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A Numerical Method for Calculating Interior Ballistics

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A NUMERICAL METHOD FOR CALCULATING INTERIOR BALLISTICS Harold L. Brode and James E. Enstrom September 1970 P-4466 A NUMERICAL METHOD FOR CALCULATING INTERIOR BALLISTICS Harold L. Brode and James E. Enstrom * * The RAND Corporation. Santa Monica, California September 1970 We shall describe a procedure which calculates the interior ballistie properties of guns by numerically integrating the partial differential equations appropriate to the dynamics of the propellant gas. The equations wh
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  ANUMERICALMETHODFORCALCULATINGINTERIORBALLISTICS HaroldL.BrodeandJamesE.EnstromSeptember1970 P-4466  ANUMERICALMETHODFORCALCULATINGINTERIORBALLISTICS * aroldL.Brodeand * amesE.EnstromTheRANDCorporation.SantaMonica,CaliforniaSeptember1970Weshalldescribeaprocedurewhichcalculatestheinteriorbal-listiepropertiesofgunsbynumericallyintegratingthepartialdif-ferentialequationsappropriatetothedynamicsofthepropellantgas.Theequationswhichdescribetheinteriorballisticsareone-dimensionalalongthegunbarrel,assumingauniformcrosssection.TheequationofmotionofthegasinLagrangianformis au -= at a am (P + Q), (1) whereuisvelocity, t istime,mismassperunitarea,Pispressureand Q istheartificialviscosityforhandlingshockwaves.lTheequa-tionofenergyconservationis aE at _(p + Q) av + B at (2) * Anyviewsexpressedinthispaperarethoseoftheauthors.TheyshouldnotbeinterpretedasreflectingtheviewsofTheRANDCorpora-tionortheofficialopinionorpolicyofanyofitsgovernmentalorprivateresearchsponsors.PapersarereproducedbyTheRANDCorpora-tionasacourtesytomembersofitsstaff.  -2-whereEistheinternalenergyperunitmassofthegas,V=lipisthespecificvolumeorreciprocaldensity,andBistherateatwhichenergyisaddedtothegasbytheburningofthepropellant.Theequa-tionofmotionoftheshellasitmovesdownthebarrelis (3) whereMshisthemassoftheshell,ushandFsharethevelocityandpressureofthegasattheshellboundary,andAisthebarrelcross-sectionalarea.Forourexampleweshallusealarge5-inch/54calibernavalgunwiththefollowingdimensions:2chamberlengt~L=100cm;cross- o 25 sectionalarea,A=130cm;shellmass,Msh=3.8x10gm;andtotalshelltraveldistance.L=600cm.Thepropellantusedis90%nitrocel-luloseand10%othervolatiles,anditisintheformofsmallcylin- 3 derswithdiameter,D=.69cm;soliddensity,p=1.5gm/cm;andtotal o 3 weigh~W=9.0x10gm.Thepropellantcontainssmalldiameterwebbingwhichweshallneglect.Theequationsofstateofthegasproductsresultingfromtheburningofnitrocellulose,mainly,CO2,CO,H20,H2,andN2,havebeendeterminedpreviously.3.4Thethermalequationofstateis P(V-b) nRTã (4) 3 wherethecovolumecorrection.b~.95cm/gm;theamountofgasformedfromagramofnitrocellulose,n~.04gm-mole/gm;andthegasconstant.R=83.l5bar o cm3/(gm-mole o OK). Thecaloricequationofstateis
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