thermal explosion of powder mixtures: numerical approach

Thermal Explosion of Powder Mixtures: Numerical Approach

Claudio Zanotti*, Alessandra Tacca, Marzio Monagheddu, Noemi Bertolino and Piero Giuliani

Istituto per l×Energetica e le Interfasi-Consiglio Nazionale delle Ricerche, Sezione di Milano, Via R. Cozzi 53,20125 Milano (Italy)

Abstract

A numerical approach to investigate the heating and theignition of powder mixtures by radiant energy is presented. Theignition study is based on the possibility of separating theinitiation transient from the propagation process, by operatingin thermal explosion mode.Binary systems such as Ti-Al, Ni-Al and Ni-Ti, characterized bydifferent compounds, were taken as case study for the exper-imental tests and the numerical simulations. Data concerningheating and ignition transients were estimated by best fitting theexperimental results. The energy stored in the pellet up to theignition temperature can be calculated. Radiant absorbancedependency on sample temperature at the laser wavelength wasevaluated for each studied system.

Keywords: Powder Mixtures, Radiant Ignition, CombustionSynthesis, Numerical Computation.

1 Introduction

Powder mixtures are generally used to produce advancedmaterials by Self-propagation High-temperature Synthesis(SHS). This methodology is characterized by two mainaspects: ignition of the chemical reactions and then selfpropagation of the reaction front [1].The ignition transient is the basic step that, controlling the

exothermal reactions rate, allows to generate two regimes:explosion or propagating mode. In the second case thepropagation rate of a SHS process depends on the temper-ature distribution in the sample at the ignition time [2]. Thesample reacts in thermal explosion mode once the entiresample is heated up to the ignition temperature and thereaction starts everywhere in the mixture. On the contrary,once the reactants are locally initiated, by an externalheating source, a combustion wave moves throughout thesample and the process becomes self-propagating [3].Thermal theory has been used in dealing with the ignition

process adopting a simple one-dimensional model [4] basedon the definition of the heated layer thickness wherechemical reactions proceed and induce the sample ignitionof the following layer. This theory assumes that, beforeignition, no phase transformations or chemical reactionstake place and thermo-physical properties are considered tobe constant in the sample.

Following this approach, theoretical studies were thendeveloped to better understand the ignition process. Someauthors based their work on the homogeneous premixed-flame theory [5].Moreover, ignitionphenomenon studies bythe heterogeneous theory, including the dependence of thereaction front propagation rate on the particle size, havebeen reported [6].Zhang et al. [7] and He et al. [8 ± 9] have elaborated a

model based on dimensionless energy and mass continuityequations. They found that the most important parameteron the ignition of a self-propagation reaction is theactivation energy, followed by the rate of local heatgeneration and the rate of surface heat loss by convectionand by radiation.Shen et al. [10] studied the sample ignition by a CO2 laser

system of NiAl mixture and approached the problem tosolve the one-dimensional governing equation. They ob-tained the analytical solution of the temperature field by amethod also used in the study of solid propellant ignition[11]. Investigations on initiation processes of Ni-Al, Ti-AlandNi-Ti systems, reacting in thermal explosionmode, havebeen carried out by Lee et al. [12]. They proposed a simplemathematicalmodelwith the aimof investigating the effectsof the heating rate on the temperature profile as well as thefraction of the reactant consumed. In this model precom-bustion, ignition, combustion temperature and precombus-tion duration have been taken into account.

2 Modeling of the Heating and Ignition Process

In the present work a model, describing the volumetricsample heating and the ignition transient of intermetalliccompounds heated by radiant energy which react in thermalexplosion mode, is developed. Computations are based onthe following assumptions:

± the radiant energy is the only external heating source,± the temperature distribution in the sample is uniform,± sample heat capacity is dependent on temperature,± phase transition of one or both reactants is accounted,± melting of the reactants is included,± ignition process is driven by the heat release of solid/solidand liquid/solid reactions,

± melted unreacted powder solidification is considered,* Corresponding author; e-mail: [email protected]

112 Propellants, Explosives, Pyrotechnics 29 (2004), No. 2

¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim DOI: 10.1002/prep.200400036

± heat losses by free convection and radiant emission areaccounted.

