NUMERICAL SIMULATION OF CONDENSATION ON A CAPILLARY GROOVED STRUCTURE

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1 Proceedngs of MECE'OO 2000 nternatonal Mechancal Engneerng Congress and Exhbton November 5-1 0, 2000, Orlando, Florda, USA 2000 MECE NUMERCAL SMULATON OF CONDENSATON ON A CAPLLARY GROOVED STRUCTURE Yuwen Zhangland Amr Faghr Department of Mechancal Engneerng Unversty of Connectcut Storrs, Connectcut Emal: zhangyl2 asme.org ABSTRACT Condensaton n a capllary grooved stzucture s nvestgated usng the Volume of Flud (VOF) model. The governng equatons arc wrtten n a. generalzed form and are applcable to both lqud and vapor phases. Condensaton on the fn top and at the menscus s modeled by ntroducng addtonal source terms n the contnuty, VOF, and energy equatons. The effects of temperature drop, conta.ct angle, su.rfa.cc tenson, and fn thckness on the condc.nsaton heat transfer are nve;tge.ted. NOMENCLATURE Cp specfc heat, J /(kgk) F body force, N h enthalpy, J /kg he beat transfer coeffcent, W/(mZ K) hjg latent heat of condensaton, J /kg le thermal conductvty, Wj(mK) L 1 one half of the fn thckness, m m mass fow rate (kg/s) m"' condensaton rate at the menscus, kg J ( m 3 s) p pressure, Pa q local heat flux, W/m 2 SAH source term, W/m 3 T temperature, K t 1 heght of the fn, m tv heght of the vapor space, m u., v velocty components n the ::, y drecton, m/ s W one half of the the groove wdth, m ::, y coordnate, m 1 Prescntly at ThcnnoB.ow, nc., 29 Hudson Rd., Sudbury, MA Greek symbols 6 flm thckness on the fn top, m 6.T temperature dfference, T,4! - To, K c volume fracton of lqud Bmen menscus contact angle µ dynamc vscosty, Pa s p densty, kg/m 3 r7 surface tenson, N /m Subscrpts 0 bottom of the fn f fn e lqud men menscus sat saturaton v vapor NTRODUCTON Vapor condensaton n capllary grooves, pores and slots s very mportant for many two-phase devces such as heat ppes, condensers, etc. The vapor condenses because the sold wall temperature s below the vapor saturaton temperature. Heat transfer durng condensaton from the lqud-vapor menscus n these structures s very mportant for predctng performance of two-phase devces. Khrustalev and Faghr (1994a) analyzed heat transfer of a mcro beat ppe wth trangular cross secton. At the condenser secton of the heat ppe, lqud resdes n the corner of the trangular cross secton due to capllary pressure. The vapor condenses at these nner surfaces of the trangle, and the condensate s sucked nto the corner of the trangle. The Copyrght 2000 by ASME

2 thckness of the flm was descrbed usng an ordnary dfferental equaton proposed by Kamotan {1976) together wth an assumpton of polynomal flm thckness varaton. Khrustalev and Faghr (1994b) nvestgated heat transfer durng evaporaton and condensaton on capllary grooved structures of heat ppes. Effects of dsjonng pressure and nterfacal thermal... esstance on the flm condensaton on the fn top surface were taken nto account n ther model. Ther numercal results ndcated that the effect of nterfacal thermal resstance s neglgble because the flm thckness s comparatvely large. The vapor was assumed to be saturated and the effect of vapor superheat on condensaton was not taken nto account. n addton, heat transfer n the lqud resdng n the grooves was modeled as purely a conducton problem by Khrustalev and Faghr (1994a.,b). Khrustalev and Faghr ( 1996) nvestgated flud fl.ow effects on evaporaton from the lqud vapor menscus. Heat conducton n the sold wall was neglected based on the argument that the thermal conductvty of the sold wall s much hgher than that of workng flud. They concluded that flud flow can be very mportant when the temperature dfference