Hong-xu GAO*,Feng-qi ZHAORong-zu HUHong-n ZHAO,Hi ZHANGSiene nd Tehnology on Comustion nd Explosion Lortory,Xi’n Modern Chemistry Reserh Institute,Xi’n 710065,Chin College of Communition Siene nd Engineering,Northwest University,Xi’n 710069,Chin Deprtment of Mthemtis,Northwest University,Xi’n 710069,Chin

1.Introduction

According to the literature,addition of 2,4,6,8,10,12-hexanitro-2,4,6,8,10,2-hexaaza-tetracyclo-[5.5.0.05,9.03,11]-dodecane(HNIWor CL-20)to propellants or explosives is expected to increase the performance parameters such as speci fi c impulse,ballistics and detonation velocity.In Ref.[1],the kinetic parameters(apparent activation energy Eaand preexponential factor A)for nonisothermal thermal decomposition reaction and adiabatic time-to-explosion(tTIad)of HNIW were presented based on Arrhenius formula k=Aexp(-E/RT)(where E and A are constants).The results show that the kinetic model function in integral form and the value of E and A of the decomposition reaction of HNIWare 3(1- α)[-ln(1- α)]2/3,155.04 kJ mol-1and 1013.55s-1,respectively.

For studying thermal analysis kinetics and thermal safety of energetic materials,the kinetic parameters of nonisothermal thermal decomposition reaction and the values of tTIadfor HNIW are calculated based on Kooijformula [2],k=A0TBexp(-E/RT),where A in A=A0TBis not strictly constant,but depends on TBaccording to collision theory,and B=0.5[3].

2.Theory and method for describing nonisothermal reaction kinetics based on the Kooij formula

2.1.Differential equation

Thebasic isothermal differentialrate equation [3,4]describing the change of the conversion degree with time based on Kooij formula is

where α,t,A,A0,B,E,T,n,R and f(α)have the usual meanings[3,4].

Setting T0as the initial temperature at which the peak on the DSC or DTG curve deviates from its baseline in K, and β as the constant heating rate in K s-1, we have

An overall equation for non-isothermal kinetics can be yielded by combining Eq.(1)with Eq.(2)

Differential of Eq.(3)with respect to T gives

when

the maximum reaction rate occurs.Eq.(4)gives

and

whereTpandαparethepeaktemperatureonDSCorDTGcurve and the conversion degree at peak temperature,respectively.

By taking the natural logarithms on both sides of Eq.(7),we can obtain

Integration of Eq.(3)with the temperature between 0 and T,and the conversion degree between 0 and α results in

we have

Eq.(6)becomes

The following equation can be obtained by combining Eq.(6)with Eq.(13),T=Tp,

From Eq.(15),we have

When

The following equation can be obtained by combining Eq.(8)with Eq.(19)

The items on both the sides of Eq.(20)can be rearranged to obtain the following differential equation[3,4]

For getting the values of E and A,the data(B,βi,Tpi,i=1,2,…,L)are fi tted toEq.(21)by the linear least-squares method.

2.2.Integral equation

The temperature integralusing Frank-Kamenetskii’s expression[6]

Integration of Eq.(3)with the temperature between 0 and T,and the conversion degree between 0 and α results in

Taking the logarithms on both sides of Eq.(23),the integral equation[Eq.(24)][3,4]can be obtained

Eq.(24)may be solved by the iterative method.Any arbitrary value may be assumed for E(E＞0),and this arbitrary value,original data(B,βi,Ti,αi,i=1,2,…,L)and any given G(α)can be used to calculate the value on the left-hand side of the expression for each data point.When a curve of the left-hand side of Eq.(24)against(1/T)is plotted by the linear least-squares method,new values of E from the slope and A0from the intercept are given.This modi fi ed value of E is used as a starting value for the next iteration which yields another modi fi ed value of E.Thus after a few iterations,the consistent values of E and A0will be obtained.

2.3.Model-free integral isoconversional non-linear equation[NL-INT-B]based on Kooij formula

According to the integral non-isothermal kinetic equation based on Kooij formula

and Eq.(25)can be written for a given conversion and a set of experiments performed under different heating rates as

we have

Eq.(27)[5,7]is known as the integral isoconversional nonlinear equation based on Kooij formula(NL-INT-B method)for calculating Eα.The value of Eαcalculated using Eq.(27)is used to check the validity of activation energy.

2.4.Adiabatic time-to-explosion equation based on Kooij formula

Under the adiabatic condition,the differential equation describing the time-temperature relation of this exothermal decomposition with Kooij’s temperature dependence is

where Cp,Q,A0and E are the speci fi c heat capacity,heat of reaction,pre-exponential constant and activation energy,respectively,R is the gas constant,T is the absolute temperature,t is the time,and α is the fraction of substance decomposed and can be expressed as a function of temperature

where Cp=a+bT.On rearranging and integrating Eq.(28),we obtain

Eq.(30)isknownastheadiabatictime-to-explosionequation basedonKooijformula.Oncethevaluesofa,b,Q,A,E,Te0and Tb,are calculated from an analysis of the DSC curves,the correspondent value of t can then be obtained from Eq.(30).

