Liquid Temperature Effect on Sound Propagation in Polymeric Solution with Gas Bubbles

Transcription

1 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 Liquid Temerature Effect on Sound Proaation in Polymeric Solution with Ga Bubble S. Levitky International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 Abtract Acoutic roertie of olymeric liquid are hih enitive to free a trace in the form of fine bubble. Their reence i tyical for uch liquid becaue of chemical reaction, mall wettability of olid boundarie, train of air in technoloical oeration, etc. Liquid temerature influence eentially it rheoloical roertie, which may have an imact on the bubble ulation and ound roaation in the ytem. The taret of the aer i modelin of the liquid temerature effect on inle bubble dynamic and ound dierion and attenuation in olymeric olution with herical a bubble. The baic ource of attenuation (heat exchane between a in microbubble and urroundin liquid, rheoloical and acoutic loe are taken into account. It i uoed that in the tudied temerature rane the interface ma tranfer ha a minor effect on bubble dynamic. The reult of the tudy indicate that temerature raie yield enhancement of bubble ulation and increae in ound attenuation in the near-reonance rane and may have a tron imact on ound dierion in the liquid-bubble mixture at frequencie cloe to the reonance frequency of bubble. Keyword Sound roaation, a bubble, temerature effect, olymeric liquid. A I. INTRODUCTION COUSTIC roertie of olymeric liquid have imortance for olymer rocein technoloy and acoutic monitorin of olymer dynamic module [1], []. Acoutic method i alo one of the well-known tool for bubble dianotic, becaue both ound eed and attenuation are hihly enitive to free a concentration [3]. Theoretical foundation of the method in the cae of olymeric bubbleliquid mixture need aroriate decrition of bubble dynamic with account for non-newtonian roertie of the liquid. Probably, firt aroach to modelin of bubble dynamic in liquid with comlex rheoloy wa formulated in [4], where effect of liquid hear-thinnin wa incororated in the oriinal Raleih model of herical cavity collae. Firt reult on bubble dynamic in vicoelatic liquid were reorted in [5], mall amlitude ulation of a bubble in Oldroyd liquid were tudied in [6]. Detailed review of further ublication in thi field, both theoretical and exerimental, can be found in the book [7]. More recent reult were reorted in [8]-1], etc. Comrehenive review of lat year ublication on bubble dynamic in non-newtonian liquid and certain relevant roblem can be found in [11], [1]. Theory of ound roaation in vicoelatic liquid with bubble wa develoed in [13], [14]. Vicoelaticity i the mot imortant roerty inherent to olymeric liquid S. Levitky i with the Shamoon Collee of Enineerin, 56 Bialik St., Beer Sheva, 841 Irael ( levit@ce.ac.il. (olution and melt. At a macrocoic level thi kind of non- Newtonian behavior i decribed by relaxational model uin time derivative of the tre tenor, or by equivalent interal equation [15]. Temerature ha eential imact on both relaxation ectrum of liquid and it vicoity, which yield temerature deendence of loe at inle bubble ulation in the wave and may caue chane in ound eed and attenuation in the ytem. The taret of the tudy i to decribe thi effect and to etimate it manifetation quantitatively for the cae of olymeric olution with fine a bubble. II. SINGLE BUBBLE DYNAMICS A. Baic Model General aroach to modellin of inle vaour-a bubble dynamic in olymeric liquid wa develoed in [16]. A ditinct to [16], the analyi reented below i limited to the cae of ure a bubble, which lead to eential imlification of the reultin relation for the amlitude of bubble ulation