PRESUMABLE CAUSE OF TORNADO EVOLUTION.

Transcription

1 P10.7 PRESUMABLE CAUSE OF TORNAO EVOLUTION. A.Gus kov, Insttute of Sold State Physcs RAS, Chernogolovka, Russa. INTROUCTION. We have obtaned the dependences of perod of eutectc pattern on a velocty of nterface, a mechansm of formng of cellular front under a crystallzaton of melts, ntercouplng between dependence of perod of eutectc pattern on a velocty of crystallzaton wth a number of atomc layers of nterface n papers Gus kov, Orlov [1-4]. These outcomes have common fault - model, used n these operatons, does not explan physcal reasons of orgn of the nterface nstablty, whch sold phase wll not vary, but n the lqud phase the pressure wll vary owng to a modfcaton of veloctes of components and mxture. Let's wrte down expresson for a dffuson flow [8] J ρ C (2) where s dffuson coeffcent. Let's substtute the equaton (2) n the dffuson equaton, whch n ths case becomes form w 2 C (3) ρ C wn N s mass flow consdered component. STATIONARY REGIMES OF IRECTIONAL PHASE TRANSITIONS. Fg. 1. Example of phase dagram. reduces n the lsted appearances. Reason s that classcal model of a drectonal crystallzaton [5, 6] s vald proposton only n range (fg. 1) of an ntal concentraton of a mxture, whch s range of a unlmted solublty of components. Now there s no theory, whch would explan behavor of the system n range of the restrcted solublty. In represented paper we try to explan behavor of the system n the range of the restrcted solublty. The classcal model of crystallzaton [5] consders statonary dstrbuton of concentraton n a lqud phase under condton of denstes equalty of lqud and sold phase. In ths case the condton of conservaton of mass flow s satsfed, f an ntal concentraton of the melt s equal to concentraton of the sold phase [5]. In the feld of the restrcted solublty the concentraton of the sold dffers from ntal concentraton of the melt. The requrement of a constancy of mass flows of components n an one-dmensonal case leads that the veloctes of the mxed flud and separate components should dffer from solvng of classcal model. The arsng addtonal dffuson flows lead to addtonal pressure between components of the mxture, whch s descrbed by the arsy equaton. dp J ρ( w w ) (1) dz here J s dffuson flow, ρ s densty, w s velocty of consdered component, w s velocty mxture. In bulk phases the pressure between components s counterposed by a mutual nfluence of dffuson flows of each of components. On nterface n the range of the restrcted solublty of such equlbrum wll not be, snce the pressure on the part of the We consder three statonary regmes of drectonal phase transtons. Frst regme s an equlbrum regme. An nterface velocty n ths regme s so small, that both phases are n equlbrum. The concentraton dstrbuton n both phases s a constant value, and value of concentratons s determned expermentally by equlbrum phase dagram. The second statonary regme s drectonal phase transton n the feld of a unlmted solublty of components. The nterface velocty s equal of such value, that the dffuson n one of phases can be neglected. The theory of such phase transtons s consdered n many monographs, for example n Pfann [9], Barton a.o. [10] for sold - lqud of the phase transton, Sedunov [11] for lqud - gas of the phase transton. In ths paper we consder a possblty of quanttatve descrpton of the statonary regmes n the feld of the restrcted solublty of components. Let's C s component concentraton on the lqud nterface, C sol s component concentraton n the sold. We want to fnd a poston C and C sol on the phase dagram for a nonequlbrum system wth arbtrary gven ntal concentraton and nterface velocty. We have a dffuson equaton, whch yelds of concentraton dstrbuton bascally at any boundary condtons. Intally n any statonary regme (except for equlbrum) the nterface concentraton s not defned. The classcal theory of phase transton supposes, that n the statonary regme C and C sol are nterrelated by the equlbrum phase dagram by a requrement of equalty of temperatures on boundary [9-11]. To fnd the concentraton dstrbuton, Burton [11] had used the classcal soluton of the statonary dffuson equaton for an nfnty nterval Vz s C( z) [ C( 0) C( ) ] exp + C( ) (4) An oblgatory requrement for use of the classcal soluton s the requrement of equalty of

