like
cleaning up the mess you made when you broke the bottle and got
champagne all
over the place. Whenever we try to compute the thermodynamic
lost power from
First Law variables, we must have some such step as pumping heat
– unless
everything is done precisely at T_{o}.
(If we make a mess at a
temperature appreciably higher than ambient, we can recover some of the
work.)

We shall now solve this problem in three slightly different ways using the Second Law exclusively.

Second
Law: _{} ,
Eq. (I-11)

where
s* is the constant ratio of mean entropy to mean number of photons for
a
macroscopic ensemble. Notice that, in this rendering of the
balance equation,
the rates of change of CV properties are isolated on the left-hand
side,
whereas the convection, conduction, and radiation terms (and only those
terms)
appear on the right-hand side, which is written in the form
‘out minus in’.
This equation is intended to be used in the steady-state case; i.e. _{}=
0, and with no convection or conduction terms. Thus,

_{}

_{}

The
denominator in the lost work term in our entropy balance is _{},
usually the temperature of the coldest heat sink we can find to which
our heat
engines may send useless exhaust heat as required by the Second Law,
represented as an entropy balance in Eq. I-11, above, where the rate of
change
of the entropy of the Earth is

_{},

the
instantaneous rate of change of the product of the mass of the Earth
(plus the
mass of its atmosphere but excluding the mass of the core of the Earth)
times
the average entropy per kilogram of the Earth, indicated by angle
brackets, as
usual. Comparison of E_{e}s*
(large) to E_{i}s*
(small) explains how the Sun
lowers the Earth's entropy. This is what allows life to exist
on Earth.

Life is
a process during which (roughly speaking) disorder is converted
to order; i.e., high entropy is converted to low entropy.
When the entropy is
lowered within a living thing, it must be raised elsewhere in the
environment.
This could not occur over a prolonged period of time unless the overall
entropy
of the Earth were lowered somehow. The average frequency of
the photons
entering the Earth’s CV from the Sun is relatively high;
therefore, since the
energy of each photon is *h*n
and the total
energy of the photons entering is approximately equal to the energy of
the
low-frequency photons leaving, each photon has more energy and,
consequently,
the photons entering must be fewer in number.

(The
quantity _{} ,
cs = e or i, the number of
photons per second, equals the total energy per second divided by the
average
energy per photon. The symbol n
represents the frequency of the
electromagnetic wave associated with a photon and *h* is
Plank's constant, a “fundamental constant of the
universe” – according to many
physicists.) Fewer photons means fewer degrees
of freedom and a much diminished opportunity for chaotic behavior
(chaotic in
the old-fashioned sense of Boltzmann not in the
“modern” sense of Feigenbaum,
Lorenz, Kolmogorov, Moser, et al. [9]), which, in turn, means lower
entropy.
On the other hand, the low-frequency photons leaving the Earth are much
greater
in number and, therefore, have many more degrees of freedom, which
means
greater entropy. With less entropy entering and more entropy
leaving, the net
effect of the Sun is to lower the entropy of the Earth.

The *irreversibility rate*
is the *thermodynamic
lost work rate*
divided by _{}.
Also, it is the *mechanical
lost work rate* divided by the
temperature of the CV – assumed isothermal, which is good
enough for our
purposes. Let us write the Second Law for the
Earth’s control volume as
follows:

_{}.

Therefore,

_{} .

Just as
a check, I shall solve the problem without using the relation _{}.

_{}_{}

A final approach involves computing the number of photons leaving and entering per unit time.

_{}

_{}

Wien’s Displacement Law,

_{}

where
μ is a micron and 1 micron = 10^{-6},
is an easy way to get the
wavelength corresponding to the maximum emission of radiative flux at a
given
temperature. It accounts for the color shift that we observe
when a metal
object, for example, is heated. Bowman was aware of the law
that gives the
average wavelength, λ_{avg}
, at any temperature, as 1.838 ∙λ_{max}.
Thus, we can write λ_{avg}
T = 5326.2
micron K. By using this modified Wien's Law for the black
body radiation from
the Sun at (effectively) 5760 K and for the infrared junk heat leaving
the
Earth’s CV at 254 K (resp.), we obtained 0.9247 microns and
20.97 microns
(resp.) for the 70% of the Sun’s radiation not affected by
the albedo and the
Earth’s far-infrared radiation (resp.), for the average
wavelength.

