PRINCIPAL STATISTICAL RATIOS APPLIED TO THE SEPARATION PROCESS IN A VERTICAL TURBULENT FLOW

Transcription

1 PRINCIPAL STATISTICAL RATIOS APPLID TO TH SPARATION PROCSS IN A VRTICAL TURBULNT FLOW ugn Barsky Jrusalm Acadmc Collg of ngnrng ABSTRACT In th framwork of a statstcal approach to two-phas flows n th sparaton mod, th noton of flow moblty has bn formulatd. Prncpal rgularts of mass transfr n zon/apparatus systm ar analyzd. Prncpal das of a cllular modl of th procss ar formulatd, and man rlatonshps for ts dscrpton ar drvd. Procdng from th cllular modl, a dpndnc dscrbng th dgr of fractonal sparaton of a narrow sz grad s found. Th analyss of ths dpndnc maks t possbl to rval th crucal rol of th Froud crtron for gravtatonal sparaton procsss. It s found mprcally that ths paramtr s a bass for th affnzaton of sparaton curvs. INTRODUCTION Procsss of pourabl matrals sparaton ar rathr wdsprad n modrn ndustrs. In th ovrwhlmng majorty of cass, pourabl matrals ar sparatd n movng flows. Thortcal prncpls of such procsss practcally hav not bn dvlopd yt. To th prsnt day, thy ar basd on th bhavor of sngl partcls n a flow, whch s not gnralzd to a mass procss. Ths prvnts th dvlopmnt of unvrsal computaton mthods, whch srously mpds th dvlopmnt of sparaton mthods. Mass thory of th procss s basd on th gravtatonal sparaton of polyfractonal powdrs by thr gran sz. Th smplst gravtatonal sparator can b rprsntd as a vrtcal pp wth a gas or flud flow fd from blow and a polyfractonal mxtur fd from abov. Dpndng on th flow vlocty, fn partcls ar carrd out upwards, whras coars ons fall down n a countr-flow. Prncpal rlatonshps of gravtatonal sparaton can b radly xtndd to cntrfugal sparaton and smlar sparaton procsss wth countr-drctd moton of partcls. Basc das of a statstcal approach to th gravtatonal classfcaton procss wr lad n Rfs. [] and []. W procd wth dscussng ths ssu usng th systm of symbols accptd n ths paprs. W analyz a stady-stat flow (a flow wth th numbr of partcls n th systm rmanng constant n tm) n a systm rprsntd n Fg.. A charactrstc fatur of ths systm s a longtudnal partton along th flow. Ths partton sparats th systm comprsng, say, partcls of th sam sz grad nto two solatd flows. Lt th frst flow b charactrzd by th paramtrs N ; J, and th scond by N ; J. Not that th dynamc condtons of th flow ar not ncssarly th sam for both parts,.., χ χ (w w ). On can asly show that for such a stady-stat statstcal systm th followng rlatonshps ar vald:

2 N N N const J J J const () If w now rmov th partton, w obtan an ntgratd systm wth a possbl ntrchang of partcls. As shown n [], th most probabl confguraton of th ntgratd systm s that wth th maxmal numbr of admssbl stats at th sam chaotzng factor of both systms. Ths maxmal numbr can b dtrmnd by th analyss of th product of th numbrs of admssbl stats for sparat systms wth rspct to ndpndnt varabls charactrzng both systms. From th rlaton th xtrmum condton can b wrttn as ( N ; J ) ( N N ; J J ) d ( ) N dn J dj N dn J dj 0 () Takng () nto account, w can wrt: Hnc, ; N N dn dn dj dj J J Dvdng both sds of () by th product φ φ and takng nto account th drvd rlatons, w obtan: 0 N N dn dj J J Ths xprsson rflcts th condton of mutual lvlng of both systms. Ths dpndnc can b somwhat smplfd: ln ln ln ln 0 N N dn dj J (3) J Apparntly, th condtons of qualzng th two systms wll b satsfd whn th xprssons n brackts acqur zro valus, snc undr qulbrum condtons th scond brackt s zro, as stablshd prvously. Thus, w obtan from (3) that H N H and N H J H J

