Optimal CB for a fixed gun length

[jimmy101]

I've been fiddling with the three combustion spud gun models seeing what interesting things can be learned.

Most spudders know of "Latke's rule-of-thumb" concerning the relationship between CB ratio and gun performance. For a given chamber, the maximum performance occurs at a CB ratio of ~0.8. This ratio maximizes both the muzzle velocity and the efficiency of the gun. A fundamental question for spudders is; does "Latke's rule-of-thumb" work if, instead of a fixed chamber size, you have a fixed barrel size? In other words, does the optimum chamber size for a particular barrel also occur at a CB of 0.8?

What about for a fixed total gun length? If both the chamber volume and barrel volume are changed (by changes in their lengths) what CB maximizes the muzzle velocity for a fixed total length? This is the question that I'll try to answer using the combustion spudgun models.

I've looked at three models; <a href="http://www.spudfiles.com/EVBEC/JSE.html">EVBEC Live 1.5</a> by <b>Boilingleadbath</b>, <a href="http://www.thehalls-in-bfe.com/HGDT">HGDT</a> v0.4.4 by <b>D_Hall</b> and my own, mostly unpublished model "JPS". These three models were developed from different starting points. EVBEC is based primarily on the measured performance of actual spudguns. EVBEC attempts to scale the measured performance to the dimensions of the gun being modeled. HGDT, primarily designed to model hybrid guns, is based on modeling the combustion process from "first principles". Though, HGDT has been "tweaked" a bit to better reproduce the performance of real spudguns, it is still basically an <i>ab initio</i> (from first principles) model. Since HGDT is an outgrowth of <a href="http://thehalls-in-bfe.com/GGDT/index.html">GGDT</a> (Gas Gun Design Tool) it pays more attention to the flow of gases than do the other two models.

My model, "JPS", is similar to HGDT in that it is a "from first principles" model. Exactly which first principles are used are not necessarily the same as HGDT. As with HGDT, various portions of the model and the model's parameters have been tweaked a bit to get better agreement with the performance of real spudguns.

So here's the problem: I want to build a combustion gun that will be fired from the hip. Given this firing mode, the total length of the gun will be limited to 5 feet. I want to use spuds as ammo so I'll use a 1.5" PVC for the barrel. (2" would also be good but it's hard to find spuds big enough to tightly seal a 2" barrel.) My local hardware store only has pressure rated pipe up to 3" so that will be the chamber diameter. For simplicity of construction, I will build an inline design instead of an over-and-under or co-axial design.

So, we have an inline gun with a 1.5" ID barrel, 3"ID chamber and a total length of 5 feet. <b><i>Given these design constraints, how should the total length of the gun be partitioned between the chamber and barrel lengths?</i></b> The 0.8 rule suggests that the chamber should be 10" and the barrel 50". The barrel length maximizes the performance and efficiency of the chamber but not necessarily the performance of the entire gun.

Enough background, on with the number crunching!

I used an Excel spreadsheet to keep track of the inputs and outputs from the three programs. The Excel sheet is <a href="http://www.inpharmix.com/jps/_images/Co ... s">here</a> if you are interested. The programs have many inputs in common (barrel and chamber length and diameter, spud mass…) but there are some inputs that are present in one model but not the others. I attempted to use consistent and reasonable values in the three models. The spud mass and friction values typical for a full length spud and a double bevelled spud cutter. Here is a screen shot from the Excel sheet summarizing the inputs for each of the models.
<img alt="FLGD_Excel1.gif" src="http://www.inpharmix.com/jps/_images/FLGD_Excel1.gif">

The calculation results are shown below in the screen shot of the Excel sheet.
<img alt="FLGD_Exce2.gif" src="http://www.inpharmix.com/jps/_images/FLGD_Exce2.gif">
This table lists the chamber and barrel lengths (which sum to 5 feet), the CB ratio, the muzzle velocities predicted by the three programs and the "efficiency" calculated by my program. (The "efficiency" should be taken with a grain of salt. The mathematics is trivial but the result is critically dependent on what value is used for the heat of combustion of the fuel.) The row for the gun with CB 0.8 is boxed. For each model, the maximum muzzle velocity is in bold face. In addition, for each gun the range of velocities which are within 95% of the maximum velocity are boxed.

Here's a graph of the predicted muzzle velocities versus the CB ratio. The vertical purple line marks CB = 0.8.
<img alt="FLGD_graph1.gif" src="http://www.inpharmix.com/jps/_images/FLGD_graph1.gif">
It appears that the three models agree reasonably well. EVBEC and JPS agree very well at CBs below ~2, and HGDT is an outlier. Above CB ~2, HGDT and JPS agree very well and EVBEC is the outlier. All three models agree that the CB ratio that maximizes the muzzle velocity is in the vicinity of 2. That is significantly higher than the Latke' rule of CB 0.8. In the image below I've expanded the most interesting part of the graph.
<img alt="FLGD_graph2.gif" src="http://www.inpharmix.com/jps/_images/FLGD_graph2.gif">
In the graph above the optimal CB values for each of the models is highlighted with a larger symbol. In this view, the agreement and disagreement between the three models is more apparent. Overall though, I'm surprised at how well the three models agree.

