Numerical simulation of combustion frozen nanosized aluminum suspension with water

Transcription

1 Nueical siulation of cobustion fozen nanosized aluinu suspension with wate V A Poyazov and A Yu Kainov Tos State Univesity, Tos, Russia E-ail: poyazov@ftf.tsu.u Absact. The pape pesents a atheatical odel fo cobustion of a fozen nanosized aluinu suspension (ALICE), taing into account the cobustion of aluinu in wate vapo, the otion of cobustion poducts, and the velocity lag of paticles copaed to gas. The odel was foulated based on Belyaev s appoach to odeling the cobustion of volatile fuels. The buning ate calculated is in ageeent with the expeiental data on the ALICE buning ate and its vaiation with pessue. Keywods: aluinu, fozen suspension, cobustion, nanosized paticles, ALICE, atheatical odeling. The linea buning ate of the fozen nanosized aluinu suspension is deteined by the ate of wate evapoation fo the aluinu suface. The aluinu paticles ae consideed to be beads, with thei size and disibution being unifo. Thee is supposed to be a cheical eaction between wate vapo and aluinu paticles, its poduct being aluinu oxide eaining on the paticle suface and hydogen ( H O O H ). The ignition of aluinu paticles occus when they each a cetain tepeatue [1]. The buning ate of aluinu paticles is descibed based on the expeiental data [, ]. The heat exchange between aluinu paticles and gas is defined by Newton s law. The aluinu paticles ae in otion due to the fiction caused by the gas. Because of the low volue concenation of aluinu paticles in the gas thei otion has no effect on the gas flow. The aggloeation of aluinu paticles on the condensed phase suface and thei inteaction with each othe in the gas phase ae not taen into consideation. The cobustion egie is supposed to be isobaic and the pessue does not depend on the distance fo the buning suface. æ T T ö T cç u l 4 pa n( T-T) è ø x u ) ( -G ( u) G H H Н О (1) () ()

2 æ T c è t T ö w x ø 4 ) ç - - GQ ( pa n T w ) ( G w w w -t n ( nw) æho ö H P RT const ç HO è H ø HO T -H H ; H In accod with [4], heat capacitance of aluinu paticle in pocess of waing, elting and cobustion of aluinu paticles follow the foula: ì c, solid, T < Tпл -DT ï L c í, Tпл -D T < T < Tпл DT ï D T ïî c, liquid, T > Tпл DT Hee, Equation (1) and Equation (4) ae the enegy equations fo the gas phase and fo the paticles, espectively. Equation () is the gas phase ass consevation equation, Equation () is the ass consevation equation fo the gaseous eaction poducts, and Equation (5) is the ass consevation equation fo the paticles. Equation (6) and Equation (7) descibe the otion of the paticles and thei nube, espectively. Equation (8) is the ideal gas equation witten fo a two-coponent gas ixtue. The coodinate x coesponds to the buning suface. The following elations ae valid on the evapoation suface, naely ass consevation of the wate flow (Equation (9)) and consevation of the ass flow fo the aluinu paticles (Equation (1)): (1- M ) 1V u (9) x ( ) M V w (1) 1 x The equality of the wate, vapo and paticles tepeatues is defined by: The tepeatue of the satuated vapo Clausius-Clapeyon equation: The initial conditions ae as follows: T T T (11) x x v Tv above the evapoation suface is defined based on the æ L ö P Aexp ç - RT è ø ; ( a), L Tv Rln PP LT O va (4) (5) (6) (7) (8) (1)

3 ( ) ; T ( x,) Tign ; ( x,) ; ( ) RT ( x,) T x T, ign HO P ( x ) ; u( x,) ; w( x,) ; ( ), H x, ; n x, (1) The heat flow fo the gas to the evapoation suface is consued to wa up the condensed phase to the ice elting tepeatue, to elt the ice, to heat the wate to its evapoation tepeatue and to evapoate it: T l V ct - ct - M V LL ( ) (1 ) ( ) 1 1 x 1 1, 1 x x The wate vapo density above the evapoation suface of the condensed phase is defined by the ideal gas equation: x HOP ; P const RT x The nube of paticles pe unit volue is deteined by the following elation: n x, x ( 4) p, whee, is the initial adius of the aluinu paticle; is the paticle density. Thee is assued to be no gaseous eaction poducts nea the evapoation suface, Н x At x the bounday condition is the following: T x x The foce of inteaction between the aluinu paticles and the gas is calculated applying the foula: t F 4p ; ( ) w u u w F CR S - - Hee, the esistance coefficient is defined accoding to the epiical foula given in [5]: 4 ( 1.15Re.68 C R ) ; Re The coefficient of heat ansfe is deteined as follows: Nul a ; Nu Nu Nu, whee l t Re.664Re.5 Nu l ; u-w h (14) (15).8 Nu t.7re [5] (16) O G n 4p, whee is assued to be constant (17)

