Per the Beta law you can only take approximately 2/3 of the energy, so I had no issue with him saying this,
I was responding to a posting some way back. Per Betz you can take a little over half (16/27, about 59.3%) and you get it when the wind leaves at a third the speed it arrived. Practical turbines at utility scale peak about 75% to 80% of this.
. My issue is the VAWT works on a 3-dimensional plane around a curved path and is influenced by any number of directions. Only perfectly linear airflow from a single direction can achieve the maximum theoretical amount.
Not really - and if it did, so what, as you only get the correct direction for a vanishingly small distance at particular points in the rotation. We're still talking a Darrieus here (for the main part of the turbine), right? Adriaan has it right.
For a Darrieus blade the important things are this:
1: At the trailing edge, the air leaves at right angles to the radius of the rotor.
2: At the leading edge, once it's spinning adequately, the air attaches at the wide range of incident angles at which the apparent wind arrives.
3: Between them, again once it's spinning adequately, the drag is low (mainly, the air moves in laminar flow.)
The forward (rotation-wise) thrust on a Darrieus blade comes from bending the airflow. It leaves straight backward (right angles to the radius), so its entire velocity is "backward". But it arrives at the same speed (neglecting drag), but usually at some other angle. So the component along the motion of the blade was originally somewhat less. Effectively the blade applied a force to the air to bend its path, "pushing" it to the rear with energy obtained by "pushing" it against crosswise motion. The reaction force from pushing it to the rear (minus the component of drag at right angles to the rotor radius) is what pushes the blade forward.
The leading edge of the blade sees the apparent wind: The vector sum of the actual wind and the motion of the blade. Neglecting, for now, any slowing of the wind by another blade's transit on the upwind side:
As the blade goes around a full circle, the wind component of the relative wind experienced by the blade also goes around a full circle (in the opposite direction). But the self-motion component of the relative wind is always from the front. So (for TSRs greater than 1) the incoming wind swings back and forth by +- arcsine(1/TSR). For a TSR of 6 that's about +- 9.6 degrees.
(For TSR 3 it's about 19.5, for TSR 1 it's 90. The formula blows up below that because it assumes apparent wind from the front and below TSR 1 the wind comes from all directions, including straight backward.)
To get this to provide working thrust you need the attachment to work from a range of angles. But the forward thrust you get depends on the amount you bend the wind. (Again neglecting drag: 1-cos(bend angle)).
With the apparent wind directly from the front (when the blade is going due upwind or downwind) you get no thrust (but you still have drag). You lose power that you have to replace with power gained in other parts of the cycle.
With the wind from the side (the middle of the crosswind traverse) the apparent wind's angle relative to the blade's motion (and the amount it becomes bent) is maximum, as is the thrust. (Drag is also a bit higher, but not drastically more). This is where you get your peak power.
Between the leading and the trailing edge you really don't care what the air is up to, as long as it doesn't substantially increase drag.
Now a Darrieus behaves like this if the blade is short compared to the diameter, so the angular error at the leading and trailing edge is very small. And if the blade is long and cambered so the centerline of the (otherwise symmetric) airfoil is at constant radius, it also behaves this way. But what happens if it's long but not cambered?
Let's look at one that's long enough to subtend three degrees, running at TSR6 so the total angular error is about a third of the apparent wind's maximum offset to one side.
We'll position the trailing edge so it's at right angles to the radius, to maximize conformity to item 1: That means the leading edge is pointed outward about 3 degrees.
- On the upwind and downwind run this pulls the air inward slightly, and thus increases drag, but only slightly.
- On the windward side crosswind run it cuts the maximum angle of the apparent wind from 9.5 to 6.5 degrees. Airflow bend is reduced, thrust is reduced, energy extracted from the wind is reduced, speed of the wind leaving this first blade interaction is higher.
- On the leeward side crosswind run it increases the maximum angle of the apparent wind from 9.5 to 12.5 degrees. Airflow bend is increased, thrust is increased, energy extracted from the wind is increased. Also: The wind is arriving with less slowing from the upwind interaction, so there's more energy to be pulled from it.
The energy pulled from the wind (including losses) is only dependent on the speeds of the wind on its way in, on its way out, and other energy in the outgoing wind from things like turbulence or spin. So if the outgoing air is about the same as with the cambered blades, and drag didn't substantially increase, at first glance it looks like all we did is move some of the energy extraction from the upwind to the downwind blade transits. This is OK. In fact it's desirable: The downwind transit sees air that is both slowed by the upwind stage and has a sidewise component added that also reduces the apparent wind. With available wind energy proportional to the CUBE of the wind speed, though the two transits might normally slow the wind by the same amount (about 1/3 at Betz optimum) the upwind transit has half again the thrust and collects a substantially larger proportion of the energy. Moving some of the collection leads to better balance and lower peak forces.
But when we look at details it's even better: The upwind transit (where the wind is at the max) would normally see the apparent wind at a greater angle than the downwind transit (where it's slowed but the blade isn't, effectively raising the local TSR and lowering the angle). But without the camber it sees it at a lower angle. At low speeds, where the angles become high enough that the wind doesn't attach for a full cycle, the fraction of the cycle where it iS attached is raised. This helps with startup.
So it seems to me that the extra expense and work of cambering the blades to conform to the circumference only matters if they're long in proportion to the diameter of the mill, and then it actually LOSES advantages you'd otherwise get from the deviations from the leading edge pointing at an "ideal" angle.
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On the other hand, if you abandon lining up the trailing edge of an uncambered blade, but line up the middle instead, you get rid of the extra drag from pumping air slightly inward - at the cost of having half the above angular error at the leading and trailing edge.
At the leading edge it doesn't particularly matter, as the air will still attach nicely. In the middle it doesn't particularly matter, as momentarily moving the air slightly inward and then back out in laminar flow won't do much, if anything, to the drag (and nothing else matters in the middle). At the trailing edge the error does matter, as directing the outgoing airstream at a slight angle to the circle's tangent means you've lost a bit of your drive. It's a cosine function, so the first couple degrees don't make much difference. But if you want to get that last bit you can do it by adding a camber to just the last part of the blade. That should be a lot less trouble than cambering the whole blade.
