Bubble growth in superheated He-II is controlled by the transfer of heat to the surface of the growing bubble by nonlinear Gorter-Mellink counterflow. The present work presents analytic formulas for the bubble radius as a function of time in the limiting cases of small and large superheats. The formulas include the effect of the inertial reaction of the surrounding liquid to the expansion of the bubble. A numerical example shows that bubble velocities of the order of meters per second are possible. A related problem, involving only heat transfer but no movement of the liquid, is the motion of the free surface of superheated He-II in a very long tube. This problem has a similarity solution. The interfacial velocity in the tube is much smaller than the bubble growth velocity. 1 ref

Dresner, L.

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a co»i«acu» ol iha U Sundw contract No D€ACO5-8*OH21400 Accotcfrngb, I t * U SCONF-8807123 — 1to pufcbsh o* roproducsth« pubfcshad lorn of iha contribuion. o*f low othwt lo do w . toi U S GowrnmaniDE39 001874Bubble Growth in Superheated He-II*Lawrence DresnerOak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, TN 37831-8054, USAAbstractBubble growth in superheated He-II is controlled by the transfer of heat to the surface ofthe growing bubble by nonlinear Gorter-Mellink counterfiow. The present work presentsanalytic formulas for the bubble radius as a function of time in the limiting cases ofsmall and large superheats. The formulas include the effect of the inertia! reaction of thesurrounding liquid to the expansion of the bubble. A numerical example showp that bubblevelocities of the order of meters per second are possible. A related problem, involving onlyheat transfer but no movement of the liquid, is the motion of the free surface of superheatedHe-II in a very long tube. This problem has a similarity solution. The interfacial velocityin the tube is much smaller than the bubble growth velocity.IntroductionSuppose we have a sample of liquid He-II at a pressure P and a temperature T& = Tt{P) 4-AT, where Ta(P) is the saturation temperature corresponding to the pressure P and ATis a small superheat. (A list of symbols used in this work is included at the end of thearticle.) Such a 6ample is thermodynamically unstable and will tend to change into vapor.The rate of this conversion depends on the density of nucleation sites initially present andthe rate of growth of ihe bubbles arising at these sites.The rate of bubble growth is controlled by the transfer of heat from the superheatedliquid to the surface of the growing bubble. In He-II this transfer is by nonlinear Gorter-Mellink counterflow (heat flux proportional to the cube root of the temperature gradient);see eq. (16). For superheats AT that are not too large, the temperature distribution inthe liquid surrounding a bubble will be very close to its steady-state distribution T =Ti, — ATRS/TS. This greatly simplifies the problem of bubble growth; but, as we shall seelater, the superheats for which the quasi-static theory is valid are quite small. As will beshown, the theory also takes a simple form in the opposite extreme of large superheats,where bubble growth is 6O rapid that the liquid layer in which the temperature changesappreciably is thin compared with the bubble radius.Bubble growth in Ee-II is explosive, bubble velocities of hundreds of centimeters persecond easily being possible. The growth of the bubbles can be studied by high-speedphotography if nucleation sites are present. A related situation, for which a completetheory is available, and which may be easier to study experimentally, is evaporation fromthe free surface of He-II in a long tube open at one end. The temperature of the liquidhelium in the tube is again T&, and at the start of the experiment the external pressure atthe open end of the tube is dropped to a pressure P so that Tb = T,(P) + AT, where AT *is the desired superheat. If no bubbles are nucleated in the bulk of the helium and phase Hchange takes place only at the free surface, the motion of the free surface is calculable. Its mvelocity is nonuniform, varying as the three-fourths power of the elapsed time, and is of'Research sponsored by the Office of Fusion Energy, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. Work performed :.n part while the author was /on assignment to the Applied Superconductivity Center. University of Wisconsin-Madison. ' /i DISTRIBUTION OF THIS MCUMENT 13 IJNLWiiTTBthe order of centimeters per second. The reason for the much slower motion of the phasefront in this case compared with the case of bubble growth is that in this case the vapordoes not impart any motion to the liquid adjacent to it.Characteristic Time to Reach Steady StateThe characteristic time to reach the steady state is