street et al

8/2/2019 Street Et Al

1/11

the exp eriments consistently yielding smaller values th an pre-dicted. This difference is probably related to a change in themercury-glass contact angle, s ince this factor contributedlargely to the second mom ent calculation.Acknowledgment

Th e des ign of th e mercury in jec t ion appara tu s was , wi th theexception of minor modifications, furnished b y the C onti nen talOil Co. Du ring th e research, d iscuss ions with B ert Ki l l ia msand John Gidley of th e Esso Produc t ion Research Co.were very helpful.literature Cited

Cranier, H., Wold, II. , J. London Math. SOC. 1, 29G (1936).Drake, L. C. , In d . Eng. Chem. 41,780 (1949).Guin, J. A , , Ph.D. dissertation, Ilepartment of Chemical Eirgi-neering, University of Texas at Austin, Austin, Tex., 1969.Hendrickson, A . R., Harris, 0. E., C odte r , A . W., J . Petrol.Technoi. 18, 1291 (1966).Rlorrow, K. . , 1969 Summer Hyniposiuiiion Flow through PorousMedia, Division of Industrial and Engineering Chemistry,ACS, Carnegie Institu te, W ashington, I). C., 1969.Purcell, W. R . , Petrol. Trans. A I j l E 186, 39 (19 49).Reigle, E. G., 11.S. thesis, University of Texas at Austin, Au s t in ,Tex., 1962.Richards, P. I. , lfaiiual of Mathematical Physics, p. 286,Pergamoii Pr es , London, 1959.Rowan, G. , J . Inst . Pe tro l. 45 , 321 (1959).Schechter, R . S., Gidley, J. L. , A. I .C h .E . J . 15,339 (1969).Williams, B. B., Gidley, J. L., Chin, J. A., Schechter, l< .S..I K D .ENG.C m w . FUSD.IM., 198 (1970).

Adamson, A . W., Physical Chemistry of Surfaces, 2nd ed.,Bucker, H. P., Felsenthal, )I., Conley, F. Ii., Petrol. Trans.Burdine. 11. T.. Gourriav. L. S.. Reichertz. P. P.. Petrol. Trans.

I


2/11


3/11

Finally, for a va riab le viscosity coefficient, v,+-i.e., for acoefficient which varies with space and time because of i t sdependence upon the l iquid temperature and concentrat ionof the d i sd v ed b lowing agent

where

( 5 )

Equa t ion 4 s derived in detail in the Appendix. The oute rradi us of t he liquid ca n be related to the bubble radius by atot al volum etric balaiice 011 the liquid mass to give

which s tates that the volume of the l iquid at any t ime equalsthe oiigiiial volume niiiiua thc volume of tilowing agent whichhas tliffiised in to th e g a s hil ible .Discussion of Mom entu m Tr anspo r t P r oces s . Thesi tuat ion of interes t in thi s paper is one in which theviscoiis forces dominate dur ing the growth I lrocess .Typically melt viscosities lire in the r : u i g of lo 3 o lo5poises.Jf-ith iicgligihle erro r, there fore, th e iiirrtial te rm s c:in beomitted! givii ig

where t h c "effertivc" noli-Sew toiiiaii viscosity is giveii by

The iiitcgrxl i l l 9 rcpreseiits the effect of viscosity variationin th e melt .Let u h focus here 01 1 the first term nil the right-halid side ofEquatioii 9. For simplicity, let the f luid be Ken toiii aii (n := 1)a nd lct t hc vi,sco.;ity gradi ents lie negligible. Th en we have th at!Jeff = p ( 1 - ;

l'hc. rffwtive viscosity of the fluid lessens as the bubble es-p i i d s a nd t1icrcfoi.e 2 s the geometr ical extent , S - R , of themelt der rewes. J1-e hav e coined the ter m "geometric thiniiiiig"to ch arncteiizr this effect 011 the viscous behavior of the f luid.