_{}_{}

Finally, we can write the First and Second Laws Combined with its radiation terms represented by photons as follows:

_{}

and

G_{e}
= _{}

where G stands for Gibbs. This leads to the incredibly terse statement:

_{} .
Eq. (I-12)

The
First and Second Laws Combined was obtained by multiplying the Second
Law by _{} and
subtracting
the product from the First Law.

Following De Nevers and Seader [10], we can make an interesting point with only a little more effort. Let

_{} ,

and

F_{e}
= _{} ,

so that we can write

_{} ,
Eq.
(I-13)

which gives _{}, the reversible
power required
to obtain the same result from this CV that is obtained by any process
with (i)
the same inputs and outputs, (ii) the same change of state inside the
control
volume, and (iii) the same heat terms transferring heat at the same
temperatures to the same thermal reservoirs – with the
exception of heat
reservoirs at the temperature of the environment _{}.
This may be
the maximum power or minimum power depending upon
circumstances. In either
case, it can be approached but never attained. It is the sum
of the
thermodynamic lost work and the net external work done by or done upon
the CV,
for any process – reversible or irreversible. If we
know (i) _{} the
change of state of the CV, (ii) the heat transfer rates to each thermal
reservoir (other than the environment) and their temperatures, and
(iii) the
net power produced or consumed by the CV, we can compute (a) the
thermodynamic
lost power, and (b) the reversible power necessary to achieve (i),
(ii), and
(iii), that is, the reversible power – or, for a batch
(non-continuous)
process, the reversible work, which represents the unattainable ideal
for that
process.

Suppose
we compare our (real) process with a reversible
process _{},
with (i) the same streams entering and leaving, (ii) the same internal
changes,
and (iii) the same heat transfer to the same thermal reservoirs other
than the
reservoirs at _{}.
Then, we see from Eq. I-12 that _{} where _{}; and,
therefore, _{} _{}. _{} .
Thus,
the lost work for Process is just the
additional heat that
must be dumped to the environment at _{} over
and above
that which must be dumped for Process _{}.

Let us
consider the first term on the left-hand side of Eq. I-13: _{} ,
where _{}.
It is the rate of accumulation of availability within the Earth and her
oceans
and her atmosphere and her biosphere, all of which are capable of
storing
availability. A particularly large storehouse of availability
is the
concentrated biomass represented by our coal, gas, and petroleum
deposits.
This took millions, if not billions, of years to accumulate; but,
during the
last two hundred years (particularly the last fifty years), we have
been
exhausting it at an alarming rate; therefore, it is virtually a
certainty that _{} is
negative.

The åf_{i}b_{i
}term would represent only the negligible amount of debris
that Earth
picks up as it travels through nearly empty space – a few
meteorites and some
space dust. Likewise the Sf_{e}b_{e}
term represents only the small amount of availability associated with
matter
lost to space, particularly availability associated with hydrogen high
in the
Earth's atmosphere. For all practical purposes this term can
be neglected too.

We have
a relatively small amount of work done *on* the
Earth by nearby gravitating objects
– primarily the moon. Most of this work comes from
the moon giving up some of
its potential energy with respect to the Earth by moving a few
centimeters away
from the Earth each year. (The reader will recall from
high-school physics
that the gravitational force exerted by the moon upon the Earth is
inversely
proportional to the square of the distance between the moon and the
Earth.)
The Earth, correspondingly, does very little work upon her
surroundings.
Therefore, we shall neglect the work terms.

