Research Article Multiphase CFD Simulation of Solid ...Research Article Multiphase CFD Simulation of Solid Propellant Combustion in a Small Gun Chamber AhmedBougamraandHuilinLu School

Research ArticleMultiphase CFD Simulation of Solid PropellantCombustion in a Small Gun Chamber

Ahmed Bougamra and Huilin Lu

School of Energy Science and Engineering Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Ahmed Bougamra ahmedbougamrayahoofr

Received 15 March 2014 Revised 29 May 2014 Accepted 2 June 2014 Published 25 June 2014

Academic Editor Jose C Merchuk

Copyright copy 2014 A Bougamra and H Lu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The interior ballistics simulations in 9mm small gun chamber were conducted by implementing the process into the mixturemultiphase model of Fluent V63 platform The pressure of the combustion chamber the velocity and the travel of the projectilewere investigated The performance of the process namely the maximum pressure the muzzle velocity and the duration of theprocess was assessed The calculation method is validated by the comparison of the numerical simulations results in the small gunwith practical tests and with lumped-parameter model results In the current numerical study both the characteristics and theperformance of the interior ballistic process were reasonably predicted compared with the practical tests results The impact of theweight charge on the interior ballistic performances was investigated It has been found that the maximum pressure and the muzzlevelocity increase with the increase of the charge weight

1 Introduction

Small guns have been used for a long time Nowadaysthey are still the most used in military sports and testsA hand gun can be modeled with two connected cylindersrepresenting respectively the combustion chamber and thelaunching tube (the barrel of the gun) We can assume thatthe two cylinders have the same diameter because for smallgun using rimless ammunition the diameters are almost thesame (see Figure 1) The breech contains the primer a smallspace filled with black powder The space defined by thecombustion chamber sealed by the projectile is filled withsolid propellant [1]

The basic interior ballistic process may be consideredas a heat engine which through combustion converts thechemical energy stored in solid propellant into kinetic energyfor the projectile [2] This process is well described byFarrar and Leeming [3] The sequence of this process canbe briefly described as follows The firing sequence beginswith the ignition of the primer This injects hot gases andincandescent particles into the propellant bed which causethe ignition of the solid propellant grains The gases andenergy liberated by the combustion of the propellant increase

dramatically the pressure and the temperature within thesealed chamber Since the burning rate of the propellant isroughly proportional to the pressure the increase in pressureis accompanied by an increase in the rate of combustionat which further gas is produced Usually slightly beforepeak pressure the projectile starts travel down-bore Themovement of the projectile causes the chamber volume toincrease and generate refraction waves which lower thepressure [4] The projectile continues to accelerate until itreaches the muzzle where the propellant gases expand thepressure falls and so the acceleration lessensThe sequence ofinterior ballistics ends when the projectile leaves the muzzleThe entire sequence takes less than 2 milliseconds for smallguns

Since computational efforts increased dramatically inthe last decade numerical simulations of interior ballisticsprocess have been established as valuable tools in researchand development of high-performance guns Besides theirsimplicity low cost and safety they provide data that cannotbe directly measured by experiment

The main purpose of computer modeling of the inte-rior ballistics process is to predict the muzzle velocity ofthe projectile the maximum pressure in the chamber and

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2014 Article ID 971808 10 pageshttpdxdoiorg1011552014971808

2 International Journal of Chemical Engineering

Combustionchamber Barrel (the exit tube)

Solid

pro

pella

nt

bed

Shot

bas

e

Bulle

t Muzzle

Bree

ch

Figure 1 Initial geometry configuration in small gun

the pressure history in the system during the process aswell Computer modeling can be accomplished by usingone-dimensional single-phase ldquolumped-parameterrdquomodelssuch as IBHVG2 [5] and STANAG 4367 [6] which arebased on the assumption that grains and the products ofcombustion constitute a well stirred mixture or by usingmultidimensional multiphase flow model [7]

Presently the most accurate modeling of the interiorballistics problem is provided by two-phase multidimen-sional computational fluid dynamics (CFD) codes such asthe American codes NOVA or XKTC [8 9] NGEN [10] andEuropean codes AMI [11 12] MOBIDIC [13 14] CTA1 [15]and FHIBS [16] These codes are elaborated precisely for thesimulation of interior ballistics process They are the resultsof many years of improvements and validation and most ofthem were developed for large guns

In this paper we have attempted to elaborate two-phaseflow interior ballistic model for small-caliber gun and imple-ment it into commercial CFD code Fluent The characteristiccurves of pressure projectile velocity and projectile travelare reproduced and the performances of the interior ballisticprocess are predicted

2 Theoretical Model

The mixture model of Fluent is used to model the unsteadyreactive two-phase flow of solid propellant and its gaseouscombustion products This model can model n phases (fluidor particulate) by solving the momentum continuity andenergy equations for the mixture the volume fraction equa-tions for the secondary phases and algebraic expressions forthe relative velocities [17]

21 Basic Assumptions The combustion of the solid pro-pellant in the gun is a complex process It is necessary toaccept some assumptions to be able to model this processusing the mixture model Some of the assumptions are thesame as those mentioned by Gokhale and Krier [18] and thenreproduced by Gollan et al [2] The basic assumptions usedare (1) the solid phase composed of solid propellant andthe gas phase composed of propellant combustion gas (2)air in the chamber which is neglected because mass of theair is very small compared with the mass of combustion gasproduced from solid propellant [19 20] (3) the two phases are

treated as interpenetrating continua but separately coupledby appropriate interaction terms which account for mass andenergy transport between the phases (4) the solid consideredas an incompressible pseudofluid (5) the solid (propellant)burning which results in loss of mass from the solid phaseand equivalent gain by the gas phase (6) the gases which areinviscid except for their action on the particles through thedrag (7) the gas viscosity assumed to be known as a functionof temperature (8) the gas phase considered as an ideal gas(9) the ignition assumed to be ideal that is all the solidpropellant grains are simultaneously and uniformly ignited(10) the walls which are adiabatic