All three models agree that the optimal CB is near 2.0. Since the peaks have broad flat tops the optimal CB changes a fair amount between the three models but the difference between the predicted velocity at a model's optimal CB and the predicted velocity at the average optimal CB of ~2 is very small. For example, EVBEC says the optimal CB is 2.67 and the velocity is 262 FPS compared to 260 FPS at CB 2.0, a difference of just 2 FPS.

From these results it is impossible to say which of the three models is "correct" or even "most correct". Perhaps none of them are correct, perhaps all three are correct (which is possible). The level of agreement between the three models suggests, but certainly doesn't prove, that they are making a valid prediction. The CB that maximizes muzzle velocity, for this particular set of design constraints, is about 2. The table below summarizes the predicted velocities at CB 0.8 and the optimal CB's velocities predicted by each of the three programs.
<table border="1" cellpadding="4" cellspacing="0"> <tr> <td>Program </td> <td>Velocity at
CB 0.8 (FPS) </td> <td> </td> <td>Optimal
CB</td> <td>Velocity at
Optimal CB (FPS)</td> <td>Increase in
Velocity (FPS)</td> <td>Increase in
Muzzle Energy</td> </tr> <tr> <td>EVBEC </td> <td>216 </td> <td> </td> <td>2.7 </td> <td>262 </td> <td>46 </td> <td>47% </td> </tr> <tr> <td>HGDT </td> <td>236 </td> <td> </td> <td>1.7 </td> <td>259 </td> <td>23 </td> <td>20% </td> </tr> <tr> <td>JPS </td> <td>217 </td> <td> </td> <td>2.0 </td> <td>256 </td> <td>40 </td> <td>40% </td> </tr> <tr> <td>Average </td> <td>223 </td> <td> </td> <td>2.1 </td> <td>259 </td> <td>36 </td> <td>36% </td> </tr></table>
The models predict that at CB 2 the muzzle velocity is greater than it is at CB 0.8. But is the difference of <u>practical</u> significance? If you actually built the guns with CB 0.8 and 2.0 could you <u>measure</u> the difference in performance?

The average velocity at CB 2.0 for the three models is 259 FPS. The average velocity for the three models at CB 0.8 is 223 FPS, a difference of 36 FPS (16%). That difference should be large enough to measure even with the typically high shot to shot variability of combustion spudguns. In addition, EVBEC and JPS predict that the muzzle energy is increased by 47% and 40%, respectively, as a result of increasing the CB from 0.8 to 2.0. HGDT predicts a more modest, but still significant, increase in muzzle energy of 20%.

<b>Conclusions</b>

1. These results suggest that the performance of <u>this</u> gun can be noticeably improved if it is built to a CB of 2 instead of 0.8.

2. The overall cost of the gun would be essentially the same for both CB ratios. The fittings are the same and the longer chamber is offset by the shorter barrel . Since you usually have to buy pipe in lengths of 12 feet, the total cost will be the same for the two designs.

3. These results <u><b>do not</b></u> suggest that the "Latke CB 0.8 rule" is incorrect. They simply point out that different design constraints can lead to different optimal CB ratios.

p.s. If I've misrepresented, misused or abused any of the model developer's programs please let me know.

[daxspudder]

There is simply one thing to say, and it would have to be, yes. all i know is a put a potato in, press a button, BOOM, and i have a airborne spud, but if i ever decide to program my own potato number cruncher, i now have a database to reference... nice work, more than i would do.

[D_Hall]

Wow. That really is impressive stuff. The agreement between the three models is fantastic, in my opinion.

[psycix]

Another set of interesting number crunching jimmy.
This is one of the things I was wondering about but couldnt find out when I built my first gun. It has a 0.8 ratio now...

0.8 might be the most powerfull ratio for a fixed chamber, but since the barrel is the long part that restricts you in length(portability), we have to know what the optimal ratio is for a fixed gun length or barrel length.

The 0.8 rule points out the most powerfull barrel you can pick for your chamber, but we want it in REVERSE.
What is the most powerful chamber for a barrel?
Now you brought us some enlightment in that, wich is great,

Now we encounter the second problem:
Advanced combustion builders (like you and people who bother about these numbers) usually have more then one spark gap!

In fact, if you double the chamber length AND double the amount of spark gaps, your combustion should occur in the same time.
And since youve got a larger chamber on the same pressure, the performance will increase. (but with diminishing returns as you approach the "salt mine chamber" theory)

[JDP12]

that was really nicely written up jimmy, extremely impressive, great job there

[D_Hall]

Now we encounter the second problem:

THere's more than two problems...

Another: Well, not really a problem; just a variable. Chamber diameter. Double the diameter of the chamber and you quadruple the chamber volume without affecting the length. The consequence of this is gong to be that Jimmy's 2.0 "optimal" CB ratio for a fixed length gun is only going to be true for THAT system. Change the chamber diameter and the number will change.