With
current technology, we can tap geothermal energy at only a few
isolated places on Earth. We shouldn't count on this energy
anyway since it is
*not*
renewable, moreover we have
no idea what the effect might be of cooling the interior of the Earth
even
slightly. Probably, the interior of the Earth is a more
important scientific
frontier than is outer space. My personal disapproval of
space research, based
on the past and current destructiveness of man’s activities
on Earth, is
discussed in my essay “On Space Travel and
Research” in Vol. II of my collected
papers [8]. Since the heat from the Sun and the heat radiated
by the Earth (at
approximately 254 K) are accounted for by the radiation terms, we may
drop the
conduction terms as well as the convection terms shown in Eq. I-13.

Finally, the lost work term represents all of the irreversibilities occurring in all of the subsystems of which the CV is composed. This includes the falling rain, the burning of fuels and forests, the turbulence of the winds and waves, the industrial activity of man, in fact, nearly every act of man and Nature. Almost every action involves some irreversibility and the only reason I say “almost” is because I want to be safe. Perhaps the reader can think of a completely reversible process that actually occurs. I cannot. Equation I-11 can now be written as follows:

_{}.
Eq. (I-14)

This version of Eq. I-7 is useful for illustrating how the Sun increases the Earth's Helmholtz availability.

If we impose our condition of steady state with respect to Helmholtz availability, Eq. I-13 becomes

_{} .
Eq. (I-15)_{ }

We may
write _{} as
_{} since
we
showed earlier that _{} , then

_{} ,
Eq.
(I-16)

This
equation is restricted to quasi-equilibrium situations, simple
compressible substances, uniform state, and uniform flow by most
authors. De
Nevers and Seader [10], however, point out that the way the First and
Second
Laws Combined is set up here it applies to both reversible and
irreversible
processes. Other restrictions may apply. Let _{} be
constant value of
the entropy to number of photons ratio. _{}is, of course, a
positive
number, in a steady-state, adiabatic CV with negligible mass crossing
the
control surface – and (for the combined law) negligible
(gravitational) work
terms. Our estimate of _{}is given above.

*Also*,
it is the
rate at which negentropy enters the CV, which is where many analysts
would
stop. But, it is the availability budget that tells us how
much reversible
work can be done and that's what we would optimize to minimize
operating costs
– unfortunately with the trade-off of infinite capital
costs. We are using the
language of capitalism, but we are doing *Economics
As If People Mattered* [E. F.
Schumacher] and we are looking at the
items of legitimate interest to people – not at failed
abstractions. The
things we really need to live and the real costs to the environment of
providing
them are best measured in emergy units [Howard Odum]. (Emergy
is based upon
availability as discussed in Chapter 2.)

The availability of the photons associated with the incoherent infrared radiation constantly radiated from Earth in all directions into space is useless to Earthlings regardless of what space nuts may imagine. Hopefully, if they try to harness that low-grade energy, they shall be restrained appropriately. If the reader wishes, he can read Wayburn’s essay “On Space Travel and Research” [8]. We find the space scientists singularly selfish, inconsiderate of the needs of the rest of us, and really quite stupid. So much for the fabled rocket scientist and his latter-day cowboy myths. The myth of space research appeals to the romantic in all of us and to the credulity of the soft-minded. Soft-minded not necessarily weak-minded.

Since
Helmholtz availability is _{} , if the
Earth is in
steady-state with respect to energy and with respect to Helmholtz
availability
too, then She is in steady-state with respect to entropy.
After we compute the
rate of lost work (power), we can compute (approximately) the increase
in
entropy we can allow because of irreversibilities. And *everything*
is irreversible. For now it is
enough to recognize that the First and Second Laws Combined will reduce
to the
same equations we used to compute the lost power using the Second Law
provided _{},
but we already know that _{} It
would be pointless to
repeat the exercise, although the reader may do it if he wishes
(presumably
with the book closed).

It is
interesting to note that the lost power can be computed as the
work entering the Earth from a Carnot engine operating between the Sun
and _{} minus
the heat leaving the Earth in the First and Second Laws
Combined. Begin with

_{} ,

where

_{},

_{} =
_{} ,

and discard all the terms except the following:

_{} .

