<h2>Part V: Adiabatic Gun Model<br> Alternate title: The Spud Finally moves!</h2> In Part IV we built a model for a closed combustion chamber. Now we will include movement of the spud in the model.<br> <h3>Spud Movement</h3> As combustion progresses the pressure in the chamber rises. At some point, the pressure in the chamber becomes great enough to start the spud moving. The spud starts to move when the net force (F<sub>net</sub>) on the projectile is high enough to overcome the force due to atmospheric pressure (F<sub>atmosphere</sub>) and the static friction (F<sub>static friction</sub>) between the spud and the barrel. <div style="margin-left: 40px;">F<sub>net</sub> = F<sub>combustion gas</sub> - F<sub>atmosphere</sub> - F<sub>static friction</sub> >? 0 (Eq. 5.1) </div> <h3>Projectile Acceleration</h3> <p>Once the spud starts to move F<sub>static friction</sub> is replaced by F<sub>dynamic friction</sub>, the frictional force retarding movement of the projectile. (As an estimate I am using a static frictional force of 30 pounds and the assumption that F<sub>static friction</sub> = 2F<sub>dynamic friction</sub>. Both quantities are inputs to the model.) Since F=ma Eq. 5.1 can be written as;<br> </p> <p style="margin-left: 40px;">F<sub>net</sub> = ma = F<sub>combustion gas</sub> - F<sub>atmosphere</sub> - F<sub>dynamic friction</sub> (Eq. 5.2)</p> <p>Since P=F/area we can rewrite for a particular time step as</p> <p style="margin-left: 40px;"> F<sub>net</sub> = ma<sub>i</sub> = F<sub>combustion gas,i</sub>(Area) - P<sub>atmosphere</sub>(Area) - F<sub>dynamic friction</sub> (Eq. 5.3)</p> Where Area = πR<sup>2</sup><sub>barrel</sub>, m is the mass of the projectile and a<sub>i</sub> the acceleration. Simplifying and assuming P<sub>atmosphere</sub> = 1 ATM, and converting pressures from ATM (absolute) to Pascal gives; <p style="margin-left: 40px;"> F<sub>net</sub> = ma<sub>i</sub> = (P<sub>combustion gas,i</sub> - 1)(πR<sup>2</sup><sub>barrel</sub>)(101325 Pascal/ATM) - F<sub>dynamic friction</sub> (Eq. 5.4)</p> <p>(Note: P<sub>atmosphere</sub> should probably be a separate input to the program. This would allow P<sub>0</sub> to be used to model hybrids and/or burst disk guns where F<sub>static friction</sub> is set to the burst disk failure pressure.)</p> <p>Solving for acceleration gives<br> </p> <blockquote> <p>a<sub>i</sub> = <sub><big><big><big><big>(</big></big></big></big></sub>(P<sub>combustion gas,i</sub> - 1)(πR<sup>2</sup><sub>barrel</sub>)(101325 Pascal/ATM) - F<sub>dynamic friction<big><big><big><big>)</big></big></big></big></sub> / m (Eq. 5.5)</p> </blockquote> <p>The F<sub>dynamic friction</sub> could be expressed in pseudo-pressure terms (pounds force/barrel area) but the current model uses Newtons.</p> <h3>Projectile Velocity and Position</h3> <p>The velocity and position of the projectile for this time step are calculated with;</p> <blockquote> <p> v<sub>i</sub> = v<sub>(i-1)</sub> + a<sub>i</sub>Δt (Eq. 5.6)<br> x<sub>i</sub> = x<sub>(i-1)</sub> + v<sub>i</sub>Δt (Eq. 5.7)</p> </blockquote> <p>Where v<sub>i</sub> is the velocity at time i and x<sub>i</sub> is the spud position (x<sub>0</sub> = 0).<br> </p> <blockquote> <p> </p> </blockquote> <h3>Correct total volume</h3> <p>The total volume (big V, little v is velocity) of the chamber increases to</p> <blockquote>V<sub>i</sub> = V<sub>(i-1)</sub> + ΔV<sub>i</sub> (Eq. 5.<br> V<sub>i</sub> = V<sub>(i-1)</sub> + Δx<sub>i</sub>(πR<sup>2</sup><sub>barrel</sub>) (Eq. 5.9)</blockquote> <p>where Δx<sub>i</sub> = x<sub>i</sub> - x<sub>(i-1)</sub>, the distance the projectile moved in this time step.</p> <h3>Correct Flame front positions and move combustion center</h3> Since the projectile has moved the total chamber volume has increased. <u>This presents a problem</u>. Since the fraction combusted is used to calculate the temperature and pressure in the chamber, increasing the chamber volume would correspond to combustion going "backwards" unless the combusted volume is also increased. Therefore, the flame fronts must be moved in order to keep the combusted fraction (f<sub>comb</sub>) constant for this time step. Exactly how this <i>should</i> be done is not clear to me. I have decided that both flame fronts move towards the muzzle but the breech end flame front moves less than the muzzle end flame front. The center of combustion is moved also. The original center of combustion (the spark position) is adjustable in