flow field around collapsing bubble trains: theoretical analysis

12
Appl. Sci. Res. 24 April 1971 FLOW FIELD AROUND COLLAPSING BUBBLE TRAINS" THEORETICAL ANALYSIS J. ISENBERG and S. SIDEMAN* Dept. of Mechanical Eng. / Dept. of Chemical Eng., Teehnion-Israel Institute of Teehnology Haifa, ISRAEL Abstract A method of successive approximations is utilized to analyze the potential flow field around freely rising collapsing bubble trains. For simplicity, the analysis is coniined to a two bubble column. An infinite series solution is obtained and the truncation errors are quantitatively analyzed. Nomenclature A, B coefficients C distance between centers of bubbles P Legendre polynomial q limit of summation R radius of bubble r radius S spherical multiplet U approach velocity of continuous medium /3 greater of either Ra/C or Rb/C velocity error at surface of bubbles 0 angle ¢ velocity potential subscrip•s a pertaining to bubble a b pertaining to bubble b k, m, n order of Legendre polynomial and spherical multiplet *) Currently on sabbatical at University of Houston, Houston (Tex.) U.S.A. -- 53 --

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Appl. Sci. Res. 24 Apri l 1971

FLOW FIELD AROUND COLLAPSING BUBBLE TRAINS" THEORETICAL ANALYSIS

J . ISENBERG and S . SIDEMAN*

Dept. of Mechanical Eng. / Dept. of Chemical Eng.,

Teehnion-Israel Institute of Teehnology Haifa, ISRAEL

Abstract

A method of successive approximat ions is uti l ized to analyze the potent ia l flow field around freely rising collapsing bubble trains. Fo r simplici ty, the analysis is coni ined to a two bubble column. An infinite series solution is obta ined and the t runca t ion errors are quan t i t a t ive ly analyzed.

Nomenclature A, B coefficients C dis tance be tween centers of bubbles P Legendre polynomia l q l imi t of summat ion R radius of bubble r radius S spherical mul t ip le t U approach ve loc i ty of cont inuous medium /3 greater of e i ther Ra/C or Rb/C

veloci ty error at surface of bubbles 0 angle ¢ ve loc i ty po ten t ia l

subscrip•s a per ta ining to bubble a b per ta in ing to bubble b k, m, n order of Legendre polynomial and spherical mul t ip le t

*) Currently on sabbatical at University of Houston, Houston (Tex.) U.S.A.

- - 5 3 - -

54 J. ISENBERG AND S. SIDEMAN

§ 1. Introduction

Collapsing bubble trains are encountered in direct contact con- densors utilizing immiscible fluids, as well as in sohble gas-liquid systems. This communication is devoted to the s tudy of the flow field associated with such buble trains. The determination of the flow field is mandatory before the temperature (or concentration) field is evaluated.

As shown by a number of workers [1-4J, the assumption of potential flow around, or near, gas surfaces is fairly accurate, and this approach has permitted the temperature or concentration field around single bubles to be determined with great accuracy in many eßses.

Solutions for ideal flow around two solid spheres [5, 61 and a row of solid spheres [7J are available in the literature. The present work deals with rectilinear flow combined with buble collapse. For simplicity of presentation, only two adjacent bubbles of different radii are treated liere, but the technique employed can clearly be extended to a rnulti-bubble column.

The mathematical method of attack employed here is similar to Herman's work [8] on two rising spheres with radial motion, dated 1887 (which, incidentally, came to our attention only after the present analysis was completed). This work may therefore, though unintentionally, be considered as an expansion of Herman's work. However, unlike Herman, the present authors, being motivated by the actual problem of condensation of multi-bubble systems, dwell upon the analysis of the accuracy of the truncated series solution required for practical applications.

§ 2. The model

A Lagrangian point of view with respect to the bubbles is un- tractable for application to the ensuing heat transfer model. There- fore, the bubbles (Fig. 1) are assumed to be stationary with respect to the continuous medium which is moving t o w a r d them with constant approach velocity, U. The approach velocity is parallel to the mutual center line of the bubbles; hence, the flow is axisym- metric. Based on physical observation of condensing bubbles trains in nonviscous media, the distance between them, C, is assumed constant. Lastly, surface tension forces are assmned to enable the bubbles to retain their spherical shape throughout the heat transfer controlled collapse.

