Understanding and calculating exhaust thrust

[Nicknick]

Calculating the exhaust thrust for a mechanically supercharged engine is impossible without some thermodynamic skills. It is relatively easy for turbocharged engines (simple rocket science) because of the stationary flow process through a simple nozzle, but in a purly mechanically charged engine the exhaust process is an open flow process. The exhaust velocity and the mass flow between the opening and closing of the exhaust valves is not even close to being constant, which makes it quite demanding for calculations. If you have access to 1-d engine simulation programs like GT Power or Boost, it is a relatively easy task. Without that it will be almost impossible to calculate manually unlike you are a super really extremely capable in integrating complex equations. However, with Mathlap, Origin or even Exel you could find a decent solution (with using a lot of assumptions and simplifications).

If somebody is drawing nice lines which show no connection to the engine power, you can be sure that he never really calculated he exhaust thrust!

So, this is your input data what you definitely need for an estimation of the exhaust thrust:

Cylinder:

p,t,V at the time the exhaust valve opens

dV/dt (cylinder volume over time/crank angle)

exhaust valve:

equivalent isentropic opening area over time/crank angle (dA/dt):

valve lift curve, flow curve

Ejector:

-Volume between exhaust valve and Laval nozzle

-cross section area

-smalles area (Laval cross section) of the Laval nozzle

-Expansion ratio of the Laval nozzle

-athmospheric pressure



The following simplifications would need to be done to make it manageable:

-exhaust gas will be an ideal gas (it is not..)

-no heat losses to the walls (which will give quite an error….)

-no gas friction losses

Thermodynamic model:

Basically, the exhaust model can be described by the cylindervolume, the exhaust valve, a fixed intermediate volume between the valve and the Laval nozzle and the Laval nozzle itself (see my pic). The exhaust valve can be described as a convergent isentropic nozzle with a varying cross section (according to the valve lift curve) and an unsteady cross section enlargement from the isentropic cross section to the cross section area of the Ejector.


The exhaust process must be split into several simplified steps. Depending on the flight height and the geometry of the ejector, the Laval conditions at the exhaust valve or the Laval nozzle can be reached in a different sequenz:

The exhaust valve opens and at that time the pressure in the ejector volume is atmospheric

The gas will reach the Laval condition in the equivalent exhaust nozzle cross section. The ejector pressure has no influence on the exhaust valve mass flow. The pressure inside the ejector will rise until the system reaches the Laval condition in the Laval nozzle (Mach=1).

• Same as 1, but with supersonic exhaust in the Laval nozzle and further increasing pressure in the ejector volume until the pressure becomes so high, that pressure drop over the exhaust valve cross section is below the Laval pressure, This goes on until (whatever comes first) the Laval pressure is no longer reached at the Laval nozzle, or the pressure inside the ejector volume fell so much, that we regained the Laval conditions at the corresponding exhaust cross section.
• 
  
• In case, the pressure relation in corresponding exhaust cross section equates to the Laval conditions again, this goes on until the pressure drop in the Laval nozzle or the corresponding exhaust cross section is less than the Laval conditions.
• 
  
• If the pressure difference over the corresponding exhaust cross section drops under the Laval conditions but the pressure ratio in the Laval nozzle is still at Laval condition, go on until it reaches the Limit of the Laval condition at the nozzle
• 
  
• Even below the Laval condition, some exhaust thrust is being produced, on this case an expansion in the Laval nozzle will slow the exhaust down, so that in many cases this will be quite insignificant.

There is no shortcut to this calculations and you should not believe that somebody did these type of calculations without explaining how he did it and which boundary conditions and tools he used.



Some thought about the ejector design goals:

For producing a maximum amount of thrust, the pressure before the Laval nozzle must be as high as possible. The downside of that, is that an increased exhaust pressure will cause losses in crank power, but you need to keep in mind, that as long as we have a Laval pressure ratio over the exhaust valve, the back pressure will not have any impact on the engine. As most thrust will be produced around BDC when the cylinder pressure is high and the exhaust valve is allready halfway open, an ejector should be designed with a Laval cross section which will rise the pressure in the ejector at that point so high that it just allows to keep the pressure drop over the exhaust valve just in Laval condition of only short time below that.

Like always a fixed geometry can only work at its optimum at one point or one operational line. With very little back pressure at great height, the emphasis should be on producing less back pressure, since the pressure ratio over the Laval nozzle will nearly always produce high exhaust velocities and the shrinking propulsion efficiency will lower the benefit of additional back pressure. A large expansion ratio of the Laval throttle will help in great height but reduce the thrust in low heights (same is true for rocket nozzles). You see, rocket science is included in combustion engine technology, but its just a small part of it….

Sleeve valve engines with their fast exhaust opening, are far more efficient in producing exhaust thrust, this is often overlooked.

 

[Aeroengineer1]

I have never looked at this problem, but if I had to start, I would likely start at looking at the m_dot of the system. I would expect that backing into an average m_dot would not be too hard (famous last words). It is past midnight here, and I would need to think on it more.

 

[HoHun]

Hi Nicknick,

However, with Mathlap, Origin or even Exel you could find a decent solution (with using a lot of assumptions and simplifications).

Thanks for creating an separate thread for this!

