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Fixed size barrel, what CB ratio max's the muzzle Velocity?For a fixed size barrel, what CB ratio maximizes the muzzle velocity?
This question is, in a sense, the opposite of the question Latke's CB studies were designed to answer (for example 15cb.html). In Latke's studies the chamber volume was held constant and the barrel volume (length) was varied. What would be the results of a study in which the barrel was held constant and the chamber volume varied? Would one obtain the same results as the Latke barrel studies, that is, the optimal CB ratio is ~0.8? Or, does a larger chamber increase the performance for a fixed barrel length? If a larger chamber increases the performance of the barrel is there a limit to how large the chamber can be and still get increasing performance? In order answer this question we need a way to scale the Latke data to different size guns. To do this scaling we will assume that; The chemical energy in the chamber is proportional to the volume of the chamber which is proportional to the kinetic energy of the spud. The actual scaling of the Latke data is done in four steps; 1. Scale the Chamber To scale the chamber we will simply take Latke's chamber volume (160in<sup>3</sup>) and multiply by factors of 1.25, 1.5 and 1.75. 2. Scale the Energy To scale the energy in the chamber we will assume that the energy simply scales as the volume of the chamber. Increasing the chamber volume by a factor 50% results in 50% more energy in the chamber. We will assume that the efficiency of the combustion process is the same for all chamber sizes. 3. Scale the Barrel Since we have scaled the chamber we will also scale the volume of the barrel. To keep things consistent we will scale the volume of the barrel by just scaling it's length. We will scale each of Latke's 12 barrel lengths. 4. Scale the Muzzle Velocity Since the chamber volume and barrel length of been scaled we also need to scale the muzzle velocity. This is the most problematic part of the scaling. We are assuming the overall efficiency of the gun does not change on scaling. Since we are assuming the KE of the round is proportional to the volume of the chamber we can use the relationship between the KE of the spud and the spud's velocity; KE =(1/2)mv<sup>2</sup> Where m is the mass of the spud and v is the muzzle velocity of the spud. Since the mass of the spud is being held constant, we can write; V2 / V1 = v2<sup>2</sup> / v1<sup>2</sup> Where V2 is the scaled chamber volume, V1 is the Latke chamber volume, v1 is the Latke muzzle velocity and v2 is the calculated velocity for the scaled gun. Since we actually want the muzzle velocity we take the square root of both sides of the equation and rearrange it to; v2 = (v1)SQRT(V2 / V1) In other words, we scale the velocity by the square root of the ratio of the chamber volumes. As an example, consider the Latke data for the 100 inch barrel (for which the CB is 0.8 ). The chamber volume will be scaled by 1.25. The barrel length (and therefore it's volume) is also scaled by 1.25 to a length of 125 inches. The muzzle velocity calculated for this new chamber and barrel combination is; v2 = (514FPS)SQRT(1.25/1.00) = 575 FPS I've repeated this process for the three scale factors (1.25, 1.5 and 1.75) and all 12 barrel lengths. A graph of the muzzle velocity versus the barrel length for the four guns is shown below. In this graph the black curve is the original Latke data; variable barrel length on a fixed size chamber. The thick vertical green line marks the constant 100" long barrel, this is the 0.8 CB barrel for Latke's chamber. The graph for the calculated gun with a 25% larger chamber (red line) shows that the muzzle velocity is predicted to increase with the larger chamber, from 514 FPS to 556 FPS. For the 50% larger chamber (blue line) the muzzle velocity also increases, from 514 FPS to 594 FPS. Interestingly, for the calculated gun with a 75% larger chamber the muzzle velocity is predicted to drop off substantially, from 514 FPS to 453 FPS for the constant barrel length. The predicted muzzle velocity is not only less than that for the next smaller chamber size, it is also less than the velocity of the original chamber size. For each of the chamber sizes we can also calculate the CB ratio for the 100 inch long barrel. A graph of the muzzle velocity as a function of the CB ratio is given below. As you can see, the scaling model predicts that, for a fixed size barrel, the maximum muzzle velocity is obtained with a CB ratio of about 1.2. If the CB ratio is increased (the chamber is made bigger) beyond 1.2 then the muzzle velocity drops off dramatically. It