deLaval Nozzles for Small Ramjets?

[larry cottrill]

It is sometimes suggested that adding a "deLaval cone" to the exhaust nozzle of a small jet will improve performance. This is to illustrate the performance of a small deLaval nozzle, using the UFLOW1D program, and test what the performance parameters of an engine will be to properly use such a nozzle. I will also show how the performance of the nozzle is affected by a few simple changes.

The nozzle bearing his name was designed by Dr Carl Gustaf Patrick deLaval (1845-1913). He originally designed it for use in steam turbines, there being no gas turbines around at the time. This very serious-looking fellow was no slouch at innovation - he is said to be "the inventor of the modern impulse steam turbine, the deLaval nozzle, the disk of equal strength, the flexible shaft, and other elements of highest importance in modern gas-turbine design." (Introduction to Gas-Turbine and Jet Propulsion Design, Norman and Zimmerman, Harper & Brothers 1948).

The nozzle under consideration is shown approximately to scale in Figure 1. Diameters go from a maximum of 64mm to a minimum of 32mm at the throat; reasonable dimensions for a small jet exhaust. The slope of the cone (i.e. the half-angle) is about 5 degrees, which is close to the limit of how "fast" we can make the cone without allowing separation of the flow from the surface. The throat was carefully made very smooth. Section 1 is 30mm in length; sections 2-9 are each 5mm long, and section 10 (the "deLaval cone") is 100mm, for a total length of 170mm. I assumed that we could supply combustion gas to the nozzle at 1200 degK in a small engine. Flow is from left to right, as usual. The dimensions of the nozzle sections are as follows:

Section 1 - 64mm to 42mm - length 30mm
Section 2 - 42mm to 39mm - length 5mm
Section 3 - 39mm to 36mm - length 5mm
Section 4 - 36mm to 34mm - length 5mm
Section 5 - 34mm to 33mm - length 5mm
Section 6 - 33mm to 32mm - length 5mm
Section 7 - 32mm to 32mm - length 5mm
Section 8 - 32mm to 33mm - length 5mm
Section 9 - 33mm to 34mm - length 5mm
Section 10 - 34mm to 64mm - length 100mm

Figure 2 shows virtually perfect flow performance in this nozzle. This kind of nozzle, working properly, is a marvel to behold: Note how the absolute pressure and density of the gas go down to nearly zero at the point of exit; the exit velocity is practically three times the speed of sound! An engine small enough to hold in your hand that would do this would be a remarkable beast, indeed.

HOWEVER -

There is a price to pay for this kind of performance. The price is that the input (left end) pressure (i.e. the combustion chamber pressure) has to be 2.7 atmospheres. That is a LOT of pressure for a small ramjet to supply! Probably, more than we can ever achieve with a reasonable subsonic design. So, the price of this level of performance is too high for us to meet. The pressure at which we must drive such a nozzle for it to work is called the 'critical pressure', and it can be calculated if we know the temperature of the exhaust gas we have available. It is always pretty high for reasonably attainable temperatures, though.

One thing that is often not understood about this type of nozzle is that its performance does not change in a smooth way. There is a critical point where it begins to work as deLaval meant it to, and below that point its performance quite suddenly becomes unremarkable. If we drop our chamber pressure to just 2.6 atmospheres (i.e. we reduce pressure by 1/10 atmosphere, just 1.5 PSI) the nozzle behavior is as shown in Figure 3. Note the sudden pressure recovery and equally sudden drop in exit velocity in front of the exit face. In reality, we no longer have a true deLaval nozzle under the gas and air conditions we're using. It's more of a "heated venturi" or something of the sort.

A similar thing occurs if we happen to make our cone just a bit too long. Figure 4 shows the length increased by just 30mm, all other parameters being equal (we are back to the 2.7 atm chamber pressure in this case). So, by lengthening the cone of a well-designed nozzle only a little, we find ourselves cursed with "too much of a good thing". Again, we no longer have a true deLaval nozzle under these conditions.

In Fig. 5, I tried shortening the nozzle by 50mm from its original length (while keeping the cone angle the same), just to see how performance would be affected. The nozzle still works as a deLaval nozzle, but final exhaust velocity is reduced - the cone is a bit too short to fully develop the potential thrust. This is a good illustration of how critical the design parameters are for really good performance, even if we have the basic conditions right to make the nozzle work.

In small engines, the most likely problem we will have is not being able to get close to the critical pressure. Fig. 6 shows the nozzle brought back to all original dimensions, but this time fed with a much more modest chamber pressure, just 1.5 atmospheres. Interestingly, we still get a supersonic 'spike' in the throat, which I did not expect to see. But, as we saw in the reduced pressure example above, performance is poor. Note that 1.5 atm would probably still be an extremely good chamber pressure for any small ramjet we could build and operate. It would probably be plenty hard to get that kind of compression at even high subsonic speed.