The graph that triggered your response was in fact, as customary for WW2 aviation, a static graph at V = 0, so the power displayed there was pure (propeller) shaft power. You're right that the exhaust thrust curve is fully generic (with the peak calibrated to yield the power equivalent indicated on the historic Focke-Wulf Baubeschreibung) and lacks the sawtooth shape it should have - I did not include that feature in my preliminary analysis since it only affects performance between the two full-throttle heights, leaving the main characteristic points of the performance graphs unaffected.

It's my impression that in WW2, the engineers when calculating aircraft performance on occasion took a few shortcuts, including approximating exhaust thrust as a constant power, and the calculation (and even measurement) of accurate engine power graphs was an evolving discipline, so not all of the historical power graphs are fully realistic to begin with.

The wartime booklet "The Performance of the super-charged Aero Engine" by Stanley Hooker et al. provides a methodology to calculate realistic engine performance, including exhaust thrust, and the authors seem to consider their work to be pioneering. They refer to the previously used engine power formula by Brookes, which they consider unsatisfactory - and there are one or two German power graphs from WW2 with remarks like "nach der Brook'schen Höhenformel", so it seems the German industry was in no better a position than the British one. My impression is that BMW at least had a pretty good methodology, as is evident from fragments posted by Junkers-Peter over on flugzeugforum.de, but unfortunately, I've not seen anything as complete as Hooker's booklet.

Here's a pretty good review of Hooker's booklet on enginehistory.org: https://enginehistory.org/members/articles/ACEnginePerfAnalysisR-R.shtml

Hooker also calculates exhaust thrust, but I couldn't quite make sense of his calculations. My attempts at applying his methods gave me unrealistically low thrust values, so I must have been doing something wrong.

Generally, I would be quite interested in even a simplified approach. A complete calculation of an entire engine wouldn't even be needed (as for many historic engines, data is hard to find), but the ability to make more intelligent assumptions on shifting operating parameters to different working points would already be quite useful for the purpose of aircraft performance analysis.

Regards,

Henning (HoHun)

 

[Nicknick]

Hi,

Interesting link, but unfortunately in doesn’t include any exhaust thrust calculations, at least I can’t find it… It might be, that there have been empiric formulas for calculating the exhaust thrust, but like all empirical approaches, these are only valid for a small range which is not too far away from the data base. The exhaust nozzle design can have an impact on the engine crank shaft power (back pressure) which makes it quite difficult to predict it significantly more precise than just by guessing. The engine design and cam timing will have a large effect to. A single exhaust valve (like in the Jumo) despite having lower heat losses, will very likely produce less exhaust thrust than two exhaust valves per cylinder, unless it want be operating with a much higher acceleration. For maximum thrust, the exhaust cross section needs to increase rapidly because using the high pressure at the beginning of the exhaust process effectively is most decisive. As said, the sleeve valve engines have surely been more effective in this regard, since they don’t need an opening ramp.

I cant prove it, but I guess there is no simple empiric formula because of to many relevant variables…

 

[Nicknick]

I have never looked at this problem, but if I had to start, I would likely start at looking at the m_dot of the system. I would expect that backing into an average m_dot would not be too hard (famous last words). It is past midnight here, and I would need to think on it more.

The problem is, that the mass leaves the cylinder extremly unsteady und with a wide range of speed ant temperature. An avarage m_dot (about 60-70 % of the time no massflow at all) and an avarage pressure (ranging from abbiant to about 8 bar) will not be represantive...

 

[Trident]

Per cylinder, sure, but if you look at it from whole-engine perspective?

 

[HoHun]

Hi Nicknick,

Interesting link, but unfortunately in doesn’t include any exhaust thrust calculations, at least I can’t find it… It might be, that there have been empiric formulas for calculating the exhaust thrust, but like all empirical approaches, these are only valid for a small range which is not too far away from the data base.

Hooker's booklet describes his approach, the link was only meant to provide the historical perspective. I think it's necessarily simplified ...here's a bit from the introduction:



Regards,

Henning (HoHun)

 

[Nicknick]

Hi,

As I understand it, this approach it more fitting to an exhaust manifold with a constant mass flow and not for individual exhaust stacks for each cylinder. At great flight height, the nozzle should be convergent/divergent and not just convergent as Hooker wrote, the pressure drop will easily enable supersonic speeds.

If you regard an even exhaust back pressure, you don’t take the high blow down pressure pulse into account. In theory, this could deliver about 30 % of the engine power (with an hypothetical 100 % efficient turbine which works perfect with all pressure and massflow conditions). For supercharged engines it is even more. I’m sure, this is the reason why Hookers formulas delivered far less exhaust thrust than real world engines could achieve.

 

[HoHun]

Hi Nicknick,

If you regard an even exhaust back pressure, you don’t take the high blow down pressure pulse into account. In theory, this could deliver about 30 % of the engine power (with an hypothetical 100 % turbine which works perfect with all pressure and massflow conditions). For supercharged engines it is even more. I’m sure, this is the reason why Hookers formulas delivered far less exhaust thrust than real world engines could achieve.

Later in the chapter, Hooker states:

"Again the agreement [between calculated and measured results] is reasonably good and the fact that the observed thrusts exceed the caclulated values can be accounted for by the pulsations in the flow."

However, the graphed comparison shows the following corner points:

Merlin III operating at 3000 rpm ... thrust for one ejector ...

