Angular velocity inside a vortex tube

[T C]

I hope many here know about vortex tube and its application. I myself have done some study but were unable to find one answer. If we know the speed of the inlet fluid, then how can the angular velocity of the fluid during its rotational motion can be determined.

 

[boneh3ad]

That depends entirely on the design of the blades and none of us will have that information.

 

[T C]

That depends entirely on the design of the blades

As far as I know, vortex tubes don't contain any kind of blade or moving part.

 

[boneh3ad]

They absolutely do. The picture you posted previously in another thread even showed them. There's no way to generate the rotation otherwise.

 

[T C]

There are many kind of vortex tubes available in market now and most of them don't have fin/blade.

 

[boneh3ad]

In which case it will depend on the design of the "swirl chamber", which we also know nothing about.

 

[T C]

Actually I am looking for answer from someone who has good idea about the working principles of vortex tube.

 

[boneh3ad]

Good luck getting answers with an attitude like that.

 

[cjl]

For what it's worth, boneh3ad is one of the members here with the most expertise in fluid dynamics, especially relating to high speed compressible flows. He would be an excellent resource to listen to.

 

[T C]

Vortex tube is a little bit different. It can be started with conservation of momentum and conservation of energy principle. In short, what a vortex tube does is converting linear velocity and momentum to angular velocity and momentum. A very well written paper researcher Raoul LIew can be found here.
The main reason behind all the functions of the vortex tube is that the fluid inside is rotating like a solid body i.e. with same angular momentum inside. For compressible fluids this gives rise to a unique phenomenon. Both pressure and temperature rises from centre to the periphery. In short, if the consider the vortex inside as a combination of one hot and one cold flow, then the cold flow is is expanding and the hot flow is being compressed by the centrifugal force. Both compression and expansion are adiabatic in nature.
In the Eqn. 6.10 of the above mentioned paper, Dr Liew gives an equation regarding pressure rise at the hot flow. Another unique feature of the vortex tube is that energy is added to the hot flow not only as enthalpy but also as kinetic energy. That means the hot flow also has higher velocity than the cold flow.

 

[boneh3ad]

Vortex tube is a little bit different. It can be started with conservation of momentum and conservation of energy principle.

This does not make a vortex tube any different from any other fluid flow device or problem. Literally all of them obey conservation of mass, momentum, and energy.

For compressible fluids this gives rise to a unique phenomenon. Both pressure and temperature rises from centre to the periphery.

This is not unique, nor is it a feature only of compressible flows. Any vortex will feature lower pressure near its center than along its edges. It's a rotational motion, after all, so there is a centripetal force required to turn the flow around the vortex. That centripetal force is provided by a pressure gradient directed outward from the center of curvature (i.e. pressure is larger near the edges than at the center), leading to a net inward force. This is true whether the flow is compressible or not. Temperature change is the part of this that is a feature of compressible flows. The temperature will not change much (if at all) in an incompressible flow.

At any rate, the key point here is that compressible flows, like any other flows, are beholden to the various theorems related to vorticity (Kelvin, Helmholtz, Crocco). The most important point here is that a flow that is initially irrotational will remain that way unless it is acted upon but some external rotational force. In other words, it won't just spontaneously start rotating. There needs to be a specific means of generating that rotation, either through vanes (like the picture you posted in the previous thread) or through the design of the inlet such as that the flow begins to rotate (such as the wikipedia link you posted). Either way, determining the rotation rate requires knowledge of the design of those vanes or the inlet, and we simply don't have that information.

 

[JBA]

Note: This was submitted before seeing the above post.

By Confromal Flow Theory the non-forced flow from a source to a sink is naturally a vortex flow; but, in reality, a deflecting feature, which can consist of a slight angle of the incoming supply flow, a turning vane or even a short spiral feature (whether intentional or not) at the entrance, is needed to insure the initiation of a vortex.

The angular velocity at any point along the vortex tube is a function of the ratio of the tube diameter at the point of interest to the angular gas velocity and diameter at the tube entrance; and, the ratio of the gas temperatures at those two points.

 

[T C]

Is there any formula available that can connect the angular velocity with the parameters you described? And from which the angular velocity inside can be calculated if we know the other parameters?

 

[boneh3ad]

I am not aware of such a formula. Fluid mechanics generally does not lend itself to simple plug and chug formulae.

 

[T C]

Is there any way to calculate that out? After all, there should be some relation.

 

[boneh3ad]

Again, that is going to be very geometry-dependent. There is no general answer. You would need to know the geometry of the device in question and either make simplifying assumptions to the governing equations that allow you to arrive at a solution analytically or you'd need to study the phenomenon via simulation.