# Environmental Calculations

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```					                                        AP Environmental Calculations
Dimensional Analysis

    Show all your work on a separate sheet of paper; partial credit is given for partial solutions to problems.
If the answer is not correct, you are not likely to receive credit for correct thinking if there is no
evidence of this process on paper.
    All problems must be solved using dimensional analysis.
    Organize your answers as clearly and neatly as possible, showing the steps you took to reach your
solution. If your reasoning can not be followed, you are less likely to receive credit for it.
    It is important to pay attention to units for quantities that have them. If you keep track of units as you do
points if the units are wrong or are missing from the answer.

1  farmer requires 1 hen/day
 1 hen eats 25 grasshoppers/day
 1,000 grasshoppers have a mass of 1 kg
 1 grasshopper requires 30 g of soy/yr
 1 human requires 600 grasshoppers/day
 dry soybeans have about 3.3 cal/g

1. Calculate the number of grasshoppers a hen needs per year.

1.                    1 hen         25 grasshoppers          365 days       =                      9,125 grasshoppers
hen  day               year                                       year

2. How many grasshoppers are needed for a year’s supply of hens for the farmer each year?
3. What is the total mass, in kilograms, of the grasshoppers needed to feed all the hens for one year?
4. How many kilograms of soybeans are needed to feed all the grasshoppers for one year?
5. Estimates of early Native American hunter-gather societies indicate that a person could collect about 90
kg (200 lbs) of grasshoppers per hour, when they are abundant. Now suppose the farmer chose to eat
grasshoppers instead of hens. How many people could the grasshoppers feed, compared to the one
person that the hen fed?
6. The farmer needs to consume 3,000 cal/day. If he ate only soybeans instead of the hens or the
grasshoppers, how many people would his soybean crop feed (see your response to Question 4)?

1 kWh = 3.41 x 103 BTU ( British Thermal Units)
1 BTU = 1,055 J (joules)
1 pound of bituminous coal = 12,000 BTU
1 barrel of oil = 5.6 x 106 BTU
1 ft3 of natural gas = 1,030 BTU
1 g 235U = 4.0 x 107 BTU
1 tire = 250,000 BTU

7. The average American uses 10,000 kWh of energy. Convert kWh to cubic feet of natural gas.
8. How much coal would be burned to provide the energy?
9. How much uranium would be needed to provide the energy?

Scholten 1
10. The cost for U3O8, the primary nuclear reactor fuel is \$0.022 per gram. What would be the cost of the
11. Coal costs about \$24.38 per ton (2,000 pounds), and the cost of natural gas for electric utilities, on the
average is about \$4.67 per 1,000 cubic feet. Calculate the cost of these two fuels to produce electricity.
12. How many used car tires would be required to supply the electricity if the tires burn at 60% efficiency?

Coral reefs are produced when corals acquire calcium ions (Ca2+ ) and carbonate ions(CO32- ) from
seawater and deposit solid CaCO3 to form their exoskeletons. Scientists are concerned that relatively
rapid decreases in ocean water pH will hinder the deposition of CaCO3. Use the following assumptions
below to perform the following calculations:

 Assume that the total global area of corals growing in reefs is 2.5 x 1011 m2.
 Assume that corals grow only vertically and that the average vertical growth rate of corals is 3 mm/year
3       3
 Assume that average density of CaCO3 in corals is 2 x 10 kg/m

13. Calculate the current annual global increase in volume, in m3 of CaCO3 in coral reefs. Show all steps in
14. Calculate the current annual global increase in kg of CaCO3 in coral reefs. Show all steps in your
calculation.
15. Because of ocean acidification, it is expected that in 2050 the mass of CaCO3 deposited annually in coral
reefs will be 20 percent less than is deposited currently. Calculate how much less CaCO3 in kg, is
expected to be deposited in 2050 than would be deposited if ocean water pH were to remain at its
current value.

A grid-connected residential PV system is placed on the roof of a 2,000-square-foot suburban
house. The PV array with an area equal to 50 square meters covers half of the south-facing part
of the roof. The power rating of this PV system is 5.0 kW, meaning that it will produce 5.0 kW
under peak sunlight conditions. The installed cost of this system is \$50,000.

