Visualising Compressible Flow

[fg37eeehuI]

Something I was looking into while back but still not really found an answer, so though I would try you lot.

I had been reading a beginners fluid mechanics book, which briefly introduced compressible flow with examples of nozzles, shock waves, Rayleigh and Fanno flow. It runs through the use of equations for basic calculations, but it doesn’t go very far with descriptions of what is physically happening with the fluid. Something I can’t seem to find with online searches either.

I think I’ve got my head round the movements and property changes to the fluid in sub/supersonic nozzles with isentropic flow. But when it looks at the effects of heat addition/removal and the effects of friction on sub/supersonic flows in a duct, I don’t seem to be able to picture/visualise what is happening that gives the strange changes in the fluid properties.

-- Below was the first stab at what is happening in Rayleigh flow, maybe someone will be able to see where I’m going wrong.--

Ideal gas with constant mass flow rate, no frictional effects, through constant cross-section duct with heat addition or removal.

Subsonic - heat added

I can see a dense gas expanding with the heat addition, the expansion pushing out giving this increase in velocity downstream, and with the relatively slow flow, acting somewhat against the upstream flow. So gives a drop in density and pressure across the heating point.

Subsonic – cooling

The flow energy is reduced with cooling and it slows (would there be a contraction?), this slower wall of gas has the upstream flow pushing on it increasing the pressure and density.

Supersonic – heat added

Having a guess here. The increase in energy is causing the molecules to become more excited and that increases the pressure, this wall of pressure is acting against the flow and slows it, but as the flow is moving very fast it doesn’t affect the upstream. It just puts a break on it as it passes the heating point. The flow backs-up downstream increasing the density and pressure further.

Supersonic - cooling

A guess. The reduction in the energy the molecules have after cooling, gives the reduction in pressure. This low pressure void that’s created allows the upstream to push through into the lower pressure downstream increasing the velocity. The molecules are moving off at a higher velocity giving the reduction in density.

-- After a reply on another forum the thought process changed to, --

Should I be thinking more towards the idea of the heating or cooling at the duct wall as creating a sort of artaficial diffuser or nozzle?

The property changes (Velocity, Density, Pressure, Temp) for supersonic heating match that of a supersonic diffuser (reduction in cross section in flow direction). And with someone mentioning it emulating friction losses its got me thinking maybe its just a reduction in the cross section of the main flow that gives the property changes.

Supersonic cooling matches a supersonic nozzle.
Subsonic heating matches a subsonic nozzle, but not the temp other than when Ma > 1/k^1/2.
Subsonic cooling matches a subsonic diffuser, but not the temp other than when Ma > 1/k^1/2.

Hopefully that makes sense.

Cheers

s34n

 

[Snickster]

I will try to answer your first round of questions first as I am not sure what you are really saying in the second revised set:

Ideal gas with constant mass flow rate, no frictional effects, through constant cross-section duct with heat addition or removal.

Subsonic - heat added

I can see a dense gas expanding with the heat addition, the expansion pushing out giving this increase in velocity downstream, and with the relatively slow flow, acting somewhat against the upstream flow. So gives a drop in density and pressure across the heating point.

Response: Right gas will expand with the addition of heat expanding the volume of gas at constant pressure (note the actual process is a constant pressure process because at any given point in the pipe dX the pressure will be constant) while also increasing the temperature and therefore internal energy at the same time increasing the velocity. So looking at the finite diffential of length dX containg that volume it appears to be just like adding heat to a non flow system volume with piston arrangement, with the difference being that you also increase velocity. So you are adding enery in the form of heat that is going into increasing the internal energy (temperature), gas expansion doing boundary work pdv and also incresing velocity/kinetic energy.

So on the inlet to the differental volume contained in dX you have u1 + (p1)v1 + KE1 plus heat added equals conditions at outlet boundary conditions;

= u2 + (p1)v2 + KE2

So how would you get subsonic flow with no friction in a constant area pipe? If the differential pressure between the source and the destination is such that there is only enough differential pressure to get the flow in the pipe inbetween to subsonic velocity at that low of a differential pressure. In such case the velocity will immediately increase at the pipe inlet to a constant velocity following an isentropic expansion process (pressure and temperature decrease isentropically) and remain constant velocity and pressure throughout since there is no pressure drop. That is until you add heat somewhere when all the enegy terms change as described above.