Ejector areas of 10 sqin per bank of cylinders:

- At +0 lbf/sqin boost: 8 lbf calculated, 12.5 lbs measured
- At +9 lbf/sqin boost: 27 lbf calculated, 35 lbf measured

Ejector areas of 4 sqin per bank of cylinders:

- At +0 lbf/sqin boost: 20 lbf calculated, 24 lbs measured
- At +9 lbf/sqin boost: 54 lbf calculated, 62 lbf measured

So he's underestimating thrust by some 20 to 30%, which probably puts the assessment of this being a "reasonable" fit into (historical) perspective.

He credits the "Experimental Section at Hucknall Aerodrome" and Mr. H. Pearson for the test flight results, pointing out that "experimental flight data is very scarce".

I think there are two NACA test stand experiments dealing with exhaust thrust, the flight test report on Spitfire exhausts linked in the Merlin exhaust thread, and another flight test report that I previously didn't know of, which another forum member linked in this thread: https://www.secretprojects.co.uk/threads/use-of-exhaust-thrust-in-radial-engines-in-late-30s.36563/

Regards,

Henning (HoHun)

 

[Nicknick]

Usually, if you calculate any form of power output under simplified conditions, your results will end up as being much too optimistic. In his case, I assume, that his simplified approach delivers only about half of the theoretical potential and the real-world losses make it look better than it is...

I read the thread about the radial engines, I think it might have something to do with difficulties in designing an efficient cowling with flaps and short exhaust stacks sticking through them. Also, you dont want to inhale exhaust gases in a single engine fighter, so that exhaust stacks in front of the cockpit might be troublesome. The AM6 "Zero" used, in some variants, such short exhaust stacks to provide thrust.

 

[HoHun]

Hi Nicknick,

Usually, if you calculate any form of power output under simplified conditions, your results will end up as being much too optimistic. In his case, I assume, that his simplified approach delivers only about half of the theoretical potential and the real-world losses make it look better than it is...

Interestingly, Hooker also compares his results to independently measured data from flight testing of a Hawker Hurricane by the Hawker company. He points out that they used slightly different assumptions regarding the charge mass flow, but their flight test results are fitting his calculations well (he calculates 119 HP @ 22000 ft ejector thrust power, they measure 127 HP @ 21750 ft). So maybe the mismatch with Rolls Royce's own measurements were aggravated by experimental difficulties as well.

Hooker notes that the exhaust thrust is directly linear to boost pressure at least up to 35000 ft. From the context, this would seem to apply to the Merlin XX in FS (high) supercharger gear, and I wonder if it also applies above full throttle height only as I'd expect the thrust to slightly increase with altitude below full throttle height, similar to brake power.

I read the thread about the radial engines, I think it might have something to do with difficulties in designing an efficient cowling with flaps and short exhaust stacks sticking through them. Also, you dont want to inhale exhaust gases in a single engine fighter, so that exhaust stacks in front of the cockpit might be troublesome. The AM6 "Zero" used, in some variants, such short exhaust stacks to provide thrust.

You might well be right ... if you look at inline engines, the BMW VI for example was often fitted with long downward exhausts that clearly were meant to keep the exhaust gases away from the cockpit. The Heinkel He 51 was one of the aircraft thus fitted. It's successor, the Me 109, had very short stubs at first, so I suspect with inline engines, there also was the desire to eliminate the drag of external exhausts before it was realized that the benefit of exhaust thrust outweighed the drag of the ejector stubs. No troublesome cowl flaps on liquid-cooled inline engines, though.

Regards,

Henning (HoHun)

 

[sienar]

Messerschmitt tried to keep the exhaust in the boundary layer which is why they are flat/squared.

Here is a neat graph that compares exhaust thrust for a turbo and supercharged 603g.

 

[HoHun]

Hi Sienar,

Messerschmitt tried to keep the exhaust in the boundary layer which is why they are flat/squared.

Is that confirmed by documentation? My impression is that at any given time, all German airframe builders had fairly similar (maybe identical) styles of exhausts, so I thought the exhausts were provided by the engine manufacturers. Of course, you might still be right, and they simply provided the exhausts requested by Messerschmitt.

From what I've read, it seems that in Germany, there often was some friction between engine companies and aircraft designers, each blaming the other for causing engine problems and performance shortcomings. However, Hooker also mentions disagreements between the two sides, so that was probably not a German phenomenon alone.

Regards,

Henning (HoHun)

 

[HoHun]

Hi Sienar,

Here is a neat graph that compares exhaust thrust for a turbo and supercharged 603g.
View attachment 691861

Thanks a lot, that's a highly interesting graph because it shows the difference between the thrust from a mechanically supercharged engine (with individual ejector stubs, undoubtedly) and from the same engine when it's equipped with a turbo-supercharger! I haven't seen a graph like that before, and while I've generally considered remanent exhaust thrust after the turbo-supercharger to be negligble, it's interesting to see that at least in the case of the DB 603G, it's still about 20% at sea level, and 40% at full throttle height.

There's a statement by Kelly Johnson that unfortunately, on the P-38 the turbo superchargers did not allow the exploitation of exhaust thrust since they wouldn't tolerate any backpressure (increase), but that was of course not a quantified statement.

