Applied Thermal Engineering 130 (2018) 492–500

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate /apthermeng

Research Paper

A reduced order model for prediction of the burning ratesof multicomponent pyrotechnic propellants

https://doi.org/10.1016/j.applthermaleng.2017.11.0081359-4311/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: jjyoh@snu.ac.kr (J.J. Yoh).

Anirudha Ambekar, Jack J. Yoh ⇑Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Republic of Korea

h i g h l i g h t s

� Reduced order analytical model for burning rate of pyrotechnic compositions.� Linear burning rate of multi-component granular porous pyrotechnics predicted.� Conductive combustion regime with primary reactions occurring on the surface.� Technique accounts for the propellant conductivity, heat of reaction, and porosity.� Case study predictions for KClO4 and KNO3 based pyrotechnics reasonably accurate.

a r t i c l e i n f o

Article history:Received 19 July 2017Revised 29 September 2017Accepted 1 November 2017

a b s t r a c t

This study reports a reduced order model for the prediction of the burning rate of pyrotechnic composi-tions. The combustion process of most pyrotechnics is primarily driven by condensed phase reactions. Apriory estimation of the burning rate of pyrotechnics with multiple components may not be possibleusing the established methods. The study provides a simplified approach based on integral analysis ofa proposed combustion wave structure for estimating the burning rate when the pyrotechniccomposition, pure component thermo-physical properties, and thermo-kinetics parameter are known.The proposed combustion wave assumes a staged combustion process where the oxidizer undergoesdecomposition in a broad reactive zone while fuel combustion occurs in a thin surface region. Thisapproach takes account of the effective thermal conductivity as well as porosity of the pyrotechnicmatrix. The pyrotechnic compositions studied here are expected to burn conductively at atmosphericpressure with little or no overpressure. The phenomenology of the combustion process of energetic mate-rials is elucidated, and the reduced order model is validated through a case study.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Pyrotechnics are a particular class of fuel-rich energetic compo-sitions that are distinct yet closely related with other energeticmaterials such as explosives and propellants. While, the word‘‘pyrotechnic” is a literal Greek translation of the word ‘‘fireworks”,military pyrotechnics are considered separate from fireworks,which symbolize recreational purposes. Through the process ofcombustion, pyrotechnics may act as a source of gas, heat, light,smoke, sound, or any combination of thereof along with significantamount of solid phase combustion products. Typical pyrotechnicscompositions consist of fine powders of oxidizer, fuel, as well asvarious additives, which are thoroughly mixed and compressedto form a porous solid charge. The additives may act as binders,

stabilizers, catalysts, coloring agents, smoke dyes, and so ondepending on the intended application. These granular heteroge-neous porous composite energetic materials find utility in a rangeof applications such as chemical oxygen generators, pyrotechnicfasteners, automotive airbag inflators, thermites, material synthe-sis through solid-solid reactions, and fireworks.

The burning rate or the rate at which the combustion front pro-gresses through the pyrotechnic material is an important charac-teristic that determines their performance and safetycharacteristics. Generally, the combustion of solid energetic mate-rials is classified in four regimes [1] based on the burning rate viz.conductive burning, convective burning, compressive burning, anddetonation. A large number of pyrotechnic applications such asilluminating flares, display pyrotechnics, gas generators, and delayfuses operate in a conductive burning regime. The factors thatdetermine the burning rate of pyrotechnics have been discussedin detail in the literature [2]. However, a change in any of these fac-tors controls one or more of the three fundamental aspects viz. the

Nomenclature

Latin alphabetA frequency factor (1/s)Cp specific heat (J/kg K)Ea activation energy (kJ/mol)gðnÞ a weak function of the reaction order that varies be-

tween 1 and 2h0f enthalpy of formation (kJ/mol)DhR heat of reaction (J/kg)k thermal conductivity (W/m K)Lm latent heat of melting (J/kg)_m00 mass flux (kg/m2 s)_m000 mass consumption rate (kg/m3 s)_m000 average mass consumption rate (kg/m3 s)Ru universal gas constant (kJ/mol)_r burning rate (m/s)T temperature (K)T average temperature (K)Y mass fraction

Greek lettersa thermal diffusivity (m2/s)d reactive zone thickness (m)

/ volume fractionc fuel to oxidizer mass ratiog catalyst factorq density (kg/m3)

Subscripts0 ambient conditionsactual measured valuec combustion zoneeff effective valueF fuelf Flamei ith speciesKClO4 potassium perchloratemax maximum theoretical valueOx oxidizers propellant surface

A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500 493

chemical activation energy, heat of reaction, and the fraction of thegenerated heat that reaches the unreacted propellant. Any tech-nique for prediction of the burning rate must account for thesethree parameters with respect to the particular composition andconfiguration under consideration.

