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Transient Heat Transfer Model of Machine Gun BarrelsRyan D. Hill a & Jon M. Conner aa National Technical Systems, Ordnance Sciences Division, Dana Point, California, USAAccepted author version posted online: 24 Apr 2012.Published online: 12 Jul 2012.

To cite this article: Ryan D. Hill & Jon M. Conner (2012) Transient Heat Transfer Model of Machine Gun Barrels, Materials andManufacturing Processes, 27:8, 840-845, DOI: 10.1080/10426914.2011.648694

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Transient Heat Transfer Model of Machine Gun Barrels

Ryan D. Hill and Jon M. Conner

National Technical Systems, Ordnance Sciences Division, Dana Point, California, USA

The desire to significantly reduce the weight of machine gun barrels while simultaneously increasing their operational life has led to the

need to develop a transient heat transfer model that accurately addresses the thermal management problem while simultaneously supporting

the execution of rapid trade studies of barrel geometry and material. To address this need, National Technical Systems Ordnance Sciences

Division has created a detailed finite-difference based numerical model using the Mathematica software, developed and marketed by

Wolfram Research, as the solution platform. This model solves the one-dimensional temperature profile through the thickness of the barrel

at a specified axial location.

Keywords Ballistics; Defense; Gun barrel; Heat transfer; Machine gun; Material trade study; Mathematica; Military; Modeling;

Numerical; PRODAS; Simulation.

INTRODUCTION

Repetitive firing of small and medium caliber car-tridges in machine guns provides a unique and challeng-ing transient heat transfer problem for evaluating theresponse of the gun barrel material. This problem com-prises three very different time scales: the first is theextremely transient temperature-time history of the car-tridge propellant reaction products flowing past theinterior wall of the barrel with each bullet fired, whichis only a few milliseconds in duration; the second isthe period of time between cartridge firings of approxi-mately 100 milliseconds during burst firing of the gunsystem; the third is the period of time between burstfirings that lasts several seconds.To address this challenging problemwe have developed

a one-dimensional transient heat transfer numericalmodel that predicts the temperature-time profile throughthe thickness of a machine gun barrel at a fixed axial pos-ition on the barrel. The model incorporates the followingfeatures: convection with variable heat transfer coeffi-cients at both the interior and exterior walls, radiativecooling at the exteriorwall, andarbitrary burst firing sche-dules in terms of rate of fire, number of rounds per burst,and time between bursts for a specified cartridge.In addition to the highly transient nature of the heat

load applied to machine gun barrels during burst firings,traditional steel alloy barrel materials, such as 4130 steel,exhibit thermal properties that are highly temperaturedependent. These effects have been incorporated in thenumerical model described here by allowing both thethermal conductivity and the specific heat of the barrelmaterial to vary with temperature.

The numerical model presented herein has been imple-mented in the Mathematica software program and maybe used to perform rapid trade studies of candidate bar-rel materials and thickness profiles for a given cartridgeand machine gun firing schedule. Inputs to the model arethe thermal conductivity and specific heat of the barrelmaterial as functions of temperature, the inner and outerbarrel diameter at a specified axial location, the firingschedule, and the temperature, density, and velocity ofthe bore gas throughout the firing cycle. Althoughimplemented in Mathematica, the methodology is port-able to other commercially available software.In order to illustrate the utility of the model and to pro-

vide a specific example of the methodology used to furnishthe input variables along with a comparison with opensource experimental data,wehave selected scheduledburstfirings of the M80 cartridge from an M60 machine gun.

