Optimal CB for a fixed gun length
Posted: Thu May 08, 2008 6:25 pm
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 and my own, mostly unpublished model "JPS".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.
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.