Propagation of Detonation Waves in Bubble Liquid with Consideration for Relative Slip of Phases

Transcription

1 Internationa Journa of Pure and Appied Mathematics Voume 119 No ISSN: (on-ine version) ur: Specia Issue Propagation of Detonation Waves in Bubbe Liquid with Consideration for Reative Sip of Phases Sergey Anato evich Lepikhin Industria University of Tyumen branch in Surgut Russia Surgut St. Enthusiasts 8 Abstract This artice investigates into dynamics of detonation waves in chemicay active bubbe iquid with consideration for reative sip of phases using numerica anaysis. The infuence of initia pressure and voumetric gas content of medium on veocity and ampitude of detonation waves has been anayzed. Key words: detonation wave bubbe iquid reative motion. 1 Introduction Occurrence of detonation waves in iquid with bubbes of fammabe gas is stipuated by energy reease in gas phase which supports the mode of bubbe detonation compensating energy oss of wave upon its propagation in medium [1 2]. The structure and properties of detonation waves are determined by physicochemica properties and parameters of bubbe gas iquid mixture its initia state [2-5]. It has been demonstrated in [6] that heat oss in detonation wave depends significanty on reative motion of phases of gas iquid system herewith dispacement of bubbes with regard to iquid phase can exceed characteristic sizes of the bubbes. This work anayzes the features of propagation of detonation waves in chemicay active bubbe medium at various initia conditions (pressure voumetric gas content) with consideration for sip of phases in the course of interphase heat exchange. 2 Formuation Of The Probem In order to sove the probem et us write the set of macroscopic equations of mass conservation amount of bubbes and puses with consideration for reative motion of phases as foows [7] : di i dn i ( i g) i i i n dt x dt x d p d d g f w g dt x dt dt a where g g g 1 4 g an. i αi υi are the density voumetric content and veocity of the i-th phase respectivey; a n are the radius and concentration of bubbes respectivey; p is the pressure of carrier iquid; w is the radia veocity of bubbes f is the force of viscous friction. The subscripts i = g denote the parameters of iquid and gas phases. The force of viscous friction can be written as foows: 1 f CD a g g 2 where υg is the reative veocity of phases. The resistance coefficient СD can be written as foows [6] : 48 CD Re 18 Re g Re 4 Re v CD Re where v is the viscosity of iquid Re is the Reynods number. For radia motion of bubbes in accordance with correction given in [8] we assume that w = wr + wa where wr is determined using the Rayeigh Lamb equation: 2 dw R 2 g w p R a wr 4 dt 2 4 a and wa is determined from the soution of spherica discharge probem on sphere with the radius a in carrier iquid in acoustic approximation: p w A 1/ C g where C is the sound veocity in pure iquid. Let us write the equation for gas pressure in bubbes as foows: d ( 1) da w q w. dt a a dt where γ is the adiabatic index for gas; q is the intensity of heat exchange or heat fow from gas to iquid per interphase unit surface area. We assume that the iquid is acousticay compressibe and the gas is caoricay perfect: 2 p p C ( ) p BT g g g where B is the reduced gas constant. Here and beow the subscript denotes initia undisturbed medium 2441

2 Internationa Journa of Pure and Appied Mathematics Specia Issue state. Let us set the interphase heat exchange by approximate finite correation [7]. Herewith it is assumed that as a consequence of reative sip of phases the bubbe surface is competey regenerated [6]. In this case the heat fow wi be determined by heat conductance of iquid: Nu 65 Pe T q Nu g Pe k a T p a k / ( c ). It shoud be mentioned that in the frames of mode without consideration of two-veocity effects the heat fow q is imited by the heat conductance of gas phase: T q Nu gg a T p a Pe g Peg 1 T a w Nu g Peg 12( 1) 1 Peg 1 T kg k / ( c ). g g g g where T = const is the temperature of iquid; Nui and Pei are the Nusset number and the Pecet number respectivey; ci λi and ki are the heat capacity heat conductivity and therma diffusivity of iquid (i = ) and gas (i = g) respectivey. In this work oxyacetyene stoichiometric mixture C2H O2 was considered as gas phase and 5% mixture of gycero and water as iquid. Gas infammation inside bubbes are considered in the frames of instant fowchart according to which the gas temperature is varied by ΔT [9] upon reaching certain critica vaue T*. Resuts And Discussion In this work the impact on stationary bubbe iquid was performed by instant pressure increase by ΔP at initia boundary (x = ) of the considered area. The parameter ΔР was seected so that to provide origination of detonation wave on initia boundary at any parameters of medium initia state. The thermophysica properties of carrier iquid were as foows: = 11 kg/m v = m 2 /s c =. kj/(kg K) λ =.42 W/(m K) С = 17 m/s T = 29 K and of the gas phase: a = 1.25 mm = 1.26 kg/m λ = W/(m K) γ = 1.5 cg = 1.14 kj/(kg K) T* = 1 K ΔT = 2 K. Figure 1 iustrates cacuated curves of ampitude of detonation wave reative bubbe radius and gas temperature in the bubbes in 8 ms after initiation of impact of boundary pressure on bubbe iquid. The soid ine corresponds to cacuation with consideration for reative motion of bubbes the dashed ine to one-veocity mode. Initia parameters of medium are as foows: р =.1 MPa αg = 4%. It can be seen in the pots that cacuations with consideration for two-veocity effects give rapid decrease in gas temperature in bubbes and pressure in iquid after detonation wave neary to initia vaues whereas cacuation according to one-veocity mode corresponds to high temperature of gas phase and increased pressure in iquid in the region after detonation wave. Fig. 1: Distribution of pressure in iquid reative bubbe radius and gas phase temperature. Increase in reative bubbe radius and voumetric gas content near initia boundary (x = ) of the considered region is reated with the fact that in the cacuations at this boundary after short pused impact on iquid the gas pressure in bubbes after detonation wave is assumed to be equa to that of initia state of bubbe iquid whereas the gas pressure in bubbes after detonation wave is somewhat higher than р. As a consequence at the boundary of the considered region there appears area of bubbe growth with its subsequent penetration in the considered region. It is known that the propagation veocity of detonation wave in bubbe iquid is higher than the equiibrium sound veocity in this medium though it is ower than the sound veocity in pure iquid. Figure 2 iustrates the speed of detonation wave D as a function of initia voumetric gas content of bubbe 2442