2.5.Critical temperature of thermal explosion(Tb)

The value of Tbcan be calculated by Eq.(31)[4,5]

It may also be expressed as

In Eq.(32),the value of Te0(=TSADT)corresponding to β→0 may be obtained by using linear regression of Teiand βias described in Eq.(33)

3.Calculation example

3.1.Calculation of nonisothermal decomposition reaction kinetics

A multiple heating method [Eq. (21)] is employed to obtain the kinetic parameters (apparent activation energy E and preexponential constant A0) by Kooij formula. From the original data [1] in Table 1, the values of E and A0 obtained by Eq.(21) are determined to be 154.40 kJ mol-1 and 1012.09 s-1,respectively. The linear correlation coefficient r is 0.9982.

By substituting the original data [1], βi, T0i, Ti, αi,i = 1,2,…,n, tabulated in Table 2 from DSC curves into Eq.(27), the values of Eα for any given value of α in Table 3 are obtained.The Eα-α curve is shown in Fig.1.It shows that the activation energy changes slightly in the range from 0.125 to 0.875,which is properly selected to calculate the nonisothermal reaction kinetics.The average value of Eαin the range of α from 0.125 to 0.875 in Fig.1 is 152.91 kJ mol-1.

Table 1 Initial temperature(T0),onset temperature(Te)and maximum peak temperature(Tp)of exothermic decomposition reaction for CL-20 determined by DSC curves at various heating rates(β).

Table 2 Data of HNIW determined by DSC.

Eq.(24)is cited to obtain the values of E,A and the most probable kinetic model function[f(α)]from each non-isothermal DSC curve.f(α)and G(α)in Eq.(24)are the differential and integral model functions,respectively.

Forty-one types of kinetic model functions[6]and the data in Table 2 are put into Eq.(24)for calculation,respectively.The values of E,A,linear correlation coefficient(r),standard mean square deviation(Q)and believable factor(d)(where d=(1-r)Q)were obtained by the linear least-squares.

The kinetic parameters,E and A and the probable kinetic model functions selected by the logical choice method and satisfying the ordinary range of the thermal decomposition kinetic parameters for energetic materials(E=80-250 kJ mol-1,lg A=7-30 s-1)together with their appropriate values of r,Q and d,obtained by Eq.(24)are presented in Table 4.These values E and lg A obtained from the each nonisothermal DSC curve are approximately in agreement with the values calculated by Eqs.(21)and(27).Therefore,a conclusion can be drawn that the reaction mechanism of main exothermal decomposition process of HNIW is classi fi ed as random nucleation and subsequent growth,n=l/3,m=3,and the most probable kinetic model function of the decomposition reaction is G(α) =[-ln(1-α)]1/3,f(α)=3(1-α)[-ln(1-α)]2/3.Substituting f(α)with 3(1- α)[-ln(1- α)]2/3,E with 152.73 kJ mol-1and A with 1011.97s-1in Eq.(34),we have

We can now establish a kinetic equation of the exothermic decomposition process of HNIW as follows

Table 3 Apparent activation energies of thermal decomposition of HNIW obtained using isoconversional method[Eq.(27)]and the data taken from Table 2.

Fig.1.Eα-α curves for the decomposition of HNIW by NL-INT-B method.

which is similar to the kinetic equation in Ref.[1]of the exothermic decomposition process of HNIW obtained by Arrhenius formula The E values obtained with the integral and differential methods,and the integral isoconversional non-linear method based on Kooij formula and Arrhenius formula[1]are close to eachother,showingthattakingthevalueofBof0.5inA=A0TBis suitable for calculating the kinetic parameters by Kooij formula.

3.2.Self-accelerating decomposition temperature(TSADT)

The values of Te0of 486.55 K and Tp0of 495.64 K are obtained by substituting the original data,βi,Tei,Tpi,i=1,2,…L,in Table 1 into Eq.(33).

3.3.Critical temperature of thermal explosion(Tb)

The value of Tbof 493.11 K is obtained by substituting the valuesofE of152,730 Jmol-1,Te0of486.55 K,R=8.314 J mol-1K-1into Eq.(32).

3.4.Adiabatic time-to-explosion(tTIad)

Table 4 Kinetic parameters obtained for the decomposition process of HNIW(B=0.5).

By substituting the values of Cp=0.2472+0.002705992 T[1], differential mechanism function f(α) =3(1- α)[-ln(1- α)]2/3,E=152,730Jmol-1,A0=1011.97s-1,decomposition heat Q=2340 J g-1,R=8.314 J mol-1K-1,α,conversion degree integral upper limit,T=Tb=493.11 K for tTIadand integral lower limit,T0=Te0=486.55KintoEq.(32),ThecalculatedvalueoftTladof HNIW is 52.01 s.

4.Conclusions

1)A differential and an integral method are proposed for estimation of the kinetic parameters and mechanism function of thermal decomposing reaction of energetic materials by use of the Kooij equation.The differential method is based on the data acquired for a reaction investigated under various heating rates and may be considered as a new multiple scan method.Taking the value of B of 0.5 in A=A0TBis suitable for calculating the kinetic parameters by Kooij formula.

2)Using differential and integral equations,the kinetic parameters and the kinetic model function in integral form for the exothermic decomposition reaction of HNIW are 3(1-α)[-ln(1-α)]2/3, 152.73 kJ mol-1and 1011.97s-1,respectively.The kinetic equation of the exothermic decomposition reaction of HNIW can be described as

3)The values of TSADT,Tband tTIadof HNIW are 486.55 K,493.11 K and 52.01 s,respectively,showing that HNIW has better thermal safety and heat-resistant ability.

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