Certain pyrotechnic materials are designed to undergo uncon-fined combustion at atmospheric pressure with little or no pres-sure difference (overpressure) between the gaseous combustionproducts and the gases inside the pores of the unburnedpyrotechnic matrix. Various theoretical and experimental studiesregarding the burning rate of gasless pyrotechnics [3–5], con-fined deflagration of porous nitramine propellants [6,7], andcombustion of metal-halocarbon pyrolants [8,9] have beenconducted in the past. However, other than certain reviews[10–12] and experimental measurements [13,14] very few stud-ies addressing the estimation of the unconfined burning rates ofa granular multicomponent pyrotechnic material can be found inthe literature.

The methods for the approximate solution of the governingequations of the theory of combustion of homogeneous gaslesssystems, which were intensively investigated during at least thelast 50 years, are well-known and have been described in variousjournal articles as well as books [15,16].

Classically, the burning rate of solid materials have beenexpressed as a function of the material properties through theoriesproposed by Zeldovich [17] as well as Khaikin and Merzhanov [18].The Zeldovich expression given in Eq. (1) while relationship pro-posed by Khaikin and Merzhanov is shown in Eq. (2).

_m00 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2qkDhR

C2pðTs � T0 þ Lm=CpÞ2

RuT2s

Ea

!Ae�Ea=RuTs

vuut ð1Þ

_r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikARuT2

c

qEaDhRgðnÞ e�Ea=RuTc

sð2Þ

Waesche and Wenograd [19] report on calculation of solidpropellant burning rates from condensed-phase decompositionkinetics. They conducted pressurized DSC and TGA experimentsin order to evaluate chemical kinetic parameters, which were in

turn utilized to determine the prevalent surface temperature atthe operating pressure. They suggest a correlation for the massburning rate in flameless regime based on the Zeldovichmethod.

Sinditskii et al. [20] have demonstrated the utility of the Zel-dovich expression for predicting the mass burning rate of compo-sitions based on ammonium nitrate. Similarly, Eq. (2) wasutilized by Swanepoel et al. [5] who studied the use of manganeseas a fuel for slow-burning gasless delay compositions. In each ofthese methods, the knowledge of the chemical kinetic parametersparticular to the given composition is necessary. These parametersmay be measured directly through a calorimetric study or inferredfrom measured burning rates. As the proportions of the con-stituents would affect the chemical kinetics of a given multicom-ponent pyrotechnic composition, accurate estimation of theburning rate would require repeated measurement of chemicalkinetic parameters or burning rates for each new formulation. Thismay prove to be cumbersome and resource intensive.

However, the thermo-kinetic parameters for pure componentshave been well characterized and a method utilizing these charac-teristics for predicting the burning rate or the effective apparentkinetic parameters would prove to be useful. In addition to suchan analytical method, the burning rate of pyrotechnics can alsobe investigated through numerical simulations. However, analyti-cal approach has the advantage of providing a relatively quick esti-mation. Furthermore, the analytical technique enables animproved understanding of the phenomenon and the dominantprocesses involved in each case.

The current study proposes a reduced order model for the pre-diction of the burning rate of the porous heterogeneous pyrotech-nics undergoing unconfined combustion. This simplified techniquerelies on finding a homogeneous analogue for the compositepyrotechnic. Subsequently, an expression obtained through analy-sis akin to the one classically applied for laminar flame speed ingaseous media was utilized to obtain the burning rates. The pro-posed method utilizes pure component thermo-physical andchemical kinetic properties along with the composition to estimatethe burning rate. This is proposed as a distinct advantage com-pared to previously reported techniques where application of the

494 A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500

homogeneous propellant assumption requires the knowledge ofthermo-physical and chemical kinetics of the specific composition.The model was validated against a priory known experimentalburning rates.

2. Reduced order model for prediction of burning rate

2.1. Description

The analysis focuses on the one-dimensional steady-state defla-gration of a granular porous pyrotechnic composition. The lack ofoverpressure along with the high metal loading of typicalpyrotechnic compositions implies that the condensed phase reac-tions may be assumed dominant. The heat generated during thecombustion process is utilized for bringing about the endothermicdecomposition of various additives, as well as heating and meltingof the unreacted propellant. The decomposition and combustionprocesses release gaseous species and a fraction of unreacted solidpropellant particles are dispersed in the gas phase.