looks like the performance would be pretty poor at a CB of 1.5. This result is fundamentally different than the Latke result. In the Latke studies the performance and efficiency of the chamber is being optimized by changing the barrel length. In this modeling study we are maximizing performance without regard to efficiency. Indeed, the gun scaled by 1.5 (which gives a CB of 1.2 on with the 100" barrel) is less efficient than that chamber would be on a longer barrel. It appears that a 50% change in the length of the chamber, which is a small change in the total length of the gun (7 inches total out of 9.5 feet), increases the muzzle velocity by 16% and the kinetic energy by 34%. To get the full benefit of the larger chamber size requires a significant increase in the total length of the barrel and gun (4.2 feet longer) but the much larger length gun only increases the muzzle velocity by an additional 7% and the KE by 16%. Conclusion This mathematical model suggests that there is indeed an optimal CB that maximizes muzzle velocity for a fixed length barrel. The optimal CB appears to be about 1.2. The velocity does drops off with an oversized chamber. Furthermore, the drop off occurs at CB's greater than about 1.3 and the drop in muzzle velocity is substantial. For more details see Optimal Chamber Volume for a Fixed Barrel Size
well that is is a whole lotta working out
i was searching for something that told me the disireable chamber size for my largest feasible barrel that could fit in my car actually! thnx
Aussie spudders unite!!
Look at the linked page. Down near the bottom there is a section called "So, Lets Design a Gun Using a CB of 1.2". There is a formula (eq. 5) to calculate the chamber length and barrel length given the chamber and barrel IDs, target CB and the desired total length of the gun.
Jimmy,
Nice work! Its about time someone brought this up, with actual calculations to back their thoughts. However, because I have been working on the same issue for a while, I have reason to suggest that your calculations for higher C:B ratios may be flawed. I quote Latke's site:
The moral of the story, Latke's 1.5" barrel tests were flawed on account of a buildup in the barrel significantly affecting the velocity output of the launcher after a certain point (probably around the 1.4:1 ratio tests). The solution? Use Latke's 3/4" barrel data (which also conveniently uses ridiculously high C:B ratios that can be used in an attempt to establish an upper limit). Original Latke 3/4" data for a 55" barrel (0.8:1 ratio) showed an average muzzle velocity of 506FPS. I will scale the chamber by 2.45, which gives a C:B ratio of 1.96:1. Latke's 3/4" data for this barrel gives an average velocity reading of 440FPS. Calculated velocity for a 55" barrel with a 245% larger chamber: (440FPS)SQRT(2.45/1) = 689FPS Let's try Latke's highest C:B ratio used in the 3/4" tests, 25.2(!):1. For this, we will need a scale factor of 35.1. Latke's velocity readings had an average of 167FPS. So, for a 55" barrel, a chamber of this scale would yield a muzzle velocity of (167FPS)SQRT(35.1/1) = 989FPS. I'm fairly certain that the muzzle velocity will continue to increase with increasing chamber volume (up to a certain, very high ratio). Honestly, someone needs to figure out a way to build a chamber that can easily and inexpensively vary its size. We need hard data on this issue. "If you have a theory, and you have real world data, and they do not mesh, which one is wrong? The theory."
People should not be afraid of their governments. Governments should be afraid of their people.
I do have a theory on how that could be done. If you could get an PVC rod that almost fitted the chamber, and cut some oring grooves into one end of that, you could put that in (oring end first), screw on the chamber end and test. Then you take the PVC rod out, cut an inch or two off the back end and repeat  adjusting the fuel mix as you go  which would slowly increase the chamber volume. Still a bit costly depending on exactly how much PVC rod will cost you, but it's the best option I can think of. It would also be very good to have information on single tube launchers, where the overall volume of the barrel + the chamber is constant.
Does that thing kinda look like a big cat to you?
ragnarok good idea... but i think it could be done much easier with a sttel chamer and a long bolt that would go almost all the way throught the chamber.... the bolt would hold the rod with an oring in preffered position... whenever you would like to change the size of the chamber you could do so by unscrewing the bolt a little
Spudblaster: Good point about the problem Latke had with the 1.5" tests.