Or if the pressure differential is high enough to impart enough energy when coverted into velocity that is enough to reach sonic velocity in the pipe. The sonic wave front will travel to the end of the pipe and form a higher pressure above the downstream pressure source right at the exit of the pipe (with any crossectional area upstream at steady flow conditions). Then it will enter the destination vessel and decrease to that lower pressure via ireverasble flow sonic shock waves dissapating that higher pressure energy. This is because based on equations of flow, once sonic flow is reached in a constant area pipe it can only be increase to a higher velocity if the area of the pipe increases like in a an expanding nozzle. Therefore in a constant area pipe the maximum flow can only be sonic and if it occurs it will occur at the end of the pipe of that constant diameter. For instance if you had a source vessel at 500 psia and a destination vessel at 50 psia then sonic velocity will exist just at the exit of the pipe and will be close to 0.5 upstream static pressure or 250 psia. The the 250 psia pressure at the exist of the pipe will dissipate in the lower pressure vessel via sonic shock waves.

In any case at any cross section of the line the pressure will be constant - that is heat is added at constant pressure.

Subsonic – cooling

The flow energy is reduced with cooling and it slows (would there be a contraction?), this slower wall of gas has the upstream flow pushing on it increasing the pressure and density.

Response: This is just the reverse case of the above. Removing heat make the density increase and therefore volume shrinks at constant pressure p1v1 < p1v2, velocity decreases v2 <v1 and internal energy/temperature decreases.

(CONTINUED)

 

[Snickster]

Lets see if I can breakdown what is going on here having done a lot of compressible flow calcs but not specifically looking at this process I will see if I can figure out what is going on:

Supersonic – heat added

Having a guess here. The increase in energy is causing the molecules to become more excited and that increases the pressure, this wall of pressure is acting against the flow and slows it, but as the flow is moving very fast it doesn’t affect the upstream. It just puts a break on it as it passes the heating point. The flow backs-up downstream increasing the density and pressure further.

Response: If the flow is sonic in the pipe it will be as described above with sonic conditions existing only at the exith of the pipe since greater than sonic flow cannot exist in the pipe length. Base on fricitional equation of flow, once sonic pressure is reached in a pipe the area would have to increase to achieve higher supersonic velocity therefore the flow choked and the sonic wave always travels to the end ot the pipe. At the end of the pipe the pressure will be exactly pressure of the downstream vessel if there is just enough upstream pressure to reach sonic flow but no more. If more than enough pressure upstream then flow will reach critical sonic flow. In other words since velocity or kinetic energy cannot increase anymore above sonic conditions at the pipe exit then the addtional pressure above that kinetic energy will basically be transfered from the upstream vessel to just at the exit tip of the pipe then decrease pressure to the lower pressure vessel via sonic shock waves.

That being said there will be sonic velocity at the exit of the pipe and critical flow pressure at the exit of the pipe at sonic Temperature = 2/k+1 times the upstream vessel stagnation pressure. Then the pressure will remain constant all the way upstream (since no friction) until just at the entrace to the pipe where the pressure increases because it has dropped from the upstream vessel due to an isentropic expansion since we are not considering friction. So in the upstream vessel the flow goes through and isentropic expansion to sonic velocity and pressure and temperature. This is very theoritical because in a pipe there is always friction so friction would cause the critical flow sonic velocity and pressure/temperature at the end of the pipe to increase the pressure and decrease the velocity as you travel up the pipe to the inlet, then you would increase pressure and decrease the velocity to zero in the upstream vessel via a reversed isentropic expansion (working backwards from the pipe exit).

In any case you have the same situation as the subsonic flow case where when you add heat along any dX the pipe you adding it at a steady flow condition of constant pressure, temperature and velocity at location.

