**Overall Availability Analysis of
Earth, Part 1**

Table of Contents to Part 1

__Why
Bother? The Technical Usefulness of the Calculation is
Minimal.__

__Computing
the Lost Work or Maximum Reversible Work for Earth’s Control
Volume__

Prof. Bowman may wish to distance himself from some of the definitions, logic, conclusions, and other statements rendered in this appendix. This section is where that should be done – unless, of course, he objects, also, to that arrangement, in which case whatever must be done will be done. Also, Dave, at some time you should come to realize how politically radical this book really is. You may find that you are (i) too conservative or (ii) justifiably concerned about your association with an extremely radical book.

In
addition, Dave and I share a difference between his view that a
photon gas is *not*
a simple
compressible substance and that it does *not*
do injection work upon the control volume and my view, which I have not
expressed previously in the following manner: It seems to me
that radiation
fits into the framework of thermodynamics a little differently than do
other
manifestations. It is *as though*
a photon carries the temperature of the Sun into the control volume
until it
strikes an ion, atom, or molecule, at which time it *does*
transfer momentum to that particle and, therefore, it *does* do
injection work to allow something *within*
the control volume, like a tree
leaf or a generic solar collector, to be an agent for the performance
of useful
work subsequently. We both agree that nothing happens until
the photon strikes
something. Enthalpy is defined to be the internal energy plus
the injection
work, therefore we both agree that it is the enthalpy that should be
computed.

We wish to make an availability balance over the entire Earth including its atmosphere but excluding its core. The exterior component of the control surface (CS) is taken to be a sphere 100 miles from the average level of the sea at the equator, which describes it more accurately than is justified by the precision of our numerical techniques. The interior of the Earth is excluded from our control volume, since, for all practical purposes, we cannot get to it unless it chooses to come to us. The control volume (CV) for this analysis is shown as the shaded region in Fig. I-5 on the next page. The thin shell (of Earth) outside the core is supposed to contain every agency of useful work - from the roots of trees to heat engines at the bottom of the deepest mine or well.

**Figure I-**1.
Control volume for the
calculation of availability accumulation on Earth due to sunlight

Imagine for a moment the Earth divided into two hemispheres by a plane parallel to the plane wave front of the incoming radiation from the Sun. At every instant in time, the hemisphere facing the Sun is accepting radiation at 6000K. In our thought experiment this will be used to do reversible work inside the control volume, therefore an imaginary heat engine will reject heat to the surroundings at 300K, which we may take to be the Earth and such thermal reservoirs as we may take advantage of such as oceans, lakes, and rivers, which, in this analysis, are imagined not to contribute further to the reversible work. Simultaneously and roughly at the same instantaneous rate that the hot hemisphere is receiving energy from the Sun, the dark hemisphere is radiating energy from the 300K thermal reservoirs, represented by the Earth’s core in Fig. I-5, to deep space, the effective temperature of which is approximately 4K. It is difficult to imagine how a reversible heat engine could be constructed to accept heat at 300K and reject heat at 4K; nevertheless, we should consider the availability associated with this radiant heat transfer.

Since the Earth rotates, the entire control volume of Fig. I-5 should be considered twice: first, with the temperature of the surroundings, namely, the thermal reservoirs, equal to 300K, and, second, with the temperature of the surroundings, namely, deep space, equal to 4K. The two calculations are imagined to be superposed upon one another and to take place simultaneously. In the sequel we shall neglect the second calculation.

We already know that the Helmholtz availability budget provided by Mother Nature is virtually astronomical. Since this might encourage people to embrace “The Myth of the Infinite Earth”, why should we go to so much trouble to provide an estimate of a number that, quite frankly, is useless for further computation or analysis?

The possibility of achieving a sustainable, one-kilowatt-per-capita, soft-energy society can be estimated with greater confidence by comparing the 10 TW that we hope to harvest from biomass to the world photosynthesis rate of about 100 terawatts [4] than by comparing human needs to the Earth’s ~150,000-terawatt total power throughput because of the numerically intractable disparity in scale between human needs and the total figure and the difficulty of computing Mother Nature’s availability budget for running the weather and other natural phenomena that make the Earth habitable. (For reasons known only to Herself, Mother Nature seems committed to making the Earth a home for living things.)