Regards,

Henning (HoHun)

 

[Nicknick]

@HoHun
Hooker only regarded purly convergent nozzels, which can only produce Mach 1. The conditions in the Laval cross section (here the outlet opening) are only depending in the pressure side and not on the athmospheric condition. In this case, it is clear that the thrust doesn't rise with lower backpressure. With a convergent/divergent type of nozzle, the exhaust could become supersonic with low back pressure.

 

[HoHun]

Hi Nicknick,

@HoHun
In this case, it is clear that the thrust doesn't rise with lower backpressure.

Thanks, that's useful information! Typically, on historic engine power/thrust graphs, there still is an increase of thrust with altitude below full throttle height.

What would you say is the background of this increase? Just increased mass flow through higher charge mass flow at constant manifold pressure and lower ambient air temperatures?

Regards,

Henning (HoHun)

 

[Calum Douglas]

Calculating the exhaust thrust for a mechanically supercharged engine is impossible without some thermodynamic skills. It is relatively easy for turbocharged engines (simple rocket science) because of the stationary flow process through a simple nozzle, but in a purly mechanically charged engine the exhaust process is an open flow process. The exhaust velocity and the mass flow between the opening and closing of the exhaust valves is not even close to being constant, which makes it quite demanding for calculations. If you have access to 1-d engine simulation programs like GT Power or Boost, it is a relatively easy task. Without that it will be almost impossible to calculate manually unlike you are a super really extremely capable in integrating complex equations. However, with Mathlap, Origin or even Exel you could find a decent solution (with using a lot of assumptions and simplifications).

If somebody is drawing nice lines which show no connection to the engine power, you can be sure that he never really calculated he exhaust thrust!

So, this is your input data what you definitely need for an estimation of the exhaust thrust:

Cylinder:

p,t,V at the time the exhaust valve opens

dV/dt (cylinder volume over time/crank angle)

exhaust valve:

equivalent isentropic opening area over time/crank angle (dA/dt):

valve lift curve, flow curve

Ejector:

-Volume between exhaust valve and Laval nozzle

-cross section area

-smalles area (Laval cross section) of the Laval nozzle

-Expansion ratio of the Laval nozzle

-athmospheric pressure



The following simplifications would need to be done to make it manageable:

-exhaust gas will be an ideal gas (it is not..)

-no heat losses to the walls (which will give quite an error….)

-no gas friction losses

Thermodynamic model:

Basically, the exhaust model can be described by the cylindervolume, the exhaust valve, a fixed intermediate volume between the valve and the Laval nozzle and the Laval nozzle itself (see my pic). The exhaust valve can be described as a convergent isentropic nozzle with a varying cross section (according to the valve lift curve) and an unsteady cross section enlargement from the isentropic cross section to the cross section area of the Ejector.


The exhaust process must be split into several simplified steps. Depending on the flight height and the geometry of the ejector, the Laval conditions at the exhaust valve or the Laval nozzle can be reached in a different sequenz:

The exhaust valve opens and at that time the pressure in the ejector volume is atmospheric

The gas will reach the Laval condition in the equivalent exhaust nozzle cross section. The ejector pressure has no influence on the exhaust valve mass flow. The pressure inside the ejector will rise until the system reaches the Laval condition in the Laval nozzle (Mach=1).

• Same as 1, but with supersonic exhaust in the Laval nozzle and further increasing pressure in the ejector volume until the pressure becomes so high, that pressure drop over the exhaust valve cross section is below the Laval pressure, This goes on until (whatever comes first) the Laval pressure is no longer reached at the Laval nozzle, or the pressure inside the ejector volume fell so much, that we regained the Laval conditions at the corresponding exhaust cross section.
• 
  
• In case, the pressure relation in corresponding exhaust cross section equates to the Laval conditions again, this goes on until the pressure drop in the Laval nozzle or the corresponding exhaust cross section is less than the Laval conditions.
• 
  
• If the pressure difference over the corresponding exhaust cross section drops under the Laval conditions but the pressure ratio in the Laval nozzle is still at Laval condition, go on until it reaches the Limit of the Laval condition at the nozzle
• 
  
• Even below the Laval condition, some exhaust thrust is being produced, on this case an expansion in the Laval nozzle will slow the exhaust down, so that in many cases this will be quite insignificant.

There is no shortcut to this calculations and you should not believe that somebody did these type of calculations without explaining how he did it and which boundary conditions and tools he used.



Some thought about the ejector design goals:

For producing a maximum amount of thrust, the pressure before the Laval nozzle must be as high as possible. The downside of that, is that an increased exhaust pressure will cause losses in crank power, but you need to keep in mind, that as long as we have a Laval pressure ratio over the exhaust valve, the back pressure will not have any impact on the engine. As most thrust will be produced around BDC when the cylinder pressure is high and the exhaust valve is allready halfway open, an ejector should be designed with a Laval cross section which will rise the pressure in the ejector at that point so high that it just allows to keep the pressure drop over the exhaust valve just in Laval condition of only short time below that.

Like always a fixed geometry can only work at its optimum at one point or one operational line. With very little back pressure at great height, the emphasis should be on producing less back pressure, since the pressure ratio over the Laval nozzle will nearly always produce high exhaust velocities and the shrinking propulsion efficiency will lower the benefit of additional back pressure. A large expansion ratio of the Laval throttle will help in great height but reduce the thrust in low heights (same is true for rocket nozzles). You see, rocket science is included in combustion engine technology, but its just a small part of it….