in a ound field, found in [16]. It i uoed that rheoloical behaviour of olymeric liquid, urroundin the bubble follow eneralized linear Maxwell model [15], which i ued uually for decrition of olymer liquid dynamic: τ t ij G( t t1 ij ( t1 dt1 + η ij 1 r 1 v v i j ij eij ( v I, eij ( + 3 x j xi Here G( t t1 i the relaxation function; τ, ij ij (1 - deviator of tre and rate-of-train tenor, reectively; η - lowmolecular olvent vicoity; v r - liquid velocity. Rheoloical equation (1 mean that only hear vicoelaticity of liquid i accounted for hereafter, and relaxation feature at bulk deformation of liquid in the wave are nelected. It wa hown elewhere [7] that the inut of liquid volume vicoelaticity in bubble dynamic i mall in a wide rane of condition, a comared with other ource of dierion and diiation. It i well known [3] that a bubble in a ound field of mall amlitude can be treated a linear ocillator with effective elaticity and diiation contant. Let denote the bubble radiu in the wave a a a + a, a / a << 1, where a i the equilibrium radiu. The reure in liquid can be decribed by the relation i t + ae ω. Then, chooin, a and equilibrium liquid denity ρ for characteritic arameter, and earchin the bubble radiu variation in the form International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16

2 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 i t e ω a / a δ, we can come [7] to the followin relation for dimenionle amlitude of the bubble, normalized by the dimenionle ound reure a a / : 1 δ δ / a [( χ i ν ] Ω Im{ G } Im{ D } T ν + + η + c f Ω Ω DT 1/ a( / ρ Pe, a Ω Ω, ( χ 4Re{ G } + Re{ D T } σ 3β γ, β iω Pe β 3(1 γ ( β cth β 1 k, a ρc c f ( t / a c, f ρ σ σ / ( a 1/ 1 T c (1 γ Ω ωt, t a ( ρ /, η η / ( t Here, ρ, T are equilibrium reure, denity and temerature of the a hae; k, c, γ - the a thermal conductivity, ecific heat caacity at contant reure and adiabatic exonent, reectively; Pe - Peclet number; η - olvent vicoity, c f - ound eed in liquid, σ - liquid-a urface tenion coefficient, G - comlex dynamic module of vicoelatic liquid: ( Ω θ F( θ ( i + Ω θ G dθ, θ θ / t 1 + ( Ωθ In (3 F ( θ i the non-dimenional ectrum of relaxation time θ. Equation ( i baed on exact olution of the roblem of bubble dynamic in vicoelatic liquid, derived in [16]. It i valid in a wide rane of frequencie becaue account not only for rheoloical loe, but alo for heat and acoutic diiation at inle bubble interaction with the liquid. When derivin (, it wa aumed that the thermal boundary layer thickne around ulatin bubble i much maller than it radiu and the reure in bubble doen't deend on the atial coordinate, that i ( t and homobaricity take lace in the a hae [17]. Parameter ν, χ in ( have the meanin of effective diiation coefficient and elatic contant of the bubble, reectively. One can ee that within the formulated model, rheoloy of the liquid influence both arameter in an additive manner. More exact analye of bubble dynamic in vicoelatic liquid [16], takin into account cro effect of liquid rheoloy, heat tranfer and liquid comreibility, ha howed that their inut in total diiation and effective elaticity of the bubble i minor and can be nelected in the frequency rane 6 ω << 1 Hz. In the cae of a dicrete ectrum (3 take the form, (3 where α (4 η 1 η k iωθ1 ( iω G z( α k 1 k α + ( Ωθ1 η i Newtonian (low-frequency vicoity of olymeric olution, and z( α i the Riemann zeta function of the ectral ditribution arameter α, introduced accordin to the Sri law [15]: θ θ1 / k α. B. Temerature Effect k The main relaxation time in the ectrum θ 1 can be etimated from the Roue formula [18]: ( η η θ1.68 M (5 φrgt where M, φ are molecular ma of the olymer and it concentration. The deendency of the