Void wave dispersion in bubbly flows

Nuclear Engineering and Design 121 (1990) 1-10 North-Holland 1 VOID WAVE DISPERSION IN BUBBLY FLOWS * J-W. P A R K , D.A. D R E W and R.T. L A H E Y Jr. Department of Nuclear Engineering & Engineering Physics, Rensselaer Polytechnic Institute, Troy, N Y 12180-3590, USA and A. C L A U S S E Centro Atomico Bariloche, 8400 Bariloche, Argentina Received 1 November 1989 A linear dispersion relationship is derived using a one-dimensional two-fluid model to investigate void wave dispersion in bubbly flows. This dispersion relationship includes generalized forms of the kinematic wave speed, the characteristics of the system of equations and the relaxation time. The relaxation time turns out to be a key parameter for the void wave dispersion. By using appropriate constitutive relations for bubbly flow, the kinematic wave speed and the characteristics are found. The Froude number is found to be the crucial parameter for void wave dispersion. That is, for two-phase flows with large slip between the phases (the small Froude number case) the dispersion effect is negligible and thus the kinematic wave approximation is valid. However, as the relative velocity decreases (the large Froude number case), void wave dispersion becomes pronounced. In the limit for zero relative velocity, void waves propagate at the same celerity as the characteristics for homogeneous conditions. The model presented herein also shows the existence of a complementary kinematic wave which is related to the kinematic wave speed and the characteristics. 1. Introduction W a v e p r o p a g a t i o n p h e n o m e n a in two-phase flow has been extensively studied since transient response and even some steady-state behavior (e.g., choking and flooding) are often controlled by the p r o p a g a t i o n of disturbances. Moreover, the propagation of void fraction disturbances (i.e., void waves) is p r e s u m a b l y responsible for flow regime transition [1]. In addition, since it has also been f o u n d that the properties of void waves strongly depend on the constitutive relations (i.e., the closure laws) used in two-fluid models [2,3], two-phase flow constitutive relations can be developed a n d / o r assessed b y investigating void waves. The concept of kinematic waves in two-phase flow has been discussed b y Wallis [4], and an extensive kinematic model for void waves was presented by Bour6 [5]. P a u c h o n and Banerjee [6] have derived an analytical expression for the characteristics of a two-fluid model which supposedly quantified the d y n a m i c behavior of void waves. Unfortunately, the available experimental data [2,6,7] have too m u c h scatter to allow one to properly assess such models. However, it is k n o w n that in two-phase systems the characteristics are basically responsible for p r o p a g a t i n g high frequency signals which decay faster than the lower frequency signals which are p r o p a g a t e d at the kinematic wave speed. Nevertheless, except for the studies of Bout6 et al. [8] and Ruggles et al. [3] no previous investigators have seriously studied frequency dependency on the void wave speed (i.e., the dispersion of void waves). In this study, a linear dispersion relationship is derived from a one-dimensional two-fluid model to investigate the void wave dispersion. This dispersion relationship includes generalized forms of the * Originally presented at the 26th ASME/AIChE National Heat Transfer Conference, Philadelphia, PA, August 6-9, 1989. 0 0 2 9 - 5 4 9 3 / 9 0 / $ 0 3 . 