I t is of interest to examine th e effective viscosity when thebubble is expanding in a thi n liquid film. To a f irst approsi-inat ion, this can be thought of as corresponding to bubbleexpansion during the filial stages of growth w hen th e bubblesinteract t o form thi n films. If the thickness of fluid is given b yA = S - R, nd th e bubble volume is much greater than theliquid volume,

A- - 1 +- here A /R


4/11


5/11

B L O W I N G VISCOSITY THERMALA G E N T COEFFICIENT EFFECTSI IDEAL CONSTANT ISOTb lERM AL2 IDEAL VARIABLE ISOTVEi(MAL400 - 3 REAL VAQIABLE ISOTHERNAL4 REAL VARIABLE h O N SOTHERMAL

-~

TIME, SECONDSFigure 2. Comparison of effects on growth rates

Tabl e 1. Basic Data fo r Phase Grow t h Ca l cu l a ti ons i n aV i s c o u s , N o n - N e w t o n i a n M e l t

Proper ty orQuan ti ty Numerical ValueJ 4 x 104 L.v."a. 1 5 4 x 104 H.V.

a1 1 . 0 X cni*/dynea2 1 . 5 x lo-* OKa3 - 3 . 6 X 10-15 O K cm4/dynes2a4 1 .8 X cni4/dynes2co 0.148 g /ccCPC 0. 51 9 cal /g OKC V G 0.312 cal /g O KC L 0 . 7 c a l / g OKD 1 x cm2/secdi 2 1 . 9 c m 3 / gda 2300 O Kk c 6 .7 X cal/sec cm O Kk L 1 . 5 X cal/sec cni OKnP o 1.31 X lo7 dynes/cm*P , 1.06 X lo 6 dynes /cm2Ro 1 x cmR (0 ) 0 .0 cm/secTO 422 OKA N 47 cal/gAv 47 cal/gP B A 0 . 4 4 g/ccP L 0 .8 g/cc

1 0 . 6 5 L.V."0 . 5 0 H.V.

P S 3 . 2 x 10-3 g/ccC 20 dynes/cm

a L.V. an d H.1'. denote melts with lo w viscosity and highviscosity, respectively, in sense of range of viscosities investi-gated i n numerical calculations. Units of viscosity coefficients,ao ,are (dynes/cm2)seco.65and (dynes/cm*)secO.5, respectively.

sho rt times or equivalently during t he initial stages of ph asegrowth. Fur thermore, i t focuses at tent ion on the var iousprocesses occurring a t th e vapor-liquid interface.Results and Discussion

The results of the numerical method outlined above havebeen compared with results published by Barlow and L anglois(1962) and l lar ique and Houghton (1962) . In all cases,

VISCOSITY VlSCOSlTYLEVEL' COEFFICIENT

1 H . V . O R I G I N A L V A L U E2 H.V. VARIABLE3 H.V . INTERFACIAL VALUE4 L . V . VARIABLE'SEE TABLE I

0 I 10 0.02 0 . 0 4 0 . 0 6 0 8 0 10

TIME, SECONDSFigure 3.on growth ratesEffect of viscosity level and viscosity variation

agreement was obtained between published results and th eresults obtained using the nunierical method presented here.Th e example calculations discussed are concerned w ith t,hegrowth of bubbles in a melt. Th e basic da ta used in th e cal-culations are presented in Ta ble I. Deviat ions f rom these da taare indicated in the subsequent f igures. The melt viscositycoefficient as a functio n of blowing age nt conce ntration andmelt temperahre is given by

The compressibility of the vapor was assumed to va ry negli-gibly with tempera ture over the tem perature range en-countered. Th e following relation was used over the pressurerange of interest:(32)where PG s in calories per cubic centimeter.Th e equilibrium relation between th e concentration of blow-ing agent a t the inter face and the bubble pressure is t akenfrom the work of D err (1964):

ct = p s (1 - 5) (ale-a*Ti)PG + (aoTi + a4)Po2] (33)Comparison of the curves in Figure 2 shows the effect ofseveral idealizing assum ptions on the theoretical prediction ofbubble growth. Curve 1 indicates the isothermal expansionof an ideal gas in a melt whose viscosity is constant^ a t t heinitia l viscosity of th e polymer-blowing ag en t solution .Cu rve 2 shows th e effect of relax ing t,he assum pt,io n of con-

st an t melt viscosity-Le., th e effect of the varia tion of meltviscosity with concentration of blowing agent and melt tem-perature. T he depletion of blowing agent in th e neighborhoodof the gas bubble causes an effective increase in th e viscosityof the melt in which the bubble is expanding. Curve 3 showsthe effect of relaxing the ideal gas assumption. H ere only th eeffect of pressure on the compressibility is t,aken into account.Finally, the assum ption of isothermality is relaxed an d theeffect of lett 'ing the bubble expand in a n onisot,hermal manneris shown. The relaxation of all these idealizing assumptionsresults for this case in a significantly different growth curvefrom curve 1 o curve 4.Figure 3 shows the effect of viscosity on th e bubble growth