2 components densty of flud and sold mxtures. All ths allows to fnd concentraton and temperature on nterface of the nonequlbrum system by geometrcal buld-up from the requrement of an equlbrum of the system on nterface n a quasequlbrum regme. Thermodynamc sense of the equlbrum state of a system s equalty of chemcal potentals of the mxtures component on the nterface. The classcal theory of phase transton [9, 10, 11] guesses, that n nonequlbrum statonary regme on nterface the equlbrum value of concentraton s fxed. Ths value, as was shown above, s assocated wth the concentraton C and C sol. Hence equalty of chemcal potentals of mxtures components s used here too. We keep n the common theory the same requrement for arbtrary denstes of components and arbtrary concentraton on exteror boundary. The theory of phase transton [9-11] consders the concentraton dstrbuton n a dffuson layer close to nterface. The concentraton value s set on outsde boundary of ths layer. The thckness δ of the dffuson layer depends on a hydrodynamc velocty of the flud [9, 10] n case of the sold - lqud phase transton or of the velocty of gas [11] n case of the flud - gas phase transton. The soluton of the dffuson equaton (2) looks lke N N w C () z ( 0) exp C + C δ δ w N N exp z + C ( δ ) ( 0 C ) + w w 2w N exp δ exp ( δ z) exp ( δ ) 1 + (5) The soluton of the dffuson equaton does not gve any nformaton on the value of nterface concentraton C. The arcy equaton gves allocaton of partal pressure. On nterface ths pressure depends on concentraton, nterface velocty and mass flow the component. We wrte down the solvng of arsy equaton n the form ( 0) ( ( 0 ),, ) p p C V N S However ths dependence does not gve the nterdependence of the nterface concentraton and the phase dagram. In the equlbrum V S 0, N 0 and the arcy equaton gves an ntegraton constant, whch s equal to fractonal pressure of equlbrum regme eq eq ( ( 0 ),0,0) p p C, (6) Together wth these solvng we consder dependence of the chemcal potental on temperature, pressure and concentraton. The common pressure s constant; therefore chemcal potental depends on temperature and concentraton. The value of concentraton on nterface s set only by equlbrum condton of the system n statonary regme, whch relates the chemcal potental of the component on the nterface to the chemcal potental of the component n the sold. The equalty of chemcal potentals of phases of the component at temperature of phase transton looks lke ( T, C ( 0 )) T, C ( 0) µ µ Sol e Sol Lq e Lq In ths equaton we know dependence ( ) T C 0, C 0 0 (7) e Sol Lq Let's consder the classcal solvng of the dffuson equaton (4). We do not know the value of concentraton on nterface C. In an experment s set only the concentraton C( ). The arcy equaton gves solvng ( _ ) ( 0) ( 0 ),, p p C V C (8) Clas Clas S Clas C( ) here s obtaned for the consdered solvng from the raton N C ρ w The dependence (8) relates fractonal pressure to concentraton on nterface. The equatons (7) and (8) contan three unknowns p, C and C sol, therefore for a solvng of the problem there should be one more equaton. We have found such equaton. Accordng to defnton of the chemcal potental t depends only on temperature, pressure and concentraton. ( ) µ T, C 0, P µ T, C 0, P Sol e Sol Lq e Lq here P s external pressure. We want to note, that the chemcal potental does not depend on fractonal pressure the component. Ths property of the chemcal potental specfes dstncton between propertes of fractonal pressure and component concentraton. The system becomes nonequlbrum, when the nterface begns to move. The movement of nterface leads to a varaton of the dffuson flow. The varaton of the dffuson flow leads to a varaton of fractonal pressure. If the velocty of the mxture w and ts densty ρ are constants, then the dstrbuton of fractonal pressure depends on the component velocty w and on hs concentraton C ρ / ρ. To assocate the varaton of fractonal pressure wth nterface concentraton we guess exstence of a onedmensonal boundary layer close to the nterface. Owng to exteror actons n the layer there s an fractonal overpressure, whch dffers from the equlbrum one. In lqud bulk the fractonal overpressure s compensated by relatve nfluence of the lqud components. On the nterface the fractonal overpressure should be compensated by fractonal pressure of the sold component. We guess that n the nonequlbrum statonary regme the fractonal pressure on the nterface has the equlbrum value. The concentraton on the nterface n ths case should vary. It becomes equal to a value, whch correspond to other equlbrum state of the system. STATIONARY REGIMES IN THE RANGE OF THE UNLIMITE COMPONENTS SOLUBILITY.