s large. The objectve of the present study s to nvestgate condensaton at the lqud vapor menscus n a capllary groove. The successful soluton of the problem requres the numercal solutons of: (1) flm condensaton on the fn top; (2) condensaton at the lqud vapor menscus; and (3) flud flow n the capllary groove. The effects of coolng temperature, contact angle, surface tenson, and fn geometry on the condensaton wll also be nvestgated.,, y, ; ,! l : :! : l l Vapor : j '! ~""%---~--;~~~~ v,... Lqud ---r--t---x j----t 2W Lo Fgure 1. Geometrc confguraton of the capllary groove fn PHYSCAL MODEL The condensaton of the vapor at the lqud-vapor menscus nterface s shown n Fg. 1. t can be consdered as a capllary groove or pore structure n the condenser. n order to solve the.problem, the followng assumptons a..re made: 1. Snce the wdth of the groove s very small, the effect of gravty can be neglected due to surface tenson domnance. 2. The number of the grooves s large enough so that symmetry condtons are applcable at the left and rght sde of the doman (Khrustalev and Faghr, 1999). 3. The temperature n the fn s unformly equal to the coolng temperature, Tw. Ths assumpton s justfed because the thermal conductvty of the fn s several hundred tmes hgher than that of the lqud workng flud (Khrustalev and Faghr, 1994b; 1996). 4. The lqud and vapor fl.ow are lamnar and ncompressble. 5. The thermal propertes of lqud and vapor arc not dependents of temperature.. Governng equatons Snce heat transfer and flud flow s a symmetrc problem1 only half of the groove needs to be nvestgated. n order to smplfy the solvng procedure, one set of the governng equatons s wrtten for both the lqud and vapor regons. The contnuty and momentum equatons are Dp (au av) -+p -+- =O Dt 8:n 8y (1) 2 Copyrght 2000 by ASME

3 Substtutng eq. (9) nto eq. (7), and obtanng the VOF equaton where F:c and F 11 are the components of the body force resultng from surface tenson at the nterface and wll be descrbed later. The flud propertes are defned as p = (1 - Ct)Pv +Ct.Pt (4) (5) (10) The thermal conductvty and specfc heat can be defned usng a smlar method by whch densty s defned (11) The specfc heat s defned usng the weght fracton of lqud and vapor: and et s volume fracton of lqud. The value of Ct s zero n the vapor phase and unty n the lqud phase. The volume fracton of water, el, satsfes the contnuty equaton for the lqud phase The total enthalpy s defned as (12) Dct ( ou av ) m"' -+et -+- =- Dt ox oy Pt where m 111 represents the mass producton rate of condensaton. Equaton (6) can be rewrtten as (6) 1 H = - ((1- c)p 11 (h,, + h1 9 ) + eplht] p 1 1 = -((1 - e)p 11 h., + ep1.ht) + -(1- e)p,,hfg (13) p The two terms n the rght hand sde of eq. (13) represent contrbutons of sensble enthalpy, h 1 and latent heat, D..H, to the total enthalpy p De1. _ m 111 (au ov) _ S f:l, ~t Dt Pt. oz 8y (7) 1 h = - [(1 - e)p,,h,, + Eptht} p (13a) Smlarly, the volume fracton of vapor, e 11, satsfes the followng equaton De 11 m Cv '\ V = - - Dt Pu Substtutng eq. (4) nto eq. (1) and consderng eq. (6), the contnuty equaton s smplfed (8) 1 D..H = - [(1 - e)p,,h,, + eptht] p (13b) the total enthalpy defned n eq. (13) can be rewrtten n terms of sensble enthalpy and latent heat H = h+ D..H (14) au+ av = - (_!_ - ~) ~ll ox oy Pv Pt (9) The energy equaton wrtten n terms of enthalpy and temperature s: 3 Copyrght@ 2000 by ASME

