Can Vmg exceed the V of a jet exhaust?
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Convergence?
I think the discussion so far is getting close. Any incorrectness in the discussion so far appears to be based on terminology and modulated to account for intuition.
The gas entering the engine must leave the engine faster: unless there is acceleration there isn't any forward thrust. The exhaust jet must move backwards compared with the swallowed air. All viewers agree: people on the ground, any people floating in the air, people on the plane.
That does means that the exhaust jet has to travel in the opposite direction to the aircraft as far as the undisturbed local air is concerned, although not necessarily at any particular speed. (Thanks to the following post for prompting the edit). Viewed from the undisturbed air, the exhaust gas must move backwards to produce a forward thrust. That's not so for a rocket.
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The extreme case of exhaust speed (efficiency) is probably the space shuttle main engine: mass flow rate at liftoff is 1000 lb/s (from a ~500s burn and ~500,000lb of fuel/oxidiser per engine); quoted thrust is 480,000 lbs/2.1 Meganewtons. 500 Kg/s mass ejected and 2.1 Meganewtons thrust implies an exhaust speed of 2,100,000/500 = 4200 m/s, which isn't going to change much with height. However, orbital speed is about 7500 m/s. So, above some point in the climb, the exhaust still moves in the same direction as the shuttle compared with the earth, by about 3 km/s when the fuel is cutoff.
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For low-speed thought experiments, you need to be careful when quoting numbers, because the speed that air hits the fan is faster than the pitot airspeed from the nose: the airflow has already been sucked faster when it enters the nacelle.
The gas entering the engine must leave the engine faster: unless there is acceleration there isn't any forward thrust. The exhaust jet must move backwards compared with the swallowed air. All viewers agree: people on the ground, any people floating in the air, people on the plane.
That does means that the exhaust jet has to travel in the opposite direction to the aircraft as far as the undisturbed local air is concerned, although not necessarily at any particular speed. (Thanks to the following post for prompting the edit). Viewed from the undisturbed air, the exhaust gas must move backwards to produce a forward thrust. That's not so for a rocket.
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The extreme case of exhaust speed (efficiency) is probably the space shuttle main engine: mass flow rate at liftoff is 1000 lb/s (from a ~500s burn and ~500,000lb of fuel/oxidiser per engine); quoted thrust is 480,000 lbs/2.1 Meganewtons. 500 Kg/s mass ejected and 2.1 Meganewtons thrust implies an exhaust speed of 2,100,000/500 = 4200 m/s, which isn't going to change much with height. However, orbital speed is about 7500 m/s. So, above some point in the climb, the exhaust still moves in the same direction as the shuttle compared with the earth, by about 3 km/s when the fuel is cutoff.
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For low-speed thought experiments, you need to be careful when quoting numbers, because the speed that air hits the fan is faster than the pitot airspeed from the nose: the airflow has already been sucked faster when it enters the nacelle.
Last edited by awblain; 29th Jun 2009 at 19:02.
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High school physics....
To keep the jet aircraft moving at a constant velocity, you need a force which balances the drag force.
That force is obtained from the engine exhaust.
F = Ma
That is, the force (thrust) developed by the engine depends on the mass of air it accelerates, and the size of that acceleration.
The force generated by the engine is in the forwards direction, so the acceleration of the air must be in the reverse direction.
So the air coming out of the back of the engine is moving faster than the air coming in the front.
Relative to the engine, the speed of the air coming in the front is the aircraft's airspeed; so the air going out the back has been accelerated relative to that velocity; so the air coming out of the engine is moving faster than the aircraft's forward speed.
For a rocket, the situation is completely different - you're just throwing mass out of the back to generate the force, and you don't care about its velocity (because it's starting velocity in the rocket's frame was zero, so any non-zero exit velocity means that the gases have been accelerated, meaning a forward thrust has been generated).
Not hard to understand, surely?
That force is obtained from the engine exhaust.
F = Ma
That is, the force (thrust) developed by the engine depends on the mass of air it accelerates, and the size of that acceleration.
The force generated by the engine is in the forwards direction, so the acceleration of the air must be in the reverse direction.
So the air coming out of the back of the engine is moving faster than the air coming in the front.
