1. IntroductionClassic interior ballistic method includes the form

function and burning rate equations of gun propellantsbased on the geometrical burning rule, equations ofprojectile motion and energy conservation equation１）－３）.Except the chemical kinetics computing methods４）, theburning rate of gun propellants is obtained by closedvessel test or gun firing, which is dependent on thepressure usually described by Vieille’s law shown inEquation (1)５）

��������� (1)

Where �� is the burning rate coefficient and � is thepressure exponent.

The main work carried out with a closed vessel consistsin pressure measurements during the combustion of apropellant, i.e. the pressure history and the pressuremaximum６），７）. These measurements at different loadingdensities allow calculating the thermodynamic propertiesof the combustion gas (covolume and propellant forceconstant), the dynamic vivacity as well as burning rate ofpropellants based on the Nobel-Abel equation of state.

The Nobel-Abel equation of state is used to determinethe amount of propellant that must have burn (massfraction) to produce the experimentally measured

Interior ballistic prediction of gun propellantsbased on experimental pressure-apparent

burning rate model in closed vessel

Zhenggang Xiao＊, Sanjiu Ying＊, Weidong He＊, and Fuming Xu＊

＊School of Chemical Engineering, Nanjing University of Science and Technology200 Xiaolingwei Street, Nanjing, Jiangsu Province, P. R. CHINA 210094Phone : +86-25-84315138†Corresponding author : xiaozhg@njust.edu.cn

Received : May 29, 2014 Accepted : September 19, 2014

AbstractA new experimental pressure-apparent burning rate model was established based on the recorded pressure-time data

in the closed vessel to study the actual burning gas generate rate of a propellant whose form function is not available. Thenew burning model can replace the Vieille’s law and form function of propellants in the interior ballistic equations topredict the interior ballistic performance of gun. As distinct from the traditional geometric form function based burningrate model, the new model introduced the concept of relative pressure impulse related to the actual burnt web thicknessof propellant statistically. The apparent burning rate was expressed by the propellant mass fraction burnt vs. relativepressure impulse curve. Based on the recorded experimental pressure time data of three kinds of propellants, such as12/1, 13/7 and 13/19 single-base propellants, numerical calculation was carried out using the computer program of thenew model. And numerical calculation results were different form those of the traditional geometric form function basedburning rate model. The new experimental pressure-apparent burning rate model is more suitable for the actualcombustion circumstances of propellants in the closed vessel, for which is ultimate independent of the various grain sizesand have taken account of the influence of chamber pressure on the actual burning rate. The interior ballistic simulationof selected gun propellants was conducted using the new burning rate model. The results predicted by the new modelshow a good agreement with the classic interior ballistic predication results though the agreement with the experimentalresults is still to be improved.

Keywords : propellant, closed vessel, burning rate models, interior ballistic, simulation

Researchpaper

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pressure. The measured pressure should be correctedconsidering the heat loss to the wall of closed vessel. Aform function related to the current dimensions of theremaining propellant is used to calculate the surface areaof propellant for each pressure-time point. The burningrate is then calculated using the derivative of the massgenerated at each time step according to the equation(2)８），９），15）

������������� (2)

Where A is the surface area of remaining propellant,�����is time derivative of propellant mass fraction burnt,i.e, burning gas generation rate.

It is well known that the derivation of propellantburning rate is based on the geometry of grains andparallel layer burning rule and expressed as Vieille’s law.The Vieille burning rate is still used as a burning model ofpropellant in gun bore although it bears no relation tothermodynamic process occurred in gun bore.

Recently, theoretical models of two-phase flow in gunbore and their solutions have been developed10）, which areable to study the space distribution of pressure and gas-flow velocity, but geometrical burning rule and fromfunction are still used as one of the main assumptions inthese models.

Obviously, geometrical burning rule is little more thanideal. According to the assumption of this rule, all grainsshould have same shape and dimensions and burn inparallel layers in the direction that is perpendicular tograins’ surfaces after ignition of all surfaces. Actually, theuniformity of burning for all propellant grains are directlyinfluenced by many factors such as the variation in shapesand dimensions from one grain to another grain, and thedifference in the ignition condition as well as thecircumstance that the burning charges. The actualburning rule should reflect a statistical property, and thereexits a deviation with the burning rate determined byform function based on geometrical burning rule.

