Paradox of a convergent nozzle fed by an electric fan

[cjl]

Oh, and as for that wind turbine, I really don't see it as being viable. The capture area is simply too small. One benefit of non-shrouded turbines is that the blades can sweep out and interact with a truly incredible quantity of air without needing an enormous amount of material themselves. On that prototype though, it looks like the capture area is maybe 100 square meters at most, while modern wind turbines are 120+ meters in diameter (leading to a capture area in the range of 10,000 square meters). I can't see how that design could reasonably be scaled up to capture wind from a 100 by 100 meter box without it ending up more complex, heavier, and more expensive than a normal, non-shrouded horizontal axis design.

 

[eudesvera3]

Dear cjL, many, many thanks to you. You has really made my day. Your post was the one I was waiting for and, unless some participant proves the opposite, the paradox of the converging fan fed by an electric fan is after all partially true. So, according to your reasoning, it would be quite possible for a 1,500 W motor to produce an airflow kinetic power of 3,848.45 W at the throat separating the converging nozzle from the diffuser, and still have an exit airflow power less than 1,500 W, so that first law of thermodynamics is preserved. Cheers!

 

[eudesvera3]

Oh, and as for that wind turbine, I really don't see it as being viable. The capture area is simply too small. One benefit of non-shrouded turbines is that the blades can sweep out and interact with a truly incredible quantity of air without needing an enormous amount of material themselves. On that prototype though, it looks like the capture area is maybe 100 square meters at most, while modern wind turbines are 120+ meters in diameter (leading to a capture area in the range of 10,000 square meters). I can't see how that design could reasonably be scaled up to capture wind from a 100 by 100 meter box without it ending up more complex, heavier, and more expensive than a normal, non-shrouded horizontal axis design.

Good points, cjl. However I firmly believe this ground-level shrouded wind turbine can be improved a lot by a new extremely efficient renewable energy machine that will hopefully be presented to the world very soon (just a matter of not more than a couple of months). I will keep you posted.

 

[cjl]

Good points, cjl. However I firmly believe this ground-level shrouded wind turbine can be improved a lot by a new extremely efficient renewable energy machine that will hopefully be presented to the world very soon (just a matter of not more than a couple of months). I will keep you posted.

It's not a matter of efficiency. It's a matter of collection area and material quantity. At the end of the day, you can't extract more than ~59% of the kinetic energy that would pass through an area equal to your collection area in the freestream. This applies whether you use a collector design, a traditional turbine, or anything else. Current horizontal-axis wind turbines are already very close to this limit, so you aren't going to make more power without increasing the collection area or the design wind speed.

 

[Dale]

This is why wind tunnels nearly always have a diffuser on them

Is the size of the diffuser typically equal to the inlet size or is it usually even larger?

 

[eudesvera3]

It's not a matter of efficiency. It's a matter of collection area and material quantity. At the end of the day, you can't extract more than ~59% of the kinetic energy that would pass through an area equal to your collection area in the freestream. This applies whether you use a collector design, a traditional turbine, or anything else. Current horizontal-axis wind turbines are already very close to this limit, so you aren't going to make more power without increasing the collection area or the design wind speed.

IMO, you are right if the wind turbine only extracts kinetic energy from the wind and nothing else as all conventional wind turbines do. In fact, all conventional wind turbines decelerate the incoming airflow considerably. Using another approach it is possible to extract other type of energy from the wind rather than KE only.

 

[eudesvera3]

Is the size of the diffuser typically equal to the inlet size or is it usually even larger?

IMO, the size (length) of the diffuser should be at least two or three times greater than the size (length) of the converging nozzle, in order to make deceleration of airflow as gradual as possible so as to reduce as much as possible turbulence losses along the diffuser before exhausting into ambient.

 

[cjl]

Is the size of the diffuser typically equal to the inlet size or is it usually even larger?

Most of the ones I've seen have a diffuser exit area similar to the nozzle inlet area, but I'm far from an expert on wind tunnel design (I mostly do wind turbine design). Also, most of the ones I've seen have the fan at the exit of the diffuser, rather than the inlet of the nozzle, but this is primarily to reduce turbulence in the test section.

 

[cjl]

IMO, you are right if the wind turbine only extracts kinetic energy from the wind and nothing else as all conventional wind turbines do. In fact, all conventional wind turbines decelerate the incoming airflow considerably. Using another approach it is possible to extract other type of energy from the wind rather than KE only.

It's not possible to extract energy from the wind other than kinetic. You can't reduce the pressure because you don't have a low pressure reservoir to dump the exhaust into, and you can't extract thermal energy because you don't have a cold reservoir to dump the thermal energy into. Every device that extracts power from wind will decelerate the incoming airflow considerably - that's where the power is coming from.

