Theoretical Analysis on Deflagration-to-Detonation Transition *

Transcription

1 heoretical Analysis on Deflagration-to-Detonation ransition * Yun-Feng Liu( 刘云峰 ),**, Huan Shen( 沈欢 ),, De-Liang Zhang( 张德良 ),, Zong-Lin Jiang( 姜宗林 ), Institute of echanics, Chinese Academy of Sciences, Beijing, 0090 School of Engineering Science, University of Chinese Academy of Sciences, Beijing, he study on deflagration-to-detonation transition (DD) is very imortant because this mechanism has relevance to safety issues in industries, where combustible remixed gases are in general use. However, the quantitative rediction of DD is one of the major unsolved roblems in combustion and detonation theory to date. In this aer, the DD rocess is studied theoretically and the critical condition is given by a concise theoretical exression. he results show that a deflagration wave roagating with about 60% Chaman-Jouguet (C-J) detonation velocity is a critical condition. his velocity is the maximum roagating velocity of a deflagration wave and almost equal to the sound seed of combustion roducts. When this critical conation is reached, a C-J detonation is triggered immediately. his is the quantitative criteria of the DD rocess. Keywords: deflagration, detonation, deflagration-to-detonation transition PACS: Pq, Rs, 8.33.Vx. Introduction It is generally known that combustion waves roagating in a remixed detonable mixture can be classified into two modes, either deflagration or detonation. he detonation wave is initiated by two ways, one is direct initiation, and the other is deflagration-to-detonation transition (DD). he direct detonation initiation needs extremely high ignition energy sulied to the remixed detonable mixture. Once the detonation is initiated directly, it will roagate self-sustainable with a Chaman- Jouguet (C-J) detonation velocity. ost of the detonation initiation is imitated by the DD rocess. he flame is ignited by a small amount of energy at the initial stage and then the flame front is accelerated by natural or artificial reasons from laminar flame with velocities at the level of some meters er second to turbulent flame with seeds at the level of a hundred meters er second. Under certain critical conditions, the DD rocess can occur and the C-J detonation is triggered immediately. here is a critical threshold inherent in the DD rocess. Sufficient evidence from the revious studies * Suorted by the National Natural Science Foundation of China (Grants 673 and 5304). ** Corresonding author:liuyunfeng@imech.ac.cn

2 indicates that the deflagration velocity has to reach some quasi-steady critical value which is relatively reeatable before the DD rocess abrut. he DD rocess has attracted considerable research interest. Over the last half century, many exerimental and numerical studies were conducted for detonation onset and DD rocess to test a wide range of initial and boundary conditions. he DD was first observed in exeriments by Brinkley and Lewis []. hen, Oenheim and his coworkers did much work and had a dee insight into DD rocess [-4]. Excellent reviews that summarize our understanding to date have been given by Lee and oen [5] Sheherd and Lee [6], and Zhao et al [7]. In the exeriments, the flame is usually accelerated by utting obstacles or siral-coils into smooth tubes, and turbulence lays an imortant role in DD [-4, 8-]. But, the mechanism of the inherent threshold in DD is not known to date. It should be noted that the DD rocess is very comlex and transient, and the comlex mechanism cannot be observed in exeriments. Numerical simulations have been done extensively in recently years [-6], but the mechanism is still unknown. Since the flame acceleration involves all mechanisms that are sensitive to different initial and boundary conditions, it seems nearly imossible to have a universal theory to describe the DD rocess [5]. It is a common oint of view that the obstacles create turbulence and the turbulence enhances the couling between turbulent flame and incident shock wave. Lee roosed a mechanism of Shock Wave Amlification by Coherent Energy Release (SWACER) [7] to exlain this henomenon. It can be observed both in exeriments and numerical simulations that the DD rocess always occur when the velocity of deflagration reaches about 50% C-J detonation velocity, which is about the sound seed of the combustion roducts [8-3]. No deflagration wave with a seed faster than 50% C- J detonation velocity has been observed in ractice. It is obviously that this is a critical state. he quantitative rediction of DD is very imortant to industries because detonation has very destructive ower. It is one of the major unsolved roblems in combustion and detonation theory. It is also an extremely interesting and difficult scientific roblem because of the comlex nonlinear interactions among the different contributing hysical rocesses, such as turbulence, shock interactions, and energy release. It is not surrising, therefore, that a quantitative criterion of DD rocess has not been yet derived. Such a criteria is necessary for various estimates of the exlosion hazard of combustible systems in ractice. In this aer, the DD rocess is theoretically analyzed and the theoretical exression about this critical condition is given.. heoretical Analysis he DD rocess in a smooth tube is analyzed in this aer. he structure of the deflagration wave is a leading shock wave (SW) followed by a flame surface. Figure gives the structure of a