4 a a a ( at) a ( at) -(puh) + -(pvh) = - k- +- le- (15) ax oy ax ax ay oy Substtutng eq. (14) nto eq. (15), one can obtan a a a ( at) a ( at) - (puh) + - (pvh) = - k- + - k- + SD.H ax ay ax ax 8y ay (16) where St:J.H = - [:x (pu6h) + :y (pv6h)] (17) t can be seen that an addtonal term appears n the rght hand sde of eq. (16). Ths term s to account for the effect of condensaton n the energy equaton. The term s zero everywhere except at the lqud vapor nterface. The rate of condensaton at the control volume cells that encloses the lqud-vapor nterface can be expressed as. ll St:;.H m =-- hjg (18) The boundary condtons at the top of the computatonal doman are (23} U = 0 1 y = lg + tv (24) (25) where Vn s vapor nlet velocty and ts value depends on the amount of vapor condensed at the fn top and the menscus. The left and rght boundares of the doman ( :z: = 0, L 1 +W) satsfy the symmetrc boundary condton, u = 0, ov/ox = O, and at/ax = 0. Consderaton of lqud-vapor nterface The conservaton of normal and tangental momentum for the control volumes at the sold-lqud nterface s automatcally satsfed because the governng equatons were Boundary condtons The temperature at the bottom of the groove s constant T =To, y = 0 (19) The tempe-ratfre n the fn s unformly equal to the temperature at the bottom of the groove T =To, 0 < y < t 9, 0 < x < L (20) The velocty at the bottom of the groove satsfes u = 0, y = 0 (21) av - 0 0y - y=o (22) Fgure 2. Water volume fracton: (a) t:::.t = 5K; (b) t:::.t:::: lok 4 Copyrght 2000 by ASME

5 J.7'1~..tl:? r~.1.13(;<-0! t"~.\.7~1:+02 :\.7lH+02.l.7:!l~+02 ' 711 ~ +0! :t7h~+o2 l,7tr+o~ o.s -Present - Khrustolev nd Faghr {l9~b) J(,tJt 1U2 :tmu~..,02 ).C.91:<-02 x{m) 3.681;-Mll,:t,()81:+02 ( ) Fgure 3. Comparson of lqud flm thckness on the fn top wrtten for the entre computatonal doman, ncludng lqud and vapor. The effect of surface tenson on pressure s modeled usng the contnuous surface model. The model nterprets surface tenson as a contnuous 3-D effect across the nterface, rather than a boundary condton on the nterface (Brackbll et al., 1992). Forces due to the pressure jump at the nterface can be expressed as volume force usng the dvergence theorem r,., : '-.1.7~1 02 J.7JE+Ol.l.72E+o2 J.721~-c02 ).71[; :w2.l.701:.+()2 3.70\;+0Z J.<.9H-t02 ).6%+11? M18llt<l2 (26) where curvature of the nterface, K, s gven by (Brackbll et al., 1992) H>4E<-02 J.641;... )l K == 2.. [(~ v) lnl - (V n)] lnl n! (27) Fgure 4. Temperature Contour: (a) t:..t =SK; (b) tj.t = lok The normal drecton of the lqud vapor nterface toward the vapor phase s n. The body force, F, obtaned by eq. (26) s substtuted nto the momentum equatons (2-3) to solve for the veloctes n the lqud and vapor phases. Lqud and vapor temperature are contnuous at the nterface. At the lqud vapor nterface, the temperature should be equal to the saturaton temperature. { y = tg + 6, Y = Ymen1 0 < x <L L1 < x <L+ W (28) by ASME

6 6 5 t.t lok 2 (m) 6 8 x L..~..._~..._~,_~~'--~..._~-:7~-:'-:~--:,~s~~,~a:----' X 10 ' (m} Fgure 5. Local heat fluxes at dfterent temperature Fgure 6. Effect of contact angle on the lqud flm thckness The sensble enthalpy correspondng to the above nterface temperature s At steady state, ths condensaton rate can be used to determne vapor nlet velocty (29) where Tref s reference temperature of enthalpy. The energy balance at the lqud vapor nterface also needs to be satsfed. Snce lqud and vapor regons are treated as one doman, the energy balance at the lqud vapor nterface s satsfed when the converged soluton for the entre doman s obtaned. NUMERCAL SOLUTON n the numercal smulaton, the nterfacal temperature and enthalpy a.re set to the values specfed n eqs. (28-29). The source term, Sfl.H, that satsfes eqs. (28-29) s determned usng eq. (16). Once the source term n the energy equaton s determned, the condensaton rate at the lqud-vapor nterface can be determned usng eq. (18). The total condensaton rate, whch ncludes condensaton