Bubble growth in a viscous Newtonian liquid

2352 u

Y

-----

-----_

_-L-_

53

S2

--

I

I

I I 53

I

, SI

52

Y

F1g 1 non-zero multiplier) IS coincident with the rate of motion unit body with energy A in the field of potenti energy

U(y)=lf(u)exp

(21

E Fig 2

r(u)dv)du

states, Ss IS an unstable

steady

state

If in (5)

A,
A = mm{Ai,

Sl

of the

(3) If a lumped system described in terms of mechamsm (l)-(4) has three steady states (whxh are possible m a wide range of temperatures and pressures[6,7J), then the shape of the function U(y) IS similar to that shown in Fig I Here 5. S, stand for stable steady

I

---------_-__

Ad

We deal Hnth the object well-known m the theory of non-bnear waves, namely, so called sobtary wave (soliton)[S, 91 This solution corresponds to the case when at 6 -* -Cm one of the stable steady states exists on tbe surface and in a certain region the state approaches another steady state (not reaching it) (Fig 2)

CONCLUSION Existence of the above propemes of system (2), (3) indicates that at simultaneous occurence of a complex catalytic reaction and surface d&sion the appearance of ordered structures, spots, clusters. etc IS possible They may be of pure macroscopic origin The problem of stability of these formations IS beyond tbe scope of this study

Computrng Center, Stbenan Branch U S S R. Academy of Scrcnccs Krasnoyarsk 66QQ49. U S S R Instrtute of Catafysysls Slbenan Branch U S S R Academy of Sciences Novosrbrrsk 630090, V S S R

A N GORBAN V I BYKOV G S YABLONSKII

RJtFJCRENCES

111 May J A, Aduan Catal 1970 21 151 I21 Bernian A D and Krylov 0 V , Nestatswnamye I nemvnovesnye pmcessy v getemgennom katalrze, pp 102-115 Nauka, Moscow 1978 131 Glensdorf P and Pngozlun I , Temwdmamrcheskaya teonya

struktury, ustorchwostt I fluktuatsrr Ma, Moscow 1973

I I, Merzhanov A G and M Barelko V V , Kurochka Shkadmsku K G , Chem Engng SCI 1979 33 805 PI Kuchaev V L and N~latushma L M , AU-Unwn Conference

on the Mectwntsm

of Hetemgeneous-cataIytu

rcactwns,

preprint 59 Moscow 1974 161 Yabkmsku Cl S , Bykov V I, Sbnko M G and Kuxnetsov Yu I, Bkf AN SSSR 1976 229 917 AN Fl Bykov V I, Yablonsku G S and Sbnko M G , fiH

SSSR 1976 229 1356

G B , Lmear and Nonlrnear Waves Bl Wthem York 1974 [93 pismen L M , J Chem Phys 1978 69 4149

Wtiey. New

Bubble growth in P viscous Newtonian liquid (Accepted 5 March 1980) The problem of diffusion-fed gas bubble growth in a laqtudIS of interest in many areas of engmeermg The parhcular case of such growth in a highly VISCOUSthud has application in polymer foam formation One method of foam formation involves growth of bubbles by dilfusion of blowing agent from an oversaturated solution of the blowing agent in the liquid surrounding the bubbles The oversaturation may be achieved by a lowering of the system pressure, or by some other means A large body of literature exists on the subiect of phase

growth Detailed bibliographies may be found in the works of Scnvenll]. Street et al 121 and Rosner and Epstem[3] Ddfusionfed phase growth in VISCOUSliquids has been treated by Barlow and Ianglo1s[4]. Street et nl [2] and Szekely and Martms[5] Barlow and Langlo~s and Sxekely and Martins treated phase growth in Newtonian liquids wlule Street et al considered growth in an Oshvald-de-Waele power law liquid The analysis of Streef et ol is further complicated by consldenng the liquid surrounding the bubble to bc firutc, the liquid viscosity to vary