So now when you add the heat you are adding energy so the same thing should happen as with subsonic flow case where the heat added goes into increasing internal energy, the pv term (pressure at that point is expanding a boundary just like in a no-flow volume piston arrangement) and KE. The difference is that sonic velocity still can only exist at the exit of the pipe. So as the gas eexpands due to the addition of heat and the velocity increases to sonic at the end of the pipe but at the higher temperature due to the increas in internal energy, so you have the same mass flowrate at higher temperature so a higher sonic velocity since c = SQRT(gkrt). Therefore if you work it out the new critical flow sonic pressure at the end of the pipe will decrease to about SQRT(T2/T1) or in other words P2/P1= SQRT(T2/T1) Where P2-T2 is the new sonic flow pressure/temperature at the tip of the pipe exist and P1-T1 was the critical flow sonic pressure/temperature at the tip of the pipe before the heat was added.

So the heat again went into increase in internal energy u, boundary work pv by expanding the hot gas against its own boundaries, and increase of velocity/KE, but the difference is at the exit of the pipe where sonic flow exists the temperature has increased and pressure decreased. but note that the heat is no longer added at constant pressure but the pressure reduces and the temperature increases as the heat is added following the same thing that is happening at the pipe exit.

I think this is about correct but may be a little confusing because I wrote it in a thinking out loud manner. I could make sound alot better but it would take me much longer time to go through and clarify. The opposite happens in the scenario below. If you have any questions let me know.

Supersonic - cooling

A guess. The reduction in the energy the molecules have after cooling, gives the reduction in pressure. This low pressure void that’s created allows the upstream to push through into the lower pressure downstream increasing the velocity. The molecules are moving off at a higher velocity giving the reduction in density.

 

[pierreick]

Hi,
On YouTube you have a very nice set of lectures (34) by Dr J.R Biddle, I encourage you to view together with the reading of frank White's book on Fluid mechanics,5th edition.
an example of VDO.

https://www.youtube.com/watch?v=clVwKynHpB0&list=PLZOZfX_TaWAGocs2k5QmTL44OKOl7rn34

Good luck
Pierre

 

[fg37eeehuI]

Hi Snickster,

This is the book Im refering to

https://imtk.ui.ac.id/wp-content/uploads/2014/02/Fluid-Mechanics-Cengel.pdf

Chapter 12, also gives property changes on a table 12-3.
Maybe I need to re-read your reply, but it seems the downstream property changes differ for the supersonic flow?
Also sort of sounds like your saying supersonic flow cant be in the duct?

Thank you for your reply Snickster.

Lets see if I can breakdown what is going on here having done a lot of compressible flow calcs but not specifically looking at this process I will see if I can figure out what is going on:

Supersonic – heat added

Having a guess here. The increase in energy is causing the molecules to become more excited and that increases the pressure, this wall of pressure is acting against the flow and slows it, but as the flow is moving very fast it doesn’t affect the upstream. It just puts a break on it as it passes the heating point. The flow backs-up downstream increasing the density and pressure further.

Response: If the flow is sonic in the pipe it will be as described above with sonic conditions existing only at the exith of the pipe since greater than sonic flow cannot exist in the pipe length. Base on fricitional equation of flow, once sonic pressure is reached in a pipe the area would have to increase to achieve higher supersonic velocity therefore the flow choked and the sonic wave always travels to the end ot the pipe. At the end of the pipe the pressure will be exactly pressure of the downstream vessel if there is just enough upstream pressure to reach sonic flow but no more. If more than enough pressure upstream then flow will reach critical sonic flow. In other words since velocity or kinetic energy cannot increase anymore above sonic conditions at the pipe exit then the addtional pressure above that kinetic energy will basically be transfered from the upstream vessel to just at the exit tip of the pipe then decrease pressure to the lower pressure vessel via sonic shock waves.