3 deflagration wave in the laboratory coordinate. he flow field is divided into three regions by the SW and flame surface. Region is the remixed detonable mixture at initial ressure and temerature at rest. Region is the reheated mixture behind the SW, and region 3 is the combustion roducts. Fig.. Structure of a deflagration wave in the laboratory coordinates his is a weak DD rocess. he flame is accelerated from laminar flame to turbulence flame gradually and slowly. Suose the temerature of the reheated mixture in region is much lower than the auto-ignition temerate and the combustion only takes lace on the flame front. No auto- herefore, the small ressure rise caused by weak combustion can roagate ustream by sonic ignition or hot sot occurs in region. At the beginning stage, the leading shock wave is very weak and the flow behind it is subsonic. waves and enhance the strength of the SW. With the strength of leading shock wave SW becoming stronger, the combustion behind it becomes stronger. he big ressure rise caused by violent combustion can roduce a series of secondary shock waves (SW ), which is shown in Fig.. he strength of SW is a function of ressure rise on flame front and the arameters in region. It should be noted that the strength of the leading shock wave SW cannot be enlarged without limit. It is limited by the energy release of combustion on the flame front. If the ach number of SW is tooo high, the secondary shock wave SW will become weaker. here is a negative feedback mechanism between the leading shock wave and the flame surface. Suose there exist a critical state that when the secondary shock wave SW catches u and merges with the leading shock wave, the thermodynamic arameters of the new shock wave are exactly equal to the thermodynamics arameters of a C-J detonation, and therefore, a DD rocess occurs. We will analyze the thermodynamic characteristics of this critical state by using theories of shock wave dynamics in the following art. In order to simlify the analysis rocess, the secific heat ratio of remixed detonable mixture is assumed to be γ.4 and kees constant. According to the normal shock wave relations, the arameters in region can be calculated by Eqs.() and (), ( ) γ γ 7 γ + 6 () 3

4 ( ) ( ) ( γ + ) γ γ γ + (7 )( + 5) 36 () where, is the ach number of the leading shock wave, γ is the secific heat ratio, and are the initial ressure and temerature in region, resectively. And and are the ressure and temerature in region, resectively. In order to determine the ressure rise after combustion, we suose that the time scale of heat release is very shot and can be neglected comared with the flow time scale, and the combustion rocess is a constant-volume combustion rocess. herefore, the ressure 3 can be calculated by Eq.(3), (3) 3 0 where, 0 is the total temerature of constant-volume combustion. In addition, we choose the ressure as the controlling arameter of the critical state, the relationshi is given by Eq.(4), where, 3 ZND (4) ZND is the nondimenstional von Newman sike of a C-J detonation. Combining the Eqs.()-(4), we can easily obtain the theoretical results for this critical condition by Eq.(5), 6 ZND (5) From Eq.(5), we can find that, for a C-J detonation whose von Newmann ressure sike is certain, the ach number of the deflagration wave is only determined by the combustion energy release. his is the key mechanism of the DD rocess. he von Newmann sike can be calculated by Eq.(6), 7 CJ 7 CJ ZND (6) 6 6 Finally, we obtain the critical criterion for DD rocess by Eq.(7), 6 7 CJ (7) 4