 

[Dale]

Every device that extracts power from wind will decelerate the incoming airflow considerably - that's where the power is coming from.

I have always wondered how that works. How can the wind be decelerated? The Mach number is low so the flow should be incompressible, but with an equal inlet and outlet area I don’t see how that works.

 

[russ_watters]

I have always wondered how that works. How can the wind be decelerated? The Mach number is low so the flow should be incompressible, but with an equal inlet and outlet area I don’t see how that works.

The wind acts like it is in an expanding duct:

http://home.uni-leipzig.de/energy/ef/15.htm

 

[russ_watters]

Thank you. Can you tell me if I can close this thread and how to do it?

There isn't a way for users to close their own threads and we generally keep them open in case something else relevant comes up.

 

[eudesvera3]

It's not possible to extract energy from the wind other than kinetic. You can't reduce the pressure because you don't have a low pressure reservoir to dump the exhaust into, and you can't extract thermal energy because you don't have a cold reservoir to dump the thermal energy into. Every device that extracts power from wind will decelerate the incoming airflow considerably - that's where the power is coming from.

It's not possible to extract energy from the wind other than kinetic. You can't reduce the pressure because you don't have a low pressure reservoir to dump the exhaust into, and you can't extract thermal energy because you don't have a cold reservoir to dump the thermal energy into. Every device that extracts power from wind will decelerate the incoming airflow considerably - that's where the power is coming from.

I am sorry to disagree. It is quite possible to extract thermal energy from the wind, according to I Hirshberg (https://patents.google.com/patent/EP1841544A4) and the evidence you can see on airplane wings due to the condensation of water vapour due in turn to the drop in temperature of the airflow. On the other hand augmented or shrouded wind turbines can theoretically achieve an efficiency up to 88,88%, according to Jamieson (https://www.researchgate.net/profile/Peter_Jamieson2/publication/238184620_Beating_Betz_Energy_Extraction_Limits_in_a_Constrained_Flow_Field/links/574dbca308ae8bc5d15bf497/Beating-Betz-Energy-Extraction-Limits-in-a-Constrained-Flow-Field.pdf). However by extracting thermal unit from air or wind the efficiency could go beyond that limit,

 

[eudesvera3]

There isn't a way for users to close their own threads and we generally keep them open in case something else relevant comes up.

Thank you. It seems a reasonable policy to me.

 

[russ_watters]

I am sorry to disagree. It is quite possible to extract thermal energy from the wind, according to I Hirshberg (https://patents.google.com/patent/EP1841544A4) and the evidence you can see on airplane wings due to the condensation of water vapour due in turn to the drop in temperature of the airflow. On the other hand augmented or shrouded wind turbines can theoretically achieve an efficiency up to 88,88%, according to Jamieson (https://www.researchgate.net/profile/Peter_Jamieson2/publication/238184620_Beating_Betz_Energy_Extraction_Limits_in_a_Constrained_Flow_Field/links/574dbca308ae8bc5d15bf497/Beating-Betz-Energy-Extraction-Limits-in-a-Constrained-Flow-Field.pdf). However by extracting thermal unit from air or wind the efficiency could go beyond that limit,

Airplanes don't utilize natural wind. Natural wind is far too slow - unless in a hurricane or tornado - for compressibility to be significant.

Also; patents are not scientific sources, they are protection for inventions. The governments that issue patents care very little if the devices actually work.

...though actually this application doesn't even appear to have been accepted anyway.

 

[eudesvera3]

Airplanes don't utilize natural wind. Natural wind is far too slow - unless in a hurricane or tornado - for compressibility to be significant.

Also; patents are not scientific sources, they are protection for inventions. The governments that issue patents care very little if the devices actually work.

Thank you for your very valuable opinion, but we have to agree to disagree on this particular point. I hope forthcoming events in the field of renewable energy will prove which viewpoint is correct, yours or mine. If it is yours, I will humbly acknowledge it.

 

[Dale]

the evidence you can see on airplane wings due to the condensation of water vapour due in turn to the drop in temperature of the airflow.

The Mach numbers are substantially higher for an airplane wing. I wouldn’t use the incompressible flow assumption for an airplane, but I would for a wind turbine.

 

[russ_watters]

Thank you for your very valuable opinion, but we have to agree to disagree on this particular point. I hope forthcoming events in the field of renewable energy will prove which viewpoint is correct, yours or mine. If it is yours, I will humbly acknowledge it.

Please be aware that PF is a mainstream science site. There is only one viewpoint accepted by mainstream science:

... the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

https://en.wikipedia.org/wiki/Bernoulli's_principle

That's 220 mph or 100 m/s. Compressibility effects can always be said to be non-zero, but at the speed wind turbines operate at, they are a tiny fraction of a percent. This is settled science.