That being said there will be sonic velocity at the exit of the pipe and critical flow pressure at the exit of the pipe at sonic Temperature = 2/k+1 times the upstream vessel stagnation pressure. Then the pressure will remain constant all the way upstream (since no friction) until just at the entrace to the pipe where the pressure increases because it has dropped from the upstream vessel due to an isentropic expansion since we are not considering friction. So in the upstream vessel the flow goes through and isentropic expansion to sonic velocity and pressure and temperature. This is very theoritical because in a pipe there is always friction so friction would cause the critical flow sonic velocity and pressure/temperature at the end of the pipe to increase the pressure and decrease the velocity as you travel up the pipe to the inlet, then you would increase pressure and decrease the velocity to zero in the upstream vessel via a reversed isentropic expansion (working backwards from the pipe exit).

In any case you have the same situation as the subsonic flow case where when you add heat along any dX the pipe you adding it at a steady flow condition of constant pressure, temperature and velocity at location.

So now when you add the heat you are adding energy so the same thing should happen as with subsonic flow case where the heat added goes into increasing internal energy, the pv term (pressure at that point is expanding a boundary just like in a no-flow volume piston arrangement) and KE. The difference is that sonic velocity still can only exist at the exit of the pipe. So as the gas eexpands due to the addition of heat and the velocity increases to sonic at the end of the pipe but at the higher temperature due to the increas in internal energy, so you have the same mass flowrate at higher temperature so a higher sonic velocity since c = SQRT(gkrt). Therefore if you work it out the new critical flow sonic pressure at the end of the pipe will decrease to about SQRT(T2/T1) or in other words P2/P1= SQRT(T2/T1) Where P2-T2 is the new sonic flow pressure/temperature at the tip of the pipe exist and P1-T1 was the critical flow sonic pressure/temperature at the tip of the pipe before the heat was added.

So the heat again went into increase in internal energy u, boundary work pv by expanding the hot gas against its own boundaries, and increase of velocity/KE, but the difference is at the exit of the pipe where sonic flow exists the temperature has increased and pressure decreased. but note that the heat is no longer added at constant pressure but the pressure reduces and the temperature increases as the heat is added following the same thing that is happening at the pipe exit.

I think this is about correct but may be a little confusing because I wrote it in a thinking out loud manner. I could make sound alot better but it would take me much longer time to go through and clarify. The opposite happens in the scenario below. If you have any questions let me know.

 

[fg37eeehuI]

Hi Pierre,

I will take a look.

Thank you

S34n

Hi,
On YouTube you have a very nice set of lectures (34) by Dr J.R Biddle, I encourage you to view together with the reading of frank White's book on Fluid mechanics,5th edition.
an example of VDO.

https://www.youtube.com/watch?v=clVwKynHpB0&li...

Good luck
Pierre

 

[Snickster]

Sorry for the confusion but my responses really are for flow with friction. With friction the flow cannot get to above sonic in a pipe and if so it will occur at the end of the pipe just as I described. I will take a look at true rayliegh flow and see if I can offer some clarity.

 

[fg37eeehuI]

No problem. Thanks Snickster.

s34n

Sorry for the confusion but my responses really are for flow with friction. With friction the flow cannot get to above sonic in a pipe and if so it will occur at the end of the pipe just as I described. I will take a look at true rayliegh flow and see if I can offer some clarity.

 

[Snickster]

S34n I will get back with you again today with further discussion. However could you clarify what you are actually having a problem with understanding.

 

[fg37eeehuI]

Hi Snickster,

It was to make sure Im understanding why we get those property changes in subsonic and supersonic cooling and heating correctly. I think mainly the not so intuituve changes in supersonic flow.
I can find the mathematics in my book and online to give numbers for the changes, but nothing giving a kind of visual picture of what is physically happening. Just looking at the numbers, you seem to jump from inlet to outlet with no real picture of what is happening inbetween.

Thanks

s34n

 

[fg37eeehuI]

Hi Snickster,

I read your deleted post.

The kind of balancing of equations, if one term reduces another must increase I understand.
Your end note, physically visuallising what is happening is I guess what Im trying to do. Like for supersonic flow with heat addition, what are the kind of physical movements happening in the fluid that cause the heat to seem to put a break on the flow?

Thanks

s34n