Fig. 1 shows the schematic of the model consisting three zonestermed as unreacted composition, reactive zone, and gas phaseflame while an arrow shows the direction of combustion propaga-tion. The combustion wave consist of a reactive zone within whichdecomposition of the oxidizer takes place and a thin surface layerwhere the fuel reacts with the gaseous oxygen liberated throughthe oxidizer decomposition reaction. The boundary between unre-acted propellant and reactive zone was taken as the referenceplane (x = 0). Thus, the reactive zone lies from x ¼ 0 to x ¼ d whilethe gas phase flame extends in positive x direction beyond x ¼ d.The coordinate system moves with the burning surface such thatthe unreacted propellant boundary is maintained at x ¼ 0. A dottedline in Fig. 1 shows the temperature variation through thepyrotechnic charge.

Furthermore, in the current reduced order model, followingassumptions have been made in order to facilitate the analysis.

i. Gases trapped within the pores do not participate in thechemical reactions and the convective flow within the poresis assumed negligible. The atmosphere within the pores of agranular propellant may consist of ambient air along withgaseous species released from the solid matrix. In a pressuredriven combustion wave, the gases generated by the flameare driven deep into the propellant matrix, accelerating thecombustion wave. In contrast, for pyrotechnics burning inconduction regime, this effect may be assumed absent. How-ever, the porosity of these granular propellants can stillaffect the burning rate by changing the overall thermal con-ductivity of the propellant matrix.

ii. The oxidizer decomposition reaction is the rate controllingprocess determining the overall burning rate. The typicalpyrotechnic oxidizers undergo an endothermic decomposi-

Fig. 1. Schematic of the combustion wave.

tion reaction that supplies the oxygen necessary for the sur-face combustion of the metallic fuel particles. Thisassumption was substantiated by established data in litera-ture [21–23] establishing the role of solid phase decomposi-tion reaction as the trigger for the pyrotechnic ignition.

iii. The heat transfer from the gas phase flame to the solid phaseis assumed negligible.

iv. The temperature within the reactive zone of the propellantmatrix is assumed to increase linearly from the initial tem-perature (T0) to the surface temperature (TS).

v. Due to the large heat release corresponding to the fuel oxi-dation at the surface, the temperature in the gas phase justafter the reactive zone reaches the final flame temperature(Tf). This is reflected by an abrupt temperature jump throughan extremely small distance in Fig. 1.

2.2. Integral analysis

The governing equations for mass and energy conservation forthis system have been given in Eqs. (3) and (4) respectively.

_m00 ¼ q_r ð3Þ

_m00CpdTdx

� kd2T

dx2¼ �

Xh0f ;i

_m000i ð4Þ

The termP

h0f ;i

_m000i in Eq (4) can be replaced to reflect the overall

heat of reaction and mass consumption rate as shown in Eq. (5).

_m00CpdTdx

� kd2T

dx2¼ � _m000DhR ð5Þ

The integration of Eq. (5) in the reactive zone and application ofappropriate boundary conditions for temperature shown in Fig. 1yield Eqs. (6) and (7) respectively.

_m00CpðTÞþ1�1 � k

dTdx

� �þ1

�1¼ �DhR

Z þ1

�1_m000dx ð6Þ

_m00CpðTf � T0Þ ¼ �DhR

Z þ1

�1_m000dx ð7Þ

The linear temperature gradient within the reactive zone maybe evaluated as shown in Eq. (8)

dTdx

¼ Ts � T0

dð8Þ

The right hand side of Eq. (7) was evaluated by replacing thespace variable with temperature variable using the linear temper-ature gradient assumption as shown in Eq. (9).

DhR

Z d

0_m000dx ¼ dDhR

1Tf � T0

Z Tf

T0

_m000dT ð9Þ

Substituting Eq. (9) in Eq. (7) and splitting the integral term intwo parts separating at the reactive zone surface where the firstintegral term within the limits of T0 to Ts indicates the oxidizerdecomposition reaction which is dominant in the reactive zonewhile the second integral Ts to Tf corresponds to the subsequentcombustion of fuel at the propellant surface.

