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34rd International Annual Conference of ICT June 24 - June 27, 2003

Karlsruhe Federal Republic of Germany

A MATHEMATICAL MODEL FOR COMBUSTION OF ENERGETIC POWDER MATERIALS

Ms. Alice Atwood (1), Ms. Eva K. Friis (2), Dr. John F. Moxnes (3),

(1) Naval Air Warfare Center Weapons Division China Lake, CA, USA Telephone: (760) 939-0203, E-mail: alice.atwood@navy.mil

(2) Nammo Raufoss AS P.O: Box 162,N-2831 Raufoss, NORWAY Telephone+4761153609 E-mail: eva.friis@nammo.com

(3) FFI (Norwegian Defence Research Establishment) P.O. Box 25, N-2007 Kjeller, NORWAY Telephone+4763807514 E-mail: john-f.moxnes@ffLno

ABSTRACT

This article presents a mathematical model of combustion which can be used in numerical codes. We also present experimental results where a convective burning front moves behind the pressure front during combustion in long tubes. The mathematical model is visualized in analytical studies of burning in a closed bomb.

The detailed kinetic cherrlical reaction mechanisms which occur during combustion of energetic materials are complicated and difficult to model from first principles. By studying the combustion mechanism in a variety of experiments, we have constructed a mathematical model based on average quantities, which can then be implemented (as a subroutine) into more general numerical computer codes (Autodyn, Hi Dyna and Hi Nike). The combustion model can be linked with granular mechanical models based on the theory of continuum mechanics. The model enables simulation of the velocity and the pressure in a combustion front, and the model can thus be used to study the corresponding mechanical effect on surrounding materials.

INTRODUCTION

This article presents a mathematical model of combustion which can be used in numerical codes. During combustion of energetic materials the velocity of the reaction front usually moves behind the mechanical pressure front originating from the burning solid material. The detailed kinetic chemical reaction mechanisms which occur during combustion are complicated and difficult to model from first principles. By studying combustion both in a closed bomb and in a long tube, we have constructed a mathematical model based on average quantities, which can be implemented into more general numerical computer codes (Le. Autodyn, Dyna and Nike). The combustion model can be linked with the granular M-O model [1] and [2], which is based on the theory of continuum mechanics.

The general idea of the burning model is to facilitate simulation of burning of energetic materials. Traditionally this is achieved by using general two-phase hydro code models. Our approach is different. By modeling the reaction kinetics on the meso scale as a subroutine in finite element codes, the burning of energetic materials can be simulated without using general two-phase models. In the subroutine each numerical cell may contain both solid and gaseous material. The subroutine calculates the different solid and gaseous masses and densities as well as the burn rate of the solid particle surface, the burn fraction, temperature, gas pressures and the propagation of the burning front. The average pressure in a cell as a result of the combustion is used as input to the general code to calculate the mechanical effect of the combustion on surrounding cells. The model can thereby be used as a tool for studying the combustion of energetic materials and the corresponding mechanical effect on surrounding materials, such as fragmentation.

In an earlier paper [3] simulations of mechanical pressure waves caused by impacting a piston into powder materials, were presented. A hot-spot model was also presented [1], [4] and [5]. The hot-spots are caused by friction between the granular particles during compaction. By firing a piston into the energetic material, experimental ignition thresholds were observed by using a high-speed camera. Good agreement was found between simulations of the experiment using the hot-spot model and the experiments. This paper addresses the burning of the energetic material after ignition.

A short outline of the paper: First the mathematical burning model is presented. Next section presents analytical studies of burning in a closed bomb. Finally, the last section presents theory and experimental results of the general situation where a convective burning front moves behind the pressure front during combustion in long tubes.

THE MATHEMATICAL MODEL

Each numerical cell may contain both solid and gaseous material. This is illustrated in figure 1.

Numerical cell Solid with volume V Gas'-+-WfIH.

Figure 1: Illustration of solid particles with gas in between in a numerical cell.

