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• # Large Scottish Pumped Storage Hydroelectric Reservoir and Dam (@ Coire Glas)

Discussion in 'Your mechanical design work' started by Peter Dow, Feb 20, 2012.

1. ### Peter DowWell-Known Member

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"Dow" equation for the power and energy output of a wind farm.

"Dow" equation for the power and energy output of a wind farm.

"The power and energy of a wind farm is proportional to (the square root of the wind farm area) times the rotor diameter".

In his book which was mentioned to me on another forum and so I had a look, David MacKay wrote that the power / energy of a wind farm was independent of rotor size which didn't seem right to me considering the trend to increasing wind turbine size.

Now I think the commercial wind-turbine manufacturing companies know better and very possibly someone else has derived this equation independently of me and long ago - in which case by all means step in and tell me whose equation this is.

Or if you've not see this wind farm power/energy equation before, then see if you can figure out my derivation!

2.
3. ### Peter DowWell-Known Member

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Wind farm power and energy equations

No takers for the derivation challenge huh? OK then.

Derivation

Assume various simplifications like all turbine rotors are the same size and height, flat ground and a rotationally symmetrical wind turbine formation so that it doesn't matter what direction the wind is coming from.

Consider that an efficient wind farm will have taken a significant proportion of the theoretically usable power (at most the Betz Limit, 59.3%, apparently, but anyway assume a certain percent) of all the wind flowing at rotor height out by the time the wind passes the last turbine.

So assume the wind farm is efficient or at least that the power extracted is proportional to the energy of all the wind flowing through the wind farm at rotor height.

This defines a horizontal layer of wind which passes through the wind farm of depth the same as the rotor diameter. The width of this layer which flows through the wind farm is simply the width of the wind farm which is proportional to the square root of the wind farm area.

Wind farm turbine formations

Therefore the width or diameter of a rotationally symmetrical wind farm is a critically important factor and arranging the formation of wind turbines to maximise the diameter of the wind farm is important.

Consider two different rotationally symmetrical wind turbine formations, I have called the "Ring formation" and the "Compact formation".

Let n be the number of wind turbines in the wind farm
Let s be the spacing between the wind turbines

Ring formation

Larger image also hosted here

The circumference of the ring formation is simply n times s.

Circumference = n x s

The diameter of the ring formation is simply n times s divided by PI.

Diameter = n x s / PI

Compact formation

Larger image also hosted here

The area of the compact formation, for large n, is n times s squared. This is slightly too big an area for small n.

Area = n x s^2 (for large n)

The diameter of the compact formation, for large n, is 2 times s times the square root of n divided by PI. This is slightly too big a diameter for small n.

Diameter = 2 x s x SQRT(n/PI)

This is easily corrected for small n greater than 3 by adding a "compact area trim constant" (CATC) (which is a negative value so really it is a subtraction) to the s-multiplier factor.

The CATC is 4 divided by PI minus 2 times the square root of 4 divided by PI.

CATC = 4/PI - 2 x SQRT(4/PI) = - 0.9835

This CATC correction was selected to ensure that the compact formation diameter equation for n=4 evaluates to the same value as does the ring formation equation for n = 4, that being the largest n for which the ring and compact formations are indistinguishable.

The CATC works out to be minus 0.9835 which gives

Diameter = s x ( 2 x SQRT(n/PI) - 0.9835) (for n > 3)

Ratio of diameters

Larger image also hosted here

It is of interest to compare the two formations of wind farm for the same n and s.

The diameter of the ring formation is larger by the ratio of diameter formulas in which the spacing s drops out.

Ring formation diameter : Compact formation diameter

n/PI : 2 x SQRT (n/PI) - 0.9835

This ratio can be evaluated for any n > 3 and here are some ratios with the compact value of the ratio normalised to 100% so that the ring value of the ratio will give the ring formation diameter as a percentage of the equivalent compact formation diameter.

Here are some examples,

n = 4, 100 : 100
n = 10, 123 : 100
n = 18, 151 : 100
n = 40, 207 : 100
n =100, 309 : 100
n =180, 405 : 100
n =300, 514 : 100
n =500, 656 : 100

As we can see that for big wind farms, with more turbines, the ratio of diameters increases.

Since the Dow equation for the power and energy of a wind farm is proportional to the diameter of the wind farm then it predicts that the power and energy of the ring formation wind farms will be increased compared to the compact formation wind farms by the same ratio.

In other words, the Dow equation predicts, for example, that a 100 turbine wind farm in the ring formation generates 3 times more power and energy than they would in the compact formation, assuming the spacing is the same in each case.

Practical application when designing a wind farm

My recommendation would be to prefer to deploy wind turbines in a wind farm in the ring formation in preference to the compact formation all other things being equal.

The compact formation can be improved up to the performance of a ring formation by increasing the turbine spacing so that the circumference is as big as the ring but then if a greater turbine spacing is permitted then the ring formation may be allowed to get proportionally bigger as well keeping its advantage, assuming more area for a larger wind farm is available.