5 0 © 1990 - Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d ) 2 J-V~ Park et al. / Void wave dispersion in bubbly flows In this study, a linear dispersion relationship is derived from a one-dimensional two-fluid model to investigate the void wave dispersion. This dispersion relationship includes generalized forms of the kinematic wave speed, the characteristics and the relaxation time. Using appropriate constitutive relations, the kinematic wave speed and the characteristics for bubbly flows are found. The influence of some important parameters, such as Froude number and virtual mass coefficient, on void wave dispersion has been systematically investigated. 2. T h e o r y 2.1. Basic equations A one-dimensional two-fluid model for adiabatic and incompressible air/water flow in a constant area duct can be written using the conservation of mass and momentum equations for both phases: aak a (1) 3-7- + ~ ( a ~ u k ) =0, [ auk auk l Op k Pk [--gi- + uk-aT-z ] = az + A p k i 00~ k ak Oz 7"ki Oa k ak Oz 1 a v Mk, - cos o + ak 4 rk., Dn ak (2) where, ML, = --MG, = a c ( F D + Fvm + FR), and, Ap< =Pk, --Pk" By convention [9] the interfacial momentum transfer term has been partitioned into the interfacial drag ( F D), the virtual mass force (F~m) and a force due to bubble pulsation ( F R). In this study, the force due to bubble pulsation was neglected since it is normally only important near bubble resonance [9]. Such phenomena are of interest for pressure waves but not for void waves. Convenient dimensionless forms of eqs. (1) and (2) are: oak at* + (3) (akut) =0' [ ouZ , o u t l = _ ~apt + apt, Oak -a , 0z* P~[ Ot* + uk a z * J m - 9~ cos O + - - k,• ak ~* k, aak 1 a a z * + a-; v* ) %*,,,, 4 (4) ak Fro, where, ut= M *ki Uk URo, t* ---- g t, URo Mk, ~-~'-- PLg ' Z* --- g U2Ro Z, Pt Pk PLU2o ' "rk q.~ = ---'--'~' PL UR~ Pk pa~ = -PL -, gDH Fr0= 2 , URo U~=(UG0--UL0 ). 2.2. Linearization When the two-phase system is disturbed about a fully developed steady-state condition, the perturbed variables satisfy the following: J- W. Park et al. / Void wave dispersion in bubbly flows 38% - 3t* - 38% 38u'~ * - -OZ* + -}- Uko [ 0Buff %o (5) =0, OZ* 0Buff ] 3 Apk* o 3 8 % 38p'~ p~[ a - r ' - + u:0 o 7 1 = 3:* %o 8M;, M *ko +-- OZ* %o 4 3Z* T* 08r=~ -~- -%o -OZ - * -[- OZ* "~*wo 1 ~- aak/. &'k*w %2 8 % -- ~gro ko %o T*o 0 8 % r~z~ r k,o * 08ak %o %o (6) J To achieve closure, the constitutive relations must be expressed in terms of the state variables, u~, u~ and a(0/= % = 1 - %). Thus, we assume that the force perturbation on the right hand side of eq. (6) can be expressed as: OF* = 0F*30/ o 80/+ ~0F* o 8u~. + 0F*3u~ oOU~. (7) Equation (7) can only be used for the algebraic interfacial and wall transfer laws. Other 'forces' such as those due to virtual mass, must be treated differently. The nondimensional virtual mass force can be modeled as [10]: rc,_--~.*_--f°u~ 3u~ 3u~ Ou~] * "vm= -vm[ "dt* + U°'dz z" Ot* - - U *L- - az* (8) w .I From eq. (8), we obtain the perturbation in the virtual mass force as, [3aug 38u~ * afv,~=Cv,..[ at* +u*Go o~* 08u~ or* u * 38u~/ - - (9) Lo az* j" If we neglect the interfacial pressure difference for the gas phase (Ap~, = 0), we obtain 8 PG* - 8 P t = 8 A P t , , (10) where the surface tension between phases is neglected (i.e.: p~, =PL,). If we subtract the perturbed momentum equation of the gas phase from that for the liquid phase and eliminate gradients in the velocity, and the phase average pressure perturbations by using the linearized phasic continuity equations and eq. (10), we obtain, 380/ 380/ 328% 3280/ 028% K 1 0 ~ - + K2 0-ff~- + K3 0~--2 + K 4 