INFLUENCE OF VARIABLE HEAT TRANSFER COEFFICIENT OF FIREWORKS AND CRACKERS ON THERMAL EXPLOSION CRITICAL AMBIENT TEMPERATURE AND TTIME TO IGNITION

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1 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks... THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp INFLUENCE OF VARIABLE HEAT TRANSFER COEFFICIENT OF FIREWORKS AND CRACKERS ON THERMAL EXPLOSION CRITICAL AMBIENT TEMPERATURE AND TTIME TO IGNITION by Zerog GUO *, Qua XIA, Peiyu YAN, ad Zhimig DU State Key Laboratory of Explosio Sciece ad Techology, Beijig Istitute of Techology, Beijig, Chia Origial scietific paper DOI: /TSCI G Aim of this study was to ivestigate ifluece of heat trasfer coefficiet of fireworks ad crackers o thermal explosio, critical ambiet temperature ad igitio time. It is proposed that heat trasfer coefficiet is power fuctio of temperature, mathematical model to describe thermal explosio, steady-state ad usteady-state of fiite cylidrical fireworks ad crackers with complex shell structures are established. Models have bee based o two-dimesioal steady-state thermal explosio theory. The ifluece of variable heat trasfer coefficiet o thermal explosio critical ambiet temperature ad time to igitio are aalyzed. Whe heat trasfer coefficiet is chagig with temperature ad i the coditio of atural covectio heat trasfer, critical ambiet temperature lesse, thermal explosio time to igitio shorte. If ambiet temperature is close to critical ambiet temperature, the ifluece of variable heat trasfer coefficiet o igitio time will be large. For firework with ier barrel, the critical ambiet temperature of propellat is K ad igitio time is s. At ambiet temperature 466 K, ad with costat heat trasfer coefficiet critical temperatue is 0.26 K lower ad time to igitio s less smaller. The calculatio results show that the ifluece of variable heat trasfer coefficiet o thermal explosio time to igitio is greater i this example. Therefore, the effect of variable heat trasfer coefficiet should be cosidered ito thermal safety evaluatio of fireworks to reduce potetial safety hazard. Key words: cylidrical fireworks ad crackers, variable heat trasfer coefficiet, critical ambiet temperature, thermal explosio time to igitio Itroductio Fireworks ad crackers are ethic characteristics ad export products of Chia, ejoyig high reputatio i the iteratioal market. The thermal safety of fireworks ad crackers i the productio, storage, trasport, ad discharge has attracted cosiderable attetio. The paret of moder theory of thermal explosio Semeov [1] itroduced mathematical methods to establish a critical coditio of thermal explosio for the first time. Frak-Kameetskii [2] established the thermal explosio model with the spatial distributio of temperature i Thomas [3] put forward a ew boudary coditio, ad a more uiversal theory was established comprisig Semeov ad Frak-Kameetakii theory i I the early 1980s, Feg studied ad summarized the domestic ad foreig research results of thermal explosio. The theory of * Correspodig author; zerogguo@gmail.com

2 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp thermal explosio was established [4]. The theory of thermal explosio was applied to study the thermal safety of fireworks ad firecrackers by Zhou [5]. There is much study about steadystate theory of fireworks ad crackers i recet years, such as criterio of thermal explosio, critical ambiet temperature, ad so o [6]. However, the amout of the study of usteady-state theory such as temperature process, thermal explosio time to igitio is less, especially the study of No class A geometries. As we kow, it is very importat that we should prevet the thermal accidets from happeig i the process of trasportatio ad storage of fireworks ad crackers, which eed scietific ad reliable thermal explosio critical data to moitor the ambiet temperature. Heat trasfer coefficiet is oe of the most importat boudary parameters i thermal explosio model, the accuracy of this parameter has a obvious effect o the simulatio results. However, it is commo that chemical reactio i system will be treated as zero order reactio whe study the thermal