2353

Shorter Commurucauons wrth blowmg agent concentration, etc A common feature of all three works IS that the final mathematmal models are qmte complex and are difficult and mconvement to solve The models of Barlow and Langlors and of Street et al mvolve nonhnear mtegroddferenual equatmns whtie Szekely and Martms used mute dfierence techmques to solve theu equatnms Because of this complexity It IS not easy to study the effect of various parameters on bubble growth The purpose of this work IS to develop a srmpllfied model for diffusion-fed bubble growth m a VISCOUS Newtoman fluid Results from this model are compared to those of previous workers and found to be in good agreement The effects of certain physical vanables on bubble growth are then examined MATHEMATICAL

IWRMULATION

We consider an Isolated stationary spherical gas bubble growmg m a stagnant hqmd of infInite extent because of dufusion from the liquid to the gas phase We assume the bqmd LS Newtoman and of constant vlscoslty, the system IS Isothermal, and that there IS thermodynamic equihbnum. gven by Henry’s law, at the gas-liquid interface The bubble IS assumed to consist only of the ddIusmg species, IS completely homogeneous and grows with complete spherical symmetry All physical properties are constant The equation of contmmty and motion for such a system may be combined to yleldfl, 43 RR+~R2+4~RIR=@o-p,-2alR)p,

(1)

Equation (1) IS obtamed by assuming that (I) the density of the gas in the bubble, po. IS Independent of time and (II) that P~~PGPGII The equation of conservation of the ddTusmg species 1s

$+,where the radial velocity

ac

Da

( ac> 2

ar 7arrFr

tr = RRZlrz under the assumpuons (1) and (II) above The uutlal and boundary condmons are gven

R(O) = 0

(3)

c(r, 0) = c(m. 1) = C, c(R, 0 = c,(t)

(8)

PC = kc,

and the gas m the bubble IS assumed po =

to be Ideal. gvmg (9)

MpdWT

The above system of equations was solved by Barlow and Langlo~s[d] by assuming that the concentratton of gas In the hqmd IS dlsturbed by the growmg bubble only In a “thin shell” surroundmg the bubble Under this assumption the equations were combmed mto a single mtegro-dtierentml equation for R which was solved by a fimte ddference method Street et al 123 took a slmdar approach for the more complex system they considered INTEGRAL METHOD We adopt an integral method slmllar to the procedure descnbed by Rosner and Epstem[3] BasmalIy tlus mvolves mtegratmg the d&uuon-convecuon eqn (2) m the r &&on over a thm concentraDon boundary layer thickness. S(I). surroundmg the growing bubble An assumed concentraDon profile 3s then substltuted mto the integral relation thus obtamed, yleldmg an expression for S(t) Our results ~111 be slightly tierent from those m 131 because our mass conservation boundary condmon, eqn (7). IS different from that of Rosner and Epstein We multiply eqn (2) by 9 and tntegrate with respect to r from r = R(t) to I = R(t) + 6(t) Usmg eqns (3) and (7) followed by one mtegratlon Hrlth respect to t gves the result

wluch corresponds used IS 133

to eqn (14) m [3] The concentration

profile

by

=o

(3) (4)

where R. IS the uutnd bubble radius, and It IS assumed that the imtial bubble growth velocity IS zero The conditions on concentration are and

would be sdcantly changed smce c, IS usually than umty aud the change m the radial velocity shght assumption of Henry’s law at the bubbk surface

rZ=R

m the lrqutd, V, IS grven by

R(0) = R.

that the results much smaller would also be Fmally, the yields

r>R+6

(11)

where 5 = [r - RWllNt) Usmg this profile in eqn (10) and assuming

(12) [3] 8/R 4 1, we obtam (13)

(5% b) (6)

which corresponds eqn (7) yields

to eqn (16) m [3] Using eqns (11) and (13) m

where c, IS the unknown, time-varying concentratlon of solute m the hqmd at the bubble surface Conservation of dtiusmg species at the bubble surface m conJunction wrath eqn (3) YeIds the condition

(14)

(7)

The gas density pa(t) and the irutml gas den&y poO may be evaluated m terms of c,,. and c, using eqns (8), (9) and (5a), reducmg eqn (14) to one m R, c, and parameters It IS further convement to define a new vanable

