Part V: Adiabatic Gun Model

Post questions and info about combustion (flammable vapor) powered launchers here. This includes discussion about fuels, ratios, ignition systems and anything else relevant to launchers powered by igniting things like hairspray or propane.

Postby jimmy » Thu Feb 01, 2007 1:18 pm

<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. 5.8)<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."
<i>Edit: Fixed bad html for special characters.
Edit: Again to fix the still screwed up special characters</i>
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Postby FLONE » Fri Feb 02, 2007 10:42 am

WOW! I look at this and realize I have forgotten more abouth math than I ever learned. That makes a negative math minded me?

RE: the single ignition point located in your spudgun. On ignition the 2 flame fronts created would expand outward, 1 to the barrel, 1 to the breech. About the time the front flame hits the spud, the rear flame is rebounding off the cleanout cap, and starts moving forward. So that flame front is not doing as much work as the other one?

RE: chamber diameter vs length. Yes, the spherical flame front mode has a faster burn time, but does that translate into higher spud speeds? The gasses expand equally in all directions with equal pressure until they are redirected toward the barrel by the confines of the chamber. So a small percentage of the gasses head directly to the spud and the majority wander around for a fraction of a second and then get to work?

I will certainly admit my limited knowledge of the subject and accept comments accordingly. Seriously!

Your model has been modified to allow for variable chamber volume. In your past posts multiple sparks have your attention, will this be a variable included in the model? Internal chamber shape? Barrel diameter?
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Postby Freefall » Fri Feb 02, 2007 11:58 am

<blockquote id="quote"><font size="1" face="tahoma,verdana,arial" id="quote">quote:<hr height="1" noshade id="quote">About the time the front flame hits the spud, the rear flame is rebounding off the cleanout cap, and starts moving forward. So that flame front is not doing as much work as the other one?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Flame fronts don't rebound, as there is nothing left to burn behind the flame front. When the flame front reaches the back of the chamber, it simply disappears.

<blockquote id="quote"><font size="1" face="tahoma,verdana,arial" id="quote">quote:<hr height="1" noshade id="quote">The gasses expand equally in all directions with equal pressure until they are redirected toward the barrel by the confines of the chamber. So a small percentage of the gasses head directly to the spud and the majority wander around for a fraction of a second and then get to work?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">At combustion speeds (significantly lower than the speed of sound), chamber pressure is fairly uniform for any given timeslice. As the combusted gases expand behind the flamefront, the uncombusted gases in the rest of the chamber are compressed.
Pressure builds in the chamber until the pressure overcomes the friction of the spud, and then the gas can begin to do work. The gas is not really "redirected", but acts equally in all directions. Since only the spud can move, only the spud does move. The gas then moves in that direction because that's where the new space is.
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Postby jimmy » Fri Feb 02, 2007 1:28 pm

deleted my duplicate post
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Postby jimmy » Fri Feb 02, 2007 1:36 pm

Well put Freefall.

Pressure equalizes at the speed of sound. The speed of sound is much faster than the spud speed. Heck the speed of sound starts at ~1100 fps and is probably two or three times faster than that in the chamber when combustion is complete. So it really doesn't matter too much which way the flame front(s) are moving. An extremely detailed model probably would take the momentum of the gases into account but my Math-Fu isn't strong enough for that.

FLone said: "<i>RE: chamber diameter vs length. Yes, the spherical flame front mode has a faster burn time, but does that translate into higher spud speeds? The gasses expand equally in all directions with equal pressure until they are redirected toward the barrel by the confines of the chamber. So a small percentage of the gasses head directly to the spud and the majority wander around for a fraction of a second and then get to work?</i>"

I think the important thing is to get the fuel burned as quickly as possible so the pressure rises as quickly as possible. You don't want to "waste" any of the barrel by having the spud moving at low pressure.

This is just an observation but...

The model say the spud starts to move at ~10% combustion and has left the barrel (a properly sized barrel with C:B ~0.8) at about 90% combustion. If you can get the fuel to burn faster the spud still starts to move at 10% and still exits at 90% but the <u>time</u> the spud spends moving though the barrel is less. Same distance in less time means higher muzzle velocity.

The challenge is getting the fuel to burn faster. One way is with a chamber shape as close to spherical as possible. Another way may be multiple sparks. Another way may be to run the fan during firing. (Can anyone think of any other ways?)