Especially, as for the deterred and multi-layer coatedpropellants, conventional burning rate calculation methodcan still employ the geometry form function method11）. Forthe deterred propellants, the deterred layer possesses acontinuously increasing burn rate from outer layer to theinner core. The distribution function of deterrent insidethe deterred layer can be determined by experiments. Itcan be assumed that the propellant force is constant foreach layer12）. A multi-layer coated propellant generallyconsists of two or three distinct coating layers. Since thecoating layers are very thin compared to the burningthickness of propellants, it can be assumed that theburning rate of coating layers is constant duringcombustion process. When the outer coating layer isconsumed, the inner part of the propellants start to burnaccording to the geometry burning rules13）. However, forthe multi-layer coated propellants which have complexshape of inner hole, if the diameter of inner hole isrelatively large, or the coating layers are brittle, perhapsthe coating layers will not finished burning

simultaneously. Thus, some surface coating layers on theinner holes will be broken, and the inner holes begin toburn in advance. In that case, it will become very difficultto calculate the burning rate of coated propellants.

Moreover, if the form function of some kinds ofpropellant is not available, it is impossible to determine theburning rate by the traditional method. Eisenreich etal14）,15） modified the Vieille’s law to account for thetemperature dependence of the burning rate or thecombustion in porous propellants by a simplified analysisof the heat flow in the solid phase. The dependence of theburning rate on initial temperature, on phase transitions,porous structure and gaseous reactions can be describedby the new burning rate model.

The theory of potential equilibrium for interior ballisticsestablished by Bao et al16）can explain the actual burningrate of propellants in gun bore by means of the point ofpotential equilibrium. Actual burning rate equation ofpropellants in the new theory of potential equilibrium wasderived from the firing data of gun directly. The closedvessel tests need not be used. The corresponding solvingmethod of interior ballistics can be derived and constitutea new interior ballistics system which was different fromthe classical system on the basis of geometrical burningrule. However, it was still too complex to be applied in theballistic prediction extensively. Obviously, simple ballisticmethod is rapid and efficient and can give useful resultsfor charge estimation and prediction of performance ofgun.

In the present work, the apparent burning rate ofpropellant was deduced from closed vessel test data basedon the concept of relative pressure impulse related to theactual burnt web thickness of propellant statistically. Thenit was incorporated into the equations of classic interiorballistic to replace the form function and equation ofburning rate to establish a novel ballistic predictionmethod for gun propellants. The new method doesn’t needthe form function calculation. It is very suitable for thepropellants whose form function is not available, such asthe coated consolidated propellants. Simulation results fora selected gun propellant charge showed a goodagreement with the classic interior ballistic predictedresults.

2. Concept of relative pressure impulseFrom the point view of form function, the most

important variable is the actual burnt web thicknessduring combustion. For a special form of propellant, thesurface area can be calculated from the actual burnt webthickness and form function. For a propellant whose formfunction is not available, we need to choose a measurablequality which is related directly to the actual burnt webthickness to replace the immeasurable nominal burnt webthickness.

Many researches17）,18） showed that the burning layerconcept can still be used to the calculation of propellantburning rate when the form function is not available. Thethickness of the burning layer distributes in a certainrange (the pressure-time (���) curve of propellant

Zhenggang Xiao et al.2

recorded in closed bomb test should reflect thick or thin ofpropellant’s burning web thickness. Thinner propellantsfinish burning faster, while thicker propellants take moretime to finish burning). Therefore, the normalizedpressure impulse is defined as

�������������������� (3)

as an independent variable, then ��� curve can beconverted into ���� curve which may be expressed asthe following cubic polynomial equation

��������������������

� (4)

and it describes the actual burning gas formation rule inclosed vessel. Equation 4 is the experimental pressure-apparent burning rate model related to the burnt layerthickness statistically.

The coefficient ��, ��, ��, ��, �� in Equation 4 can besolved by least-squares fitting method based on theexperimental data points. The formula of solving isshowed as follows�����������������������������������������������������������������������������������������������������������������������������������

�������������(5)

where�is the different data point number in the curve of����.