«

FLOW FIELD AROUND COLLAPSING BUBBLE TRAINS g5

IIIu

Sphere 0 //?

Sphere b

Fig. 1. Schemat ic presenta t ion of two rising, collapsing bubbles.

Employing the velocity potential ¢, the governing equation for the flow field is the spherical Laplacian, V2¢ = 0. For convenience, ra -- 0a coordinates will be used with relation to sphere a, and rb -- Ob coordinates wi th relation to sphere b. The t ime-dependent bubble radii at a and b will be denoted Ra and Rb, respectively. Similarly, the corresponding velocities of collapse associated with these bubbles will be designated as Ra and Rb. The appropriate boundary conditions are

8¢ Ra at ra = Ra (1) ~ra

Bp = __ Rb at rb = Rb (2) ~Yb

~ra o r

- - U c o s 0 a and ~ 0 a

Ura sin 0a at ra ~ c~ (3)

- - U cos 0b and -- Urb sin 0b at rb --> oo ~rb ~)0b

The general solution to the spherical Laplacian is given by

c o

¢ = E Anr-n-lPn(cOS O) + BnrnPn(cos0) (4) n=0

which m a y be related to either the ra -- 0a or rb -- Ob coordinate systems.

56 J. ISENBERG AND S. SIDEMAN

The solution to the specific Neumann problem offered here, is obviously those combination of coefficients, An and Bn which satisfy the boundary conditions. Here, a method of successive approximations is suggested, which yields the required solution.

We start by assuming that the flow field around each bubble is independent of its neighbor, i.e., the bubbles are separated by an infinite distance. The flow field is then given by the sum of the following basic velocity potentials:

¢o = UraPl(cos 0a) = U[rbPl(COS Ob) -k C] Rectilinear flow (5)

R~R~ ¢ 1 - - Sink at a (6)

ra

R~Rb ¢2 -- Sink at b (7)

/'b

UR~ PI(cos 0a) ¢3 = 2 r~

- - ~ Doublet at a (8)

UR~, Pl(cos Ob) ¢4 -- 2 r~ Doublet at b (9)

Once the dlstance between the bubbles becomes finite, (5) to (9) still satisfy (3), but ¢1 and ¢3 violate (2), while (1) is violated by ¢2 and ¢4. The functions ¢1, ¢2, ¢3, and ¢4 must now be modified to yield the desired solution, while still eonforming to (3). Hence, all successive approximations must be confined to those solid spherical harmonics which vanish at ra -+ oo and rb -+ 0% i.e., the general multiplets

Sn,a = r~~-lPn(cos Oa), Sn,b = r~»-*Pn(cos Ob) (10)

Evaluation of ¢1 around sphere b, in terms of rb and Ob, is facili- tated by the generating function

, 1 ~ (r~~~ ra - c ~=o\ c } P~(¢os0b) (11)

where C > rb. This yields

¢1 = Ra2/~a ~ (~'b ~m ])m(COS0b). (12) C ~ ~ o \ C /

F L O W F I E L D A R O U N D C O L L A P S I N G B U B B L E T R A I N S 57

Equation (12) yields a radial velocity

~91 _ 2" oo /r \ ~ - i RaRa ~ ~ - ) Pm(cos Ob) (t3)

Drb C 2 ~t~ l m

which, by (2) taust be neutralized at rh = Rb. This is accomplished by the addition of the negating harmonics:

2 " o o

R~Ra E C 2

to ¢1, yielding

2R [ 1

~~ R~ m+ l Pm(cOS Ob)

.~= 1 m + i C m-1 r~~+ 1

oo p9ù.~+1 Pm(cos Ob) ] X/~ ~ ~q4 ~~b

,~=1 m + 1 C m+l r~n+l J" (14)

However, the series term in (14) yields a radial velocity at ra = Ra violating (1). In order to evaluate (and negate) this velocity, we have to express the general multiplet Sn,a in terms of the rb --- Ob coordinates and Sn,b in terms of r a - 0a. To accomplish this, point P (Fig. 1) is revolved on to the axis of symmetry where 0a = ~ and Ob = 0. (Since Sn,a forms part of the general spherical harmonics of the r b - Ob coordinate system given by (4), the coefficients An and Bn may now be evaluated). Equation (10) reduces to :

Sn,a---- (--1)nr2~~-I at 0a = ~ (15)

and, for Ob = 0, (11) yields :

