btrettel's Nerf blog

random thoughts about the ballistics of foam darts

Archive for the ‘Misconceptions’ 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

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

My challenge to Nerf rifling believers

Posted by btrettel on April 17, 2011

Do you believe most Nerf darts can be made more stable by spinning? If so, this challenge is for you!

The challenge is simple: Provide objective and statistically significant evidence that spinning darts improves stability, precision, or range.

Some notes

Rifling barrels for Nerf has already been thoroughly debunked. To say it in brief, the vast majority of Nerf darts are already stable, so they  have nothing to gain by spinning. In fact, they have a lot to lose, and this is often ignored by rifling believers.

So far, rifling believers have singularly failed to provide strong evidence to support their claims. Whenever they do, their results are shown to actually not be statistically different than what you would expect from smoothbores.

Also, tests are very often completely subjective. The results of many tests are simply “it looked better.” A subjective test like examining the dart’s trajectory could potentially be useful, but only when the experiment is properly designed. The tests must be blinded to have any shred of objectivity.

I don’t care how the spinning is done as long as the darts can be verified to be spinning too.

Posted in Exterior ballistics, Misconceptions | Leave a Comment »

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 »

Rifling: Helpful, harmful, or ineffective?

Posted by btrettel on January 7, 2011

By making analogies with real guns, some Nerfers have proposed that rifled barrels may be beneficial for Nerf blasters. But is this true? I will examine the two most popular claimed benefits of rifling, that rifling increases range and improves accuracy, and conclude that rifling as implemented thus far has had no significant effect on range or accuracy and it is not likely to have any effect under any circumstances.

First, the reader must realize that these claims are made most often without any backing. The hypothesis that rifling improves accuracy or range is often made based on misunderstandings of what rifling does. Spinning projectiles do not have less drag. Projectiles are spun to improve stability, as I will explain.

Stability of projectiles

A projectile is stable if it flights straight without overturning. This is desirable as the overturning motion reduces accuracy and range.

Rifling is used to improve the stability of a projectile’s flight. But can the stability of a Nerf dart be improved? In general, the answer is no because Nerf darts get their stability from static rather than dynamic characteristics of the dart.

The simplest way to make a stable projectile is to put the center of gravity far in front of the center of pressure. Details as to why this is stable will be later written in the Wiki. Most Nerf darts get their stability in this way; this is why darts are weighted at their nose.

But, most real bullets are made of a single material and they do not have this desirable weight distribution. Spinning the bullet around its longitudinal axis (as rifling does) can stabilize bullets in this case.

So, by simple examination of the mechanisms involved, we can conclude that rifling won’t have any significant effect on darts with the right weight distribution. Those darts are already very aerodynamically stable. There is no reason to rifle as there will not be any real benefit.

Some benefit from rifling seems plausible for very light darts that do not have the right weight distribution. But this is not an argument for rifling necessarily; adding weight to the front is by far the easiest way to stabilize these projectiles. However, this may not seem to be an acceptable choice for some Nerfers. Very lightweight darts may be desirable for safety reasons, however, there are other ways to improve safety of a dart (like reducing the muzzle velocity) that are far simpler than rifling.

Potential disadvantages of rifling

There are many potentially significant disadvantages to rifling that most proponents of the idea are unaware of. I detail the disadvantages that come to mind below.

  • Increased friction – If done poorly, the rifling could increase friction in the barrel and potentially reduce performance as a consequence.
  • Leaks around projectile – If done poorly, the rifling grooves could allow for air to leak around the projectile, reducing performance.
  • Increased complexity of building – Smoothbore barrels are simpler.
  • Less translational KE – To have a spinning dart, some of the energy that would have been put into translational kinetic energy and have contributed to range is instead put into rotational kinetic energy. Rifling is beneficial when this trade-off improves stability such that range or accuracy is improved satisfactorily. However, the reduction in translational KE may not be acceptable in all cases.
  • Reduction of stability – Poorly made darts may not have their weight distributed evenly around the longitudinal axis of the dart. Spinning could destabilize these darts and reduce range and accuracy.

Examining the accuracy claim with data

In 2009, a Nerfer who went by the handle Landru did some tests to see what effect spinning a dart had on accuracy. He used a setup with a spinning barrel. It is believed that this spinning barrel provides a way to control the spinning without making multiple rifled barrels. The test did not address rifling directly, rather, it addressed the question of whether spinning darts could even improve accuracy.

Landru posted some data that he claimed showed that the standard deviation of the locations of darts spun at 2000 RPM was lower than that from no spinning.

However, Landru neglected any sort of statistical analysis. I made a brief post that demonstrated his methods were flawed. I used an f-test to see whether there was any statistically significant difference between the two groups. Assuming a sample size of 20, I found critical f-values of 0.46 and 2.12 for\alpha = 10%. The f-value of was 1.49. As this was between the critical values, the differences were not statistically significant and consequently we can not determine if they were due to the rifling or random chance.

Landru made no follow-up tests.

Examining the range claim with data

Back in perhaps 2003 or 2004, a Nerfer who went by the handle Vassili tested rifled PETG barrels. He found that the average range of rifled PETG was higher than that of smoothbore PETG. Thankfully, Vassili didn’t claim rifling improved range directly. He only offered a tautology: “When it worked, it worked.” But did it work? Can we attribute any of the differences to the rifling and not random chance?

No, we can not. A t-test suggests the two data sets are statistically the same at the \alpha = 5% level. The critical t-value is 2.65. The t-value of the test for the mean is 1.21. As this is within the bounds we would expect at the 5% level of error, we can confidently state that rifling did not increase range in this case.

However, it can be shown that rifling increases the standard deviation of the range with an f-test (data to be added later). This should lead to a decrease in precision due to a decrease in repeatability (each shot is more variable). It also shows that more shots will have lower range with rifling. These two disadvantages are significant.


Based on the implausibility of the explanation for the benefit for rifling and the lack of evidence to suggest that rifling provides any benefit for Nerf darts, I conclude that rifling is ineffective at best and harmful at worst for Nerf.

2014-09-05: Comments disabled due to spam.

Posted in Exterior ballistics, Misconceptions | 19 Comments »