olution Newtonian vicoity, η, on the olymer concentration φ at not too lare value of φ i decribed uually by the Martin relation: η (1 + βex( k β η M. Here k i the Martin contant, M β φ [ η] - reduced olymer concentration, [ η] - characteritic vicoity of olution, R i the univeral a contant and T G the abolute temerature. The value of [ η ] can be etimated from the Mark-Houwink relation: b [ η ] KM (6 where K and b are contant for a iven olymer-olvent air at a iven temerature over a certain rane of molecular ma variation. The arameter b (the Mark-Houwink exonent lie in the rane.5 to.6 for olution of flexible chain olymer in thermodynamically bad olvent and in the rane.7-.8 for ood olvent [1]. For the former one the contant K 1 (if the intrinic vicoity [ η ] i meaured in cm 3 3 /, while for the latter K 1. Temerature deendence of the relaxation characteritic of olymeric liquid i decribed by the time-temerature ueroition (TTS rincile [18], which tate that with chane in temerature the ectrum of relaxation time hift a a whole in a elf-imilar manner alon t axi, accordin to the value of the temerature-hift factor a T, defined by: ρ( T T ( η ( T η ( T at ρ( T T ( η ( T η ( T where ρ i the liquid denity. With Ω at for an arument it become oible to lot temerature-invariant curve Re{ G1 ( Ω a T } and Im{ G 1 ( Ω a T }. The temerature deendence of vicoity i decribed by activation theory: (7 η η ex[ E ( R T ( T / T 1] (8-1 G International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16

3 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 η η ex[ E ( R T ( T / T 1] -1 G International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 where E, E are activation enerie of the olution and the olvent, reectively; η η ( T, η η ( T. The E value i uually about 1 to kj/mol. For low-concentrated olution of olymer with moderate molecular mae, the difference between thee two activation enerie, E E E, doe not exceed uually 1 kj/mol [1]. For low-concentrated olution of certain olymer in thermodynamically bad olvent neative E value were reorted [19]. Relation ( wa tudied numerically with account for the temerature deendence of rheoloical arameter, accordin to (4, (5, (7, (8, which were written in the dimenionle form a: ( ( ( ( θ.68η Aex( k β, A [ η] M / R T 1 M G θ ( T a θ, a ( ρt η η / η η 1 1 T 1 T η η ex[ E ( T 1], η η ex[ E ( T 1] (9 1 1 ρ +, f f ( T / f ( T, α1 α1t f [1 α1( T 1] ρ ρ ρ { η, η, η, η} { η, η, η, η}/ ( t { E, E} { E, E}/ ( RGT, T T / T, θ1 θ1( T Here α 1 i the liquid thermal exanion coefficient. Plot of the non-dimenional normalized bubble amlitude δ1 δ veru frequency Ω and temerature T were obtained for different value of reduced olymer concentration β. The value of the ytem arameter, choen for imulation, are collected below; they were etimated for air bubble in olution 6 of olymer with M ~ 1 [9]: Pa, η.1 Pa, ρ 1 k / m (1 c f 15 m /, km.4, E 6.6, E 4.9 A 5, α1.3, σ.3 The non-dimenional value of activation enerie correond to E 1 kj / mol, E 16 kj / mol ; the tudied temerature rane i equal aroximately to 6K. The Sri ditribution arameter α wa choen equal to (Roue ditribution. Reult of imulation are reented on Fi For all lot 4 a 5 1 m, the tudied non-dimenional frequency lot 4 a 5 1 m, the tudied non-dimenional frequency rane for the choen arameter value correond aroximately to dimenional frequency1.5 < f < 11 khz, f ω / π. The 3D lot on Fi. 1 wa calculated for β 4. Fi. 1 Liquid temerature and frequency deendence of the bubble amlitude in olymeric liquid A it follow from Fi. 1, bubble amlitude row with temerature. The reult i exlained by the dynamic vicoity reduction. It can be een alo that the temerature ha minor effect on the reonance frequency