Published: July 05, 2008 06:44 pm         

Try ‘back of envelope’ energy calculations

Your body, gasoline, coal and sun

One of the most powerful ways to use basic mathematics is in back of envelope calculations, where one uses rough approximations to arrive at answers to difficult problems that are usually only a factor of two off.

The term “back of the envelope” refers to a calculation that only needs a small amount of paper, perhaps the back of a used envelope. In doing back of envelope calculations, it is helpful to know how to handle numbers written in scientific notation, where numbers are the product of a standard number (coefficient) between 1 and 9.999 and a power of 10.

There is a new book on back of envelope calculations titled “Guesstimation” by Weinstein and Adams, published this year by Princeton University Press. I would like to spotlight some surprising results from “Guesstimation” that would be helpful in understanding our difficulties relating to energy.

Assume that an average person consumes 2500 food calories per day. Over the course of a year, that would be about a million calories. Since a calorie is about 4,000 joules (metric energy unit), we consume 4 billion (thousand million) joules per year. Now a watt is a joule per second. So if we consider the number of seconds in a year (about 32 million), our bodies take in energy at the rate of about 100 watts continuously.

Now gasoline has a very high energy density of 30 million joules/Liter. There are about four liters in a gallon, giving you 120 million joules per gallon that you pump in your gas tank. Now an average car gets 20 miles per gallon. An average U.S. driver puts 10,000 miles on his/her car per year. This results in a consumption of 500 gallons of gasoline. This is 2,000 liters, each containing 30 million joules. This results in an average car consuming 60 billion joules per year, about 15 times the energy of the food that we eat.

Half of the U.S. electricity is generated by coal burning power plants. A large power plant generates 1 billion electrical watts (often stated as 1 gigawatt). This amount of power could supply a million modest homes, each consuming an average of 1,000 watts through the year. (The electrical power need increases in the winter and summer (when heating and cooling increase electrical demand) and decreases in the mild months.)

A typical coal burning plant is about 1/3 efficient, so it must generate three times as much thermal energy as the electricity it actually provides. The total energy that must be provided by the coal is 3 billion watts times 4,000 joules times number of seconds in a year, which comes to about 10 to the 17th joules (this is the number 1 followed by 17 zeroes).

Now each kilogram of coal when burned releases 20 million joules per kilogram. (Kilogram is the metric mass unit and on the Earth’s surface has a weight of 2.2 pounds.) This analysis shows that a 1 gigawatt power plant burns 5 million tons of coal each year. (Each year, the U.S. consumes a billion tons of coal.)

The sun’s total power can be determined by sampling the sun’s rays above our atmosphere; we find that a black collector can absorb 1,400 watts per meter squared. By knowing the distance of the Earth from the sun and the area of a sphere that would trap all the sun’s heat, we can determine that the sun’s energy is 4 x 10 to the 26th watts. From knowing the Earth’s size, we can calculate that the Earth intercepts only billionths of the sun’s energy. Yet this amount of solar energy is thousands of times the energy that humans generate.

Dog Days return

The sun is now in the star group Gemini and will pass into the star group Cancer in late July where it remains till Aug. 10. Our sun is now closest to the star groups, Canis Major (Big Dog) and Canis Minor (Little Dog). It is during these days that we have high temperatures, long sunlight hours with the sun very high in the mid day sky.

Call (301) 687-7799 to request a free planetarium/Science Discovery Center bookmark (shows and tours resume on Sept. 7).

Bob Doyle is available for brief talks and discussions for clubs or groups (call above phone number). He invites comments and questions from readers; his email is rdoyle@frostburg.edu .