_m00CpðTf � T0Þ ¼ �dDhR1

Tf � T0

Z Ts

T0

_m000OxdT þ

Z Tf

Ts

_m000F dT

� �ð10Þ

Using the definition of average burning rate and rearranging theterms yields

_m00 ¼ �DhRdCp

ðTs � T0Þ þ ðTf � TsÞcðTf � T0Þ2

_m000Ox

" #12

ð11Þ

Table 3Individual component properties for KClO4 based pyrotechnics.

Component Cp;i (J/kg K) ki (W/m K) qi (g/cm3)

Potassium perchlorate 811.26 [33] 0.148 [34] 2.52 [35]Magnesium 1024.48 [33] 156 [35] 1.74 [35]Strontium carbonate 551.38 [33] 11.7 [36] 3.5 [35]Polyvinyl chloride 1046.75 [37] 0.17 [37] 1.4 [37]Chlorinated Rubber 1674.80 [37] 0.126 [37] 1.5 [37]Copper oxide 531.74 [33] 9.21 [38] 6.31 [35]Barium nitrate 579.26 [33] 1.17 [35] 3.24 [35]Sulfur 706.25 [33] 0.266 [39] 2.07 [35]Cryolite 1057.56 [32] 1.85 [40] 2.21 [40]

A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500 495

where c indicates the mass ratio of fuel and oxidizer for a givencomposition. Assuming the reactive zone thickness is controlledby the thermal diffusivity of the condensed phase and using thecontinuity relationship as shown in Eq. (3) gives an approximateestimation of the reactive zone thickness as shown in Eq. (12).

d ¼ a_r¼ k

Cpq_r¼ k

Cp _m00 ð12Þ

Substituting Eq. (12) in Eq. (11) the correlation for mass burningrate for the granular pyrotechnic propellant as shown in Eq. (13).

_m00 ¼ �DhRk

C2p

ðTs � T0Þ þ ðTf � TsÞcðTf � T0Þ2

_m000Ox

" #12

ð13Þ

The average mass consumption rate of the oxidizer may bedefined by using Eq. (14) where the rate of decomposition of KClO4

is assumed to be described by Arrhenius relation.

_m000Ox ¼

1Ts � T0

Z Ts

T0

_m000OxdT ¼ qYOxAe

ð�Ea=Ru�TÞ ð14Þ

where T is defined assuming that most reactions occur at the end ofthe reactive zone.

T ¼ Ts þ 0:5ðTs þ T0Þ2

ð15Þ

2.3. Thermo-physical and chemical kinetic properties for the mixture

The reduced order model requires the effective solid phasespecific heat capacity (Cp), thermal conductivity (k), and density(q) for the heterogeneous propellant mixture. The theoretical den-sity of the propellants was calculated using Eq. (16).1

qmax¼XYi

qið16Þ

The theoretically calculated density is the maximum possibledensity, which is typically observed to be higher than the actualmeasured density of a given heterogeneous mixture. Thus, the cor-responding void fraction for each component of the mixture and airtrapped within the pores was calculated using equation:

/i ¼ Yiqactual

qið17Þ

Table 1Composition of the KClO4 based pyrotechnics.

Component Formula Red

Potassium perchlorate KClO4 45%Magnesium Mg 20%Strontium carbonate SrCO3 20%Polyvinyl chloride (C2H3Cl)n 8%Chlorinated Rubber C10H11Cl17 7%Copper oxide CuO –Barium nitrate Ba(NO3)2 –Sulfur S –Cryolite Na3AlF6 –

Table 2Calculated volume fractions for the KClO4 based pyrotechnic compositions.

Component Red Green

Potassium perchlorate 0.32 0.11Magnesium 0.21 0.19Strontium carbonate 0.10 –Polyvinyl chloride 0.10 0.10Chlorinated Rubber 0.08 0.10Copper oxide – –Barium nitrate – 0.28Sulfur – –Cryolite – –Void space 0.18 0.23

Subsequently the effective conductivity and specific heat of themixture was calculated using Eqs. (18) and (19) respectively. Thesame values were assumed to prevail in the unreacted propellantas well as the reactive zone.

k ¼X

k/ii ð18Þ

Cp ¼X

YiCp;i ð19ÞThe heat of reaction for the pyrotechnic composition was calcu-

lated theoretically using Eq. (20).