22 Governing Equations The balance equations for massmomentum and energy of mixture flow in propellant cham-ber and gun barrel are given by

120597

120597119905(120588119898) + nabla sdot (120588

119898sdot V119898) = 0 (1)

120597

120597119905(120588119898V119898) + nabla sdot (120588

119898V119898V119898)

= minusnabla119901 + nabla sdot [120583119898(nablaV119898+ nablaV119879119898)] + 120588

119898119892 +

+ nabla sdot (

119899

sum

119896=1

120572119896120588119896V119889119903119896

V119889119903119896

)

(2)

120597

120597119905

119899

sum

119896=1

(120572119898120588119898119864119896) + nabla sdot

119899

sum

119896=1

(120572119896V119896(120588119896119864119896+ 119901))

= nabla sdot (119896effnabla119879) + 119878119864

(3)

where the subscript 119896 denotes the phase 119896 For primary phase119896 = 0 and for secondary phase 119896 = 1 119899 where 119899 is thenumber of the secondary phase present in the flow 120572

119896is the

volume fraction of phase 119896In (1) V

119898is the mass-averaged velocity and 120588

119898is the mix-

ture density V119898and 120588119898are given by (4) and (5) respectively

Consider the following

V119898

=sum119899

119896=1120572119896sdot 120588119896sdot V119896

120588119898

(4)

120588119898

=

119899

sum

119896=1

120572119896sdot 120588119896 (5)

The momentum equation (2) can be obtained by summingthe individual momentum equations for gas and solid phaseswhere is a body force 120583

119898is mixture viscosity and V

119889119903119896is

the drift velocity for the solid phase 119896 120583119898and V119889119903119896

are givenby (6) and (7) respectively Consider the following

120583119898

=

119899

sum

119896=1

120572119896sdot 120583119896 (6)

V119889119903119896

= V119896minus V119898 (7)

The first term on the right-hand side of energy equation (3)represents energy transfer due to conduction where 119896eff is the

International Journal of Chemical Engineering 3

effective conductivity while the last term 119878119864accounts for the

energy exchange between the solid phase and the gas phaseThe last equation needed for the model is the volume

fraction equation for the secondary phases which can beobtained from the continuity equation for secondary phase119901

120597

120597119905(120572119901120588119901) + nabla sdot (120572

119901120588119901V119898)

= minusnabla sdot (120572119901120588119901V119889119903119901

) +

119899

sum

119902=1

(119902119901

minus 119901119902)

(8)

where subscript 119901 denotes primary phase and subscript 119902

denotes secondary phase 119902119901

and 119901119902

accounts for masssource exchange between solid phase and gas phase

In order to fully describe the system additional equationsare needed to determine some terms in previous equationsand to implement the physical process of the interior ballisticsinto Fluent

23 Constitutive Laws We use constitutive laws to determinethemass and energy exchanged between the phases the inter-phase drag the propellant combustion rate the propellantgrain size calculation and the projectile motion

231 Mass and Energy As the volume of solid propellantdecreases by propellant combustion the volume of the gasincreases The rate of decomposition of the solid propellantin the control volume is given by

= (1 minus 120572) 120588119901

119878119901

119881119901

119903 (9)

where 120572 is the porosity 119878119901is particle surface 119881

119901is particle

volume 120588119901is the solid density and 119903 is propellant linear

burning rate which is modeled by Vieillersquos law (also knownas St Robertsrsquo law) [21]

119903 = 119860119901119899

+ 119861 (10)

where 119860 119861 and 119899 are constants for a given propellantmaterial they are determined from closed vessel (alsocalled manometric bomb) experiments The experiments aredescribed in STANAG 4115 [22] and by Mickovic [23]

The energy exchanged between the gas and solid phasesthe last term in (7) is given by

119878119898

= ℎex (11)

where ℎex is specific heat of propellant combustion [Jkg]The mass and energy terms (9) and (11) were imple-

mented into Fluent as source terms via User Defined Func-tion (UDF)

232 Particle-Size Calculation The particle diameter canbe calculated as the average diameter of all particles at theconsidered location however this assumption is far fromaccurate In fact the movement of the projectile creates

a pressure gradient inside the chamber According to Vieillersquoslaw propellant grains under high pressure will burn morequickly and then produce more gases This phenomenonmaintains the pressure gradient which induces a spatialdistribution of propellant grains size In order to take intoaccount this spatial distribution in the determination of theparticle size a similar method to Spaldingrsquos [24] ldquoShadowrdquomethod [25] is used The method consists to solve an addi-tional transport equation which expresses the conservationof number of grains per unit of volume Consider

120597119873

120597119905+ nabla (119873V) = 0 (12)

where 119873 [1m3] is the number of grains per unite of volumeand V [ms] is the velocity vector The implementation ofthis equation into Fluent is accomplished by using the UserDefined Scalar (UDS)

Equation (12) is used to calculate the volume of the solidparticle in the considered location and by assuming that thesolid particle has the shape of sphere the particle diameter inthe considered location is given by

119889119901= 2 sdot

3radic

3 sdot 119881119901

4120587 (13)

233 Interphase Drag The interphase drag in the chamberis given by [7 20] For spherical particle the interphase draghas the following form

119891119904=

1 minus 120572

119889119901

120588 (119906 minus 119906119901)10038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119891sc (14)

where 119891sc and Re are given by

119891sc =25

Re008141198911199040 Re =

12058810038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119889119901

120583 (15)

234 Projectile Motion The motion of the projectile whichis a translation along the gun barrel axe is updated accordingto Newtonrsquos second law