determined by the Gorter-Mellinkconductance parameter K (W m~5^3 K~X^J), the heat capacity of He-II per unit volumeS (J m~s K"1), the superheat AT (K), and the instantaneous bubble radius R (m). Theonly time that can be made out of these four variables is r = S(AT)2'iR*'*/K. Forthe quasi-static theory developed here to be valid, the condition Rr <C R must hold. Insuch a case, the bubble radius will change little during the time it takes to reach thermalequilibrium, and the temperature distribution will be a quasi-static steady state.Quasi-static Heat Balance for the Growing BubbleThe quasi-static heat balance for the growing bubble can be writtenFrom eq. (1) and the preceding expression for r , we find at once thatRT 51^3SAT~R~= PvLso that the allowable superheats AT must satisfyAt T, = 1.8 K, P, = 1.64 kPa, and pv (estimated from the perfect gas law) is4.38 xlO~4 g/cms. Using 5 = 0.408 J cm"3 K"1 and L = 23 J/g, we find that ATmust be -C14.4 mK. According to the Clausius-Clapeyron equation, the slope (dP/dT)tof the saturation line at 1.8 K is 5.60 kPa/K. Thus, the limiting superheat of 14.4 mKcorresponds to a pressure reduction of 80.6 Pa = 0.612 torr, which is extremely small.The heat balance equation [eq. (1)] can be integrated with respect to time to giveassuming the initial bubble radius is very small.Transient Limit—Large SuperheatsWhen the superheat is much larger than the limit of eq. (26), the temperature distributionis never close to the steady distribution. But we can approximate it with the help ofthe assumption, justified later, that the layer in which the temperature changes is thincompared with the radius of the bubble. Then we can neglect the curvature of that layer.The temperature distribution in that layer in a coordinate system moving with the bubblesurface is given by(4a)whereX = z(AT)^2(S/Kt)3^ (46)and z is the distance outwards from the bubble surface. The solution of eq. (4) 'R thesolution to the damped-temperature problem.1 It is applied in the moving coordinatesystem because the thin liquid film in which the temperature changes appreciably is beingtransported outwards by the expanding vapor.The inward heat flux at the bubble surface is given by^ ^ ' 1 ' 4 (5)The heat balance equation for the growing bubble is4wR2RPvL = K(6T/dz)lJJ04TrR2 (6a)or~ \ 8 ) PvL { 'This can be integrated at once to give_ / 8 \ K'S{&T)WR = I —y=. I (7a)\3/3/ PvL 'or1/4(76)which allows easy comparison with eq. (3).Tha thickness of the temperature transition layer will be small compared with theradius given in eq. (7) if the value of X in eq. (46) calculated for z = R is large comparedwith 1. Substituting eq. (7a) into eq. (46), we findwhich will be much greater than 1 if AT greatly exceeds the limit given by the right-handside of eq. (26).As an example, we take:T. = 1.8 K K = 11.6 W cm"5/3 K"1/3 (1.85 K)AT = 0.1 K pv = 4.38x10"* g/cm3Tb = 1.9 K < = l sL = 23 J/g 5 = 0.533 J cm"' K~l (1.85 K)Then, according to eq. (7), JR = 209 cm and R = 157 cm/s. At t = 0.1 s, R = 37.2 cm andjR = 279 cm/s. Such rapid bubble growth can be described as little Ies6 than explosive.Consideration of Mechanical TermsThe expanding bubble pushes the surrounding liquid out of its way, and, in doing so, itdoes work on the liquid that must be subtracted from the energy available for vaporization.Furthermore, the inertial reaction of the liquid to being accelerated causes a rise in pressureat the bubble surface that increases the saturation temperature there. Wt estimate theseeffects now.To calculate the pressure rise at the bubble surface, we treat the liquid as incompress-ible. Then the velocity field surrounding a bubble is v = RR7/r2. The pressure rise at thebubble can then be calculated from Euler's equation1 8P dv dv ,n.-7^=di+VTr (9)A short calculation givesP{R)-P. = pt(RR+lR2^ (10)According to eq. (76), R ~ <s/4, in which case eq. (10) can be writtenThe power W expended by the expanding vapor in pushing the liquid out of its way is(12)If we divide this by the left-hand side of eq. (6a), we getR\/t*)4nR2RpvLIn the numerical example considered at the end of the preceding section, this ratio is 0.294at t = 0.1 s and 0.204 at t = 1 s. Thus, not all the heat transported to the bubble isavailable for vaporization. We can estimate the effect of this on bubble growth by usingthe complement of the ratio of eq. (13) as an ad hoc correction to K. Then the radiiachieved at t = 0.1 s and t = 1 s are 28.7 cm and 176 cm, respectively.If we use the pressure rise given in eq. (11) to correct the superheat in eq. (7b), wefind, after rearrangement,where i?» is the as yet uncorrected value of R given by eq. (76). Here AT is still T& — T,.From eq. (14), it follows at once that^ - ( 1 + a ) , a - 3 2In the numerical example we have been following, a = 1.42 and 0.534 at t = 0.1 s and 