rcoopster said:
The rotational velocity of the wheel is determined by the potential flow rate of how fast air can fill a vacuum, but the real question that I asked previously, is what will the pressure differential be on the wheel fin submerged in the airflow. By the time the wheel fin enters into the airflow, the flow already has substantial velocity and forward momentum, which is created by what I am deeming to be called “atmospheric pressure, and I believe that this forward velocity and momentum, where the air flow is already moving with a velocity which is close to the wheel fin, greatly reduces the net pressure difference exerted on the wheel fin to a pressure well below atmospheric pressures difference compared to a vacuum.
Okay, let's see if I can answer some of these questions. Let's start with the misconceptions inherent in your statement above.
The first obstacle is that the vacuum wheel as you have drawn it cannot work. Consider the air flowing into the entrance. The way you have drawn it, the flow approaches a sharp corner with vacuum on one side, and then continues merrily in the direction it was flowing, ignoring the vacuum next to it, and I'm guessing you think that somehow this momentum has been "earned" and that the flow won't significantly change direction. Compressible flows absolutely do not behave like that. In a convergent passage like you have drawn, with anything less than half an atmosphere of pressure downstream, the flow will accelerate to Mach 1. When that flow exhausts into a vacuum, it expands around the corner in what is called a "
Prandtl-Meyer fan". The Mach number continues to rise as the flow turns the corner. Going from M=1 to M=infinity, the flow goes from axial and turns 130 degrees, i.e. it flows back upstream into the vacuum channel.
Now, the way you have depicted the wheel working, that accelerated flow runs straight into the vane coming out of the vacuum channel. This hypersonic flow rams into the vane, and what you would see would be a complicated set of shocks, reflecting and intersecting and moving with the vane, with a subsonic layer next to the vane material and the hub, because these strong shocks would decelerate the hypersonic flow back to a subsonic condition, generating a lot of entropy in the process. It's very difficult to compute the result of this moving set of shocks and expansions.
As the vane passes by the entrance channel, the flow field gets more complicated. Part of the flow slams into the vane from behind, generating another set of compression shocks and sneaking flow upstream into the vacuum channel again. It's hard to tell, exactly, but I think it is likely the flow will reach the next vane after shocking off the outer curved walls and expanding around the inner curved walls before that vane has a chance to move appreciably. The other part of the flow expands around the vane and generates a set of expansions and shocks. Meanwhile, the curved main channel that you intended the flow to go down will also throw off a set of shocks because supersonic flow doesn't just change direction without shocking (there is one exception, for highly uniform flow at a fixed Mach number you can design a curved compression surface that will isentropically compress the flow, but this surface (a) is not circular, and (b) only works for one particular Mach number).
So for these reasons the wheel will not provide anything approaching a steady vacuum to draw flow into the device. It will stall and sputter as the flow intermittently fills the vacuum channel as the vanes pass the entrance.
That's why it's practically impossible to provide a reasonable answer to how much power the device will demand. But let's push on some of your other questions. For example you ask
rcoopster said:
For example what would the net pressure differential on a 1 sq ft plate moving 650 ft/sec in the same direction as a 645 ft/sec air flow? Would it be similar to a 5 ft/sec flow?
Again, compressible flows don't behave like this. Even in a one-dimensional setting, if the vane is moving at 650 ft/s, the flow behind and ahead of the vane will be moving at 650 ft/s. So your question doesn't make any sense. It's not like a vane or propeller in water, where there is a possibility of cavitation. The flow will accelerate or decelerate to match the boundary conditions. And the boundary conditions are of primary importance. You cannot calculate the pressure on a surface just knowing the speed of the surface.
rcoopster said:
How much power would it take to rotate the wheel at 1800 rpms?
How much power can be produced by the airflow which will naturally flow through the wheel spinning at 1800 rpms?
Maybe the two independent calculations will be exactly the same, I don’t know, but I bet that if you honestly and independently calculated the two numbers the later would be higher than the first.
No, it won't. And in fact once you take into account all the entropy-generating shocks in the system, it takes considerably more power to pump the flow than you can extract from it.