The ring formation may be best if there is a large obstacle which can be encircled by the ring, such as a town or lake where it would not be possible or cost effective to build turbines in the middle of it and so a compact formation with larger spacing may not be possible there.

Where it is not possible to install a complete ring formation then a partial ring formation shaped as an arc of a circle would do well also.

4. ### Peter DowWell-Known Member

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Reservoir bed drain

Reservoir bed drain

The high pressure of water which is deeper than 100 metres has the potential to induce seismic activity or earthquakes in susceptible rock in which a new reservoir has been constructed.

Coire Glas/SSE/92 m

Hopefully, reservoir induced seismicity was an issue considered by the SSE when selecting Coire Glas for their hydro dam project.

I am speculating that this issue may be why the SSE have limited their dam to a height and their reservoir to a depth of 92 metres?

I would note however that the pressure in the head race tunnels which supply water from the reservoir to the turbines would be proportional to their depth below the surface of the reservoir and this could be as much as 500 metres deep, so there would seem to be some potential for water to penetrate the bed rock from the high pressure water tunnels and induce seismic activity even in the SSE case.

This is an issue which ought to have been addressed in the many previous pumped-storage hydro scheme projects, most of which seem to have a difference in head of more than 100 metres.

Given that "understanding ... is very limited" according to Wikipedia, though, I do wonder if the reservoir induced seismicity issue has not always been properly addressed in all previous dam and reservoir construction schemes where the great depth of water and susceptible geology ought to make it a relevant concern?

Coire Glas/Dow/317+m

I am proposing measures to counter the reservoir induced seismicity effect in the case that the geology of Coire Glas is susceptible to it and in the general case.

I propose the construction of a large reservoir surface drain to cover the whole reservoir bed and the reservoir sides too to try to stop the penetration of water under high pressure into fractures in the bedrock and so thereby stop this high pressure water from widening and extending bedrock fractures.

To illustrate my "reservoir bed drain" concept, I have drawn a diagram comparing the usual no drain on the left, with my proposed reservoir bed drain on the right.

Image also hosted here.

So my idea is that the top layer of the drain is as impermeable as practical, using perhaps a layer of reinforced asphalt concrete.

In engineering practice I believe that impermeable reservoir bed layers have used clay or clay with asphalt or even rubberised asphalt mixed with sand.

My basic idea is to construct an impermeable layer and to use whatever material is best for that.

Then working downwards, the permeable drain layers are increasingly bigger loose particles, with sand at the 2nd top then beneath that grit, then gravel, then small stones and finally below all those a layer of large stones.

The higher layers support the top impermeable layer which is under high pressure from the reservoir water and the lower permeable layers provide many small channels for any (hopefully tiny amounts of) water which forces its way through the supposedly impermeable top layer to drain down the slope of the reservoir bed out under the dam.

The bottom layer is another impermeable layer to try to make doubly sure that the relatively low pressure water that gets into the drain will find its way out under the dam by following the course of the drain.

These kinds of layers of different sized loose particles have previously been used to make simple narrow drains and impermeable layers have been added to reservoir beds before now but whether professional dam engineers have ever covered the entire reservoir bed and sides with one large drain I don't know. If not, this could be named the "Dow drain" solution to reservoir induced seismicity!

5. ### Peter DowWell-Known Member

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Dam base drain pipes

My second take is to use drain-pipes through the base of the dam which now extends all the way down to the bed rock with the drain-pipes built in, instead of continuing the bed drain under the dam as I had at first.

The large embedded image in the above post is remotely hosted on my forum so I was able to change that there. However, it is too late now to edit the above post to change the small image and the link to the postimage.org host.
So I am posting the new versions now.

Image also hosted here.

Why not add a simple impermeable layer to the reservoir bed?

I think the additional complexity and expense of a bed drain (and drains for the sides too) is better than simply adding an impermeable layer.

Consider the fault condition of the two possible solutions.

If a simple impermeable layer fails, if it cracks or ruptures or disintegrates under the pressure changes, how would anyone know? It may look fine but be leaking high pressure water into the bedrock and inducing seismicity which OK the engineers would notice any earthquakes but so would everyone else, the earthquakes could cause damage or loss of life and it could lead to a loss of confidence in the project and in the engineers who built it. They could go to jail!

If the top impermeable layer of the bed drain fails then there would be some water pouring out of the drainpipes through the base of the dam when at most it should only be a tiny trickle of water. So the engineers would know there was a problem with the bed drain and they'd know to drain the reservoir and fix or replace the top supposedly "impermeable" layer, fix the bed drain so that it operated as it should.

So failure with the bed drain is noticed right away and it is not a catastrophic failure. Whereas failure with the simple impermeable layer may not be noticed until a catastrophic earthquake happens.

So this is why I think the bed drain is worth the extra complexity and expense. It is a more fault tolerant engineering solution.

6. ### durraniMember

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v impressive work sir, what is ur field of engineering. is it civil.

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