# Paradox of a convergent nozzle fed by an electric fan

• I
Let us assume we have a cylindrical wind tunnel having a 0.5 m diameter fed by an electric fan. The cross-sectional area of the wind tunnel would be A1 = (PI/4) D1^2 = 0.196349541 m2. Let us suppose the motor driving the fan has a power rating of 1,500 W. At this stage, let us assume that the electric motor is consuming its maximum power rating, and that the measured airflow velocity is V1 = 20 m/s, just at the output of the wind tunnel. The kinetic power P1 of the airflow just at the output of the wind tunnel can be calculated as by applying the formula P1 = (1/2) (density of air) (A1) (V1^3) = 962.11 W. If we viewed the wind tunnel plus the fan and the electric motor as a system (System 1), we could say that the efficiency of this system 1 would be System 1 Efficiency = 100 P1/1500 = 64.14%.

At this stage, let us consider a new system (System 2), formed by system 1 plus a 2:1 ratio convergent nozzle attached to the output of system 1. Now the output of system 2 would have a cross-sectional area A2 given by A2 = 0.09817477 m2. If now the electric motor is ran at its maximum power rating as before, can we expect the velocity of the airflow just at the output of the convergent nozzle to be V2 = 40 m/s?

If that were the case, the kinetic power P2 of of the airflow just at the output of the convergent nozzle would be P2 = (1/2) (density of air) (A2) (V2^3) = 3,848.45 W. Notice that now the power of the output airflow would be greater than the input power to the system (1, 500 W), which would apparently be a violation of first law of thermodynamics. Moreover, the efficiency of system 2 would be System 2 Efficiency = 100 P2/1500 = 256.56%, which apparently constitute a violation of second law of thermodynamics also.
Is there a flaw in this analysis?

## Answers and Replies

russ_watters
Mentor
At this stage, let us assume that the electric motor is consuming its maximum power rating, and that the measured airflow velocity is V1 = 20 m/s, just at the output of the wind tunnel....

At this stage, let us consider a new system (System 2), formed by system 1 plus a 2:1 ratio convergent nozzle attached to the output of system 1. Now the output of system 2 would have a cross-sectional area A2 given by A2 = 0.09817477 m2. If now the electric motor is ran at its maximum power rating as before, can we expect the velocity of the airflow just at the output of the convergent nozzle to be V2 = 40 m/s?...

Is there a flaw in this analysis?
Welcome to PF.

The flaw is that you didn't do any real analysis of the fan's performance; you just multiplied the outlet velocity by 2!

Real fans have performance curves that must be used to analyze the fan's performance when attached to a given system.

• eudesvera3
.Scott
Homework Helper
The nozzle will create back pressure on the fan - and that will slow the airflow and the fan.

• eudesvera3 and russ_watters
Welcome to PF.

The flaw is that you didn't do any real analysis of the fan's performance; you just multiplied the outlet velocity by 2!

Real fans have performance curves that must be used to analyze the fan's performance when attached to a given system.

Thank you russ_waters. You are certainly right, I did not actually do any real analysis of the fan's performance. Sorry about that. However I just wanted to point out the issue whether it was possible at all that a system formed by an electric fan, a wind tunnel and a convergent nozzle could produce an airflow power greater than the motor power. But your opinion is certainly very valuable to me.

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• russ_watters
The nozzle will create back pressure on the fan - and that will slow the airflow and the fan.
Scott, I appreciate very much your opinion. It seems very reasonable to me that the back pressure on the fan will not allow the airflow power to go beyond the motor power. In other words an output airflow velocity of 40 m/s would never be possible in this particular system and hence no violation of thermodynamic laws could occur.   Dale
Mentor
2020 Award
If now the electric motor is ran at its maximum power rating as before, can we expect the velocity of the airflow just at the output of the convergent nozzle to be V2 = 40 m/s?
Clearly not, as your subsequent analysis shows.

Is there a flaw in this analysis?
Yes, the non physical assumption that a nozzle increases the power of a fluid flow.

• eudesvera3
Clearly not, as your subsequent analysis shows.

Yes, the non physical assumption that a nozzle increases the power of a fluid flow.

Well, the purpose of my post was to learn from your opinions and to try to clarify this apparent paradox, because I am sure you will admit that in the case there was not an artificial airflow but a wind flow as it happens in the Invelox wind turbine, if the velocity of the wind flow at the entrance of a 2:1 ratio convergent nozzle is 20 m/s, then the velocity at the output of the CN will be 40 m/s, and the output wind power would be 4 times the wind power at the entrance of the CN. So the convergent nozzle in this case does increase the power of the wind flow. Thank you for your most valuable opinion.  Dale
Mentor
2020 Award
if the velocity of the wind flow at the entrance of a 2:1 ratio convergent nozzle is 20 m/s, then the velocity at the output of the CN will be 40 m/s,
Assuming an incompressible flow that is always the case. But that does not mean that your conclusion follows.