Relative to the engine, the speed of the air coming in the front is the aircraft's airspeed; so the air going out the back has been accelerated relative to that velocity; so the air coming out of the engine is moving faster than the aircraft's forward speed.
For a rocket, the situation is completely different - you're just throwing mass out of the back to generate the force, and you don't care about its velocity (because it's starting velocity in the rocket's frame was zero, so any non-zero exit velocity means that the gases have been accelerated, meaning a forward thrust has been generated).
Not hard to understand, surely?
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Out of the mouth of babes....
In this case I'm strictly referring to the post title... "high school physics".
Well done, Bryce!
Go and stand on a skateboard with a bag of sand.
Throw it backwards, and you'll move forwards. Rocket engine.
Now stand on your skateboard and have somebody throw a bag of sand towards you. Catch it and you'll move backwards. Put your own energy into it and throw it backwards faster than it arrived. You'll move forwards. Jet engine.
CJ
In this case I'm strictly referring to the post title... "high school physics".
Well done, Bryce!
Go and stand on a skateboard with a bag of sand.
Throw it backwards, and you'll move forwards. Rocket engine.
Now stand on your skateboard and have somebody throw a bag of sand towards you. Catch it and you'll move backwards. Put your own energy into it and throw it backwards faster than it arrived. You'll move forwards. Jet engine.
CJ
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Out of interest I ran the figures for the CF6 through NASAs educational software and it came up with the following
It should be noted that the air meeting the compressor face is not at free stream velocity (speed of the aircraft). The inlet is designed to slow the airflow so that it is presented to the compressor at a speed of .5M or less, to avoid choking the compressor. The ram drag above, is the drag associated with slowing down the free stream air as the air is brought inside the inlet. Even the SR-71 at 3.2M has the air delivered to the compressor at approx .5M.
Of interest also is the J-58 installed in the SR-71. At 3.2M, 54% of the thrust is provided by the differential pressure between the internal and external surfaces of the inlet spike. Of the remainder, 17% is provided by the engine and 29% by the ejector (the afterburner, which is in ram jet mode at this speed - air being tapped off the 4th stage compressor, diverted around the turbine, and injected into the afterburner)
I've tried reformatting with little luck - hope you can make sense of it.
Code:
ALTITUDE SPEED CORE FUEL FLOW GROSS RAM NET (MPH) FLOW (LB/S) (PPH) THRUST DRAG THRUST 0 0 254 13820 45,171 0 45,171 0 170 262 14062 47,415 8,716 38,699 3,000 282 251 13511 47,256 14,146 33,109 35,000 .85M 103 6501 23,183 11,382 11,801
Of interest also is the J-58 installed in the SR-71. At 3.2M, 54% of the thrust is provided by the differential pressure between the internal and external surfaces of the inlet spike. Of the remainder, 17% is provided by the engine and 29% by the ejector (the afterburner, which is in ram jet mode at this speed - air being tapped off the 4th stage compressor, diverted around the turbine, and injected into the afterburner)
I've tried reformatting with little luck - hope you can make sense of it.
Last edited by Brian Abraham; 30th Jun 2009 at 02:42.
OK, I am convinced now! The post from Brian Abraham above shows quite clearly how a forward speed increase at sea level directly and signifcantly reduces net thrust, which concurs with the principle that exhaust speed cannot be less than forward speed. So I am happy to admit I have learned something, which in retrospect I should have been able to figure out.
Thank you to those posters who so patiently took the trouble to explain it and to those who were able to resist the temptation to be condescending.
Most importantly, I thank the original poster for initiating the discussion and I hope he/she enjoyed the postings.
Thank you to those posters who so patiently took the trouble to explain it and to those who were able to resist the temptation to be condescending.
Most importantly, I thank the original poster for initiating the discussion and I hope he/she enjoyed the postings.
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Ok, so now that we have got that sorted out, let’s look a bit more closely at the propelling nozzle.
Most texts tell us that the nozzle produces forward thrust by accelerating the air rearwards.
But if we examine it closely we will find that the aerodynamic force on the nozzle is actually trying to tear it off the jet pipe and push it rearwards.
Anyone who does not believe this can try the following simple experiment.
Buy an ice cream cone and throw away the ice cream (OK you can eat it if you really must)
Now bite off the pointed end to leave a convergent nozzle.
Put the narrow end between your lips and blow through it.
