On phases volume flows evaluations in upward gas liquid flow

Transcription

1 On phases vome fows evaations in pward as iqid fow V.P.Drobkov Evaation of phases vome fows in vertica cyindrica channes based on the measrements of oca fow conditions is possibe ony sbject to adeqate notions on the two-phase fow strctre correctness of the correspondin comptationa mode and correctness of averain oca veocity vaes and fow as content over the channe cross-section. In this case satisfactory rests can be obtained for bbbe fow and as piston fow. Bbbe fow. In case of a bbbe fow a matter of interest is a coection of independent measrements of fow parameters in two pipe sements connected in series and differin in cross-section area. The vome as content of the fow in the i-th section (i ) averaed by pipe cross-section is F ϕ i F i i () where F i ( π / 4 )D i is a pipe cross-section area at the i-th sement F i Q / i is cross-section area occpied by as. The correspondin cross-section area occpied by iqid is F i (ϕ i )F i Q / i. Here i and i are as and iqid veocities respectivey. Usin in addition the foowin reationships F i Q / i redc Q / i redc redc i / ( ϕ i ) i i + re. i as we as a dependency correspondin to re the experimenta data [] i / ( ϕ i ) where is emersion veocity of sine as bbbes in infinite medim it is possibe to derive the foowin expression for iqid vome fow: Q F i [ i ( ϕ i ) ]. () By eiminatin vae from () we wi et: 85

2 F ( ϕ ) ( ϕ ) () Q where F / F. From eqations () and () we can derive a checkin expression for averae emersion veocity of sine as bbbes: ϕ ( ϕ ) ( ϕ ). ϕ (4) If we introdce notations [4] η c d b c σ ρ η c ρ η m. η.48 v c d exp(.4 D b ) the foowin expression can be sed for cacated vae of emersion veocity of bbbes with diameter d п : 5 6η η η 44η ; η if η 67 7η η 44η η m (c) 94η if η 67 ; v -5 (6) 7η ηm η 8 if η 67 ; 5 8 if η 67 77η η 94η if η 67. Vae η η m is correspondin to maximm emersion veocity of bbbes (on condition that η m < ). Fire presents a dependence between dimensioness veocity U (с) /ν and reative bbbe radis η / c for varios η vaes. (5) 86

3 Fire. U( ηη ) (c) / v η 75 η η 5 η η/ 4 c Averae as bbbe radis vae is determined from forma: P() r r () r dr (7) where r * ( r ) ½ P(r) is distribtion density of probabiity that a bbbe sticks on sensor at a distance r from its center. Assmin P(r) we sha et the foowin expression for determinin the averae vae of radis r bein fixed: r r dr r r + arcsin r π (8) Hence the acta bbbe radis wi be eqa to: ( 4 / π ) r.7r. (9) In view of a imited measrement accracy of the vome as content ϕ and ϕ a vae may distincty differ from a cacated vae (c) determined with forma (6). In this case it is necessary to make corrections of measred vaes ϕ ϕ and incded in expression () for Q sin the foowin scheme: + ˆ ( ) c ( ) () 87

4 ˆ ϕ ϕ ˆ () Q ( ) ( ϕ ) ˆ ( ˆ ϕ ) F () ˆ ( c) () ˆ ϕ ϕ ˆ (4) Q ( ) ( ˆ ϕ ) ( ϕ ). F ˆ (5) expression: The vae of a corrected iqid phase vome fow is defined by Q ( ) ( ) ( Q + Q ). (6) Gas piston fow. A distinctive featre of the as piston fow is sccessive trave throh a channe of are as bbbes (ss) occpyin a sinificant part of its cross-sectiona area and so caed iqid ps ocated between as ss (see Fi.). 88

5 Fire. At a channe section in the spot of as s movin there is an opposin motion of phases: a as s moves pward throh an effective channe with diameter D o Dδ and a iqid fim fows downward on the channe was. In so doin in case of bbbin there wi be eqaity of the vome fows of bbbin as Q s F o and fowin iqid fim Q fim F fim where F fim F F o F (π/4)d F o (π/4)(dδ). The as s rise veocity in the enera case (takin into accont the inertia viscos and srface forces actin on the s) can be described by expression []: s [ D( ρ ρ )] (7) k ρ where 9 k 45 e e 4 5 n 69 ( ) D Eo && ( ρ ρ ) σ 7 4Bo Eo & n 56 (8) (9) () [ D ρ ( ρ ρ )] () 89