The ordinary differential equation describing the wholephenomenon can be written as:

mCpdTdt

� �1Pa � �2Pr � �3Ps� � � Pc � Pe � �4Pm� � �1�

with the initial boundary condition:

T(t� 0)�Ta

and where:Pa ��(T)I0 is the power absorbed at 10.6 �m, �� a� bT isthe sample surface absorbance, Pc� hAl(T�Ta) is thepower loss by free convection, Pe� ��(T)Al(T4�T4

a) is thepower loss by radiation, ��AA�A(T)�AB�B(T) is the pellettotal emissivity, Pm �Qm/�tm is the power absorbed formelting, Ps�Qs/�ts is the power released during solid-ification,Pr is the thermal power release by reactions.� is thefunction selecting the contribution of each term, defined as:

�1 � 1 t � tl�off

0 t � tl�off�2 � 1 Treac � T � Tmax

0 T � Treac�T � Tmax

��

�3 � 1 tisol � t � tfsol

0 t � tisol� t � tfsol�4 � 1 timelt � t � tfmelt

0 t � timelt� t � tfmelt

��

The heat capacity of the green pellet, Cp, was evaluatedtaking into account the stoichiometry of themixture and thephase transitions of the reactants.The heating, ignition and reaction phenomena are

simulated considering that the thermal convective coeffi-cient of the gas, surrounding the suspended sample, waschosen as function of temperature at the operating pressure.The first order differential equation (1) is integrated using

a fourth order Runge-Kutta method. The whole process,including the reactions and the cooling transient after thelaser cut off, was best fitted varying the surface absorbancethat is considered a function of the pellet temperature.When the temperature of the pellet reaches a critical

value (Treac), themodel takes into account the power releasedue to solid/solid and/or solid/liquid reactions as a functionof the temperature.Once the experimental temperature history is numeri-

cally reproduced, the energy accumulated in the pellet canbe estimated as:

E*��t�

t� 0

[�1Pa �Pc�Pe��2Pr� �4Pm]dt (2)

Changing the upper limit time, t*, it is possible to calculatethe energy associated to different aspects of the process.Moreover, the model permits to simulate the heating andmelting of the pure elements to obtain data useful to analysethe behaviour of the powdermixtures. Finally, the heating orsustained reaction can be stopped at every defined sampletemperature, generating the cooling transient.

3 Experimental Tests and Numerical Results

Different powder mixtures: Ni :Al� 1 :1, 1 :3, 3 : 1,Ti :Al� 1 :1, 1 : 3, 3 :1, Ti :Ni� 2 :1, 1 :1, 1 : 3were compactedin pellets with a green density of 2.2 ± 5.1 g/cm3 for Ni-Al,1.8 ± 2.8 g/cm3 for Ti-Al and 3.8 ± 5.4 g/cm3 for Ni-Ti.The experimental tests were carried out irradiating, by a

CO2 laser, small cylindrical pellets of 6 mm in diameter and0.8 ± 1.2 mm in height, characterized by a thermal conduc-tivity ranging into 10 ± 100 W/mK.The long heating times aswell as the sample characteristics (see Tab. 1) permit to havea uniform temperature distribution in the whole sample andthe ignition can be considered occurring in thermal explo-sion mode.A type S thermocouple with a bead size of 50 �m was

imbedded in the middle of the sample during the powdermixtures pressing. The pellets were suspended by thethermocouple wires to make negligible any heat losses byconduction. The experiments were carried out in a vacuumchamber in Ar environment (pressure of 100 ± 200 Pa). Thisexperimental configuration assures that the tests areperformed in accordance with the assumptions of thenumerical approach. Details on the experimental apparatusand results can be found in Refs. [13, 14].As shown in themodeling section, the knowledge ofmany

parameters and their dependence on temperature is neces-sary to integrate Eq. (1). However, the radiant absorbance,related to the tested powder mixtures, and the freeconvection coefficient dependency on temperature are notalways available in literature for the operating condition.Therefore, it is important to evaluate these parametersbefore approaching the numerical simulations concerningthe whole process. The effect of the free convection, as thedependence of the h coefficient on temperature was esti-mated by simulating the cooling transients of pure elementpellets characterized by no phase transition in the testedtemperature range. Among the powders used in this work, Tiwas selected because of its high solid-solid transition temper-ature (�� at 1170 K). Several heating and coolingtransients were performed imposing different heating ratesand a typical result is reported in Figure 1. A comparisonbetween the experimental and the numerical results permitstodefine the free convection influenceon thewhole transient.Once evaluated h(T), the radiant absorbance depend-

