Yet more flame acceleration ponderings....

[D_Hall]

Today I was looking at Jimmy's Technical Spudgun page (LINK). I was looking for calibration data for HGDT's new combustion model. In any event, I was struck by the results of shots numbers 8 and 11.

For those not aware, shots 8 and 11 represent well mixed combustion using 1 and 3 ignition points respectively. Given a 12x3 chamber and ignition points centered axially and spaced correctly longitudinally (as they appear to be)...

Shot 8:
Flame must travel (6^2 + 1.5^2)^0.5 = 6.18"
Time to burnout = 0.0539 s
=> Average flame speed = 6.18/.0539 = 114.7 in/s = 2.91 m/s


Shot 11:
Because shot 11 used multiple ignition points, the flame would not have to travel as far before burnout...
Flame must travel (2^2 + 1.5^2)^0.5 = 2.5"
Time to burnout = 0.0394 s
=> Average flame speed = 2.5/0.0394 = 0.205 m/s

The first thing that hits me is the speed for shot 11. 0.205 m/s? That seems insanely slow. It's less than half of the initial burn rate velocity found elsewhere on Jimmy's page (0.43ish, IIRC). Further, given that the flame front will be accelerating (also evidenced by shot 8 results), it implies a doubly insanely low initial speed. WTF?

Perhaps I'm not interpreting Jimmy's tabulated data correctly? Jimmy, care to comment?


My goal is to come up with an idea of how the flame front accelerates... but I've got to get off TDC.

[jimmy101]

D_Hall:
I think there is a problem with the way you calculated the average flame speed. And, I know there is a problem with the dimensions of the gun. The image on my page is accurate, though somewhat misleading when it comes to the dimensions.

Below is a better-dimensioned drawing. The chamber is roughly 15" long, not 12" (the 3" pipe is 12" but you need to add the length of the fittings), and the spark gaps aren't perfectly spaced (I wanted to avoid the end of the fittings and the fan etc.). Even the dimensions shown in the drawing below aren't perfect since the threaded plugs, reducers etc. make it difficult to exactly define the length of the chamber. From the <a href="http://www.inpharmix.com/jps/Chamber_Te ... ">acoustic temperature study</a> I got an effective chamber length of 14.7".

I believe the average flame speed should be calculated as the maximum distance the flame front has to travel divided by the time to burnout. I don't think the function you used is correct. Is the sqrt(6^2+1.5^2) supposed to be the "RMS" distance? Shouldn't there be another factor of 1/2 in there? In any case, the chamber has not yet burned out when the flame front has propagated the "RMS" distance. Using (max distance)/(burnout time), I get;

Shot 8 (single central spark):
Flame must travel the maximum distance to end of chamber = 7.5"
Time to burnout = 53.9mS
=> "Average" flame speed = 7.5"/0.0539S = 11.6FPS = 3.5m/s

Shot 11 (three sparks):
Flame must travel the maximum distance to end of chamber = 3.5" (right hand gap to the end of the adaptor)
Time to burnout = 39.4mS
=> "Average" flame speed = 3.5"/0.0394S = 7.4FPS = 2.3m/s

So, the "average" flame speed is greater than S_L0~0.4m/s for both configurations.

There is still the apparent anomaly of the 3-spark chamber giving a lower "average" flame speed than the single-spark chamber. I believe this is caused by the arbitrary way of doing the math. Since the flame speed is an exponential function, the distance/time is a very poor measure of the burn dynamics. In addition, the "average flame speed" is really kind of an irrelevant parameter. The "average" speed may be slower but since the flame has a shorter distance to travel the time to burnout is less. Therefore, I would think the key point is that the 3-spark chamber burns faster (by 27%) than does the 1-spark chamber.

It is tempting to think that if you go from one spark to three that the chamber burn time should be cut by 2/3. I don't think that will be the case. In the limit of a very large number of sparks (on the chamber's central axis) the maximum flame speed (and the minimum burn time) will asymptotically approach the "speed limit" of how fast the spherical flame front can reach the nearest chamber wall. I would think that this would be very close to the behavior of the "optimal" spherical chamber.

