# Conservation of Angular Momentum on a rotating disc

• CullenWillett
In summary: When gamma is high, the tangential velocity initially decreases from the inlet (r2) up to a certain value of r/r2 at which it attains its minimum value. At lower values of r/r2, it increases again as the effect of the angular momentum conservation starts to dominate.
CullenWillett
I have a disc that is rotating due to air being blown at its outer radius. The incoming relative velocity of the air is high, therefore the effect of friction supersedes the effect of conservation of angular momentum. The tangential portion of this velocity decreases due to the friction as it travels (swirls) towards the inner radius of the disc. When the tangential velocity reaches its minimum, the tangential velocity then begins to increase due to the conservation of angular momentum.

How does tangential velocity increase due to conservation of angular momentum and not just stay at its minimum value?

Thank you.

CullenWillett said:
supersedes the effect of conservation of angular momentum.
You cannot 'supersede' that conservation. Momentum is Transferred. Are you asking where, on the disc, most of the transfer takes place?

I think a photo or diagram would help.

sophiecentaur said:
You cannot 'supersede' that conservation. Momentum is Transferred. Are you asking where, on the disc, most of the transfer takes place?

The supersede part makes sense to me when considering the angular momentum of the entire system. But could you explain how momentum transfers in this scenario? My understanding is that the momentum from the air flow transfers to the disc via friction, thus slowing down the flow to a minimum value while the disc gains rotational speed. But I do not understand how flow could then increase due to conservation of angular momentum. I feel like flow would remain at this minimum value until it exits the disc.

CWatters said:
I think a photo or diagram would help.

I am sorry I should have included this in the main message. When I say disc I am referring to the discs within a tesla turbine. I have attached an image of the discs, the air enters at the outer radius r2 and exits the turbine through the central exit at r1.

The specific paper that I am referencing is "A theory of Tesla disc turbines" by Sayantan Sengupta and Abhijit Guha. My question derives from the statement:
"For a high value of gamma=Vtheta/(angular velocity*r2), the relative tangential velocity is high, therefore the effect of friction may supersede the effect of conservation of angular momentum. This is why, when gamma is high, tangential velocity initially decreases from the inlet (r2) up to a certain value of r/r2 at which it attains its minimum value. At lower values of r/r2, it increases again as the effect of the angular momentum conservation starts to dominate."

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CullenWillett said:
angular momentum conservation starts to dominate.
This concept is basically flawed. If it appears to you that angular momentum is not conserved then you need to approach the problem differently. The disc is not isolated and neither is the gas around it so you can't look for or apply conservation. There will be a velocity profile of gas with distance from the disc surface, Whenever there is a difference in velocity between the disc surface and the gas flow, momentum will be transferred.

CWatters
CullenWillett said:
"For a high value of gamma=Vtheta/(angular velocity*r2), the relative tangential velocity is high, therefore the effect of friction may supersede the effect of conservation of angular momentum. This is why, when gamma is high, tangential velocity initially decreases from the inlet (r2) up to a certain value of r/r2 at which it attains its minimum value. At lower values of r/r2, it increases again as the effect of the angular momentum conservation starts to dominate."

As I read it, that passage discusses whether friction or momentum is dominant in different regions. It does not imply that momentum is not conserved. A passage like that is difficult to interpret in any language. It should be accompanied by equations and curves to make the point.

CullenWillett and Merlin3189
anorlunda said:
whether friction or momentum is dominant in different regions.
But they are two different quantities. How can one dominate the other? The brakes on a car will produce a Force which, when applied for a certain time (the Impulse) will change the Momentum of the car. The only difference here is that it's angular momentum being discussed, rather than linear momentum. The interaction between the gas and the turbine will be due to transfer of momentum by friction. I think this has to be true because there are no obstructions (blades) in the axial direction so the only transfer can be through lateral forces over the surfaces of the discs - that can only be a frictional effect as friction is the only force on the disc surfaces..
Any vortices formed in between the discs will involve angular momentum and that can transfer to the discs by friction. So I think your question could be modified to ask when the transfer is greater or less by straight tangential force from free flowing gas or by a tangential couples from the gas vortices. Presumably the vortices will increase as the angular velocity of the discs increases.

CullenWillett said:
have a disc that is rotating due to air being blown at its outer radius.
If air is being blown at the outer radius, there is an external torque acting on the disk. If an external torque is acting on the disk, its angular momentum is not conserved.

kuruman said:
its angular momentum is not conserved.
And who would expect it to be? The angular momentum of the whole system is what is conserved. The disc is not isolated so conservation of the disc's angular momentum is not a meaningful concept. This is my point. The disc accelerates because it gains angular momentum, transferred from the moving gas.