the model. <p><b>Combustion center location:</b></p> <blockquote> <pre>New center location = (Previous location)*(corrected burn volume/uncorrected burn volume)<br></pre> </blockquote> <p><b>Rear flame position:</b></p> <blockquote> <pre>IF(flame front position<G_Chamber_Diameter/2,<br> (flame front position)*(corrected burn volume/uncorrected burn volume)^(1/3),<br>ELSE<br> (flame front position)*(corrected burn volume/uncorrected burn volume))</pre> </blockquote> <p><b>Forward flame position:</b></p> <blockquote> <pre>IF(Forward h<(G_Chamber_Length - combustion center),<br> IF(Forward h<G_Chamber_Diameter/2,<br> (flame front position)*(corrected burn volume/uncorrected burn volume)^(1/3),<br> ELSE<br> (flame front position)*(corrected burn volume/uncorrected burn volume)),<br>ELSE<br> (flame front position)+(projectile position - previous projectile position))</pre> </blockquote> <h3>Correct T<sub>i</sub>, P<sub>i</sub>, P<sub>max</sub> and T<sub>max</sub></h3> <p>Since the volume of the chamber has changed the current temperature and pressure must be corrected. This is done assuming an adiabatic, isentropic expansion.<br> </p> <blockquote>T<sup>'</sup><sub>i</sub> = T<sub>i</sub>(V<sub>1</sub>/V<sub>2</sub>)<sup>(γ-1)</sup> (Eq. 5.10)<br> P<sup>'</sup><sub>i</sub> = P<sub>i</sub>(V<sub>1</sub>/V<sub>2</sub>)<sup>γ</sup> (Eq. 5.11)<br> </blockquote> <p> </p> <p>Where V<sub>1</sub> is the volume before projectile movement and V<sub>2</sub> is the volume after projectile movement. Since the total volume has changed we must also correct the P<sub>max</sub> and T<sub>max</sub> values. These are corrected in the same way as P<sub>i</sub> and T<sub>i</sub>. (Remember that the P<sub>max</sub> and T<sub>max</sub> values were initially set equal to the GasEq values, 2600K and 9.3 ATM for stoichiometric propane at 1 ATM). <br> </p> <p>The value of γ (gamma) in equations 5.10 and 5.11 is an input parameter of the model. Currently, I am using a value of 1.31, the average of the GasEq calculated γ's for propane + air and the combustion products. Some selected values for γ are;</p> <blockquote> <table border="0" cellpadding="1" cellspacing="0"> <tbody> <tr> <td align="center" height="6" valign="middle"> <b>γ</b> </td> <td height="6" valign="middle"><b>For:<br> </b></td> </tr> <tr> <td align="center" height="6" valign="middle"> 1.67 <br> </td> <td height="6" valign="middle">ideal monoatomic gases (e.g., helium or argon)</td> </tr> <tr> <td align="center" height="6" valign="middle"> 1.40 <br> </td> <td height="6" valign="middle">ideal diatomic gases (e.g., N<sub>2</sub>, O<sub>2</sub>, and air)</td> </tr> <tr> <td align="center" height="6" valign="middle"> 1.37 <br> </td> <td height="6" valign="middle">propane + air at 300K and 1 ATM (from GasEq)<br> </td> </tr> <tr> <td align="center" height="6" valign="middle"> <b>1.31 </b><br> </td> <td height="6" valign="middle"><b>average of propane+air and combustion products<br> </b></td> </tr> <tr> <td align="center" height="6" valign="middle"> 1.25 <br> </td> <td height="6" valign="middle">combustion products, N<sub>2</sub>, CO<sub>2</sub> and water vapor at 2630K and 9.28 ATM (from GasEq)</td> </tr> </tbody> </table> </blockquote> <p>Instead of using the average γ, it might be better to use a variable γ<sub>i</sub>. The variable γ<sub>i</sub> could be scaled between the initial and final γ's (1.37 and 1.25) based on the combusted fraction. </p> <h3>New Flame Front Form Function Case</h3> Since the spud has moved the geometry of the "chamber" has changed. To account for this we need to add a new case to the form function of the forward flame front. <p><b>Case 4:</b> Barrel<br> <img alt="case 4" src="http://home.earthlink.net/%7Ejimsluka/_images/case_4.gif" height="86" width="394"> </p> <blockquote>volume = volume of chamber cylinder + volume in barrel<br> = π(Chamber Diameter/2)<sup>2</sup>*(chamber length - flame center)<br> + π(Barrel Diameter/2)<sup>2</sup>*MINIMUM(r + flame center - chamber length, spud position)</blockquote> <p>Currently case 4 has a minor geometry problem in that the volume in the barrel is treated as a cylinder with flat, instead of domed, end.