If we could harvest 10% of the 100 TW stored (continuously) in the biosphere by photosynthesis without doing major ecological damage in a world such that the population had stabilized at ten billion ‘souls’ – to be as optimistic as we can be, we could enjoy a one-kilowatt-per-capita economy that (for intelligent creatures) should be sufficient to live well in harmony with Nature. One kilowatt per capita is more than thrice Bangladeshi consumption.

The
lost power term in the Combined First and Second Law Availability
Balance around Earth represents the maximum rate at which work can be
done
inside the Earth’s control volume. Despite the lack
of technological
usefulness, we have three reasons to compute an estimate of the lost
power term
_{} in
Eq. I-7, the First and Second Laws Combined:

_{} =_{} .

Eq. (I-7)

First
and foremost, we said we would put the Combined First and Second Laws
to use
immediately and this is a grand use indeed. So, we have done
this to show that
it can be done. Second, when _{}, the value of _{} represents a* hard limit on human activity*
in a
sustainable society, which proves the finiteness of such a
limit. And, third,
good scholars do not suppress relevant information uncovered during
their
research. Every true scientist would prefer to report that
his theory is false
or to report “no results” than to report an
instance of an experiment that
proceeded according to the predictions of his theory. A true
scientist should
be his own most severe critic. Frequently, we learn more from
falsifications
than from corroborations of our conjectures.

For
example, we know that the upper bound we are computing is
astronomical and is likely to persuade those who are vulnerable to
unjustified
optimism that human society has *more*
than enough sustainable high-grade
‘energy’. These are people who, like
ourselves,
have no idea of the recovery costs (both direct and indirect) for *any* type
of sustainable energy – firewood
excepted. Moreover, they have no idea how much energy is
likely to be consumed
by turbulence in the air of the atmosphere and the water of the
seas. They
have not studied the energetics or the effects upon various living
things of
the shifting of the tectonic plates, volcanism, forest fires, storms,
earthquakes, El Niño, the Gulf Stream, normal circulation of
oxygen, nitrogen,
carbon dioxide, water, and other elements that consume availability on
a
periodic basis, and catastrophic weather in the sense of sudden
–discontinuous
– changes. In short, they don’t know if
150,000 terawatts
(150,000,000,000,000,000 watts) is a large amount of availability or a
small
amount of availability on a global basis – anymore than we
do. We, at least,
are trying to assess the likelihood of scarcity or abundance beginning
with
scientific methods, which we shall employ until they fail and must be
abandoned
– probably in favor of prudence.

Dan Wilkins, an astronomer at the University of Nebraska, introduced Wayburn to Dave Bowman, a physicist at Georgetown College, who has provided nearly every theoretical idea worth considering in this attempt to get an availability balance around the entire Earth. This section is dedicated to Dan Wilkins, a gentleman and a scholar. Others whose help should be acknowledged are Jim Green, John Mallinckrodt, Leigh Palmer, and Brian Whatcott. These are Internet friends. Gentlemen, we thank you all.

In
the Bowman calculation of lost work inside the Earth’s CV at
steady state, a
number of standard constants, such as the radius of the Earth, were
employed.
Wayburn has changed these to the data that appear in the references on
his
shelves. This has almost no effect on the answers
obtained. Also, Dr. Wayburn
derived all of Prof. Bowman’s equations and computed
essentially the same
values as did he. The creative leaps, however, were due to
Bowman. Bowman
calculated the solar constant by applying the ratio of the
Sun’s surface area
to the surface area of a sphere centered at the Sun with the Earth on
its
surface, i.e., by the *inverse
square law*.
Wayburn took his solar constant from Häfele [4]. We
shall use Bowman’s
calculated K_{solar}
to harmonize with the numbers from which he calculated it. We
shall employ
Wayburn’s standard data (except for K_{solar})
without mentioning the minor differences.

a , the
Earth’s albedo, the fraction of power, P^{a}_{sun,in},
that is immediately
reflected and, for practical purposes, has no effect on our
CV. We computed
the value a = 0.3 from Fig. 2-9 in Chapter 2, which was taken from
Häfele [4].

_{}

h
, Plank’s constant, 6. 626 ∙ 10^{-34}
J sec [5, p.
xviii].

k
, Boltzmann’s constant, 1. 380 ∙ 10^{-23} J
/ K [5, p. xviii].

r = T_{CV}
/ T_{sun}
, the ratio of the effective
temperature of the far-infrared radiation from the Earth to the
effective
temperature of the (nearly) blackbody radiation from the Sun.

r_{CV},
radius of Earth at equator plus 100 miles =
6,539,000
meters [6].

r_{earth},
radius of Earth at equator = 6,378,000 meters [6 (back cover)].

r_{earth,avg},
average of equatorial and polar radii = 6,367,500 meters [6
(back cover)].

r_{sun},
radius of Sun = _{} meters [7, p. 116].