BACKGROUND

Traditionally small caliber (12.7mm or less) machinegun barrels have been fabricated from either monolithic4000 series alloy steels or a combination of steel andanother metal alloy, such as Stellite, used as an innerwall lining material due to its retention of mechanicalproperties at high temperature and resistance to chemi-cal erosion as discussed in the U.S. Army Material Com-mand Pamphlet on Gun Tubes [1]. Advances inmanufacturing technology have opened up the possi-bility of manufacturing precision machine gun barrelsfor small arms from a very broad range of pure metals,alloys, and composites, one of the key topics of therecent symposium on Gun Tubes hosted by the U.S.Army Research Laboratory [2]. The challenge for gundesigners is to select the right combination of barrelmaterial properties to insure optimum performance ofthe design with respect to resisting the effects of gun tubeerosion and retaining the mechanical stiffness of thebarrel assembly when subjected to the thermal loading

Received July 8, 2011; Accepted August 9, 2011

Address correspondence to Ryan D. Hill, National TechnicalSystems, Ordnance Sciences Division, 28 Monarch Bay Plaza,Suite F, Dana Point, CA 92629, USA; E-mail: ryan.hill@nts.com

Materials and Manufacturing Processes, 27: 840845, 2012Copyright # National Technical SystemsISSN: 1042-6914 print=1532-2475 onlineDOI: 10.1080/10426914.2011.648694

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associated with high rate of fire burst schedules typicalof small caliber machine gun weapon systems.The mechanisms associated with erosion of the inner

surface of gun barrels due to repeated firings has beenreviewed in detail by Johnston [3] and Ebihara andRorabaugh [4]. Both have described the erosion processas consisting of three major factors: thermal, chemical,and mechanical. Of these, thermal factors are knownto contribute to both the chemical and mechanicalaspects of gun tube erosion.Heating of a machine gun barrel assembly due to

burst firing will also lead to a reduction in the accuracyof the gun system if the bulk temperature of the barrelmaterial reaches a level that results in a loss of mechan-ical stiffness, or elastic modulus, of the material therebyreducing its normal mode frequency of vibration to alevel which approaches the rate of fire. This phenom-enon is also discussed in [1].Predicting the transient temperature profile through the

thickness of a gun barrel at a given axial position whensubjected to burst firing requires the solution of the com-plete time dependent conservation equations whichdescribe the individual cartridgefiringheat input to thebar-rel and the response of the barrel material as described inHeiney [5] and Gerber and Bundy [6]. One-dimensional,transient heat transfer models which neglect spatial gradi-ents of temperature in the axial direction of the gun barrelare suitable tools to apply to this problem since the tem-perature gradient in the radial direction is at least an orderof magnitude greater than the temperature gradient in theaxial direction of the barrel. Although solution of the com-plete two- and three-dimensional time dependent conser-vation equations is possible using current state-of-the-artfinite element engineering computer programs, it is theauthors experience that this effort requires a number ofcalendar days or weeks to complete for a single barrelgeometry=material combination using current state-of-the-art multiprocessor computers and is, therefore, notsuitable for the performance of rapid trade studies. Thiswas the primary motivation behind the current work.

MODEL ASSUMPTIONS

The basis for the heat transfer model used here isdescribed by Gerber and Bundy [6], in which the follow-ing assumptions apply:

1. The temperature gradient in both the axial and theazimuthal directions are negligible when comparedto the gradient in the radial direction.

2. All interior ballistics properties (gas pressure, tem-perature, velocity, etc.) are assumed to be inde-pendent of barrel temperature and are identical foreach round.

3. Heating due to friction is neglected.4. Thermal expansion of the barrel is neglected, and the

density of the barrel material, qs, is held constant(7,845 kg=m3 for 4130 steel).

5. The thermal conductivity, k, and the specific heat, c,of the barrel material are treated as functions of thebarrel temperature, T, as shown in Fig. 1 [7]. Note

that the large specific heat anomaly in 4130 steel isdue to the alpha-gamma phase transition that occursin carbon steels at about 1,000K.

6. The heat transfer coefficient between the bore gasand the inner wall, hint, and the temperature of thebore gas, Tgas, each represent their respective meancross-sectional values. Similarly, the heat transfercoefficient between the ambient air and the outerwall, hext, and the ambient air temperature, T1, eachrepresent their respective mean values. These exteriorvalues are assumed to have negligible gradients inany direction.