My tentative conclusion is that we would generally be wasting effort trying to 'enhance' exhaust nozzles by applying deLaval cones to them. Without having sonic speed air available at the intake, the critical pressure is practically impossible to reach, at least in small engines (and. I would guess, probably in any subsonic ramjet of conventional design). So, we might as well keep things simple. This doesn't necessarily apply to our rocketeer brethren, of course, since with their wild chemical reactions they don't have much trouble getting really high chamber pressures.

Comments / criticisms?

L Cottrill

[pezman]

Nice work! I have use UFLOW a bit to design design venturis, but it is great to see such an extensive example and the effort that you spent relating it to a theoretical design.

[larry cottrill]

Nice work! I have use UFLOW a bit to design design venturis, but it is great to see such an extensive example and the effort that you spent relating it to a theoretical design.

Thanks! UFLOW1D is a great tool for analyzing (and illustrating!) problems like this. You just have to be careful that you set up parameters that represent realistic conditions.

I thought there were a lot of interesting things to observe in this exercise:
- The deLaval design is only a nozzle (i.e. producing a velocity gain from a pressure drop) if the pressure difference is high enough
- The deLaval design is not a nozzle just because you manage to get supersonic flow in the throat! (This one surprised me.)
- There is a VERY sharp "break" in performance as the critical pressure is reached
- Small changes in pressure and dimensions have far more impact than we might intuitively expect
- At low pressures, this type of "nozzle" is far inferior to a simple convergent nozzle!
- For subsonic ramjets (or at least for small ones) this design will never be useful because the chamber pressure to make it work is practically unattainable

That's really a lot of information obtained from a few minutes of playing around with a couple of variables.

L Cottrill

[db]

Larry has anyone used a deLaval nozzle in place of the 'pipe' in a 'snorkel' type Jam Jar..??
The 2.7 bar may be briefly available on expulsion and the deLaval may not interfere too much on inhalation..
I assume that the performance would be quite different to that of a conventional 'snorkel'...
Cheers.. db

[pezman]

It's cool to superimpse an image of the nozzle over the Mach Number curve adn observe how sonic choking limits the speed at the narrowest point of the nozzle to Mach 1, and then see how the speed jumps up as the nozzle expands.

Sorry for the crummy image, but you get the idea ...

I remember reading an old article in Scientific American about how Gary Settles created a small super-sonic wind tunnel and schlieren imaging system for a high school science project. I got tired just reading it. At any rate, the tunnel had a similar profile.

[Bruno Ogorelec]

Larry has anyone used a deLaval nozzle in place of the 'pipe' in a 'snorkel' type Jam Jar..??
The 2.7 bar may be briefly available on expulsion and the deLaval may not interfere too much on inhalation..
I assume that the performance would be quite different to that of a conventional 'snorkel'...
Cheers.. db

Not a good idea. De Laval nozzle is useful only for steady (or slowly changing) flows. It acts as a wave reflector, so that it would simply blow the exhaust pulse of a jam-jar back. Never put constrictions in the path of a pulsating flow unless you want your pressure wave to bounce back.

[larry cottrill]

It's cool to superimpse an image of the nozzle over the Mach Number curve adn observe how sonic choking limits the speed at the narrowest point of the nozzle to Mach 1, and then see how the speed jumps up as the nozzle expands.

Yes, that's really a beautiful picture of the forces of nature in action. Just like it ought to be, smooth and efficient. That cone really has a lot of "leverage" on the gas, once the conditions are all set up right. It's no wonder the method has such appeal. And in rocketry, where it's easy to get high and fairly constant pressure, it can certainly work well if correctly designed.

I remember reading an old article in Scientific American about how Gary Settles created a small super-sonic wind tunnel and schlieren imaging system for a high school science project. I got tired just reading it. At any rate, the tunnel had a similar profile.

He must have had a heck of a fan! At the A/E firm where I used to work, they helped design a supersonic wind tunnel that was driven by a HUGE high pressure cylinder of compressed air. I don't remember where that was for sure, it might have even been up at Iowa State in Ames. Anyway, it was like a three story cylinder rated at 3000 PSIG or something, with a huge fast-opening valve at the base, then a long flow straightener leading the air to the nozzle section and the test chamber. It was only good for a fraction of a second, but it gave you multiple Mach air speeds, and with modern photographic methods the split second was all you needed. I'll bet that didn't exactly run "whisper quiet", either!

