As much as possible would probably mean a 600 m high turbine with 300 m blades made of pure carbon fibre or something like that. That could cost hundreds of millions and would be a useless monument.
Look, it's a simple cost trade - currently the most economical turbines are in the 1 to 3 megawatt range and roughly 100 m rotor diameter.
Tower cost is probably exponentially related to height for example. so the things balance out at that point.
Tower cost is probably exponentially related to height
Exponentially? So that every additional 50 feet, say, doubles the cost? I seriously doubt it. I would expect a quadratic relationship: the incremental cost to make it taller is proportional to the current height. Maybe I'm missing something and the relationship is cubic, but it's certainly not exponential.
(I know, maybe you didn't mean "exponentially" literally. But we're engineers here :-)
In a traditional shrouded turbine every additional foot of tower height would require, at a minimum, 2*pi feet of additional circumference to the shroud (in the case of the design that started this discussion it appears that the shroud is the main structural element so it is a bit different, but similar principles apply.) The shroud has weight. As the shroud circumference increases it will require both a stronger tower and stronger internal supports to handle the ever-increasing weight of the shroud. It is not hard to see that as the weight of the shroud increases most of the tower and most of the structural mass of the shroud becomes dedicated to holding up the shroud itself and an ever increasing proportion of the tower/shroud mass is dedicated to holding up the extra mass that is only necessary to keep the rest of the shroud from collapsing (e.g. structural mass to hold up the structural mass that is keeping the shroud up), with an appropriate increase in the cost of the tower. The exponent is probably closer to 1.1 than to 2 or more, but exponential is the proper term here.
Let's say you need 1000 kg for 10 meters of tower supporting another 1000 kg of load. A mass ratio of 2.
To extend that another 10 m, you need to support the above 2000 kg, so you need more beefy stuff for the next 10 m below, 2000 kg of tower.
Now you have 4000 kg to support for the next 10 m so you have to use 4000 kg of tower, 8000 for the next etc.
That's exponential.
Of course, in reality the base is less than two every 10 meters, steel is stronger per weight than that.
Though yes, on the other hand, the bending moment grows linearly only with height, and different buckling things are only power things. I don't know then if structural frequencies etc start coming in at some point.
I'm confused with the math used here. I think exponential is something like you double the height, and quadruple the costs. And that doesn't sound unrealistic to me.
I'm sure nontechnical people use the term "exponential" loosely, but this being HN, I think clarification is in order.
Consider the function
f(x) = x ^ 2
(where the caret stands for exponentiation, of course). This is called a quadratic function. Here are some example values:
0 1 2 3 4 5 6 7 8 9
0 1 4 9 16 25 36 49 64 81
This is the kind of relationship you have described: when x is doubled, f(x) is quadrupled.
Now consider this function:
f(x) = 2 ^ x
This is an example of a function that is properly called exponential. Here are some example values:
0 1 2 3 4 5 6 7 8 9
1 2 4 8 16 32 64 128 256 512
Here, every time x increases by 1, the value is doubled. See how much faster it grows?
In very practical terms, the difference between a "power law", as functions of the form x ^ k are called, and an exponential, of the form k ^ x, is massive. I grant that the terminology may be a little confusing, but this is not a pedantic distinction!
I'm sorry I didn't get that right. I should studying where I study.
Anyhow I don't think "50 more and double the cost's" is exponential?
// And I still would not think exponential is impossible function for the cost's. When you get about one kilometer high, the stuff just gets shit expensive. I mean humankind-scale expensive.
Nope. When something is exponential, a fixed increase in the input causes a multiplier in the output. So, for example, if I have the exponential y = 2^x, an increase of 1 in x, from x to x+1, increases y by a factor of two.
Look, it's a simple cost trade - currently the most economical turbines are in the 1 to 3 megawatt range and roughly 100 m rotor diameter.
Tower cost is probably exponentially related to height for example. so the things balance out at that point.