I used my combustion model to compare the flame position and speed for two closed chambers. One chamber is 15" long and the other 5" long. The 5" L chamber is a crude estimate of the behavior of a 15"L chamber with three spark gaps. I've attached graphs of the flame speed and position as a function of time.

The "average" and the "instantaneous average" flame speeds are predicted to be slower for the smaller chamber even though the smaller chamber burns out sooner than the larger chamber. The "average" flame speed are similar to the experimental numbers; one spark (blue line) ~4.2m/s, "three spark" (red line) 2.3m/s.

In the second graph, the maximum flame speeds are nearly identical for the two chambers, which is to be expected. But, the smaller chamber has an "average" flame front acceleration of about twice the larger chamber.

The burnout times are similar between my model and my data. One spark in a 15" chamber; calc. burnout at 45mS, measured burnout 54mS. Three spark burnout (calc as a chamber of 1/3 length with a single spark) 27mS, measured for 3 sparks in 15: chamber 39mS. The agreement isn't exactly spectacular but, given the complexity of the combustion dynamics, I think it is pretty good.

The graphs can be a bit misleading, you have to be careful about how you interpret them. Looking at the flame speed versus time graph you might think that the high flame speed is highly significant. In actuallity, you probably need to look at flame speed versus fraction of the fuel burned to get a better idea of what is going on. (Or perhaps energy released per unit time.) How much of the fuel was actually burned at the very high flame speed? The low flame speed region lasts a long time but how much of the fuel was actually burned in that time period?

I've always been amazed at the length of time after ignition when essentially nothing detectable happens in the chamber.

[willarddaniels]

I love your work, Jimmy.

I am trying hard to believe that multiple, evenly spaced spark gaps will create the most muzzle velocity, but even after reading all that you have done here and on your site, I cannot bring myself to fully accept that.
If a pipe full of propane is ignited on one end the flame front will continually accelerate until DDT occurs. (I know DDT is not simply just just adding speed up, but that is a different topic.) Would it not stand to reason that one single spark in the breech end would produce a faster flamefront and therefore a higher muzzle velocity than several ignition sources?

Another item I am aware of is the fact that the pressure caused by the single, breech-end spark->combustion, is that the flamefront will be pushing the unburnt fuel in front of the "wave", possibly losing that potential energy down the barrel.

I plan on testing muzzle velocities out with measured ammo and a crony in my advanced combustion this summer (July), but until then, What do you think?

[jimmy101]

Willard: You have to be careful when you go from things like flame speed to how that affects muzzle velocity. It is possible that all this flame speed stuff actually has minimal affect on muzzle velocities. My model says there is an affect. D_Halls's model also probably says there is. But if the affect is small enough it might not be of practical significance. The same can be said for DDT. It is possible that DDT that occured late in the combustion process would have essentially zero affect on the performance of the gun. A 1,000 PSI pressure spike that lasts for a few microseconds won't have much affect on the velocity of the ammo.

In any case ...

If a pipe full of propane is ignited on one end the flame front will continually accelerate until DDT occurs.

That is not true. The flame front will accelerate until one of several different things happens (in no particular order);
1. DDT.
2. The end of the pipe (or the butt of the spud) is reached.
3. Rate of heat loss equals, or exceeds, the rate of heat generation.
4. The spud leaves the barrel

In a typically sized spudgun chamber DDT is unlikely. I suspect that in a typically sized spudgun DDT is basically impossible, you can't keep the spud in the barrel long enough to build up the shock needed to trigger DDT.

One of the amazing things about combustion spudguns (non-hybrid, non-burst-disk) is that the spud starts to move after only a small amount of fuel has been burned. The percent fuel burned at spud movment depends mostly on the static friction between the spud and the barrel. For typical sized guns, and a tight fitting spud, the spud starts to move when only 10~20% of the fuel has been burned. The gun is now in a "race state". What happens first? Does the fuel finish burning and release all of it's energy or does the spud exit the barrel and leave unburned fuel?

Would it not stand to reason that one single spark in the breech end would produce a faster flamefront and therefore a higher muzzle velocity than several ignition sources?