at fn top and menscus, s then expressed as m= L n"' 6V O<e<l (30) m Un = p.,(l1 + W) (31) The effectve condensaton heat transfer coeffcent s defned as follows he = (L+ W)(T,.., - To) (32) Here, only the steady-state soluton of the condensaton problem s nvestgated. t s mpossble to solve the steady state problem drectly, because the donor-acceptor model used n the VOF method works only for unsteady state problems. A false transent method (Basu and Srnvasan, 1988) s employed. Wth ths methodology, the false transent terms are ncluded n the governng equatons and steady state s obtaned when the condensaton length does not vary wth the false tme. The overall numercal soluton procedure for a partcular tme step s outlned below: L Assume a value for the mass producton rate, m 111 and by ASME

7 6,. ' Omf'n- 84 l l x(m) x 10 ~ o.so::--0:':2:------=ol-. --o:':.s:----::o"=".s--'-1---',2-- l<(m) 1""" -_ s--1L-.e-_;2 10 Fgure 7. Effect of contact angle on the local heat fluxes Fgure 8. Effect of surface tenson on the lqud flm thckness compute the source term for the VOF eq. (10) and the contnuty eq. (9). 2. Calculate vapor nlet velocty, Vn, based on eqs. (30-31). 3. Solve the VOF eq. (10). 4. Solve the contnuty eq. (9) and momentum eqs. (2-3). 5. Solve for the temperature dstrbuton from eq. (16). For the control volumes ncludng the nterface, the temperature s set to the saturaton temperature and eq. (18) s used to determne the mass producton rate, mm. 6. Compute t~e 1!9Urce term for the VOF equaton (10) and the contnuty equaton (9). 7. Go back to step 2 untl the relatve resduals for the pressure correcton equaton, momentum equatons, and enthalpy equaton are wthn the lmt. After the soluton for the current tme step s obtaned, the computaton for the next tme step s performed. The heat transfer and flud flow s solved usng the SMPLE Algorthm (Patankar, 1980). The convecton-dffuson terms are dscretzed usng a power law scheme. After the grd number test, a non-unform grd wth 44 grd ponts n x drecton and 66 grd ponts n y drecton was used n the numercal soluton. After the grd sze and the tme step test, the problem s solved usng a non-unform grd of 44(:z:) x 66(y). 1:'he- grd near the top of the fn s very fne n order to smulate flud fl.ow and heat transfer n the thn lqud layer. The grd near the rght hand sde of the fn surface s very fne n order to smulate the effect of contact angle and surface tenson on the condensaton. After several numercal tests, the false tme step used n the computaton s D.t = o- 1 s. RESULTS AND DSCUSSON Numercal smulatons are performed for water at the saturaton temperature of 373K. The other geometrc parameters are: L 1 = 0.02mm, W = 0.06mm, t 9 = 0.24mm, t" = 0.54mm and Omen = 84. The vapor s assumed to be saturated and therefore there s no heat transfer n the vapor phase. Fg. 2 (a) and (b) show the contour of the VOF of water for temperature drops of D.T = 5K and lok respectvely. t can be seen that the lqud flm thckness on the fn top s very small, and t s therefore expected that heat condensaton manly occurs on the fn top surface. Snce the dstance between most parts of the lqud vapor menscus and the cold wall s much larger than the flm thckness on the fn top, t s expected that, as becomes evdent later, the contrbuton of condensaton on the menscus to the overall heat transfer of the grooved structure s not mportant. Fg. 3 compares the lqud flm thckness obtaned by the present numercal soluton and the soluton of Khrustalev and Faghr (1994b), who assumed tha.t the heat transfer by ASME

