What are the key concepts and formulas for Rayleigh and Fanno flows?

[Urmi Roy]

There doesn't seem to be any proper online source that describe these flows properly.

Please explain the following points that i read about on wikipedia:

1. "The heat addition causes a decrease in stagnation pressure, which is known as the Rayleigh effect and is critical in the design of combustion systems"...why is this so important?

2. " unlike Fanno flow, the stagnation temperature is a variable"-- is this because we're considering uniform cross-sectional area and no mass change, so due to heat addition/removal the net energy at any point will vary?

3. Can anyone tell me where I could find a derivation of the formulae related to Rayleigh flow found in the wikipedia article http://en.wikipedia.org/wiki/Rayleigh_flow?...similarly for Fanno flow.

Thanks a lot!

 

[Urmi Roy]

No replies?? :-( Please help!

 

[boneh3ad]

For your first question, imagine you don't add heat. Note te static pressure. You then isentropically slow the flow to zero and note the stagnation pressure. Now imagine adding the head, you are essentially slowing the flow without affecting the static pressure, so if you take this heated flow and isentropically bring it to rest, it will not have as much to slow down before reaching zero and the pressure won't increase as much as the unheated flow. In other words, the stagnation pressure is lower.

For the second question, the stagnation temperature is a variable because you are adding or removing heat from the system, meaning you are increasing (or decreasing) the stagnation enthalpy and thus stagnation temperature for a calorically perfect gas. In Fanno flow there is no heat added.

You can find these equations derived in many compressible flow books. I know I originally learned it from Oosthuizen and Carscallen, but you can find it in other books. It is helpful to know the derivations of things such as the isentropic flow relations before attempting Rayleigh flow or Fanno flow.

 

[Urmi Roy]

For your first question, imagine you don't add heat. Note te static pressure. You then isentropically slow the flow to zero and note the stagnation pressure. Now imagine adding the head, you are essentially slowing the flow without affecting the static pressure, so if you take this heated flow and isentropically bring it to rest, it will not have as much to slow down before reaching zero and the pressure won't increase as much as the unheated flow. In other words, the stagnation pressure is lower.

...When we heat a gas,we agitate it..so that it gains more energy and so it should take a greater effort to bring it to rest and hence the stagnation pressure should be greater?...but that doesn't match what you said...

 

[boneh3ad]

You don't have to agitate a gas to heat it. Additionally, if you look at Rayleigh flow, adding heat actually slows the flow down.

 

[Urmi Roy]

You don't have to agitate a gas to heat it.

Actually I meant that heating automatically agitates a fluid, and always increases the entropy.

Additionally, if you look at Rayleigh flow, adding heat actually slows the flow down.

My original question is rather why it is in Rayleigh flow that heat slows the flow down, as intuitively, I expect the opposite, as I explained above.I didn't find this point explained at all in any source.

 

[boneh3ad]

Actually I meant that heating automatically agitates a fluid, and always increases the entropy.

Not true. Entropy and agitation are also disjoint concepts. True, heating a fluid often agitates it, such as when you heat a pot of water and the resulting temperature gradient causes the water to circulate in a process called Rayleigh-Bénard convection. However, in the case of Rayleigh flow, there is already so much movement in the fluid that any gradients introduced by heating it are going to be so much smaller than what actually exists in the flow that it isn't actually going to agitate the flow. You shouldn't look at the two as one in the same. They can often be related, but they are not necessarily so.

My original question is rather why it is in Rayleigh flow that heat slows the flow down, as intuitively, I expect the opposite, as I explained above.

You should have said this in the original post then. The best description I can think of off the top of my head is that increases in entropy due to heating tend to bring the flow closer to Mach 1, so they will accelerate a subsonic flow and decelerate a supersonic flow. A good, physical explanation escapes me at the moment, but mathematically it is fairly easy to show that it does happen.

 

[Urmi Roy]

Not true. Entropy and agitation are also disjoint concepts. True, heating a fluid often agitates it, such as when you heat a pot of water and the resulting temperature gradient causes the water to circulate in a process called Rayleigh-Bénard convection.

Rayleigh benard convection is the 'normal' convection that we talk about? Whereas Rayleigh flow is at supersonic speeds?

The best description I can think of off the top of my head is that increases in entropy due to heating tend to bring the flow closer to Mach 1, so they will accelerate a subsonic flow and decelerate a supersonic flow. A good, physical explanation escapes me at the moment, but mathematically it is fairly easy to show that it does happen.

Is this what you are referring to :
(this is specifically about what happens in a converging nozzle, but I think you are referring to a similar concept...I mean the whole density change thing)

For supersonic flows,M > 1, as the area decreases velocity also decreases, and as the area increases, velocity also increases. We can explain thisbehaviour like this. In response to an area change all the static properties change. At subsonic speeds changes in density are smaller. The velocity decreases when there is an increased area offered (and vice versa). But in case of a supersonic flow with increasing area density decreases at afaster rate than velocity. In order to preserve continuity velocity now increases (and decreases when area is reduced). vice versa).

 

[boneh3ad]

Rayleigh benard convection is the 'normal' convection that we talk about? Whereas Rayleigh flow is at supersonic speeds?

No, convection is a much broader term. Rayleigh-Bénard convection is just one manifestation of convection, particularly in a fluid heated from below. It is unrelated to Rayleigh flow. Rayleigh just did a lot of stuff.

Is this what you are referring to :
(this is specifically about what happens in a converging nozzle, but I think you are referring to a similar concept...I mean the whole density change thing)

For supersonic flows,M > 1, as the area decreases velocity also decreases, and as the area increases, velocity also increases. We can explain thisbehaviour like this. In response to an area change all the static properties change. At subsonic speeds changes in density are smaller. The velocity decreases when there is an increased area offered (and vice versa). But in case of a supersonic flow with increasing area density decreases at afaster rate than velocity. In order to preserve continuity velocity now increases (and decreases when area is reduced). vice versa).

I suppose that would be one convoluted way of looking at it. If you just look at the derivation of the equations though, you can see how the two cases (subsonic and supersonic) differ as a result of the governing equations (which, after all, are just the physics translated into equations). The bottom line is that in a compressible flow, there are simply more variables that can change, including density, so the traditional wisdom that one develops when studying incompressible flow doesn't hold at all.

 

[Urmi Roy]

One more question...in Fanno flow, Boneh3ad said that there is neither heat or work transfer...but there is friction in Fanno Flow, so how come we do not consider it in the energy equation for fanno flow and thus, why is the stagnation temperature constant throughout the flow?

 

[Urmi Roy]

Is it because friction and viscosity are not counted as external forces?