DhR ¼X

YPh0f ;P �

XYRh

0f ;R ð20Þ

The apparent activation energy for linear combustion of a sub-stance has been reported by Steinz [24] to be half of the puredecomposition values despite the chemical activation energy forthe individual molecule remaining the same. Furthermore, thecorresponding pre-exponential factor must be determinedindependently. Similarly, the apparent chemical kinetics of decom-position of the oxidizer in the pyrotechnic matrix was postulatedto be affected by the concentration of the oxidizer, spatial distribu-tion of the oxidizer particles, and concentration of the any catalystswithin the matrix. This effect was incorporated for estimating theeffective activation energy as well as pre-exponential factorthrough a multiplication factor based on the volume fraction ofthe oxidizer and catalyst factor based on the concentration ofcatalysts as shown by Eqs. (21) and (22). The catalyst factor is unity

Green Blue Yellow Silver

16% 48% 40% 58%18% 2.5% 18% 5%– – 12% –8% 6% 7% 6%8% 8% 5% –– 23% – 10%50% – – 14%– 12.5% – 7%– – 18% –

Blue Yellow Silver

0.34 0.28 0.380.03 0.18 0.05– 0.06 –0.08 0.09 0.070.10 0.06 –0.07 – 0.03– – 0.070.11 – 0.06– 0.14 –0.28 0.19 0.34

496 A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500

for the un-catalyzed reactions while it assumes a value less thanunity for catalyzed reactions indicating a reduction in the activa-tion energy.

Table 4The mass averaged properties for each KClO4 based pyrotechnic along with the measured

Red Green

Cp (J/kg K) 908.49 848.63k (W/m K) 6.85 6.49q (kg/m3) 1796.94 1793.04

Table 5Calculation of heat of reaction for KClO4 based pyrotechnic compositions.

Reactants YR h0f ;R (kJ/mol)

Red compositionKClO4 0.45 �430.12Mg 0.20 0.00SrCO3 0.20 �1220.10C2H3Cl 0.08 �94.60C10H11Cl7 0.07 �395.00P

YRh0f ;R (kJ/kg) �3243.89

DhR (J/kg) �4.62E+6

Green compositionKClO4 0.16 �430.12Mg 0.18 0.00Ba(NO3)2 0.50 �988.00C2H3Cl 0.08 �94.60C10H11Cl7 0.08 �395.00P

YRh0f ;R (kJ/kg) �2591.14

DhR (J/kg) �2.90E+6

Blue compositionKClO4 0.48 �430.12Mg 0.03 0.00CuO 0.23 �157.30C2H3Cl 0.06 �94.60C10H11Cl7 0.08 �395.00S 0.13 0.00P

YRh0f ;R (kJ/kg) �2119.04

DhR (J/kg) �130.72E+3

Yellow compositionKClO4 0.40 �430.12Mg 0.18 0.00SrCO3 0.12 �1220.10C2H3Cl 0.07 �94.60C10H11Cl7 0.05 �395.00Na3AlF6 0.18 �3316.91P

YRh0f ;R (kJ/kg) �5234.60

DhR (J/kg) �4.33E+6

Silver compositionKClO4 0.58 �430.12Ba(NO3)2 0.14 �988.00CuO 0.10 �157.30Mg 0.05 0.00C2H3Cl 0.06 �94.60S 0.07 0.00P

YRh0f ;R (kJ/kg) �2618.34

DhR (J/kg) �2.47E+6

Eaeff ¼ Ea� /Ox � g ð21Þ

Aeff ¼ A� /0:715Ox ð22Þ

density.

Blue Yellow Silver

821.82 947.05 770.495.73 7.57 5.921798.07 1749.16 1660.67

Products YP h0f ;P (kJ/mol)

SrO 0.14 �592.04MgO 0.23 �601.24KCl 0.24 �421.79CO2 0.06 �393.52H2O 0.02 �241.83HCl 0.09 �92.31

CO2 0.13 �393.52Mg 0.06 0.00C 0.02 0.00P

YPh0f ;P (kJ/kg) �7864.48

BaO 0.29 �548.10MgO 0.22 �601.24KCl 0.09 �421.79CO 0.10 �110.53H2O 0.02 �241.83HCl 0.10 �92.31

NO 0.11 90.29Mg 0.04 0.00C 0.01 0.00P

YPh0f ;P (kJ/kg) �5490.50

Cu2O 0.21 �170.71MgO 0.04 �601.24KCl 0.26 �421.79CO 0.11 �110.53H2O 0.02 �241.83HCl 0.09 �92.31S(g) 0.13 276.98

O2 0.14 0.00PYPh

0f ;P (kJ/kg) �2249.76

SrO 0.08 �592.04MgO 0.28 �601.24KCl 0.22 �421.79CO2 0.04 �393.52H2O 0.02 �241.83HCl 0.07 �92.31NaF 0.07 �546.20

