Exponential Curve 4
Arithmetic, Population and Energy, Part 4
For the love of the human
race.
Wednesday, November 19, 2014
Our Thesis
We are greatly indebted to Dr. Albert Allen Bartlett (1923-2013),
former emeritus professor of physics at the University of Colorado, Boulder.[1] These are Dr. Bartlett’s ideas, we are merely
reporting them. We have performed a
lengthy analysis of Dr. Bartlett’s “arithmetic” elsewhere.[2]
One cannot investigate either energy policy or energy theory
without a thorough understanding of the exponential curve.[3]
Arithmetic, Population and Energy, Part 4
Dr. Bartlett opens this part with a juicy quote from a very
important Texas oil man, Michel T. Halbouty
(1909-2004), “There is still as much oil to be found in the U. S. as has
ever been produced.”[4]
1.
What time is
it?
A. It’s 11:59, 1 minute before noon.
A. It’s 11:59, 1 minute before noon.
Coal
Dr. Bartlett immediately turns the conversation to coal with
this quote, “We’re sitting on half of the
world’s known supply of coal, enough for over 500 years.” A report to the Senate committee said this, “At
current levels of output and recovery, these American coal reserves can be
expected to last more than 500 years.”[5]
“There is one of the most dangerous statements in the
literature. It’s dangerous because it’s
true. It isn’t the truth that makes it
dangerous, the danger lies in the fact that people take the sentence apart:
they just say coal will last 500 years. They
forget the caveat with which the sentence started. Now, what were those opening words? “At current levels.” What does that mean? That means, if and only if we maintain zero
growth of coal production.”[6]
Dr. Bartlett’s discussion continues. The Demonstrated Coal Reserve Base is around:
Rb = 470 G-tons according to Dr. Bartlett’s charts. The Recoverable Reserve is around: Rr
= 240 G-tons (1991). The significance of
these two numbers is that we cannot practically reach over 50% of the total
coal with existing technical equipment and methods. Vastly improved technical equipment and
methods can nearly double the realistic coal supply.
As with oil, Dr. Bartlett’s data (ca 1970) are hopelessly
out of date. So we will turn to fresh
data studies in separate documents.
Extraction Time and Peak Production
Extraction time is the length of time we may expect a
resource to last. Peak Production is the
highest level that can be reached with a given model. The model that best fits reality is the
result of business and political decisions.
Science has little or no influence on such decisions, it merely seeks to
analyze and explain these decisions with clarity.
The Zero
Growth Model
If we are evaluating a zero growth model the extraction time
is simply the amount of the resource remaining, A, divided by its
present rate of consumption, y0. Te ≡ A / y0.
The peak production rate is y0,
since the present rate of production is not growing.
This produces a rectangular graph, which is Te
wide and y0 tall.
The area of the rectangle A / y0 x y0
is equal to the amount of the resource remaining, A.
Te at zero growth is the figure
most commonly reported in public records, which conveniently ignore the fact
that production rates can increase, or decrease, as well as remain constant. Contemporary business and political attitudes
mandate that production rates will increase.
The Positive
Growth Model
If we are evaluating a positive growth model the extraction
time is more complicated to calculate because we must account for the changing production
rate. We reason that the amount of the
resource remaining, A, doesn’t change from one model to another,
so the area under the exponential curve must also be A. We need calculus to calculate this area, and
the new Te under growth conditions. Here is the result.
Te
= 1 / ln (b) * ln [ ln (b) *
A / y0 + 1 ]
We may approximate this time by remembering from part 1 that
b = 1 + r, and that for small r (less than 0.10), ln
(1 + r) ≈
r.
Te
≈ 1 / r * ln [ r * A / y0 + 1 ]
The peak production rate is now:
y
= y0 * bTe
In this model the peak production rate is immediately
followed by a catastrophic and instantaneous crash to zero. It is the perfect “arithmetic” model for the
Titanic. Full speed ahead when you run
into an immovable obstacle, killing nearly everyone on board.
The Negative
Growth Model
If we are evaluating the negative growth model, specifically
known as Sustained Availability, we may use the basic equation y(t) = y0 * e-kt.
k
= 1 / Te
Theoretically a declining rate of k
will make a resource last forever. In
actual practice we may expect somewhat less than this. Instead of doubling time we calculate
half-life: t = ln(2) / k,
which is our old friend the Rule of Seventy.
As a practical rule of thumb we will be doing very well if we are able
to extend the life of a resource to five times the half-life.
The peak production rate
is y0
since the rate is declining steadily.
So, for example, if we have a 50 years
supply of a resource; we may attempt to conserve that resource indefinitely by
reducing consumption by 1 / 50, or 2% annually.
This will result in a half-life of approximately 70 / 2, or 35
years. We will be able to extend the
life of that resource from 50 to 175 years.
If reductions greater than k are practical, we can make a resource last
even longer. We will return to this
topic under Sustainability.
The Gaussian
Models
We will address the Gaussian models in
part 5.
Utopian Mythology
Clearly, “We [do not] have coal coming out of our ears.”[7] To achieve a mythical 500 year reserve time
we have to throttle back to zero growth and develop technology that will use
100% of the Demonstrated Coal Reserve Base.
Using the 2008 data, we have at best 223 years of coal left, and very
possibly as little as 30 years left.
Since the 1991, by 2008, a mere seventeen years, we have dropped from
between 242/475 years of coal to 223 years of coal.