A_{proj,e},
projected area of Earth as seen by plane wave from Sun
= 127.4 ∙ 10^{12}
m^{2}.

A_{proj,cv},
projected area of Earth’s CV as seen by plane wave from
Sun = 134.3 ∙
10^{12}
m^{2}.

A_{sphere,e},
area of Earth’s spherical surface = 509.6 ∙ 10^{12
}m^{2}.

A_{sphere,cv},
area of outer surface of our CV = 537.3 ∙ 10^{12}
m^{2}.

D ,
average distance between the Sun and the Earth = 1497 ∙ 10^{8}
meters [7, p. 116].

K_{solar}
or K_{s},
solar constant (calculated by Bowman as Stephan-Boltzmann flux times
inverse
square ratio, ψ) implies that K_{s}
= 1350.7 W/m^{2},
but looked up by Wayburn in
Häfele [4, p. 151], which offers 1353 W/m^{2}.
In this computation we use
Bowman’s number.

R_{b}[T],
radiant energy flux or rate of radiant energy flow per unit area from a
blackbody. This is determined by the Stephan-Boltzmann
Equation:

_{}

T_{sun},
effective temperature of Sun’s surface as a
blackbody radiator, 5760 K [4,
p. 151] .

T_{earth,surf},
average temperature at surface of Earth, 288 K .

T_{cv,IR
rad} or, to save typing, T_{cv},
effective black body
temperature for radiation of far-infrared (IR) junk heat from
Earth’s CV, 254
K was computed from energy balance.

T_{0},
the thermodynamic ‘temperature of the
surroundings’,
which we take equal to T_{earth,surf}
= 288 K.

ε_{cv},
emissivity of the Earth’s control surface, one minus the
albedo

ε_{IR},
average emissivity of the Earth’s outgoing
far-infrared (far-IR) waste flux. The Earth is well
approximated as a
blackbody regarding its outgoing IR flux, so we use ε_{IR}
= 1.0 .

ε_{sun
}= 1.0 as the Sun is normally treated as a blackbody whose
effective
temperature is 5760 K.

η, Carnot efficiency

_{}

ξ_{inv.
sq.}, inverse square law for
surface of Earth to top of
atmosphere

_{}

σ, the Stephan-Boltzmann constant

_{}

where

_{}

_{}.

Notice
that the 1/4 is already included in this variable. _{} .

Ψ_{inv.sq.},
inverse square law for surface of Sun to Earth

_{}

Φ_{emit,earth} =
ε_{IR}∙σT_{cv}^{4}
= 236 W/m^{2},
the average far-infrared flux
emitted from the Earth’s upper atmosphere. (The
straightforward calculation
from an Earth’s surface at 288 K gives a
disappointing 390.1 W/m^{2}
. Even applying the
inverse square law to the datum reduces it to 369.9 W/m^{2}
only. It is fair to say, though,
that much of the radiation emitted from the Earth does *not* come
from the surface.)

Φ_{abs,earth} =
¼∙(1-a) ∙K_{solar}
= 236 W/m^{2},
the average solar
flux crossing the Earth’s outer control surface.
(The use of Häfele’s [4]
choice of _{} =
1353 W/m^{2 }rounds
off to _{} since
the data used to compute 254 K is that which gives a solar constant of
1301.7
W/m^{2},
which
is the value we shall use throughout this paper – making
changes wherever
necessary.)

To
write the Combined First and Second Law over the Earth’s CV
showing
the radiation terms, the rates _{} (resp.) in photons per
second
at which photons enter and leave (resp.) the CV could be computed and
multiplied by the corresponding average value of the ratio of the mean
total
enthalpy (in joules) divided by the mean total number of photons and
the
constant value of the ratio of mean total entropy (in joules per
Kelvin)
divided by the mean total number of photons. Photons do not
have mass *per
se*, but they do carry pressure, *p*,
internal energy, *u*,
enthalpy, *h*,
and entropy, *s*.
Also, photons have a Gibbs availability function,* b*, which
would be zero for a photon gas whose (constant)
temperature happened to be _{}.
That is, in
the case of T = _{} ,
the Gibbs availability
function equals the Gibbs free energy, which is zero for a photon
gas. Thus,
we might do well to remember that H = TS for
photons. Alternatively, we can
compute the thermal power in and out, the enthalpies, and the entropies
for the
entire ensemble in one piece. We do both.