The obvious extreme case of this would be a chamber that is (say) 10" in diameter but only 1" long. I'm gonna go out on a limb and say that such a system would display an optimal ratio in the 0.8 range.

[jimmy101]

Thanks to everybody for there comments.

Another: Well, not really a problem; just a variable. Chamber diameter. Double the diameter of the chamber and you quadruple the chamber volume without affecting the length. The consequence of this is gong to be that Jimmy's 2.0 "optimal" CB ratio for a fixed length gun is only going to be true for THAT system. Change the chamber diameter and the number will change.

Yep, that is why I was careful and said, a couple times, that the results are for the particular set of parameters; overall gun length, barrel and chamber diameters, ammo, ... Change any one of those parameters and you have to redo the calculations. The choice of the parameters were based mostly on what the typical spudder can get at the local hardware store. A 10" diameter chamber would be great, but it's pretty hard to get the pipe and fittings.

The obvious extreme case of this would be a chamber that is (say) 10" in diameter but only 1" long. I'm gonna go out on a limb and say that such a system would display an optimal ratio in the 0.8 range.

Hmmm, that's not what I would guess. I would think that if there is an "unusual" chamber diameter it would be diameter = chamber length. A very short but fat chamber would be ineteresting though. The flame front area would always be increasing. But the chamber would have a relatively high surface/volume ratio.

psycix: HGDT does multiple sparks, it would be interesting to see what affect HGDT says they have. I'm not sure how confident D_Hall is about that part of the model.

It cetertainly is expected that more sparks, assuming they are well spaced, will increase burn rate. How big of an affect that'll have on muzzle velocity is less clear. If you imagine a gun with an infinite number of sparks along the chamber's central axis (or a single big ass spark from end to end) then the burn speed reduces down to how long the flame takes to reach the nearest wall. That begins to look like D_Hall's suggestion of a short-fat chamber (but without the large surface area problem).


Both my model and HGDT calculate the flame front positions and burn out times. I wonder if there is a relationship between the optimal CB and the flame front burn out time(s)? Is CB~2 the ratio that completely burns the fuel at about the time the spud exits the barrel?

That's one of the cool things about having models, even if the exact accuracy of the models is uncertain. You can ask questions about things that you can't measure, or would be extremely difficult to measure, in a real gun.

[D_Hall]

psycix: HGDT does multiple sparks, it would be interesting to see what affect HGDT says they have. I'm not sure how confident D_Hall is about that part of the model.

As long as the sparks are well spaced, I see no reason to believe that increasing the number of ignition points affects HGDT's accuracy. My combustion model is a bit hokey (long story that I don't even want to get into), but it was selected precisely because it would be able to reasonably handle multiple ignition points with ease.

Again, for emphasis.... It does assume evenly spaced ignition points. In (say) a 12" chamber with 2 ignition points separated by 1"? You're probably better served just with using 1 ignition point in HGDT.



edit: And for the records, if you tell HGDT that you've got 2 ignition points, it assumes that they are located at 3" and 9". Why? Well....

- Flame from I3 travels 3" to hit aft end of chamber.
- Flame from I9 travels 3" to hit forward end of chamber.
- Flames from either igniter travel 3" to meet each other.

Symmetry is a good thing.

[psycix]

Ofcourse HGDT assumes the gaps are evenly spaced because that is the most efficient setup (fastest burn rate).
Placing two spark gaps next to eachother has no reason.

But now I just got an idea!
Barrel combustion and unburned fuel.
When the air enters the barrel, so does fuel and it may not burn in time.
So what if we would put an additional spark gap right at the front, near the breach.
This causes a flamefront in the front end of the chamber so no unburnt fuel will be sucked in.



But back on the topic, it would be awesome if someone could build a program (or some addition to HGDT) to do the calculations for finding the most powerfull chamber for a specific barrel.

[mark.f]

Very interesting jimmy.

I've wondered since a while back if 0.8 C:B ratios weren't for optimizing a fixed chamber volume ONLY. The decrease in latkes velocity data after the C:B was increased always seemed to me as if it was only because the barrel length was decreasing, and not because of a significant portion of unburnt fuel by the time the projectile exited the barrel.

In short, I think we need a another test that uses a fixed barrel length and variable chamber volume. Only downfall to this sort of test is that you would need some sort of telescoping chamber... or a big checkbook to buy multiple chambers in the correct increments.

EDIT: BTW are you actually designing and building this gun or is this the usual hypothetical mumbo jumbo? Because if you want to fire from the hip gun length can be extended quite further than 5 feet. For my next cannon that I'm building now (all that's left is to solvent weld a male adapter for the barrel and install a female Q.D. for propane) the overall length is about 110" total, plus fitting length (96" barrel plus 14" chamber). I made a sling system that supports the chamber at the back, front, and the barrel at the back cam-lock and about 2' down the actual barrel itself. It's not the most comfortable thing to shoot in the world but I'm hoping muzzle velocity is reasonably high (which I would expect from an 8' golfball barrel using a reasonably efficient combustion {{3 spark gaps spaced in a pattern to produce three spherical flame fronts, and a chamber fan}}).