Cheh, H.Y.

Tobias, Charles W.

eScholarship - University of California

Lawrence Berkeley National LaboratoryRecent WorkTitleON THE DYNAMICS OF HEMISPHERICAL PHASE GROWTH IN NONUNIFORM TEMPERATURE FIELDSPermalinkhttps://escholarship.org/uc/item/9s61529nAuthorsCheh, H.Y.Tobias, Charles W.Publication Date1967-03-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaUCRL-17415 .-University· of. CaUfornia '< ' ~ /Ernest·~O.:Lawrence Radiatian· labaratory • , _ -< ' , a > ~ , '\ ;i l-ON THE DYNAMICS OF HEMISPHERICAL PHASE / .. , GROWTH IN ,NONUNIFORM TEMPERATURE FIELDS , .', " ~ / " < ~ ,'" ~. ~. , c." TWO-WEEK LOAN COPY This is a Library Circulating Copy which may be borrowed for two weeks. For a personal retention copy, call Tech. Info. Diuision, Ext. 5545 Berkeley~, Califom.ia . i DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. Submitted to: Physics of Fluids UNIVERSITY OF CALIFORNIA Lawrence Radiation Laboratory Berkeley, California AEC Contract No. W-7405-eng-48 ON THE DYNAMICS OF HEMISPHER!CAL PHASE GROWTH IN NONUNIFORM TEV~ERATURE FIELDS H .. Y. Cheh and Charles W. Tobias March, 1967 UCRL-17415 . Preprint ON THE DYNAMICS OF HEMISPHERICAL PHASE GROHTH IN NONUNIFORM TEMPERATUPE FIELDS H. Y. Cheh* and Charles W. Tobias Inorganic Materials Research Division, Lawrence Radiation Laboratory,and Department of Chemical Engineering University of California, Berkeley March, 1967 UCRL-17415 The dynamics of vapor phase growth yields important information to the understanding of nUcleate boiling. The history of the growth of a vapor bubble may be divided into three stages known as the initial, transition and asymptotic stages~ In the initial stage, the bubble is very small and the growth is slow due to the high surface force that ar:tests its radial expansion. The transition ~rom this slow growth to the asYE?-( -2 ) 1 totic growth occurs in a very short time approximately 10 sec. In the asymptotic stage, the bubble has reached a size where the inert1e., viscous and surface forces can all be neglected in comparison with the pr~ssure force •. This is the stage that provides practical interest. . 2 Plesset and Zwick , using the thin thermal boundary layer approxi-mation, obtained a solution for the asymptotic bubble growth ina unifo~mly supersaturated liquid. Birkhoff, Margulies and Horning3 and also Scriven4 obtained an exact solution to this problem using the mettod of similarity transform. The exact solution agrees completely with PIes set and Zwick's calculation at large rate of growth. Skinner and Bankoff5 .. using a similar approach as that of PIes set and Zwick, developed a parametric solution fO,r bubble grO'.'lth in general temperature fields. * . . Present address: Bell. Telephone Laboratories, Murray Hill, N. J. - 2 - UC~17l;.15 In this paper, we shall apply this solution. to two practical cases, i.e .. , 6 the constant wall temperature and the constant heat flux cases. The parametric solution for the hemispherical bubble srowth by .Skinner and Bankoff can be expressed as 00 (L3(S) 3J J gem) ·m = erfc ---y dm , 2~2 ° and Cl.£t ~ -r(S) J . d~' =-= ° tt4 (~.~ ) , £2 where gem) is a dimensionless initial condition, 7T . J [f(m)-Ts ] sin e de gem) = _0-=--_____ _ 7T' , . J [f(O)-T ] sin e de s ° and f(m) is the initial condition. For the case .of' constant wall temperature, assuming no convection, the initial condition for bubble growth is7 f(r,e) or, gem) where * r cos e = T - (T -T ) erf'c £ ' co w 00 T -T , w t><) m = T -T w s m can be expressed more .generally by interface temuerature - bulk temuerature m= + * interface temperature - saturation temperature (1) (2 ) (4 ) Therefore, ~O signifies the uniform superheat case. ~<l m~ans th~t the liquid is superheated, hoy/ever a gradient of temperature from the b'.:lk to the interface exists. ttFl means the bulk liquid is at its boiling ~empera­ture. W>l is the qase of sub cooled boiling. - 3 - UCRL-174l5 For the case of constant heat flux with no convection, the initial condition is7, f( e) T ~ i f r cos e r" = ~+_q er c £ ' or, where T /J:; CJ.) = ---",q -' --T T + -.9.