8 6 7 'rr~2rro ~ ',,, '1=ao.S ~s ~E ~ ~4,',.,.' 2,, 2 O.o.l l.s,.,,,.,,,(~... O.o<.... ~,.,, {,... j o.s 2 x(m) ' x(m) Fgure 9. Effect of surface tenson on the local heat fluxes Fgure 10. Effect of fn thckness on the lqud flm thckness across the lqud flm on the fn top s due to conducton only. t can be seen that the present results agreed very well wth that of Khrustalev and Faghr ( l 994b), whch suggests convecton has lttle effect on the flm thckness on the fn top. Fg. 4 (a) and (b) show temperature contours for temperature drops of 6.T :;:: SK and lok respectvely. t can be seen that the temperature gradent across the thn lqud flm on the fn top s very large. On the other hand, the temperature gradent at most parts of the menscus n the fn s very small. The dfferent behavors of the temperature gradent on the fn top and the menscus suggest that condensaton occurs ma.nly on the fn top. Fg. 5 shows the local heat flux on the fn top and the lqud- vapor menscus for dfferent temperature drops. t can be seen that local heat flux at the fn top surface s hgher than that at the menscus for two order of magntude. The majorty of heat s transferred through condensaton on the fn top surface. Ths s because the thckness of the lqud flm on the fn top surface s much thnner than that n the menscus. When the temperature drop s ncreased from SK to lok, the local heat flux at both fn top and menscus s sgnfcantly ncreased. Fg. 6 shows the effect of contact angle, Om~n, on the lqud flm thckness on the fn top. The lqud flm thckness at the centerlne of the fn decreases wth the ncreasng contact angle. On the other hand, the lqud flm thckness at the corner of the fn ncreases wth the ncreasng contact angle. n other words, ncreasng the contact angle causes the lqud flm on the fn top to become more fat. The effect of the contact angle on the local beat Bux s shown n Fg. 7. The heat flux s more unform when the contact angle s ncreased from 84 to 88. The effect of surface tenson on the lqud flm thckness on the fn top for the temperature drop of lok s shown n Fg. 8. n addton to the result for surface tenson at ts normal value, uo, two curves wth changed surface tenson are also plotted n Fg. 8 for comparson. t can be seen that the lqud flm thckness on the fn top decreases wth ncreasng surface tenson. The reason that flm thckness thns wth ncreasng surface tenson s that the condensaton phenomenon always tends to mnmze the surface energy at the nterface. Wth ncreasng surface tenson, ths energy ncrease-. n order to reduce ths energy, the rad of curvature has to be ncreased and that means lqud flm thckness bas to be decreased. Fg. 9 shows the local heat fluxes for varyng surface tenson. The maxmum heat Bux s obtaned from the case wth the hghest surface tenson because the lqud flm thckness s thnnest for hghest surface tenson. The effect of the fn thckness, L 1, on the lqud flm thckness of the fn top s shown n Fg. 10. The groove wdth, W, s decreased by the same value as the ncrease of the fn thckness to mantan a constant L 1 + W for both 8 Copyrght@ 2000 by ASME

9 .. x ,...c..;.~~~~~~~~~~--,.~~~~~--r-~~~~~~ s L 1 c 0.02mm ' ' '' 1 ' L1=0.03mm X10' r--~ ,~---r----r--~-~ \ \ ' \,.\. ' \ ' ' \ ' ' ' '!:: ~ 1.3 ~ ~ <(mj 6 0.9' '----'----'---'---'---'-- --'-----'----' 1 6.:T(K) Fgure 11. Effect of fn thckness on the local heat fluxes Fgure 12. Varaton of heat transfer coeffcent wth temperature drop cases. n other words, the number of grooves s not changed when the fn thckness, L1 s ncreased. As can be seen from Fg. 10, the lqud flm thckness ncreases sgnfcantly wth ncreasng fn thckness. Fg. 11 shows the effect of fn thckness on the local heat fluxes. t s seen that the local heat flux s sgnfcantly lower for larger fn thckness, because the lqud flm s thcker. The effect of fn thckness, however, on