Sleeve valve engines with their fast exhaust opening, are far more efficient in producing exhaust thrust, this is often overlooked.

This stuff was all covered in by M. Pohl at the DVL in Germany (Forschungsberichte Nr. 1986, 10th Aug 1944). When I have some time I might put it on my website, but its a bit..."messy" and needs a lot of curation before its digestible. He discusses Laval geometry.

The study appears rather more in depth than the work in Hookers booklet from earler in the war, it is of course possible that by 1944 the work being done at RR was equally rigourous. (But, personally, I would suspect not)

 

[sienar]

Hi Sienar,

Messerschmitt tried to keep the exhaust in the boundary layer which is why they are flat/squared.

Is that confirmed by documentation? My impression is that at any given time, all German airframe builders had fairly similar (maybe identical) styles of exhausts, so I thought the exhausts were provided by the engine manufacturers. Of course, you might still be right, and they simply provided the exhausts requested by Messerschmitt.

From what I've read, it seems that in Germany, there often was some friction between engine companies and aircraft designers, each blaming the other for causing engine problems and performance shortcomings. However, Hooker also mentions disagreements between the two sides, so that was probably not a German phenomenon alone.

Regards,

Henning (HoHun)

I'm not sure how they split work exhaust design, but this implies it was done by Mess.

Regarding the comparison - its from a report comparing the merits of turbocharing to supercharing. It comes to the conclusion that supercharging offers superior total thrust. I'm a bit dubious of this report as it assumes a lower prop efficiency for the turbo engine. The graph I posted is from earlier Hirth performance estimates used in the report and should be good.

 

[Nicknick]

@HoHun

With a constant charge pressure, the mass flow will be constant too (not taking into account an temperature changes). In combination with a purely convergent nozzle, the constant mass flow with constant temperature will always exit with Mach 1, so the thrust (force) is constant as well.
With a convergent/divergent type of Laval nozzle, the exhausts will further expand in the divergent section and will become supersonic. The degree of expansion (which defines the exhaust speed) is depending on the back pressure (atmospheric pressure) and geometric expansion in the divergent part of the nozzle. It is quite similar to rocket nozzles, if you make use of the full potential in cruise height, you will over expand at lower flight (causing a sonic shock wave) height so that you need to find a compromise. A variable nozzle geometry which alters its shape during every exhaust stroke and for every atmospheric pressure would be the ideal solution. This would have been a nice approach and developing this in 1944 would have saved you against fighting at the front….

@Calum Douglas

Sounds very interesting, but I have to admit that the old engineers of that area were much more used to complex math than me…. This could be pretty hard stuff to read!

@sienar

Why should anyone try to keep the exhaust gases in the boundary layer??? When the hot gases are coming into contact with the fuselage, they would produce a lot of drag because the relative speed is increased compared to the surrounding air and hot, turbulent gases will cause a lot of friction (higher viscosity).

 

[HoHun]

Hi Calum,

This stuff was all covered in by M. Pohl at the DVL in Germany (Forschungsberichte Nr. 1986, 10th Aug 1944). When I have some time I might put it on my website, but its a bit..."messy" and needs a lot of curation before its digestible. He discusses Laval geometry.

I'm looking forward to that!

The study appears rather more in depth than the work in Hookers booklet from earler in the war, it is of course possible that by 1944 the work being done at RR was equally rigourous. (But, personally, I would suspect not)

Hooker et al. make a quite interesting statement in their introduction:

"Whatever the truth of our calcluations, it is to the credit of the Executive and Technical Director of Rolls-Royce Ltd. that their full support has always been given, even though our theories have apparently substandially reduced the altitude performance of all our products as calculated by the present approved R.D.E. formulae. On the other hand, by a logical application of our results, we claim that the actual performance of Rolls-Royce engines can and will be improved, and as witness to this we would cite the case of the Merlin XLVI."

Have you ever come across these "R.D.E. formulae" (which in the following, Hooker states were laid down by the Air Ministry)? While these apparently lack consideration of ram effect and exhaust thrust, it would be interesting to know them as it might help to determine if any set of historic engine curves were based on these somewhat flawed calculations. (It seems they were also know as "Air Ministry Correction Formulae".)

Specifically, Hooker mentions that these formulae were fine for the engines for the 1920s, but underestimate full throttle height of modern engines. They also have an inconsistency in that they offer two ways of calculating power for a working point, either from constant boost performance or from full throttle performance, which don't agree on the results. Additionally, the results are not really independent of the test stand conditions, in particular the simulated altitude for the bench test results on an altitude test stand.

(Minor historical detail also revealed by Hooker: At the time of his writing, March 1941, altitude tests stands available in Britain were limited to 25000 ft.)

Regards,

Henning (HoHun)

 

[HoHun]

Hi Nicknick,

With a constant charge pressure, the mass flow will be constant too (not taking into account an temperature changes).

I believe the temperature drop in the atmosphere is in fact the main driver of the constant-charge-pressure power increase at increasing altitudes which we see in the historic documents. As far as I know, Jumo was the only company to develop a compensating gouvernor in their "Füllungsregler", but I figure that was only possible because they used a swirl throttle which increased their supercharger's efficiency. The soviet AM-38 had the same throttle, but I'm not sure it regulated to constant charge mass like the Jumo 213 did.