1 s,respectively, 6O that R = 18.5 cm and 142 cm, respectively.Since R* ~ i5/*, it follows from eqs. (13) and (15) that the corrections will be largefor small t and small for large t. Thus, the theory presented here is an asymptotic theorycorrect for large enough t.Evaporation from a Free SurfaceLet the tube containing the liquid He-II extend in the z-direction, and let the initial positionof the free surface be z = 0. At t — 0, let the pressure above the free surface be droppedsuddenly so that the liquid at temperature T\, becomes superheated by a temperaturedifference AT. The vapor produced at the free surface exhausts in the negative z-direction,while the free surface advances in the positive z-direction. The temperature distributionin the quiescent liquid obeys the Gorter-Mellink diffusion equationsubject to the following boundary and initial conditions:T(z,0) = r6 (17a)T(oo,*) = r6 ' (176)r[z(o,«] = r. (i7c)PtLZ = K(dT/8z)lJJz (I7d)Equations (17a) and (176) state that the initial liquid temperature and the liquid tem-perature at the far end of the tube are both T&. Equation (17c) says that the liquidtemperature at the free surface, whose instantaneous position is z = Z(t), is T,. Equation(17d) is a heat balance equation that states that the heat transported to the free surfaceby Gorter-Mellink counterflow is expended in vaporizing liquid.When the thermophysical properties K, S, and L are independent of temperature, theproblem stated in eqs. (16) and (17) has a similarity solution, the details of which are givenin Appendix 1. The final result for the displacement Z of the free surface isZ = A(/f/5)5/4(Ar)-1/2f3/4 (18a)whereA = (8/3v/3)1/2B(l - £ )~ 1 / : (186)andB = SAT/ptL (18c)When B < 1 , which is usually the case,8Equation (18) differs from eq. (7a) only in the appearance of pt (which is 0.145 g/cm3) inplace of pv. With the data given previously, B = 0.0159; eq. (18d) then gives Z = 0.630 cmand Z — 0.473 cm/s at the end of 1 s.Reference1 Dresner, Lawrence Transient heat transfer in superfluid helium—Part II. AdvCryog Eng (1984) 29 323-333.Appendix 1In solving eqs. (16) and (17), we find it convenient to setc = Tb-T (A-lc)and work in special units in whichK/S = 1 (A-16)AT = Th - T, = 1 (A-lc)Then (16) and (17) becomedc=d_(dc\1/3 u _ 2 )dt &z \ dz J ic(z,0) = 0 (A—3a)c(oo,<) = 0 (A-36)c{Z,t) = l (A-3c)(ptL/S)Z = -{dc/dzfJJz {A-Zd)These equations have a similarity solution of the formc = y(z/t3/*) (A-4a)Z = At3'* (A-46)where y is a function and A is a constant yet to be determined.If we substitute eq. (A-4a) into eq. (A-2), we find the following ordinary differentialequation for y:The boundary and initial conditions becomey(oo) = 0 (A-60)y{A) = 1 {A-Qb)(3/4){PtL/S)A = -y1 /3(A) {A-6c)Equation (16) can be integrated readily if we set 0s = dy/dx. We find(A-7)where o2 is a constant of integration. If we integrate eq. (A-7) from A to 00, we have, inview of eq8. (A-6a) and (A-66),f°° {"K \ ~"s/2/ I±x2 + a2\ dx = l {AWith the help of eq. (A-7), eq. (A-6c) becomesV - 1 6 B 2 (ABy using the identity J dz(l + z2)"3/2 = z(l + z2)~1 / 2 , we can rewrite eq. (A-8a) asoV3/2 = 1 - (V3A/2a)(l + 3A2/4a2)-1/2 (A-Sc)If we set £2 = 3A2/4a2 and rj = a2y/Z/2, eqs. (A-8fe) and (A-8c) becomei7«l + £2)1/2 = B (A-9a)and <.From eqs. (A-9a) and (A-96), we find easily thati = 5(1 - 2B)~1'2 (A-lOo)»; = (1 - 25)(1 - B)-1 (A-106)from which follows eq. (186).If we now convert to ordinary units, B becomes SAT/ptL, which is dimensionless.Thus, 60 is A. Then, eq. (A-46) becomes eq. (18a). The function y can easily be calculatedfrom eq. (A-7), but we have no need for its explicit form here.Symbolsa constant of integration introduced in eq. (A-7)A constant in eq. (18a) and eq. (A-46) for the displacement Z of the free surfaceB S&T/piLc Tb-TK Gorter-Mellink conductance parameterL heat of vaporizationP pressureP, saturation pressurer radial coordinateR bubble radiusR dR/dtS heat capacity per unit volumet timeT temperatureTf, bath or ambient temperatureT, saturation temperatureAT superheat Th - T,v radial velocity of liquidW power expended by expanding vapor/3'*/X quantity defined in eq. (4b)y similarity solution introduced in eq. (A-4a)z distance from bubble surface; also length coordinate along the tubeZ displacement of the free surface of the liquidZ dZ/dta dimensionless quantity defined in eq. (15)£ 3A3/4o2pv vapor densitypi liquid densitye ?/*T characteristic time to reach temperature equilibriumDISCLAIMERThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use wLjld not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any igencv thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.

Bubble growth in superheated He-II

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Bubble growth in superheated He-II

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