safety of fireworks ad crackers i previous work. I the eergy coservatio equatio temperature affect othig but the reactio rate. I fact, whe the fireworks has a larger temperature differece betwee the chemical reactio system ad the eviromet or a loger spotaeous combustio time(such as a few hours to a few days), variable heat trasfer coefficiet chages as the temperature chages observably, which elargig the error of results. Thus, the effects of variable heat trasfer coefficiet should be cosidered i the study of thermal safety of fireworks ad crackers. El-Sayed [7] examied how the criterio for criticality for a system with uiform temperature (Semeov case) is affected whe the heat trasfer coefficiet is ot costat, but depeds o temperature. Wag ad Du [8] cosiderig the heat trasfer coefficiet as the fuctio of temperature, studied the ifluece of variable heat trasfer coefficiet o critical ambiet temperature of Semeov system ad aalyzed the thermal safety margi. A ew method for theoretical aalysis of the self-heatig kietics for moomolecular exothermic reactios has bee proposed by Filimoov [9] ad the detailed phase trajectories aalysis has bee performed for the case of first-order reactios. But o temperature process ad thermal explosio time to igitio ivolved. The relatioship betwee heat trasfer coefficiet ad temperature is aalyzed i this paper. Accordig to two-dimesioal thermal explosio theory, Mathematical thermal explosio steady-state ad usteady-state model of fiite cylidrical fireworks ad crackers with complex shell structures ad variable heat trasfer coefficiet were established, simulatig ad aalyzig the ifluece of variable heat trasfer coefficiet o critical ambiet temperature ad thermal explosio time to igitio. The relatioship betwee heat trasfer coefficiet ad temperature I literature [5] boudary heat trasfer mode of the fireworks ad crackers have bee studied ad clarified. Cosiderig that the shell is exposed to air ad that there is o forced covectio i the productio, storage ad trasport process, the exteral heat trasfer coefficiet is atural covectio heat trasfer coefficiet. The heat trasfer coefficiet is the fuctio of temperature [4]: χ = χ h( θ) (1) 0 2 where, χ 0 is the heat trasfer coefficiet with costat, θ = ( T Ta)/(R Ta / E) the dimesioless temperature, ad h(θ) stads for the fuctio of heat trasfer coefficiet chagig with temperature. The value of h(θ) i covectio heat trasfer is h(θ) = θ [10]. Thus: χ = χθ 0 (2)

3 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks... THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp Whe = 0, it meas that heat trasfer coefficiet does ot chage with temperature. 0 represets for heat trasfer coefficiet chagig with temperature. Here χ 0 is the heat 2 trasfer coefficiet of the surface of shell at θ = 1 ( T = R Ta / E+ Ta). Whe > 0, the situatio of heat trasfer coefficiet icreases with temperature ca be simulated. Coversely, heat trasfer coefficiet decreases with temperature whe < 0. Particularly, whe = 0.25, it is the atural covectio heat trasfer for most materials [11]. Thermal explosio model ad boudary coditio treatmet Take the cylidrical fireworks as subject ad the physical fiite cylider model ca be see i fig. 1. Accordig to literature [12], mathematical thermal explosio steady-state ad usteady- -state model of fiite cylidrical fireworks ad crackers with complex shell structures are established based o eergy coservatio law, which cosider the chage of heat trasfer coefficiet with a 0 radius ad H legth diameter ratio (Model hypothesis is ot metioed here). Figure 1. Physical fiite cylider model The coservatio equatio of thermal explosio steady-state model: 2 2 T 1 T T exp E k + + QA = R1 R1 R1 R2 RT The coservatio equatio of thermal explosio usteady-state model: (3) 2 2 T T 1 T T E σ cv = k + + QAexp (4) t R1 R1 R1 R2 RT The dimesioless parameters were itroduced [4]. The dimesioless coservatio equatio of thermal explosio steady-state model was obtaied: 2 2 θ 1 θ 1 θ θ δ exp = (5) ρ1 ρ1 ρ1 H ρ2 1+ εθ The dimesioless coservatio equatio of thermal explosio usteady-state model: δ = θ + + θ + δ exp θ (6) τ ρ1 ρ1 ρ1 H ρ2 1 + εθ Cosiderig the relatioship betwee heat trasfer coefficiet ad temperature, effective Biot umber ca be writte: 0 Bi = a = Bi (7) k χθ 0 0 Igitio poit is ukow because of differet coolig coditios of the upper ad lower surface. I order to solve the equatio, assumig igitio poit locatio for x, the cylider is divided ito two zoes by igitio poit. The dimesioless boudary coditios of zoe 1: θ