I will post details later

russ_watters
Mentor
Well, the purpose of my post was to learn from your opinions and to try to clarify this apparent paradox, because I am sure you will admit that in the case there was not an artificial airflow but a wind flow as it happens in the Invelox wind turbine, if the velocity of the wind flow at the entrance of a 2:1 ratio convergent nozzle is 20 m/s, then the velocity at the output of the CN will be 40 m/s...
That's true, but do you see the difference between this and the scenario you first proposed?
...and the output wind power would be 4 times the wind power at the entrance of the CN.
That isn't true. You are neglecting to account for pressure. Conservation of energy applies. This is the point of Bernoulli's principle.

Pro tip: if you are ever not sure if conservation of energy applies, the answer is that it does apply.

• bob012345 and eudesvera3
That's true, but do you see the difference between this and the scenario you first proposed?

That isn't true. You are neglecting to account for pressure. Conservation of energy applies. This is the point of Bernoulli's principle.

Pro tip: if you are ever not sure if conservation of energy applies, the answer is that it does apply.

I agree, conservation of energy always apply. But within the nozzle the increase of kinetic energy at the narrower end of the nozzles comes about at the expense of a reduction of the enthalpy of the flow at that end in such a way that the total energy at that end is exactly the same as the total energy at the entrance of the nozzle. So energy conservation is kept within the nozzle as it should outside. However, how do you reconcile the apparent fact that the power of the airflow coming out of the narrow end at 40 m/s is 4 times the power of the airflow coming into the wide end at 20 m/s? There seems to be an incongruence here, but the only explanation I can find here is that the total energy entering the nozzle is composed of kinetic energy and thermal plus pressure energy (enthalpy), and the sum of these two must be preserved at all times. Nevertheless, the kinetic energy does not require preservation, Thus it is perfectly possible for the kinetic power at the output of the nozzle to be 4 times greater than the kinetic power at the input of the nozzle, and that would not constitute a violation of conservation of energy on account of the fact that the thermal energy (or more exactly, the enthalpy) of the airflow is reduced (i.e., a drop in temperature and pressure takes place in the very exact amount as the kinetic energy increases. So I contend unless I am proven wrong that the kinetic power at the output of a 2:1 ratio convergent nozzle is 4 times greater than the kinetic power at the entrance of the CN. Thank you for your opinion and for reading my post.

Dale
Mentor
2020 Award
However, how do you reconcile the apparent fact that the power of the airflow coming out of the narrow end at 40 m/s is 4 times the power of the airflow coming into the wide end at 20 m/s?
The power of the airflow at the exit is not 4 times the power at the inlet. That is where you are going wrong. The KE is 4 times greater, but the hydraulic energy (pressure * volume) is reduced.

As @russ_watters mentioned, this is covered by Bernoulli’s principle. For an incompressible horizontal flow any increase in ##\frac{1}{2}\rho v^2## is accompanied by an equal decrease in ##p##. So the total energy is unchanged, as it must be by a device that does no work.

kinetic power at the output of a 2:1 ratio convergent nozzle is 4 times greater
The term “kinetic power” is not one that I have heard before, but assuming that it means what I think it means then “kinetic power” is not the same as “power”. So the above objections stand as worded, but similar statements specifying “kinetic power” would be non controversial beyond possible semantic objections. Certainly it is no paradox.

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• eudesvera3
The power of the airflow at the exit is not 4 times the power at the inlet. That is where you are going wrong. The KE is 4 times greater, but the hydraulic energy (pressure * volume) is reduced.

As @russ_watters mentioned, this is covered by Bernoulli’s principle. For an incompressible horizontal flow any increase in ##\frac{1}{2}\rho v^2## is accompanied by an equal decrease in ##p##. So the total energy is unchanged, as it must be by a device that does no work.

The term “kinetic power” is not one that I have heard before, but assuming that it means what I think it means then “kinetic power” is not the same as “power”. So the above objections stand as worded, but similar statements specifying “kinetic power” would be non controversial beyond possible semantic objections. Certainly it is no paradox.

Thank you for clearing this up for me. Probably I did not express myself very well. What I meant to say was precisely what you has just pointed out, i.e., that the KE at the exit is 4 times greater than the KE at the inlet.