Many
readers may be surprised to learn that a photon gas is a simple
compressible substance that behaves like an *ideal
or perfect gas*. As in ideal gases,
photons do not interact with
each other. They interact with atoms, molecules, and other
objects, but not with
each other [5]. A photon gas, as an example of what is known
in statistical
mechanics as a partially degenerate Bose-Einstein gas, does not have an
equation of state like p = _{}(T,V) with two independent
variables. But rather, p = _{}, which is not a legitimate
equation of state since pressure is a function of one variable
only. The
densities for every thermodynamic property can be written as functions
solely
of T. For example, the energy density of a photon gas in
equilibrium with the
walls of a black body cavity that contains it is _{} = _{} _{} = _{} _{} , where _{}is a special
symbol used (in this appendix only) to notify the reader that the
object within
the primary parentheses is a density.

If j(U(T)) is energy flux from the photosphere of the Sun to the Earth, where

_{}

one can write down a generic formula for the total influx rate of Property X to the Earth’s CV in terms of the flux of X leaving the photosphere of the Sun or the density of X in the photosphere (of the Sun):

_{}

This
is the solar constant, K_{solar}
or K_{s}
= 1350.7 W/m2 calculated by Bowman. It is the
number we use in these calculations.

_{}

[The identifier j is not expected
to indicate a flux
outside of this appendix; however, some of us use it in e-mail because
of its
typographical feasibility. For example the flux of entropy
from the Earth
might be written j_e (S, cv) = 4/3_{*}sigma_{*}T(cv)^3,
where
the under-bar ( _ ) is used to indicate subscript and the caret ( ^ )
indicates
exponents (as in BASIC.) This is *not*
quite standard notation. Perhaps, e-mail typography will
become barely
adequate soon.]

The
Stephan-Boltzmann equation permits us to compute the flux of
photons irradiated from what is known as a black body as a function of
the
temperature alone. Without going into details, we shall
present this
fundamental and important equation in a primitive form that employs a
constant
called _{} (simply
*a*
in Yourgrau et al. [7], however
we have used *a*
for albedo) but
from which the Stephan-Boltzmann equation, in its normal form, can be
derived
easily. _{} is
the energy flux or rate of radiant energy flow per unit area from a
black body
– hence the subscript *b*.
A black
body is a perfect emitter as well as a perfect radiator. _{} is a function only
of the (equilibrium) temperature of the radiating black body.
It is easy to
show that

_{} Eq.
(I-8)

where

_{}

*k*
is Boltzmann’s constant, *h* is
Plank’s constant, and *c* is the
speed of light. Far from a
fundamental constant of the universe, *k*
is precisely the ideal gas constant divided by Avogadro’s
number. Boltzmann’s
constant is _{}. Plank’s
constant is _{} . In the
cavity of a blackbody at 5760 K = T_{sun},
which is surprisingly well-approximated by the
photosphere of the Sun, we have for the photon density

_{}

and for the photon flux

_{}

where

_{}

and

_{}

In a
little ‘disquisition’, distributed over a physics
‘list server’,
Dr. Bowman has shown that most of the so-called ‘fundamental
constants of the
universe’ are no better than conversion factors. If
energy were measured in
inverse seconds, Plank’s constant would be one –
with no units! If energy were
measured in Kelvin, entropy would be dimensionless, which, as an
accounting for
the *amount*
of information
deficiency, it should be! The speed of light in vacuum, c, is
3.0∙10^{8}
meters /sec . It
is really a conversion factor between time and length, both of which
should be
called *interval*
from special
relativity. If we measured time in years and distance in
light-years, the
speed of light would be one and length could be measured in years,
which might
be slightly inconvenient in nano-technology. Finally, the
Stephan-Boltzmann
constant is 5.6697∙10^{-8} J∙m^{-2}∙sec^{-1}∙K^{-4}.
Equation I-8 is a very important equation; however, it applies only to
photons
in equilibrium with a black body from which they are radiating, e.g.,
the inner
walls of a blackbody cavity.

