A Spy's Guide to Estimating Power by Paul Dougherty

Want to figure out how much energy your neighbor's trash could generate? Or how many watts you can get out of a solar cell from the corner electronics shop? Here are the keys to estimating the power of several alternative energy sources. All you need are some measurements a little simple math. Your power estimates should be between a half and double the actual power.

Solar Cells

You can pick up a solar cell from any store that sells consumer electronics. These cells will convert sunlight into electric power with about 10% efficiency.

We represent efficiency as "e". In this case e = 10%, or .1.

Sunshine provides about 1000 watts of electric energy per square meter. This amount of energy is called the "solar constant," represented as "C".

Estimating the power generated is simple: the solar constant tells you how much light energy falls on your solar cell, and the efficiency tells you how much of that light energy will be converted. The only other thing you need to know is the area of the cell, which you can get by multiplying the length by the width.

This equation requires area in meters, so you have to measure your solar cell with a metric ruler.

Now, if we say that "A" stands for area, then you can plug your numbers into this equation:

Power=ACe

Let's say your solar cell array is 1 square meter. Then this is how the numbers would plug in:

P = 1 x 1000 x 0.1 = 100 watts.

That's enough power to light up one bright lightbulb. But it would have to be while the sun is shining, unless you have a battery to store the energy.

Wind Turbines

Once again it starts with area, A, the area swept out by the blades of the wind turbine.

Let's say we have a wind turbine with propeller blades that reach out 10 m from the central hub. The area they sweep out will be everything within the circle that the blades make as they spin in the wind.

We can measure that by multiplying the radius squared by pi, or 3.14 (here, we estimated pi to be 3), or using the equation:

A = (pi)r ² = (3)10 ² = 300 m ²

We can represent the power of the wind that blows through this area with the letter P. We measure power by dividing the amount of energy that's generated by the time it takes to generate it. In equation terms:

P = E/t

When it comes to windmills, E is the energy of the wind. We can determine that from knowing how fast the wind is blowing, and by using the standard energy equation:

E = 1/2 mv ²

where m=mass and v=velocity.

Wait...mass? Yes, the air going through the windmill has a mass. We can find that by multiplying the density of air by the speed of the wind and the amount of time passing. Mathematically:

m=dvt

The density of air is easy. It's 1kg/meter ³ .

The speed of the wind is whatever your meteorologist says it is. Let's just say it's 10 meters/second.

The time, well, we want to know how much energy goes through the windmill in a second. So we'll just say the time is...1 second.

Now, go back to our equation for energy, E=1/2 mv ² . We've figured out that m=dvt. So put that into the energy equation, and you get:

E=1/2 dvt (v ² )

Now that we know the energy, we can find the power. Remember, P=E/t. Because we're saying time=1 second, we don't actually need to divide by anything.

But we only want to know the amount of power going through the area swept out by the blades, so we multiply the energy in the equation by the area. That leaves us with:

P= A 1/2 d t v3 watts

or, putting the numbers in:

P = 300 x 1/2 x 1 x 1 x 103 = 100,000 watts

That means the turbine in this example would be a a 100 kilowatt wind turbine. That's enough to power 1000 bright light bulbs.

Hydroelectric

This one is a lot like the windmills. We've got the same equations for the power of falling water:

P = E/t

And again, we find the energy by knowing the mass. Here, though, we find the mass by knowing the weight of the water and the height from which it falls, and multiply that by the acceleration of gravity. Mathematically speaking:

E = (mass)(gravity)(height)

We'll figure mass in terms of kilograms of water falling per second. Knowing that one cubic meter weighs 1000 kilograms, or one metric ton, estimate the amount of water falling by watching as it flows through the plant. For this example, let's say your estimate is that there are 100 kg of water falling per second.

Gravity is easy -- that's 10 m/sec ²

And let's say the water is falling from a height of 10 meters.

And remember, like the windmill, we're letting time=1 second, which means that we can find the power by simply finding the energy. So, with the numbers plugged in, we get:

P = m g h /t = 100 x 10 x 10 / 1 = 10,000 watts.

Which makes our example a 10 kilowatt hydroelectric plant.

Biomass

The key thing to remember is the dieter's law: every bit of dry carbohydrate contains 100 calories per ounce. In metric units this is about 15 Joules per gram or 15,000 J/kg.

To estimate the power of biomass you have to estimate the dry mass, m, burned per second, t, and then multiply that by the dieter's law, or 15,000 J/kg.

P = m/t x E

Let's say you estimate that about one ton, or 1000 kg. That's about a cubic meter of stuff. And let's say it's burned per hour or m/t = 0.3 kg/s.

Then P = 0.3 15,000 = 5 kW.

This power is then converted into steam which turns a turbine to produce electric energy. Unfortunately, this conversion is done with less than 50% efficiency.

So we'll say it produces about 2 kW per dry ton of biomass burned per hour.