3 The equlbrum regme s a regme at a small nterface velocty. In ths regme the dffuson has tme to reduce both phases n the equlbrum state. The classcal solvng (4) can be transformed to equlbrum by a requrement of equalty of nterface concentraton and concentraton n the nfnty pont. In ths case the sum n brackets s equal to zero and for the equlbrum regme we obtan expressons N C ( z) and (6). Let's remark that n one-dmensonal statonary regmes always N const. Let's consder dstrbuton of concentraton and of fractonal pressure on an nfnte nterval at V S 0. The solvng of the equatons (1) and (3) n ths case look lke N wz N C ( z) C ( 0) exp + N wz p ( z) ρ C ( 0) exp p κ + eq At nterface the fractonal expresson of pressure has form ρ N p ( z) C ( 0) + p κ eq (9) To balance ths pressure between rgd and flud phases we should change concentraton C so that the bracket n expresson (9) converted to a zero. Hence we obtan for concentraton the equaton C ( 0) N Hence concentraton on boundary n a flud of an equlbrum condton should be equal to concentraton an nfnte pont of a classcal condton. But the concentraton at the nfnte pont of the nonequlbrum regme s equal to concentraton n the sold accordng to the equaton of balance of mass flow. We have obtaned the value of nterface concentraton of the nonequlbrum regme. The Burton a.o. theory [10] uses the classcal dffuson equaton. Ths theory guesses that the one-dmensonal dffuson layer s located close to nterface. The boundary condton n [10] s set on a fnte nterval. Concentraton dstrbuton looks lke [ ] C ( z) C 0 CSol Vz V s δ s exp exp + C ( δ ) Ths expresson concdes wth the classcal soluton (4) qualtatvely. It contans the same exponental curve and constant tem. If we shall substtute δ n ths solvng, t becomes equal to a classcal solvng (4). Hence value of parameters n ths case can be found as well as for the classcal solvng. STATIONARY REGIMES IN THE RANGE OF THE RESTRICTE COMPONENTS SOLUBILITY eut Let's consder the calculaton of values C and C Sol for solvng (5) n the feld of a restrcted solublty. Intal the expresson for concentraton on nterface s N w C ( 0) 2 exp C δ δ + 2N 2w C ( 0) exp δ + C ( 0 un un ) (10) 1 2w exp δ + 1 Ths equaton s obtaned as follows. We consder two statonary regmes. The concentraton C un regme Ι s n the nterval of the unlmted solublty 0 - C eut (for example n the nterval for the phase dagram of the fg. 1). The concentraton C regme ΙΙ s n the nterval of the restrcted solublty C eut - 1 (0.5-1 for the phase dagram of the fg. 1). When the system goes out the equlbrum, the values of component concentraton at frst are ncreasng n the range of the unlmted solublty up to C eut, and then go n range of the restrcted solublty, n whch C > C eut. The correspondng equlbrum values of component concentraton n the sold n both regmes Ι and ΙΙ are n the concentraton nterval of the left-hand lne of a lqudus ( for the phase dagram of the fg. 1). All parameters of the regme Ι we fnd how t s descrbed n the prevous tem. To fnd parameters of regme ΙΙ we use equalty of concentratons n the sold n both regmes. We guess that f the concentratons n the sold on the nterface are equal, then the fractonal pressure on the nterface of regmes Ι and ΙΙ are equal also. It follows from the conservaton law of the mpulse flow, because the arcy equaton s an approxmaton of the conservaton equaton of the mpulse flow [12]. We equate fractonal pressure of the component on the nterface of regmes Ι and ΙΙ. The obtaned equaton gves expresson (10) for concentraton on the nterface of the regme ΙΙ. The nterface velocty V S, thckness of the dffuson layer δ and concentraton on exteror boundary C (δ) are gven ndependent parameters n ths problem. Physcal propertes dependng from a composton of a materal are the densty of flud and sold mxtures and dffusvty. The veloctes of mxtures depend on the common gven nterface velocty V S. Three parameters C, C un (δ) and C Sol are unknowns n ths problem. The equaton (10) descrbes relatve movement of components of the mxed flud n the dffuson layer. Ths movement gves rse a addtonal dffuson flow, whch leads to addtonal fractonal pressure of components n the regme 2. On nterface ths addtonal pressure s equalzed by the change of components concentraton. The value of ths concentraton s obtaned from the equlbrum condton of sold and flud on the nterface. The obtaned here statonary regmes are unstable and can not be observed n experments. They lead to arse of nonunform structures. Specal