The following equation apply for the mass of the solid and the mass of the gas:

de! Ms(t) +M g (t) = M = Ms(tO) +M/to) (1 )

where,

M s (t) : Total amount of solid in the volume V at time t

M g (t) : Total amount of gas in the volume V at time t

M : Total amount of solid and gas

to : Initial time zero

The different volumes and densities are defined in equation (2) - (7):

de! V =V +V : V is the volume of the solid and V is the volume of the gas (2)s g s g

de! M (t) Ps(t) = s : density of the solid (particles) (3)

Vs(t)

de! M (t)Pg (t) = g : density of the gas (4) Vg(t)

_ de! M (t)Ps (t) = s : average density of the solid in the total volume V (5) Vet)

de! M (t)P (t) = g : average density of the gas in the total volume V (6)g Vet)

" .

If the cell is subjected to high external pressure the solid particles will get compacted and thus the density Ps (t) will increase. The following relation is proposed in order to model this phenomenon:

mod ~ (Ps (t)) _ Ps(t) = Ps(tO)+ (_ ) Ps(t) (7)

K Ps(t)

where Ps (to) is the initial density of the particles, ~ (Ps (t)) is the average solid pressure as -

measured according to the continuum mechanics approach and K(ps (t)) is the elastic bulk modulus.

The mathematical model is equipped with an equation of state for the gas and the solid of the form:

mod _ mod _ Pg = f g (pg ,Tg ), P s = fs(Ps,Ts ) (8)

where Ts : average temperature measured according to the continuum mechanical approach.

Observe the asymmetry. The equation of state for the gas is given as a relation between the pore pressure and the density inside the pores, while the equation of state for the solid is given as a relation between the average pressure and the average solid density.

Consider a cell, which is burning. The burn fraction F(t) and the burn fraction rate are defined in equation (9). The burn 'fraction of the solid material describes the fraction of the solid material that has reacted.

F(t)~ Ms(to)-Ms(t) =1- Ms(t) , (9) M s(to) M s(to)

The burning velocity, b(t) , of a burning front normal to the burning particle surface is given by a function H which typically is dependent of the gas pressure:

. mod de! bet) = H(fJPg ), where fJ = 1/Pa (10)

In general the following model is assumed:

\

de! where fJ = 1/ Pa (11 )

where H (~Pg ) expresses the velocity of the burning front normal to the burning surface, and where G(F(t expresses the size and shape of the burning surface.

This model is not obvious, although it is easily proven when the particles are identical spheres (see later in this article).

Assume that the gas in the cell decompose, then if the heat conduction between the solid and the gas is neglected, the following equation is descriptive during a short time interval:

~(cvgMg (t)Tg (t) = cvgTgO~g (t) - Pg (t)~V = cvgT~~g (t), (12) ~ (cvsMs (t)~ (t) ) =cvs~ (t)~s (t) - div (r(t)~ii; (t) ) =cvs~ (t)~ s (t)

where ~V and ~ii; (t) is zero for a non deforming closed cell. Also cvg [J / kg / K]: Heat capacity of the gas pr unit mass

cvs [J / kg / K]: Heat capacity of the solid pr unit mass

TgO [K]: Decomposition temperature for the gas

~ii; (t) : Lagrangian displacement of the boundaries of a cell during a short time interval r(t): Lagrangian stress tensor

(13)

Observe that if the initial temperature of the gas equals the decomposition temperature, the temperature of the gas equals the decomposition temperature for all times.

BURNING OF IDENTICAL SPHERICAL PARTICLES IN A CLOSED BOMB

Consider now, as an example, the burning of a granular press loaded explosive consisting of particles described as identical spheres. Assume that the volume of the cell is constant (closed bomb). The following equation is then valid if the burning rate of the outer surface is given by bet) = -f(t) :

mod Ms (t) = N 41l" r(t)2 Ps (t)f(t) (14)

where N is the total number of particles within the total cell volume V, and b(to) =a is the initial radius of the particles. The reaction ratio F(t) is now given from (9) and (14) as

(15)

To simplify, assume that the densities of the particles are constant, Le. Ps (t) = const. = P (to). It then follows from (15) that s

F(t) =1- r(t)3 3 ::::> ret) = aR./(l- F(t) (16) a

Inserting (16) into (15) gives that

3(1- F(t))2/3 F(t) = f(t) (17)

a

Compare the relation in (17) with the more general relation in (11). It then follows that