3 Th scond condton s known, - t s rducd to χ = χ,.. th valus of chaotzng factors n both parts of th systm ar qualzd. Th frst condton s nw. W ntroduc a notaton: H ( ) (4) N J whr τ s a paramtr sgnfyng a moblty factor corrspondng to th flow rat squard n a spcfd pont of th apparatus cross-scton. (5).., two systms that can xchang partcls gt qulbratd whn th rato of thr moblty factors to th chaotzng factor bcom qual. W hav assumd that th chaotzng factor s th man flow rat squard,.. χ = w. Th moblty factor charactrzs partcls and apparatus dsgn, but ts dmnson should b that of vlocty squard. Such a unqu charactrstc of a partcl s a magntud multpld by th sparaton factor, whch compltly dtrmns all th aspcts of a partcl bhavor wth rspct to a spcfc flow. Apparntly, ths paramtr s proportonal or qual to th flow rat of partcl n concrt pont w. Takng ths nto account, w pass to th analyss of a systm that comprss a constant numbr of partcls N α n a statc stat. At a dfnt flow rat, lftng factor of ths systm s charactrzd by th magntud J α. W convntonally dvd ths systm nto two parts and call th largr part an apparatus and th smallr part a zon. Th zon mpls a part of th vrtcal channl volum havng a modrat hght and covrng th ntr cross-scton of th channl. Th zon hght s accptd to b small, suffcnt for holdng a larg numbr of partcls, but nsuffcnt for apprcabl changs n th composton, concntraton and othr paramtrs of th flow. For th sak of convnnc, w locat th slctd zon on th uppr bordr of th systm, although n prncpl, t can b sngld out n any part of th apparatus, whch wll not affct th valdty of our drvatons. W ntroduc anothr lmtaton of th zon hght. It s chosn so small that all th partcls movng upward lav th lmts of th systm undr study,.. ar rmovd out of th apparatus. At th sam tm, th zon and th apparatus ar xamnd n a statonary stat wth a constant numbr of partcls n thm. W xamn statstcal proprts of such a zon takng nto account ts contact wth th apparatus. Th lattr mans that th flow rats thrn ar qual or, at last, rgdly connctd, whch s stpulatd only by th rato of corrspondng opn flow aras. Bsds, on should accpt th qualty of chaotzng factor and moblty factor. Th apparatus xchangs partcls wth th zon. If th numbr of partcls n th zon quals N (N << N α ), thr numbr n th apparatus quals (N α - N). If a systm posssss a lftng factor, thn th apparatus wll possss ths paramtr (J α ). Usng th rsult of [], w dtrmn th probablty for th zon to b found n th -th stat wth th lftng factor and to comprs N partcls at a gvn obsrvaton. -3

4 Th probablty P( ; N) s proportonal to th numbr of admssbl stats of th apparatus, and not of th zon, snc f th zon stat s fxd, th numbr of admssbl stats of th ntr systm s proportonal to th numbr of admssbl stats of th apparatus,.. P(N; ) s proportonal to ( N N);( J ). In ths rlatonshp th proportonalty coffcnt s unknown. W apply a mthod usually usd by th rsarchrs for ovrcomng ths dffculty dtrmn th rato of th probablts for th zon to b n two stats: P( N; ) P( N ; ) ( N N; J0 ) ( N N ; J ) 0 (6) From th dfnton of ntropy for th ntr apparatus, w can wrt: ( N ; J ) H N J ( ; ) Takng th lattr nto account, th xprsson (6) can b wrttn as P N P N ; ; H H( N N)( J ) H ( N N )( J ) (7) Ths xprsson can b xpandd nto Taylor s srs: H H H( N N; J ) H( N ; J ) N... N J W can wrt th ntropy dffrnc to wthn th frst ordr as follows: J N ( ) ( ) ( J ) ( J ) H N N N N H N J H J N H H ( N N ) ( ) N J (8) Usng th dfnton of th nwly ntroducd factors H J ; N H N J W rwrt th xprsson (8) as H ( N N ) ( ) (9)