ences on temperature (defined as �� a� bT), could beestimated by best fitting the heating transients up to themelting point of the pure elements. In this way, it waspossible to obtain information about the radiant absorbancechange in presence of a liquid phase, if any. In particularpure elements, having theirmelting points close to amixtureignition temperature, were considered. Figure 2 depicts theresults when heating and melting of a pure Al pellet issimulated. In this case, in order to reproduce the exper-imental results, the absorbance of the liquid Al must beincreased at about 30%with respect to the value of the solidjust before the melting temperature.The knowledge of h(T) and �(T) allows calculating the

radiant absorbance for the powder mixtures. The data

Thermal Explosion of Powder Mixtures: Numerical Approach 113

¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim Propellants, Explosives, Pyrotechnics 29 (2004), No. 2

obtained by this procedure are used to simulate the heating/cooling sequences of samples heated up to a temperaturelower than the ignition one. Figure 3 depicts the comparisonbetween the experimental results and simulations of theTi :Al� 1 :3 mixture, showing how this numerical approachcan reproduce the experimental curve. In order to explorethe behavior of the Ti :Al� 1 :3 mixture during the Almelting, some experimental tests were carried out imposingthe laser cut off before the Almelting completion. A typicalresult is reported in Figure 4, showing the presence of awideplateau corresponding in the first part to theAlmelting (thelaser is on). After the laser cut-off, the solidification occursalong the final part of the plateau, followed by the samplecooling. As noticeable in the picture, the amount of thesolidified Al is lower than the melted one, suggesting that

some solid /liquid reaction occurred subtracting somereactant Al. In fact, XRD analysis [14] showed smallamount of TiAl3 formation, although the reaction heatreleased was not enough to start any self-sustained process.The Ti-Al system compounds can be prepared by combus-tion synthesis [15 ± 17], but the combustion process needssome pre-heating to ignite. In our experiments the radiantenergy heated the sample during the whole experiment,sustaining the reaction completion; thus, it was possible toquench the reaction at any time by cutting the laser off.Figure 5 shows the whole temperature history of a Ti :Al�1 :3 sample and indicates that the reaction kept going untilthe laser cut off. It is possible to identify many regions,associatedwith the specific phenomenaoccurringduring theexperimental test, which have specific terms in the differ-

Table 1. Sample characteristics and parameters featuring the heating, ignition, reaction processes. Density of the green pellet is alsoexpressed by its relative value �r� �/�th ¥ 100, where �th� �A ¥ �A� �B ¥ �B.

Stoichiometry �p �r I0 tig Tig dT/dt Eheat Eig a b ¥ 105 �m

g/cm3 % W s �C K/s kJ/molat kJ/molat ± 1/K ± -

Al [19] ± 100 ± ± ± ± ± ± 0.016 5.5 0.04Al 1.57 58 58 ± ± 37 17.65 28.1 0.02 3.2 0.04Ti [19] ± 100 ± ± ± ± ± ± ± ± 0.15Ti 2.69 60 20 ± ± 40 ± ± 0.13 7.5 0.15Ni 5.29 59 59 ± ± 36 ± ± 0.15 1 0.16Al :Ni� 1 : 1 3.4 59 50 7.5 595 108 ± 16.6 0.145 6 0.18

3.3 57 40 9.2 598 64 ± 16.5 0.1 6.7 0.143.6 62 30 10.5 590 55 ± 16.7 0.1 6 0.143.15 54 25 12.9 582 38 ± 16.5 0.105 2.5 0.122.9 50 20 24.1 582 24 ± 16.5 0.085 5 0.11

Al :Ni� 1 : 3 5.1 69 50 7.06 617 82 ± 17.5 0.115 3.4 0.134.8 65 50 7.15 614 83 ± 17.2 0.11 3.4 0.135.1 69 50 7.55 623 80 ± 17.3 0.112 1 0.12

Al :Ni� 3 : 1 2.6 61 50 8.7 630 67 17.2 25.4 0.135 4.6 0.162.2 52 50 10.4 628 60 16.9 24.7 0.11 3.6 0.13