NaAlF4 0.11 �1840.96CO 0.09 �110.53Mg 0.01 0.00P

YPh0f ;P (kJ/kg) �9567.74

MgO 0.08 �601.24KCl 0.31 �421.79BaO 0.08 �548.10Cu2O 0.09 �170.71NO2 0.05 33.10CO2 0.08 �393.52H2O 0.02 �241.83

HCl 0.03 �92.31SO2 0.14 �296.84O2 0.11 0.00P

YPh0f ;P (kJ/kg) �5093.09

Fig. 2. Comparison of experimental and theoretical burning rates of KClO4 basedpyrotechnics.

Table 6Composition of the KNO3 based pyrotechnics.

Component Formula A B C D E

Magnesium Diboride MgB2 15% 18% 21% 24% 30%Potassium nitrate KNO3 82% 79% 76% 73% 67%Polytetrafluoroethylene C2F4 3% 3% 3% 3% 3%

Table 7Calculated volume fractions for the KNO3 based pyrotechnic compositions.

Component A B C D E

Magnesium Diboride 0.11 0.13 0.14 0.16 0.20Potassium nitrate 0.70 0.67 0.63 0.61 0.55Polytetrafluoroethylene 0.02 0.02 0.02 0.02 0.02

A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500 497

3. Case study

3.1. KClO4 based pyrotechnics

The proposed model was implemented for pyrotechnic compo-sitions with known experimental burning rates [25,26]. Thedetailed composition of these propellants, which create red, green,blue, yellow, and silver colored flames, has been specified by masspercentage in Table 1. In these pyrotechnic formulations KClO4 isthe primary oxidizers while the additives Ba(NO3)2, and CuO actas secondary oxidizers as well as coloring agents. The primary fuelin these compositions is magnesium while the carbon from polyvi-nyl chloride and chlorinated rubber also contribute to a smallextent.

Temperatures for the exothermic decomposition of pure KClO4

was found to be span over 510–620 �C. Polyvinyl chloride and chlo-rinated rubber undergo endothermic decomposition accompaniedby evolution of HCl, various hydrocarbon, and hydrogen in thetemperature range 200–350 �C. The left over carbon or char is oxi-dized in presence of oxygen at high temperatures [27]. Hussain andRees [28] have reported a simultaneous decomposition of KClO4

and oxidation of the carbon at 535 �C in the presence of sulfurwhich acts as a catalyst for KClO4 decomposition. The decomposi-tion of SrCO3 has been reported to be endothermic and occurs overa temperature range of 900–1150 �C [29]. Pourmortazavi et al. [21]also report a thermo-gravimetric study of pure copper oxide pow-ders. The study confirms that pure copper oxide shows no thermalevent at low temperatures and undergoes sharp exothermicdecomposition at to 1026 �C. Pure barium nitrate was observed[30] to melt at approximately 592 �C while its decompositionoccurs over a temperature range of 580–700 �C. Cryolite meltingtemperature has been reported by Dolejs and Baker [31] to be1011.4 �C and complete endothermic decomposition may beassumed to occur at approximately 1000 �C [32].

Based on these thermo-gravimetric observations, during thecombustion of these pyrotechnics, initially the polyvinyl chlorideand chlorinated rubber can be expected to undergo decompositionreaction. This will be followed by melting and decomposition ofsolid KClO4, which may be catalyzed by sulfur in certain cases.The oxygen released by decomposing KClO4 would lead to exother-mic oxidation of the solid phase magnesium. This surface combus-tion of magnesium would in turn release sufficient energy toinitiate the decomposition of rest of the additives.

3.1.1. Estimation of thermo-physical characteristicsThe volume fractions of each component of the pyrotechnic for-

mulations evaluated using Eq. (17) are shown in Table 2. The effec-tive solid phase specific heat capacity (Cp) and thermalconductivity (k) for the pyrotechnics under considerations werecalculated using Eqs. (18) and (19).

Table 3 lists the individual component properties while Table 4shows the calculated effective properties for the mixture alongwith the measured density of the propellants. With an approxi-mate single-step chemical reaction, a reaction mechanism wasassumed based on stoichiometric calculations such that the freeoxygen generated through decomposition of oxidizer is completelyconsumed in the surface reaction. Considering the fuel rich natureof most of these compositions, the excess fuel components remainin the combustion products.