119898 sdot119889V119889119905

= sum119865 (16)

where119898 is the mass of the projectile 119889V119889119905 is its accelerationand sum119865 is the sum of all forces acting on it

There are two opposing forces acting on a projectilewithin the gun the propelling force 119901pr due to the high pres-sure propellant gases pushing on the base of the projectilewhilst the frictional force 119901fr between the projectile andbore which includes the high resistance during the engravingprocess opposes the motion of the projectile [3] Thus (16)can be written as

119898 sdot119889V119889119905

= (119901pr minus 119901fr)119860 (17)

where119860 is gun tube cross-sectional areaWhen the propellingforce becomes great than the frictional force the projectilestarts moving


Wall (breech)

Axis

Wall (shot base)

(a) (b)

Figure 2 Computational domain mesh (a) initial mesh generated by Gambit (b) Final mesh after the end of the simulation by Fluent

In Fluent the projectile is represented by a moving wallThe ldquoDynamic meshrdquo module is used to update the meshduring the calculation

3 Method of Solution

Finite volume method by Patankar [26] was used to solvethe governing equations The differential equations were dis-cretized by a second-order upwind differencing scheme overthe finite volume and solved by the commercial CFD packageFluent V63 produced and owned by Fluent IncThe velocity-pressure linkage was handled through the PISO algorithmThe simulations were performed in 2D axisymmetric gridA 2D mesh was produced in Gambit containing 70 cells ofwidth 0001m as shown in Figure 2(a) which is believed tobe fine enough to simulate this process When the projectilestarts moving Fluent creates new cells with the same width(0001m) until the projectile leaves the barrel By the end ofthe simulation the computational domain is composed of 895cells as shown Figure 2(b) As the whole process occurs inless than 1 millisecond a very small adaptive time step (1 times

10minus7 s to 1 times 10

minus9 s) with 20 iterations per time step was usedThe numerical convergence criterion (as defined in Fluent63 [17]) is set to 1 times 10

minus6 for energy and 1 times 10minus3 for other

scaled residual components Underrelaxation factor which isequal to 1 is adopted for all flow quantities The materialsproperties and the operating conditions are summarized inTable 1

4 Results and Discussions

There are a considerable number of numerical and experi-mental works dedicated to large-caliber guns which makethe validation of numerical simulation of those types of gunseasy however a limited number of works have been done forsmall-caliber guns For our case (small gun 9mm) we couldnot find experimental or numerical data in the literatureThis motivates us to do some experiments in the futureworks Fortunately for the current work we can compare ourresults especially the interior ballistic performance of thegun with test results delivered by ammunition companies[28ndash30] and with the results of the thermodynamics interiorballistic model with global parameters code (STANAG 4367)[6]

For all the simulations we consider a ldquoperfect ignitionrdquothat is the solid propellant begins to burn at 119905 = 0 s in theentire volume of the chamber Thus we do not simulate theigniterWe replace the effect of the igniter by raising the initialpressure to 2MPa [31]

00 01 02 03 04 05 06 07 080

50

100

150

200

250

Cham

ber p

ress

ure (

MPa

)

Time (ms)

08 mm mesh10 mm mesh

Figure 3 Chamber pressure-time history for119898119901= 0245 g without

barrel friction

41 Comparison with Firing Test Data by [28] for a ChargeWeight of 0245 g A 9mm cartridge is usually filled witha solid propellant mass varies between 018 g and 050 gThis charge weight depends on the type and density of thepowder as well as on the type mass and manufacturer ofthe projectile [28ndash30] For a fast burning low density doublebase ball propellant powder and an Ares lead projectiletype RNBB of mass 80 g the solid propellant weight variesbetween 0190 g and 0245 g [28] In this section the interiorballistic process inside 9mm gun is simulated for chargeweight of 0245 g Table 1 lists the parameters used for thissimulation

First the interior ballistics is simulated assuming thatthe friction pressure inside the barrel equals zero except for119871 = 0 where the friction pressure equals engraving pressure(Figure 9) The resultant characteristic curves are shown inFigures 3ndash8

Figure 3 shows the time history of the mean pressureinside the combustion chamber for tow grids with differentmesh sizes It is clear that themesh refinement has no effect onthe pressure-time curve therefore a 1 times 1mm cell size meshis fine enough for this ballistic simulation The combustionchamber pressure profile is similar to that known for interiorballistics in firearmsThewhole process happens in less than 1millisecondwhich is a good prediction for the time of interiorballistics process in handguns

Figure 4(a) shows the time history of the mean pressureat the breech and the base of the projectile and Figure 4(b)


Table 1 Materials properties and operating conditions

Parameter ValueGeometry

Length of combustion chamber 119871 ch 1355mmCamber diameter 119863ch 9mmLength of barrel 119871bar 150mmBarrel diameter (caliber) 119863bar 9mm

Physical propertiesProjectile mass119898proj 800 gPropellant mass119898

1199010245 g

Initial propellant particle diameter1198631199010

0300mmPropellant density 120588

119901600 kgm3

Propellant heat of combustion ℎex 1135MJkgSpecific heat of propellant 119888V 12032Thermal conductivity of propellant 120582

11990100164WmK

Specific heat of gas at constant pressure 119888119901

1858 JkgKSpecific heat ratio of gas 119896 124Viscosity of gas products 120583

1198920134064 (119879298)15(119879 + 110) kgms [7]

Conductivity of gas products 120582119892

005225 (119879700)07 JsmK [27]Constitutive laws

Coefficient in burning law 119860 20 times 10minus9msPa119899

Exponent in burning law 119899 102Coefficient in burning law 119861 0Engraving pressure 119875fr(119871bar=0) 40MPa