2 = 0.89 - 0.53 Po

58 Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971


6/11

I6 O I22 -9 80 -m

n = 1.0 ( ISOTHERM AL1

0 0 .01 0.02 0 . 0 3 0.04TIME, SECONDS

0 I0 0 .01 0.02 0 . 0 3 0.04

TIME, SECONDSFigure 4. Effect of shear thinning on growth rates

TIME, SECONDSFigure 5.rates Effect of blowing agent diffusivity on growth

process in two ways . Curves 1, 2, an d 3 show the bubblegrow th in a high viscosity material for, respectively,A melt of con stan t viscosity whose viscosity is th at of t heor iginal mixtu re.A melt of variable viscosity whose viscosity is influencedby the concentrat 'ion and tempe rature prof iles in the melt .A melt of high viscosity in which the bubble sees a liquidwhose viscosi ty is evaluated at the vapor-melt inter face.This viscosi ty is a n increasing function of t ime d ue to thedeplet ion of blowing agent in the neighborhood of the bubble

and the decrease in inter facial temperatu re .These results indica te th at dur ing the ini t ia l s tages ofgrowth the bubble essentially sees only the low-viscositymelt conta ining the original dissolved blowing agen t.Cur ve 4, n the other han d, shows growth in a low viscositymater ial . I n this case the viscosi ty is taken to be var iable ,as in curve 2. T he very high ini t ia l grow th rate is app aren t .Th e curve is extended only out t o abo ut 0.02 second, becausethe thin-film approximation limits the accuracy of the solu-t ion a t greater t imes.F igur e 4 how s th e effect of exp one nt 71 ( in the power-lawmodel) on the growth rate for the base case given in Tab le I.As a general quali ta t ive rule , the greater the spread of th e

5 200 I

C, - 0 06 g r am c cI I0 02 0 04 0 06 0 8 0 I0TIME SECONDS

Figure 6.growth ratesEffect of initial blowing agent concentration on

0 I , I I 1T I M E , SECONDS

Figure 7. Variat ion of interfacial blowing agent con-centration with time during phase growth

molecular weight distr ibutioii, the lower the value of n-thatis , the more the f luid is l ion-Newtonian. This curve, there-fore, indicates th e effect of m olecular weight distr i but ionon the growth curves a t a con stan t visco.;ity coefficient.The lower values of n engender a very high ini t ia l growthrate , as opposed to th e Yewtonian f luid.Figures 5 an d 6 show , respectiv ely, th e effect of diffusiv ityand ini t ia l blowing agent concentrat ion on t 'he bubble growthprocess. These results clearly indicate that the mass transferprocess is a factor controlling the rates of bubble growth.Grow th t imes m ay be s ignificantly lengthened b y using l)low-ing ag ent s possessing a low diffusivity. The model predicts astroiig depend ence of grow th ra te o n conce iitratioii of blowingage nt. Th is is due t o t he natiire of the melt viscosity-bloivingigagent concentration function used, which incorporated astrong (exponential) depend ence of viscosity o n blowing agentconcentrat ion.Figures 7 , 8, a n d 9 show concentrat ion and temperatureprofiles at t he interface and in the polym er melt, respectively.Figure 7 indicates that a t shor t t imes the inter facial concen-trat ion (and therefore the bubble pressure) rapidly d rops to aquasi-steady value after about 0.02 second. Thu.;, the coii-cen tratio n profiles in the neighborhood of th e hubb lc arequickly established so th at a near-steady flux of ma teria linto the bubble is maintained dur ing the la t ter s tages ofbubble growth. F igure 8 is of interest, in th at it show s th atthe gas bubble temperature reaches a mii i imuin dur ing thegrowth process. Initially, cooling of liquid and vapor due tothermod ynam ic expansion of the bubble an d blowing agentvaporization causes n decline in the bubble temperature