$ (PC@) = 3R2pLD($),=,

It must now be expbcltly stated that whereas m the derwauon of eqns (I) and (3) the gas density, po, 18 considered a constant, it IS left as a v-able m eqn (7) Thus treatment paraIlels that of Barlow and LangIois[4] and of Street et al [Z] although these authors do not specdhzaIly point it out Sxekely and Martms[5] assumed po to be constant m eqn (7) as well If the probkm LS ngorously formulated for varymg po, eqns (1) and (3) must be mod&d and eqn (7) would have a factor (1 -c,) m the denominator on the r&t hand sldef3] However, it IS unbkuly

t = R’c, and to non-duuensionahxe

(15)

eqn (14) to obtam

(16) where 5+ = R3c,,,/&‘c, IS a dnnenslonless vanable related to the mass of solute m the bubble, R* = R/h IS the dunensmnless

2354

Shorter Commumcatmns

radms. t* = p 29t2T2Dt/M2k2Rs2 IS the dunenstonkss tune, 8 = M2k2R~21p~%#T2D bemga reference trme In eqn (f), p. may be dmunated m terms of c,,. usmg eqn (S), yleldmg, III dunenslonless form,

(17) where Re = RJ4Ve IS the Reynolds number, PO= kc,821pLRo2 IS the dunenslonlcss mrtml bubble pressure, P, = pmt92/pLR02IS the dunenslonless amblent pressure, and We = pLRo3/2d2 IS the Weber number Equatmns (16) and (17) must be solved s~mukaneously to obtam R*. p and hence c,,, as a funchon of t* The tmtral condltlons are obtamed from eqns (3)-(6) as P(O) = 1

(18) (19)

Fv-3 = 1

ml

It may be seen that the problem has been reduced to the solution of an u&al value problem for a pau of ordinary dIfferenbal equattons SUbJeCl to the restncuon that 8/R * 1 Thus restnction IS unphclt m previous works employing the thm shell approfimation Whde these equations are non-hnear, they can be readily solved numermally using any of the vaflous computer codes for sets of 8rst order ordmary ddferenhal equations Equatmn (16) IS smgular at I* = 0. but this dlflicutty can be ctrcumvented by the transformation V/z = 5* - 1, z(0) = 0

(21)

It IS also seen that four dimensionless groups determine the growth rate of the gas bubble For highly VISCOUS hqtuds, such as polymer melts, the Reynolds number IS very small so that the merha terms m the equatmn of motion can be neglected Equatmn (17) then reduces to !!$

= AFiRe

- BR* - C

where A, B and C are the dunensionless products of respectively PO, P, and We-’ untb Re, C bang the inverse of the capdhuy number Inihal condmon (19) IS no longer apphcable and the growth rate IS controlled by only the three dunensionless parameters A, E and C

1.

RESULTS AND DISCUSSION

to htghly viscous flmds such as polymer melts, so that eqn (22) IS used throughout The model was 8rst tested agamst the results of Barlow and Langlois[4] These authors analyzed the uienhcal physrcochenucal system described above so that duect compansons are possible In contrast, Street et al [2] solved a more rcal~shc problem and thereby introduced many more compkxlhes, so that only general compansons can be made In Fig 1 we show bubble growth as a functmn of hme for several values of the mtlal bubble sme. I&, from Barlow and Langlo1s(4] and from the preseut work (The same values for physnxd propertms were used as m (41) The agreement IS remarkable. wrth devmhons no greater than about 5% over most of the range of & and f For the four largest values of & (0 3 to 1 Frn) the maxunum value of 8/R attamed was 0 55, whereas for the two smaller III&A radu. #R exceeded 1 at certam values of t In spite of these large values of 6/R. the predmnons of the model are exceUent Conservahvely, therefore, one may use the model results wtth con8dence as long as S/R CO 5. even though m theory 5/R IS restricted to be much less than umty It IS unprachcal to present here all possrble results of the present model smce there are three controlhng parameters. yleldmg a large number of combmatlons However, d we restrmt our attentmn to growth of bubbles m hqmds sundar to polymer melts, estimates of the values of the three parameters, A. B and C can be made and some useful results may be obtamed Usmg the property values grven m Table 1 of Street el al [2], (and additionally takmg k = 109dynes/cm’, Y = 10s cm21sec). it IS found that A IS much iarger than either B or C and therefore one m&t suspect that the parameter A WIU dommate the soluhon Numerical tests showed that thm was the case The solution R*(t) IS quite msenslhve to the value ofC, doubling the value of C produces no change Somewhat more change IS obtamed by doubling B-roughly IOto 15% at I* = IO A sm&r change in A produces large changes m R* Thus the growth of the bubble IS determmed prmclpally by the value of A, less so by the value of B and IS vutually unaffected by C for growth m such systems The computed growth rate showed general agreement with those reported by Street et al [2] Figures 2 and 3 may be used for systems charactenstm of polymer melts These 8gures tuve two sets of dunenslonless plots of R* vs I* wth A as a parameter, each for a Merent value of B and the same value of C Growth rates at ddferent conditions may be obtamed by mterpolahon between these two sets of curves TrmJs usmg such a procedure gave preddlons wlthm 5% We confine