According to the classic interior ballistics, if the burningrate of propellant is proportional to the pressure, it can beexpressed simplify by

������� (6)

Actually, when the pressure in the combustion chamberis above the pressure of 100 MPa, the above Equation 6 istotally approximate to the actual combustion. Then theabove equation can be wrote as

������� (7)

Whereis the burnt web thickness. Integrating Equation7

����������� (8)

Where � is the pressure impulse at the pressure-timepoint. It is the area under the pressure-time curve alongthe time axis from the zero point to the instant pressure-time point. When the propellant is finished burning atburnt instant��

�������������� (9)

Where � is the initial web thickness. � is the pressureimpulse at the burnt point �. It is a constant for a specificpropellant with certain web thickness.

From the Equation 8 and 9, the relative pressureimpulse �� is a measurable quantity and proportional tothe burnt web thickness when the burning rate of

propellant is proportional to the pressure. Actually, even ifthe burning rate of propellants is described by theexponent equation, the relative pressure impulse is stillstatistically corresponded to the actual burnt webthickness under the same test conditions. Therefore, thepoint of maximum pressure of propellant in the closedvessel

����,����

will act as the boundary condition of Equation 4 to show astatistically property during closed vessel combustion.

The concept of relative pressure impulse based on thepressure-time data in the closed vessel can avoid theerrors resulted from the variation of shape and dimensionsof propellants. The new experimental pressure-apparentburning rate model is directly derived based on theconcept of relative pressure impulse and recordedpressure time data in closed vessel, which is very differentfrom the traditional method of form function. The newburning rate is not dependent on the form functioncalculation of propellants. Meanwhile, the influences of thepressure on burning rate during combustion process isconsidered, which can result in the synergetic effect ofpressure and burning circumstance to increase the gasgeneration rate in closed vessel. Therefore, it is morepractical to calculate the apparent burning rate, especiallysuitable for the propellants whose form function is not ordifficult to available.

3. Interior ballistic equations based on theexperimental pressure-apparent burningrate model

There are many interior ballistic models historically３），10）.Three levels of interior ballistic models employed are theIBHVG2 lumped-parameter interior ballistic code ; theXKTC one dimensional (1-D), two-phase flow interiorballistic code ; and NGEN3 multidimensional, two-phaseflow interior ballistic code.

In China, the classic interior ballistic model similar to theIBHVG2 is extensively employed in prediction of the gunperformance. It can do a better job of gun performancethan the two-dimensional, two-phase flow models. Itcontains equations representing the physics of the guninterior ballistic process. In general, the equations of theconservation of mass, momentum, and energy can be usedto describe the physical process.

These equations are presented as follows.Equation of form function

������������� �������

������� �� ������� (10)

where �is the relative burnt thickness. � is the relativeburnt thickness for multi-peroration propellant at all burntpoint.�,�,�are the characteristic coefficients related onlyto the geometrical shape and dimensions of grain.�,�arethe characteristic coefficients of multi-perforationpropellants in splintery burning stage.

Burning rate equation

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�������

���� (11)

Equations of projectile motion

��������� (12)

����� (13)

where� is projectile weight,is the velocity of projectile,� is the bore area,�is the coefficient of secondary worksin terms of mean pressure in the bore. �is the projectiletravel.

Energy equation

������� ������

���� (14)

where

����� (15)

����� ���

����������� � (16)

�������

�(17)

���

��(18)

������

(19)

Where, the variable �is the specific heat, �� is the borevolume, � is the propellant charge weight (kg), � is theloading density of charge in the bore, �� is the traverse ofprojectile in the bore, �� is the propellant density, �is thecovolume,��is the resistance coefficient.

The mass burning rate of propellant in classic interiorballistic is mainly based on the assumption that the rate ofburning is the same on all surfaces of grain, which resultsin the acceptance of geometric form function. However, ashas been previously discussed, it is not true in the actualcombustion process. Furthermore, we will show adeviation example in the section 3 for three kinds ofpropellants which has almost the same neutral burningbehaviours. The equations of from function and burningrate of propellant in the classic interior ballistic arereplaced by the experimental pressure-apparent burningrate model described in this paper. The new equations forinterior prediction method are as follows.