A variable nozzle geometry which alters its shape during every exhaust stroke and for every atmospheric pressure would be the ideal solution. This would have been a nice approach and developing this in 1944 would have saved you against fighting at the front….

Hehe, neat idea! I'd suggest to operate the variable nozzle by thrust from a tiny pulse jet added to each exhaust jet, which gets ignition from the peak pulse of the cylinder exhaust. Trying to make *that* work should keep me safe for quite a while!

Regards,

Henning (HoHun)

 

[HoHun]

Hi Sienar,

Regarding the comparison - its from a report comparing the merits of turbocharing to supercharing. It comes to the conclusion that supercharging offers superior total thrust. I'm a bit dubious of this report as it assumes a lower prop efficiency for the turbo engine.

I believe that's from a report on the Fw 190 ultra-high altitude fighter with an 18 m wing span? I would think it's entirely plausible, since that report is considering flight at altitudes of up to 17 km (from memory). Up there, direct exhaust thrust would probably be much more valuable than shaft power converted by a propeller operating at the ragged edge of the sensible envelope. I believe the report also pointed out that the propeller diameter was limited by groud clearance requirements, so the turbo-supercharged engine, which had to transfer more brake power, could not be equipped with an aerodynamically optimized propeller.

Regards,

Henning (HoHun)

 

[Nicknick]

I believe the temperature drop in the atmosphere is in fact the main driver of the constant-charge-pressure power increase at increasing altitudes which we see in the historic documents.

The increased exhaust thrust is surly almost completely caused by the higher expansion ratio with increasing height.

Most German engines didn’t use a charge cooler, so even with lower ambient temperature/pressure, the rising required boost ratio will have caused an increase in temperature. At least this is true, if the boost pressure is regulated by rotational speed (hydraulic clutch) and probably also for a swirl throttle (more part load losses). With a pre charger throttle (like it was used on most allied planes), the pressure ratio over the charger will be constant which will result in much higher charge temperatures which will decline with flight height (until the charger switches to the next gear/stage). This might be one reasons, why charge air coolers have been much more common on allied than on Germans planes.

 

[Nicknick]

somthing new to this debate:

https://www.secretprojects.co.uk/threads/messerschmitt-me-329.19997/post-575410

this was posted quite recently and it show in intresting exhaust system. As we can see in here:

https://www.secretprojects.co.uk/threads/understanding-and-calculating-exhaust-thrust.40891/post-577705

this type of arrangement could be as good as the free exhaust stack despite having much more losses in all these pipings. The increases mass flow and possibly an afterburner like secondary combustion (when running rich) which might have resulted in a kind of Lorin jet could explain that.

 

[HoHun]

Hi Nicknick,

The increased exhaust thrust is surly almost completely caused by the higher expansion ratio with increasing height.

Hm, how does that align with the "thrust doesn't rise with lower backpressure" statement? I just noticed that I might have confused backpressure and ambient pressure.

This might be one reasons, why charge air coolers have been much more common on allied than on Germans planes.

I believe charge coolers were only used on Allied planes that had more than one stage of compression.

There weren't many two-stage superchargers on German planes, but there was a single-stage supercharged variant of the Jumo 211 that had a charge cooler. If I remember correctly, "Deutsche Flugmotore und Strahltriebwerke" states that the Jumo 211 was a disappointment because the charge cooler caused aerodynamic losses. I think it was a bit ambiguous if they meant the internal losses in the heat exchanger or the drag of the ambient air flowing through the heat exchanger.

The swirl throttle only came in late in the war, and I believe the Jumo 213 was actually the only mass production engine to use it. I believe when a charge-cooler equipped version of this engine was compared to water-alcohol injected version without a charge cooler for use on the Fw 190 or Ta 152, the version without charge cooler was actually preferred as the absence of charge cooler drag more than made up for the higher weight of the MW50 injection system with its liquid supply. I'm not entirely sure where I read that, though ... might be in Hermann's "Ta 152", but I can't find it right now.

Regards,

Henning (HoHun)

 

[Nicknick]

Did I write "thrust doesn't rise with lower backpressure"? if so, I need to read it in context. Unfortunately I used backpressure for the stack (=ambient pressure) and the engine (backpressure = pressure after the exhaust valve and befor the nozzle). The thrust will rise with lower back pressure, when ou have a convergent/divergent nozzle, but not significantly with a purely convergent type (Hooker only regarded this type).

In the purely convergent type, only Mach1 at the nozzle can be reached, idependently of the ambient pressure. Only after the pressure befor the nozzle droped below the Laval pressure, the lower ambient pressure might increase the thrust a little bit. Most of the push forward will happen during the early phase of the exhaust stroke when the pressure is high.

 

[HoHun]

Hi Nicknick,

Did I write "thrust doesn't rise with lower backpressure"? if so, I need to read it in context. Unfortunately I used backpressure for the stack (=ambient pressure) and the engine (backpressure = pressure after the exhaust valve and befor the nozzle).

I quoted it from here (for full context, click on the arrow) ... but from your explanation, it seems I understood it just as you intended:

In this case, it is clear that the thrust doesn't rise with lower backpressure.