</p> <h3>Results for this Adiabatic Model</h3> So, how well does this model work for a typical spud gun? <br> <br> I have chrono data for my standard gun;<br> <blockquote> <table border="0" cellpadding="0" cellspacing="0"> <tbody> <tr> <td valign="top">- chamber: 3"D x 11"L</td> <td valign="top"> <br> </td> <td valign="top">- stoichiometric propane in air<br> </td> </tr> <tr> <td valign="top">- chamber volume 77in<sup>3</sup></td> <td valign="top"><br> </td> <td valign="top">- chamber fan for mixing</td> </tr> <tr> <td valign="top">- barrel: 2"D x 30"L</td> <td valign="top"><br> </td> <td valign="top">- 0.08Kg spud (actually an apple)</td> </tr> <tr> <td valign="top">- C:B ratio 0.8:1</td> <td valign="top"><br> </td> <td valign="top">- muzzle velocity = <b>~330 FPS (101 m/s) (at 30")</b></td> </tr> <tr> <td valign="top">- single spark gap at ~center<br> of the chamber</td> <td valign="top"><br> </td> <td valign="top"><br> </td> </tr> </tbody> </table> </blockquote> Here is a screen shot of the Excel spread sheet showing all the input parameters (in the blue boxes);<br> <blockquote><img alt="screen_shot_1.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/screen_shot_1.gif" border="1" height=""> <br> </blockquote> The simulations were run with the default set of values for T<sub>max</sub>, gamma, etc. The length of the barrel was set to be very long so the optimal C:B ratio would be calculated. In the current model the calculation actually assumes the barrel is infinitely long, the simulation just continues to model the spud's movement until the end of the Excel table is reached. <br> <br> The model does not take into account the mass of the air in front of the spud. For muzzle velocities that do not approach the speed of sound this seems reasonable. The mass of the column of air in a 2"D by 30" barrel is only about 1.5 gram, significantly less than the 80~120 grams of the spud.<br> <br> The graph below shows the calculated spud velocity and fraction combustion versus the spud position in the barrel.<br> <blockquote><img alt="Pos_vel_a.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/Pos_vel_a.gif" border="1" height="" width=""><br> </blockquote> <b>At the barrel length of 30" (0.76m) the calculated velocity is 287 fps (87.5 m/s). This is less than the measured muzzle velocity (330 fps) by ~40 fps, roughly 12%.</b><br> <br> The graph above indicates that the optimum barrel length for this chamber is predicted to be about 7.2 feet (2.2m). At that barrel length, the velocity is 368 fps (112 m/s) and the <u><b>C:B ratio is 0.29</b></u>. Interestingly, this is about the C:B ratio that GGDT predicts to be optimal for a compressed air gun of the same dimensions pressurized to 120 PSIG.<br> <br> A graph showing the chamber pressure and dP/dt curves is shown below. The dashed lines mark the time at which the spud started to move (54mS) and when it would have left a 30" long barrel (75mS).<br> <blockquote><img alt="Time_pressure_a.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/Time_pressure_a.gif" border="1" height="305" width="483"><br> </blockquote> <br> The sharp spike in the dP/dt curve at 67mSec is when the forward flame front reached the barrel. As I mentioned earlier, the form function for the flame front in the barrel is not quite correct. The flame front is modeled as a dome (or sphere) when it is in the chamber but I wimped out and went to the much simpler planar flame front model for the flame front in the barrel. The spike in dP/dt is the sudden jump in combusted volume that occurs when the model switches from a domed front to a planar one. Though the spike looks significant on the dP/dt plot, it is only a single data point wide (20 uS) and has only a small affect on the pressure versus time curve.<br> <br> The peak pressure on the graph is ~3.6 ATM, which corresponds to ~38 PSIG. This is significantly lower than the P<sub>max</sub> value of 9.28 ATM for a closed chamber. Part of the difference between 3.6 and 9.28 ATM is simply the change in volume of the chamber as the spud moves through the barrel. In addition, the observed peak pressure will be lower than the closed chamber P<sub>max</sub> value if the spud leaves the barrel before combustion is complete.<br> <br> Below is a graph showing the fraction reaction and d(reaction)/dt curves.<br> <blockquote><img alt="time_reaction.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/time_reaction.gif" border="1" height="312" width="475"><br> </blockquote> I've marked some important points on the graph, for example, when the spud starts to move, and the flame front transitions. Remember that the model does not explicitly treat the spud leaving the barrel. The sharp drop in the <font color="#ff0000">dReact/dt</font> curve near the point that the "spud leaves the barrel" is due to the rear flame burning out and not to the spud leaving the barrel. It may be that the close correspondence between the rear flame burnout time and the time at which the spud has moved through a chamber volume corresponding to a C:B ratio of 0.8 is significant.