3(1-F(t))2/3 . G(F(t)) = in this case when bet) =-f(t)

a

Case with Constant Particle Burn Velocity, f(t) : As a first almost trivial test, a case where f(t) =constant = Vo is calculated analytically. In this case the time development of the burn fraction is independent of any assumed equation of state of the solid or the gas. The position of the burning front is then given by

f(t) = -vo ::::> ret) = a -vot, when t ~ a/va (18)

Inserting (18) into (17) and (16) gives:

2/3 . 3vo (1- F(t) ) vot 3F(t) = , ~ F(t)=l-(l--) , whent~alvo

a a (19) F(t) =1, when t > alvo

The result of an analytical calculation of the burn fraction as a function of time for a cell with

constant volume, when a = 10-4 m, ret) =constant =Vo =1ml sand Ps (t) =const. =Ps (to) is shown in figure 2.

~0.6 - _._... ., ; ---- -- .. " L .

0+------+------1------+------1

O.OE+OO 3.0E-05 6.0E-05 9.0E-05 1.2E-04 Time [5]

Figure 2: Analytical calculation of the burn fraction as a function of time for a cell with

constant volume when a = 10-4 m, ret) =constant =Vo =1ml sand Ps (t) =const. =Ps (to).

Calculation of the Different Densities and the Gas Pressure

Assuming again burning of en energetic material in a closed bomb. From equation (1) - (6) and (9) it follows that: 15 (t) =Ms(t)IV = (1- F(t))Ms (t)IV =(1- F(t))(M - M g(to)IVs

=(1- F(t))p(t) - (1- F(t))Mg(to)IV, M -M (t)15g(t) =M g(t)IV = V s = pet) - 15 (t) =p(t)F(t) + (1- F(t))Mg(to)IV, (20)s

Ps (t) =Ps (to) +. (:: ) p,(t)K Ps (t) () M ()I M -Ms(t) = p(t)F(t)+(1-F(t))Mg(to)IV

Pg t = g t Vg =V[l- V IV] [1- 15 (t)1 Ps (t)]s s

Ms+Mgwhere pet) =------:::....

V

If the initial mass of the gas in the pores is very small, Le. M g (to) :::::: 0, and the particles as

such are not compacted, Le. Ps (t) = constant equation (20) gives (21 )

Equation (20) and (21) then gives

-Ps(t) =(1- F(t))p(t) =(1- F(t)) Ps(to), -P (t) =F(t)p(t) =F(t) Ps (to),g

(22)-

P (t) = p(t)F(t) = Pg (t) g [1-(I-F(t))p(t)/Ps(t)] (1-Ps(t)/Ps(to))

Ps (t) =Ps (to)

Figure 3 shows an analytical calculation of the four densities as a function of time in a case

where pet) =p(to) = 1.5g / cm 3 and Ps (t) = Ps (to) =1.82g / cm 3

2 -,----------,------.......,----.--------, 1.8 """''''''-''''''. .-." .." " ,"" . 1.6 - -.; - ./1s(O' .

1.2E-049.0E-OS

75 '(t) _ .. / .. ,s ..

3.0E-OS

t'5' 1.4 E Co) 1.2-C);:1

;t:::

~ 0.8 Q)

0.6

6.0E-OS Time [5]

Figure 3: Analytical calculation of the four densities as a function of time in a case where

pet) =p(to) =1.5g / cm 3 and Ps (t) =Ps (to) =1.82g / cm 3

The equation of state of the gas may be given as:

(23)

where n' , Rand Pg max are constants. The unit for n' is mol/kg.

Figure 4 shows analytical calculations of the gas pressure, Pg , as a function of time for the

same case as shown in 'figure 3. P max is assumed to be 1.82 g/cm3 . In addition g

P (t)(I- Ps(t)]as given in equation 28 is plotted. This expression is the effective gas g Ps (t) pressure, as explained later in the paper.