Al :Ti� 1 : 1 1.86 52 20 23.2 654 39 17.6 22.8 0.08 4.2 0.101.99 55 30 12.0 655 71 17.9 23.1 0.047 3.9 0.071.86 52 40 11.8 659 75 18.0 23.2 0.07 2.8 0.092.5 69 50 8.2 641 102 17.6 22.8 0.055 4.3 0.081.93 54 50 6.0 640 135 17.6 22.8 0.045 3.1 0.06

Al :Ti� 1 : 3 2.76 68 30 6.2 647 110 17.9 20.6 0.15 0 0.152.15 53 40 5.1 651 127 17.7 20.3 0.115 0 0.1152.15 53 50 4.9 660 137 17.9 20.9 0.11 0 0.112.82 70 60 3.7 661 173 17.9 20.5 0.102 1.5 0.11

Al :Ti� 3 : 1 2.03 64 20 ± q ± ± ± 0.046 1.45 0.051.9 60 30 38.7 650 20 17.8 25.6 0.04 4 0.061.65 52 40 29.4 655 32 17.3 25.2 0.025 2.7 0.042.05 65 50 37.9 640 25 17.6 25.4 0.03 1 0.041.9 60 60 11.8 654 57 17.8 25.7 0.04 1.9 0.05

Ni :Ti� 1 : 1 4 60 25 18.1 940 52 ± 26.1 0.112 3.5 0.134.84 72 30 12.25 930 77 ± 26.7 0.115 3.2 0.134.8 72 40 6.36 931 146 ± 26.8 0.095 4.9 0.123.81 57 50 6.18 920 145 ± 26.3 0.144 1 0.154.31 64 60 5.29 920 166 ± 26.2 0.097 4 0.12

Ni :Ti� 3 : 1 4.85 62 20 ± q ± ± ± 0.115 1.3 0.125.43 70 35 17.5 940 53 ± 26.5 0.105 2 0.124.77 61 40 18.69 890 48 ± 25.3 0.085 1 0.094.8 61 54 6.92 904 131 ± 25.3 0.09 3 0.11

Ni :Ti� 1 : 2 4.38 73 30 ± q ± ± ± 0.09 3.8 0.114.88 82 30 ± q ± ± ± 0.065 5.2 0.104.94 83 40 12.1 945 79 ± 27.3 0.12 2.5 0.134.95 83 54 5.62 880 156 ± 27.2 0.122 1.9 0.13

The symbol q is used to indicate quenched reactions.

114 C. Zanotti, A. Tacca, M. Monagheddu, N. Bertolino and P. Giuliani

Propellants, Explosives, Pyrotechnics 29 (2004), No. 2 ¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim

ential equation (1). During the heating transient (0� t�26 s) the pellet temperature increased up to the Al meltingvalue. Then the sample melting process occurred (26� t�38 s) at constant temperature, and its radiant absorbanceincreased.As pointed out above, some liquid/solid reactionsbegan and this event was taken into account in the relevantsimulations. Once the Al melting is completed the pellettemperature increased again (38� t� 49 s) up to themaximum value where the laser was cut off. This transientis characterized by fast reactions but the amount of the heatreleased is not enough to reach the Ti melting temperature,thus, during the explosive reaction the pellet was composedby solid Ti, solid TiAl3 and liquid Al. After the laser cut off,during the cooling transient, Al solidified (60� t� 67 s),releasing its latent heat of fusion and keeping the pellettemperature constant. The cooling took the solid pellet

temperature down to ambient temperature.During all theseprocesses, even after ignition and reaction, the pellet did notchange its geometry, therefore it was possible to use thesame h(T) function to simulate the entire temperatureprofile.In Figure 5 are also reported numerical results obtained

by using different radiant absorbance dependencies ontemperature. In particular, curve A was calculated consid-ering that no reactions occurred and the absorbance did notchange when Al started melting. Also curve B is calculatedwithout considering reactions, but the absorbance changesince Al started melting. Finally, in curve C liquid-solidreactions were activated and the absorbance changed whenAl started melting.

Figure 1. Heating/cooling profile of a pure Ti sample: exper-imental and simulated curves.

Figure 2. Heating ramp and melting plateau of a pure Alsample: experimental and simulated traces.

Figure 3. Heating/cooling profile of a Ti :Al� 1 :3 powdermixture pellet: experimental and simulated curves. No reactionoccurred.