7. Heat transfer to other parts of the gun (stock, butt,scope, etc.) is neglected.

PROBLEM STATEMENT

For symmetry, we will use the cylindrical coordinates:r, h, and z. The gun barrel is centered along the z-axis,with the projectile base at z 0 at time t 0. We definerRi and rRo to be the inner and outer walls of thebarrel, respectively. In three dimensions, the barrel tem-perature distribution T (r, h, z, t) is represented by thegeneral heat diffusion equation:

qsc@T

dt r krT : 1

Since we are holding z and h constant, their respectiveterms will vanish when we expand the divergence andthe gradient into cylindrical coordinates. The tempera-ture profile is simplified to T (r, t), and heat diffusionequation inside the barrel becomes the one-dimensionaldiffusion equation:

1

a@T

@t @

2T

@r2 1

r

@T

@r 1k

dk

dT

@T

@r

2; 2

where a k=qs c is the thermal diffusivity of the metal.The third term on the right-hand side of this equation

FIGURE 1.Variation of thermal properties of 4130 steel with temperature.

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arises from the fact that we are considering k as a func-tion of temperature, and therefore dk=dT is nonzero.On the interior surface of the barrel, we consider con-

vective heat transfer between the gas and the barrel(ignoring radiation on the inside wall) in accordancewith Newtons law of cooling:

@Q

@t hintA T Tgas

; 3

where Q is the heat transfer from the barrel to the gasand A is the surface area of the barrel wall. For conti-nuity [6] at the boundary, we equate the rate of heattransfer from the surface to the gas with the rate of heattransfer from the barrel volume to the surface:

1

A

@Q

@t k @T

@r; r Ri; 4

and by substitution, the boundary condition for the bar-rel temperature at the inside wall becomes

k@T

@r hint T Tgas

; r Ri; t > 0: 5

We treat the exterior boundary condition in a similarmanner, except here we allow for cooling by both con-vection and radiation:

k@T

@r hext T1 T er T41 T4

; r Ro; t > 0;

6

where r is the StefanBoltzmann constant, and e is theemissivity of the barrel surface, a dimensionless numberbetween 0 and 1, with e 1 representing a perfect black-body radiator. Note also that at the inside wall r isincreasing toward the barrel, while at the outside wallr is increasing away from the barrel; this accounts forthe reversal of the temperature terms in Eqs. (5) and (6).The initial condition at time t 0 defines the tempera-

ture throughout the barrel to be equal to the ambienttemperature:

T r T1; t 0; Ri r Ro: 7

INPUT DATA

To solve the heat transfer equations at the inner andouter walls of the barrel in Mathematica, we must firstobtain respective functional representations for the heattransfer coefficient at each surface and for the tempera-tures of the bore gas and of the ambient air.

Interior Wall Boundary Condition

For the interior wall, the conditions are treated as aninput in our calculations, so the gas temperature andheat transfer coefficient in the bore are treated solelyas functions of time. The input data for the boundary

condition at the interior wall was computed from aninterior ballistics model of the M80 cartridge usingPRODAS, a software tool developed and marketed byArrow Tech Associates, Inc. Using PRODAS to calcu-late the interior ballistics of a single round via theBaerFrankle method [8], we are able to obtain a tabularoutput of several variables, namely, projectile displace-ment and velocity, average gas temperature, gas pressureat the projectile base and the breech, and the percentageof propellant burned, all as functions of time.According to Robbins and Raab [9], an approxi-

mation for the heat transfer coefficient in the bore is

hint h0 kNqcpv; 8

where h0 is the free convective heat transfer coefficientfor the air inside the bore (here held constant at10W=m2K), v is the mean cross-sectional flow velocityof bore gas at the fixed axial position, q is the mean den-sity of the bore gas, and cp is the specific heat of the boregas. The Nordheim friction factor, kN, is a dimensionlessconstant which is approximated by

kN 13:2 4 log 200Ri 2: 9

The specific heat of the bore gas is not definitivelyknown as a function of temperature, pressure, and therelative amount of each reaction product at each timeand each axial position. Therefore, we held cp constantwith respect to time and axial position at a value withinthe range of the specific heat values for all of theexpected reaction products [10, 11]. The value wasadjusted to provide a roughly equal error at the interiorand exterior surfaces when compared to the availableexperimental data. The value used for the example calcu-lations presented herein was 1,050 J=kg K.The interior ballistics model of the M80 cartridge