<br> <br> The table below lists some of the key times during the combustion process for this chamber along with the spud's velocities and other results.<br> <blockquote> <table border="1" cellpadding="2" cellspacing="0"> <tbody> <tr> <td align="center" valign="middle"><small><b>Time<br> (mSec)<br> </b></small></td> <td align="center" valign="middle"><small><b>Event<br> </b></small></td> <td align="center" valign="middle"><small><b>Velocity,<br> fps (m/s)<br> </b></small></td> <td align="center" valign="middle"><small><b>Position,<br> feet (meter)</b></small></td> <td align="center" valign="middle"><small><b>%<br> Combustion</b><br> </small></td> <td align="center" valign="middle"><small><b>Comments<br> </b></small></td> </tr> <tr> <td align="center" valign="middle"><small>53.6</small></td> <td align="left" valign="middle"><small>Spud starts to move</small></td> <td align="center" valign="middle"><small>0<br> </small></td> <td align="center" valign="middle"><small>0<br> </small></td> <td align="center" valign="middle"><small>8%<br> </small></td> <td valign="middle"><br> </td> </tr> <tr> <td align="center" valign="middle"><small>59.0<br> </small></td> <td align="left" valign="middle"><small>Spherical to domed flame front transition<br> </small></td> <td align="center" valign="middle"><small>29<br> (8.9)<br> </small></td> <td align="center" valign="middle"><small>0.066<br> (0.020)<br> </small></td> <td align="center" valign="middle"><small>18%<br> </small></td> <td valign="middle"><small>spud has moved ~1 inch when flame front reaches<br> the chamber wall</small><br> </td> </tr> <tr> <td align="center" valign="middle"><small>67.0</small></td> <td align="left" valign="middle"><small>Flame front reaches barrel</small></td> <td align="center" valign="middle"><small>148<br> (45)</small></td> <td align="center" valign="middle"><small>0.72<br> (0.22)<br> </small></td> <td align="center" valign="middle"><small>48%<br> </small></td> <td valign="middle"><small>spud has moved ~9 inches when<br> flame front reaches the barrel</small></td> </tr> <tr> <td align="center" valign="middle"><small>68.1<br> </small></td> <td align="left" valign="middle"><small>Peak chamber pressure (3.6 ATMa)<br> </small></td> <td align="center" valign="middle"><small>167<br> (51)<br> </small></td> <td align="center" valign="middle"><small>8.89<br> (0.27)<br> </small></td> <td align="center" valign="middle"><small>53%<br> </small></td> <td align="left" valign="middle"><small>spud is accelerating at 19,000 fpss (590 G,<br> 5800 m/s<sup>2</sup>)</small><br> </td> </tr> <tr> <td align="center" valign="middle"><small>75.0</small></td> <td valign="middle"><small>Spud at 30" (C:B 0.8:1)</small></td> <td align="center" valign="middle"><small>287<br> (88)</small></td> <td align="center" valign="middle"><small>2.5<br> (0.76)<br> </small></td> <td align="center" valign="middle"><small>87%<br> </small></td> <td valign="middle"><small>the spud has reached the end of a 30" barrel<br> </small></td> </tr> <tr> <td align="center" valign="middle"><small>76.6</small></td> <td align="left" valign="middle"><small>Rear flame front burnout</small></td> <td align="center" valign="middle"><small>305<br> (93)</small></td> <td align="center" valign="middle"><small>2.9<br> (0.88)<br> </small></td> <td align="center" valign="middle"><small>94%<br> </small></td> <td valign="middle"><small><br> </small></td> </tr> <tr> <td align="center" valign="middle"><small>81.3</small></td> <td valign="middle"><small>Flame front reaches spud</small></td> <td align="center" valign="middle"><small>351<br> (107)</small></td> <td align="center" valign="middle"><small>4.6<br> (1.4)<br> </small></td> <td align="center" valign="middle"><small>100%<br> </small></td> <td valign="middle"><small>combustion complete, spud has moved<br> 4.5 feet (1.37m)<br> </small></td> </tr> <tr> <td align="center" valign="middle"><small>88.3<br> </small></td> <td align="left" valign="middle"><small>Spud at 7.2' (C:B 0.29)<br> </small></td> <td align="center" valign="middle"><small>367<br> (112)</small></td> <td