 

[russ_watters]

That's 220 mph or 100 m/s. Compressibility effects can always be said to be non-zero, but at the speed wind turbines operate at, they are a tiny fraction of a percent.

Let me be more specific about the numbers:

• At 10m/s, a healthy wind speed for a wind turbine, the velocity pressure from Bernoulli's equation is 61 pa or 0.06% of atmospheric; a tiny fraction of a percent.
• At 100m/s, a common speed for a jet airliner soon after takeoff, the velocity pressure is 6,100 pa or 6% of atmospheric. That's enough to start to have some significance. That's why it's the commonly cited cutoff for incompressible vs compressible flow.

 

[cjl]

I am sorry to disagree. It is quite possible to extract thermal energy from the wind, according to I Hirshberg (https://patents.google.com/patent/EP1841544A4) and the evidence you can see on airplane wings due to the condensation of water vapour due in turn to the drop in temperature of the airflow. On the other hand augmented or shrouded wind turbines can theoretically achieve an efficiency up to 88,88%, according to Jamieson (https://www.researchgate.net/profile/Peter_Jamieson2/publication/238184620_Beating_Betz_Energy_Extraction_Limits_in_a_Constrained_Flow_Field/links/574dbca308ae8bc5d15bf497/Beating-Betz-Energy-Extraction-Limits-in-a-Constrained-Flow-Field.pdf). However by extracting thermal unit from air or wind the efficiency could go beyond that limit,

That patent is incorrect. Specifically, this statement is complete nonsense:

A method of converting air internal energy into useful kinetic energy is based on air flowing through substantially convergent nozzle, which accelerates the air as the cross section of the nozzle decreases thus increasing the air kinetic energy. The increment of the kinetic energy equals to the decrement of air internal energy, i.e., air temperature

In a low-speed nozzle, the vast majority of the increased kinetic energy comes from pressure drop, not temperature drop. In addition, the exhaust of the system is exhausting to ambient, so you're constrained in what the conditions in the system can actually be. You can't use the trick I talked about above either, since you're extracting energy at the minimum area point (which prevents you from being able to recover as much in the diffuser). I would also point out that whoever wrote that patent has absolutely no idea what they're talking about with wind turbines. Case in point:

When the wind turbine propeller rotates, only fraction of the flowing air within the circle created by the propeller tips is actually flowing close enough to any of the propeller blades in order to generates aerodynamic lift on that blade. These lift forces (actually their component that lies within the propeller rotating plane and tangent to circle created by the blade segment that generates said lift component) distributed along the propeller blades create rotational moments around the propeller axis. The lift forces multiplied by their respective distance from the propeller rotating axis accumulated to a certain amount of torque, which rotate the propeller blades. Since considerable amount of air is flowing between the propeller blades, this air doesn't contribute any lift or torque to the propeller. This is one reason why such a propeller uses only about 20% of the kinetic energy of the air

Most modern wind turbines extract about 45-50%, and the theoretical limit is 59 regardless of design. Despite the large gaps between the blades, the blades are moving much faster than the wind, enabling them to meaningfully interact with basically 100% of the air that flows through the disk.

Another example:

Another inherent flaw of these wind turbines is their limit to operate on strong winds. This is because the propeller blades are heavy- about 11 tons thus the centrifugal forces at high rotation speed becomes huge and there is no economic justification to design these blades to winds more than 25 meter per second.

They do not operate in strong winds because consistent winds above 25-32 m/s (the usual range of cutout speeds for modern turbines) just don't happen often enough to make a meaningful difference in the annual energy production. Centrifugal forces do not play into it at all, since modern designs have complex control systems to regulate speed and they hit full rotor speed at something like 8 m/s.

That isn't the end of the errors in the patent either, but I suspect going through and debunking everything in it would just be a waste of both my time and the time of anyone reading this, so I'll stop there.

As for that 88% number, you're misunderstanding the paper. If the flow is constrained such that it cannot flow around the turbine but must flow through it, you can achieve that 88% number. However, a turbine with a nozzle/diffuser structure does not meet this criterion. If you try to extract too much power, there will be excessive backpressure within the structure and wind will just flow around the nozzle rather than into it. This is the same mechanism that limits conventional wind turbines, and it means that the Betz limit of 59% still applies (and it will be based on the overall collecting area, not the turbine area).

 

[Dale]

Despite the large gaps between the blades, the blades are moving much faster than the wind, enabling them to meaningfully interact with basically 100% of the air that flows through the disk.

Cool, I didn't know that! Is that true even close to the center, or is it only on the tips where the blades are moving so fast?