4 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp ρ = 0, = 0; 0 x ( ρ ) 1 2 ρ1 ρ = 0, = 0; 0 1 ( ρ ) 2 1 ρ2 ρ = 1, + Bi θ = 0; 0 ρ x 1+ 1 r 2 ρ1 ) (8) (9) (10) ρ = x, + HBi θ = 0; 0 ρ 1 ( ) 1+ 2 z1 1 ρ2 The dimesioless boudary coditios of zoe 2: ρ = 0, = 0; 2 0 ( x ρ ) 1 2 ρ1 ρ = 0, = 0; 0 ρ ρ2 ρ = 1, + Bi θ = 0; 2 ρ 0 ( x ) 1+ 1 r 2 ρ1 ρ = x, + HBi θ = 0; 0 ρ z2 1 ρ2 The iitial coditios of thermal explosio usteady-state model: The umerical solutio ad ifluece aalysis ) ) (11) (12) (13) (14) (15) τ = 0, θ = θi (16) The umerical solutio The mathematical model thermal explosio steady-state is described by equatios (5), (8)-(15) ad usteady-state model by equatios (6), (8)-(16). Five poit differece method was used to discretize the model by equal step legth. The model was solved by programmig based o Matlab with Newto-homotopy method [13]. For the comprehesive ad cocise, the differetial equatio is ot metioed here. The critical parameter of zoe 1 ad zoe 2 ca be calculated. If the value of zoe 1 is equal to zoe 2, solutio is obtaied; If ot, the calculatio should be repeated by movig x by oe step legth util the two results are equal. Ifluece of variable heat trasfer coefficiet o critical ambiet temperature (Thomas system) The boudary coditios of thermal explosio model vary from characteristic expoet. Whe determied, critical ambiet temperature ad thermal explosio time to igitio of fireworks ad crackers ca be obtaied by solvig o-liear equatios. Usig expoetial approximatio ε = 0, Bi z1 = Bi r = Bi z2 ad the legth diameter ratio H = 1. I order to aalyze the ifluece of variable heat trasfer coefficiet o critical ambiet temperature accurately,

5 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks... THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp the rage of Biot umber ad legth diameter radio H take for study is same to the literature (similarly hereiafter). The calculatio results are listed i tab. 1. Table 1. The ifluece of Biot umber ad o δ cr, θ 0,cr Bi = 10-2 Bi = 10-1 Bi = 1 Bi = 10 Bi = 10 2 Bi δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr * Whe Bi z1 = Bi r = Bi z2 = 10, the calculatio results are show i tab. 2. The results show that the error betwee the umerical solutio ( = 0) ad literature solutio [14] ( = 0*) is less tha This proves that the umerical method used i this paper is accurate ad reasoable See from tab. 1 ad fig. 2, criterio of thermal explosio δ cr, temperature rise of system ceter δ 0,cr icrease ad ted to costat as Biot umber icreases. The effect of variable heat trasfer coefficiet o δ cr first becomes larger the decreases with the icrease of Biot umber. Whe 10 0 < Bi < 10 2, Biot umber has greater effect o δ cr. That is because the larger Biot umber, the better heat dissipatio, the higher critical temperature rise of system. The ifluece of heat trasfer coefficiet decreases with the icrease of Biot umber. Because heat dissipatio is good eough ad the ifluece of variable heat trasfer coefficiet ca be igored. From tab. 2 that the smaller the legth diameter ratio, the greater the ifluece of Criterio of thermal explosio = 0.25 = 0 = 0.25 Ifiite variable heat trasfer coefficiet o δ cr. This is because the smaller legth diameter ratio, the bigger the specific surface area of cylider whe feature size a 0 is certai. Thus, surface heat trasfer has great effect o system. lg (Bi) Figure 2. The ifluece of Biot umber ad o δ cr Table 2. The ifluece of H ad o δ cr, θ 0,cr H = 0.2 H = 0.5 H = 1 H = 5 H δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr δ cr θ 0,cr * Accordig to tabs. 1 ad 2, i practical applicatio, the atural covectio heat trasfer coditio ( = 0.25), cosiderig heat trasfer coefficiet chagig with temperature, the criterio of thermal explosio (critical ambiet temperature) is lower.