4.00E+06 --,-------,-..------..--~-------,

3.50E+06

........ 3.00E+06 co a.. '(; 2.50E+06

~ :::J ~ 2.00E+06 Q)

~

en c- 1.50E+06 co (!) 1.00E+06

5.00E+05

O.OOE+OO """1'--------i---------r---------.-------l O.OE+OO 3.0E-05 6.0E-05 9.0E-05 1.2E-04

Time [5] Figure 4: Shows an analytical calculation of the gas pressure, Pg , as a function of time for

the case shown in fjgure 3. Pg max = 1.82 g/cm3, nRTgPg(t) =1100 Pa. The other curve shows the effective gas pressure as given in equation 28.

CONVECTIVE BURNING

Convective Combustion Experiment

A schematic of the convective combustion experiment is shown in Figure 5. The experiment was designed to resemble the configuration used in the experiments which incorporated mechanical initiation [3]. The energetic material is confined in a 2D3.2-mm (eight-inch) Lexan tube of 25.4-mm (one-inch) internal diameter and 76-mm (3.D-inch) outer diameter.

'

Piezoelectric transducers were placed at the top and bottom of the tube as indicated in the figure.

The pressure data were measured using Kistler model 607e3 dynamic gages, and Kistler model 504E4 amplifiers (Type 545A1 filters). The frequency response of the pressure transducer is reported as 50 kHz, with a 1.5-psec rise time. A Nicolet, Multipro digital oscilloscope was used to collect the pressure time history data as well as the temporal fiducial record. The digitized Nicolet records were imported into Igor Pro for conversion of the signal from voltage to pressure.

Red Dot smokeless powder, shift 2, lot 733-12/94, ignited by a Reynolds, SO-80 igniter, was used to ignite the porous bed. The Red Dot was housed in a stainless steel basket, as shown in the Figure 5. The igniter was separated from the porous bed by a 1.27-cm (0.5-inch) ullage. This combination of igniter aid and free volume were selected to provide minimal bed disturbance and maintain ignition.

SO-80 HEAD

TRANSDUCER

IGNITOR BASKET

POROUS BED

Figure 5: Schematic of Convective Combustion Apparatus.

High-speed motion picture photography was used to evaluate the ignition and combustion event down the length of the tube. Two high-speed Photec IV motion picture framing cameras viewed the experiment 1800 from each other. The cameras operated at approximately 8000 pictures per second in the quarter framing mode resulting in an overall framing rate range between 15,500 and 36,000 pictures per second. This framing rate resulted in an interframe picture time of 28 to 65 psec for these experiments. Both cameras were 'fitted with a 150-mm telephoto lens and operated with an aperture setting of f8.

The combustion rate data were determined from the high-speed motion picture film with the aid of a Vanguard Motion Analysis System. A Reynolds header mounted at the base of the porous bed arrangement was fired with closure of the firing circuit and provided a temporal fiducial for data collection.

Flame propagation and pressure data for the HMX based explosive PBXN-5 at 74 percent of theoretical maximum density (TMD) are plotted in Figure 6 a) and b). The pressure data are appended with the flame location data. A maximum flame propagation rate of 553 m/s was measured, while the pressurization rate, as determined 'from the two pressure signals was 1050m/s. These data indicate that the pressurization wave is moving nearly two times faster than the luminous burn front.

PBXN-5 at 74tfc, TMD OOl019E

200 1001019 PBXN-5 @ 74tfc, TMDI 160

I

I 1553m1~1/

I

150

~ 120

~ 100 l8 @ a.

50

o 4.3 4.5 4.7 4.9 5.1 5.3

Tube leak/break - top gage - bottom gage

o-l--l------a;;;;;:;==:.--~

TIM:E-~ 2.5 3.0 3.5 4.0 4.5 5.0 Time - msec

a) b)

Figure 6: Flame propagation and pressure data for PBXN-5 at 74% TMD.