Experimental pressure-apparent burning rate model

������� ���� ���� �

(4)

Relative pressure impulse equation

��������������� (20)

Equation of projectile motion

��������� (12)

Equation of projectile velocity

����� (13)

Energy equation

������� ������

���� (14)

4. Comparison with traditional burning ratemodel

4.1 Experimental propellantsTable 1 lists the size of three kinds of single-base

propellant grains in closed vessel test19）. For theconvinence of study, the three kinds of single-basepropellant are designed with special size of roughlyconstant burning surface, i.e. nearly neutral burning. Intable 1, � is outside diameter, �is inside diameter, ��� isweb size of propellant, �� is propellant density, and ��ispropellant length. The experimental loading density inclosed bombs is 0.2 g·cm－３. The igniter is 2# Nitrocellulose(NC) of 1.0 g.

4.2 Deviation of propellant mass fraction burntIn order to compare the effect of shape of propellant on

the mass fraction burnt, the deviation in the wholecombustion procedure calculated using the model in thispaper and using traditional form function method isshowed in Figure 1. Figure 1 (a), (b) and (c) are �� �curves of 12/1, 13/7 and 13/19 single based propellants,respectively. The curve 1 in the figure is �� curvescalculated by the traditional form function method. Thecurve 2 in the figure is �� � curve calculated byexperimental pressure-apparent burning rate model.

As seen in Figure 1, the three kinds of single-basepropellant showed a roughly constant burning surface.However, there is a significant deviation of burning gasgeneration rate between the traditional method and thenew method in this study. For example, for 12/1 tubularsingle-base propellant, when it burned, the regressiveburning properties of burning surface was minimum, andthe���curve calculated by the traditional form functionmethod was similar to a diagonal line. With the increasingof the relative burnt thickness, the gas generationgradually increased to the maximum. However, the

Table１ Size of three kinds of single based propellants grain.

Propellant �[mm] �[mm] ���[mm] � [mm] ��[g · cm－３]

12/1 single-base tubular grain 3.09 0.58 1.25 3.99 1.59613/7 single-base cylindrical grain 7.61 0.63 1.43 9.59 1.583

13/19 rosette single-base cylindrical propellant 10.05 0.48 1.27 11.51 1.593

Zhenggang Xiao et al.4

propellant mass fraction burnt calculated by the newburning model indicates the rapid increase in the initialburning stage, then gradually to the maximum earlierthan that calculated by the traditional form functionmethod.

Figure 2 shows the����curve of three kinds of singlebase propellants using the experimental pressure-apparent burning rate model.

As seen in Figure 2, it indicates that at the beginning ofpropellant combustion, as the increase of relative pressureimpulse, the gas generation increases, and then there is aplatform of gas generation with no more increasing later.At the later combustion phase, the gas generation ratebecomes slow. These numerical results are consistent withthe actual burning circumstance of propellant in the closebomb. Obviously, influence of the pressure on the burningrate of 12/1 single-base tubular propellants is much biggerthan the other two kinds of multi-perforation cylindricalpropellants.

5. Interior ballistic simulation based on theexperimental pressure-apparent burningrate model

Base on the above comparative research results, thenew experimental pressure-apparent burning rate modelwill be used to predict the interior ballistic performanceduring the actual gun firing. In order to validate the newinterior ballistic model incorporated with newexperimental pressure-apparent burning rate model, agun interior ballistic simulation compared with the classicinterior model based on the same gun and actual chargedata was conducted.

Given the following data of 100mm gun, Bore area : 81.8cm３ ; Case volume : 7741 cm３ ; Projectile travel : 474.30cm ; Projectile weight : 15.6 kg ; Propellant chargeweight : 5.6 kg.

The propellant charge is composed of double-basetubular propellant grains. Its size is listed in Table 2.

Figure２ ���� curve calculated by experimental pressure-apparent burning rate model.(1) : 12/1 single-base tubular grain ; (2) : 13/7 single-base cylindrical grain ; (3) 13/19 rosette single-basecylindrical propellant

(A)

(B)

(C)Figure１ Comparison of ���� curve of three kinds of

propellants.(A) 12/1 single-base propellant (B) 13/7 single-basepropellant (C) 13/9 single-base propellant calculatedby two kinds of different models : 1-Traditional formfunction method ; 2-experimental pressure-apparentburning rate model.