6 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp Ifluece of variable heat trasfer coefficiet o thermal explosio time to igitio (Thomas system) Usig expoetial approximatio ε = 0, Bi z1 = Bi r = Bi z2 = 10 ad H = 1. Now ambiet temperature is the major ifluece factor o thermal explosio time to igitio. Its dimesioless form ca be expressed as: δ = a0qeσaexp( E/R Ta)/ λrt. Study the effect of 2 2 a o thermal explosio time to igitio with selectig differet values of δ. The data i tab. 3 is dimesioless thermal explosio time to igitio τ ig. Table 3. The ifluece of δ ad o τ ig Dimesioless thermal explosio time to igitio δ * = 0.25 = 0 = 0.25 Frak-Kameetskii umber Figure 3. The ifluece of δ ad o τ ig delay of thermal explosio time to igitio. Example aalysis Take Dragoes Voladores, a kid of fireworks with ier barrel, as example. The propellat is black powder ad the effector is fulmiatig powder. The height of the tube is 290 mm, the iside diameter is 30 mm, ad the thickess of the side shell is 3 mm. The thickess of the mud shell is 25 mm. The mass of black powder is 3.7 g, the desity of the charge is 0.4 g/cm 3 ad the height of the charge is 13 mm. The height of the effector is 120 mm, the iside diameter is 22 mm, ad the thickess of the effector is 3 mm. The thickesses of the upper, ad dow mud shell of effector are both 20 mm. The mass of fulmiatig powder is 15 g, the desity of the charge is 0.33 g/cm 3 ad the height of the charge is 80 mm. It ca be see from tab. 3 ad fig. 3, thermal explosio time to igitio τ ig decreases with the icreases of δ. It meas that the higher the ambiet temperature, the more proe to thermal explosio. Whe the value of δ is small but greater tha δ cr, the ifluece of variable heat trasfer coefficiet o thermal explosio time to igitio is large. Comparig case whe = 0 with case 0, > 0, heat trasfer coefficiet icreases whe temperature icreases, which ca ehace the heat dissipatio capacity. However, thermal explosio time to igitio shorte. The reaso is that thermal explosio time to igitio starts from θ = 0, ad the temperature of chemical reactio system rises with the accumulatio of heat. Before θ reaches 1, χ is always less tha χ 0 which results i the rapid accumulatio of heat ad short of thermal explosio time to igitio. Coversely, whe < 0, heat trasfer coefficiet decreases as temperature icreases. Before θ reaches 1, χ is always greater tha χ 0 which results i Mud shell Thi shell Paper shell Effector Propellat Figure 4. Simplified structure of the fireworks Dragoes Voladores with ier barrel

7 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks... THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp Table 4. Physical ad chemical parameters of various materials Reactat Activatio eergy E [kjmol 1 ] Pre-expoetial factor A [s 1 ] Reactio heat Q [kjkg 1 ] Chargig desity σ [kgm 3 ] Thermal coductivity coefficiet [Wm 1 K 1 ] Propellat/black powder Effector/fulmiatig powder The thermodyamics ad thermophysical parameters are listed i tab. 4. Activatio eergy, expoetial factor ad reactio heat are obtaied by usig TGA-DSC Aalysis. The thermal coductivity are obtaied by the DRY Thermal Coductivity Coefficiet Tester. The thermal coductivity coefficiet of paper shell, sealig powder ad mud shell are , 0.95, 0.95 W/mK, respectively. The value of air atural covectio heat trasfer coefficiet is usually betwee 1~10 W/m 2 K. The value of χ 0 is 5 i literature [15]. As a result, the effective Biot umbers ' of the propellat are Bi = r 1.04, ' ' Bi z 1 = 0.08, Bi z 2 = 0.36, respectively, ad the effector s are ' Bi = r 0.53, ' ' Bi z 1 = 1.69, Bi z 2 = The critical ambiet temperature of propellat ad effector obtaied by solvig the steady-state equatio are K ad K, respectively. If cosiderig the chage of the heat trasfer coefficiet, the critical ambiet temperature of propellat ad effector chage to K ad K, respectively. Take the lowest critical ambiet temperature K as assessmet criteria of thermal safety evaluatio. I the study of thermal explosio time to igitio, oly propellat eed to be cosidered i this case, so the choice of ambiet temperature should be higher tha the critical ambiet temperature. Because the shells of fireworks ad crackers are made up of kraft paper ad straw board, characteristic expoet is 0.25 uder the coditio of atural covectio heat trasfer. Thermal explosio time to