You will find that a slow warm airstream comes out of it. (because it is a divergent duct)
Now put the wide end between your lips and blow through it.
This time you will get a faster colder airstream. (because it is now a convergent duct)
The above results are exactly what Bernouli would lead us to expect.
Now release your grip on the cone a little bit and blow really hard until you no longer hold the nozzle.
If the nozzle is producing thrust it will force its way into your mouth.
But it will actually fly out and away from you.
This is because the aerodynamic force on the nozzle is drag acting downstream.
So if the force on the nozzle is drag, how does it increase the thrust?
This question is not unrelated to the subject of this thread.
Most texts tell us that the nozzle produces forward thrust by accelerating the air rearwards.
But if we examine it closely we will find that the aerodynamic force on the nozzle is actually trying to tear it off the jet pipe and push it rearwards.
Anyone who does not believe this can try the following simple experiment.
Buy an ice cream cone and throw away the ice cream (OK you can eat it if you really must)
Now bite off the pointed end to leave a convergent nozzle.
Put the narrow end between your lips and blow through it.
You will find that a slow warm airstream comes out of it. (because it is a divergent duct)
Now put the wide end between your lips and blow through it.
This time you will get a faster colder airstream. (because it is now a convergent duct)
The above results are exactly what Bernouli would lead us to expect.
Now release your grip on the cone a little bit and blow really hard until you no longer hold the nozzle.
If the nozzle is producing thrust it will force its way into your mouth.
But it will actually fly out and away from you.
This is because the aerodynamic force on the nozzle is drag acting downstream.
So if the force on the nozzle is drag, how does it increase the thrust?
This question is not unrelated to the subject of this thread.
No ! (or is this a wind up ??)
Pressure forces air out of the cone. It is that pressure acting on the cone which propels it away. In a jet, what ever force that accelerates the air must have a reaction, ie thrust. For a rocket you integrate the pressure over the internal space, same would work for anything else, as it must unless you want an argument with Mr Newton.
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Ummm... Keith - without trying to be offensive: your question, though stated reasonably enough, is not actually very reasonable.
You say 'most texts state ...' then go on to discount the information in those texts.
Maybe you meant to say 'I don't really understand this, please explain', but it reads as 'the texts are wrong - explain that!'.
You're not (I expect) denying that there is thrust (I mean, we've all seen that jet aeroplanes fly, right?) - just questioning where in the engine it appears. Which surfaces, that is, are acted upon by the exhaust to propel the aircraft?
I don't know. I would send you back to the texts you mention - if they're decent books, they'll tell you, or tell you which other book to read. From my reading, Cumpsty and Kerrbrock's books are decent starting points.
If you want a guess, based on having read a bit about how gas turbines work (but by no means being an expert of any kind) - any compressor fan (large effect in high bypass engines), plus the nozzle (small effect in high bypass engines).
You say 'most texts state ...' then go on to discount the information in those texts.
Maybe you meant to say 'I don't really understand this, please explain', but it reads as 'the texts are wrong - explain that!'.
You're not (I expect) denying that there is thrust (I mean, we've all seen that jet aeroplanes fly, right?) - just questioning where in the engine it appears. Which surfaces, that is, are acted upon by the exhaust to propel the aircraft?
I don't know. I would send you back to the texts you mention - if they're decent books, they'll tell you, or tell you which other book to read. From my reading, Cumpsty and Kerrbrock's books are decent starting points.
If you want a guess, based on having read a bit about how gas turbines work (but by no means being an expert of any kind) - any compressor fan (large effect in high bypass engines), plus the nozzle (small effect in high bypass engines).
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Mr Optimistic
I have no argument with Mr Bernouli.
And as far as I can see I have no argumnet with anything that you have said.
Agreed
Agreed.
But it is an aerodyanmic force acting rearwards so we (usually) call it drag.
Agreed, provided the acceleration is rearward.
Agreed.
Now let's get back to what I actually said.
Many people believe that the force acting on the nozzle is thrust. They are wrong it is actually drag.
It is made up of friction forces, pressure forces and inertial forces acting on the forward facing (internal) surfaces of the nozzle. But the overall force acting on the nozzle is in a rearward direction.
You are correct in saying that the thrust is the Newton3 reaction to the pressure that is accelerating the air through the nozzle. But this thrust force is not acting on the nozzle.