Regarding the term "kinetic power", I am sorry for my poor choosing of words as I am still in the learning process of the English language. What I meant to say was that the rate of change of KE at the exit is 4 times the rate of change of KE at the inlet. Since these two physical quantities I believe are measured in Watts hence the use of the term kinetic power to describe them.

Now, going back to the system formed by the electric motor, the fan, the wind tunnel and the 2:1 ratio convergence nozzle I wonder if the power contained in the KE of the airflow at the exit could be greater than the motor power. Scott has already said that is not possible because of the increase in pressure that takes place on the fan blades and reduces the airflow velocity. Kindly could you give me your opinion about that?

Dale
Mentor
2020 Award
I wonder if the power contained in the KE of the airflow at the exit could be greater than the motor power.
No (obviously). This is a direct result of Bernoulli’s principle, which is simply conservation of energy.

Scott has already said that is not possible because of the increase in pressure that takes place on the fan blades and reduces the airflow velocity. Kindly could you give me your opinion about that?
Both @.Scott and @russ_watters are correct. You can use Bernoulli’s principle to calculate the pressure at the inlet of the nozzle and from thence the energy required by the fan.

• eudesvera3
No (obviously). This is a direct result of Bernoulli’s principle, which is simply conservation of energy.

Both @.Scott and @russ_watters are correct. You can use Bernoulli’s principle to calculate the pressure at the inlet of the nozzle and from thence the energy required by the fan.
I am most grateful to you for your prompt and clear response. Now I can see there is no paradox at all.

@eudesvera3 please read about Bernoulli's Principle: https://en.wikipedia.org/wiki/Bernoulli's_principle
Thanks for the suggestion. I think I understand Bernoulli's principle. It's just that for any given motor power the airflow velocity at the inlet of a convergent nozzle can not be arbitrarily assumed as I was wrongly doing. If this airflow velocity is arbitrarily assumed then the airflow velocity at the nozzle exit can be wrongly calculated although Bernoulli's equation is properly used.

• Dale
russ_watters
Mentor
As best as I can tell, this is a scenario like the OP describes. Here is a random fan curve: If not connected to a system, the fan is allowed to free-blow, and the fan curve tells us the airflow is 2550 CFM, or a velocity of 3072 fpm.

However, the outlet velocity in the selected configuration is 1802 fpm (1550 CFM) and the static pressure 4.0" w.g. Velocity pressure is often ignored when it is a small fraction of static pressure, but in this case it is 0.2". If the only thing attached to this fan is a nozzle, a nozzle ratio of 4.55:1 will provide a final velocity of 8208 fpm, using up all that 4.2" of available total pressure (outlet static pressure is 0/atmospheric).
• This is how Bernoulli's principle/the Venturi effect works, for incompressible flow, with no losses.
• This is how real fans work.
Note, @.Scott in this case the fan is turned by a 3-phase motor and the rpm is fixed. For some motors/fans, the rpm will drop if you add a nozzle, as you said.

#### Attachments

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We were discussing incompressible flows here. This is a reasonable approximation for low Mach numbers, and here we are dealing with around Mach 0.1, so the incompressible flow is a good assumption.

I don’t believe this. Please cite the scientific reference that you believe teaches this.

If I may, I would like to try to reach some conclusions for this thread, and please correct me if I am wrong.

Conclusion 1. When a convergent nozzle is attached to a fan driven by an electric motor it is impossible due to the back pressure generated by the nozzle for the kinetic power of the airflow at the nozzle exit to be greater than the motor power.

Conclusion 2. If a convergent nozzle is used to accelerate wind before applying it to a wind turbine, the kinetic power of the wind flow at the exit of the nozzle will always be greater than the kinetic power of the wind flow at the inlet of the nozzle. As an example, if a 2:1 ratio convergence nozzle is used, the outlet kinetic power will be exactly 4 times greater than the inlet kinetic power.

For both conclusions, it is assumed that the flow is inviscid, incompressible, steady and horizontal. In addition, by kinetic power it is meant the following physical quantity:(1/2) times (cross-sectional area seen by airflow) times (velocity of the airflow at such area to the cubic power). In other words, kinetic power of a flow is simply its kinetic energy per unit time.

Dale
Mentor
2020 Award
Looks good to me, although I didn’t check the math on that last part.

• eudesvera3
Looks good to me, although I didn’t check the math on that last part.

Thank you Dale. I feel rewarded with a better understanding of convergent nozzles thanks to your contribution to the discussion and that of the other participants. It is a shame that convergent nozzles do not seem to be very productive for raising airflow speed without increasing back pressure on the fan. If that difficulty could be sorted out somehow, then convergent nozzles would be a great help to increase airflow velocity.