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[　　]

[ms－1 ]

[　 ]

The pressure and projectile velocity vs. time curve aspredicted by the new interior ballistic method and classicinterior ballistic method respectively is given in Figure 3.

The fitted mass fraction burnt function of double-basetubular grain is as follows.

������������������������������������

(21)

The experimental peak pressure is 325.0 MPa, and therecorded muzzle velocity 900.0ms－１. The classic interiorballistic prediction of peak pressure is 314.0 MPa, and thepredicted muzzle velocity 907.7ms－１. However, the newinterior ballistic prediction based on the experimentalpressure-apparent burning rate model in closed vessel ofpeak pressure is 305.8 MPa, and the predicted muzzlevelocity 903.9ms－１. Obviously, the two kinds of interiorballistic methods indicate almost the same predictedresults, although there are still some errors with theexperimental peak pressure and muzzle velocity. Theinitial gas generation pressure increases more rapid thanthe predication by the classic interior ballistic. However,the peak pressure coming up later. Simulation resultsshow that the new methods based on the experimentalpressure-apparent burning rate model can be used topredict the peak pressure and muzzle velocity of gunpropellant charge.

The error of new interior ballistic method based on theexperimental pressure-apparent burning rate model isgoing to be discussed. From the equations of mass burningrate and pressure impulse, obviously, these rules arederived from the closed vessel which is just approximateto the gun bore combustion. Another important factor isthe pressure impulse introduced in this model. Although�� is a constant for a specific propellant under a certaintest conditions in closed vessel, it is not the same as thevalues in the gun. In the closed vessel,�� is the area underthe pressure-time curve along the time axis from the zero

point to the maximum pressure-time point. However, inthe gun, the actual �� is difficult to determine in advance.Therefore, the possible choices are controlled by fitting toobserved ballistic data.

6. Conclusion(1) A new experimental pressure-apparent burning rate

model is established which are different from thetraditional method of form function. The new model baseson the concept of relative pressure impulse. Thepropellant gas generation rate in the new model ischaracterized by ���� curve. By this new model, thecomplicated form function calculation is unnecessary, andthe deviation of web size from one grain to another graincan be avoided. Meanwhile, the influences of the pressureon burning rate during combustion process is considered,which can result in the synergistic effect of burningcircumstance to increase the gas generation rate in closedvessel and during the combustion process.

(2) Numerical calculation is conducted using theexperimental pressure-apparent burning rate model basedon the test data of three kinds of propellants. Meanwhile,the obtained numerical simulation results are comparedwith the calculated result using the traditional formfunction method. There is a significant deviation of the gasgeneration rate between them. The traditional calculationmethod depends on the uniform dimensions and shape ofpropellant, without considering the influences of thepressure on burning rate during combustion process.Actually, the grain web size varies from one grain toanother grain. The new method is not dependent on theform function calculation, and considers the ssynergisticeffect of pressure on the gas generation rate in closedbomb and during the combustion process. Therefore, it ismore practical to calculate the apparent burning rate.

(3) Interior ballistic prediction results based on the newexperimental pressure-apparent burning rate modelshowed good agreement with the classic interior ballisticprediction though the agreement with the experimentalresults is still imperfect. The new interior ballistic methodis not necessary to calculate the form function ofpropellant. Therefore, it is easy to be applied to predictsome special gun propellant charge while the classicinterior ballistic can’t be employed.

AcknowledgementThis work was supported in part by the National

Science Foundation of China (No. 51376092).

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Table２ Size of grain in 100mm gun propellant charge.

Propellant [mm] �[mm] ���[mm] ��[mm] � [g.cm－３]

18/1 double-base tubular grain 4.95 1.75 1.60 40.75 1.6

Figure３ Mean pressure and projectile velocity vs. timecurves as predicted by classic interior ballistic andnew interior ballistic method based on experimental-apparent burning rate model.

Zhenggang Xiao et al.6

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