solar powered hurricanes and power densities

I was walking in our woods over the lunch hour thinking about the superstorm almost exactly a year ago.  There is still a considerable amount of damage including our part of the grid which is patched together rather than robustly upgraded.   It is bright and sunny - enough to make you think about how solar energy drives our weather.   It is a good excuse to chat a bit about energy and power.

 

For a quick exercise I worked out the power associated with the hurricane - at least a quick back-of-the-envelope approximation as more accurate work requires much more information than I have along with sometime more than the simple models I came up with. 

 

There are several ways to look at the problem.  Most people associate hurricanes with their winds, so you can calculate the kinetic energy of the air mass.  I'm omit the details here, but a category 3 storm is in the vicinity of 2 x 10¹² watts - two terawatts.¹  The total electricity generating capacity of the United States is a bit over a terawatt and we use much less than our capacity.  An enormous amount of power, but how do you harvest it?

 

It turns out the kinetic energy of the wind is much smaller than the energy released by rain and cloud formation.  A quick estimate of the power released by the average category 3 hurricane is 6 x 10¹⁴ watts - 600 hundred terawatts and several hundred times greater than the total electricity generation capacity of humanity..  

 

Huge gee-whiz numbers and we need some way to make sense of them.

 

A few years ago I was looking at ways to express some of these concepts and scales in a way that made sense and could easily be compared with personal experiences.  I came to the conclusion that people think in terms of power (the rate of energy transfer) rather than energy.  We buy energy - calories of food, kilowatt hours of electricity, gallons of gasoline - but we are usually more aware of how quickly it is used.  The average person gets about 100 watts of power from their food (and gives off most of it as heat).³  You might say a person really does light up the room. 

 

To compare power from different sources we use a power density and define it as power divided by area.⁴  The power density of a person's metabolism would be about 50 watts/m².

 

Congratulations for hanging in there.  Now we're all set to make a few comparisons and even find some natural limits.

 

The power density of the wind in our hurricane is 2 x 10¹² watts/(60km)² or about 550 W/m² for the core of the storm.

 

The Sun powers supplies most of the energy on earth (geothermal and nuclear energy excepted).  Just outside the Earth's atmosphere it delivers an average of just under 1350 W/m².  If you correct for the Earth's rotation, shape and absorption and reflection, an average of about 170 W/m² makes it to the surface. About three times that of a person and a third that of the wind energy of the core of a hurricane.

 

Green plants are the most important mechanism for collecting sunlight and turn it into stored energy - at least as far as plants and animals who eat them are concerned.  It turns out most plants are only 0.1 to 0.5% efficient - the 170 W/m² of solar energy is turned into a few tenths of a W/meter² of stored plant energy.⁵   A family getting all of their energy from plants would need about 10,000 m² of crop area to stay alive - a hectare.  Figure several times as much to allow for crop failure, the energy required to harvest and so on.  Eating animals is a very inefficient way to get your solar energy as animals only turn 5 to 20% of the energy they take in as food into food a carnivore can eat - so multiply the amount of land required by a factor of about ten to account for the animal part of your diet.

 

We can look at electricity from solar power.  Skipping the details current photovoltaics average about 5 W/m² in cloudy places like Germany and about four times as much in the deserts of the American Southwest.  The average home in San Diego uses 1000 watts of electricity, so you are looking at a minimum of 50 square meters to switch to solar power.⁶

 

Fossil fuels represent millennia of stored layers of solar energy - a slow concentration of a relatively diffuse source.  The process is extremely inefficient (well under a thousandth of a percent in most cases), but that doesn't matter to us as we're just mining what has been built up over the years.  Of course it gets increasingly expensive to mine as we exhaust the easy sources, but it is a very convenient fuel.  

 

Moving away from alternate energy, power densities can be very high.  Since it is the anniversary of the big storm my mailbox has been filled with ads for fossil fuel powered electric generators.  A one kilowatt generator can comfortably fit on a square meter, so it has a power density of over 1,000 W/m² larger power plants are more impressive.  You can make estimates for any type of power production and use.  

 

And now for a beautiful chart.  This is from David MacKay - an English physicist.  He plots something like power vs population density (changed around a bit to match some units he likes).  The diagonal lines are power density.  He has labeled some of the more important generation methods (fortunately some of my back of the envelopes are close enough to his:-)  You can see what forms of generation might be practical in various locations, the wide variation in power use for different countries and so on.  One striking feature that you can work out is the average power consumption per unit of land area for the planet is currently about 0.1 W/m² (we are currently using about 17 terawatts of power, divide by land area).