Table 5 shows the reactants and assumed combustion productsalong with the individual heats of formation for each species, massfraction of each species and the calculated heat of reaction.

3.1.2. Estimation of the burning rateIn order to predict the burning rate of the compositions the

reactive zone peak temperature (Ts) was assumed to be equal to

600 �C and the gas phase flame temperature (Tf ) was assumed tobe equal to the adiabatic flame temperatures for the compositionsobtained from equilibrium calculations [25]. The thermal decom-position activation energy and pre-exponential factor was reportedby Lee et al. [41] as well as Glasner and Weidenfeld [42]. The acti-vation energy of 275.3 kJ/mol and a pre-exponential factor of 16.06may be attributed to pure KClO4 decomposition. The effectivechemical kinetics parameters were evaluated using two separateapproaches. Initially, an activation energy of 137.65 kJ/mol, whichis half of the pure component value, was assumed for the decom-position reaction of KClO4. The corresponding pre-exponential fac-tor value was selected to be 11 for obtaining a reasonable matchwith the experimental data. However, a far better match betweenthe experimental and theoretical burning rates was obtained whenthe chemical kinetic parameters were evaluated using Eqs. (21)and (22). The catalyst factor for compositions containing sulfurwas estimated to be 0.93 based on the ratio of the sulfur catalyzeddecomposition temperature of 535 �C to the un-catalyzed decom-position temperature of 575 �C for KClO4 as reported by Hussainand Rees [28].

Table 9The mass averaged properties for each KNO3 based pyrotechnics along with themeasured density.

A B C D E

Cp (J/kg K) 1274.83 1266.52 1258.20 1249.89 1233.26k (W/m K) 3.23 3.33 3.43 3.54 3.78q (kg/m3) 1800.00 1790.00 1760.00 1760.00 1740.00

Table 10Calculation of heat of reaction for KNO3 based pyrotechnic compositions.

Reactants YR h0f ;R (kJ/mol)

Composition AMgB2 0.15 �91.96KNO3 0.82 �494.00C2F4 0.03 �828.23P

YRh0f ;R (kJ/kg) �4555.52

DhR (J/kg) �4.83E+06

Composition BMgB2 0.18 �91.96KNO3 0.79 �494.00C2F4 0.03 �828.23P

YRh0f ;R (kJ/kg) �4469.00

DhR (J/kg) �5.76E+06

Composition CMgB2 0.21 �91.96KNO3 0.76 �494.00C2F4 0.03 �828.23P

YRh0f ;R (kJ/kg) �4382.48

DhR (J/kg) �6.19E+06

Composition DMgB2 0.24 �91.96KNO3 0.73 �494.00C2F4 0.03 �828.23P

YRh0f ;R (kJ/kg) �4295.96

DhR (J/kg) �6.24E+06

Composition EMgB2 0.30 �91.96KNO3 0.67 �494.00C2F4 0.03 �828.23P

YRh0f ;R (kJ/kg) �4122.91

DhR (J/kg) �6.32E+06

Table 8Individual component properties for KNO3 based pyrotechnics.

Component Cp;i (J/kg K) ki (W/m K) qi (g/cm3)

Magnesium Diboride 1039.74 [44] 25.00 [45] 2.55 [46]Potassium nitrate 1316.92 [35] 0.80 [47] 2.11 [35]Polytetrafluoroethylene 1300 [48] 0.25 [48] 2.20

498 A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500

The experimental [26] burning rates have been compared inFig. 2 against the burning rates predicted using constant Ea andvariable Ea. Where the constant Ea indicates the half of the purecomponent value while variable Ea indicates the activation energyobtained using Eq. (21). The discrepancies between the experimen-tal and predicted burning rate may be attributed to the inaccuracyin estimating the flame temperature and heat of reaction throughequilibrium calculation, effect of the particle dispersion from reac-tive zone, lateral heat loss from the high temperature material, andheat flux contribution from the gas phase.

3.2. KNO3 based pyrotechnics

The proposed model was also utilized for predicting the burningrates of pyrotechnic compositions based on KNO3 as an oxidizer[43]. The detailed composition of these pyrotechnics based onMgB2, KNO3, and Polytetrafluoroethylene (PTFE) has been tabu-lated by mass percentage in Table 6.