Mathematical Model

As seen in the experiment, during combustion (not detonation) of energetic materials the velocity of the reaction front usually moves behind the mechanical pressure front originating from the burning solid material. The ability to model this phenomenon is important if the main physical mechanisms are to be captured in the mathematical model. The velocity of tile burning 'front is modeled according to the following relation

(24)

where y and Pmax and Ps bulk are constants. p s bulk is the bulk density of uncompacted

powder. Observe that

Vi CPs ,Pg ) ~ H(/lPg ), when 15s ~ 00, (25)vJ15s' Pg ) = (1 + y)H(/lPg ), when 15s =15sb'

The first case reflects a situation for a highly compacted material with only few scattered

pores, like in a cheese. The burning will then be like a cigarette. The second case reflects the situation for a more porous material, where the burning mechanism is convective.

Let now LU be the distance between two neighbor cells where one is ignited. This is

illustrated in figure 7.

has ignited

cell x

Figure 7: Illustration of the distance LU between two neighbor cells where one is ignited.

Based on the calculated velocity vi CPs' P ) of the ignited cell and the distance LU betweeng this cell and the not ignited neighbor cell, one can calculate an ignition time delay:

T(t) =LU / Vi (15s (t), P (t) ) (26)g

We assume that for a given cell it is possible to track the time of ignition of the cell and keep

it as a variable tig . The ignition criterion for the neighboring cell is then given by:

t -tig ~ T(t) = LU/Vi (15s(t), Pg (t)) (27)

".,,

The hot-spot model [1], [4] and [5] is used as an additional ignition criterion. If the hot-spot temperature in a cell, due to rapid compaction of the cell, is higher than the ignition temperature, the cell also gets ignited.

By assuming that the burning of the particles in a cell contribute significantly more to the gas pressure than any gas coming from neighbor cells, the gas pressure in convective burning may be calculated as for the closed bomb. The relation is given by equation (23).

The pressure that is used to calculation of the cell deformation is given by equation (28) as

Pc (t) = Ps(t) + P (t) (1- ,os (OJ (28)g Ps(t)

where the mechanical average solid pressure and the effective gas pressure contributes to the cell deformation additively. The gas pressure and the effective gas pressure are illustrated in Figure 4.

CONCLUSIONS

This article presents a mathematical model of combustion w~lich can be used in numerical hydro codes. We also present experimental result where a convective burning front moves behind the pressure front during combustion in long tubes. The mathematical model is visualized in analytical studies of burning in a closed bomb.

By studying the combustion mechanism in a variety of experiments, we have constructed a mathematical model based on average quantities, which can then be implemented (as a subroutine) into more general numerical computer codes (Autodyn, Hi Dyna and Hi Nike). The combustion model can be linked with granular mechanical models based on the theory of continuum mechanics. The model enables simulation of the velocity and the pressure in a combustion front, and the model can thus be used to study the corresponding mechanical effect on surrounding materials.

The model is currently being implemented into the explicit numerical code Hi Dyna2D. Comparison between simulations and experimental results will be presented in future papers.

REFERENCES

[1] Diep, Q. B., Friis, E. K., Moxnes, J. F., Str0mgard, M., 0degardstuen, G.:"Simulation of the Compaction of Energetic Materials", 33rd International Annual Conference of ICT, Karlsruhe, Federal Republic of Germany, June 25 - 28, 2002

[2] Friis, E., Moxnes, J. F.: "Establishing Material Data of Pyrotechnic Powder Materials by use of Inverse Modeling", 28th International Pyrotechnincs Seminar, Adelaide, Australia, November 4 -9,2001

[3] Atwood, A., Curran, P., Moxnes, J. F., 0degardstuen, G.: "Mechanical Properties of a Porous Material Studied in a High Speed Piston Driven Compaction experiment", 30th International Annual Conference of ICT, Karlsruhe, Federal Republic of Germany, June 29 - July 2, 1999

[4] Moxnes, J. F., 0degardstuen, G.: "Ignition of a Pyrotechnic Powder by "Hot-Spots" and Ordinary Adiabatic Compression", Karlsruhe, 32nd International Annual Conference of ICT, Federal Republic of Germany, July 3 - July 6,2001

[5] Friis, E.K., Moxnes, J.F., 0degardstuen, G.,: "Simulation of Hot-Spots in Energetic Granular materials Created During Launching of Ammunition" 20th International Symposium on Ballistics, Orlando, Florida, USA, September 23 - 27, 2002