In the purely convergent type, only Mach1 at the nozzle can be reached, idependently of the ambient pressure. Only after the pressure befor the nozzle droped below the Laval pressure, the lower ambient pressure might increase the thrust a little bit. Most of the push forward will happen during the early phase of the exhaust stroke when the pressure is high.

Ah, I think I understand it now. Is this the explanation for the following quote, too?

The increased exhaust thrust is surly almost completely caused by the higher expansion ratio with increasing height.

Regards,

Henning (HoHun)

 

[Nicknick]

The divergending Laval nozzle is the key, without that, the ambient pressure has nearly no effect. A divergent nozzle enables the gases to expand further than to the Laval pressure relation and gain more speed (>Mach 1).

Hooker only regarded the pureley konvergent type, but as we can see fro other graphs (e.g. no 12), there must have been nozzles of the convergent/divergent type which resulted in increased thrust by lower ambient pressure. It would have been a suprise, if no one would have used convergent/divergent nozzels because the Laval nozzle theory was well knowen back than.

Hookers theory wasn"t very sophisticated, he very much oversimplified the boundary conditions, to get some simple formulas which didn"t fit to the reality, especially when planes began flying ever higher and faster.

 

[HoHun]

Hi Nicknick,

The divergending Laval nozzle is the key, without that, the ambient pressure has nearly no effect. A divergent nozzle enables the gases to expand further than to the Laval pressure relation and gain more speed (>Mach 1).

Hooker only regarded the pureley konvergent type, but as we can see fro other graphs (e.g. no 12), there must have been nozzles of the convergent/divergent type which resulted in increased thrust by lower ambient pressure.

According to Hooker as well as to Hawker calculations, ambient pressure did actually have a marked effect on exhaust thrust ... here's the graph showing a comparison:



(That was the background of my earlier question ... if Laval nozzle give approximately constant thrust regardless of ambient pressure, how can the variation in thrust be explained. It would probably have been clearer had I posted that graph along with the question!)

Regards,

Henning (HoHun)

 

[Nicknick]

As I see it, the thrust value on this diagram is only given in power, not in force, therefore I assume, that these values are here simply rising with the increasing speed and the force may very well stay almost constant.

Despite that, whenever the exhaust is flowing subsonic out of the exhaust stack, we will have an impact of the ambient pressure.

 

[HoHun]

Hi Nicknick,

As I see it, the thrust value on this diagram is only given in power, not in force, therefore I assume, that these values are here simply rising with the increasing speed and the force may very well stay almost constant.

I cross-checked this with the speed diagram also included in Hooker's booklet, and the thrust power difference is much larger than only the speed-dependent difference. (Top speed at 21750 ft is 338 mph, and speed at 17500 ft is 330 mph, while thrust in the above diagram is 120 HP @ 21750 ft and 95 HP @ 17500 ft.)

The same general shape is also shown on other graphs that show thrust as force, for example on the BMW graph you posted:

https://www.secretprojects.co.uk/threads/engine-power-considerations-in-performance-calculations.40874/#post-576248

Regards,

Henning (HoHun)

 

[Nicknick]

With an convergent/divergent type, the BMW charts fullfills the expectations.

Could you please give me the boundary conditions of Hookers calculations? If he assumes a subsonic flow, the power would rise with lower backpressure.

 

[F119Doctor]

Hi Nicknick,

This might be one reasons, why charge air coolers have been much more common on allied than on Germans planes.

I believe charge coolers were only used on Allied planes that had more than one stage of compression.

There weren't many two-stage superchargers on German planes, but there was a single-stage supercharged variant of the Jumo 211 that had a charge cooler. If I remember correctly, "Deutsche Flugmotore und Strahltriebwerke" states that the Jumo 211 was a disappointment because the charge cooler caused aerodynamic losses. I think it was a bit ambiguous if they meant the internal losses in the heat exchanger or the drag of the ambient air flowing through the heat exchanger.

The swirl throttle only came in late in the war, and I believe the Jumo 213 was actually the only mass production engine to use it. I believe when a charge-cooler equipped version of this engine was compared to water-alcohol injected version without a charge cooler for use on the Fw 190 or Ta 152, the version without charge cooler was actually preferred as the absence of charge cooler drag more than made up for the higher weight of the MW50 injection system with its liquid supply. I'm not entirely sure where I read that, though ... might be in Hermann's "Ta 152", but I can't find it right now.

Regards,

Henning (HoHun)If you look at the top Reno Unlimited P-51 racers, they have eliminated the intercooler due to its flow restriction when running at very high boost levels of 130+ in Hg. Of course, they are also using a huge quantity of water injection to cool the charge flow.

 

[F119Doctor]

The exhaust flow from a single cylinder is very unsteady. When the exhaust valve opens, there is a rapid blow down thru the exhaust valve into the exhaust passage over approximately 10-20 degrees of crank rotation, followed a relatively low pressure pump out over the remaining 160 degrees of the exhaust stroke.