yields the projectile displacement and velocity as func-tions of time. In order to estimate v, the gas at the pro-jectile base is assumed to have a velocity equal to that ofthe projectile, and the gas at the breech is assumed tohave zero velocity. Then a linear interpolation is appliedto the gas velocity for all axial points between the breechand the projectile base [12]. Similarly, since the interiorballistics model of the M80 cartridge provides the per-centage of propellant burnt as a function of time, andthe initial mass of the propellant is known, we calculateq by simply dividing the mass of propellant burnt by thevolume from the breech to the projectile base. While theprojectile is still in the barrel, the interior ballistics modelsupplies the average temperature of the gas in the bore(we treat the gas in the bore as not having any tempera-ture gradient at any given time).However, the interior ballistics calculation ends once

the bullet reaches the muzzle. After this point, we useda model of the reaction product gas venting out of arocket motor nozzle (computed using the PRODASrocket motor simulator) to calculate the pressure insidethe bore as it decays back to the ambient pressure. Todo this, we first defined a cylindrical rocket motor with

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the same length and diameter as the gun barrel, and wetreated the muzzle like a nozzle. We then manipulatedthe properties of the rocket propellant until the burnoutpressure at the nozzle matched the muzzle pressure ofthe gun barrel obtained from the interior ballistics calcu-lation. In this way, when the data from the two analysesare combined, the pressure inside the barrel is continu-ous at the time of muzzle exit.After the projectile leaves the muzzle, we fit an expo-

nential decay to this pressure curve and assume Tgas andq decay at the same rate as the pressure. We hold v con-stant with respect to time after the muzzle exit.The axial position under consideration, zc, is always

greater than zero for this model, and we have definedthe displacement of the projectile base, z(t), such thatz(0) 0. We assume here that the barrel at zc will nothave any heat input until tc, which we define to be thetime at which z(t) zc. In order to account for this, wehave set TgasT1 and hint h0 for all t tc.For the example provided, we have chosen zc to be 6

inches from the breech, and tc is about 1.7 milliseconds.(Note that the propellant ignition is delayed by 1 ms toensure steady-state conditions at t 0. The projectileactually arrives at zc 0.7ms after ignition.) At this location,Ri is 3.88mm and Ro is 15.2mm, and Fig. 2 shows a plotof Tgas and hint as functions of time for a single round.

Exterior Wall Boundary Condition

The conditions at the outer wall are very mild incomparison to the inner wall, so here we hold the air tem-perature constant at 298K (about 77F). The heat trans-fer coefficient at the outer wall is calculated from thedefinition of the average Nusselt number, NuD, aroundthe circumference of the barrel, which is proportionalto the Rayleigh number [11], RaD, for the buoyancydriven flow. The heat transfer coefficient is then

hext kairD

NuD kairC

2RoRanD; 10

where kair is the thermal conductivity of the ambient air,D is the local outer diameter of the barrel, and C and n

are constants which depend on RaD, taken to beC 0.48 and n 0.25 for the case of a long, horizontalcylinder [11]. Figure 3 shows an example plot of the heattransfer coefficient as a function of temperature at theoutside wall of the barrel, 6 inches from the breech.

CALCULATION

The final calculation involves seeking a solution to theset of differential equations stated in Section 4. To dothis, we make use of Mathematicas built-in numericaldifferential equation solver, NDSolve (NDSolve begansupporting partial differential equations with the releaseof Mathematica 5.0 in 2003; we are using Mathematica7.0.1 for the calculations herein). This solver allowsmany user-options such as the desired precision, stepsize, solution methods, etc.For the time steps, we set a maximum step size of 10

milliseconds (to increase calculation speed betweenrounds), but we did not set a limit on the number ofsteps, allowing Mathematica to choose a sufficientlysmall time step to meet our desired precision duringthe transient section of the firing cycle. We used 250equally spaced points in the radial direction.In an attempt to optimize the efficiency of the calcu-