igitio is calculated as followed with several differet ambiet temperature. From tab. 5 it ca be see that whe the eviromet temperature is much higher tha the critical ambiet temperature (such as 490 K, 500 K), the differece i time to igitio is less tha 10 1, betwee the case whe heat trasfer coefficiet chage with temperature ad case with costat heat trasfer coefficiet. It ca be igored i thermal safety evaluatio of fireworks ad crackers. However, whe the eviromet temperature is close to the critical temperature, variable heat trasfer coefficiet has a great ifluece o thermal explosio time to igitio. For example, whe the ambiet temperature is 466 K, cosiderig heat trasfer coefficiet chagig with temperature, thermal explosio time to igitio is s less. Moreover as the eviromet temperature is closer to the critical temperature, the differece is greater. It ca be see from fig. 5, at 470 K, the heatig rate of the ceter is faster whe cosiderig the chage of the heat trasfer coefficiet. This is because lower temperature rise lead to smaller heat trasfer coefficiet, more heat will be rapidly accumulated. As a result, for the complicated eviromet, it s Cetral temperature [K] Time [s] Figure 5. Temperature rise of system ceter of propellat at 470 K

8 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp Table 5. The results of thermal explosio time to igitio of propellat T a [K] s s s 730.9s 403.2s 145.3s 58.7s s s s 712.3s 398.1s 145.0s 58.7s more reasoable to cosider the ifluece of variable heat trasfer coefficiet whe calculatig the thermal explosio time to igitio. Coclusios y The ifluece of variable heat trasfer coefficiet o the boudary coditio of heat dissipatio of fireworks ad crackers are aalyzed. The mathematical thermal explosio steady-state ad usteady-state model of fiite cylidrical fireworks ad crackers which cosiderig the heat trasfer coefficiet as the power fuctio of temperature are established for the first time. Whe = 0, the error betwee umerical solutio ad literature solutio is less tha Therefore the model is correct ad the calculatio procedure meets the practical computatioal requiremets. y Whe heat trasfer coefficiet is the power fuctio of temperature ad i the coditio of the atural covectio heat trasfer, for Thomas system, there are several coclusios: Firstly, the critical ambiet temperature of cylidrical fireworks ad crackers lesse. Meawhile, Biot umber has great ifluece o the critical ambiet temperature withi the rage of 10 0 to10 2 ad the smaller the legth diameter ratio, the greater the ifluece. I example aalysis, the critical ambiet temperature of propellat is K, 0.26 K less tha without cosiderig the chage of heat trasfer coefficiet. Secodly, the thermal explosio time to igitio of cylidrical fireworks ad crackers shorte. For firework with ier barrel, the thermal explosio time to igitio is 78.6 s less at 470 K. However, the thermal explosio time to igitio is s less at 466 K. Thus, whe ambiet temperature is slightly higher tha the critical ambiet temperature, variable heat trasfer coefficiet has greater effect o the thermal explosio time to igitio. The calculatio results also show that the ifluece of variable heat trasfer coefficiet o thermal explosio time to igitio is greater tha the ifluece o the critical ambiet temperature i this example. I practice, this paper has a great applicatio value for the thermal safety desig ad evaluatio of fireworks ad crackers. It provide basic data ad theory guidace for safe storage ad use of pyrotechic products. The thermal explosio critical ambiet temperature ad time to igitio of ay cylidrical fireworks ca be obtaied by solutios used i this paper. The ambiet temperature ca be moitored by the temperature sesor, etc. Oce the ambiet temperature higher tha the critical ambiet temperature, relevat ad effective measures ca be take based o time to thermal explosio. Obviously, thermal explosio critical data will be scietific ad reliable with cosiderig the effect of variable heat trasfer coefficiet. Therefore, the effect of variable heat trasfer coefficiet should be cosidered ito thermal safety evaluatio of fireworks to reduce potetial safety hazard. Ackowledgemet The authors would like to express their special gratitude to the ZDKT12-01 Project of the State Key Laboratory of Explosio Sciece ad Techology for the great fiacial support i the study.

