The most efficient nozzle shape

[T C]

TL;DR Summary

I want to know what is the shape of a convergent nozzle to get best result.

We all know that a convergent nozzle can increase the velocity of a fluid at the throat/exit than the inlet. But there are different type of nozzles available in market now like parabolic nozzles, straight nozzles and elongated nozzles. I want to know which shape/geometry is the best to get the maximum efficiency for a fixed inlet to throat ratio. Suppose the velocity would be in the subsonic level throughout the process.

 

[anorlunda]

Efficiency of what? Define what you mean.

 

[T C]

Efficiency means the velocity at the exit would be as close to the theoretical value as possible. Suppose we have a nozzle having 5:1 ratio i.e. the inlet to throat area ratio is 5:1. That means the velocity at the exit would be five times that of inlet. I want to know with wihich geometry, the real velocity at the exit/throat would be as close to 5 times that of the inlet as possible. That I will consider to be the most efficient.

 

[russ_watters]

Efficiency means the velocity at the exit would be as close to the theoretical value as possible. Suppose we have a nozzle having 5:1 ratio i.e. the inlet to throat area ratio is 5:1. That means the velocity at the exit would be five times that of inlet.

That only applies if your fluid is incompressible at the conditions of your flow. Is it? If that's the case, then every nozzle will satisfy your requirements and provide 100% efficiency by that definition of efficiency. Conservation of mass demands it. Usually, though, efficiency in such a case is measured by energy loss, which in that case come from static pressure loss.

That aside, the basic answer to your question is that there is no single answer to your question. That's why there is more than one nozzle design.

For more on when the flow is compressible:
https://www.learnthermo.com/T1-tutorial/ch08/lesson-C/pg07.php

 

[boneh3ad]

Suppose the velocity would be in the subsonic level throughout the process.

This is literally always the case in a converging nozzle. The flow cannot be anything other than subsonic.

Efficiency means the velocity at the exit would be as close to the theoretical value as possible. Suppose we have a nozzle having 5:1 ratio i.e. the inlet to throat area ratio is 5:1. That means the velocity at the exit would be five times that of inlet. I want to know with wihich geometry, the real velocity at the exit/throat would be as close to 5 times that of the inlet as possible. That I will consider to be the most efficient.

If that's the case, then every nozzle will satisfy your requirements and provide 100% efficiency by that definition of efficiency. Conservation of mass demands it.

@russ_watters is correct in pointing out that your 5:1 comment is only true for an incompressible flow. However, he is not correct in stating that every nozzle will satisfy your requirements. Conservation of mass demands that the stated efficiency metric be satisfied on average when you compare the inlet and the exit. In the real world, viscosity causes the actual contraction ratio to be different from the one set by the nozzle. Conservation of mass states that the mass flow into the nozzle equals the mass flow out. Mathematically, this is
$$0 = \int_{A_{\mathrm{out}}}\rho \vec{u} \cdot\;d\vec{A} - \int_{A_{\mathrm{in}}}\rho \vec{u}\cdot \;d\vec{A}$$
If $\rho$ and $u$ are constant at the inlet and exit, then your statements both old.

In a real situation, even for an incompressible flow where $\rho$ is constant, $u$ is not constant. A boundary layer exists that is going to make both the inlet and the exit effectively smaller than the physical geometry dictates. This effect will be more pronounced at the exit, so the real contraction ratio will tend to be larger than what you would measure from the dimensions of your nozzle. Your outlet flow would be a bit faster. Viscosity also dissipates energy.

Usually, though, efficiency in such a case is measured by energy loss, which in that case come from static pressure loss.

That aside, the basic answer to your question is that there is no single answer to your question. That's why there is more than one nozzle design.

Honestly, for his criterion, the basic answer is simply "the biggest nozzle you can find" because that would mean the constrictions from the boundary layer are minimized relative to the physical dimension of the nozzle. Otherwise, the answer usually revolves around whichever nozzle minimizes separation in the boundary layer, which is a much more complicated question to answer.

EDIT: Fixed my equation to include vector notation since I left it off before and it therefore wasn't technically correct.

 

[T C]

Honestly, for his criterion, the basic answer is simply "the biggest nozzle you can find" because that would mean the constrictions from the boundary layer are minimized relative to the physical dimension of the nozzle. Otherwise, the answer usually revolves around whichever nozzle minimizes separation in the boundary layer, which is a much more complicated question to answer.

Many many thanks for your detailed answer. Just one more question. You have mentioned "as big as you can nozzles". I just want to know with a practical (applicable) point of view that what sizes of nozzles can be considered to be "big" here.

 

[boneh3ad]

Many many thanks for your detailed answer. Just one more question. You have mentioned "as big as you can nozzles". I just want to know with a practical (applicable) point of view that what sizes of nozzles can be considered to be "big" here.

$$D \gg \delta$$

 

[T C]

D is diameter here, but what is

δ

here?

 

[boneh3ad]

Boundary layer thickness

 

[T C]

Boundary layer thickness

And how to find it out?

 

[boneh3ad]

Well that's the million dollar question. There are various approximate methods as well as full blown CFD. None are particularly beginner friendly.

 

[T C]

Seems to be tough to understand. Kindly just tell me what's the average layer thickness for market available nozzles. And to be more precise, let's assume that we have a 15 cm radius inlet and 3 cm throat nozzle. What can be the possible thickness in such a case?

 

[boneh3ad]

Seems to be tough to understand. Kindly just tell me what's the average layer thickness for market available nozzles. And to be more precise, let's assume that we have a 15 cm radius inlet and 3 cm throat nozzle. What can be the possible thickness in such a case?

That's not a thing that I or probably anyone else is going to be able to answer for you. The whole point was that your criterion was a poor one and that just making your nozzle gigantic would maximize your definition of efficiency.

Also, if you are saying to assume a given inlet and exit diameter already, you've already effectively set the size. Now the difference in $\delta$ comes down to the actual shape, which is not a trivial thing to answer. You'd need, as I said before, a CFD package or to use some lower fidelity estimation process, but neither is something suitable for someone without a decent amount of background knowledge of fluid dynamics.

 

[jedishrfu]

This is a question that is better posed to professional engineers working for companies making this type of nozzle to insure that you get the best soltuon for your project.

Having said that, its time to close this thread.