Numerical dataNumber of initial cells119873

11989470

Time step Δ119905 10minus07minus10minus09 s

00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

Breech pressureShot base pressure

(a)

00 01 02 03 04 05 06 07 080

1

2

3

4

5

Diff

eren

tial p

ress

ure (

MPa

)

Time (ms)

(b)

Figure 4 (a) Time history of breech and base pressures for 119898119901= 0245 g without barrel friction (b) Differential pressure-time history for

119898119901= 0245 g without barrel friction


00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)

Shot base pressureShot velocityShot travel

Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)

Figure 5 Time history of shot base pressure shot velocity and shottravel for119898


000 005 010 0150

50

100

150

200

250

Shot base pressureShot velocity

Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)

Figure 6 Shot pressure and shot velocity against shot travel for119898119901= 0245 g without barrel friction

00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

Figure 7 Solid volume fraction-time history for 119898119901

= 0245 gwithout barrel friction

00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas

Mixture at shot base Gas at shot base

Figure 8 Mach number time history for 119898119901

= 0245 g withoutbarrel friction

minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction

Engraving pressureFriction

0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)

Figure 9 Friction pressure inside the barrel

shows the time history of the pressure differnce (breechpressure-base pressure) At the early stage of combustion (119905 lt

015ms) the breech pressure equals the shot base pressureThis is because a perfect ignition is considered When therate of reaction becomes more important a pressure gradientestablished between the breech and the shot base Thisgradient becomes more important with the motion of theprojectile and reaches its maximum value (about 5MPa)slightly before the peak pressure time In the detent phasethe breech and the shot base pressures decrease with time aswell as the difference between them

Figure 5 represents the shot base pressure velocity andtravel of the projectile as functions of the time The velocityof the projectile and its travel inside the barrel directlycorrespond to the pressure at its base The projectile startsmoving when the pressure at its base is greater than the


friction pressure which in this case is equal to 40MPaFigure 5 reveals that the projectile movement starts at about0165ms but the projectile velocity is very small so that theprojectile travel is almost zero From 025ms the velocity ofthe projectile starts to increase rapidly as a direct result of thepressure augmentation behind it this allows the projectile totravel more rapidly inside the barrel until it leaves the muzzleof the weapon at 076ms

Figure 6 represents the base pressure and the velocityas functions of the shot travel down the barrel of length119871 = 015m It can be seen that the major acceleration ofthe projectile happens in the first one-sixth of the barrellength where the pressure behind the projectile is higher than100MPa At 119871 = 0m the stationary projectile seals thecombustion chamberThe combustion of the solid propellantproduces high pressure gases which dramatically increasesthe pressure inside the closed volume of the chamber Assoon as the pressure behind the projectile is high enough toovercome the friction pressure the projectile starts movingdown the barrel creating an increasing volume to be filedwith the high pressure gases produced by the combustionThe pressure inside the chamber continues to rise untilthe peak pressure is reached (about 208MPa) then it fallsdown as the projectile moves forwards and the volumeincreases However the projectile continues to accelerate forthe remaining travel in the barrel

From Figure 7 which shows the time history of solid vol-ume fraction inside the combustion chamber we can see thatthe consumption of the solid propellant happens in the first03ms thus when the projectile reaches 000828m length ofthe barrel the solid propellant is almost all consumed and theacceleration of the projectile is done only by the expansion ofthe gases down the barrel At the projectile exit time 076mspractically there is no solid propellant (solid volume fractionequals 26 times 10

minus07)Figure 8 displays the average Mach number history for

the mixture and gas phase in the combustion chamber andthe Mach number for the mixture and gas phase at the baseof the projectile The mixture is composed of the gas phaseand solid phase therefore the evolution of mixture Machnumber is different from that of the gas phase Howeverafter 119905 = 03ms the solid phase is almost consumed andthe mixture is practically composed of just the gas phasethus the Mach number is the same Before 119905 = 0165msthe Mach number is equal to zero because at that stage thepressure is low the projectile is immobile and the gas velocityinside the chamber is very small After 119905 = 0165ms atwhich the projectile starts moving as a result of the raising ofpressure inside the chamber theMach number following therapidly accelerated projectile quickly increases until about119905 = 04ms Beyond that point the combustion is finishedand the acceleration of the projectile is moderate Thus theMach number shows a low increase until the end of processAt the end of process theMach number equals 036The fluidis subsonic during the whole process

The previous results show that the present simulationrepresents qualitatively adequately themain interior ballisticbehavior and characteristics for small-caliber gun

00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

STANAG 4367Present code

000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

Figure 10 Shot base pressure-time history for 119898119901

= 0245 g withbarrel friction

Comparison of computational and practical tests results[28] for the interior ballistic performance is given in Table 2

This comparison reveals that the present simulationunderestimates the maximum average pressure (measuredat the mouth of the case) by 1064 and overestimates themuzzle velocity by 1118 These results can be amelioratedby taking into account the barrel friction

The case is resimulated taking into account the barrel fric-tion which is shown in Figure 9 and the new interior ballisticperformance is presented in Table 2 A net improvement inthemaximum average pressure and themuzzle velocity of theprojectile is achievedThe pressure is underestimated only by234 and themuzzle velocity is overestimated just by 082Therefore the barrel friction is an important parameter in thesimulation and has a great influence on the estimation of theinterior ballistic performance especially the muzzle velocity

42 Comparison with a Lumped-Parameter Code The sameprevious case (mass weight 0245 g with the same parameterlisted in Table 1 and taking into account the barrel friction)is simulated using a lumped-parameter code which usesthermodynamic interior ballistic model with global param-eters known as STANAG 4367 [6] The chamber pressure-time history for present code and STANAG 4367 are showntogether in Figure 10 and the interior ballistic performanceof STANAG 4367 is presented in Table 2