align="center" valign="middle"><small>7.2<br> (2.2)<br> </small></td> <td align="center" valign="middle"><small>100%<br> </small></td> <td valign="middle"><small>pressure behind spud has dropped below<br> P<sub>atm</sub> + dynamic friction and spud starts decelerating<br> </small></td> </tr> </tbody> </table> </blockquote> <h3>Validation</h3> To check if the equations describing the movement of the spud are correct, I set the static friction in the combustion model to just slightly less than the P<sub>max</sub> value of 9.28 ATM. This essentially models a burst disk gun with a disk that ruptures at P<sub>max</sub>. This also uncouples the combustion process from the movement of the spud. For a comparison, I modeled this same chamber in GGDT using a gas temperature of 2600K (4200F) and a burst disk valve with flow coefficient of 44% (the ratio of the chamber area to barrel area).<br> <blockquote><img alt="model_v_GGDT.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/model_v_GGDT.gif" border="1" height="323" width="482"><br> </blockquote> As you can see, the two models agree fairly well for the dynamics of the spud movement. The short vertical line marks the length of the modeled gun's barrel, which is a C:B ratio of 0.8. The difference between the "combustion model of a burst disk" and GGDT at this barrel length is 9 fps (422 versus 413 fps). So it appears the equations describing the spud movement are OK. Any deficiencies in the model must be due to other factors such as incorrect flame front speeds or heat loss. <h3>Some studies using the current model:<br> Affect of S<sub>L0</sub> on Velocity</h3> The timing of the model's combustion processes is critically dependent on the starting laminar flame front speed (S<sub>L0</sub>) and the exponents used in the power law. Literature values for S<sub>L0</sub> range from 0.32 to 0.5 m/s for propane in air. Currently, I am using an S<sub>L0</sub> value of 0.43 m/s. The table below gives the velocity of the spud for a 30" barrel length as a function of S<sub>L0</sub>. <blockquote> <table border="1" cellpadding="2" cellspacing="2"> <tbody> <tr> <td align="center" valign="middle"><b>S<sub>L0</sub><br> (m/s)</b></td> <td align="center" valign="middle"><b>fps at 30"<br> barrel length</b> </td> </tr> <tr> <td align="center" valign="middle">0.32 </td> <td align="center" valign="middle">245 </td> </tr> <tr> <td align="center" valign="middle"><b>0.43 </b> </td> <td align="center" valign="middle"><b>287 </b> </td> </tr> <tr> <td align="center" valign="middle">0.50 </td> <td align="center" valign="middle">314 </td> </tr> <tr> <td align="center" valign="middle">0.55<br> </td> <td align="center" valign="middle"><b><font color="#cc0000">330</font></b><br> </td> </tr> <tr> <td align="center" valign="middle">0.60 </td> <td align="center" valign="middle">343 </td> </tr> </tbody> </table> </blockquote> As you can see, only a small change in S<sub>L0</sub> is needed to get the predicted muzzle velocity to the 330 fps velocity actually measured for this gun. It is also possible that the flame front is not laminar throughout the combustion process. A turbulent flame front would be expected to propagate faster, and have a larger form function, than a laminar flame front. <h3>Affect of γ (gamma) on fps at 30"</h3> As mentioned earlier, there is some question as to what the proper value of γ is for the combustion process. (γ is used in equations 5.10 and 5.11.) Currently, I am using the average value for the starting and ending materials, γ=1.31. The affect of γ on the muzzle velocity with a 30" barrel is shown in the table below. <blockquote> <table border="1" cellpadding="2" cellspacing="2"> <tbody> <tr> <td align="center" valign="middle"><b>γ</b><sub> </sub></td> <td align="center" valign="middle"><b>Gas Composition</b><br> </td> <td align="center" valign="middle"><b>fps at 30"<br> barrel length</b><br> </td> </tr> <tr> <td align="center" valign="middle">1.37<br> </td> <td align="center" valign="middle">starting materials<br> </td> <td align="center" valign="middle">279<br> </td> </tr> <tr> <td align="center" valign="middle"><b>1.31<br> </b></td> <td align="center" valign="middle"><b>average<br> </b></td> <td align="center" valign="middle"><b>287<br> </b></td> </tr> <tr> <td align="center" valign="middle">1.25<br> </td> <td align="center" valign="middle">ending materials<br> </td> <td align="center" valign="middle">295<br> </td> </tr> </tbody> </table> </blockquote> <p>The calculation appears to be fairly insensitive to the gamma value used. The velocity at 30" changed by about 5% from the lowest γ to the highest γ. The average γ gives muzzle velocities within ~2.5% of both the high and low values. </p> <h3>Velocity at 30" versus spark position</h3> The spark position is an input parameter in the current model. The model assumes that the spark is located along the central axis of the cylindrical chamber. The graph below shows the velocity at 30" as a function of the spark position. The center of the chamber has a spark position of 0.5, smaller values move the spark closer to the breech of the gun.