Uncharted Territory

November 11, 2009

Pissing in the Wind, Part 1

Filed under: Books/resources, Climate change, Energy, Energy policy, Global warming, Wind — Tim Joslin @ 11:51 am

When I worked as part of a team made up of nationals of several different European countries, we’d be fond of swapping phrases from different languages (all translated into English). Most would make Hank Paulson blush, and this is a family blog. But one I liked was the equivalent of the English phrase “to make a mountain out of a molehill”. In Holland (or was it Greece?), you’d say instead “to make an elephant out of a mouse”. So, of course, we combined the two and made elephants out of molehills and mountains out of mice. My most notable contribution was the phrase “pissing in the wind“.

What’s bugging me is the question of the potential for generating energy from wind-power. In what’s fast becoming the Bible for such matters, Sustainable Energy Without the Hot Air (SEWTHA), David MacKay asserts that you can only practically generate around 2W of wind power per m2 on or around the UK.

David therefore concludes (page 216) that the UK could feasibly build 35GW of onshore capacity and 29GW of offshore, total capacity 64GW, producing on average 4.2kWh/day/person and 3.5kWh/d/p, 7.7kWh/d/p in total. (Other energy plans for the UK including more or less wind energy are discussed elsewhere in SEWTHA).

Sorting out the units

One man’s sensible units are another man’s bizarre eccentricity. I want to convert David’s units for comparison with other, even more eccentric, sources. Personally I’d like to divide by 24 to get rid of both the hours and the day – David’s wind totals 7.7kWh/day per person, that is 7700/24W per person – call it 300W. Now we’ve got to something I can relate to! And I don’t know, but 300W seems not a lot more than the lights and the TV to me! Maybe we’re going to discover the wind won’t save us…

Anyway, figures are often given in TWh/year for the UK. Strange but true.

I assume MacKay bases his estimates on 60m people. So 7.7kWh/d/p is 7.7*60m*365kWh/yr for the UK or 7.7*60*365GWh/yr = ~170TWh/yr.

How much wind do we need for 1 million jobs?

David MacKay is now an energy advisor to the UK Government, so his view counts. But I keep reading higher figures for the potential for the UK to generate wind energy than 170TWh/yr.

For example, on Saturday I picked up a booklet One million climate jobs NOW! which notes on p45:

“In 2008 the total UK supply of electricity was 401TWh. 7TWh of that came from wind. In 2008 the UK had 3.4GW of installed wind power. So approximately 2TWh of electricity were produced that year for each [G]W of installed capacity. [So far so OK: cf David’s 170/64 or a bit over 2.5TWh/yr/GW installed capacity]. 150GW of installed capacity should produce 300TWh, three quarters of current electricity production.”

Obviously, if there is not enough wind for 150GW of capacity and/or for 300TWh/yr, the whole 1 million jobs plan starts to unravel.

Sorting out the units again

One man’s sensible units are another man’s bizarre eccentricity… What does “150GW capacity” mean? Let’s work instead in terms of average output, because we’re going to be considering average wind-speeds (really we should be considering average power in the wind, which is different, but, hey, the modern Principia will have to wait!). Let’s go back to the energy needed of 300TWh/yr. What average power output do we need to achieve this?

What a pretty pass we’ve come to when we’re calculating in Watt-hours per year!! We want Watt-years per year, in other words, simple Watts!! There are roughly 24*365 = 8760 hours in a year, so 300TWh/yr = 300,000/8760GWyears/year = 35GW, rounded up a tad.

To create 1 million jobs we need to build enough wind-turbines to give an average power output of 35GW.

Is there enough wind?

Now we can finally start to make comparisons. How much wind is really out there? And how much of that do we need?

What’s been bothering me for some time now is that MacKay bases his figures (all derived from the 2W/m2 power density) on wind-turbines having to be spaced in a grid 5 times their diameter (5d) apart, as described in his Technical Chapter B, p.265.

This argument seems to apply to current technology only, but is also somewhat counter-intuitive as you would have thought you could simply put taller wind-turbines in between the ones you’ve already got and they wouldn’t interfere. If you only used 2 heights you’d double up to 4W/m2 and we could create our 1 million jobs, moreorless.

In fact, the idea that you can only extract the same amount of energy per unit land area whatever the diameter of the wind turbines is somewhat paradoxical. Surely 1cm turbines spaced 5cm apart is not going to be as good a solution as 100m turbines spaced 500m apart! All very odd: MacKay’s Paradox, perhaps!