Now the question becomes, "So if the thrust isn't acting on the nozzle, what is it acting on"?
BRYCEM
I'm not saying that the texts are wrong. Some cover this aspect of the subject and some do not.
I'm saying that many people draw the wrong conclusions from the texts. The majority of people walking away after completing a JAR ATPL Theory course (and a good many other courses) probably think that the force on the propelling nozzle is thrust.
I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject.
I have no argument with Mr Bernouli.
And as far as I can see I have no argumnet with anything that you have said.
Pressure forces air out of the cone.
It is that pressure acting on the cone which propels it away.
But it is an aerodyanmic force acting rearwards so we (usually) call it drag.
In a jet, what ever force that accelerates the air must have a reaction, ie thrust.
For a rocket you integrate the pressure over the internal space, same would work for anything else,
Now let's get back to what I actually said.
Many people believe that the force acting on the nozzle is thrust. They are wrong it is actually drag.
It is made up of friction forces, pressure forces and inertial forces acting on the forward facing (internal) surfaces of the nozzle. But the overall force acting on the nozzle is in a rearward direction.
You are correct in saying that the thrust is the Newton3 reaction to the pressure that is accelerating the air through the nozzle. But this thrust force is not acting on the nozzle.
Now the question becomes, "So if the thrust isn't acting on the nozzle, what is it acting on"?
BRYCEM
I'm not saying that the texts are wrong. Some cover this aspect of the subject and some do not.
I'm saying that many people draw the wrong conclusions from the texts. The majority of people walking away after completing a JAR ATPL Theory course (and a good many other courses) probably think that the force on the propelling nozzle is thrust.
I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject.
Last edited by Keith.Williams.; 1st Jul 2009 at 13:46.
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I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject. Today 13:12
I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject. Today 13:12
Keith - some people (not me, obviously) might construe that as you 'wasting their time', and general clever-dickness.
If you already know the answer to the question, please supply it; otherwise, what exactly are you trying to achieve, except education by irritation?
On modern (high bypass) engines, the bulk of the thrust is obtained at the fan.
I'm done with this, BTW.
I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject. Today 13:12
I could of course go and read more books, but my purpose in asking the question is not get an answer. It is to get the readers to think a bit more about the subject.
If you already know the answer to the question, please supply it; otherwise, what exactly are you trying to achieve, except education by irritation?
On modern (high bypass) engines, the bulk of the thrust is obtained at the fan.
I'm done with this, BTW.
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Originally Posted by Keith.Williams
Many people believe that the force acting on the nozzle is thrust. They are wrong it is actually drag.
On Concorde (like the SR-71 quoted earlier) most of he thrust came from... yes, the intakes.
On the SR-71, above Mach 3 under certain circumstances, the engine itself actually produced drag, evidenced by it moving backwards on the suspension points.
CJ
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Now the question becomes, "So if the thrust isn't acting on the nozzle, what is it acting on"?
It's past midnight here so I'll sleep on it and get back to this in the morning.
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FE Hoppy,
Most of the maths just deals with a jet engine as a black pipe (rather than a black box
) with a hole at the front and a hole at the back. The slightly more subtle maths adds front and back surface and local pressure at the "interface" into the equation.
We now are getting into the internal dynamics.... I'll leave that to the experts. About the only thing I'm familiar with in that field is the classic rocket engine with a con-di nozzle with Mach 1 in the throat. Took me enough time to get my mind round it at the time, when actually "playing" with real ones.
CJ
Most of the maths just deals with a jet engine as a black pipe (rather than a black box
) with a hole at the front and a hole at the back. The slightly more subtle maths adds front and back surface and local pressure at the "interface" into the equation.We now are getting into the internal dynamics.... I'll leave that to the experts. About the only thing I'm familiar with in that field is the classic rocket engine with a con-di nozzle with Mach 1 in the throat. Took me enough time to get my mind round it at the time, when actually "playing" with real ones.
CJ
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Thrust mechanism
Origin of most of the thrust in subsonic engines: the fan bearings.
The fan blades accelerate the air backwards, and Sir Isaac pulls the fan blades forward, dragging the aircraft along with them.
In a rocket/turbojet/ramjet, it's the combustion chamber pressure, pushing on the closed front end of the chamber, reduced by viscous forces as the exhaust streams along the nozzle, and in some cases a turbine.