When gasoline and natural gas fail, their energy demand will
be thrust upon coal. Houses and
factories will install coal burning furnaces as were common, even in the
1940’s. Automobiles and trains will be
powered by coal generated electricity, or by external combustion steam engines,
ala 1850. The demand on coal will
roughly triple, quadruple, or even quintuple.
The reserve of 223 years will evaporate before our eyes to 223/3 less
than 75 years; perhaps as little as 223/5, equals 45 years. Under current economic plans this change will
not be exponential, it will be sudden and catastrophic.
Wake Up and Smell The Reality
Don’t say it won’t happen.[8] When the oil crisis struck in the 1970’s,
there was no warning. Overnight, the
price of gasoline doubled. Our leaders
have repeatedly employed a policy that it is best to crash the system. When the coal is gone we will turn to our
forests. Our children and grandchildren
and great-grandchildren will be forced to cope with this disaster.
“In the
1970’s there was great national concern about energy, but this concern
disappeared in the 1980’s. The concerns
about energy in the 1970’s prompted experts, journalists and scientists to
assure the American people that there was no reason to be concerned.”[9]
There is no question that additional reserves of coal can be
found. However, there is little
incentive to pursue such reserves as long as gasoline and natural gas are the
fuels of choice. Nevertheless, it is
likely that these new reserves of coal will be found under forests. This means that we probably cannot get the
coal without destroying the forests.
The continued devastation of forests will further deplete
our life-giving supply of oxygen.
Moreover, in certain cases deep soil carbon provides the nutrients for
vigorous and long sustained plant growth.[10] It seems to me that if we are going to mine
carbon out of the earth, we ought to be looking for judicious ways to put
carbon back into the earth. Terra-preta
may well be a critical and essential ingredient in revitalizing our infertile
and chemically dependent American soils, the answer to world hunger, and the
solution to the oxygen depletion problem.
It may also enable us to grow healthier, more robust forests.
“Don't believe any
prediction of the life expectancy of a non-renewable resource until you have
confirmed the prediction by repeating the calculation [yourself].
“Corollary, The more
optimistic the prediction, the greater is the probability that it’s based on
faulty arithmetic or on no arithmetic at all.”9
Our Conclusion
Once again Dr. Bartlett’s “arithmetic” speaks with deadly
accuracy. We have examined his theorems
with rigor, and proved them using standard mathematical methods. We have lived with a system of utopian false
hope: but at what cost? Everybody likes
positive attitudes and hopefulness. The
reality is that this, Titanic like, will cost many lives.
We also verified two very handy formulas that allow us to
reevaluate the remaining time under any kind or percent of growth model.
Te
= 1 / ln (b) * ln [ ln (b) *
A / y0 + 1 ]
Te
≈ 1 / r * ln [ r * A / y0 + 1 ]
By putting this result for Te back
into the basic exponential equation we calculate a maximum limit for peak
production. Peak production may be lower
than this number, but it can only be made higher by increasing the growth rate
or changing the data.
y (t) = a * bTe
[1]
http://en.wikipedia.org/wiki/Albert_Allen_Bartlett
[2] http://swantec-ep.blogspot.com/2014/11/energy-policy-analysis-4-ra.html
[3] http://www.albartlett.org/presentations/arithmetic_population_energy_video1.html
[4]
Halbouty, Michel, Time Magazine, October 29, 1990, Halbouty
(1909-2004), a Texas wildcatter is responding to the question, “But haven’t many of our bigger fields been drilled
nearly dry?” http://en.wikipedia.org/wiki/Michel_T._Halbouty
[5]
American Electric Power Company: This quote is possibly sourced in a report to the committee on Interior and Insular Affairs of
the United States Senate.
[6]
This is Bartlett’s profound and telling analysis. It deserves our utmost attention. In our “sound-bite” world we, typically, do
not pay attention to what is actually said, we frequently drop conditional clauses;
especially, when they are placed at the beginning of sentences and not
emphasized. We suspect that such a
phenomenon is being deliberately used by expert communicators to manipulate
readers; if conditional clauses are correctly positioned, they will almost
always be overlooked. Thus the public is
played. School reading programs often
fail to teach the need to dissect sentences and analyze them. Even with training, the rush of information
is so great that critical information slips by us. We desperately need to understand Dr.
Bartlett’s method here, and learn to employ it ourselves. The words, “if
and only if,” are a standard mathematical expression indicating that the truth
of the statement works both ways: if we have enough coal for over 500 years, we
must be at zero growth conditions.
Mathematical logic statements are not usually true in both directions.
[7]
Falkie, Thomas, Energy Fuels Corporation, June, 1976. http://www.aimehq.org/programs/award/bio/thomas-v-falkie
[8] I
started out in the steel business. We
were still using iron ore from the Mesabi Range. Mining continues there, but for all intents
and purposes, the once great steelmaking operations of the United States are
gone. Nobody was watching. Today, most American steel is made from
melted scrap. When I was in engineering
school in Wyoming (1965), CF&I was going strong as the tenth largest steel
producer in the United States and biggest west of the Mississippi.
[9]
Dr. Bartlett
[10] http://en.wikipedia.org/wiki/Terra_preta,
and http://news.nationalgeographic.com/news/2008/11/081119-lost-cities-amazon_2.html
[11] If you have been blessed or helped by any of these meditations,
please repost, share, or use any of them as you wish. No rights are reserved. They are designed and intended for your free
participation. They were freely
received, and are freely given. No other
permission is required for their use.
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