Products YP h0f ;P (kJ/mol)

KBO2 0.54 �994.96N2 0.11 0.00MgO 0.13 �601.60KF 0.07 �568.61

BO2 0.00 �300.40CO2 0.03 �393.52K 0.01 0.00O2 0.11 0.00P

YPh0f ;P (kJ/kg) �9381.80

KBO2 0.54 �994.96N2 0.11 0.00MgO 0.16 �601.60KF 0.07 �568.61

BO2 0.05 �300.40CO2 0.03 �393.52O2 0.04 0.00P

YPh0f ;P (kJ/kg) �10224.45

KBO2 0.52 �994.96N2 0.11 0.00MgO 0.18 �601.60KF 0.07 �568.61

BO2 0.09 �300.40CO2 0.03 �393.52B 0.01 0.00P

YPh0f ;P (kJ/kg) �10577.24

KBO2 0.49 �994.96N2 0.10 0.00MgO 0.21 �601.60KF 0.07 �568.61

BO2 0.07 �300.40CO2 0.03 �393.52B 0.03 0.00P

YPh0f ;P (kJ/kg) �10532.57

KBO2 0.44 �994.96N2 0.09 0.00MgO 0.26 �601.60KF 0.07 �568.61

BO2 0.03 �300.40CO2 0.03 �393.52B 0.08 0.00P

YPh0f ;P (kJ/kg) �10443.23

Fig. 3. Comparison of experimental and theoretical burning rates KNO3 basedpyrotechnics.

A. Ambekar, J.J. Yoh / Applied Thermal Engineering 130 (2018) 492–500 499

3.2.1. Estimation of thermo-physical characteristicsThe volume fractions of each component of the pyrotechnic for-

mulations evaluated using Eq. (17) are shown in Table 7. The val-ues of Cp and k for these pyrotechnics were calculated using Eqs.(18) and (19).

Table 8 lists the pure component properties while Table 9shows the calculated effective properties and the density of themixtures. Based on approximate single-step chemical reactions,the balanced chemical reactions yielding reactant and productmoles were formulated. Table 10 shows the reactants and assumedcombustion products along with the individual heats of formationfor each species, mass fraction of each species and the calculatedheat of reaction.

3.2.2. Estimation of the burning rateTemperatures for the exothermic decomposition of pure

KNO3 was found to be span over 550–790 �C. In this case, thedecomposition of KNO3 was followed by the exothermic reac-tions between fuel and oxidizer. In order to predict the burningrate of the compositions the reactive zone peak temperature(Ts) was assumed to be equal to 670 �C and the gas phase flametemperature (Tf ) was assumed to be equal to the adiabaticflame temperatures for the compositions obtained from equilib-rium calculations [25]. The thermal decomposition activationenergy and pre-exponential factor was reported by Pouretedaland Ebadpour [49]. The activation energy of 223.5 kJ/mol anda pre-exponential factor of 15 was attributed to pure KNO3

decomposition. The effective chemical kinetics parameters wereevaluated using Eqs. (21) and (22). In this case, the catalyst fac-tor was assumed unity.

The experimental [43] burning rates have been compared inFig. 3 against the burning rates predicted using reduced ordermodel and effective activation energy. The discrepanciesbetween the experimental and predicted burning rate may beattributed to the inaccuracy in estimating the flame temperatureand heat of reaction through equilibrium calculation, effect ofthe particle dispersion from reactive zone, lateral heat loss fromthe high temperature material, and heat flux contribution fromthe gas phase.

4. Conclusions

The study presents a new reduced order model for prediction ofthe burning rate of granular pyrotechnics through integral analysis.Notably, the technique incorporates the effect of propellant con-ductivity, net heat of reaction, and porosity in the burning rate pre-diction. The effective chemical kinetic partakers for theheterogeneous mixture were correlated with the known pure com-ponent values. The technique is based on a simplified combustionwave structure and average thermo-physical properties. The com-bustion is assumed to progress in two steps where in first step thesolid oxidizer undergoes melting and decomposition in a relativelybroad reactive zone. Subsequently, the fuel component of thepyrotechnic burns at the surface in an extremely thin surface reac-tion zone. Utilizing the assumption of average mixture properties,single step reaction mechanism, and steady state combustion thetechnique was found to predict the burning rate of KClO4 andKNO3 based propellants with reasonable accuracy. This techniquemay be extended to predict the burning rate of similar composi-tions based on a range of oxidizers.

Acknowledgments

Authors thank the financial supports from Agency for DefenseDevelopment contracted through the Institute of Advanced Aero-space Technology at Seoul National University. Additional supportfrom the Next Generation Space Propulsion Research Center atSeoul National University is warmly acknowledged.

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