3 Internationa Journa of Pure and Appied Mathematics Specia Issue iquid. The medium parameters are the same as in Fig. 1. The symbos correspond to experimenta data [4] the soid ine corresponds to cacuations with consideration for two-veocity effects the dashed ine to cacuations according to one-veocity mode. It is seen that the obtained resuts of veocity of detonation wave on the basis of two-veocity mode agree better with experimenta resuts. Fig. 2: D(αg) as a function of gas content. Symbos are experiments ines cacuations. The properties of detonation waves occurring in bubbe iquid aso depend on initia state of gas iquid system on initia pressure in particuar. With decrease in initia pressure in bubbe iquid the ampitude and veocity of detonation waves decrease as a consequence of accompanying decrease in caorific capacity of the system. Figure iustrates veocity and ampitude of detonation wave as a function of initia pressure of medium. The symbos denote experimenta data [4] the ines cacuations according to two-veocity mode. Points 1 4 in Fig. a correspond to initia voumetric content of gas phase αg =.5% points 2 5 αg = 1% a) b) points 6 αg = 2%. It can be seen that the veocity of detonation wave as a function of initia pressure of bubbe system is neary inear and at ow voumetric gas content (αg =.5%) the veocity of detonation wave approaches the sound veocity in iquid. The ampitude of detonation wave as a function of initia pressure of bubbe iquid is aso neary inear and increases with increase in voumetric gas content (Fig. b). Line 1 corresponds to initia voumetric content of gas phase αg =.5% ine 2 to αg = 1% ine to αg = 2%. Fig. : Veocity (a) and ampitude (b) of detonation wave as a function of initia pressure of bubbe iquid. Symbos are experiments ines cacuations. 4 Concusion This artice anayzed the infuence of twoveocity effects on pecuiar feature of propagation of detonation wave in bubbe iquid. It is demonstrated that simuated cacuations with consideration for reative motion of phases provide resuts which correspond to experiments to a higher extent than those without consideration for two-veocity effects. In particuar gas temperature rapidy decreases in bubbes and pressure drops in iquid after detonation wave neary to initia state of the medium. In addition it is demonstrated that cacuation of veocity of detonation wave provides resuts which are coser to experimenta. The infuence of initia pressure of bubbe medium on veocity and ampitude of detonation wave has been anayzed. It has been detected that with the increase in pressure of initia state of gas iquid system the veocity and ampitude of detonation wave increase neary ineary which aso agrees with experimenta resuts. References [1] Sychev A.I. (1985). Vona detonatsii v sisteme zhidkost` - puzyr`ki gaza [Detonation wave in 244

4 Internationa Journa of Pure and Appied Mathematics Specia Issue iquid-gas bubbes system]. Fizika goreniya i vzryva 21() pp [2] Pinaev A.V. and Sychev A.I. (1986). Struktura i svoistva detonatsii v sistemakh zhidkost` - puzyr`ki gaza [Structure and properties of detonation in iquid-gas bubbes systems]. Fizika goreniya i vzryva 22() pp [] Pinaev A.V. and Sychev A.I. (1987). Viyanie fiziko-khimicheskikh svoistv gaza i zhidkosti na parametry i usoviya vozniknoveniya detonatsionnykh von v sistemakh zhidkost` - gazovye puzyr`ki [The infuence of physicochemica properties of gas and iquid on parameters and conditions of origination of detonation waves in iquid-gas bubbes systems]. Fizika goreniya i vzryva 2(6) pp [4] Sychev A.I. (215). Viyanie nacha`nogo daveniya puzyr`kovykh sred na kharakteristiki von detonatsii [The infuence of initia pressure of bubbe mediums on properties of detonation waves]. Zhurna tekhnicheskoi fiziki 85(4) pp [5] Gimatdinov I. K. Kucher A. M. (214). Detonation waves in a muticomponent bubbe iquid. High Temperature 52() pp [6] Shagapov V.Sh. and Abdrashitov D.V. (1992). Struktura von detonatsii v puzyr`kovoi zhidkosti [Structure of detonation waves in bubbe iquid]. Fizika goreniya i vzryva 28(6) pp [7] Nigmatuin R.I. (1987). Dinamika mnogofaznykh sred [Dynamics of mutiphase mediums]. Vo 1. Moscow: Nauka. [8] Nigmatuin R.I. Shagapov V.Sh. and Vakhitova N.K. (1989). Proyavenie szhimaemosti nesushchei fazy pri rasprostranenii von v puzyr`kovoi srede [Compressibiity of carrying phase upon wave propagation in bubbe medium]. Dokady Akademii nauk SSSR 4(5) pp [9] Shagapov V.Sh. and Vakhitova N.K. (1989). Vony v puzyr`kovoi sisteme pri naichii khimicheskikh reaktsii v gazovoi faze [Waves in bubbe system during chemica reaction in gas phase]. Fizika goreniya i vzryva 25(6) pp

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