of bubble in the tudied rane. Thi effect i clearly een on the lot reented on Fi., which were calculated for different concentration of olymer: the curve 1, 1' correond to β. ; the curve, ' - to β 4 ; and curve 3, 3' to β 6. The reonance amlitude of bubble dro with olymer concentration and for each value of β the amlitude-frequency curve hift lihtly to the left with temerature. The effect i exlained by reduction of vicou loe and liquid elaticity with rowth of T, accomanied by narrowin of the relaxation ectrum, and i more ronounced for larer olymer concentration. Reonance frequency of bubble i increaed with β due to increae in the liquid elatic module (the effect wa decribed alo in [] for free ulation of bubble in vicoelatic fluid. Fi. Liquid temerature effect on bubble amlitude at different concentration of olymer: Solid line - T 1, dahed line - T 1. Growth of the bubble reonance frequency with olymer concentration i reonible for another reult, illutrated by lot on Fi. 3, where the influence of liquid temerature on bubble amlitude wa tudied earately at ub-reonant and uer-reonant value of Ω. Deite all curve demontrate amlitude rowth with temerature, the lot, correondin to ub-reonant frequency (olid line 1 at mall olymer International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16

4 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 concentration β. (lihtly vicoelatic olution with η., η.4, i located above the dahed line 1, calculated for uer-reonant frequency. For larer value of β the relative oition of olid and dahed line i ooite. Thi reult i exlained alo by hift to the riht of the reonance curve becaue of chane in reonant frequency of bubble with rowth of olymer concentration. Fi. 3 Liquid temerature effect on bubble amlitude at ub-reonant ( Ω 1.8, olid line and uer-reonant ( Ω., dahed line frequencie: Curve 1-3 correond to β., 4, 6 III. SOUND PROPAGATION IN POLYMERIC LIQUID WITH BUBBLES A. Baic Model Sound roaation in olymeric liquid with bubble i decribed within homoeneou aroach [1], baed on the followin aumtion: bubble in the mixture are herical; their radii are much maller than the tyical ditance between the bubble, while the latter i maller than the ize of an averain volume, within which a rereentative number of bubble can be found. Liquid denity in the wave and it equilibrium value ρ, ρ, reectively, are related to the reure in the wave by the tate equation + c ( ρ ρ. f Here c f i the ound eed in a ure liquid without bubble, and, rereent the intant reure in liquid and it equilibrium value, reectively. Solution of the roblem i earched in the form of a wave, travellin in the oitive direction of x - axi, which i decribed in a non-dimenional form by a tandard multilier ex[ iω( τ x / c m ]. Here τ, x, c are the non-dimenional time, lonitudinal m coordinate and comlex ound eed in the mixture, reectively. It i well known that bubble, even a minor amount, are the baic ource of diiation at ound wave roaation, and that ound abortion in a ure liquid i mall, a comared with diiation ariin from radial ocillation of bubble in the wave [3]. The hyical reaon for thi reult i that diiation at ound roaation in bubbly liquid i located mainly in thin herical liquid layer with lare velocity radient around ulatin bubble [7]. Therefore, when formulatin the dierion equation for the mixture, the liquid rheoloy term can be retained olely in equation decribin bubble dynamic, and they may be omitted from the momentum balance equation for the bulk of liquid. Such aroach ha ained wide accetance when modellin wave roaation in ure vicou bubbly liquid [1]. For the cae of vicoelatic liquid, thi aroach wa verified in [14], where dierion equation for vaor-a bubble