Figure 4. Heating, melting/solidification plateau and coolingprofile of Ti :Al� 1 :3 powder mixture pellet: experimental andsimulated curves. No explosive reaction occurred.



Some similaritieswere foundwhenmore reactive systems,such as Ni-Al, were studied. Different Ni-Al compositionswere tested [13], but only the Ni :Al� 1 :3 mixture (therichest in Al) showed a melting plateau, corresponding tothe lowest eutectic in the phase diagram. The ignitiontemperatures are close to the Al melting one (see Table 1),as depicted in Figure 6 for theNi :Al� 1 :1 composition, butdifferent for each composition. These system mixtures arefeatured by a phase transition during the heating transientdue to the Ni phase transition occurring at 630 K [18]. Thissolid-solid transition involves the NiCp change that must betaken into account. In Figure 6, the experimental trace iscompared with the simulated profiles, showing that the bestnumerical result was calculated by considering the Ni phasetransition.Different behavior is noticed when the studied system is

composed by elements having melting temperatures con-siderably higher than the ignition one such as Ni-Ti

mixtures. The Ni transition phase is important, also for thissystem, as well as some solid/solid reactions playing animportant role during the heating transient. In fact, as shownin Figure 7, the temperature profiles of the Ni :Ti� 1 ± 1mixtures, heated by different laser powers, indicate that attemperatures around 750 �C, solid-solid interactions be-tween Ni and Ti release a small amount of energy. After theignition, the thermal profile can not be simulated becausethe pellet changes noticeably its geometry. However, theheating transient and the ignition point were well distin-guished for all experimental tests, whatever compositionsand laser power impinging. The experimental temperaturehistories were compared with the calculated curves. InFigure 7, simulations with and without reaction beforeignition are reported, pointing out the importance of thesepre-reactions and their effect on the heating ramps.Data concerning the experimental results and the param-

eters obtained from some simulations are reported inTable 1 for each system and composition. Ignition temper-ature and ignition energy are tabulated and these values areconstant for each stoichiometry, independent of the laserpower. Two energies were calculated for the simulations ofthe compositions characterized by a melting phase beforeignition: Eheat is the heating energy until the melting pointand Eig is comprehensive of ignition and melting energies.For theNi-Ti systemEig includes the contribution of heatingand pre-ignition solid-solid reaction. Coefficients definingthe linear dependence on temperature of the radiantabsorbance, obtained by simulations, are compared withsome data found in literature [19].

4 Conclusions

A numerical model supported by a suitable experimentalmethodology for the investigation of the ignition phenom-ena in a SHS process was proposed. This approach may be

Figure 5. Heating, reaction and cooling of Ti :Al� 1 :3 powdermixture sample: experimental and simulated (A, B and C) curves.

Figure 6. Effect of the Ni transition on the heating of Ni :Al�1 :1 sample: experimental and simulated curves. The differencebetween the numerical results is due by considering or nor the Nitransition phase.

Figure 7. Heating transient of different Ni-Ti powder mixtures:experimental and simulated curves. Solid-solid interaction be-tween Ni and Ti need to be taken into account to reproduce theexperimental profile.

116 C. Zanotti, A. Tacca, M. Monagheddu, N. Bertolino and P. Giuliani

Propellants, Explosives, Pyrotechnics 29 (2004), No. 2 ¹ 2004 WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim

considered as a first step to decouple the initiation transientfrom the self-propagation process. Temperature historieswere experimentally measured for some binary intermetal-lic systems (Ti-Al, Ni-Al, andNi-Ti) andwell reproduced bynumerical simulations. Ignition temperatures were identi-fied as well as the heating and ignition energies wereestimated. It was verified that these parameters have typicalvalues characteristic for the system and the stoichiometry,but are independent on the laser power applied. Numericalsimulations permitted to define the dependence of theradiant energy absorbance on temperature for differentpowder mixtures. Finally, this numerical work pointed outthe parameters influencing the ignition phenomena andgave feedback information on the experimental configura-tion to perform an ignition study.

5 References

[1] J. J. Moore, H. J. Feng, Combustion Synthesis of AdvancedMaterials: Part II. Classification, Applications and Modelling,Prog. Mater. Sci. 1995, 39, 275.

[2] V. N. Vilyunov, V. E. Zarko, Ignition of Solids, Elsevier,Amsterdam, 1989, pp. 1.

