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random thoughts about the ballistics of foam darts

Archive for the ‘Interior ballistics’ Category

Notes about valves and flow rates

Posted by btrettel on March 20, 2013

Comparing valves for compressed gases can be difficult for several reasons. Even if you are familiar with how valve performance is specified, you can easily make mistakes. Valve performance is specified in several different sloppy and confusing ways, almost always in non-metric units. The worst part is that valve performance can be specified in obvious ways, but people working in the industry don’t do that.

Flow rates

The confusion starts immediately with the definition of the flow rate. Rather than specifying a mass flow rate (i.e., something like g/s), most valve manufacturers instead specify a volumetric flow rate (i.e., something like m3/s). Why? I’m not entirely sure. It’s probably a combination of tradition (e.g., “that’s all I was taught”), familiarity, laziness (e.g., the flow meter gives volumetric, and they don’t want to convert), or ignorance (e.g., they don’t realize it matters). I suspect the practice was held over from liquid valve flow, where the fluid density doesn’t vary much and the volumetric flow rate is just proportional to the mass flow rate.

This is problematic because volume can vary considerably over a range of temperatures/densities for the same mass flow rate. To use a volumetric flow rate (by converting it to a mass flow rate to use in a mass conservation equation) requires an engineer to know the temperature and pressure at which it applies.

With this in mind, some engineers developed the unit called “standard cubic feet per minute” or SCFM for short where the temperature and pressure are supposed to be standard, i.e., known. Unfortunately, there is no standard, though usually the reference pressure is 1 atm and the reference temperature is 70F.

Unfortunately, some companies simply treat SCFM as if it were the actual measured flow rate! It’s as if they don’t understand what the point of the standardization was or even that the S stands for standard. Clippard is one of these companies (I asked by email. Their reference pressure is the pressure they did the test at and their reference temperature is 70F.). If you are using a company’s valve, I suggest asking what their reference conditions are. If they don’t know, then don’t assume anything about their products. I’ve found that some companies are completely ignorant about the performance of their products. Some companies have no idea where the numbers they print on their spec sheets come from. They sometimes don’t even know the units to the numbers. This is extremely sloppy and it’s rather scary that a large number of engineers work this way.

The relationship between volumetric flow rate and mass flow rate is \dot{m} = \rho Q where \dot{m} is the mass flow rate, \rho is the mass density, and Q is the volumetric flow rate.

We can find \rho with the ideal gas law (\displaystyle P = \rho \frac{\overline{R}}{M} T) and convert between volumetric and mass flow rates. P is the pressure (in absolute units), T is the temperature (again, absolute temperature), \overline{R} is the universal gas constant, and M is the molecular mass of the gas (for us this is air). So, rearranging the equation above yields \displaystyle \rho = \frac{P M}{\overline{R} T}.

The standard uses the standard temperature and pressure. So, for example, for convert from a volumetric flow rate of 20 SCFM to a mass flow rate in g/s, we can do the following:

\displaystyle \dot{m} = \rho Q = \frac{P_\text{s} M Q}{\overline{R} T_\text{s}}
\displaystyle = \frac{101325~\text{Pa} \cdot 28.97~\text{g/mol} \cdot 20~\text{ft}^3\text{/min}}{8.3145~\text{J/(mol K)} \cdot 293.15~\text{K}} \left(\frac{3.28~\text{ft}}{\text{m}}\right)^3 \frac{1~\text{min}}{60~\text{s}} = 11.4~\text{g/s}

Valve flow coefficients and flow models

Often valve performance is specified with a valve flow coefficient. In the US, this usually is called C_\text{v} where v refers to valve. (This is not the heat capacity at constant volume. Again, this is another confusing point.)

Like the valve flow rate, C_\text{v} is defined in terms of liquid, not gas flows. C_\text{v} is defined as the volumetric flow rate in gallons per minute (GPM) for a pressure drop of 1 psi. There are some mainly empirical equations (scroll down to the section “Flow Coefficient – Cv – for Air and other Gases”) that can be used to calculate the volumetric flow rate (and mass flow rate from there). The “derivation” of this model is available in this book.

For more information on this subject, I suggest reading my notes about Nerf engineering, which contain many references to other materials. In particular, I detail a more accurate empirical valve model that can be used in computer simulations.

Accuracy of manufacturer data

You should always question the accuracy of valve manufacturer data (assuming that you understand what it means, which, as I’ve explained, isn’t always clear). In my experience, the flow rates given seem to be optimistic. It’s not easy to quantify how optimistic they are, but you should keep this in mind.