Now, we can simplify the equation, but that might obscure what is going on. Let us plug numbers into it as it is.

_{} .

Of course, what has happened is that we could have discharged heat at 254 K, but we were required to use the 288 K heat sink, which increased the lost power by about 13%.

The
average solar flux absorbed by the Earth’s surface
is _{} =
236 W / m^{2 },
which is the same as the average far-infrared flux emitted from the
Earth’s
surface or upper atmosphere , _{} = 236 W
/ m^{2}.

Since

_{} ,

_{} = _{}

I have no idea how useful or accurate such a grey body number might be. We include it for style only.

The solar
constant, _{} was
computed by Bowman as
follows: _{}

_{}_{} .

In none
of Wayburn’s eight (engineering) thermodynamics texts could
be
found a decent treatment of radiative heat transfer. Normally
it is not
mentioned. A ninth book, from the library, *Exergy
Analysis of Thermal, Chemical, and Metallurgical Processes*
by
Szargut et al. [11] devotes a few pages to radiative heat transfer, but
appears
to be a hopeless muddle. Most of the equations have the name
for the
right-hand side on the left-hand side; i.e., they are really no better
than
definitions and tautologies. Moreover, we am gradually
getting the idea that
every radiative process in the book must have a source or sink out of
or into
which heat is transferred by conduction. With all due respect
and a touch of
hyperbole, may we suggest to the authors of [11] that B is not the only
letter
in the Latin alphabet and that, when every variable is named B (for
availability) and different variables are distinguished by 6-point
subscripts,
the authors strain the patience of older readers
considerably. (Almost)
nothing makes one feel like an old man so much as reading with a
magnifying
glass.

Wayburn
just reviewed some articles on *thermodynamic*
*lost work*
and *exergy*,
which is sometimes called *availability*,
although it’s not exactly *our*
availability, and sometimes called *essergy*.
It seems that a minor war has
been raging between the supporters of rival techniques.
Apparently, though,
each methodology has its uses. We are employing the lost work
methodology in
this appendix; however, in Chapter 2, Wayburn employed exergy quite
profitably. The reader should be aware that, when
we use the expression
availability balance, we mean the use of Eq. I-7, which contains the
term for
the change in the value of the Helmholtz availability function within
the CV.
We are not referring to the availability that is defined to be the same
as
exergy. We apologize in advance for any confusion that arises
due to the
existence (in the literature) of two distinct concepts both referred to
as
availability. Most of this appendix had been written when
Wayburn found out
that exergy is sometimes called availability. We certainly
wish no one would
call it that. We shall call our availability the Gibbs
availability function
or the Helmholtz availability function except when those locutions give
the
sentence or the paragraph awkward metrical characteristics *and* no
ambiguity can possibly arise.

In this brief introduction to thermodynamics, we avoided power cycles other than the imaginary Carnot cycle. Also, in a course in a chemical engineering department, a large chunk of time will be devoted to vapor-liquid equilibrium. This is the most difficult scientific information to obtain when designing sensitive separation processes. (One of the authors has seen with his own eyes a book of such data produced from photocopies and not exceeding 300 pages by much if any that was for sale at the time for $2,500 and sales were satisfactory. Normally, liquid-liquid equilibrium is treated too but not in quite so much depth. Many thermodynamics textbooks analyze mechanical and chemical equipment from pumps to reactors. This is way too nuts-and-boltsy for us. Finally, thermodynamics is the key to chemical reaction equilibria and, normally, a chapter is devoted to chemical reactions. In mechanical engineering departments, flow through turbines and nozzles, including supersonic flow and shock waves, is studied. Theoretical chemists and physicists seem to pay more attention to the fundamental mathematical relations and special relations among the distinguished partial derivatives derived from them. I believe I made in clear that the Second Law can be studied from the viewpoint of impossible processes, namely, the Clausius Statement of the Second Law and the Kelvin-Plank Statement. We don’t bother with these at all; balance equations are so much more useful.