of actually Growth

our dlscusslon

computed values rates mcrease with

decreasmg viscosity

mcreasmg A, showmg that gwes faster growth rates as does higher

sacs

Fig 1 Bubble radius vs time, comparison of present model vvlth results of Barlow and Langlo1s[4]

2355

shol?er CommunIcaaons

a

B-20

loo0 600

400

200

102

IO0

*

a 50

Z

IO

I

'0

I

02

I

I

I

04

06

'0II

I

08

i0

02I

04I

tmme and

concentratrons of blowmg agent Relatively msoluble gas, characterized by large values of Henry’s law constant, k, also mves faster growth Growth rates are decreased somewhat wrath mcreasmg B lmplymg that higher values of p.. and, consequently, lugher saturation solubdltles, depress growth rates Growth IS unaffected by changes m C, ~mplymg that the value of surface tenslon need not be very accurately known In conclusion. a simple model has been developed for ddFuslon-fed bubble growth m viscous hqmds Growth rates can he accurately computed from a relatively simple set of ordmary dlfferentlal equations, m contrast to the comphcated mtegrocbfferentlal equation of Barlow and Langlo1s[4] For very VISCOUS hqmds characterlstlc of polymer melts, Figs 2 and 3 may bc used to compute growth rates Although only a Newtoman fimd has been consldered here, no d&culty can be foreseen m extendmg the model to non-Newtoman fluids Acknowledgment-1 am grateful preparmg the figures

to T Chen for hrs assistance

Department of Chemrcal Engrneenng Polytechnrc Inshtute of New York Brooklyn, NY 11201

RUlTON

D

m

PATEL*

USA NOTATION

A B C

PORe dImensIonless P, Re dlmenslonless

Re We-* caplllary

group group number

*Present address Exxon Research Florham Park, NJ 07932, U S A

Fa

3 hmenstonless

10 I

radms of bubble vs dlmenslonless paramet&AforB=20,C=02

mass fraction of solute m the hquld dtiuslon coefficient k Henry’s law constant Au molecular we&t of solute P dnnenslonless pressure defined after eqn (17) P pressure mstantanaous bubble radms R”r R/J?,, donenslonless bubble radms 17t gas contant Re Reynolds number defined after eqn ( 17) radutl distance T’ absolute temperature t time t+ t/O dImensIonless time radial velocity m the hqmd WI Weber number defined after eqn (17) z transformed vanable eqn (21)

A

Greek symbols cdncentration boundary layer thickness c,,J8’ concentration vanable c,JZ’lc,R,-,’ &menslonless vmable reference tune defined after eqn (16) kmemabc vlscostty of the hqmd &stance varmble, eqn (12) density surface tenslon