"a'~+l - - , » = 0 \ C - ] ] at Ob = 0 (16)

which is essentially an exponentiation of a geometric series. Thus, the An coefficients are identically zero, and the Bn coefficients may be evaluated yielding the equation of translation:

Sn,a = ran-lpn(cos 0a) =

(~) = (--1)ne-n-1 Y~ (n 4-m)! m m--0 n!ml Pm(cos Ob) (17)

where C > rb. Similarly, the equation for a general multiplet in the r b - Ob

58 J. ISENBERG AND S. SIDEMAN

system is"

Sn,b = rg~-lPn(cos Ob) ~-

oo = C - n - 1 ~ ( - -1 ) m

~~,~ 0 (18)

where C > ra. Equations (i7) and (18) could also have been derived by means of an integral representation, as outlined by Morse and Feshbach [91.

Equation (18) is now utilized to express the series term in (13) in terms of the r a - 0a coordinate system:

m -~- 1 Rb 2~~+1 Pm(cos Ob) 2 ' R~Ra

~ . m (m ÷ })' p~,~+l • " b r~P»(cos 0a) : R~Ra N • (--1)k C 2m+~+2

m=lk=o m-I- 1 ra!k!

(19)

DiIferentiation of this double series with respect to ra gives the radial velocity component in the region of the sphere a. Evaluating this radial velocity component at ra = Ra yields a radial flow that must also be neutralized. This is accomplished by the addition of the following double series to ¢1, in (I3).

ex~ cx~ f]4k(m + Æ) l. *'où/~2k+ lp2ro,+ l ~ ~ b Pk(cOS 0a) 2" R~Ra Z Z ( - 1 p

m=l k=l (m @ 1)!(k -}- 1)! C 2m+k+2 R~ +1

The procedure outlined entails correction 0I the assumed primary flows by utilizing adjusting spherical harmonics on one sphere. This, in turn, induces a radial flow on the nëighbouring sphere, which must now be neutralized. This is accomplished by adding adjusting spherical harmonics to ¢. This again affects radial flow at the first sphere, which must be similarly neutralized. Clearly, the accuracy of the obtained solution depends upon the number of such steps employed, with the error limited to the last non- neutralized velocity potential• The flow field obtained after a two step or second order successive approximation is given by:

¢0 ~- UraPl(cos Oa) = U[rbPl(cos Ob ~- C~ (20a)

FLOW FIELD AROUND COLLAPSING BUBBLE TRAINS 59

. [ 1 oo m ~,2,~+1 P,»(cos Ob) ¢1 + E *-b + RaRa t. ra ,,~= 1 m @- 1 C m+l r~t,+ 1

oo , m k ( m + k)! .,~~~k+tP2'»+l..b Pk(cos 0a) ] + ,~=El k=E1 (--1)» (m + 1)t(k + 1)I. C2rn+~+~ *~'k+l fl (20b)

¢~ = R~Rb [ 1 ~ m R 2m+1 Pm(cos 0a) rb ,,,--~ m + 1 C.~+1 rm+l

oo ~~,~+ip2k+l P~(cos Ob) -] • *~& ~~b ( 2 0 C ) + E E m k ( m + k ) )

k+l J m,= 1 k= 1 ('I/H, @ 1 ) !(Æ 4- 1) • C ~m+k+2 r b

p~m+l P~(cos Ob)

2 L r~ .,:1 r b

c~ ~ ~~¢k(I,b @ Æ) l 7p2k+1p2m+1 P k ( e o s 0a) - - E ( - - 1) ]~ . - j (20d)

1 /c=l 1~t!(k q- 1)) C 2ra+k+3 .k+~ ) ~ " Y a

¢ 4 - URb~ V -Pl(c°S 0b) ~ R ä +1 Pm(cos 0a/ 2 L y2 @ m= ~ 1 ( - 1)m ~ß C ~ t + 2 ga m+l '

~ m k ( m + k)) /~2m+lp2k+1 pk(COS Ob ) 1 + E E • ~'a ~'b

'~'~= 1 k= 1 "q~ !(k @ 1) [ C 2m+»+a r~+ * J

÷

(20e)

where q)2, ¢a, and ¢4 are derived in a similar procedure to that outlined above for ¢1. The analysis of the truncation error involved in practical application follows in the discussion.