Furthermore, it would seem the proximity of other wind turbines is only a problem downwind. Perpendicular to the direction of the wind it might even be better for the turbines to be next to each other as, like New York skyscrapers, the resistance of one would force air towards its neighbour. In many locations useful wind will normally come from one direction (the west near the UK). If only the downwind turbines have to be 5d apart, then you should be able to generate 5 times as much energy, 10W/m2. Now we’re talking!

But I don’t want to stop here. With different designs, e.g. turbines at different heights or funnelling air towards turbines, you might be able to do even better than that. In principle you should be able to capture a proportion of all the energy in the wind up to whatever height you could engineer. How much energy is this?

Problem

MacKay (Chapter B, p.263 ff) only considers the kinetic energy of the wind passing through a single turbine.

But we know that the wind turbines interfere with each other, otherwise we could put them right next to each other and there’d be no 5d rule of thumb. What I’d like to answer are questions such as:
– what proportion of the energy in the air does a large field of wind-turbines extract?
– can we do better than extract 2W/m2 with better technology?
– are we likely to hit any limits, i.e. can we extend a field of wind-turbines indefinitely without weakening the wind?

Obviously this is just a blog (but, hey, what might it lead to?), not a scientific treatise on the subject. Nevertheless, we can take a stab at answering these questions.

Thought experiment

Let’s work out the kinetic energy of the entire mass of air up to the top of the atmosphere passing between two imaginary poles a metre apart across the 6m/s wind direction. A quick calculation shows that this column of air – 15psi (sorry, pounds – 2.2/kg – per square inch – ~2.5cm2 – you can do the calculation yourself – OK, the conversion is 15psi = ~15/2.2*40*40kg/m2) in old units – weighs ~10000kg. Wow!

If the wind speed all the way to the top of the atmosphere is an even 6m/s (a conservative assumption as it moves faster higher up, we’ll try to come back to this), then the kinetic energy of the air passing between the poles every second is, by the formula 1/2mv2, with 6m of air passing every second, 1/2*6m*10000kg*(6m/s)^2 = ~1 million Joules, that is, (1 Joule per second =1 Watt) we have 1MW of power every metre across that there gentle breeze. Wow, again!!

This is rather different to the figure of 140W/m2 (note the different units) David MacKay calculates because he only considers the energy in a cross-section of the air, the 1.3kg/m3 that actually passes through a 1 m2 cross-section of wind-turbine. The wind goes up a long way and (by these back of an envelope calculations) only 140/1 million = 0.00014 or ~1/7000th of it passes through a 1 m2 cross-section near the ground! (The calculation by mass of air considered, i.e. 1.3/10000, gives roughly the same answer).

But the wind comes from somewhere. If you had many rows of wind turbines, part of the energy will be extracted by each row. The wind for the later rows will have to come from somewhere or we’d be becalmed. The answer is it comes from the other 9998.7kg of air above the wind turbines!

This rather explains MacKay’s Paradox, since we have to suppose air can only fill the lee (downwind) side of the wind turbine from above or below or even from the sides (so perhaps we can’t put our turbines right next to each other after all) at a limited rate (mostly from above). When a wind turbine creates a partial vacuum, the engineers’ rule of thumb used by MacKay is that a “hole” 1m in diameter is filled in 5m, 100m in 500m and so on.

OK, not all the air will necessarily be moving in the same direction (otherwise the weather system we know & love wouldn’t operate as observed), but if even half the mass (remember the air is less dense the higher you go) is, we have 5000kg of air and 500kW/m2 to play with.

Even if we can only extract 1% of this energy, that works out at 5kW/m2.

We can’t keep extracting 1% of the energy, though, from row after row of wind turbines, so maybe we should consider the air-mass to be a wall of wind, from which we could extract, say, 10% of the energy in total, that is, 50kW per metre length of the wall. This is equivalent to funnelling all the air through 100% efficient wind-turbines, that is, extracting all the energy in the wind, up to a height of 50,000/140 = ~350m (the 140W/m2 is David MacKay’s power per unit area of wind-turbine at a wind speed of 6m/s).