5 It s notworthy that ΔH rfrs to th apparatus, whras N ; N ; ; to th zon. Thus, th changs takng plac n th zon prdtrmn th ntropy chang of th ntr apparatus. Ths s ntutvly clar, snc vrythng that lavs th zon and only that prdtrmns th magntud of fractonal sparaton dgr that w ar skng. Takng ths nto account, th dpndnc (9) gvs an xtrmly mportant rato from th standpont of a statstcal approach to th problm: P N P N ; ; N xp( ) N xp( ) (0) Th structur of ths dpndnc s smlar to th rato obtand by Gbbs whn studyng th thrmodynamcs of lmntary partcls of an dal gas. Although t comprss absolutly dffrnt paramtrs dtrmnng th procss undr study, w wll call t Gbbs factor for a two-phas flow. Anothr wll-known rato n thrmodynamcs s calld Boltzmann s factor. It can b obtand from Gbbs factor at a fxd numbr of partcls,.. undr th condton N = N. In ths cas, an xprsson smlar to Boltzmann s factor s wrttn as P ( J) P ( J ) () Th obtand rsults allow us to mak furthr stps n drawng an analogy btwn th procss undr consdraton and thrmodynamcs. W consdr som mor paramtrs whos manng s xcptonally mportant. If w summarz th dpndnc charactrzng Gbbs factor ovr all th stats of th zon and ovr all partcls, w drv an xprsson calld a larg statstcal sum: M ( ; ) N () N Such a sum s a normalzaton factor transformng rlatv probablts nto absolut ons,.., t plays th part of th prvously unknown proportonalty coffcnt. In chmcal kntcs, a larg statstcal sum s oftn xprssd usng so calld absolut actvty paramtr. In our cas, t s wrttn as: and w call t, by analogy, absolut moblty of th systm, and th larg sum n ths cas s

6 M N N (3) Usng ths das, w can stablsh that th probablty for th zon to b n th -th stat s dtrmnd as ( Lt Z N / ) (, ). N P( N ; J ) M Thn at a fxd numbr of partcls, th avrag valu of th lftng factor n th zon amounts to J Z Z ln Z Z (4) Avragng s accomplshd hr ovr th nsmbl of stats of th zon that s n contact wth th apparatus, but comprss a constant numbr of partcls n a statonary procss. To undrstand mass xchang wth th cll, w xamn a lmtng cas, whr only on partcl of a crtan fxd sz grad s constantly locatd n th zon. Thn w pass to th xamnaton of N ndpndnt dntcal partcls of th sam class. Lt us dtrmn a statstcal sum for on partcl. vdntly, on partcl has only two possbl stats wth th vlocty orntd upward or not. In th cas of non-orntaton upward, lftng factor s zro. For ths two possbl stats J Z (5) Th avrag lftng factor valu for on partcl s: 0 Z (6) Ths rlatons lad us to th ncssty of ntroducng anothr lmnt of th modl undr study ts cll. Howvr, ths wll b don somwhat latr, and now w wll try to dtrmn ntropy from ths vwpont. Lt us fnd th logarthm of th xprsson P( ) xp( / ) / Z Hnc, ln P( ) ln Z

7 (ln P ln Z) (7) Ths s vald for systms stablzd n tm. Th lftng factor xpctaton amounts to J P whr P s th probablty of a systm to b n th -th stat. Th magntud of P s dtrmnd by Boltzmann s factor. Takng nto account th fact that and (7), w can wrt dj dp Pd dh dp (ln P ) P ln Z dp Howvr, th probablts ar normalzd to a unty,.. P ; thrfor, dp 0. Hnc, w can obtan It can b shown that dh (ln P ) dp d ( P ln P) (ln P) dp dp ln( P) dp Takng ths nto account, w can wrt dh dp d( P ln P ) W obtan th followng xprsson for th chang n ntropy: and ntropy dh d P ln P (8) H P ln P (9) For a partcl orntd upward P =, othrws P = 0. Thrfor, H = - ln. Hnc, t s clar that no addtonal constants appar at th transton from (8) to (9). Not that (9) s calld ntropy dfnton accordng to Boltzmann. If a zon posssss φ quprobabl admssbl stats, thn s vald for ach of thm, and P Hnc, P ln P ln (ln ln ) ln H ln ln