4 cases of these structures can be for example eutectc pattern or spontaneous condensaton resultng n formaton of atmospherc vortex. The area of a supercooled lqud s formed n bulk lqud phases before nterface accordng to the obtaned solvng. Ths area s smlar to the classcal concentraton supercoolng [5,6]. It s formed as a result of the value of component concentraton before nterface more than eutectc concentraton. Such supercooled layer can be cause of a plate-lke crystallzaton,.e. alternaton of a sold plates located of parallel of the nterface. At condensaton of gas ths layer leads to the accelerated transton of the nterface, whch n a result restrcts szes of nascent drops or snowflakes. Fgures 2 and 3 has shown concentraton dstrbuton and nterface movement velocty at 10-8 m/s, ρ ρ un, statonary nterface velocty V S 10-5 m/s, δ m, and for phase dagram of fgure 1. Concentraton on outsde of dffuson layer The one-dmensonal movement of the components wth dfferent veloctes n the range of the restrcted solublty can exst only n a lmted range of the system parameters. Really, the partal veloctes of components are equal to the velocty of the mxture n the regme of the unlmted solublty on any velocty of nterface. The stuaton vares n the range of the restrcted solublty. Paddng dffuson flows arse n ths range. Estmaton of exstence of lamnar one-dmensonal stream of monopropellant lqud s the Reynolds number. We can not estmate range of exstence of the onedmensonal lamnar stream of components of the mxture wth dfferent partal veloctes because the hydraulc theory of mxtures now does not exst. It s possble to do estmates of the exstence range of such stream only on the bass of expermental observatons. Narrow nterface velocty range, n whch the eutectc pattern s formed, demonstrates that the range of exstence of the one-dmensonal Fgure 2. δ s 0.3 (1), 0.42 (2), 0.45 (3), 0.47 (4), 0.5 (5). Accordng the calculatons the nterface s stable n the condtons of curve 1, and t s unstable n condtons 2 and 3. It s explaned to that n the condtons 1 value C(δ) s less, than the concentraton of the quas-equlbrum regme Ι n the same pont. The equlbrum dstrbuton coeffcent for all curves shown on fgures 2 and 3 s less than one. But the curves 1 correspond to slope of the lqudus m < 0, and curves 2-5 of m > 0. Hence, as shown 666 at parameters of the curve 1 the nterface s stable, and at parameters of curves 2-5 the nterface s unstable. These results explan physcal sense of the nstablty, found n papers [1-4]. We remark that statonary regme of curves 2-5 can not be observed n experments because t s unstable. If on the nterface the concentraton wll ncrease, temperature of phase transton of the flud wll ncrease too. The ncrease of temperature of nterface leads to ncrease of the knetc supercoolng and to ncrease of the nterface movement velocty. But the ncrease of the nterface movement velocty leads to the further ncrease of concentraton on the nterface. The smlar unstable process wll be the result and at spontaneous decrease of concentraton. Reason of nstablty s the rse of a dsperson of the components velocty of the mxed flud and, as a corollary, addtonal dffuson flows. UNSTABLE CONENSATION IN ATMOSPHERE. Fgure 3. components stream of the flud wth dfferent partal veloctes s small. Intutvely ths outcome s clear. The fluds have a large densty and consequently the partal velocty of components can not be large - streams become three-dmensonal. However there are phenomenons, n whch the consdered streams can be rather ntensve. The condensaton of a flud n an atmosphere s such phenomenon. Now we shall show, that the consdered nstablty well explans the mechansm of formaton of whrlwnds n the atmosphere - of cyclones, hurrcanes, and tornado. Most strkng example s of tornado formaton. We now shall show, that the requrements for a possblty of rse of unstable condensaton are fulflled, and that the offered mechansm of tornado formaton does not contradct any sngulartes of ths phenomenon. The unstable condensaton can arse f the followng requrements are fulflled. 1. The atmosphere n an area of condensaton s the mxture of components, whch have the restrcted solublty. The composte chemcal composton of the atmosphere s descrbed for example n the monographc [13]. The condensaton of drops s consdered there as condensaton of composte solutons. 2. The processes of condensaton should be descrbed by the transport equatons. Agrees [13] drops measure from 1 µm n clouds, almost up to 1 mm n a shower. The knetc exposton s appled only to drops by the sze of untes mcron