lation, we have broken the firing cycle into a two-stageprocess, where we use a slightly different technique ineach stage. The first stage runs for the first 10 millise-conds of the cycle, well beyond the time of the muzzleexit, and the second stage runs for the remaining 90milliseconds. In the first stage, the equations tend tobe stiff due to the rapid changes in the conditions ofthe bore gas. We consequently force Mathematica touse an explicit RungeKutta method here, which causesthe calculation to be more stable than allowing Mathe-matica to choose a method. For the second stage, how-ever, the boundary condition at the inside wall is verycalm, and Mathematicas automatic method selectionis sufficient.In order to simulate burst firing schedules, we have

also added a third stage using nested for-loops. We firstspecify the number of rounds in each burst, and allow

FIGURE 2.Temperature of the bore gas and heat transfer coefficient for a

single round at the inside wall, 6 inches from the breech.FIGURE 3.Heat transfer coefficient for a long, horizontal cylinder in

atmosphere at 298K, with a diameter of 30.4mm.

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the first two stages to repeat until that burst is complete.Then the third stage, which is set up exactly as thesecond stage, allows for long-term cooling until the nextburst begins. Thus we can arbitrarily set the number ofrounds per burst, the time between bursts, and the totalnumber of bursts in the simulation.

EXAMPLES AND RESULTS

The M60 machine gun features a barrel length of 22inches and a firing rate of 600 rounds per minute, or afiring period of 100 milliseconds. In a transient tempera-ture measurement experiment, Moeller and Bossert [13]used type K (chromel-alumel) thermocouples, whichare stated to have a response time on the order of amicrosecond, to measure the temperatures of the interiorand exterior surfaces of an unlined M60 machine gunbarrel. We will compare the results of our model withtheir experimental data for both the interior and exteriorsurfaces of the barrel at an axial position of 6 inchesfrom the breech.The temperature-time history for the inside wall of the

barrel for continuous firing is shown in Figs. 4 and 5.Figure 4 includes the transient behavior of each individ-ual round for the first 20 rounds, while Fig. 5 is shownwith the transient peaks filtered out in order to improvethe clarity when showing long term trends. The dataappearing in both figures, along with the curve inFig. 5, represent the initial temperatures at the insidewall immediately preceding each round. After 200rounds the model predicts temperature of 682K(769F) at the inside wall, while the measured valuefor this time is 647K (705F), which is a difference of10.0% when compared with the ambient temperature.Figure 6 shows the temperature of the outside wall of

the barrel for a 125 round burst, which lasts for 12.5 sec-onds. This plot shows that, due to the heat diffusioninside the barrel, the temperature at the outside wallcontinues to rise for a few seconds after the burst hasended. After 1.2 seconds of cooling, the model predicts

an external surface temperature of 488K (419F), whilethe measured value is 512K (462F). This is a differenceof 11.2% when compared with the ambient temperature.The model can also show the instantaneous radial

temperature profile through the thickness of the barrel,as shown in Fig. 7. These two plots show the profile0.5 seconds and 1.2 seconds after firing a 125 roundburst.Finally, Fig. 8 shows both the interior and exterior

wall temperatures for a series of 10 round bursts in 6second intervals, for a total of 100 rounds in one minute.Since the transient temperature spikes like those in Fig. 4are too narrow to be individually resolved on thistimescale, each 10 round burst appears as a thick bandin this plot.As shown in the examples, this model is in good agree-

ment with experimental data. The largest sources oferror are likely that this is a one-dimensional modeland that we neglect heat transfer to the gun assembly.In a real gun barrel, heat will flow axially down thelength of the barrel and into other parts of the gun,

FIGURE 4.Transient temperature-time history of the inside wall for 20

rounds.

FIGURE 5.Temperature trend of the inside wall immediately preceding

each round for 200 rounds.

FIGURE 6.Temperature-time history of the outside wall for a 125 round

burst. Experimental data compared 1.2 seconds after the end of the burst.