4 éæ O ö ù ê, - ç ( 4 ) pn ú êëè ø O úû 1 ; é - ë ( ) O ê, When deiving Equation (18), we assued fo siplification that the aluina eained on the paticle suface as a spheical laye. This assuption is inoduced to deteine the actual adius of the aluinu bead and has no effect on its buning ate since the latte is taen based on the expeiental data. The buning ate of the aluinu paticle was calculated accoding to the expeiental data on the tie of aluinu paticle cobustion in wate vapo []. In calculations, the ignition tepeatue fo the aluinu paticles of the size given was consideed to be 14 K [6]. Nueical analysis fo cobustion of the fozen suspension of nanosized aluinu powde in wate was caied out at the ass concenation of aluinu powde coesponding to the stoichioeic ixtue of aluinu with wate. The following values of the theophysical paaetes wee taen in calculations: g, M.499, c1 cwat ( 1- M ) cm, wat 4185 ( ), c, solid 9 J ( g K), c, liquid 19 J ( g K), cho 6 J ( g K) 14 ( ) 6 g, R 8.1 J ( ol K),. Pa s c J g K c J g K, H l 1W ( K), ù ú û 1 (18), h, s, T1, 7 K, Q 15. MJ g [8], L 4 J g, L J g, c HOcHO H c, H.7 g ol, O.16 g ol,. g ol,,.4 4 n, P 4 MPa. H The set of equations (Equations (1)-(8)) with the initial and bounday conditions (Equations (9)- (15)) and the elations fo the ight-hand tes was solved nueically using the pocedue descibed in [7]. The aluinu paticle adius was consideed to be 4 n, the pessue above the buning suface vaying within the ange of 4. P. MPa, accoding to [5]. The ignition tepeatue fo the aluinu paticles of the size given was taen equal to 14 K. In [, ] thee is a foula fo the buning tie of the aluinu paticle in wate vapo (its diaete is within the ange of 15- μ), which defines the coefficient used in Equation (19). Calculations with the coefficient given in [] showed oveestiated values of the buning ate fo the fozen suspension of nanosized aluinu powde copaing with the expeiental data given in [8], so in the atheatical odel (Equations (1-)) the coefficient is consideed as the only one fo atching calculations with the expeiental data [8]. The expeiental data on the effect of pessue on the buning ate fo the fozen suspension of nanosized aluinu in wate [8] ae given in the ange of 4. P. MPa. The coefficient is selected so that the buning ate calculated atches its expeiental value [6] at the pessue of 1 MPa. Note, the authos [8] appoxiated the.5719 expeiental data obtained using the following foula: Ve.77P ss, whee pessue was -1 easued in MPa. The value of 1. 1 s obtained afte the atching descibed was used in ou calculations of the buning ate at othe pessue values. Figue 1 shows how the linea buning ate fo cobustion of the fozen nanosized aluinu suspension in wate depends on pessue. The dotted line pesents the expeiental data appoxiated.5719 in [8] by the following foula: Ve.77P ss (the pessue is easued in MPa). The diffeence between the buning ate calculated and the expeiental one does not exceed 8 %.

5 Figue 1. Linea buning ate fo cobustion of the fozen nanosized aluinu suspension in wate depends on pessue. The dotted line pesents the expeiental data appoxiated in [7]. A atheatical odel fo cobustion of a fozen nanosized aluinu suspension in wate has been developed. The nueical esults obtained fo the buning ate ae in good ageeent with the expeiental data fo the dependence of buning ate on pessue fo the cobustion of fozen nanosized aluinu suspension in wate. Acnowledgeent The epoted study was patially suppoted by RFBR, eseach poject А. Refeences [1] Belyaev A F 198 On cobustion of nioglycol Cobustion Theoy of Powdes and Explosives ed O I Leipunsiy and Yu V Folov (Moscow: Naua) pp 1 4 [] Becstead M W 5 Data analysis on the aluinu paticles buning tie Cobustion Explosion and Shoc Waves 41 (5) [] Yagodniov D A 9 Ignition and Cobustion of Powdeed Metals (Moscow: Bauan Moscow State Technical Univesity Publishing) [4] Poof ev V G and Solyaov V K Nonstationay cobustion odes of gasless systes with an easily-elting inet coponent Cobustion Explosion and Shoc Waves 8 () [5] 1987 A Handboo on Heat Exchanges (Moscow: Enegoatoizdat) vol 1 [6] Guevich M A, Lapina K I and Ozeov E S 197 Citical conditions fo aluinu paticles ignition Cobustion Explosion and Shoc Waves 6 () [7] Poyazov V A, Kainov A Yu and Kainov D A 15 Siulating the cobustion of N powde with added finely divided aluinu J. of Engineeing Physics and Theophysics 88 (1) 9 11 [8] Poupoint T L, Wood T D, Pfeil M A, Tsohas J and Son S F 1 Feasibility study and deonsation of an aluinu and ice solid popellant Int. J. of Aeospace Engineeing 87476