I havn't worked through the other Latke datasets so I don't know if the "1.2 magical CB" is consitent with other ammo and barrel diameters. But, if you look at all of latke's performance graphs you see that velocity versus barrel length always rises to a peak and then drops off. All scaling function that I can think of will give a graph that looks the same as my first one. As the chamber volume increases the peak top gets higher, the peak location moves right (to longer barrels), the peak width broadens a bit. It seems to me that there will be a range where slightly large chambers will perform better on a fixed barrel. There will also be a point at which a too large chamber performs worse. (It did surprise that the "mine sized" chamber occurs at a CB of only about 1.4. Be interesting to see if Latke's other data sets give the same result.) I'll need to cogitate on the rest of your post for a while. One thing I've been wondering about is if Latke's velocity versus barrel length is really reliable. He did a good job collecting the data but he never actually measured the velocity versus position in the barrel in a single shot (hit technique doesn't allow it). His graph (and mine) could be obtained with a sharp velocity versus position graph if the graph varried significantly from shot to shot. In other words, two shots under otherwise identical conditions have different optimal CBs. The variablity in optimall CB could be caused by subtle difference in the ammo (mass, friction etc.) or by general inconsitency in the combustion process.
I think the easiest way to do the variable chamber study would be to make multiple chambers.
The problem with filling in part of a chamber with something is (1) moving the fan and (2) moving the spark(s). If you were to mount the fan and the spark electrodes on the cleanout cap (with the electrodes extending to near the center of the chamber) and used a removable barrel, then all you would need for the each chamber is a length of pipe, a cleanout adaptor, chamber to barrel reducer, and threaded coupling. The fan and spark gaps would be moved from chamber to chamber with the threaded cleanout plug. Someone (don't remember who) suggested a set of threaded chamber sections that could be added and subtracted from a barrel. That also sounds plausible. I would worry that getting the large ID threaded fittings to seal might be a challenge. You would definetly have to come up with a way to pressure test the chamber. Another thing to consider might be to partially fill the chamber with water and then fire the gun straight up so the water is at the breech end. Of course, you would have to come up with a way to protect the fan and the spark gaps. This might be the cheapest and simplest way to do it.
SpudBlaster15: Your analysis for the 3/4" data appears to be correct (except 284.7MPH is 418FPS not 440FPS). For the 3/4" data the velocity is still rising even up to a CB of 4.78 (as high as I checked).
I probably should run Latke's other two datasets through the same analysis as I did for the 1.5" spuds. "If you have a theory, and you have real world data, and they do not mesh, which one is wrong? The theory." Actually, all you can say for certain is that at least one, and perhaps both, are wrong. Besides, in this case the theory looks correct (or at least there is no data saying it is wrong), it looks like it is Latke's 1.5" data that is suspect.
Yeah, must have been looking at the ratio above 1.96:1 in the table (1.58:1), which has an average reading of 300MPH (440FPS).
Good point; however, this statement was made in reference to the potential for all of our theories to be disproved by someone with access to a variable volume chamber, and the patience to perform a test similar to Latke's. The theory is sound though, so I don't suspect that real world tests would yield much different results.
People should not be afraid of their governments. Governments should be afraid of their people.
I've analyzed Latke's 0.75" gasket slug data the same way I did the 1.5" spud data.
The results indicate (and as SpudBlaster suggested) that for this data set there is no drop off in muzzle velocity for oversized chambers. The curves for the various chamber scalings never cross. Looking back at the original post, the Latke data at barrel lengths less than about 60" (CB > 1.3) appears to be incorrect. Presumably this is because of the technical problem with buildup in the barrel as SpudBlaster posted. The steep drop off in muzzle velocity at short barrel lengths is the entire reason why, with scaling, it is possible to get the curves to cross one another. Without this steep drop off the curves never cross and there is no predicted optimal chamber size for a fixed barrel size. It looks like that if there is an optimal chamber size, then it will require either experimental data, or a much more complex scaling model, to figure out what that optimum is.
 
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