CullenWillett
It might have been more clear if the author had distinguished between the tangential force on the gas due to friction versus the tangential force on the gas due to Coriolis. It is the use of the term "angular momentum" (and especially to its non-conservation) to refer to the same effect as Coriolis which is off-putting to me.

sophiecentaur said:
This concept is basically flawed. If it appears to you that angular momentum is not conserved then you need to approach the problem differently. The disc is not isolated and neither is the gas around it so you can't look for or apply conservation. There will be a velocity profile of gas with distance from the disc surface, Whenever there is a difference in velocity between the disc surface and the gas flow, momentum will be transferred.

If I were to blow a finite amount of air into the turbine and no more, and then say that my control volume contains the discs and the finite amount of air, could I apply the conservation of momentum to that cv?

I believe that in this cv, the air transfers its momentum to the discs via shear force, which retards the air flow until it reaches a minimum point. At the minimum point there is no more momentum transfer since there is no longer a gradient between the flow and the disc. From this point on there are no momentum transfers.

anorlunda said:
As I read it, that passage discusses whether friction or momentum is dominant in different regions. It does not imply that momentum is not conserved. A passage like that is difficult to interpret in any language. It should be accompanied by equations and curves to make the point.

In your opinion, does it make sense that the flow velocity could increase due to a dominate momentum?

I am starting to think that the author is considering flow velocity to increase after its minimum value due to the increase in pressure at this minimum point, when the flow slows down to its minimum it will have a build up of pressure so then the flow will want to travel towards the center (if it has a lesser pressure at the center than the minimum point).

sophiecentaur said:
Any vortices formed in between the discs will involve angular momentum and that can transfer to the discs by friction. So I think your question could be modified to ask when the transfer is greater or less by straight tangential force from free flowing gas or by a tangential couples from the gas vortices. Presumably the vortices will increase as the angular velocity of the discs increases.

Is there a possibility that these vortices could somehow cause the fluid to speed up near minimum fluid speed (aka maximum disc speed) or would the tangential force from free flowing gas and tangential couple from gas vortices create the same outcome that is speeding up the rotation of the disc? The concept of vorticity in a turbine like this is quite challenging to think about.

CullenWillett said:
In your opinion, does it make sense that the flow velocity could increase due to a dominate momentum?
sophiecentaur said:
But they are two different quantities.

I said it poorly. I should have said that the passage does not suggest that momentum is not conserved.

sophiecentaur
anorlunda said:
I said it poorly. I should have said that the passage does not suggest that momentum is not conserved.
Any motion involving gases in containers cannot be analysed in terms of momentum conservation because there is always an external force due to the container walls. Some momentum will always be transferred out of the container. The wording is poorly chosen and the OP has latched onto the conservation idea and associated it with a different quantity. I think that starting from scratch would be a good idea with this problem, with the right terms used.

sophiecentaur said:
Any motion involving gases in containers cannot be analysed in terms of momentum conservation
In this case it is even worse. We have external transfers of angular momentum from the gas entering the system, from the gas leaving the system and from the torque produced by the turbine.

jbriggs444 said:
In this case it is even worse. We have external transfers of angular momentum from the gas entering the system, from the gas leaving the system and from the torque produced by the turbine.
Also , the only interaction between the gas and the sides of the disc has to be frictional because the walls are smooth / flat.

sophiecentaur said:
Any motion involving gases in containers cannot be analysed in terms of momentum conservation because there is always an external force due to the container walls. Some momentum will always be transferred out of the container. The wording is poorly chosen and the OP has latched onto the conservation idea and associated it with a different quantity. I think that starting from scratch would be a good idea with this problem, with the right terms used.

What if we disregard the solid walls and the flow on the outside of the two discs and are just looking at the flow in between the rotating discs, nowhere else. In this case the concept of conservation could be applied to the discs and fluid, correct?

Now the model is simplified to exclude any friction via bearings, container walls, and more and only include flow through a microchannel that is rotating and the discs. I believe in this case conservation of momentum is applicable to the system albeit a rather simplified model but atleast a starting point.

CullenWillett said:
What if we disregard the solid walls and the flow on the outside of the two discs and are just looking at the flow in between the rotating discs, nowhere else. In this case the concept of conservation could be applied to the discs and fluid, correct?

Now the model is simplified to exclude any friction via bearings, container walls, and more and only include flow through a microchannel that is rotating and the discs. I believe in this case conservation of momentum is applicable to the system albeit a rather simplified model but atleast a starting point.