5 [11]. 3. In the area of the dffuson layer there should be the boundary on whch the partcular value of concentraton s supportng durng condensaton. Hence condtons whch are descrbed by the boundary value problem of the dffuson on a fnte nterval should exst. Ths boundary value problem was solved above. Agrees [11] ths requrement also s fulflled. urng relatve movement of the drop and ar the Prandtl number s close to unt. Hence dffuson and hydrodynamc layers practcally concde. From here follows, that the value of the component concentraton equal to concentraton of the atmosphere s supported at dstance of hydrodynamc layer n lmts of the dffuson layer durng relatve movement of the drop and ar. 4. The sze of drops, whch are formed as a result of unstable growth, should have restrctons, whch gve the sze observed n experment. Precomputatons smlar [1] for parameters of an atmosphere have shown, that the growth ncrement (the calculatons are made n an approxmaton of flat boundary) δ 10 6 ([1]). If to guess, that the growth velocty s restrcted to the sound velocty, the range estmate of the drop szes gves 1 mm. Let's remark, f not t s taken nto account of atmosphere hydrodynamcs, that theoretcally velocty of movement of ar s restrcted to the sound velocty. We obtan process to smlar vacuum exploson. In contrast to chemcal explosons, n the nsde area of drops shapng, n ths case the pressure wll be cushoned by a restrcted sze of drops and by emergent hydrodynamc flows between them. 5. Comparson wth other known mechansms of rse tornado. Agrees [14] the physcal mechansm tornado completely yet s not ascertaned. The basc hndrance to an explanaton of the mechansm of tornado rse s the mpossblty to explan an energy source necessary for exstence tornado. Ths energy on exstng estmates should have magntude approxmately 3.5 MW. We have set the requrement, that durng condensaton such energy s got. Then the estmaton of veloctes of front of spontaneous condensaton area of drops was done (not front of phase transton, but the front of atmosphere area, n whch happens condensaton). Accordng to ths estmaton the velocty of area boundary of spontaneous condensaton should be equal 2.3 m/s. It s the small velocty compared wth observable veloctes of movement of the atmosphere. Ths velocty also s small n comparson wth restrcton obtaned n tem 4. Hence phenomenon of condensaton has enough energy for mantanng tornado. Now we want to show, that the offered theory explans observable durng the acton of tornado the facts. These facts are enumerated n [14]. Basc of them the followng are. Tornado s lowered from the top downward and cause damage. Accordng to our theory the unstable condensaton develops n clouds. It spontaneously sezes the next areas of the atmosphere and works as a vacuum core. Ths core absorbs the atmosphere contanng water vapor. The restrctons on movement drecton of area of negatve pressure are not present. 2. Tornado forms the core crcled by whrlwnd n a horzontal secton. 3. Constantly there s a consderable swng pressure between the core and perpheral part of tornado. The offered theory explans phenomenon of tems 2 and 3. Hence any of the basc observable sngulartes tornados does not contradct the surveyed mechansm of tornado rse. REFERENCES. 1. A.P.Gus'kov, 1999, Physcs - oklady, v.366, N4, A.Gus kov, A.Orlov, ependence of perod of macro structures on knetc parameters under drected crystallzaton. Computatonal Materals Scence 24, A.P.Gus kov, 2003, ependence of the Structure Perod on the Interface Velocty upon Eutectc Soldfcaton. Techncal Physcs, Vol. 48, No5, A.P.Gus'kov. 2002, Physcs - oklady, v.387, N2, Bruce Chalmers. 1966, Phrncples of Soldfcaton. John Wley & Sons, Inc., New York. 6. Physcal metallurgy. 1983, Edted by R.W.Cahn and P.Haasen, North-Holland physcs publshng. 7. R.I.Ngmatuln. Bass of heterogeneous medums mechancs. 1978, Moscow, (Russa). 8. I.O.Protodjakonov, N.A.Marzulevch, A.V.Markov. 1981, Transport phenomenas durng a chemcal process engneerng. Lenngrad, (Russa). 9. G.Pfann. 1966, Zone meltng, John Wley & Sons Inc., New York. 10. Burto, J.A., Prm R.C., Slchter W.P. 1955, J. Chem. Phys., 21, S.S.Sedunov. 1972, Physcs of formaton of drops lqud n atmosphere.. Lenngrad, (Russa). 12. A.N.Konovalov. 1988, Problems of fltraton of multphase ncompressble flud. Scence, Novosbrsk, (Russa). 13. Peter Brmblecombe. 1986, Composton and Chemstry. Cambrdge Unversty Press. 14. Adran E. Schedegger. 1975, Physal aspects of natural catastrophes. Elsever scentfc publshng company.