16. The PV system is operating in a location where the annual average daily incident solar energy (the
insolation) on the array equals 5.0 kWh/m2/day. Calculate the average amount of solar energy incident
on the PV array each day in kWh/day.
17. The efficiency of the PV system equals 10 percent. Calculate the daily average electric energy produced
by this system in kWh/day.
18. Over the next 20 years, United States annual electric energy consumption is projected to increase by 1.5
trillion kWh/year. How many rooftop PV systems would be needed to supply just 10 percent of this
19. Assuming the electric energy produced by these PV systems is worth \$0.10 per kWh, these residential
systems would generate electric energy worth \$15 billion/year. Calculate the simple payback period in
years for these PV systems. (Payback period is the time it takes for a system’s net benefits to equal its
cost)

Scholten 2
Consider a wind turbine that is rated at 1.5 MW. This means that with sufficiently high winds, it
will produce 1.5 MW or 1,5000 kW of power. The installed cost of this turbine is \$1.5 million. A
single turbine would produce enough energy for 1,000 homes for a year.

20. If this turbine runs at its rated power 100 percent of the time for a full year, how much energy would it
produce in a year?
21. This wind turbine has a capacity factor equal to 0.38. This means that over a year, it will produce only
38 percent of its theoretical maximum energy production. How much energy does this turbine actually
produce in a year (in million kWh/year)?
22. Calculate the cost of installing these wind turbines to meet the energy needs of an area with 1,000,000
homes.
23. Assuming the electric energy produced by these turbines is worth \$0.05 cents per kWh, these turbines
would generate electric energy worth \$7.5 billion per year. Calculate the simple payback period for
these turbines. (Payback period is the time it takes for a system’s net benefits to equal it cost.)

1 calorie = 4.186 joules
1 Btu = 252 cal
1 therm = 100 ft3
1 ft3 = 1000 Btu
1 kWh = 3.6 x 106 joules

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30. If a new 80 percent efficient furnace could be installed at a cost of \$4,000 how long would it take to pay
back the cost of this furnace, assuming gas prices remained the same?
31. The annual average solar flux in Tucson, AZ is 250 W/m2. Suppose 10 m2 of solar electric panels
operating at 10 percent efficiency were installed on a home in Tucson. How many kWh of electricity
could be collected by these panels in one year?
32. What fraction of the annual electrical requirement of 10,000 kWh for the average home does this
represent?
33. How many square meters of solar panels would be required to supply 10,000 kWh per year?

Scholten 3
34. With moderate winds, a modern large wind turbine can generate about 250 kW of electricity, whereas a
large nuclear power plant can generate 1,000 MW. How many wind turbines would be required to give
the same output as one nuclear power plant?

Scientists say microalgae are the most efficient organisms at converting sunlight to energy. In fact, they
beat other oil crops for production per acre. 70% of this oil can be recovered by pressing the algae; over
90% can be recovered by solvent extraction. The resulting oil can be used for heating, for electricity
generation, or for making other fuels, like biodiesel.

35. Calculate the number of acres required to produce 1,000 gallons of oil in one year from (a) microalgae
and (b) soybeans.

The city of Fremont operates a municipal solid-waste landfill. As represented in the diagram below, the
annual precipitation in Fremont is 200 mm/year: 50% of this water infiltrates through the landfill cover soil
into the waste, and 50% drains off the landfill. A drainage system withdraws 90% of the leachate generated
within the landfill for treatment. The rest of the leachate travels through the bottom liner of the landfill into
the surrounding soil. Most of the cadmium disposed of in the landfill remains in the landfill; the leachate
withdrawn from the landfill by the drainage system has an average cadmium concentration of 2.0 g/m3.
Pumped to a treatment station, the leachate is treated at a cost of \$10/m3.

Scholten 4
36. Calculate the volume, in m3 of each of the following: (a) The water infiltrated through the landfill per
year (b) The leachate that is treated per year (c) The leachate that is not treated per year.
37. Given that the cadmium concentration in the water draining from the landfill is 2.0 g/m3, calculate the
mass in kg of cadmium that is released into the surrounding soil per year.
38. What is the annual cost of treating the leachate from the drainage system?

The Cobb family of Fremont is looking at ways to decrease their home water and energy usage.
Their current electric hot-water heater raises the water temperature to 1400 F, which requires
0.20 kWh/gallon at a cost of \$0.10/kWh. Each person in the family of four showers once a day
for an average of 10 minutes per shower. The shower has a flow rate of 5.0 gallons per minute.

39. Calculate the total amount of water that the family uses per year for taking showers. Show all your work
40. Calculate the annual cost of the electricity for the family showers, assuming that 2.5 gallons per minute
of the water used if from the hot-water heater. Show all your work and include units with your answers.
41. The family is considering replacing their current hot-water heater with a new energy efficient hot-water
heater that costs \$1,000 and uses half the energy that their current hot-water heater uses. How many
days would it take for the new hot-water heater to recover the \$1,000 initial cost?

Use the topographic map below to answer the following question.

Scholten 5
B

A

42. Calculate the slope of the road leading from” Site A” to “Site B”.

Scholten 6

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