the total heat transfer s not as sgnfcant as ts effect on the local heat flux, because the condensaton area s larger for thcker fns. The heat transfer coeffcent defned by eq. (32) s a very mportant._ pyameter for the desgn of the grooved structure. The 'effect of the temperature drop en the heat transfer at dfferent contact angles s shown n Fg. 12. t s seen that the heat transfer coeffcent decreases wth the ncreasng temperature drop for all contact angles. When the contact angle s ncreased from 84 to 88, the heat transfer coeffcent s ncreased for all temperature drops. Ths ncrease s more sgnfcant for smaller temperature drops. When the contact angle s ncreased to 89, heat transfer at lower temperature drops s ncreased, but t s decreased for larger temperature drops. n order to nvestgate the effect of contact angle on the heat transfer coeffcent, the varaton of heat transfer coeffcent wth contact angle s plotted n Fg. 13. ncreasng the contact angle results n an ncrease of heat transfer coeffcent untl t reaches a maxmum value. Then heat transfer decreases when the contact angle ncreases further. CONCLUSONS Condensaton n a capllary groove has been nvestgated usng the Volume of Flud (VOF) model. Condensaton on the fn top and menscus s modeled by usng approprate source terms n contnuty, VOF, and energy equatons. The results show that convecton n lqud has an nsgnfcant effect on the flm thckness on the fn top. The majorty of the heat s transferred through condensaton on the fn top surface, because the thckness of the lqud flm on the fn top surface s much thnner than that n the menscus. The lqud flm on the fn top becomes flatter and the beat transfer coeffcent on the fn top becomes more unform when the contact angle s ncreased. When the surface tenson s ncreased, lqud flm on the fn top thns a.nd the local heat flux on the fn top s hgher. ncreasng the fn thckness causes the thckness of the lqud flm to ncrease sgnfcantly. For all contact angels, heat transfer coeffcents decrease wth ncreasng temperature drop. There s a contact angle at whch the heat transfer coeffcent s maxmum. 9 Copyrght 2000 by ASME

10 ... ~ 10* 1,5;:.._..::'----~----~---~-~-----~, L1" :::: --- "~ --- ~ \ - ~ pp Khrustalev, D., and Faghr, A., 1996, Flud Flow Effects n Evaporaton from Lqud-vapor Menscus, ASME J. Heat Transfer, Vol. 118, pp Khrustalev, D., and Faghr, A., 1999, Coupled Lqud and Vapor Flow n Mnature Passages wth Mcro Grooves, ASME J. Heat Transfer, Vol. 121, pp Patankar, S.V., 1980, Numercal Heat Transfer a.nd Flud Flow, McGraw-Hll, New York o.s'----'---'-----'-----_._-,.~-... J Fgure 13. Varaton of heat transfer coeffcent wth contact angle ACKNOWLEDGMENT Fundng for ths work was provded by NASA Grant NAGJ-1870 and NSF Grant CTS REFERENCES Basu, B., and Srnvasan, J., 1988, Numercal Study of Steady State Laser Meltng Problem, nt. J. Heat Mass Transfer, Vol. 31, pp Brackbll, J. U., Kothe, D.B., and Zemacb, C., 1992, A Contnuum Method for Modelng Surface Tenson, Journal of Computatonal Physcs, Vol. 100, pp Faghr, A., 19~5, Hea.t Ppe Scence a.nd Technology, Taylor and Francs, Washngton, DC. Kamotan, Y., 1976, Analyss of Axally Grooved Heat Ppe Condensers, AAA paper No Khrustalev, D., and Faghr, A., 1994a, Thermal Analyss of a Mcro Heat Ppe, ASME J. Heat Tra:nsfer, Vol. 116, pp Khrustalev, D., and Faghr, A., 1994b, Heat Transfer Durng Evaporaton and Condensaton on Capllary Grooved Structures of Heat Ppes, Proc ASME Wnter Annual Meetng, Chcago, Nov., ASME HTD-Vol. 287, pp Khrustalev, D., and Faghr, A., 1995, Thermal Characterstcs of Conventonal and Flat Mnature Axally Grooved Heat.Ppes, ASME J. Heat Trensfer, Vol. 117, 10 Copyrght 2000 by ASME