Figure 10 shows that the evolution of the pressure duringthe process has the same trend for both codes Also theoverall time of the process is almost the same However thepressure rise predicted by STANAG 4367 is faster than thatpredicted by the present code For STANAG4367 the pressurereaches its maximum at 0212ms while for the present codethe pressure reaches it maximum about 0066ms later Thisdiscrepancy is due to the early stage of the process STANAG4367 takes into account the ignition phase so we can note afast rise of pressure to 5MPa only after 0004ms while the


present code does not take into account the ignition phaseand it takes about 0066ms to reach 5MPa

Table 2 reveals that the present code which is two-phase flow local-parameter code predicts more accuratelythe performance of the interior ballistic than the one-phaseglobal-parameter code

43 Effect of Weight Charge In order to see the effect ofthe weight charge on the interior ballistic performance andverify the ability of the code to simulate the interior ballisticsprocess for small gun 9mmwith different porosity six otherssimulations were carried out Five weight charges 0190 g0201 g 0212 g 0223 g 0234 g and 0245 g (simulated pre-viously) were used to cover all practical range of the chargeweight for 9mm ammunition used in this simulation Anadditional charge of 0256 g is simulated to show the effectof an overcharge on the performance The results of these sixcases are represented together with the results of the 0245 gcase in Figure 11 which shows the time history of the shotbase pressurewith respect to the chargeweight and Figure 12which shows the time history of the projectile velocity withrespect to charge weight

Figures 11 and 12 clearly denote that the larger the solidpropellant charge is the higher pressure is produced andthe higher muzzle velocity is achieved Naturally a largeamount of solid propellant releases a large amount of energytherefore a higher pressure is produced inside the chamberwhich propels the projectile down the barrel with highervelocity

Figure 13 represents time histories of the solid volumefraction for all cases When the initial solid volume fractionis high a high number of solid propellant particles will bein the chamber which increases the surface of combustionAccording to the combustion law in (10) the rate of thecombustion increases thus the consumption of the solidpropellant accelerates and the whole time of the processdecreases

Table 3 reveals that higher muzzle velocity can beachieved by using a higher mass of solid propellant howeverthis leads to higher pressure in the gun which may cause arisk of damage for the gun if it exceeds the pressure limitsupported by the weapon

The muzzle velocity against the solid propellant charge isdepicted in Figure 14 It can be seen that for a charge weightbetween 0190 g and 0245 g the present simulation predictsthe muzzle velocity within about 5 error The discrepancybetween the firing data and the simulation results mainlyattributed to the difference between the numerical modelingand the real case The present simulation only takes intoaccount the effect of the combustion of the solid propellantand ignores other parameters such as the heat transfer andthe form of the projectile body In fact in the real casethe performance of the interior ballistics depends on theefficiency of the solid propellant as well as on the efficiencyof the projectile

Regarding the great complexity of the phenomena andthe assumptions considered the results presented above showthat the code simulates adequately the process of interior

00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


isin [0190 gndash0256 g]

00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g

Figure 12 Shot velocity-time history for119898119901isin [0190 gndash0256 g]

ballistics in small gun using 9mm cartridge especially theprediction of the performance of the process such as thechamber pressure the muzzle velocity and the time of theprocess

5 Conclusion

The interior ballistics simulations in 9mm small gun cham-ber were carried out using two-phase flow dynamics codeof two-dimensional axisymmetric calculationmethod imple-mented into CFD software Fluent The model includesconservation equations of the Fluent mixture model withappropriate constitutive laws


Table 2 Comparison of interior ballistic performance for charge weight of 0245 g

Computed values (present code) Lumped-parametercode

(STANAG 4367)Firing data test [28]No friction With friction

Value Error () Value Error ()Maximum average chamber pressure (MPa) 20782 22749 1753Maximum average breech pressure (MPa) 20884 22889 1761Maximum average mouth case pressure (MPa) 20731 minus1064 22656 minus234 232Maximum average shot base pressure (MPa) 20644 22549 1737Shot muzzle velocity (ms) 3669 1118 33272 082 31092 330Shot exit time (ms) 0776 0802 0788 lt1

Table 3 Effect of weight charge on interior ballistic performance

Computed valuesSolid propellant mass (g)

Min charge Max charge Overcharge0190 0201 0212 0223 0234 0245 0256

Maximum average chamber pressure (MPa) 16954 18099 1923 20367 21528 22749 23932Maximum average breech pressure (MPa) 17007 18165 19313 20467 21641 22889 24083Maximum average mouth case pressure (MPa) 16913 18052 19176 20304 21453 22656 23821Maximum average shot base pressure (MPa) 1685 17991 1911 20226 2136 22549 23714Shot muzzle velocity (ms) 28476 29535 30509 31448 32354 33272 3412Shot exit time (ms) 0957 0920 0887 0857 0829 0802 0779

00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g

Figure 13 Solid volume fraction-time history for for119898119901isin [0190 gndash

0256 g]

The comparisons of simulations with the available prac-tical tests results showed that the obtained interior ballisticsprocess performance results namely the average maximumpressure the muzzle velocity and the projectile exit time arein good agreement Unfortunately the lack of experimentaldata did not enable us to validate the velocity of the projectileand the pressures histories However the profiles showedthe general form observed in such simulations The results

019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)

Solid propellant mass (g)

Firing testPresent codeminus5 firing test

Figure 14 Muzzle velocity against propellant weight charge

given by this simulation are better than that given by lumped-parameter code (STANAG 4367)The code adequately repre-sents main interior ballistic parameters and characteristics inthe limit of the considered assumptions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] J Nussbaum P Helluy J-M Herard and B Baschung ldquoMulti-dimensional two-phase flow modeling applied to interior bal-listicsrdquo Journal of Applied Mechanics vol 78 no 5 Article ID051016 9 pages 2011