 

 Wind power provides about 1 to 3 W/m² of land area.  You can see that some parts of the world can't use wind power alone ..  In theory the best biofuels are about 2 W/m² (current fuels are far from that) - telling you a better use of farming is probably to make food.  A bit of staring at the chart and using production numbers from elsewhere is illuminating.

 

I'm a big fan of solar energy - sunlight is a good thing.  Ultimately we will probably be using it directly or through other alternate energies (wind and hydro for example). It turns out there is more than enough, but you have to be careful about where and how you do it.  We've been spoiled by the thin layer of fossil fuels that have slowly been building near the Earth's surface.

 

All of this is very high level - if there is interest I can go into greater depth in areas of interest. 

 

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¹ The dissipation of kinetic energy due to drag per unit area will be (air density) x (drag coefficient) x (windspeed)³.  You then integrate over the radius of the storm.  I don't information of sufficient detail, so I make an approximation.  Most of the punch will be in the inner radius.  I'll consider a rotating cylinder that is about 10% the radius of the entire storm and neglect the eye since this is just an approximation anyway.  The 10% is the part of the storm where you get hurricane force winds and above on an average sized hurricane.  The rest can be neglected as the windspeed term is cubed.  45 m/s winds (about 101mph) and air density and a drag coefficient approximation (from a journal article in Nature) gives the answer.

² This one is a more straightforward.  Just use the latent heat of condensation and calculate the volume of rainfall.  A source on hurricanes suggests the average category 3 storm has a 660 km radius and produces 1.5 cm of rain/day (very light for most of the hurricane and much much more in the inner core).  

³ Energy can be measured using many scales and you can convert from one to another.  It turns out the 2,000 or so nutritional calories an average person eats is close to 2,400 watt-hours, so the power from metabolizing that food obtained by dividing by 24 hours in a day giving us 100 watts.

⁴ The terminology gets a bit tricky.  Energy density is easy - energy/mass or energy/volume.  It is customary in physics to talk about power density in terms of energy flux - the amount of energy indecent per area on a surface per unit time - or power/area on a surface. (historically heat flux).  It is usually measured in watts/area where the area is the collection area.  We make a more general use to look at the power/area that can be produced by some area on the planet.  So the area under a power plant, of a forest, or even a city.  To compare it to a person we consider the surface area of the person, which is close to 2 m²

⁵ Some plants are much more efficient - up to four or five percent, but most of those are not suited for human consumption.  Area is taken to be the most efficient planting area rather than the surface area of plant leaves.

⁶ Lots of asterisks here, but this is just a back of the envelope.  You need to store solar energy somehow and that has varying degrees of inefficiency depending on technique.  There are also additional losses such as transmission.

 

 

Comments

[Jheri]

can you estimate how many people can be supported by growing plants?

[steve]

Yes Jheri, but the details are complex. You have to worry about the crop mix, climate, water for irrigation and so on. For a vegetarian world the number is usually between 11 and 14 billion people. If you add animal based food it drops. If everyone ate like Americans it would be under 3 billion.

There is a lot to do with the efficiency of farming and distribution. Currently about a billion people are hungry and a larger number are malnourished.

[David]

Thanks great blog post.

That graph by David MacKay is a great way of thinking about energy production and consumption. Is the difference between "before conversion" and "Solar PV parks" mostly a relationship between efficiency of turning photons into electrons, number of hours of sun a day and W/m2?

Do you think the "steady" progress of PV technology since say 1970 will continue for decades to come?

[steve]

Hi David. The 1350 W/m^2 is the power density of sunlight just outside our atmosphere. It is projected onto a sphere the radius of our orbit, so the trick is distribute it over the Earth's surface area. The difference is a factor of 4, so the average spread over a day is about 335 W/m^2. The 170 number is because the atmosphere and clouds attenuate roughly 50% of that on average. In a high desert directly under the Sun the numbers go up to about 250 W/m^2, but only when the Sun is directly overhead. These numbers are before the conversation to some other form of energy. Thermal for a concentrating solar plant, electricity for PVs...

The charts author is picking some average figures for locations in his home country (England). They assume English weather, seasons and day and night except where noted (very few deserts in England:-)

Inexpensive PVs are about 15% efficient. More efficiency can be had at a cost, but this is close to a sweet spot. In the US PV cell costs are lower than mounting and permitting costs - you could easily drop the installed cost by a factor of two with streamlined laws and more clever construction techniques - the Germans have shown that and it balances for their relatively cloudy weather.

Long term projections on cost is tricky. It is clear there will be reductions - how close to the current curve (which has seen the impact of overproduction recently) is not as clear. There are also some important infrastructure technologies that will impact solar adoption - notably storage.