Thrust from shuttle engine again: about 400,000lb. The chamber pressure quoted is about 2800 psi, the throat of the exit of the chamber is roughly 24 inches across (about 430 square inches), which would give 1,200,000lb: so the price of stabilizing (by slowing and expanding) the flow out of the nozzle seems to be 800,000lb of thrust. The nozzle is thus pulled off the engine with 800,000lb of force, while the combustion chamber is pushed up into the engine by 1,200,000lb, giving a net 400,000lb. This suggests that the link between the chamber and the nozzle need to be tougher than that between the engine and its mount. I think that makes sense. Rocket scientists?
The fan blades accelerate the air backwards, and Sir Isaac pulls the fan blades forward, dragging the aircraft along with them.
In a rocket/turbojet/ramjet, it's the combustion chamber pressure, pushing on the closed front end of the chamber, reduced by viscous forces as the exhaust streams along the nozzle, and in some cases a turbine.
Thrust from shuttle engine again: about 400,000lb. The chamber pressure quoted is about 2800 psi, the throat of the exit of the chamber is roughly 24 inches across (about 430 square inches), which would give 1,200,000lb: so the price of stabilizing (by slowing and expanding) the flow out of the nozzle seems to be 800,000lb of thrust. The nozzle is thus pulled off the engine with 800,000lb of force, while the combustion chamber is pushed up into the engine by 1,200,000lb, giving a net 400,000lb. This suggests that the link between the chamber and the nozzle need to be tougher than that between the engine and its mount. I think that makes sense. Rocket scientists?
Last edited by awblain; 1st Jul 2009 at 21:26.
Taking the simple approach - the expanding burning gases will take the path of least resistance, which in this case is out the rear of the engine, since the front is 'blocked' by the highly compressed air entering the engine. The difference in pressure at the nozzle exit (and fan-bypass exhaust) is the thrust produced.
The amount of fuel burnt is proportional to the additional energy imparted to the airflow (less internal friction loss, etc).
-GY
The amount of fuel burnt is proportional to the additional energy imparted to the airflow (less internal friction loss, etc).
-GY
Moderator
So if the force on the nozzle is drag, how does it increase the thrust?
I think where Keith is going is to look at
you integrate the pressure over the internal space
A common presentation is to look at the pressure forces throughout the engine. Especially for the FJ folk, a lot of the net thrust comes from careful and clever intake design (the airflow has to be slowed down to subsonic flow at the engine intake face) and then there are bits and pieces throughout the engine where local net forces are forward or aft.
The bit coming out the back is only part of the equation ...
I think where Keith is going is to look at
you integrate the pressure over the internal space
A common presentation is to look at the pressure forces throughout the engine. Especially for the FJ folk, a lot of the net thrust comes from careful and clever intake design (the airflow has to be slowed down to subsonic flow at the engine intake face) and then there are bits and pieces throughout the engine where local net forces are forward or aft.
The bit coming out the back is only part of the equation ...
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BRYCEM
If you manage to control your emotions for a moment you will find that over-dosing on grump is clouding your vision.
Your statement that
is ture but not relevant to the question
We are talking about the nozzle at the back end of the jet pipe.
By the time the air reaches this nozzle it has already passed through the low pressure turbine so its effect on the fan is almost zero (back pressure effects only).
Cristiaan J
We are not talking about a condi nozzle, just a plain old convergent one.
FE Hoppy
If we were taliking about a condi-nozzle then some of the thrust would be acting on that part of the nozzle that is downstream of the throat.
Awblain.
Your statement that
is along the right lines.
JT
You are of course correct (as usual).
But I think that your statement about
probably leaves many readers none the wiser.
The best answers (those from which we learn the most) are often not the ones that we get from books, but the ones that we get from within our own heads. Spending a little bit of time pondering this type of question can often be very useful (even if the questions are sometimes rather irritating).
If you manage to control your emotions for a moment you will find that over-dosing on grump is clouding your vision.
Your statement that
On modern (high bypass) engines, the bulk of the thrust is obtained at the fan.
Now the question becomes, "So if the thrust isn't acting on the nozzle, what is it acting on"?
By the time the air reaches this nozzle it has already passed through the low pressure turbine so its effect on the fan is almost zero (back pressure effects only).