[2] R J Gollan I A Johnston B T OFlaherty and P A JacobsldquoDevelopment of Casbar a two-phase flow code for the interiorballistics problemrdquo in Proceedings of the 16th Australasian FluidMechanics Conference (16AFMC rsquo07) pp 295ndash302 CrownPlazaGold Coast Australia December 2007

[3] C I Farrar and D W Leeming Military Ballistics a BasicManual Battlefield Weapons Systems and Technology vol 10Royal Military College of Science Shrivenham UK 1983

[4] M J Nusca and P J Conroy ldquoMultiphase CFD simulations ofsolid propellant combustion in gun systemsrdquo in Proceedings ofthe 40th Aerospace Sciences Meeting and Exhibit 2002 AIAAPaper no 02-14289

[5] R D Anderson and K D Fickie ldquoIBHVG2-A userrsquos guiderdquoUSArmyBallistics Research Laboratory Technical Report BRL-TR-2829 Aberdeen Proving Ground 1987

[6] ldquoThermodynamic interior ballistic model with global parame-tersrdquo STANAG 4367 North Atlantic Council ED 3 2009

[7] M J Nusca and P S Gough ldquoNumerical model of multiphaseflows applied to solid propellant combustion in gun systemsrdquo inProceedings of the 34th AIAAASMESAEASEE Joint PropulsionConference and Exhibit Cleveland Ohio USA July 1998 AIAApaper no 98-3695

[8] P S Gough ldquoThe XNOVAKTC coderdquo US Army BallisticsResearch Laboratory Contract Report BRL-CR-627 Aberdeenproving ground 1990

[9] P S Gough ldquointerior ballistics modeling extensions to theXKTC code and analytical studies of pressure gradient forlumped parameter codesrdquo US Army Research Laboratory Con-tract Report ARL-CR-460 Aberdeen proving ground 2001

[10] M J Nusca ldquoComputational fluid dynamics model of mul-tiphase flows applied to solid propellant combustion in gunsystemsrdquo in Proceedings of the 18th International Symposium onBallistics pp 262ndash261 San Antonio Tex USA November 1999

[11] R Heiser and E Meineke ldquoMultidimensional interior ballistictwo-phase flow and the chambrage problemrdquo Journal of Propul-sion and Power vol 7 no 6 pp 909ndash914 1991

[12] R Heiser and D Hensel ldquoAMI A General Gas dynamic Modelof Internal Ballistics of Gunsrdquo Tech Rep E 786 Fraunhofer-Institut fur Kurzzeitdy-namik (EMI)Weil am Rhein Germany1986

[13] C Cuche M Dervaux M Nicolas and B Zeller ldquoMOBIDICa French interior ballistics code based on a two-phase flowmodelrdquo in Proceedings of the 6th International Symposium onBallistics Orlando Fla USA October 1981

[14] B Longuet P Della Pieta P Franco et al ldquoMOBIDIC NGa 1D2D code suitable for interior ballistics and vulnerabilitymodellingrdquo in Proceedings of the 22nd International Symposiumon Ballistics pp 362ndash371 Vancouver Canada November 2005

[15] C R Woodley ldquoModelling the internal ballistics of mortarsusing the one-dimensional code CTA1rdquo in Proceedings ofthe 20th International Symposium on Ballistics pp 354ndash360Orlando Fla USA September 2002

[16] C R Woodley S Billett C Lowe W Speares and E ToroldquoThe FHIBS internal ballistics coderdquo in Proceedings of the 22ndInternational Symposium on Ballistics pp 322ndash329 VancouverCanada November 2005

[17] USerrsquoS Guide Fluent Incorporated 2006 Fluent 63[18] S S Gokhale and H Krier ldquoModeling of unsteady two-phase

reactive flow in porous beds of propellantrdquo Progress in Energyand Combustion Science vol 8 no 1 pp 1ndash39 1982

[19] H Miura and A Matsuo ldquoNumerical simulation of solidpropellant combustion in a gun chamberrdquo in Proceedings of theAIAAASMESAEASEE 42nd Joint Propulsion Conference pp6048ndash6067 July 2006 AIAA Paper 2006-4955

[20] H Miura and A Matsuo ldquoNumerical simulation of projectileaccelerator using solid propellantrdquo in Proceedings of the 44thAIAA Aerospace Sciences Meeting pp 17355ndash17372 January2006 AIAA Paper 2006-1439

[21] J Nussbaum P Helluy J-M Herard and A Carriere ldquoNumer-ical simulations of gas-particle flows with combustionrdquo Journalof Flow Turbulence and Combustion vol 76 no 4 pp 403ndash4172006

[22] ldquoDefinition and determination of ballistic properties of gunpropellantsrdquo STANAG 4115 North Atlantic Council 1997

[23] DMickovic ldquoMethod for Determination of Propellant BurningLaw fromManometric Bomb Experimentsrdquo Report VTI-02-01-0159 1988

[24] D B Spalding ldquoThe ldquoShadowrdquo method of particle-size calcula-tion in two-phase combustionrdquo in Proceedings of the 19th Sym-posium on Combustion pp 491ndash951 The Combustion InstitutePittsburgh Pa USA 1982

[25] S Jaramaz D Mickovic and P Elek ldquoTwo-phase flows in gunbarrel theoretical and experimental studiesrdquo International Jour-nal of Multiphase Flow vol 37 no 5 pp 475ndash487 2011

[26] S V Patankar Heat Transfer and Fluid Flow HemispherePublishing Corporation Washington DC USA 1980

[27] G Lengelle J Duterque and J F Trubert ldquoCombustion ofsolid propellantsrdquo in RTOVKI Special Course on InternalAerodynamics in Solid Rocket Propulsion pp 27ndash31 RTO-EN-023 Rhode-Saint-Genese Belgium 2002