6 Here is dimensioness viscosity Еö Eotvos nmber Во Bond nmber. For Eotvos nmbers Еö > 7 the second parenthesis in expression (8) is near nity (for exampe when D.6 m ρ k/m σ 5 - N/m Еö 4) so that when ρ «ρ we can assme: k 45 e 9. () Work [] in the framework of a two-ayer fow mode ives derivation of the approximated reationships for stabiized vaes of the thickness and veocity for a iqid fim fowin over a vertica srface the crvatre of which can be neected. The thickness of a aminar fim the veocity profie of which is near nity eqas δ am a ρ () where (τ w /ρ ) ½ (δ ) ½ a.5.6 [5]. Assmin δ am /δ m we wi et the foowin expression for the fowin fim thickness: a δ ρ m 5 m. ρ (4) Liqid fim thickness in a aminar-trbent fow area (m < ) is determined throh the iqid's vome fow in the fim Q by expression: δ Q /P fim where fim is the averae fim fow veocity P the perimeter of srface over which it is fowin. In case of a rond pipe with diameter D the perimeter P πd. Let s introdce into consideration a dimensioness parameter γ Q ρ /a P. Then the maximm aminar fim thickness δ δ max am wi correspond to vae m and in this case γ max max am am δ am fim ρ /a a/.8. 9

7 eynods nmber cacated for the averae iqid fim fow veocity am fim max γ am a /ρ δ max max am am e am fim ρ δ max max am / aγ am 44.. For the reation of m and γ vaes in a rane of.8 < γ < 4 we can derive the foowin expression: 898 m 44 exp( 59 γ ). (5) As we aready mentioned above in case of bbbin when a bbbe (s) is movin throh an arbitrary cross-section of the channe the foowin condition shod be met: s F fim F fim. Hence we et the foowin reation between the averae iqid fim fow veocity and as s movement veocity: fim F F fim s ( ) 4 ( ) s (6) Here δ /D is the dimensioness fim thickness and the s rise veocity takin into accont expression () can be written as foows: 9 45 e D. s (7) Besides from the iven reationships foows that 75 γ e 9 ( ). (8) We can derive the foowin expressions for dimensioness fim thickness and its dimensioness veocity fim fim /(D ) ½ : 4 exp fim [ 69 ( ) ] 567 [ ] 8 ( ) exp 4 86 ( ) (9) () 9

8 The above formas can be sed for iqids the dimensioness viscosity of which fas within a rane of -5 < < -. Fire presents dependences between dimensioness fim thickness dimensioness fim veocity and oarithm of inverse iqid dimensioness viscosity. Fi.. δ/d n(/*) n(/ ) n(/ *) * fim n(/ *) 9

9 In this case (in s fow reime) iqid and as vome fow vaes can be evaated sin the foowin formas: ( c) F Q ϕ ( ) ϕf + s. Q c () () Here - redced iqid veocity averaed by time interva and referred to f pipe cross-section; ϕ ϕ F /F ϕ as content vae averae by time interva and referred to cross-section F occpied bbbe and eqa to F π / 4 (D δ ). Since idea s fow reime doesn t occr in varios technica devices vaes and ϕ can be expressed by fnctions of measred parameters and ϕ satisfyin known experimenta vaes Q and Q if they are pt into the reationships () and (). 9

10 Concsions. Method and reationship for determination iqid vome fow are worked ot and proposed for bbbe fow reime. They are based on measrement of oca vaes of as veocity and of vome as content in two pipe sements connected in series and differin in cross-section area.. Easy in practice niversa expression for cacation of emersion veocity of bbbes varyin in diameter in iqids with different physica properties is proposed.. The scheme for correction of measred vaes ϕ ϕ and the expression for Q sin approbated anaytica reationships for presented. (c) is 4. Formas for comptation dimensioness fim thickness and its dimensioness veocity fim for s fow reime are proposed. They can be sed for iqids the dimensioness viscosity of which fas within a rane of -5 < < eationships for determination of iqid and as vome fow vaes are derived sin cacated fim thickness in s fow reime. 94

11 eferences 6. Уоллис Г.Б. Одномерные двухфазные течения: пер. с англ. М.: Мир Полянин Л.Н. Вопросы теплофизики водоохлаждаемых энергоустановок. М.: Энергоатомиздат Полянин Л.Н. Дробков В.П. Относительные скорости движения фаз в газожидкостном потоке на подъемных участках элементов энергетического оборудования // Тяжелое машиностроение Peebes F.N. Garber H.J. Stdies on the motion of as bbbes in iqids. // Chem. En. Pror. 95 V.49.. Лойцянский Л.Г. Механика жидкости и газа. М.: Наука