The literature provides us with many useful formulas for photon gases. The difficulty comes in deciding if the photons are in equilibrium with their surroundings in any useful way. In the photosphere surrounding the Sun, the photons behave somewhat as photons in a blackbody cavity in equilibrium with the walls of the cavity.

[**Note.**
But, even though the photosphere is
not in equilibrium, its steady-state behavior allows it to act as if it
were in
equilibrium (except for the turbulence and convective effects induced
in the
unstable gas because the temperature is greater below –
nearer the center of
the Sun – than above). Even though the
photosphere’s turbulent gases are not
equilibrated, the electro-magnetic radiation is much better behaved and
*acts*
as if it *were*
in equilibrium. – Dave Bowman (9-3-97)]

We shall treat these photons as blackbody radiation at an effective temperature of 5760 Kelvin. As we shall see, photons obey uniform and simple laws when they travel in rays to the Earth, therefore, they might just as well be in equilibrium with the Sun. The formulas we use give the correct values for thermodynamic properties even though the photons are not truly in equilibrium. The photons behave as though they were in equilibrium, which allows these calculations to be made.

If the photons were emitted with a given S/N ratio in a state of (effective) equilibrium, then, as they propagate through space their entropy does not change (even though their propagation through space is no longer an equilibrium situation). The photons propagate adiabatically through space. The reason for this is that, for the energy, entropy, and number densities, we have:

_{}

_{}

_{}

using the above formulas relating ρ(X,space) to ρ(X,sun); and

_{}

the usual equilibrium as employed in electromagnetism, thermodynamics, and statistical mechanics; and, therefore,

_{}

where T_{entering
CV} = T_{sun}.

Similarly, for the photon number we have

_{}

and, therefore,

_{}

Let

_{}

Then, let

_{}

Notice
that the ¼ is already included in this variable φ,
which is equal to
0.0541∙10^{-4}.
The density of U, S, and N might be represented as ρ(X,space) =
φ
∙ ρ(X,photosphere), which transforms densities in the
photosphere of
the Sun to space – but, especially, to the Earth’s
CV. This simple relation is
supposed to apply to X = U, S, and N. Presumably, it would apply to H,
and the
potentials, whether one calls them F and G, or A and B.

The constant value of the ratio of the entropy of any ensemble of photons divided by the number of photons is obtained as follows: We use

_{}

the regular formula for getting S from U. This is valid in space, too. Also, for the photon number, we get

_{};

and, for space, we get

_{} .

Thus,

_{}

We see
that the equilibrium relationships among U, S, and N are
preserved for their corresponding fluxes in space because the density
for each
one gets mapped from its equilibrium value in the photosphere to its
outer
space version (out at the Earth's orbit) according to the *same*
mapping ρ(X,space) = φ_{inv.sq.}
∙
ρ(X,photosphere) or ρ(X,space) = 0.0541∙10^{-4} ∙
ρ(X,photosphere). So we
end up with the result that S/N = 3.602_{}k for both the
quasi-equilibrium situation in the photosphere *and*
for the nonequilibrium situation of solar fluxes in outer space.

Given
the internal energy, U, the formulas for other properties of
photons are easy to write in terms of U. For example, H = _{}, _{}, and the
Helmholtz availability function A = U - T_{o}S.
Since, for natural reasons,
the CV can be assumed to be in energy steady state – global
warming
notwithstanding; i.e., _{} = 0; and, since T_{o}
is taken to be constant
in this rough calculation; and, further, since the assumption
upon which this
exercise is based is a steady-state Earth with respect to Helmholtz
availability; i.e., _{}_{} - _{} = 0: we may conclude
that the Earth’s CV is in steady state with respect to
entropy as well, that
is, _{}.

The value of T_{o},
the temperature we should use
for the surroundings of the entire Earth, is intended to be the lowest
temperature ‘infinite’ heat sink available, usually
the atmosphere or a large
body of water, whichever we think we can get away with
polluting. The heat
sink is infinite in the sense that any amount of waste heat we dump
into it is
too small to change its temperature measurably. Our
choice of T_{o}_{ }corresponds
to the nominal
temperature of the Earth’s surface, namely, 288 K.
We are not likely to be
able to do better than that. If one of these space fanatics
suggests that we
build a heat engine that exhausts to the background temperature 3.7 K
of deep
space, we should do our best to discourage him for all the reasons
mentioned in
Wayburn’s essay on space [8].