Ind. Eng. Chern. Fundam., Vol. 10, No. 1 , 1971 59


7/11

2" I /~ D, c ~ ' , i e c R M i C R O N S

I //- TO-TG' ; p p io -T i0. 02 0.04 0 06 0.08 0 . 1 0

TihlE, SECONDSFigure 8.tures with timeVariat ion of interfacial and bubble tempera-

1l 7

?A D'US, ,:CP ON8Figure 9. Variat ion of polymer melt temperature andblowing agent concentration with radius

which is compenw ted fo r eventually by coiidiictioii from thesurrounding mel t . Figure 9 shows that, the thickncss overwhich the concentration gradients are sigiiifirnnt, is ahout 16microns for the test case or aliout 5% of t h e final liiil.)l)le rarliiis.Th e temperature gradients are significxnt out t ,o a m uchlarger radius. H owever, the effe ct on viscosity of the fewdegrees change in mel t tempera ture resdts in only inillorcnhaiiges i i i 1)ul)hle growth ra tes a n d , tlierefore, canses 110serious limitation of the thin-film approxiin a t ' 0 1 ~ .Conclusions

A theoretical model describes the initial s tages of bubblegrowth in a non-Newtoniaii fluid. Xulnericwl results indicatetha t the in i t ia l bubble growth ra tes a re de termined b y t h emass and momentum-transport processes. The most s igriifi-ran t parameters controll ing the growth ra te a re the d iffus ivi tyan d con centr ation of blowing agen t, th e visccmit,g level of th emelt , and the extent to which th e liquid is shear thiniug---i.e.,the extent to which the expoiient in the power law model isless tha n 1. A siibsequeiit paper (RIcCormick arid Street, 1971)will treat the bubble g rowth process in non-X ewtoniaii fluidsof finite and iiifiiiite exte nt over relativ ely long periods of tiiiie.

AppendixDerivation of Momentum Transport Equation. T h es phe r i c a l ly s ymme t r i c fo rm o f the c on t inu i ty e qua t ion

yields (Barlow and Langlois , 1962; Scriven, 1959)RZRv = - r 2

The momentum conserva t ion equat ion in the s t ress tensorform of Bird et al. (1960) gives

( A 4where the normal stresses are related by

7 8 8 = T q p (A-2a)a nd

7 7 r + roe t rqv = 0 (A-2b)For a n Ostwald-de W aele (power law) fluid,

Tr7 = 4a (->"where

(A-3)

(A-3a)Subs t i tu t ion of 14-1 a nd >1-3n A-2 an d qiihqpquent simplifica-tion yields

dP1.2 r i d r-PL (R: + 2Rf i2 - R 4 R 2 ) - --.-

3a 1 daran br(R*7?)nryn 1 (1 - 1) t - - (A-4)

A-4 is iiow multiplied throiigh by d r and in tegra ted over rfrom R to S. he iu tegrat ionr ra n he iexdilp carried out an dthe re wl t c a st i l l t h e foiw

The condi t ion tha t the to ta l norm:d strev in the radial direc-tion be continuouq a t the inner and o uter vapor-liquid inicr-faces requires tha t

2aP ( R ) + 7r7(R)= Po - --R (A-6a)a n d

(A-6b)uP ( S ) + 7 7 7 ( S ) = P, + swhere

(A-7a)

(-4-7 )

60 Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971


8/11

subst i tut io n of Equation A - i in -4-6 allows eydic it cspressioiisfo r P ( R ) and P ( S j to be ob ta ined which a r e su bs t i t u t d inA-5 to give an equation which can be caqt in the formRp,[RRtl - R,S) + R 2 ( - 2 -s 2 s -t

n R br ( r /Rj3"

Csing the def ini t ion Equatio n 2 of v S , A43 can be readily cast.in the form of Equ ation 4.A point worthy of discussion regards the nature of th erelation between the interface velocity, R , and liquid velocityat the inter face, v ( R ) . t has been implici ty assumed in thedevelopment that v(R) = R. That this is a i l approximationcan be seen f rom a mass balance at the inter face as car r iedout by Chanibr e ( 1956) :(+ aRapG) = 47rR2PL[R v ( R ) ]dtWhence

and if we define

then we can wr i teR dev( Rj = R ( 3- 3 &)