It may well produce a faster flame front but flame front speed isn't the whole equation. A central spark produces two flame fronts. The amount of fuel burned per unit time may be greater with two fronts, each of which is moving slower, than it is with one faster moving front. With multiple sparks you get multiple flame fronts and the individual fronts can move even slower but still get the fuel burned in less time.

To make modeling things even more difficult, movement of the spud affects the flame front speed. Simple minded physics suggests that the flame front slows down as the spud moves. More complex physics suggests the flame front accelerates. The flame front might even accelerate more with a moving spud then it does in a closed chamber.

Another item I am aware of is the fact that the pressure caused by the single, breech-end spark->combustion, is that the flamefront will be pushing the unburnt fuel in front of the "wave", possibly losing that potential energy down the barrel.

This is the kind of questions that gets beyond what I can conceptualize. There are so many factors affecting the burn speed (and more importantly the chamber pressure) that changes in any single factor is difficult to translate into changes in spud velocity. A breech end ignition event gives you a flame front that has to chase the fuel down the barrel. I would think that would decrease the overall burn rate. A "barrel end" flame front would tend to push the unburned fuel up against the breech plug. The "trapped" fuel has nowehere to go but into the flame front. But now you get into the "difficult to conceptualize" domain. The flame front movement and the gas movement are related in a complex way. The gases can move in the same direction as the flame front or in the opposite direction of the flame front. The relative directions gets into the problem with how you define the flame front speed (relative to the chamber or to the gases?). Which gases are moving? Unburned fuel+air or combustion products? This is where I usually go "crap, I have no idea". Then recover a bit and say "it probably makes a difference but the difference is so small you can't detect it in a typical spudgun".

Another amazing thing about propane+air combustion is how slow it starts out. In our ever popular "mine sized chamber" a person could light a match, turn and start walking and "outwalk" the flame front for quite a long time (well for several seconds at least). The initial flame speed is ~0.4m/s (~0.9 MPH), which is only about half of a person's normal walking speed. That's amazingly slow.

It seems to me that the various models of combustion spudguns have reached their limits. A little reliable data could answer a lot of questions.

Any data you can collect on a well characterized ammo and gun would be of great use.

[D_Hall]

The chamber is roughly 15" long, not 12" (the 3" pipe is 12" but you need to add the length of the fittings), and the spark gaps aren't perfectly spaced (I wanted to avoid the end of the fittings and the fan etc.).

Ah. Thanks.

I believe the average flame speed should be calculated as the maximum distance the flame front has to travel divided by the time to burnout.

Agree

I don't think the function you used is correct. Is the sqrt(6^2+1.5^2) supposed to be the "RMS" distance?

No, that's not RMS distance. It's the max distance the flame has to travel in a 12x3 chamber with mid-point ignition. IE, the flame must travel 6" up/down the chamber and 1.5" up (ie, to the wall of the chamber).

So using your 15" length I now get....
Shot 8:
sqrt(7.5^2+1.5^2) = 7.65"
Time = 0.0539
=> 141.9 in/s = 3.60 m/s.

Shot 11:
sqrt(3.5^2+1.5^2) = 3.808"
Time = 0.0394 s
=> 96.64 in/s = 2.45 m/s.

So, the "average" flame speed is greater than S_L0~0.4m/s for both configurations.

Agree. Ah, what a difference a couple inches in chamber length make.

There is still the apparent anomaly of the 3-spark chamber giving a lower "average" flame speed than the single-spark chamber. I believe this is caused by the arbitrary way of doing the math. Since the flame speed is an exponential function, the distance/time is a very poor measure of the burn dynamics. In addition, the "average flame speed" is really kind of an irrelevant parameter. The "average" speed may be slower but since the flame has a shorter distance to travel the time to burnout is less. Therefore, I would think the key point is that the 3-spark chamber burns faster (by 27%) than does the 1-spark chamber.

The 27% number is an acceptable way to handle the problem if you're simply going to use lookup tables to handle other scenarios (say... 8 igniters?). But what I'm trying to do is (very crudely) characterize how the flame front accelerates. If I can do that, then I can extrapolate to untested scenarios with a higher degree of confidence. Now, is that time/distance relationship DIRECTLY useful? No. But it does allow me to examine the validity of theories. If they don't produce data in the same ballpark; out they go!