[3] Z. A. Munir, U. Anselmi-Tamburini, Self-Propagating Exo-thermic Reactions: The Synthesis of High-TemperatureMaterials by Combustion, Mater. Sci. Reports 1989, 3, 277.

[4] A. G. Merzhanov, A. E. Averson, The Present State of theThermal Ignition Theory: An Invited Review, Combust.Flame 1971, 16, 89.

[5] V. V. Barzykin, Initiation of SHS-systems, Pure Appl. Chem.1992, 64, 909.

[6] A. Makino, Fundamental Aspects of the Heterogenous Flamein the Self-Propagating High-Temperature Synthesis (SHS)Process, Prog. Energy Combust. Sci. 2001, 27, 1.

[7] Y. Zhang, G. C. Stangle, Ignition Criteria for Self-PropagatingCombustion Synthesis, J. Mater. Res, 1993, 8, 1703.

[8] C. He, G. C. Stangle, The Mechanism and Kinetics ofNiobium-Carbon Reaction under Self-Propagating High-Temperature Synthesis-like Conditions, J. Mater. Res. 1995,10, 2829.

[9] C. He, G. C. Stangle, A Micromechanistic Model of theCombustion Synthesis Process: Modes of Ignition, J. Mater.Res. 1998, 13, 135.

[10] P. Shen, Z. X. Guo, J. D. Hu, J. S. Lian, and B. Y. Sun, Studyon Laser Ignition of Ni-33.3 at % Al Powder Compacts, Scr.Mater. 2000, 43, 893.

[11] A. A. Zenin, C. Zanotti, and P. Giuliani, Transport Pheno-mena in Combustion Book, Taylor & Francis, London 1996,pp. 1509.

[12] S.-H. Lee, J.-H. Lee, Y.-H. Lee, D. H. Shin, and Y.-S. Kim,Effect of Heating Rate on the Combustion Synthesis ofIntermetallics, Mater. Sci. Eng. 2000, A281, 275.

[13] M. Monagheddu, N. Bertolino, P. Giuliani C. Zanotti, and U.Anselmi Tamburini, Ignition Phenomena in CombustionSynthesis. An Experimental Methodology, J. Appl. Phys.2002, 92, 594.

[14] N. Bertolino, M. Monagheddu, A. Tacca, P. Giuliani, C.Zanotti, and U. Anselmi Tamburini, Ignition Mechanism in

Combustion Synthesis of Ti-Al and Ti-Ni Systems, Interme-tallics, 2003, 11, 41.

[15] H. Y. Yi, A. Petric, and J. J. Moore, Effect of Heating Rate onthe Combustion Synthesis of Titanium- Aluminum Interme-tallic Compounds, J. Mater. Sci. 1992, 27, 6797.

[16] Y. D. Hahn and Y. T. Lee, Combustion Synthesis of Ti-AlIntermetallic Compounds, in J. M. Capus, R. M. German(Eds), Series MPIF, Princeton, NJ, 1992, pp. 309.

[17] W. C. Lee, K. C. Hsu, and S. L. Chung, Combustion Synthesisof Ti-Al Intermetallic Materials, Int. J. Self-Prop. High-Temp.Synthesis 1995, 4(1), 95.

[18] Smithells Metals Reference Book , 7th ed., in E. A. Brandes,G. B. Brook (Eds), Butterworth-Heinemann, Oxford UK1992, pp. 8 ± 2.

[19] A. Sala, Radiant Properties of Materials, Elsevier, Amsterdam1986, pp. 246.

List of Symbols

AA, AB Irradiating area % of A and BAl Surface area of the sampled DiameterEheat Heating energyEig Ignition energyh Free convection coefficientI0 Laser powerQm Heat of fusionQs Heat of solidificationt Timetfmelt End of melting timetfsol End of solidification timetig Ignition timetisol Start of solidification timetl.off Laser off timetimelt Start melting timeT Sample temperatureTa Ambient temperatureTmax Maximum temperatureTreac Reaction temperatureTtrA, TtrB Transition temperature of A, Bm Pellet mass�tm� tfmelt� timelt Melting time�ts � tfso� tisol Solidification time�A, �B Emissivity of A, B�� Average absorbance� Pellet density�� Relative pellet density� Theoretical pellet density� Stefan-Boltzman constant

(Received December 4, 2002; Ms 2002/056)



thermal explosion of powder mixtures: numerical approach

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