Since for the c a w s studied here pG 0, the approximation, v (R) = R, should not l imitthe analys is . For the isothermal expansion of an ideal gaswith no mass transfer , R dee + - - = ]3 dRand for the a diabatic expansion of an ideal gas with no masstransfer,

wherey = C P O / C I C > 1

Finite Difference Equations. T h e d e r i v a ti v e > en -counte r ed in s e t I ar e r ep laced by ap1) r opr ia te t r u i ica tedforms of a Taylo r s e r ie s expans ion . T he s i iigu la r it i es in t heintegral ids of t he integrals in 26 thr ough 30 requi r e con-s i s ten t appr oximat ions of the in tegr a i ids in or de r toper f or m ana ly t ic in tegr a t ions . The gener a l me thod ofso lu t ion of the appr o pr ia te f in i te d i f fe r ence f or ms of se t Iis as fol lows:

1. A t a given value of t ime, the bubble radius at the endof the next t ime s tep is calculated b y means of the monientumequation and the temperature and concentrat ion data eval-uated a t the given value of time.2. Vsing the updated bubble radius , the bubble tempera-ture an d pressure, interfacial temp erature , and blowing agentconcentration are calculated by means of the vapor and in ter -facial energy balances, the vapor bubble material balance,and the interfacial equilibrium relation.

3. Using the iiiforniatioii from steps 1 an d 2, the l iquidconcentration and temperature profiles are calculated usingthe l iquid energy and mass transfer equations.4. Steps 1, 2, aiid 3 are then repeated. The coinputat ionsare starte d by prespecifying th e initial sta te of sy,qtein.Th e finite difference forms used in th e a b o w (Joiiipiitatiorial~IOMI:STUMqu,i'rIos (ISI:RTIAI,ORCK-:J~:YIFICAST)

scheme are a s follows:

wheret = j A t , r = iAra n , = a[R( t ) ]n S , = a [ S ( t ) ]

( Y \ . R ) ] + l = v m ( f + A f )( A V . ~ ) ~v g ( r + Ar , t ) - v,? ( r , t j

r ( = r 4- r / 2This equa t ion i s of the form

( . L l O )

whereAn i = n ( r + &,tj - a(r , t ) (-4-13)

This equation is of the formR ~ + I Rj(1 + a r A t ) (A-14)

aiid is solved esplicitly.M as s Transfer to G as Bubble. In or der to ca lcu la te theintegrals ii i 28 t h r o u g h 30, th e coiicei i t ra t ioi i ant1 tenipe ra-tu re prof iles a t t he in ter face are represeii te t l i n th e fol low-ing mann er ( wr i t t en he r e f or concent r a t io i i otily) :ci = yj0 + 6 j E l - AOj 5 O 5 E j

j = 1 ,2 , , , , (X-lt5)where y j a n d a j , j = 1,2, . . , , are constants.performed analytically, yieldingSubsti tut ion of A-15 in 28 allows the i i i t ey tn t i on t o he

Ind. Eng. Chem. Fundam., Vol. 10, No. I , 1971 61


9/11

where Secondly,

a n dA01 = Ri4At

(A-1 a )

(il-17b)It is poss ible that the l inear approximation to the concen-trati on profile gives insufficient accuracy in th e representa tionof th e integral (A-16) a s T + 0. However , a quad rat ic approxi-mat ion nea r T = 0 yields a correction of a second order ofimpor tance.

T o improve the s tab ility properties of t he finite differenceequations , the following su bst i tut io n in .L16 i s made :

where the two der ivat ives , calculated us ing Equa tion 20 , areeva lua ted a t P o , , a nd T i + Subs t i tu t ion of -1-16 an d -4-18 in28 yields

where

(A-20)This equation is of the form

Vapor Phase and Interface Energy Transfer. T h e f o l -lowing f ini te dif ference representat ions of 29 a n d 30 a r eobta ine d in a s imi la r man ner :

Equa t ion A-22 is of the form

where ZT is given by E quatio n A-20 with T i , t eplacing Ci,k,This equation is of the formalsCi,j+l+ ~ s T i . j + 1 a i , T ~ , j + i+ 018 = 0 ('4-25)