[28] Lovex ldquoReloading guide 2013rdquo Explosia Corporation Pardu-bice Czech Republic 2013 httpwwwexplosiacz

[29] Winchester group Reloaderrsquos Manual Olin Corporation EastAlton Ill USA 15th edition 1997

[30] Vihtavouri Reloading Guide for Centerfire Cartridges EurencoVihtavuori Oy Corporation 11th edition 2013

[31] J Nussbaum P Helluy J M Herard and A Carriere ldquoNumer-ical simulations of reactive two-phase gas-particle flowsrdquo inProceedings of the 39th AIAA Thermophysics Conference 2007AIAA paper 2007-4161

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Combustionchamber Barrel (the exit tube)

Solid

pro

pella

nt

bed

Shot

bas

e

Bulle

t Muzzle

Bree

ch

Figure 1 Initial geometry configuration in small gun

the pressure history in the system during the process aswell Computer modeling can be accomplished by usingone-dimensional single-phase ldquolumped-parameterrdquomodelssuch as IBHVG2 [5] and STANAG 4367 [6] which arebased on the assumption that grains and the products ofcombustion constitute a well stirred mixture or by usingmultidimensional multiphase flow model [7]

Presently the most accurate modeling of the interiorballistics problem is provided by two-phase multidimen-sional computational fluid dynamics (CFD) codes such asthe American codes NOVA or XKTC [8 9] NGEN [10] andEuropean codes AMI [11 12] MOBIDIC [13 14] CTA1 [15]and FHIBS [16] These codes are elaborated precisely for thesimulation of interior ballistics process They are the resultsof many years of improvements and validation and most ofthem were developed for large guns

In this paper we have attempted to elaborate two-phaseflow interior ballistic model for small-caliber gun and imple-ment it into commercial CFD code Fluent The characteristiccurves of pressure projectile velocity and projectile travelare reproduced and the performances of the interior ballisticprocess are predicted

2 Theoretical Model

The mixture model of Fluent is used to model the unsteadyreactive two-phase flow of solid propellant and its gaseouscombustion products This model can model n phases (fluidor particulate) by solving the momentum continuity andenergy equations for the mixture the volume fraction equa-tions for the secondary phases and algebraic expressions forthe relative velocities [17]

21 Basic Assumptions The combustion of the solid pro-pellant in the gun is a complex process It is necessary toaccept some assumptions to be able to model this processusing the mixture model Some of the assumptions are thesame as those mentioned by Gokhale and Krier [18] and thenreproduced by Gollan et al [2] The basic assumptions usedare (1) the solid phase composed of solid propellant andthe gas phase composed of propellant combustion gas (2)air in the chamber which is neglected because mass of theair is very small compared with the mass of combustion gasproduced from solid propellant [19 20] (3) the two phases are

treated as interpenetrating continua but separately coupledby appropriate interaction terms which account for mass andenergy transport between the phases (4) the solid consideredas an incompressible pseudofluid (5) the solid (propellant)burning which results in loss of mass from the solid phaseand equivalent gain by the gas phase (6) the gases which areinviscid except for their action on the particles through thedrag (7) the gas viscosity assumed to be known as a functionof temperature (8) the gas phase considered as an ideal gas(9) the ignition assumed to be ideal that is all the solidpropellant grains are simultaneously and uniformly ignited(10) the walls which are adiabatic

22 Governing Equations The balance equations for massmomentum and energy of mixture flow in propellant cham-ber and gun barrel are given by

120597

120597119905(120588119898) + nabla sdot (120588

119898sdot V119898) = 0 (1)

120597

120597119905(120588119898V119898) + nabla sdot (120588

119898V119898V119898)

= minusnabla119901 + nabla sdot [120583119898(nablaV119898+ nablaV119879119898)] + 120588

119898119892 +

+ nabla sdot (

119899

sum

119896=1

120572119896120588119896V119889119903119896

V119889119903119896

)

(2)

120597

120597119905

119899

sum

119896=1

(120572119898120588119898119864119896) + nabla sdot

119899

sum

119896=1

(120572119896V119896(120588119896119864119896+ 119901))

= nabla sdot (119896effnabla119879) + 119878119864

(3)

where the subscript 119896 denotes the phase 119896 For primary phase119896 = 0 and for secondary phase 119896 = 1 119899 where 119899 is thenumber of the secondary phase present in the flow 120572

119896is the

volume fraction of phase 119896In (1) V

119898is the mass-averaged velocity and 120588

119898is the mix-

ture density V119898and 120588119898are given by (4) and (5) respectively

Consider the following

V119898

=sum119899

119896=1120572119896sdot 120588119896sdot V119896

120588119898

(4)

120588119898

=

119899

sum

119896=1

120572119896sdot 120588119896 (5)

The momentum equation (2) can be obtained by summingthe individual momentum equations for gas and solid phaseswhere is a body force 120583

119898is mixture viscosity and V

119889119903119896is

the drift velocity for the solid phase 119896 120583119898and V119889119903119896

are givenby (6) and (7) respectively Consider the following

120583119898

=

119899

sum

119896=1

120572119896sdot 120583119896 (6)

V119889119903119896

= V119896minus V119898 (7)

The first term on the right-hand side of energy equation (3)represents energy transfer due to conduction where 119896eff is the





120597

120597119905(120572119901120588119901) + nabla sdot (120572

119901120588119901V119898)


) +

119899

sum

119902=1

(119902119901

minus 119901119902)

(8)



and 119901119902





= (1 minus 120572) 120588119901

119878119901

119881119901

119903 (9)


119901is particle



119903 = 119860119901119899

+ 119861 (10)



119878119898

= ℎex (11)