L Cottrill

[larry cottrill]

Here's something that would be a serious problem in a flying ramjet with a nozzle like this, even if you could get the proper design AND adequate pressure at full speed: When you trottle back, the engine essentially switches suddenly into the "thrustless" mode of the nozzle. Once the plane slows down a bit, you can no longer obtain the chamber pressure that will bring the nozzle "critical" again, because you can't get adequate compression at the tail end of the diffuser at the reduced airspeed. So, you're stuck in a low-efficiency mode. Your only way out would be to dive to high speed, which you could only do if you had a lot of altitude.

Of course, I don't really know anything about this kind of design, but it seems to me that a usable supersonic ramjet design (with throttleability) could only be accomplished using variable geometry in different parts of the duct. Note, for example, that simply "straightening" the cone to a cylinder would greatly improve performance when the nozzle is in its low-efficiency mode, because it would at least prevent the "pressure recovery" phase from taking place aft of the throat. It would be like a simple convergent nozzle with a straight extension. With the right electronics, a variable-geometry tail cone could be made to respond automatically to changing conditions, optimizing its internal angle accordingly.

L Cottrill

[pezman]

The title says it all ...

Here is a link that describes the tunnel that Gary Settles built back in the mid 60's:
http://www.spiegl.org/1966/10/1966-10-body.html

[larry cottrill]

pezman -

Well, that's very impressive. Doing it by pulling from the low pressure side would probably never even have occurred to me, but apparently this has been done historically. Yes, that certainly should have brought some rewards to the guy.

This reminds me of something that I though of a few days ago when working on the graphics for my original post: We could experiment with supersonic ramjets, if we built them small enough! This is because (as our old friend Ivar found out) you can get supersonic flow right out of a nozzle at the end of a hose connected to an ordinary compressor! Of course, the mass flow will be very small, and your engine models would need to be built with fairly high precision to work. Fuel would probably need to be hydrogen or acetylene, though you could try liquids like gasoline or ether, I guess - it might work. The main trick would be to get your diffuser good enough to efficiently get the speed down to a decent value at the chamber entrance. I wonder if one of those "air amplifiers" would be good for the air nozzle (I mean on the end of your compressor hose), or whether you would have to craft something special. Obviously, however you do it, the pressure drop that gives you supersonic flow (e.g. a well-wrought deLaval nozzle) would have to be right there at the outlet end of the hose. Getting enough regulated pressure would be no problem at all from an ordinary compressor, as long as your nozzle was made small enough that the flow requirement could be maintained. A little engine like that would be pretty exciting to see in action!

L Cottrill

[larry cottrill]

Of course, I don't really know anything about this kind of design, but it seems to me that a usable supersonic ramjet design (with throttleability) could only be accomplished using variable geometry in different parts of the duct. Note, for example, that simply "straightening" the cone to a cylinder would greatly improve performance when the nozzle is in its low-efficiency mode, because it would at least prevent the "pressure recovery" phase from taking place aft of the throat. It would be like a simple convergent nozzle with a straight extension. With the right electronics, a variable-geometry tail cone could be made to respond automatically to changing conditions, optimizing its internal angle accordingly.

Nothing new in this offering - moveable tail bullets have been used in turbojets since the 1950s - but I wonder how effectively something like this would work? This seems simpler than a variable cone, and might even be workable on a small amateur-built engine. The movement of the bullet would have to tie in to changes in airspeed and fuel flow to the chamber, of course.

L Cottrill

[pezman]

Man, you are prolific!

Seems like something similar could be done at the front end as well. All of the Marquard supersonic designs (D-21, BOMARC etc.) have bluffs for the intake.

BTW, Bruno's comment on reflection got me thinking -- what if you had a "quasi-valveless" that had a short intake w/ a bluff on a spring. When it inhales, the bluff is drawn into the chamber a bit. When it exhales, the bluff is forced up into the throat a bit, making it into a deLaval nozzle -- the high-presure side of the nozzle ought to rectify the pulse a bit. Whatever goes out the "intake" can be mechanically added to the thrust in the normal way (external rectifier, rear-facing intake etc.).

The bluff would not need to be so light that it reacts to a single pulse -- I think that the spring/mass assembly just needs to be resonant at a frequency that is reasonably close to the pulse frequency. After enough pulses, it would be oscillating far enough to do its thing.

On a related note, a resonant valve approach might be workable on conventional valved designs too. If so, it would be a simple way to combat the tendency for valves to burn up -- just put in a heavy valve with a stiff spring.

[larry cottrill]

Man, you are prolific!

I am very quick. I partially compensate for this by being so often wrong.

L Cottrill

[Bruno Ogorelec]

I am very quick. I partially compensate for this by being so often wrong.

Well, I am slow and often wrong. I guess quick is better.

[Eric B.]

From what I recall, a supersonic nozzle is only required at supersonic speed. What would be the purpose of installing one on a subsonic ramjet?

-Eric