 

[cjl]

Right at the center, there's really not much power to be extracted, but I'd say as soon as you get perhaps 20% of the way out on the blade, you can achieve about 50% extraction of the kinetic energy of the air passing through the swept area. Usually, the tip speed ratio is about 10 or a bit less for modern designs (this is a measurement of the tip speed divided by incoming air speed, and is one of the common design parameters used), so even 20% out on the blade, the tangential speed from rotation is about twice as high as the incoming wind speed.

 

[Dale]

The wind acts like it is in an expanding duct:

So the blades must slow it down and deflect it outwards.

 

[cjl]

The outwards deflection is more of a consequence of slowing the flow down than an intentional thing done by the blades (since the density doesn't meaningfully change, and mass must be conserved), but yes, that's what happens.

 

[cjl]

Let me be more specific about the numbers:

• At 10m/s, a healthy wind speed for a wind turbine, the velocity pressure from Bernoulli's equation is 61 pa or 0.06% of atmospheric; a tiny fraction of a percent.
• At 100m/s, a common speed for a jet airliner soon after takeoff, the velocity pressure is 6,100 pa or 6% of atmospheric. That's enough to start to have some significance. That's why it's the commonly cited cutoff for incompressible vs compressible flow.

Interestingly, extremely locally, you can make a reasonable argument that air doesn't act entirely incompressibly with wind turbines. Near the tip of a modern wind turbine, the relative airspeed can be in the range of 90m/s, and the blade can be making a local Cl of around 1.5. This means there are almost definitely some significant density differences between the air on the suction and pressure sides of the airfoil out there. That having been said, this only matters locally, and in terms of the global flow, you can absolutely assume incompressibility for the sake of analysis.

 

[eudesvera3]

That patent is incorrect. Specifically, this statement is complete nonsense:

Thanks for this information. If I remember correctly in a previous post you suggested that it is possible to circumvent the problem of the back pressure on the fan caused by a convergent nozzle by just adding a divergent nozzle acting as an exhaust system and following a nozzle that in turn follows thew convergent nozzle. I wonder if by taking your suggestion and for the sake of clarifying further the matter I could pose the following problem:

Let us assume a system formed by a convergent nozzle (Inlet), a throat, a fan, and a divergent nozzle (Exit), as shown schematically in Figures (a) and (b) below. The cross-sectional areas of the convergent nozzle, the throat, and the divergent nozzle as seen by the airflow are A1, A0 and A2, respectively. Additionally, let us suppose that A1 = A2 = 1780 cm2, and A0 = A1 / 2 = 890 cm2. The fan is driven by an electric motor whose power rating is PM = 1500 W. When the motor is operating at its maximum power rating, let us assume an airflow is exhausted to the atmosphere with a velocity Vφ2 = 20 m/s. Therefore, the power of the airflow just at the exit of the divergent nozzle is Pφ2 = 872.20 W. I would like you help me to find the answer to the following questions:

Q1. Which is the best position for placing the fan in the system, as in Fig. (a), or as in Fig. (b), in such a way as to obtain minimum pressure on the fan, and minimum fan noise?

Q2. If the exhaust velocity is Vφ2 = 20 m/s, can we infer that the airflow velocity in the throat would be Vφ0 = 40 m/s?

In a low-speed nozzle, the vast majority of the increased kinetic energy comes from pressure drop, not temperature drop. In addition, the exhaust of the system is exhausting to ambient, so you're constrained in what the conditions in the system can actually be. You can't use the trick I talked about above either, since you're extracting energy at the minimum area point (which prevents you from being able to recover as much in the diffuser). I would also point out that whoever wrote that patent has absolutely no idea what they're talking about with wind turbines. Case in point:
Most modern wind turbines extract about 45-50%, and the theoretical limit is 59 regardless of design. Despite the large gaps between the blades, the blades are moving much faster than the wind, enabling them to meaningfully interact with basically 100% of the air that flows through the disk.

Another example:
They do not operate in strong winds because consistent winds above 25-32 m/s (the usual range of cutout speeds for modern turbines) just don't happen often enough to make a meaningful difference in the annual energy production. Centrifugal forces do not play into it at all, since modern designs have complex control systems to regulate speed and they hit full rotor speed at something like 8 m/s.

That isn't the end of the errors in the patent either, but I suspect going through and debunking everything in it would just be a waste of both my time and the time of anyone reading this, so I'll stop there.

As for that 88% number, you're misunderstanding the paper. If the flow is constrained such that it cannot flow around the turbine but must flow through it, you can achieve that 88% number. However, a turbine with a nozzle/diffuser structure does not meet this criterion. If you try to extract too much power, there will be excessive backpressure within the structure and wind will just flow around the nozzle rather than into it. This is the same mechanism that limits conventional wind turbines, and it means that the Betz limit of 59% still applies (and it will be based on the overall collecting area, not the turbine area).