9 Guo, Z., et al.: Ifluece of Variable Heat Trasfer Coefficiet of Fireworks... THERMAL SCIENCE: Year 2016, Vol. 20, No. 5, pp Nomeclature A pre-expoetial factor, [s 1 ] Bi r Biot umber of the lateral surface, [ ] Bi z1 Biot umber of the upper surface, [ ] Bi z2 Biot umber of the lower surface, [ ] c v pyrotechic heat capacity, [JK 1 ] E activatio eergy, [kjmol 1 ] H legth diameter ratio, [ ] k thermal coductivity of pyrotechic, [Wm 1 K 1 ] characteristic expoet, [ ] Q reactio heat, [kjkg 1 ] R uiversal gas costat, [Jmol 1 K 1 ] T temperature of system, [K] T a ambiet temperature, [K] x dimesioless value of the distace from the igitio poit to the upper surface ragig from 0 to 2, [ ] Greek symbols δ Frak-Kameetskii umber, [ ] δ cr criterio of thermal explosio, [ ] ε dimesioless activatio eergy, [ ] θ dimesioless temperature rise, [ ] θ 0,cr dimesioless temperature rise of system ceter, [ ] ρ 1 dimesioless co-ordiate variables i the r directio, [ ] ρ 2 dimesioless co-ordiate variables i the z directio, [ ] σ pyrotechic desity, [kgm 3 ] τ dimesioless time, [ ] τ ig thermal explosio time to igitio, [s] χ heat trasfer coefficiet, [Wm 2 K 1 ] χ 0 heat trasfer coefficiet at θ = 1, [Wm 2 K 1 ] Refereces [1] Semeov, N. N., To Theory of Combustio Proces (i Germa), Z Phys., 48 (1928), 7-8, pp [2] Frak-Kameetskii, D. A., Calculatio of Thermal Explosio Limits, Acta Physicochimica (USSR)., No. 10, p. 365, 1939 [3] Thomas, P. H., Some Approximatios i the Theory of Self-Heatig ad Thermal Explosio, Trasactio of the Faraday Society, 56 (1960), 56, pp [4] Feg, C. G., Theory of Thermal Explosio (i Chiese), Sciece Press, Beijig, 1988 [5] Zhou, G. W., Thermal Safety Study of Fireworks ad Crackers (i Chiese), M. Sc. thesis, Beijig Istitute of Techology, Beijig, 2010 [6] Liu, H. Y., Qia, X. M., et al., Thermal Explosio Model ad Calculatio of Sphere Fireworks ad Crackers, J Therm Aal Calorim, 110 (2012), 3, pp [7] El-Sayed, S. A., Critical Coditios i Uiform Temperature Systems with Variable Heat Trasfer Coefficiet i Thermal Explosio Theory, Combustio ad Flame, 98 (1994), 3, pp [8] Wag, P., Du, Z. M., Ifluece of Variatio of Heat Trasfer Coefficiet o the Critical Ambiet Temperature of Exothermic System (i Chiese), Acta Armametarii, 30 (2009), 5, pp [9] Filimoov, V. Y., Thermal Modes of Moomolecular Exothermic Reactios: Two-Dimesioal Model, Combustio ad Flame, 160 (2012), 3, pp [10] Boddigto, T., et al., Thermal Explosio, Times to Igitio ad Near-Critical Behaviour i Uiform-Temperature Systems. III. Effects of Variable Heat-Trasfer Coefficiets, J. Chem. SOC.,Faraday Tras., 2, 80 (1984), 9, pp [11] Boddigto, T., et al., Thermal Explosios, Criticality ad Trasitio i Systems with Variable Heat Trasfer Coefficiet: Uiform Temperatures, J. Chem. SOC., Faraday Tras. 2, 79 (1983), 10, pp [12] Guo, Z. R., et al., Modelig ad Calculatio of Thermal Explosio Time to Igitio of Cylidrical Fireworks ad Crackers, J. Therm. Aal. Calorim., 115 (2014), 1, pp [13] Su, Z. Z., Numerical Solutio of Partial Differetial Equatios (i Chiese), Sciece Press, Beijig, 2012 [14] Wag, L. Q., et al., Limited Space Explosio ad Igitio i Theory ad Experimet (i Chiese), Beijig Istitute of Techology Press, Beijig, 2005 [15] Ji, L. L., Study of the Critical Temperature ad Thermal Explosio Time to Igitio of Cylidrical Fire- Works ad Crackers (i Chiese), M. Sc. thesis, Beijig Istitute of Techology, Beijig, 2012 Paper submitted: August 15, 2014 Paper revised: April 16, 2015 Paper accepted: April 16, 2015