Cristiaan J
We are not talking about a condi nozzle, just a plain old convergent one.
FE Hoppy
If we were taliking about a condi-nozzle then some of the thrust would be acting on that part of the nozzle that is downstream of the throat.
Awblain.
Your statement that
The nozzle is thus pulled off the engine with 800,000lb of force, while the combustion chamber is pushed up into the engine by 1,200,00lb, giving a net 400,000lb.
JT
You are of course correct (as usual).
But I think that your statement about
integrating the pressures......
The best answers (those from which we learn the most) are often not the ones that we get from books, but the ones that we get from within our own heads. Spending a little bit of time pondering this type of question can often be very useful (even if the questions are sometimes rather irritating).
Last edited by Keith.Williams.; 1st Jul 2009 at 19:13.
Moderator
But I think that your statement about
Quote:
integrating the pressures......
probably leaves many readers none the wiser.
Keith's wisdom supersedes my writing with the brain only partly engaged ..
Folk should recall that fluid flows are about pressures. Pressure gradients and deltas generate fluid flow. Static and flowing fluids generate forces on things like engine bits (and wings, tails, fuselages, etc.) by exerting pressure on surfaces.
Pressure acting over a surface (area) results in a force. Depending on the net orientation of the bit of surface in question (in, say, an engine) the force associated with the fluid pressure will have a net forward or aft direction (and, mostly, a lot laterally .. with which we are not terribly interested).
When one adds up all the bits of forces so calculated, one gets a net thrust (forward or aft).
"Integration" is a mathematics buzzword which really just means "adding up all the little bits and pieces" in the calculation. It comes from the integral calculus which is a really neat way of doing a lot of this stuff but it's not necessary for the pilot folk to have any competence in the mathematics per se.
Quote:
integrating the pressures......
probably leaves many readers none the wiser.
Keith's wisdom supersedes my writing with the brain only partly engaged ..
Folk should recall that fluid flows are about pressures. Pressure gradients and deltas generate fluid flow. Static and flowing fluids generate forces on things like engine bits (and wings, tails, fuselages, etc.) by exerting pressure on surfaces.
Pressure acting over a surface (area) results in a force. Depending on the net orientation of the bit of surface in question (in, say, an engine) the force associated with the fluid pressure will have a net forward or aft direction (and, mostly, a lot laterally .. with which we are not terribly interested).
When one adds up all the bits of forces so calculated, one gets a net thrust (forward or aft).
"Integration" is a mathematics buzzword which really just means "adding up all the little bits and pieces" in the calculation. It comes from the integral calculus which is a really neat way of doing a lot of this stuff but it's not necessary for the pilot folk to have any competence in the mathematics per se.
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Thanks, John.... you've now probably confused even more of the readers even more....
Integrating from intake to exhaust is a nice idea but, more often than not, you don't have enough pressure data to do more than an approximation.
To me, T = m* x deltaV does it every time, with the small refinements where needed.
Otherwise, I just like looking at shock diamonds, or at the plume during a shuttle launch, when the air pressure becomes less than the pressure at the nozzle end.
CJ
Integrating from intake to exhaust is a nice idea but, more often than not, you don't have enough pressure data to do more than an approximation.
To me, T = m* x deltaV does it every time, with the small refinements where needed.
Otherwise, I just like looking at shock diamonds, or at the plume during a shuttle launch, when the air pressure becomes less than the pressure at the nozzle end.
CJ
hmm #2
The high pressure in an engine or jet would tend to push the back end off: nothing to do with drag force on the nozzle surface, just pressure acting on the rear surface. Nozzle turns pressure into flow speed. Blow up a balloon, turn it around so it faces away from you and let go of the untied end. Would it fly away from you owing to drag over the aperture or into your face because of the thrust ? Put a weak joint between the nozzle of a rocket motor and the case and see what happens (owing to the pressure inside the case) - from half a mile away that is. And the adiabatic expansion or compression on flow through a cone would cause no discernable heating or cooling (convergent at subsonic speeds would in any case heat). Don't need de laval nozzle theory here. NB look at most rocket plumes, they bulge out ie exit pressure is still above ambient. Small drag losses in a nozzle I suspect are just accounted for by an 'engineering factor' in practice.