<br> <blockquote><img alt="spark_position.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/spark_position.gif" border="1" height="281" width="399"><br> </blockquote> There appears to be an interesting relationship between the spark position and the velocity of the projectile. The optimum spark position is shifted slightly towards the breech of the gun, with a spark position in between 0.4 and 0.45 being optimal. The muzzle velocity drops off fairly quickly if the spark is moved too far towards the breech end of the chamber.<br> <h3>Affect of chamber diameter and length</h3> The standard gun being modeled has a 3"D by 11"L chamber with a total volume of 77in<sup>3</sup>. What happens if a shorter and fatter chamber with the same total volume is used? The chamber shape that would give the fastest burn time would be a sphere. Since a sphere is not a practical shape for a spud gun then the next best thing would be a chamber with a diameter equal to it's length. This relation between diameter and length will give 67% of the combustion process in the faster spherical flame front mode. The standard 3"D by 11"L chamber has just 18% of the combustion occurring in the spherical flame front mode.<br> <br> A 4.62"D by 4.62"L chamber has the same volume as the standard gun's chamber. The predicted muzzle velocity at 30" for this chamber is 371 fps. That is 29% faster than the 287 fps predicted for the standard chamber. The predicted muzzle kinetic energy is 66% greater. It appears then that a short and fat chamber will significantly outperform a longer, skinnier chamber.<br> <h3>Affect of friction on muzzle velocity</h3> I really don't have a good estimate of the frictional force for the spud. The 30 pounds of static friction force is really a WAG. So, just how sensitive is the model to changes in the frictional force? The graph below shows the predicted 30" velocities as a function of the static friction force for the standard gun. The blue curve was calculated using a dynamic friction force of half the static friction force. The red curve was calculated with the dynamic friction set to zero. <blockquote><img alt="friction.gif" src="http://home.earthlink.net/%7Ejimsluka/_images/friction.gif" border="1" height="318" width="447"></blockquote> <p>Increasing static friction increases the velocity of the spud, up to a point. The red curve suggests that a burst disk gun with a disk that ruptures at ~P<sub>max</sub> (equivalent to ~380 pounds of force or ~120 PSIG), and a low dynamic friction, would give optimal performance in this adiabatic model.</p> <h3>Problems with the model<br> </h3> The current model makes interesting predictions about how the various characteristics of a combustion gun affect the performance. Clearly though, this model has significant problems.<br> <ol> <li>Predicted velocity at the 30" barrel length is about 12% low. If this was the <u>only</u> problem then I would consider the model a success.<br> </li> <li>The predicted optimal C:B ratio is much too low and is similar to the optimal C:B ratios calculated for compressed air guns.</li> </ol> The most obvious characteristic that the current model ignores is heat (energy) transfer from the combustion gases to the gun. However, including a heat loss term (a non-adiabatic model) will only aggravate the low predicted muzzle velocity problem. This suggests that there is at least two problems with the model.<br> <br> It appears we will need to include heat loss and perhaps increase the starting laminar flame front speed in order to get the model to accurately predict the performance of real spud guns.<br> <ol> </ol> Stay tuned for Part VI: A Non-Adiabatic Model<br> Alternate Title: "You have to take into account that it is hot enough to burn your face off."

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