3 t * 0z* + K s Oz*2 - 0 , where, K1 1(1 ( 1-% 1 - % 3u~ 1 1_ 3Ow % 1 % 0u~ K2 1 )411(1 ) % au~ o +ff~0 ~ u*G° 0Fr~ 0.) a o 3u~ 4 1 [ O,~wl U*Lo O,¢w +ff~o ~ / ~ ] o + 1-a----~ Ou~ o 1 (o~& I 1 -% 0u~ 1 3rdw % 3u~ o 1 [ 0FO[ + u*L° 3Fr~' o 1 - % o / 30/ Io 1---a o 3u~ %[ (lla) u*Lo O,&o 00/ ]o + 1-%---o 3u[ (11b) F *Do )2 (1-a o Uoo*0,tw ) % u*(3o O~w / % % 3u~ o 0u~ o / Lwo ~* 0u~ o] + ( 1 - % ) 2 -I- Gw° (11c) 4 J-W. Park et aL / Void wave dispersion in bubbly flows K3 - - 1 + - - p~ + 1 - ao ao U l__~0 K4=2 Gym ao(1 _ a o ) 2 ' U* Cv m [ UI*0 + p~--~o° + 1 - - a o ~ l _ a o a0 (lle) ' ,.2 ( u# Lo + . Go + Cvm 1-a------o P6 a o 1-aotl-ao+-~o. * T* T* ~(--ApLi--TzzL +'rzzG) + Oa a0 / u.:) . Go ApL~o + * zzL T* + G ) T- Lio + TZZL0 + 1-a o --T,* + T* Gio q'ZZGo 0~0 T* U*Lo ~(--ApLi--'rZZL +'rzTG) I0 + 1 - a~o U*Oo 3 ( _ A p L , * - * -- T* 0(--ApL, + _ _1 1 - ao Ou~ T* 0u~ u.2 T* TZZL+TZzG) ] u*Go + ao %:0) 1 ~ ( __ APL, . - ~':~L T* + K5 (lld) Ou~ 0 O" (11f) Ou~ Equation (11a) can be rewritten in more convenient form as: 03a 08a Ot-----~+ a * -ff~-c + T .( 0 r. 0__3_i( 0 r.~] -~-¢ + - Oz . ] ~ -O-~ + + 0 z , ] 6 a = 0 , (12) where, a+*= a +/URo = K z / K 1, (13) T * = K3/Ka, (14) r:~= uRo 2K3 - 4- K--;3] - ~ " (15) It can be shown [11] that a*+, r2 and T * are the dimensionless forms of the kinematic wave speed, characteristics and the relaxation time, respectively. It is interesting to note that eq. (12) is similar to, but more general than, previous results [12]. In summary, the assumptions used to obtain eq. (12) were: (1) All nonlinear effects are neglected. (2) Interracial momentum transfer due to bubble pulsation is neglected. (3) The surface tension between phases and the interracial pressure difference for the gas phase are neglected. 2.3. Dispersion relationship The dispersion relationship can be obtained by assuming a solution of eq. (12) in the form: ~Ot = or' e i(~*z*-°'*t*). (16) Inserting eq. (16) into eq. (12), we obtain the following linear dispersion relationship: i(w* - a ' x * ) + T * ( w * - r * x * ) ( w * - r ' K * ) = 0. (17) J- IV. Park et aL / Void wave dispersion in bubbly flows 5 Cot Cox wave + a+ F+ .................. *¢ a+ r+ ........................................ • ~ a_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C~ wave r* ........................... C ~x w a v e / .............. _~_._...................... a* ~ [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 ~R 0 ; "- r-OR ! a)]i' ; " a t- Ir ! + J Co~ w a v e ",r a+ ~ I + C o~ w a v e I r+ +°" s+'<_,+ T 0 . . . . . . il! , , O j "",, "t". . . . . . . . . . -:72 . . . . . . . . . . . . . . I ! ~ I !'",, -1 2¢ I i 2 T* C~ w a v e ! ', i / C/x w a v e I Fig. 1. Plot of eqs. (18) and (19)/(21) for r* < a* < r* (stable). "',,,, Ii '.1 Fig. 2. Plot of eqs. (18) and (19)/(21) for a* > r* (unstable). If we consider the region where eq. (12) is hyperbolic (i.e., where r~ are real), the solution of eq. (17) for traveling waves (i.e., where k* is real) can be found by solving the following coupled equations. 1 [a:-, ~-~" ~0~' = 2 T * ] 1 , ~0~2_ C*= [ ( a * - ?)2 - (C* - ?) 2 - 4 T ' 2 ( C * ---r )_-: 2-77:~ (C~. .r *.) (.C.