The
industrial nations of the world have not even begun to consider the
possibility that we ought to stop deficit spending of availability over
every
other priority. I suppose, then, that a steady-state Earth
with respect to
availability is a theoretical idealism. Nevertheless, let us
impose the
condition _{} as
part of our demand for a sustainable society. The possibility
of running an
availability *surplus*
is
unthinkable just now. Nevertheless, under these restrictions
we shall get a
very rough idea of just how much availability we can spend without
running an
availability deficit.

Let us discuss the First Law and the meaning of some of the terms in it for photons. We shall compute the rate of lost work using a First Law energy balance in connection with additional Second Law concepts.

First Law:

_{}

where the rate of change of internal energy of the Earth is

_{}

the mass (in kilograms) times the
(average) internal
energy per kilogram – not per cubic meter. Of the
other variables, only the
mass flow rate, f; the heat transfer rate, R; the rate at which work
crosses
the control surface, P; and the photon rate, E, are
nonstandard. Therefore, if
the quantity of interest is denoted X, the rate at which it
enters the Earth’s
CV will be R_{i}(X)
and the rate at which it leaves will be R_{e}(X)
.

In our
problem, we shall have steady state with respect to energy
within the control volume, which gets rid of the left-hand side _{} . Neither
do we have to consider material flowing into and out of our CV,
therefore the f_{x}h_{x}
can be neglected. Although
radiation is frequently treated as a form of heat transfer (radiant
heat
transfer), we shall not have to treat it as having anything to do with
heat
within our First Law setting, so the Rs go too. We are left
with

_{}

where

_{}

Therefore,

_{}

Before
we proceed further, we must take a careful look at the various
categories of thermal flux that contribute to the energy balance around
the
Earth. Of course, we know that energy is not a substance;
and, therefore, *thermal
flux* is an abuse of language, but
we encourage the reader to take a tolerant view of our cavalier use of
metaphor. Also, we shall compute the rate of entry and egress
of thermal
energy from our control volume. This will require the
calculation of the
effective temperature of the far-infrared ‘junk
heat’ leaving the Earth.

The
solar constant, K_{solar},
is about 1350.7 W/m^{2 }.
That means that where the straight line joining the center of the Sun
to the
center of the Earth crosses the outermost layer of the Earth's
atmosphere, 100
miles above sea level, say (since we wish to calculate an *upper*
bound on _{}), the mean energy
flux due to radiation from the Sun is about 1350.7 Joules per second
per square
meter. The solar constant isn't quite constant, but we shall
use it for very
rough calculations; so, we may safely ignore the variations.
The atmosphere is
a very thin shell surrounding the Earth and whatever we take to be the
outermost layer has a radius less than 100 miles (160,934 meters)
greater than
the (equatorial) radius of the Earth, which we shall take to be
6,378,000
meters. Further, we shall assume that the sunlight strikes
the Earth in a
plane wave. Actually, the Sun is more like a point source in
the sky
subtending just over a half of a degree, but the calculation of the
total
energy entering the Earth's system is simpler if we assume all of the
photons
are traveling in paths parallel to the aforementioned line connecting
the
centers of the two bodies, which gives a value for the total power from
the Sun
entering the Earth's system that is slightly high. (The
Earth's system is all
of the mass contained within a large (concentric) sphere just beyond
the
furthest reaches of the Earth's atmosphere and outside the smaller
(concentric)
sphere representing the depth beneath the Earth's surface that is, for
all
practical purposes, unreachable. It is the CV depicted
symbolically in Fig.
I-5 on p. 144.) The projected area of Earth, then, is

_{}

Thus, the power input from the Sun may be estimated as follows:

_{}

in
excellent agreement with the 178,000 TW that Häfele [4]
obtained before
correcting for the Earth’s albedo, *a*,
the fraction of power that is immediately reflected and, for all
practical
purposes, has no effect upon our CV. We computed the value
0.3 from Fig. 2-9
in Chapter 2, which was taken from Häfele [4]. We
could have done a fancy
calculation using integral calculus to account for the actual spherical
nature
of the wave fronts of the Sun’s radiation, but it wouldn't
have affected our
answer by more than a few percent.

Let P_{i}^{+}
(= P_{sun})
be the total
radiation from the Sun that strikes the Earth’s CV per unit
time. It is
composed of two parts. The first part is

_{}

the portion that is reflected from the Earth’s CV or, perhaps, enters and leaves without having any effect upon the control volume according to a fundamental assumption of this study. It cancels out of our balance equations. The remainder of the solar power must be analyzed: