ES 181 (Fall 2005)
Problem Set #1
Due: 9/30
This problem set is designed to provide you with a few very simple exercises to practice interconversions among units and to familiarize you with some of the big picture issues and some of the basic
concepts of thermodynamics. Conversion factors are provided in Table A.1 of the text (p. 653) and in
the handouts, "An Introduction to Energy and Thermodynamics" and "Fundamental Physical
Constants".
Please review the “homework policy” section of the course syllabus before starting this assignment.
1. Energy Reserves
Make sure to state explicitly all assumptions that you make in coming up with the estimates below.
a) The United States currently has an energy consumption rate of approximately 1.0 x 1017 Btu/yr. This
consumption is fueled from a number of sources: most importantly, petroleum (40%), coal (23%), and
gas (23%), plus other minor sources.
At the end of 2004, proven reserves of these energy resources within the United States are
estimated to be:
oil: 22 x 109 barrels; coal: 2.5 x 1011 tons; natural gas: 1892 x 1012cubic feet.
(1 standard oil barrel = 42 US gallons)
The energy densities of these materials are as follows:
oil: 5.2 x 106 Btu/barrel; coal: 30 x 106 Btu/ton; natural gas: 103 Btu/cubic foot.
Assuming a constant rate of consumption, estimate how long these reserves would last if the US
were no longer able to import any of these energy sources: which would be the first to run out,
and when? Assuming that once this happened we made up the shortfall by increasing our
consumption of the other sources (discuss the feasibility of this proposition), when would we
“run totally dry”?
b) In 1956, Hubbert fit a bell-shaped curve to US oil production vs. year and predicted that US oil
production would peak in the early 1970s; his predictions were largely ignored until US oil production
peaked in 1970. World oil production since 1910 is provided in a file you can download from the
course web site at http://courses.fas.harvard.edu/~es181/handouts/world-oil-production-digitized.txt
Fit a Gaussian curve a exp ⎡ − ( t − t0 ) / σ2 ⎤ to these data and predict roughly when the peak in world oil
⎣
⎦
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production occurs. Also predict the total recoverable oil (the integral under the curve) and the fraction
of this still underground. Compare your results to recent predictions available in the literature. To see
how predictions vary with time, repeat your analysis for just the 1910-1976 data and just the 1910-1990
data. Note: more sophisticated analyses can incorporate more information, such as a fitted curve
constrained to have a pre-determined integral based on other information, such as the estimated total
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ES 181 (Fall 2005)
Problem Set #1
Due: 9/30
recoverable oil content of the earth. Your analysis is more simplistic but should get the point across.
You should be able to use this in Mathematica (use the FindFit command) or in a crude way in Excel by
taking the natural logarithm of the production rates and fitting a parabola using the "Add Trendline"
command. You will get different results each of these methods because a least-squares fit weights the
data differently on a logarithmic vs. a linear plot. Make sure you explain clearly the method you have
chosen. Also do not assume that all graders "speak" Mathematica, so an explanation in plain English is
required.
2.
Is hot water fattening?
A remnant of the old Caloric Theory of heat is the unit of energy known as the calorie which
measured the quantity of heat needed to raise the temperature of 1 gram of water by 1˚ Celsius (because
this quantity actually varies with ambient temperature, a full definition should include “... from 14.5 ˚C
to 15.5 ˚C” but we can neglect the temperature-dependence at this point in the course). This is a very
small quantity of heat and it is more convenient to use the kilocalorie ( = 1000 calories) which, just to
confuse things, was originally called the Calorie with a capital “C”. The latter is the unit of
measurement typically used when measuring the energy content of food and is what you find commonly
listed on the side of a cereal box.
When we eat food, our bodies burn the raw materials in oxygen in order to extract the chemical
energy which – amongst many other uses – regulates the body’s temperature. If we drink a cup of hot
water (or soup, or tea, or coffee...), in addition to the calories we gain from the nutritional content of the
drink, we also gain calories just from the thermal energy content of the drink! Is this intake sufficient to
ruin a diet?
• Estimate the thermal energy content of a hot cup of coffee above and beyond that which it
would contain at human body temperature. To see whether this effect is important, contrast the number
you calculate with typical values for cream or sugar found by inspecting packages in your local cafeteria
or dorm room or a store in Harvard Square. Also compare the result with the tabulated average daily
caloric intake from your handout “An Introduction to Energy and Thermodynamics”.
3. A can of beer at room temperature is put into the refrigerator so that it will cool. Would you model
the can of beer as a closed system, an open system, or an isolated system? What about the refrigerator
itself, assuming it’s air tight and well insulated? Explain.
4. Sonntag #2.76.
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ES 181 (Fall 2005)
Problem Set #1
Due: 9/30
5. A proposed (http://www.fe.doe.gov/programs/powersystems/futuregen/) approach to "CO2
sequestration", i.e. the removal of the CO2 combustion product of fossil fuel burning and its long-term
storage in a form that won't leak into the atmosphere for centuries, is to remove it from the exhaust
stream of coal-fired power plants, liquefy it, and bury it underground. To produce the 10 TW of power
anticipated from coal in the year 2050 (see figure on p. 10 and table on p. 11 of "an introduction to
energy and thermodynamics"), how large a volume of CO2 needs to be buried per year? Compare your
answer to the volume of water in Lake Superior. Assume the density given by Table A.4 of your
textbook.
---end of problems---
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