mixture in olymeric olution wa obtained by the multile catterin method, and rheoloy term were retained in both the momentum balance equation for the bulk of liquid and the equation decribin inle bubble dynamic. In the cae of a monodiere mixture of a bubble the reultin dierion equation i imlified eentially and can be written in the form (the detail may be found in []: (1 3 ϕ (1 ϕ ( cm c f Ω χ iω ϕ ϕ ν (11 Here ϕ i the equilibrium volume concentration of free a, and effective elaticity and diiation coefficient of a bubble are defined by (. B. Temerature Effect Reult of imulation for the cae of air bubble with equilibrium radiu 4 a 5 1 m and volume concentration 5 ϕ 1 are reented on Fi. 4 in the form of 3D lot of non-dimenional ound eed C Re{ c m } veru dimenionle frequency Ω and temerature T. The data correond to olymeric olution with β 4 ; the ret of arameter are the ame a for Fi. 1. Fi. 4 Sound eed in the bubbly mixture veru frequency and liquid temerature A it i tyical for bubbly mixture [1], the low and hihfrequency dierion (for Ω << and Ω >>, reectively i neative, while in the vicinity of the reonance frequency the dierion in i chaned for an ooite. Stron chane of ound eed in thi narrow frequency rane aear even at very mall a concentration. The dierion i enhanced with the temerature rowth. Effect of liquid temerature on the ound eed i well een on Fi. 5, obtained by rojection of 3D lot C C( Ω, T on C Ω lane for two bubbly mixture with different olymer concentration. The data reented on Fi. 5 correond to β. (lot 1 and β 6 (lot, reectively; the vertical arrow indicate direction of International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16

5 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 the temerature raie acro the lot on two branche of the dierion curve. Fi. 5 Temerature effect on the dierion rane A it follow from Fi. 5, the temerature influence on the ound eed in liquid with bubble i located in the bubble reonance rane, in accordance with reult, dicued in the reviou ection. The effect of liquid temerature i mall far from the tron dierion reion. The lat fiure i aimed to demontrate the liquid rheoloy effect on ound attenuation at different temerature. Curve, ' were calculated for ure vicou liquid with vicoity η, equal to Newtonian vicoity of olymeric olution with β 4 ; the curve 1, 1' correond to the ame liquid with account for it vicoelaticity. Fi. 6 Temerature effect on ound attenuation in ure vicou (curve, ' and vicoelatic liquid (curve 1, 1' For the line 1, - T 1; for 1', ' - T 1. The data reented on Fi. 6 indicate that temerature raie yield attenuation enhancement (both in vicou and vicoelatic liquid. However, it take lace only in the rane of tron ound dierion, were temerature chane lead to eential increae in the bubble ulation amlitude. Far from thi reion (for ub- and uer-reonant frequencie temerature raie lead to attenuation reduction. Note that in thi reion attenuation of ound in vicoelatic olution of olymer i le than in an equivalent ure vicou two-hae mixture. IV. CONCLUSION The tudy ha revealed eential imact of liquid temerature on inle bubble dynamic and ound roaation in olymeric liquid with bubble. The main effect of the temerature i localized in the vicinity of the bubble reonance frequency. Growth of temerature yield increae of the amlitude of acoutically induced ulation of bubble and lead to the ound dierion enhancement. The cale of the effect i enitive to olymer concentration it rowth i reonible for the bubble amlitude and ound dierion reduction. Dierion curve for more concentrated olution of olymer hift to the rane of larer frequencie, which i exlained by the imact of liquid elaticity on the natural frequency