120597119873

120597119905+ nabla (119873V) = 0 (12)



119889119901= 2 sdot

3radic

3 sdot 119881119901

4120587 (13)


119891119904=

1 minus 120572

119889119901

120588 (119906 minus 119906119901)10038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119891sc (14)


119891sc =25

Re008141198911199040 Re =

12058810038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119889119901

120583 (15)


119898 sdot119889V119889119905

= sum119865 (16)



119898 sdot119889V119889119905

= (119901pr minus 119901fr)119860 (17)



Wall (breech)

Axis

Wall (shot base)

(a) (b)












00 01 02 03 04 05 06 07 080

50

100

150

200

250

Cham

ber p

ress

ure (

MPa

)

Time (ms)



barrel friction










1199010245 g



119901600 kgm3


11990100164WmK



1198920134064 (119879298)15(119879 + 110) kgms [7]






11989470


00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


(a)

00 01 02 03 04 05 06 07 080

1

2

3

4

5

Diff

eren

tial p

ress

ure (

MPa

)

Time (ms)

(b)




00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)


Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)



000 005 010 0150

50

100

150

200

250


Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)


00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)



00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas




minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction


0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)












00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120597

120597119905(120572119901120588119901) + nabla sdot (120572

119901120588119901V119898)


) +

119899

sum

119902=1

(119902119901

minus 119901119902)

(8)



and 119901119902





= (1 minus 120572) 120588119901

119878119901

119881119901

119903 (9)


119901is particle



119903 = 119860119901119899

+ 119861 (10)



119878119898

= ℎex (11)





120597119873

120597119905+ nabla (119873V) = 0 (12)



119889119901= 2 sdot

3radic

3 sdot 119881119901

4120587 (13)


119891119904=

1 minus 120572

119889119901

120588 (119906 minus 119906119901)10038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119891sc (14)


119891sc =25

Re008141198911199040 Re =

12058810038161003816100381610038161003816119906 minus 119906119901

10038161003816100381610038161003816119889119901

120583 (15)


119898 sdot119889V119889119905

= sum119865 (16)



119898 sdot119889V119889119905

= (119901pr minus 119901fr)119860 (17)



Wall (breech)

Axis

Wall (shot base)

(a) (b)












00 01 02 03 04 05 06 07 080

50

100

150

200

250

Cham

ber p

ress

ure (

MPa

)

Time (ms)



barrel friction










1199010245 g



119901600 kgm3


11990100164WmK



1198920134064 (119879298)15(119879 + 110) kgms [7]






11989470


00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


(a)

00 01 02 03 04 05 06 07 080

1

2

3

4

5

Diff

eren

tial p

ress

ure (

MPa

)

Time (ms)

(b)




00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)


Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)



000 005 010 0150

50

100

150

200

250


Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)


00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)



00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas




minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction


0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)












00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Wall (breech)

Axis

Wall (shot base)

(a) (b)












00 01 02 03 04 05 06 07 080

50

100

150

200

250

Cham

ber p

ress

ure (

MPa

)

Time (ms)



barrel friction










1199010245 g



119901600 kgm3


11990100164WmK



1198920134064 (119879298)15(119879 + 110) kgms [7]






11989470


00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


(a)

00 01 02 03 04 05 06 07 080

1

2

3

4

5

Diff

eren

tial p

ress

ure (

MPa

)

Time (ms)

(b)




00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)


Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)



000 005 010 0150

50

100

150

200

250


Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)


00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)



00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas




minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction


0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)












00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














1199010245 g



119901600 kgm3


11990100164WmK



1198920134064 (119879298)15(119879 + 110) kgms [7]






11989470


00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


(a)

00 01 02 03 04 05 06 07 080

1

2

3

4

5

Diff

eren

tial p

ress

ure (

MPa

)

Time (ms)

(b)




00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)


Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)



000 005 010 0150

50

100

150

200

250


Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)


00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)



00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas




minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction


0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)












00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00 01 02 03 04 05 06 07 080

50

100

150

200

250

Pre

ssur

e (M

Pa)


Time (ms)

050100150200250300350400

Velo

city

(ms

)

000002004006008010012014016

Trav

el (m

)



000 005 010 0150

50

100

150

200

250


Shot travel (m)

Pres

sure

(MPa

)

0

100

200

300

400

Velo

city

(ms

)


00 01 02 03 04 05 06 07 0800

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)



00 02 04 06 08000

005

010

015

020

025

030

035

040

Mac

h nu

mbe

r (mdash

)

Time (ms)

Mixture Gas




minus002 000 002 004 006 008 010 012 014 016

0

5

10

15

20

25

30

35

40

45Fr

ictio

n pr

essu

re (M

Pa)

Barrel length (m)

No friction


0000 0002 0004 0006 0008 001005

1015202530354045

Fric

tion

pres

sure

(MPa

)Barrel length (m)












00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















00 02 04 06 080

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)


000

001

002

003

004

005

006

0

2

4

6

8

10

Pre

ssur

e (M

Pa)

t (ms)

















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















00 02 04 06 08 100

50

100

150

200

250

Pres

sure

(MPa

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



00 02 04 06 08 100

50

100

150

200

250

300

350

Proj

ectil

e vel

ocity

(ms

)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g



5 Conclusion











00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















00 02 04 06 08 1000

01

02

03

04

05

Solid

vol

ume f

ract

ion

(mdash)

Time (ms)

0190 g0201 g0212 g

0223 g

0234 g0245 g

0256 g


0256 g]


019 020 021 022 023 024 025280

290

300

310

320

330

340

Proj

ectil

e vel

ocity

(ms

)








References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Multiphase CFD Simulation of Solid ...Research Article Multiphase CFD Simulation of Solid Propellant Combustion in a Small Gun Chamber AhmedBougamraandHuilinLu School