Subscripts and superscripts G gas phase

and

08 1 tf

t*

FIN 2 Dnnenslodess radms of bubble vs dunenslonless parameter A for B=O4, C=O2

06 1

Engmeenng

CO,

L w 0 00

hqmd phase property at bubble surface att=O far from the bubble surface d( )Idt

time and

Shorter Commumcauons

2356 PEFERENCES

[l] Scnven L E , Chem Engng SCI 1959 10 1 121 Street J R , Fncke A L and Relss L P , Znd Engng Chem Fundls 1971 10 54

]3] Rosner D E and Epstein M , Chem Engng .Sa 1972 27 69 141 Barlow E J and Langlois W E , IBM J Res Da, 1962 6 329 [Sl Szekely J and Martins G P , C’hem Engng Scr 1971 26 147

Chemrcd Ewmmw Smnce Vot 35,DP2356-2358 PcrsamonPressLtd 1980 Pnnted,n GreatBntazn

Bubble frequency in gas-liquid slug flow in vertical tubes (Recewed 31 August 1978, accepted 7 March 1980) A recent analysts showed that mterphase mass transfer m concurrent gas-hqurd slug flow m verhcal pipes depends on the frequency of the gas bubbles or the slug hquld slugs[l] The only published mformatlon on bubble frequency m slug flow IS the empulcal correlatron by Gregory and Scott[2] which LS not applicable to the problem under conslderatron since It has been derived for flow m honzontal tubes Due to the lack of mformatton on thrs parameter, and the need for it m predlctmg gas-hquld mass transfer rates, an expenmental mvesugauon was undertaken EXPJCRKMENTAL EQUIPMENT AND PROCEDURE

The expenmental equipment consisted of a verbcal glass tube 94 cm high and 0 8 cm m mslde diameter An annular inlet device was used at the bottom of the tube to introduce the gas and the hqmd mto the test section Pure carbon dloxlde. selected as the gas phase, entered through the core of the inlet device, while water, selected as the liquid phase, entered through the annulus The gas and liquid flow rates were regulated for maintaining slug flow regune m the tube If the velocity of the gas bubble IS V, (this ~111be the velocity of the hqmd slug as well), then the average bubble (or slug) frequency, Gi, can be determined from the followmg equation

average lengths of the bubbles and the slugs and then the frequency was determined through eqn (1) In using eqn (1). the bubble velocity, Vb, can be determined from the elllstmg results m the hterature[3,4] For the specific conditions existing m this work, V, was calculated using the correlations of White and Beardmore[5], because of theu apphcablhty to wade ranges of vanables RESULTS

A total of 48 expenments corresponding to dtierent values of gas and hquld velocltles were performed Superliclal velocltles of the gas and the hqmd were vmed from 17 4 to 103 4 and 5 0 @ 45 8 c-m/s respectively The expernnental results obtained for Lb and L, and the calculated values for frequency are presented m Table 1 From the data of Table 1, we see that with some exceptions, the average frequency generally decreases Hrlth the gas superficial velocity, but increases with the liquid superficial velocity Based on ths observation, the parameter (5 V&V,, was arbitranly selected as a basis for correlating the results Except for very low values of V,,, this parameter was found to be almost mdependend from the bquld super&al velocity Thus, for each V,, the average value I@( V,)/( V,,)l., was calculated and plotted against VSG as shown m Fig 1 Least mean square method was used to obtam the followmg linear correlation

I I 6-

where & and L, are the bubble and slug lengths respectively, p is the number of gas bubbles (or lrqutd slugs) along a gven section of the pipe, and bar mdlcates average quantity When the bubble vetoctty was small enough to pernut visual detection, the frequency was determmed by countmg the number of bubbles passing a certain section of the tube in a Bven time Otherwise, the test section was photographed to obtain the

VSO =0 WV,,+65 V SL cl”

when velocmes are expressed m cm/s and frequency in Using eqn (2). G was calculated for dfierent values of V, VS~.and compared Hrlth the measured quantities as shown m 2 The comparison IS satisfactory, mdlcatmg the effectiveness the correlation The correlation shows that, as a result of the change m &

V95 +65

6 4

f

-

:E

40

10

v&*

Ftg

1 Correlation

70

SC

loo

80

cm/s

of the averaged

frequency

data

110

HZ and Fig of and