* - r * ) (18) ] , (19) where, ~* = ~a~ + ivan, (20) c * = ,o~/K*, (21) ~= (r* + r* )/2. (22) One finds from eq. (19) that void wave dispersion is pronounced for large values of the relaxation time (T * ), since the wave speed ( C * ) is strongly dependent on angular frequency when relaxation time is large. It is interesting to plot eqs. (18) and (19). As shown in figs. 1 and 2, two speeds of propagation are possible (i.e., C+ and C~- waves) for a specified frequency. The larger one (C+) is easily recognized as the 6 J-W. Park et al. / Void wave dispersion in bubblyflows predominant speed of propagation since it is close to the kinematic wave speed (a+*). However, a complementary void wave (C~-) is also present. The dispersion relationship implies that the Ca wave is slower and has relatively high damping, as shown in figs. 1 and 2. The complementary kinematic wave speed, the counterpart of the classical kinematic wave speed, (i.e., C~- at zero frequency) can be found from eq. (19) as: lira C * = a _* -- r +* + r _* (23) a+* . - tOR~0 The well-known stability criteria [11] for C~+ waves can be easily found by examining the solid lines in figs. 1 and 2 as: (24) r*<a+ 3. Solutions and discussion Appropriate constitutive relations must be used to quantify the properties of void waves. The drag force, FD, for bubbly flow can be modeled as: L FD = PL~--~n(UG -- UC)lug -- uL l, (25a) where the interfacial friction factor for undistorted bubbles is given by [13]: D H 1 + 0.1 Re°~75 Rez, f, = 18 Db (25b) with, Re2 = L(l -- PL(UG + UL)Db + , D b : bubble diameter. (25c) The interfacial pressure difference for the gas phase has been neglected (ApG ' = 0), while for the liquid phase, ApE ` = --'r/Pc(U G -- UL)2. (26) The value ~/= ~ can be derived using potential flow theory. It is good assumption that the interfacial shear and the Reynolds stress for the gas phase is negligible (~-Gi= ~rTo= 0). However, for the liquid phase, we assume that the interfacial shear is equal to the bubble-induced Reynolds stress: % , = ZTzzL = - - k a O L ( UG -- UL) 2" (27) Biesheuvel et al. [14] have found that, k = ½. Since the gas phase is dispersed in the liquid phase, the wall shear due to the gas phase is negligible (rG,~ = 0). The wall shear due to the liquid phase is, .rL,~ = f w p L (28) u2. Using these constitutive relations and eqs. (13) and (15), we can obtain the dimensionless forms of the kinematic wave speed and the characteristics in a frame referenced to the liquid phase velocity: A~_ a + - UL0 -= 1-URo nao, (29a) J- W. Park et aL / Void wave dispersion in bubbly flows 7 where, +aO)r~_woDb/4.5ODn]Re2, o [ao'r~woDb/2.25DHU~o] Re2, o 4.5 +0.3375 Re°752,o+ [(1 1 + 0.1750 Re2°,7o5+ (29b) and r_+-- ULo G - -u~ = V* + v~f~-/~'*, (30a) where, V * zx (1 - or0) [ g y m - ~J - ko~ 0 + p~(1 - a0)] = %(1 - % ) + Cvm (30b) ' r * & %(1 - % ) + Cvm + pa(1 - %)2, P* ~ ( 1 - - O~0) 2 (30c) . 2 [Cvm - ~J - ko¢ 0 + P G ( 1 -- Or0) ] . . . . . . . . . + [%(1 -%) + 2(1 - 0/0)2(7/- -'~---~"~-2 Cvm+ 06(1- %) ] Cvm/2) -- 1 0 ; ( 1 +ao(1-ao)(n+k-Cvm) (30d) -- o¢0) 3. It should be noted that eqs. (30) reduce to the results of Pauchon and Banerjee [12] if we let: fw = 0, p ~ = 0 , C~m= ½, r/=-~ and k = ~ . If we neglect the wall shear stress in eq. (29b), we find that 1.93 _< n _< 4.5 for all values of Re2, o. To bound the possibilities, the dimensionless kinematic wave speeds given by eq. (29a) for n = 1.93 and n = 4.5 are shown in fig. 3 with the characteristics. According to the stability criteria given by eq. (24), the kinematic wave is stable over a wide range of Re2%. More specifically, since Re2¢o0 is proportional to relative velocity, n is increased as the relative velocity is decreased. Thus, the kinematic wave can be stabilized by reducing the phasic slip in the steady flow. It should be noted that since Re2, ° is also proportional to the size of bubbles, small bubbles stabilize the kinematic wave. The dimensionless relaxation time can be found by eq. (17): T*- gV_ FroRe2,o [ao(1-ao)+Cvm+P~(1-ao) 2] Up,o 18DH/Db 11 + 0.175 Re °7'2q~o+(aOrtwoDb/2.25DHU¢o)Re2~0] (31) " As can be seen in eq. (31), the Froude number (gDH/U2o) strongly influences the void wave relaxation time. As shown in fig. 4, the void wave is nondispersive for Fr o = 2.7 (UR0 = 30 c m / s ) which means that void wave propagation in a stagnant pool of liquid can be well described by the kinematic wave approximation. However, as can be noted in eq. (18), as the Froude number increases void wave dispersion becomes pronounced and damping (or amplification) decreases. It should also be noted that since relaxation time is not very sensitive to void fraction, the mean void fraction does not strongly influence void wave dispersion. As mentioned earlier, the dispersion relationship presented herein has a complementary kinematic wave (a_). Using the constitutive relations previously discussed and eq. (23), we find that the dimensionless speed of the complementary kinematic wave is: A*_ a - ULo - -)t*+X*_-A*. URo (32) J- W. Park et al. / Void wave dispersion in bubbly flows 8 / . A_ (n=4.5) / 1.0 / A+, A+ 0.5 50 x+ 7- o r 0 ::: O0 A* (n=1.95) T i b! LJ 27 ] I f \ -0.5 I /./ I \\ A*+ (n=4.5) 05 Fig. 4. D i m e n s i o n l e s s relaxation time (Cvm = 0.5, D H = 2.54 cm, D b = 1 cm). Fig. 3. Characteristics and kinematic w a v e speeds (Cvm = 0.5, T/= 0.25, k = 0.2, r~w° = 0, ~ = 0). 0.5 0 i i k i of i i~ o C rqq/'Is CrN J~ co I (I/~) J j" I0 c m / s -10 Fig, 5. T e m p o r a l d a m p i n g of the a _ w a v e (Cvm = 0.5, T~w° = 0, Dr~ = 2.54 cm, D b = 1 cm). h*wo=0, J- W. Park et al. / Void wave dispersion in bubbly flows 9 As shown in fig. 3, the coalescence of kinematic wave speeds (i.e., A*+ - A * ) occurs at a lower void fraction than where the characteristics coalesce (i.e., X* = ?,*) if the corresponding a+ wave is stable. If we use eqs. (18) and (32), we can obtain the damping of the a_ wave as (.01 : 18 urn[1 +0.175 Re°75 + (ao'r~woDb/2.25Dn)Re2,o] 2~° Db[a0(1 -- % ) + Cvm + 0~(1 -- Or0) 2] Re24, o (33) The damping of the a_ wave given by eq. (33) is plotted in fig. 5 for different UR0. AS can be seen, the a wave has large temporal damping when the phasic slip is large. Thus, based upon the constitutive relations used herein, it appears to be possible to observe the a_ wave when phasic slip is small. The unique features of the a wave (i.e., the complementary kinematic wave) - can be summarized as: (1) Its speed is determined by both the kinematic wave speed and the characteristics. (2) It is always stable independent of the stability of kinematic waves (i.e., a + waves). (3) It may have large temporal damping. 4. Summary and conclusions It has been shown that the linear dispersion model appears to be able to quantify void wave dispersion. According to our model, void wave dispersion is pronounced for large values of the relaxation time, which is determined by Froude number, the virtual mass coefficient and the void fraction. Since the relaxation time is small for large values of the relative velocity (i.e., where the Froude number is small), void waves in a stagnant pool of liquid can be successfully described by a kinematic wave model. However, since relaxation time increases as relative velocity decreases (i.e., the Froude number becomes large), the void waves become more dispersive when slip is reduced. The virtual mass effect also promotes void wave dispersion since the relaxation time is increased as the virtual volume coefficient is increased. The stability of the kinematic wave was analyzed based on our model. The kinematic was shown to be stable