A convergent nozzle will attain sonic velocity at the throat with a pressure ratio of approximately 2:1, while a convergent divergent nozzle can utilize a pressure ratio in excess of 2:1 for supersonic expansion. If the pressure ratio is less than 2:1, a convergent nozzle will still accelerate the flow, but a divergent section will slow the subsonic flow. During the blowdown portion of the exhaust stroke, I could easily see a pressure ratio high enough for con-div nozzle supersonic expansion. But during the remaining portion of the exhaust stroke, will the pressure ratio at the nozzle be high enough supersonic expansion, or will the divergent section just be drag on the flow? The convergent throat can be made smaller to increase the upstream pressure for a higher pressure ratio, but now there are higher pumping losses. It is quite possible that a convergent only exhaust nozzle is the best compromise for maximizing exhaust thrust and engine shaft power.

The thought about exhaust burning as it exits the exhaust nozzle under high power rich mixture conditions is an interesting one. It would seem that this could increase the impulse thrust of the exhaust, but is it really wasted because the exhaust flow is already expanded to atmospheric and the addition heat can't be efficiently utilized? Maybe you could tap pressurized air from the supercharger and feed it into the exhaust prior to the exhaust nozzle throat for nice afterburner effect? I'm guessing that could work, but at the expense of greater complexity / cost / weight.

 

[Nicknick]

Well, I think we almost agree completely, it is fully correct what you wrote. Without detailed analysis it is hard to nail down, if the divergent type was usefull for most applications or not, but keep in mind, that the athmospheric pressure at 10.000 m is down to about 0,25 bar, so in high flying planes almost all of the exhaust will be discharged above the Laval pressure. With mechanical supercharging this is even more the case (see above, the in-cylinder pressure is around 7 bar when the valves opens). The most important part in hinsight to thrust is the early part of the exhaust stroke and this is were the convergent/divergent type really gives an advantage.

The burning exhaust thing was somthing which acually existed, the primary funktion was to hide the flames, but it worked as efficient as normal exhaust stacks, according to the quotation (with much more aerodynamic drag). At least, there must have been some benefit which compensated the parasitic drag. The exhaust system of the M329 is basically shaped like a Lorin jet, with a diffusor in the front, the comustion chamber and an exhaust which is slightly bigger than the intake. I don"t think, the exhaust pulse is beeing wasted, it is acclerating the incomming fresh air, which might even increase the propulsion efficiency (more mass, less speed).

 

[HoHun]

Hi Nicknick,

With an convergent/divergent type, the BMW charts fullfills the expectations.

I don't think the BMW 801 has convergent/divergent nozzles. It's hard to tell though since I found no clear photographs ... basically, the exhaust tubes seem to be circular in cross section, then steadily transition to a flat oval. Difficult to say anything about the cross section area, but if I had to guess, it looks to me as if the gradient of area change is pretty much constant.

Could you please give me the boundary conditions of Hookers calculations? If he assumes a subsonic flow, the power would rise with lower backpressure.

The first paragraph on p.16 states "In order to obtain the high speed of ejection the exhaust manifolds are fitted with convergent nozzles, and sufficient area is allowed to keep the gas speed below that of the local velocity of sound", so you might be right.

On p.44, he actually notes:

"The effect of the ejector manifolds is to cause an additional back-pressure on the engine and in this way to restrict the charge flow", which in turn decreases thrust.

Regards,

Henning (HoHun)

 

[Nicknick]

Hooker himself used a convergent nozzle for his calculations (as can be seen in your first quotation about his formulas), so at least at one point in his live, he was promoting them.

The sentence on page 44 is only partially true, as long as the pressure drop over the valve is larger than the Laval ratio, the piston doesn’t know anything about the backpressure. The aim of a perfect ejector geometry would be to increase the exhaust back pressure only significantly, when the mass flow and pressure drop over the exhaust valve is very high (“blow down” phase) and dropping during the “push out” phase. The divergent part will in fact decrease the exhaust velocity when the pressure drops below the Laval ratio, but than it will also lower the pressure inside the stack. So with the right geometry and the perfect operation point, such a stack would increase the back pressure when the engine doesn’t feel it and lower it afterwards. As said, with just about 0.3 bar atmospheric pressure at around 10,000 m altitude, there almost always be enough pressure difference for a supersonic Laval nozzle, but for low flying planes it is different.
We should keep in mind, that such a perfectly working design can only be achieved for one operation point, therefor it would have been much easier to design a very efficient nozzle for an high flying bomber than for a fighter plane

 

[sienar]

Some interesting things about the Republic Rainbow utilizing exhaust thrust.

Instead of dumping the waste gate exhaust off to the side like on the P-47, the waste gate gas was fed into the turbo exhaust. In this way all of the exhaust gas came out the same exit giving a pretty significant boost at lower altitudes. Here the dotted line is the dumped exhaust and solid the recombined exhaust.

A variable area exit was used to further boost the exhaust thrust.

Lastly, a comparison between the Rainbows installation and a 'typical' one. There is a typo in the net BHP on the right column; it should be 2409 not 4209.

source; https://www.emerald.com/insight/content/doi/10.1108/eb031588/full/html

 

[Nicknick]

I found these graphs very intresting, unfourtunally I can only partially read it, could you please explain what is written there exactly?

Thanks!

I had the idea of using the waste gate flow for trust since quite a long time, but I felt allways sure, there would have been somebody else before that. For that purpose, the waste gate needs to have a variable Laval nozzle, like they used for some prototypes with variable exhaust area.

 

[Foo Fighter]

Calculating the exhaust thrust for a mechanically supercharged engine is impossible without some thermodynamic skills. .......

Thanks for this, very interesting and useful insight.