-T ~.J; 5 Eqs. (1) and (2) with the two initial conditions, Eqs. (5) and (8) are solved numerically. The results are given in.:Figure 1 and Figure 2. The results show the same qvalitative feature as those obtained by . Skinner and Bankoff for the cases of a linear temperature profile and an exponential temperature profile. Hmlever, the present calculation should yield more precise information for vapor bubble growth in liquids. Ac knowledgement The authors wish to thank Mr. J. P. Earhart for his assistance during the numerical calculation for this work. This w0rk was performed under the auspices of the United Stated Atomic Energy Commission. Nomenclature f: An initial temperature field for bubble growth g: 7f sin e de J [f(m)-Ts ] o A dimensionless temperature = -7f~--------J [f(O)-Ts ] o sin e de J: L: ,,: m: R: - 4 -K. " Jakob numberfor heat transfer = --=--P "LCl-" Latent heat of vaporizatio~ A characteristic length" 'IT J [f(o,e)-Tsl ° 1 r3_R3 A dimensionless Lagrangian coordinate = - --;:::--3 £3 Radius ofa bubble ~: A dimensionless radius = R/£ T: Temperature t: Time Cl-: Thermal diffusivity of a liquid t UCRL-174l.5 sin e de ~: Cl-£ A dimensionless Lagrangian coordinate = ~ I 4 R (t) dt \"i ° t K: Thermal conductivity of a liquid p: Density of a fluid ~: A dimensionless time ro: A dimensionless temperature = Interface temperature - bulk temperature Interface temperature - saturation temperature Subscripts £: Refers to properties in the liquid phase q: Refers to a characterist.ic qUantity for the· constant heat flux case s: Refers to quantities at equilibrium, (or saturation) w: Refers 'to quantities at the wall 00: Refers to quantities in the bulk medium Functions erfc x - 5 - UCRL-17415 00 ierfc x = j'erfc t dt )( x r(a,x) =. J e-t t a - 1 dt 0 References 1. L. A. Waldman and G. Houghton, Chern. Eng. Sci. 20, 625 (1965). 2. M. S. P1esset and S. A: Zwick, J. Appl. Phys. 25, 493 (1954) .. 3. G.Birkqoff, R. S. Margulies and W. A. Horning, Phys. Fluids ~, 201 (1958). 4. L. E. Scriven, Chern. Eng. Sci. 10, 1 (1959). 5. L. A: Skinner and S. G. Bankoff, Phys. Fluids ~, 1417 (1965). 6. H. Y. Cheh, "On the Mechanism of Electrolytic Gas Evolution," Ph.D. TheSiS, UCRL-17324, University of California, Berkeley, 1967. 70 H. S. Cars1aw and J. C. Jaeger, "Conduction of Heat in Solids," 2nd ed., Oxford, 1959. "'-~-.-'--' ~ .- ... _ .. _-_.- ---- 6 - U.CRL~ 17415 1.8 1.6 1.4 1.2 If) .... T 1.0 -'J r<> - 0.8 ~ 0.6 l.6 1.8 2.0 XBL671- 80 --------.-.--~--.---- ------.--.--- --. -'.--" -.- .. - -. --- ._ .. -- -.- . -.- - -~ -.- ..... , ... FigUre 1. The Case of Constant Wall Temperature. - 7 - UCRL-17415 . 1.8 1..6 1.4 rt) 1.2 ...... 1- 1.0 ~ r<) -~ 0.8 0.6 1.4 1.6 4 8 12 16 20 r(3J)4/3 XBL671 - 81 . . FigUre 2. The Case of Constant Heat Flux. This report was prepared as an account of Government sponsored work. Neither the United States, nor the Com-mlSSlon, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, appa-ratus, method, or process disclosed in this report may not infringe privately owned rights; or 8. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any infor-mation, apparatus, method, or process disclosed in this report. As used in the above, "person acting on behal f of the Commission" includes any employee or contractor of the Com-mission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his empioyment with such contractor. 

ON THE DYNAMICS OF HEMISPHERICAL PHASE GROWTH IN NONUNIFORM TEMPERATURE FIELDS

Abstract

Similar works

Full text

Available Versions

eScholarship - University of California