§ 3. Discussion

Truncation of the series solution creates a deviation between the exact and approximate solutions. This deviation in itseK may be expressed in terms of spherieal harmonics that a r e a so]ution of V2¢ = 0. The deviation must, therefore, fulfill the maximum principle for nonconstant solutions of elliptic partial differential equations Il• 1. By this prineiple, the deviation in ¢ as well as in V¢ can aehieve a maximum only on the boundaries, i.e., the surface of the spheres. These deviations are evaluated here.

As shown above, each subsequent appro×Jmating term requires an infinite series to neutralJze it. In praetice, only a limited number of series terms can be employed. The initial, or zero-th order approximation to the flow field, expressed by the primary terms propagates a zero-th order error, d (°), in the velocity at the surface

6 0 J . I S E N B E R G A N D S. S I D E M A N

of the neighboring sphere. This error is partially neutralized by the finite number of terms in the series obtained in the first order approximation. However, these terms in themselves propagate a first order error, 6 (1), which can be partially neutralized by generat- ing a finite neutralizing series for each term. This in turn yields a second order approximation and hence a subsequent second order velocity error, d rs). The highest order of the error is equal to the order of the approximation. The highest order, d ("), will obviously be very small, but nevertheless nonneutralized.

The magnitude of all these errors is best interpreted in relation to the collapse velocities of the spheres, Ra and Rs as well as the approach velocities, U. Ideally, these ratios should approach zero.

For simplicity, we confine the analysis to the error obtained from a first order approximation. This yields zero-th and first order errors, the latter being nonneutralized. Starting with ¢1 (20b), we assume the flow to be approximated by the primary terms and a finite number of terms q of the first order neutralizing series. Since the radial velocity induced by the primary term at the neighboring sphere, b, is not completely neutralized, the zero-th order error of ¢1, is given by:

~~o, ( ~ ; ~ (»;_~ - E m P,,,(cos Ob) (21)

/ ~a m = q+ 1

Letting Ob = 0 and denoting/3 = Ra/C or Rb/C, whichever larger, yields

1,max __ E mfl ra+l- ~ m~ m+l (22) Ra m=q+l (1 - - Æ)2 m = l

which corresponds to the maximum error at sphere b; hence, to the maximum error in the flow field evolved by the zero-th order error of ¢1. This maximum error is depicted in Fig. 2 as a function of fl for various values of q.

The q terms will also induce a first order error, d~l) in the ex- pression for the flow field at sphere a. This velocity is obtained by expressing each of the q terms in terms of ra and Oa, by means of (18) and evaluating the derivative with respect to ra, at ra ~ Ra.

FLOW FIELD AROUND COLLAPSING BUBBLE TRAINS 61

lO0 I I [ I

I0" q=01

ed ,o-3

to-

to 0.1 0 2 0.5 0.4 05 Oimensionless Radius, /3

Fig . 2. M a x i m u m z e r o - t h o r d e r e r r o r of oB1 or ~2.

The resulting veloci ty of the first order error is given by"

8~ 1) q - Z • ( -1)e

R a m = l k = l (m+l ) tk! \ c / \ c / P~(cos0~)

For 0a = 7:, (23) reduces to : (23)

6(1) q ~o m k ( m @ k) ! q mf i 2m+a 1 , m a x Z Z fi2m+k+2 = Z (24) Ra m=lk=l (m-j- 1)!k! m=l (1 __ il)m+2

I t is no tewor thy tha t for the limiting case of q -+ oo, this m a x i m u m error becomes

,~(1) f15 lim vl'm~x -- (25) q-~~ R a (1 - - ô)(1 - - /~ - - f l2)2"

Equat ions (24) and (25) are depicted in Fig. 3 for q = 1 and q -+ oo. The rapid convergence of the series (24) needs no elaboration.

By a similar procedure it m a y be shown tha t for ~2, (22) is identical ly equal to 8 (°) ~~ and (23) m a y be shown to represent 2, max/ lkb, Ô(1) /~)

2, max/l~-b"

62 j . ISENBERG AND S. SIDEMAN

I0 °

5

2

iO-I q~m

q=l

IC)"

I0 0 - - (11 0.2 0.5 0.4 0.5 Dimensionless Rodius,

Fig. 3. Maximum first order error of ~ 1 o r ~2.