Another way would be to use acetylene, or hydrogen, and pure oxygen and try to get to the deflagration to detonation transistion. But that is not a good idea for a PVC based gun.
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Postby jimmy » Fri Feb 02, 2007 1:58 pm

<blockquote id="quote"><font size="1" face="tahoma,verdana,arial" id="quote">quote:<hr height="1" noshade id="quote">Originally posted by FLONE
[br]Your model has been modified to allow for variable chamber volume. In your past posts multiple sparks have your attention, will this be a variable included in the model? Internal chamber shape? Barrel diameter?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The model already allows the chamber length and ID and the barrel ID to be changed. I'll probably leave the chamber shape as a cylinder since there are few (none?) combustion guns with any other shape for the chamber.

I really want to include multiple sparks but that may not be possible using an Excel spread sheet. With multiple sparks you have to deal with a variable number of sparks and the flame fonts meeting each other.

The other feature that the model currently can't handle is a spark that is not on the long axis of the chamber. A lot (all?) of Latke's data was obtained with a spark strip. IIRC the spark strip is stuck to the chamber wall. The form function for that situation is significantly different than for a centrally located spark. My current WAG is that for the Latke type spark strip two sparks on the strip are roughly equivalent to a single centrally located spark. (That assumes the sparks on the strip are well seperated. If they a close together then the multiple sparks really don't do much.)
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Postby FLONE » Fri Feb 02, 2007 3:05 pm

Total spark was 12 mm for the triple spark strip setup, so 4mm each. One would think igniting the edge of the mixture would be less uniform than along the axis of the chamber.
And I am curious about the short fat chamber concept, if anyone has built a short 6" instead of a longer 4". Yes it is a lot more expensive.
Jimmy, are you going to try a longer barrel on your spudder to verify the calcs?
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Postby Freefall » Fri Feb 02, 2007 6:23 pm

<blockquote id="quote"><font size="1" face="tahoma,verdana,arial" id="quote">quote:<hr height="1" noshade id="quote">I really want to include multiple sparks but that may not be possible using an Excel spread sheet. With multiple sparks you have to deal with a variable number of sparks and the flame fonts meeting each other.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Since two flame fronts meet each other symmetrically, you can treat the interaction as if they were both meeting the end of the chamber, located at a distance halfway between the two sparks.
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Postby FLONE » Mon Feb 05, 2007 4:31 pm

Regarding the pressure required to get the spud going. If a person had a pressure guage attached to the chamber and could somehow slowly pressurize the chamber until the spud moved, an approximate number would be had?
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Postby jimmy » Tue Feb 06, 2007 11:30 am

FreeFall: Very good point, in fact, it might be easiest to model multiple sparks as a single spark in a short chamber, then just replicate the results.

Even easier might be to model the multiple sparks as a single event in a shorter chamber then just change the relationship between the chamber volume and barrel volume. For example, three sparks in a 10 inch long chamber would be modeled as a single spark in a 3.33" chamber but the relationship between the chamber volume and barrel volume would have an extra factor of 3. The pressure in the chamber would fall only 1/3 as fast as it should as the spud moves through the barrel.

So that is a great idea FreeFall, much easier than adding multiple columns to model the sparks.

Of course, this approach limits the placement of the sparks. Symmetrically placed sparks would be easy. Asymmetric sparks, for example, two sparks in a 10" long chamber with the sparks located at 2.5" and 5", might be tricky.
<hr>
Flone: Yes, that should work, but you would need a combustion gun with a schrader valve and a pressure gauge. (Come to think of it, my standard gun has a schrader valve as the fuel port.) If I wasn't so lazy, I would just take a spud and shove it halfway into a barrel then invert the barrel and ram rod and push it down onto a bathroom scale and try to measure the static and dynamic friction.

<i>"Jimmy, are you going to try a longer barrel on your spudder to verify the calcs?"</i>
I can't change the barrel on the "standard gun", I would have to build a new gun.

<i>"And I am curious about the short fat chamber concept, if anyone has built a short 6" instead of a longer 4". Yes it is a lot more expensive."</i>
That would be a great experiment wouldn't it? The model suggests that the difference in performance should be pretty significant. It would be cheaper if you went the other way. Instead of building 6"D and a 4"D guns build 4"D and 2"D guns (with the same chamber volume).
<hr>

<font size="+3" color=blue>Indianapolis Colts: Superbowl Champions!</font>

So much for;<ul><li>"Dome teams can't win the SB"</li><LI>"Dome teams can't win outdoors in crappy weather"</li><LI>"Defense wins championships"
(unless you figure the Colts have a great defense, which they did in the playoffs).</li><li>Peyton can't win "the big one"</li></ul>
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