An aside

I could insert a long rant about how some engineers don’t understand what they are doing and treat some equations like magic black boxes which give them “the answer”. They don’t understand the assumptions or development of the equation. They don’t try to improve the equation to be more general, more useful, or less confusing, e.g., by eliminating confusing “standard” volumetric flow rates and moving to obvious mass flow rates. This practice creates a lot of confusion for people who do know what they are doing. If you are in engineering, please, don’t be one of these folks.

Posted in Interior ballistics, Math, Misconceptions, Pneumatics | Comments Off on Notes about valves and flow rates

Nerf Engineering notes

Posted by btrettel on July 8, 2012

I’ve started organizing my writings and thoughts on Nerf ballistics into a set of notes (link corrected). Readers of my blog might be interested.

For the moment my notes are very rough and incomplete. I’ll update the PDF as I make changes. I’d appreciate any feedback, especially corrections and ideas.

Edit (2012-10-02): I’ve removed the notes because they need a lot of work before I’m comfortable distributing them.

Edit (2013-01-19): I’ve readded the notes as I’ve revised them somewhat. They are still a work in progress.

Posted in Design, Exterior ballistics, Interior ballistics, Math, Pneumatics, Safety, Springers, Terminal ballistics | 2 Comments »

boltsniper’s optimal barrel length formula

Posted by btrettel on June 11, 2011

Whenever there is a thread about one of the holy grails of Nerf ballistics, an equation for the barrel length that maximizes performance, someone is bound to mention the results of some tests boltsniper did in 2005 when he was designing the FAR. boltsniper, for the uninitiated, is one of the few engineers in the Nerf hobby, so his words have some authority behind them. People generally misrepresent what he said and overstate this formula’s abilities. What boltsniper actually said is written below. The emphasis is mine.

I did some experimentation to determine what would be the optimal barrel length for a given plunger size.  The goal was to find the barrel length for which the dart would exit the barrel as the plunger reaches the end of the plunger tube.  I started off by matching the volume of the plunger to the volume of the barrel.  I knew that this was going to produce too long a barrel but it was a good place to start.  This would assume that the air inside the plunger and barrel is incompressible and that there are no leaks.  In the real world this is not the case.  I reduced the barrel length until I had found the length at which the dart was leaving the barrel as the plunger was reaching its stop, coinciding with the maximum attainable range.  Experimentally the plunger volume seems to be about 4 times that of the barrel.  The relation for barrel to plunger size can be summed up in the following equation,

4 \pi r_b^2 l_b = \pi r_p^2 l_p

where r_b is the barrel radius, r_p is the plunger radius, l_b is the barrel length, and l_p is the plunger length.  For Nerf applications the barrel is almost always 1/2″ PVC or CPVC.  r_b can then be set as a constant at 0.25″ and removed from the equation.  Since we are trying to solve for the barrel length with a given plunger size, the equation can be rearranged and simplified to:

l_b = D_p^2 l_p

This simple equation makes it easy to roughly but quickly size a barrel to a given plunger.  The equation could also be used to size a plunger for a given length barrel.  This equation is based on experimental data and is not perfect.  Four is not the golden number.  This produces the optimal barrel length for the situation I was testing.  The type of dart, dart-barrel friction, and total system volume will likely effect the optimal ratio.  Nevertheless, the above equation can be used as a starting point.

The last paragraph seems to be completely ignored by most people who use this formula. At best it’s a starting point for further testing. The equation only applies to the FAR as that was all that he tested.

boltsniper later expanded on the restrictions on the use of this formula at NerfHaven. (Again, the emphasis is my own.)

I derived that empirically and more importantly it was derived for the specific situation I intended on using it for: a plunger weapon. It will not work for a compressed air system. One of the big factors I used to come up with that was the lack of compressibility. I later factored that in with a constant that was derived empirically. My tests were with a setup exactly like I was going to use on a the finished product. If you scale the system down that magic constant may not hold true.

There are too many variables to analytically design the optimal barrel length. If you are going to build or mod a spring gun the equation I provided may be a good starting point. That equation gives a barrel length that is slightly too long, so to obtain the optimal length you are going to have to go shorter.

The only real way to do it is experimentally.

The short message is that this equation only applies for the situation he was testing for.

But does it even apply for that situation? I’d argue no. boltsniper wasn’t testing for optimal barrel length. In his own words (which I emphasized above), boltsniper’s “goal was to find the barrel length for which the dart would exit the barrel as the plunger reaches the end of the plunger tube.” This does not coincide with when performance peaks based on my understanding of the interior ballistic processes.