Equa t ions A\-21,A-23, an d A-25 are solved qimultaneously bya Jacobi- type i terat ion procedure.Liquid Temperature and Concentration Profiles. T oobta in accur a te r epr esen ta t ion s of th e in tegr a l s appear in g

in Equa t ions 26a and 26b , the in te r f ac ia l co i icen t r a t ion( or t emper a tur e ) f unc t ion i s expanded in a power se r ie sab ou t 0-Le.,bCi* b2Ct* A

d0 bo2 2!C ~ * ( O A ) = c,*(e)- -- A + -- - - . (-1-26)Taking the f ir s t two terms of the expan4on and identifyingA with h2/4Du2,we obtain

(A-27)2 dC*iCf*(O- A) = C,*(O) - -- ~4Du2 boSubsti tut ion of -1-27 in 26a and in tegration yieldsC(h,O) = C,*(O) + [ h 20 C i * ]be erfc (T$z) -

bC *exp (-h2/4D0) 2 A-28)deSimilarly,T*(h,O) =: T,*(O) +

ex p (-h2/4aO) bT~ - (A-29)a0

Afore terms in the expansion -1-26 could have been readilyincorporated into the integral, but for the purposes here-Le.,determination of the viscosity profiles in the liquid-thefirst two term s are sufficient.The dynamics of bubble growth are obtained by solvingth e sim ultaneou s set of f inite-difference (Eq uat ion s 33, .\-9 orA-12, A-19, A-22, A-24, -4-28, and -1-29.) These equa t ionsare denoted a s set 11.Stability and Convergence Properties of Finite Differ-

ence Equations. S t u d i e s of t h e s t n b il i t 'y a n d c o n v e r g en c eprop er t ie s of t he sys tem of f inite dif ference equ ation s indi-c a te t h a tS e t I1 is not uncondit ionally s t ' able; ra ther , the s tabil i tyappea rs to be control led by t 'he rat io

which l imits the m aximum At permissible for a given systemspecification.S e t I1 under s table computat ion is convergent in the senseth at a decrease in the time a nd radial increments yields resultswhich converge to a limit as th e increm ent sizes approachzero. It is, of course, assume d th at th is limit is th e solution toth e differential equatio ns.62 Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971


10/11

STABILITYOKSIDERATIOKS.n examination of set I showsthat the fol lowing character is t ic t ime constants control ther a tes of the s imultaneous transpor t processes occurr ing:

Vapor Phase Mass R 2 / DTr ans f e r (A-30b)

( A - 3 0 ~ )n te r f ac ia l Ener gy (A , + A.U)COTr ans f e r ( hoT0 )Vapor Phase Ener gy R 2 / DTr ans f e r

(A-30d)(A-30e)

(A-30f)CVQPQhaExper ience has ind ica ted t ha t the t im e cons tan t , R * / D , sthe s ignif icant parameter control l ing th e s ta bil i ty of t he

equations representing the growth process . The maximumincremental s ize for s table comp utat ion ap pears to l ie in therange

D At10-2


11/11

C L = liquid heat capacity, cal/g O KCVG = vapor constant-volume heat capacity, cal/g OKC P G = vapor constant-pressure heat capacity, cal/g O Kdi,2 = constantsD = dsus iv i ty , cn i2 / sech = Lagrangian distance coordinate, r 3 - R 3 ,cm 3hc = heat transfer coefficient from interface to vaporHGf = entha lpy of vapor enter ing bubble, cal /gko = vapor therm al conductivi ty, cal/sec cm O Kk L = liquid thermal conductivity, cal/sec cm O K111 = blowing agent molecular weight, g/g molern = con stan t in definition of lion-N ewton ian vis-n = exponent in power law modelPo = pressure of va por bubble dyn es/cm2P = pressure, dynes/cm 2P , = ambie nt pressure, dynes/cm2Po = ini t ia l bubble pressure, dynes/cm 2T = radius, cmR = bubble radius, cmR, ,R i = initial bubble radius, cm= gas con stant , cal /mole O K= outer l iquid radius , o m

= initial outer liquid radiu s, cm,t = time, secTG = vapor bubble t emper a tur e , O KT f = tempera ture a t vapor- liquid inter face, O KT = l iquid temperature , I

street et al

Documents