8 whch totally concds wth th orgnal dfnton of ntropy []. Lt us rvrt to th cllular modl. W assum that th ntr volum of th apparatus s subdvdd nto rctangular clls n such a way that th volum of ach cll allows t to hold no mor than on partcl. If th sz grad undr consdraton comprss N partcls, w can assum that N clls n th apparatus ar occupd, and all th rst ar fr. Thus, t s assumd that th numbr of clls gratly xcds th numbr of partcls. Lt us xamn a systm consstng of on cll. It s mpld n ths cas that th apparatus rprsnts all rmanng clls occupd by (N -) partcls. It follows from th dfnton of a larg sum for on call that M (0) Th frst summand corrsponds to th cas of a non-occupd cll wth a zro lftng factor. Th scond summand corrsponds to an occupd cll whr M= and J =. Avrag occupancy of a cll amounts to n( ) x () Ths notaton gvs th avrag numbr of partcls of th class undr study pr on cll. Its magntud always vars from zro to a unty. It s notworthy that for coars partcls th paramtr of avrag occupancy of a cll concds wth th probablty, snc a cll can b thr occupd or fr. W dfnd th absolut moblty paramtr as. Substtutng ths xprsson nto (), w obtan n( ) () W dnot ths rlaton for th avrag occupancy of a cll by f ( ) n( ) (3) Ths functon shows th avrag numbr of partcls wth th paramtr orntd upward pr on cll. Th valu of f() always vars from zro to a unty. For th avrag occupancy of a cll, th dpndnc (3) s obtand. Takng nto account th fact that, by dfnton, for an solatd partcl w can show that gd

9 w 0. 5 whr w 0 s th partcl hovrng vlocty, and hovrng vlocty s proportonal to.5 th partcl sz,.. w0.5 gd. Avrag cll occupancy n th zon charactrzs th fractonal sparaton dgr: f ( ) (4) w 0. 5 w w Takng ths nto account, w analyz th dpndnc of th typ f ( ) f ( d) Fgur shows an approxmat flow structur pattrn. In dffrnt ponts of th crossscton w s dffrnt. If w consdr th pont whr w = w 0.5, n ths pont f ( d), whch corrsponds to th physcal manng of th sparaton procss, bcaus t shows th condtons of optmal sparaton. In th ponts whr w > w 0.5,.. for coarsr partcls, th xprsson w w s vald,.. n th xprsson x 0.5 magntud of x < 0, and th dpndnc f(d) > ½. In th ponts whr w < w 0.5,.. for fnr partcls, n th xprsson x th magntud of x > 0, and th dpndnc f(d) < ½. On th othr hand, th rato w 0.5 gd s proportonal to th dpndnc Fr w w. It was mprcally found that th Froud crtron s a dtrmnng paramtr for th class of procsss undr study. W hav managd for th frst tm to dmonstrat th ffct of ths factor from th vwpont of a statstcal approach. All ths furnshs a clu to furthr dvlopmnt of th procss thory. Lt us rvrt to th dpndnc (4). It can b wrttn as follows: th f ( ) w w w0. 5 w Th frst multplr n th dnomnator dtrmns th flow structur,.. gomtrcal paramtrs of th apparatus constructon. Ths shows that ths rato n a statonary procss n dtrmnd only by structural paramtrs.

10 Th scond multplr n th dnomnator dtrmns th st of flow paramtrs. In fact, w 4gd( ) c gd w 3 w w 0 cfr, whr c s a st of constant paramtrs. Th aformntond has a rlabl xprmntal confrmaton. RFRNCS [] Barsky., Barsky M., Statstcal prncpls of gravtatonal sparaton n a two-phas vrtcal flow.(manuscrpt) [] Barsky., Barsky M., ntropy of vrtcal two-phas flow n th sparaton mod.(manuscrpt) [3] Barsky, M. Fractonatng of Powdrs, Ndra, Moscow, 980. [4] Boltzmann, L. Lcturs on th Thory of Gass, Gostkhzdat, Moscow, 946. [5] Brlloun, L. Scnc and Informaton Thory, Acadmc Prss Inc., Nw- York, 956. [6] Chambadal, P.P. voluton t applcatons du concpt d ntrop, Dunov, Pars, 963. [7] Kttl, C. Thrmal Physcs, John Wly and Sons, Nw York, 977. Fg.