of bubble. Temerature raie alo yield the attenuation enhancement in the near-reonance rane. For uband uer-reonant frequencie the temerature raie lead to reduction of the ound attenuation. ACKNOWLEDGMENT The tudy wa uorted by the Shamoon Collee of Enineerin (SCE. REFERENCES [1] R. Kazy, L. Mazeika, R. Sliteri, and R. Raiuti, "Meaurement of vicoity of hihly vicou non-newtonian fluid by mean of ultraonic uided wave," Ultraonic, vol. 54, , 14. [] C. Glorieux, J. Decheemaeker, J. Vandenbroeck, J. P. Groby, L. Boeckx, P. Khurana, et al., "Temerature and frequency deendence of the vico-elaticity of a oro-elatic layer," Al. Acout., vol. 83, , 14. [3] T.J. Leihton, The Acoutic Bubble, Academic Pre, San Dieo, [4] W.J. Yan, and H.C. Yeh, "Aroximate method for the determinin of bubble dynamic in non-newtonian fluid", Phy. Fluid, vol. 8, , [5] J.R. Street, "The rheoloy of hae rowth in elatic liquid", Tran. Soc. Rheol., vol. 1, , [6] S.P. Levitky, and A.T. Litrov, "Small ocillation of a a-filled herical chamber in vicoelatic olymer media", J. Al. Mech. Techn. Phy., vol. 15, , [7] S.P. Levitky, and Z.P. Shulman, Bubble in Polymeric Liquid. Dynamic and Heat-Ma Tranfer. Lancater, USA: Technomic Publihin Co., Inc., [8] E.A. Brujan, "A firt-order model for bubble dynamic in a comreible vicoelatic liquid", J. of Non-Newtonian Fluid Mech., vol. 84, , [9] J.S. Allen, and R.A. Roy, "Dynamic of a bubble in vicoelatic fluid. I. Linear vicoelaticity", J. Acout. Soc. Am., vol. 17, ,. [1] S.P. Levitky, and Z.P. Shulman, "Bubble dynamic and boilin of olymeric olution", in Handbook of Solvent, nd ed., ch. 7., G. Wyych, Ed. Toronto: ChemTec Publihin, 14, [11] J. Naude, and F. Mendez, "Periodic and chaotic acoutic ocillation of a a bubble immered in an Uer Convective Maxwell fluid", J. Non- Newtonian Fluid Mech., vol. 155,. 3 38, 8. [1] E.A. Brujan, T. Ikeda, and Y. Matumoto, "Dynamic of ultraoundinduced cavitation bubble in non-newtonian liquid and near a riid boundary", Phy. Fluid, vol. 16,. 4-41, 4. [13] Z.P. Shulman, and S.P. Levitky, "Sound dierion in a relaxin olymer fluid with bubble", J. En. Phy. Thermohy., vol. 48,. 5-54, [14] S.P. Levitky, and Z.P. Shulman, "Proaation of ound wave in olymeric liquid with vaor-a bubble", Soviet Phy. Acout., vol. 31,. 8-1, [15] R. B. Bird, R. C. Armtron, and O. Haaer, Dynamic of Polymeric Liquid. Vol. 1: Fluid Mechanic. John Wiley & Son, International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16

6 International Journal of Mechanical, Aeroace, Indutrial, Mechatronic and Manufacturin Enineerin Vol:9, No:7, 15 [16] Z.P. Shulman, and S.P. Levitky, "Heat/ma tranfer and dynamic of bubble in hih-olymer olution II. Ocillation in a ound field", Int. J. Heat Ma Tranfer, vol. 35, , 199. [17] R.I. Nimatulin, Dynamic of multihae media. Wahinton: Hemihere, [18] A.Y. Malkin, "The tate of the art in the rheoloy of olymer: Achievement and challene," Polym. Sci. Ser. A, vol. 51,. 8-1, 9. [19] V. Budtov, Phyical Chemitry of Polymer Solution. St Peterbur: Chemitry, 199. [] Z.P. Shulman, and S.P. Levitky, "Heat/ma tranfer and dynamic of bubble in hih-olymer olution I. Free Ocillation", Int. J. Heat Ma Tranfer, vol. 35, , 199. [1] G.K. Batchelor, Wave in a bubble uenion, in Fluid Dynamic Tranaction, vol. 4, Fizdon, W., Kucharczyk, P. and Pronak, W.J., Ed. Warzawa, 1968, [] S. Levitky, R. Berman, and J. Haddad, Fluid rheoloy effect on wave roaation in an elatic tube with vicoelatic liquid, containin fine bubble, J. Non-Newtonian Fluid Mech., vol. 165, , 1. International Science Index, Aeroace and Mechanical Enineerin Vol:9, No:7, 15 waet.or/publication/16 International Scholarly and Scientific Reearch & Innovation 9( cholar.waet.or/1999.8/16