Simi la r ly , for ~a a n d ~4

(5(o) ,~(0) 3, l l lBX ' a4 , Ill&X « m(m + 1)

U U ra=c+1 2

Ba q m(m + 1) Z t~~ +2 (26)

( ] __ f i )3 m ~ l 2

w h e r e a(o) a n d ~(o) ~3,m~x ~~,m~x are e v a l u a t e d a t t he sphe res b a n d a,

r e s p e c t i v e l y , a n d

Ô(1) A(1) q ~ mk(m + k) ! fi2m+k+a

U U ~n=l k=l ra!k!

q m(m ~- 1) f12m+4 = Π. (27)

,~=1 4 (1 - fi)m+z

w h e r e dä 1) a n d d~l) a re e v a l u a t e d a t sphe re s a a n d b, r e spec t ive ly . I t is aga in n o t e w o r t h y t h a t for the l imi t ing oase of q - + oo, (27)

y ie lds :

~(1) a(1) fi6 l im Oa'max - - l im -4,m~:~ _ (28) q~¢o U q-:.oo U 2(1 - - fi - - fl2)a

F L O W F I E L D A R O U N D C O L L A P S I N G B U B B L E T R A I N S 5 3

d ~'«i

I0° I I I

B D I(?;

OI o2 0.4 Dimensionless Radius, ~ DirnensloNess Radius,

Fig. 4. Maximum zero-th order Fig. 5. Maximum first order error of ¢3 or ¢4. error of Ca or ¢4.

Equation (26) is plotted in Fig. 4 for various values of q. Equations (27) and (28) are graphically compared in Fig. 5. In comparing the figures, we note that ¢3 and ¢4 are higher order multiplets and therefore more accurate than either ¢1 or ¢2 for the same number of q terms.

Velocity of approach and distance between bubbles are assumed to be constant in order to keep the problem tractable. As shown by Moalem's E111 study of collapsing bubbles, this assumption is fairly accurate.

The assumption of a spherical bubble, rather than the more realistic oblate spheroid, is consistent with the recent work of Wittke and Chao E12~ who obtained excellent correlation for bubble collapse utilizing a spherical boundary for oblate spheroidal bubbles. This is further substantiated by the analytic work of Lochiel and Calderbank ~131 who showed that the flattening of the bubble surface does not appreciably affect the transfer rates.

It is obvious that the problem posed here could be solved utilizing bispherical coordinates as well. However, this was considered disadvantageous since the formulation would become untenable

6 4 F L O W F I E L D A R O U N D COLLAPSING B U B B L E TRAINS

for the desired moving boundary heat transfer problem. Also, it is suspected that, for the same number of terms in the truncated series solution, the use of bispherical coordinates would result with larger errors than those shown here.

A c k n o w l e d g e m e n t

The financial support of the Israel Council for Research and Development is greatly appreciated. This article constitutes part of the work of J. Isenberg toward the degree of D. Sc. at the Technion - Israel Insti tute of Technology.

Received 21 May 1969 In final form 30 Oetober 1970

R E F E R E N C E S

Il] BOUSSINESQ, M., J. Math. Pure Appl. 1 (1905) 285. [2] LEVICH, V. G., Physiochemical Hydrodynamics , Prentice Hall, New York, 1962. [31 RUCKENSTEIN, E., Chem. Eng. Sei. 10 (1959) 22. [4] SIDEMAN, S. and Y. TAITEL, Intern. J. Heat Mass Transfer 7 (1963) 273. [5] HIcKs, W. M., Phil. Trans. 171 (1880) 455. I6] BASSET, A. B., Proc. Lond. Math. Soc. 18 (1887) 369. [7] MICHAEL, P., Physics of Fluids 8 (1965) 1263. E8] HERMAN, R. A., Quart. J. Math. 22 (1887) 204. E9] MORSE, P. M. and H. FESHBACH, Methods of Theoretieal Physics, MeGraw-Hill,

New York, 1953. [10] MILNE-THOMSON, L. M., Theoretieal Hydrodynamics , Macmillan, London, 1962. [11] MOALEM, D., M.Se. Thesis, 1970, Dept. Chem. Eng., T e c h n i o n - Israel Inst. Teeh. [12] WITTKE, D. D. and B. T. CHAO, ASME J. Heat Transfer 89 (1967) 17. [13] LOCHIEL, A. C. and P. H. CALDERBANK, Chem. Eng. Sei. 19 (1964) 471.