It is tempting to think that if you go from one spark to three that the chamber burn time should be cut by 2/3. I don't think that will be the case.

Clearly. This is an obvious byproduct of the knowledge that the flame fronts accelerate. Your data quantifies it, however.

[jimmy101]

Agree. Ah, what a difference a couple inches in chamber length make.

That's what she said.


(sorry, couldn't resist)

Well duh on the formula you used, I should'a been able to figure out that is the hypoteneus of a right triangle. Makes sense, large aspect ratio and the hypot' is roughly the length, low aspect ratio and the difference is significant.

[D_Hall]

OK, here's a thought....

Assumption: The flame propogation (not to be confused with mass burn rate!) is linear as a function of distance already traveled. That is to say that if I assume that the speed of the flame relative to quiescent air follows ye ol' y=mx+b.

If I use the data from my eariler post, IE...

Shot 8:
sqrt(7.5^2+1.5^2) = 7.65"
Time = 0.0539
=> 141.9 in/s = 3.60 m/s.

Shot 11:
sqrt(3.5^2+1.5^2) = 3.808"
Time = 0.0394 s
=> 96.64 in/s = 2.45 m/s.

Then...

Speed = 0.3d + 1.31

(Admittedly I'm using bastardized units. Speed is in m/s while distance (d) is in inches.)

The 1.31 goes against our assumed Vo of 0.4ish, but bare with me....

Mach1 is approximately 358 m/s.

So.... DDT should occur after 1192 in = 100 feet = 397 diameters.

That doesn't sound very likely. Too long of a run-up. Too fast of a Vo. OK, off to run the quadratic!

[willarddaniels]

Thanks for the reply.
My DDT comparison was not necessarily in a spud gun, but an undetermined length of pipe, just for academic purposes.

I will be able to do some accurate testing this summer on muzzle velocities. I plan on using golfballs (weighed out,) and first changing the variable of spark gaps, fan(s), etc. That is much later though.

Again, thanks for the feedback and candid answers. I enjoy reading/half-participating in the comments of two (Jim, DH) great minds as you discuss this.

[D_Hall]

More musings....

If one has a planar flame front and a constant velocity flame, then one would expect the pressure increase in the chamber to follow a second order curve with respect to time....

Fuel burns. Pressure increases. As pressure increases, density increases. With increased density, a flame that burns X fps burns an increased amount of fuel per unit time. Repeat cycle.

If the flame front accelerates (as everybody agrees that it does) then similar things happen but everybody gets to add one (or more!) to the exponent. If the acceleration is linear, the flame speed is 2nd order, and the pressure trace becomes cubic (3rd order).

If the flame front accelerates as a second order? Hey, now our pressure trace is 4th order.


The only problem with all this is that our flame fronts are NOT planar. They start out roughly spherical and evolve from there.

Still... Do we have any *measured* pressure traces to look at?

[clide]

A bit smaller scale than we usually deal with, but I found this site that has some interesting pressure vs time graphs for a closed chamber. It even has low hybrid mixtures.
http://www.ae.gatech.edu/labs/comblab5/ ... nstVol.htm

[D_Hall]

Whoohoo!

Just stumbled across this gem (emphasis mine)...

Large-scale explosions of initially quiescent methane-air and propane-air mixtures at atmospheric pressure are reported, in which the flame speed of a hemispherical flame is measured up to radii just beyond 3 m. A cellular flame is developed fairly soon and thereafter the flame speed increases continually with the square root of the time.

[Lentamentalisk]

Ok, so a month or so ago I went to CAL day (I live just a mile away) and happily spent my entire day down in the combustion labs.
They have this one device called the "BOOM TUBE" which is just a clear piece of pipe with one end capped, with a methane and air inlet, and at the other end they have a glowing cigarette lighter. They can adjust the fuel/air ratios to examine the effects of running too lean or rich. They have several infrared sensors along the tube to measure flame propagation rates.
Long story short, it was awesome!

Relevance:
When the gas and air mix reach the igniter you can watch the flame front propagate. It starts out as a laminar flame, sticking to the top of the tube (heat rises) until it gets to about 2/3 of the way down the tube. All of a sudden turbulence picks up, and the flame goes to deflagration (i think.) There is the slow, laminar flame, and then a really loud honk.