It's reasonable to say that some or most of the net momentum of the incoming gas will be transferred to the disc but I would say that the turbulence which provides coupling of that aspect of the angular momentum between gas and disc is bound to be lossy and that some angular momentum has to be lost. The other coupling, between the linear momentum of the gas and angular momentum of the disc is also relevant. I think your original question is basically about the relative amounts of each. But each involves friction - there is no other mechanism to make the disc spin and that's why I questioned your original dichotomy. You are right to try to apply the momentum conservation principle, as with any collision problem. You need to include the momentum of the exit gas too, because the mass flow rate in and out must be the same - apart from any slight effect of pressure change with speed.

sophiecentaur said:
It's reasonable to say that some or most of the net momentum of the incoming gas will be transferred to the disc but I would say that the turbulence which provides coupling of that aspect of the angular momentum between gas and disc is bound to be lossy and that some angular momentum has to be lost. The other coupling, between the linear momentum of the gas and angular momentum of the disc is also relevant. I think your original question is basically about the relative amounts of each. But each involves friction - there is no other mechanism to make the disc spin and that's why I questioned your original dichotomy. You are right to try to apply the momentum conservation principle, as with any collision problem. You need to include the momentum of the exit gas too, because the mass flow rate in and out must be the same - apart from any slight effect of pressure change with speed.
This post puzzles me.

First, it is not true that the momentum of the incoming gas will be transferred to the disk. Note the conditions specified in #19. No torque at the bearings. No drag from the walls. That means that in steady state, zero net angular momentum is transferred to the system from the mass flow. This presumably means that the turbine is spinning fast enough that any angular momentum carried in by the input stream is carried out by a very rapidly spinning vortex in the output stream.

Next, there is no loss of angular momentum due to any sort of coupling. Angular momentum is a conserved quantity. It cannot be lost. You can lose rotational kinetic energy. But you cannot lose angular momentum.

Next, the mass flow rate in must match the mass flow rate out. Full stop. Pressure changes do not enter in. Those would affect the volumetric flow rate.

Finally, the most relevant factor for pressure change is likely not Bernoulli but rather centripetal acceleration. You have high pressure for the high velocity gas at the rim and low pressure for the low velocity gas in the middle -- the opposite of Bernoulli.

CullenWillett
jbriggs444 said:
This post puzzles me.
Yes, it's a bit muddled -sorry.
1. I meant that any effect must be due to friction.
2. There is a linear force produced by the straight passage of gas. This provides a couple, directly.
3. Once the gas gets turbulent (due to interaction with the moving discs) the gas has a different distribution of angular momentum and that couples - again by friction - with the discs.
I imagine that, as the side speeds up, the transfer by a direct couple will be less because the speeds of the outer bits of the discs will be similar to the gas speed.
The only links I could find are quite complicated but it seems that the reduced pressure in the gap produces the vortices. The matter of the Momentum is difficult and you must be right that a constant disc speed implies no momentum transfer. But the behaviour of an engine with no load is not straightforward - same for an electric motor which has an infinite speed limit in the absence of resistive losses because the back emf is never quite the same as the supply volts. Presumably there is an equivalent to back emf and losses in this turbine which will limit the top speed. How useful or practical is it, to do the no-load analysis? As soon as the turbine does work, conservation doesn't apply because it is outputting Impulse on its shaft. I wonder if the extra condition in #19 doesn't take things out of the realms of usefulness. Turbines are all about momentum transfer to a load. ( Post #19 was added a bit late in the day, perhaps.)

Yes, Mass flow rate will be constant if the turbine is enclosed. I'm not sure that the Bernouli effect doesn't cause the vortices.

CullenWillett
Thank you very much for all the help. You responses have been very beneficial.

## What is conservation of angular momentum?

Conservation of angular momentum is a fundamental principle in physics that states that the total angular momentum of a system remains constant as long as there are no external torques acting on the system.

## How does conservation of angular momentum apply to a rotating disc?

In the case of a rotating disc, conservation of angular momentum means that the total angular momentum of the disc remains constant as long as there are no external torques acting on it. This means that any changes in the rotational speed of the disc must be accompanied by an equal and opposite change in the distribution of mass within the disc.

## What is the equation for angular momentum?

The equation for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. This equation shows the relationship between the amount of angular momentum in a system and the mass distribution and rotational speed of that system.

## How does conservation of angular momentum affect the motion of objects on a rotating disc?

Conservation of angular momentum affects the motion of objects on a rotating disc by causing them to move in a circular path with a constant tangential velocity. This is because, as the object moves closer to the center of rotation, the distribution of mass in the system changes, causing the object to speed up to maintain the constant angular momentum.

## What are some real-life examples of conservation of angular momentum?

One example of conservation of angular momentum is the motion of a figure skater spinning on the ice. As the skater pulls their arms in, they decrease their moment of inertia, causing them to spin faster to maintain their angular momentum. Another example is the rotation of planets around the sun, which is a result of the conservation of angular momentum in the solar system.

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