On the other hand, the usefulness of convergent nozzles for increasing the power generated by wind turbines seem to be out of question, although they also increase the drag power. Nevertheless, since current augmented wind turbines can not exceed the theoretical Jamieson's limit on efficiency (88.89%), convergent nozzles can do nothing to improve this efficiency, unless some modification of these turbines can be made, but this has not occurred yet, to the best of my knowledge. Perhaps the research team to which I belong could suggest something soon in this respect.

• Dale
russ_watters
Mentor
If I may, I would like to try to reach some conclusions for this thread, and please correct me if I am wrong.

Conclusion 1. When a convergent nozzle is attached to a fan driven by an electric motor it is impossible due to the back pressure generated by the nozzle for the kinetic power of the airflow at the nozzle exit to be greater than the motor power.

Conclusion 2. If a convergent nozzle is used to accelerate wind before applying it to a wind turbine, the kinetic power of the wind flow at the exit of the nozzle will always be greater than the kinetic power of the wind flow at the inlet of the nozzle. As an example, if a 2:1 ratio convergence nozzle is used, the outlet kinetic power will be exactly 4 times greater than the inlet kinetic power.
These are correct, though I want to add a quick caveat to make sure about a critical issue with #2, since you didn't acknowledge it before: While the kinetic energy is 4x greater at the outlet than inlet, that should not be construed to believe the inlet kinetic energy is equal to a situation where there nozzle wasn't there to begin with. In other words, the inlet kinetic energy is lower with the nozzle in place than without it.

....your next post seems to imply you understand that, but I wanted to make it clear.
For both conclusions, it is assumed that the flow is inviscid, incompressible, steady and horizontal. In addition, by kinetic power it is meant the following physical quantity:(1/2) times (cross-sectional area seen by airflow) times (velocity of the airflow at such area to the cubic power). In other words, kinetic power of a flow is simply its kinetic energy per unit time.
Looks like you just forgot the density of the air, but otherwise ok.

• eudesvera3
These are correct, though I want to add a quick caveat to make sure about a critical issue with #2, since you didn't acknowledge it before: While the kinetic energy is 4x greater at the outlet than inlet, that should not be construed to believe the inlet kinetic energy is equal to a situation where there nozzle wasn't there to begin with. In other words, the inlet kinetic energy is lower with the nozzle in place than without it.

....your next post seems to imply you understand that, but I wanted to make it clear.

Looks like you just forgot the density of the air, but otherwise ok.
These are correct, though I want to add a quick caveat to make sure about a critical issue with #2, since you didn't acknowledge it before: While the kinetic energy is 4x greater at the outlet than inlet, that should not be construed to believe the inlet kinetic energy is equal to a situation where there nozzle wasn't there to begin with. In other words, the inlet kinetic energy is lower with the nozzle in place than without it.

....your next post seems to imply you understand that, but I wanted to make it clear.

Looks like you just forgot the density of the air, but otherwise ok.

Right, I forgot the density of air, which is very important. Now, regarding your caveat I find most interesting your comment that the inlet kinetic energy is lower with the nozzle in place than without it and I wonder if that is due to the drag although small introduced by the nozzle. If that is the case, the difference between both energies should not be large, I think, and the introduction of the nozzle will certainly be advantageous and advisable. I also wonder if there is a limitation as to the maximum area ratio the convergent nozzle should have. I would dare to say that the only limitation on this ratio should not be other than keeping the airflow velocity under 0.3 Mach.

Another aspect I would like your opinion on is regarding the maximum nozzle slope that is suitable to use with wind turbines if we aim to reduce the length of the whole system. Thanks in advance.

russ_watters
Mentor
Right, I forgot the density of air, which is very important. Now, regarding your caveat I find most interesting your comment that the inlet kinetic energy is lower with the nozzle in place than without it and I wonder if that is due to the drag although small introduced by the nozzle. If that is the case, the difference between both energies should not be large, I think, and the introduction of the nozzle will certainly be advantageous and advisable.
It's because the added back pressure causes air to spill out around the funnel instead of going through it. And for a wind turbine, since wind has no static pressure to convert to kinetic energy, conservation of energy demands that the funnel won't help at all.

It's because the added back pressure causes air to spill out around the funnel instead of going through it. And for a wind turbine, since wind has no static pressure to convert to kinetic energy, conservation of energy demands that the funnel won't help at all.

So, according to your appreciation, the use of a funnel in wind turbines like Sheerwind Invelox turbine (, https://www.sciencedirect.com/science/article/pii/S0360544214002837) is just baloney? And would you please explain why the wind do not have static pressure, despite being a fluid? Is it because it is not a steady flow?