Performance is maximized when acceleration slows to zero. If the plunger is at the end of the plunger tube, the pressure is approximately maximized. This corresponds to maximum acceleration because the force is maximized, not maximum velocity. The ideal barrel length is definitely longer in this case.

(I’ll mostly ignore the question of how he knew the plunger struck the end of the tube when the dart left. I seriously question how he determined that. The entire process occurs in a fraction of a second. He’d need a high speed camera with a clear plunger tube and barrel, some other optimal system, some acoustic system, some similar combination, or something I’m not considering to actually determine this with accuracy.)

In summary, this formula should not be used for general purpose design to approximate ideal barrel length. I suggest using a chronometer, ballistic pendulum, or some other device or procedure to measure the muzzle velocity or where it stops increasing as the barrel length is changed. Alternatively, range can be measured, but please note that drag can cause range to not increase from increases in muzzle velocity, the performance parameter that we’re examining.

If more general-purpose approximations are wanted, I have developed approximate equations for ideal barrel length of pneumatics and springers based on adiabatic process relationships. These equations apply when the pressure in the barrel approximately equals the pressure in the gas chamber or plunger tube. For pneumatics, this is valid for very fast and high speed valves and very heavy projectiles. For springers, this is valid for very heavy projectiles. How heavy “very heavy” is depends on the situation, and I have not fully developed a criteria to determine this. The link contains an approximation I developed a year ago.

Posted in Design, Interior ballistics, Math, Misconceptions | Comments Off on boltsniper’s optimal barrel length formula

Banding rubber bladders

Posted by btrettel on January 30, 2011

When a Nerfer wants to increase the pressure of a bladder, they generally turn to “banding,” the layering of rubber bands on the bladder. I came into this hobby from the similar water gun hobby, and I’ve often wondered why this method is so prevalent in Nerf but not water guns.

It’s not that layering rubber bands doesn’t work. I’m sure it works fine. It’s that layering rubber bands is surely far more difficult than the alternatives that will work even better. In water guns people generally layer bike tubes over bladders to increase the pressure. This is not a new idea—it probably originated in 2000 or 2001. (Water gunners generally call this modification “Colossus” as that’s what it’s traditionally called. See SSC for more information.)

When I asked about this on #nerfchat, the only salient point I heard was that kids are more likely to find rubber bands than bike tubes. I’d like to contest that point. Old bike tubes are not difficult to come by. My father had a ton of them that he kept for various odd projects. Many bike shops give them away for free. Consider the number of tubes needed too. One or two 26 inch bike tires should be more than enough to increase the pressure of a bladder substantially. But to get a similar amount of rubber over a bladder with rubber bands would require tens, maybe even hundreds, of rubber bands. How many rubber bands are people likely to have at home?

Additionally, I previously mentioned that banding is more difficult and time consuming than using bike tubes. Bike tubes generally can be rolled up and slid on a bladder quite rapidly unless one has many layers on the bladder (at which point the modifier should probably stop anyway). Banding requires a lot more effort and time, especially if one wants to ensure that the rubber bands are applied evenly over the bladder so that one part does not inflate while the remainder does not.

I hope my point is clear: Banding should be avoided as layering bike tubes is far easier. Why bike tubes are so foreign to Nerfers but well known to water gun people is not completely clear, but I suspect tradition and a lack of creativity from most people in the hobby has a lot to do with it.

Of course, there are other alternatives as well. Someone could layer latex tubing over the bladder or simply use latex tubing as a bladder. Someone could use a homemade hard pressure vessel with a regulator. There are many options.

Posted in Interior ballistics, Misconceptions, Pneumatics | 2 Comments »

Pneumatic gun model on NerfEngrWiki

Posted by btrettel on January 17, 2011

While what I have posted is far from complete, I have started posting a derivation of a relatively simple model of a pneumatic gun on NerfEngrWiki. Feel free to use the comments section to ask questions, etc., if you have any. I don’t suspect many people will have questions, but I do know a few folks are carefully examining what I have developed.

I also have started working on a page about valves with a focus on modeling the flow through valves for pneumatic gun models. So far I have used this page to derive in detail (something I have not seen before) one of the valve flow equations that I have seen.

More pages will be developed. I’d appreciate any contributions that can be made, too.

Posted in Interior ballistics, NerfEngrWiki | Leave a Comment »