5 In the following art, we will give two examles to demonstrate this theoretical result briefly. he first examlee is from reference [9], where the detonable mixture is a CH 4 +O mixture at 8.kPa and 300K. he comarison results are given in able and Fig.. he second examle is from reference [3], where the detonable mixture is stoichiometric ethylene-air mixture at 00kPa. he comarison results are given in able and Fig.. We can find that the theoretical results are in excellent agreement with exerimental results. At the critical condition, the velocity of the leading shock wave is about 60% of C-J detonationn velocity, and it is also close to the sound seed of the combustion roducts. able. Parameters of DD rocess of CH 4 + +O [9] Parameters SW(m/s) P (kpa) (K) a (m/s) P (kpa) (K) P ZND (kpa) 0 (K) CJ D CJ (m/s) a CJ (m/s) SW/D CJ Values % Fig.. Comarisons of theoretical results with exerimental results of CH 4 +O mixture e [9] 5

6 able. Parameters of DD rocess of stoichiometric ethylene-air mixture [3] Parameters SW(m/s) P (atm) (K) a (m/s) P (atm) (K) P ZND (atm) 0 (K) CJ D CJ (m/s) a CJ (m/s) SW/D CJ Values % Fig.. Comarisons of theoretical results with exerimental results of ethylene-air mixtur re [3] 3. Conclusionn In this study, the hysical model of deflagration-to-detonation transitionn is ut forth, the mechanism is analyzed and the theoretical criterion is derived. he theoretical criterion is CJ. When the ach number of the deflagration wave reaches this critical value, a 0 C-J detonation will be triggered immediately and DD rocess occurs. his is also the maximum 6

7 roagating velocity of a deflagration wave. his analysis reveals that a C-J detonation can be considered as a suerosition of two shock waves roagating in the same direction and this is a unique solution. References [] Brinkley S R, Lewis B 959 Seventh Symosium (International) on Combustion he Combustion Institute, Pittsburgh,.807 [] Oenheim A K, Laderman A J, and Urtiew P A 96 Combustion and Flame 6 93 [3] Oenheim A K, Laderman A J, and Urtiew P A 963 Ninth Symosium (International) on Combustion he Combustion Institute, Pittsburgh,.65 [4] Urtiew P, Oenheim A K 966 Proc. Roy. Soc. Lond. Ser. A 95 3 [5] Lee J H S, oen I O 980 Prog. Energy Combust. Sci [6] Sheherd J E, Lee J H S 99 ajor Research oics in Combustion (Sringer, New York).439 [7] Zhao F, an H, Wu Q, et al. 009 Physics (in Chinese) [8] Obara, Kobayashi, Ohyagi S 0 Shock waves 67 [9] Lin W, Zhou J, Fan X H, et al. 05 Chin. Phys. B [0] Wang J, Duan J Y, Huang W B, et al. 0 Chinese Journal of High Pressure Physics (in Chinese) [] Han X, Zhou J and Lin Z Y 03 Chin. Phys. B 470 [] Goodwin G B, Houim R W, Oran E S 07 Proceedings of the Combustion Institute [3] Valiev D, Bychkov V, Akkerman V, Law C K, Eriksson L E 00 Combustion and Flame 57 0 [4] Kessler D A, Gamezo V N, Oran E S 00 Combustion and Flame [5] Heidari A, Wen J X 04 International Journal of Hydrogen Energy [6] Chao J, Otsuka, and Lee J H S 005 Proceedings of the Combustion Institute [7] Lee J H S, Knystautas R, and Yoshikawa N 978 Acta Astronaut [8] Zhu Y J, Chao J, Lee J H S 007 Proc. Combust. Inst [9] Saif, Wang W, Pekalski A, Levin, Radulescu I 07 Proceedings of the Combustion Institute [0] Kuznetsov, Alekseev V, stsukov I, Dorofeev S 005 Shock Waves 4 05 [] aeda S, inami S, Okamoto D, Obara 06 Shock Waves [] Radulescu I, Lee J H S 00 Combustion and Flame 353 [3] Wang C, Zhao Y Y, Zhang B 06 Journal of Loss Prevention in the Process Industries