Point being, that would explain why almost nothing happens for so long in the chamber. This time could probably be greatly reduced by running the fan, as there would already be turbulence.

Hope the helped.

[jimmy101]

Fuel burns. Pressure increases. As pressure increases, density increases. With increased density, a flame that burns X fps burns an increased amount of fuel per unit time. Repeat cycle.

The gas density doesn't change during the combustion process in a closed chamber. There may be a density gradient but the total mass/volume is a constant. In a gun, the density decreases once the ammo starts to move.

So, shouldn't it be;
Fuel burns
Temp increases
Pressure increases (because of temp increase and the very small affect of more moles of gas if you're talking about propane combustion)
Using; S_max = S_L0 (P_max/P₀)^beta(T_max/T₀)^alpha
The increased pressure slows down the flame speed by small amount (beta=-0.17).
The increased temperature greatly increases the flame speed (alpha=2.13)
Things progress in this way for a while until something triggers turbulent flow. Could be caused by the combustion process itself, or by movement of the ammo.
Turbulent flow gives a turbulent flame front and the flame speed of a turbulent front "can be up to ten times faster than" than a laminar front. (from a paper I read long time ago, I'll see if I can dig it up).

You can use the adiabatic combustion temp to estimate the maximum laminar flame speed in a closed chamber.
S_max = S_L0 (P_max/P₀)^beta(T_max/T₀)^alpha
For propane in air;
alpha= 2.13
beta= -0.17
P_max= 9.28 atm
T_max= 2631 K
P₀= 1 atm
T₀= 300K
S_L0= ~0.4m/s
(Pmax and Tmax from GasEq for adiabatic propane + air).

S_max = (0.4m/s) (9.28/1)^-0.17(2631/300)^2.13
S_max = 30.4m/s

10X for the maximum affect of turbulence gives 300m/s. That is near the speed of sound at 300K, but is only Mach ~0.3 at the 2600K temperature in the chamber.

[D_Hall]

Fuel burns. Pressure increases. As pressure increases, density increases. With increased density, a flame that burns X fps burns an increased amount of fuel per unit time. Repeat cycle.

The gas density doesn't change during the combustion process in a closed chamber. There may be a density gradient but the total mass/volume is a constant. In a gun, the density decreases once the ammo starts to move.

I'm talking about the density of the unburned air/fuel. Your "density gradient" is my density increase. TomAto. TomAHto. It's all in how you do your accounting.

My point is that even if flame velocity and burn area are absolutely constant at [pick fixed numbers], propellant consumption rate is NOT constant due to the existance of the density gradient (to use the accounting methodology you prefer).

[jimmy101]

Ya, I get that the fuel consumption rate changes if the unburned fuel is compressed by unburned fuel. If the unburned fuel is compressed by combustion products (density increases because of hot N2, CO2 and water ejected from the combusted volume) that won't necisarily increase the burn speed, at least not because of changes in density. The big change in the temperature would be expected to be much more important than the smaller change in density (or pressure).

Increasing the pressure of the unburned fuel slows down the flame front slightly. I don't know how to equate the slightly slower flame velocity with the increased fuel density to get at the net affect on moles fuel burned per unit time. The mass burn rate might go up or down.

the flame speed increases continually with the square root of the time
Only problem is that you don't know what the proportionality constant is. And, is the constant the same for all chamber sizes and geometries or does the constant change with the chamber volume and dimensions?

Assumption: The flame propogation (not to be confused with mass burn rate!) is linear as a function of distance already traveled.

I don't think there is a general relationship between flame speed and the distance it has already traveled. I believe the flame speed is a function of, among other things, the fraction of the fuel that has been burned up to that particular point in time. I believe the maximum flame speed in a closed chamber is, to a first approximation, the same for all chamber sizes. A 3ci mini chamber reaches about the same maximum flame speed as a 100ci chamber. The details of the chamber geometry, such as the aspect ratio, tweak the velocity somewhat. Movement of the ammo in a typical combustion gun tweaks the flame speed a lot.

The relationship might break down if you get to DDT conditions but those conditions are difficult to reach and require that several factors are in perfect balance.