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.