Or, perhaps more practically, we could extract around 1/3 of the energy (MacKay suggests 50%, but I’m going to be a bit less optimistic) in the air up to 1000m, one kilometre. (Note that this doesn’t allow for air density decreasing with height, but then again I’m not yet making any allowance for the fact that the wind-speed increases with height).

To obtain the 35GW average power output we need for our 1 million jobs would therefore need a wall of such wind-turbines 35GW/50kW = 700,000m or 700km long. Ouch!

Or perhaps, since we’re talking about the UK, we could have a 1400km wall of wind turbines 500m high, which sounds a bit more practical.

Implications

My 1400km wall of wind-turbines 500m high is very roughly equivalent to (say) a field of large wind-turbines (100m+ diameter) 1400km, that is, 14,000 wind-turbines long (i.e. around the whole length of the UK), right next to each other, but only 5 wind-turbines, that is, with 5d spacing, 2 km across.

The “wall of wind” is therefore equivalent to ~14,000*5 = ~70,000 wind-turbines in total, implying an average output of of 35GW/70,000 or 0.5MW at a wind-speed of 6m/s. Wind turbines are normally quoted in capacity. The 35GW average output was based on a capacity of 150GW and empirical rather than theoretical figures relating average output to capacity. Anyway, my calculations suggest the wind-turbines each have a capacity of 150GW/70,000 = ~2MW, which is a little bit low for such large devices, but in the right ballpark. In particular, I’ve estimated cautiously for the efficiency of the turbines and have made no allowance for a higher wind speed at a higher altitude.

This higher wind-speed is absolutely crucial, because what I hope I’ve demonstrated is that a field of wind-turbines actually extracts energy from higher up in the atmosphere. A field deep enough would actually slow the entire air flow. What happens is that the first row of wind-turbines slows the air, creating a partial vacuum downwind. This is filled mostly from above, slowing the air higher up.

Consider the graphs of wind-speed against height and power density of wind against height David gives here. They’re astonishing. The wind power at a 10m height is around 100W/m2 for a 6m/s wind at that level, but at 100m where the air flows faster it’s nearly 250W/m2 and at 200m where it flows faster still we’ve got over 300W/m2 to play with!

What’s actually happening, of course, is that all the other things on the ground – water, trees and so on – are already capturing the energy in the lowest part of the atmosphere, which fills from above.

Or, to look at it another way, the wind is created by a high pressure mass of air essentially collapsing into a low pressure area, which literally fills, as the weather-men say.

Bearing all this in mind, it seems to me that we’re pissing in the wind in the first place building wind turbines near ground level. We should start 100m or 200m or even 300m up.

Conclusion

There is (significantly) more than 500kW/m or 500MW/km of kinetic energy in a flow of air – 100s of kms across – moving towards the British Isles at an average speed of 6m/s, creating what we call a (west) wind. If we could extract all this energy we’d “only” need a 70km wall of wind turbines for an average output of 35GW.

The limit of 2W/m2 only applies to the technology we are using just now to extract energy from the wind. At this stage in the development of the industry, there are plenty of sites and it’s the technology that’s expensive. This will change over time, and there will be an incentive to design machines to extract more of the energy from the wind, particularly higher in the atmosphere.

It may be possible to extract significantly more than 2W/m2 by building turbines closer together across the wind direction and (as, to be fair, David MacKay points out), much taller.

However, maybe we have to bear in mind that we might not be able to build row after row of giant wind-turbines indefinitely. From a British Isles (UK and Republic of Ireland) point of view this might not be too much of a problem, since we are on the western seaboard of Europe. But eventually if we build turbines along the west coast, perhaps along the spine of the country and in the North Sea, we could just conceivably start to affect the very wind itself – the Danes and Germans might not be so pleased!

To determine whether this hypothesis is true, we have to look at other aspects of the energy in the wind. The kinetic energy arises from the potential energy of different pressures of different air-masses. And we need to look at how that potential energy itself is generated.

In other words: how renewable is the wind?

Another time, maybe.

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1 Comment »

  1. […] I included the first couple of paragraphs for other interest – 313.4 billion kilowatt hours (why, oh why can’t journos use units in a sensible fashion? – what next? “million MWh”?) is 313.4TWh, i.e. about 10 times the UK’s electricity consumption (around 400TWh/yr, according to the source I used in a previous post). […]

    Pingback by China’s Energy Profligacy « Uncharted Territory — November 20, 2009 @ 6:31 pm


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