when the slip a n d / o r the bubble-size are reduced. The dispersion model presented herein yields a complementary kinematic wave speed which is determined by the kinematic wave speed and the characteristics. The complementary kinematic wave (i.e., the a_ wave) propagates in the flow direction in a frame of reference fixed to the liquid phase with damping if the corresponding kinematic wave is stable. It should be noted that, to date, no experimental verification of the presence of the complementary void wave ( a _ ) has been made. However, since the coalescence of the eigenvalues (i.e., ~ * = ~,*) signals the threshold for ill-posedness, and eq. (32) implies the following relationship at that point, A*= 2?t*-A~. We see that the onset of ill-posedness may be measurable if A* can be measured. Thus, it appears that further research on the complementary void wave may yield important insights into such phenomena as flow regime transition. It is hoped that this paper will help stimulate such research. References [1] A. Tournaire, D&ection et l~tude des Ondes de Taux de Vide an l~coulement Diphasique a Bulles Jusqu'a la Transition Bulles-Bouchons, Docteur-lngrnieur, Thesis, L'Universit6 Scientifique et Mrdicale et L'lnstitut National Polytechnique de Grenoble (1987). 10 J-I'lL. Park et al. / Void wave dispersion in bubbly flows [2] Y. Mercadier, Contribution a L'Etudes des Propagations de Perturbations de Taux de Vide darts les Ecoulements Diphasiques Eau-Air a Bulles, Docteur Engenieur, Thesis, L'Universite Scientifique et Medicale & L'Institut National Polytechnique de Grenoble (1982). [3] A.E. Ruggles, R.T. Lahey, Jr and D.A. Drew, An analysis of void wave propagation in bubbly flows, Proceedings of the Fifth Miami International Symposium on Multiphase Transport and Particulate Phenomena, Miami, Florida (1988). [4] G.B. Wallis, One-Dimensional Two-Phase Flow (McGraw-Hill: New York, 1969). [5] J.A. Bourr, Kinematic models, void fraction waves and other propagation phenomena in two-phase flows, Proceedings of the Ninth U.S. National Congress of Applied Mechanics, Ithaca, New York (1982). [6] C. Pauchon and S. Banerjee, Interphase momentum effects in the averaged multifield model, part I: Void propagation in bubbly flows, Int. J. Multiphase Flow 12 (4) (1986). [7] R.N.J. Bernier, Unsteady two-phase instrumentation and measurement, Report E200.4, Division of Engineering and Applied Science, California Institute of Technology (1982). [8] J.A. Bout6 and A. Matuszkiewicz, Stability of void wave fraction waves and bubble-slug transition, European Two-Phase Flow Group Meeting, Rome, Italy, June 19-21 (1984). [9] L-Y. Cheng, D.A. Drew and R.T. Lahey Jr, An analysis of wave propagation in bubbly two-component two-phase flows, J. of Heat Transfer 107 (1985). [10] D.A. Drew and R.T. Lahey Jr, The virtual mass and lift force on a sphere in rotating and straining inviscid flow, Int. J. Multiphase Flow 13 (1) (1987). [11] G.B. Whitham, Linear and nonlinear waves (John Wiley & Sons, New York, 1974). [12] C. Pauchon and S. Banerjee, Interphase momentum effects in the averaged multifield model, part II: Kinematic waves and interfacial drag in bubbly flows, Int. J. Multiphase Flow 14 (3) (1988). [13] M. Ishii and N. Zuber, Relative motion and interfacial drag coefficient in dispersed two-phase flow of bubbles, drops and particles, AIChE Paper #56a, 71st Annual Meeting of AIChE, Miami, Florida (1978). [14] A. Biesheuvel and L. Van Wijngaarden, Two-phase flow equations for a dilute dispersion of gas bubbles in liquid, Journal of Fluid Mechanics 148 (1984).