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and free convection will cause azimuthal temperaturegradients on both the interior and exterior surfaces ofthe barrel. The advantage of the one-dimensional model,however, is that it greatly increases the speed of calcu-lation in order to facilitate rapid trade studies of barrelmaterial and thickness.

CONCLUSION

The method outlined here can be used to conduct rapidtrade studies of barrel geometry or material. For ourM60example, if an alternative barrel material is desired, theonly new input data required for this model would bethe temperature-dependent thermal properties for thatmaterial. Then the barrel temperature trends for variousfiring schedules can be quickly calculated for the new

material to determine if the material would be an effectivereplacement. Additionally, depending on the strength ofthe material, there may be a desire to decrease the barrelthickness, which can be done quickly in this model bychanging the outer radius at the axial position of interest.Due to the highly efficient nature of the numerical finitedifference solver schemes embedded in the Mathematicasoftware and the approach used to divide the transientheat transfer problem into three stages, the modeldescribed here can perform machine gun barrel materialand geometry trade studies efficiently.

REFERENCES

1. AMC Pamphlet, AMCP 706252. Research and Developmentof Material, Engineering Design Handbook, Guns Series, GunTubes; U.S. Army Material Command, 1964, AD830297.

2. Proceedings of the Gun Tube M3 Symposium: Modeling, Mate-rials, and Manufacturing Technology; U.S. Army ResearchLaboratory, Aberdeen Proving Ground: Aberdeen, Mary-land, July 79, 2010.

3. Johnston, I.A. DSTO-TR-1757, Understanding and PredictingGun barrel Erosion, Weapon Systems Division; DefenseScience and Technology Organization: Commonwealth ofAustralia, 2005.

4. Ebihara, W.T.; Rorabaugh, D.T. Mechanisms of gun-tubeerosion and wear. In Gun Propulsion Technology, Vol. 109of Progress in Astronautics and Aeronautics. Stiefel, L. Ed.;AIAA: Washington, D.C., 1988; Chapter 11, 357376.

5. Heiney, O.K. Ballistics applied to rapid-fire guns. In InteriorBallistics of Guns, Vol. 66 of Progress in Astronautics andAeronautics; Krier, H. Ed.; AIAA: Washington, D.C., 1979;87112.

6. Gerber, N.; Bundy, M. BRL-MR-3984, Effect of VariableThermal Properties on Gun Tube Heating; U.S. Army BallisticResearch Laboratory: Aberdeen Proving Ground, MD, 1992,ADA253066.

7. Hoyt, S. (Ed.) ASME Handbook, Metals Properties; McGrawHill Book Company, Inc.: New York, 1954.

8. Baer, P.G.; Frankle, J.M. The Simulation of Interior BallisticPerformance of Guns by Digital Computer Program; BallisticResearch Laboratory: Aberdeen Proving Ground, MD, 1962.

9. Robbins, F.W.; Raab, T.S. BRL-MR-3710, A Lumped-Parameter Interior Ballistic Computer Code using the TTCPModel; Aberdeen Proving Ground, MD, 1988, ADA203295.

10. Bailey, A.; Murray, S.G. Explosives, Propellants andPyrotechnics; Brasseys (UK): London, 1989; 1718.

11. Incropera, F.P.; DeWitt, D.P. Fundamentals of Heat andMass Transfer, 4th Ed.; John Wiley & Sons, Inc.: New York,1996; 501504, 839843.

12. Corner, J.A. Theory of the Interior Ballistics of Guns; JohnWiley & Sons, Inc.: New York, 1950; 339343.

13. Moeller, C.E.; Bossert, A.J. Measurement of TransientBore-Surface Temperatures in 7.62mm Gun Tubes; NationalTechnical Information Service: Springfield, VA, 1973,AD780938.

FIGURE 8.Transient temperature-time history of both the inside and out-

side walls for a burst schedule of 10 rounds each 6 seconds for 100 rounds

total.

FIGURE 7.Radial temperature profile through the thickness of the barrel

shortly after a 125 round burst.

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