International Journal of Heat and Mass Transfer 43 (2000) 1391±1398

Transcription

1 Intenational Jounal of Heat and Mass Tansfe 43 (2) 1391±1398 Condensation on the outside suface of a small/mini diamete tube fo vapo owing though a hoizontal annulus suound by an adiabatic concentic tube Bu-Xuan Wang*, Xiao-Ze Du Themal Engineeing Depatment, Tsinghua Univesity, Beijing 184, People's epublic of China eceived 23 Apil 1999; eceived in evised fom 25 June 1999 Abstact An analytical model, simultaneously accounting fo gavity, vapo shea along axial diection and suface tension e ects on the condensate lm laye, is poposed fo ow condensation pocess in hoizontal annulus with small/mini diamete inne tube and taditional industial-used oute adiabatic tube. The analysis shows that, the e ect of suface tension can not be neglected fo small/mini tube of diamete less than 3 mm o.d., especially in low vapo quality zone. The shea stess and suface tension in uence the ow condensation mainly by means of distibuting condensate lm unifomly along tube cicumfeence in hoizontal annulus. Expeiments wee conducted to examine the analysis. # 2 Elsevie Science Ltd. All ights eseved. Keywods: Flow condensation; Outside of small/mini tube; Theoetical analysis; Expeimental examination 1. Intoduction Flow condensation is widely adopted in pocess industies, efigeation equipment, and powe system. As a esult of technological pogess, it is essential to have compact and e cient condenses in many applications, such as automotive ai conditiones o lifesuppot systems in space, and also the ecent advances fo mico mechanical system (MMS). It is theefoe logical to use small o mini-diamete tubes. We can deduce that moe e cient condensation will take place inside o outside a pipe of small/mini diamete owing * Coesponding autho. Tel.: ; fax: addess: bxwang@mail.tsinghua.edu.cn (B.-X. Wang). to lage speci c suface aea fo heat tansfe. In the pesent study, an assessing of the potential of such a hoizontal annulus as condense with inne small/mini diamete tube is taken. Up to now, the Nusselt analysis [1] fo lamina lm condensation is still used as the fundamental basis fo solving poblems connected with condensation. Howeve, thee ae many cicumstances whee Nusselt theoy undepedicts the heat tansfe ate. This esults fom the basic assumptions Nusselt obtained his solution on plain sufaces fo the case of non-moving vapo and neglecting the shea stess on vapo±liquid inteface. Many attempts [2±4] have been done to impove Nusselt's analysis by taking into account the e ect of shea stess. Howeve, few have taken into consideation the suface tension e ects on the condensate lm in thei analysis. ohsenow [5] put fowad that fo tubes of diamete lage than 3 mm, the con //$ - see font matte # 2 Elsevie Science Ltd. All ights eseved. PII: S17-931(99)228-8 转载

2 1392 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391±1398 Nomenclatue c f D d g h lv m. Nu p q e T u v x z fiction coe cient inside diamete of oute tube outside diamete of inne tube gavitational acceleation latent heat of condensation mass ow ate Nusselt numbe pessue heat ux outside adius of inne tube eynolds numbe adial coodinate tempeatue velocity in axial diection velocity along cicumfeence vapo quality axial coodinate Geek symbols a heat tansfe coe cient d thickness of condensate lm l themal conductivity m y s t dynamic viscosity pola angle density suface tension shea stess Subscipts inlet cool cooling wate exp expeimental i inne tube l liquid o oute tube atio of liquid and vapo popeties s satuation state v vapo w tube wall d vapo±liquid inteface Supescipt - aveage o mean value densate lm can be teated as on a plain suface, othewise, the bending e ect of condensate lm can not be neglected. Some eseaches have epoted the impact of suface tension on the cuved condensate suface, pimaily with espect to leading edge e ects [6±8]. Kupiczka [9] and Buznik et al. [1] consideed the vaiation in cuvatue of condensate lm, accounted fo a pessue gadient tem due to suface tension. Jacobi and Goldschmidt [11] extended Nusselt's analysis to discuss the Maangoni e ect on lmwise condensation heat tansfe aound a cylinde in coss ow. Most of these studies quoted [6±11] focused on the condensation fo vapo coss ow on a cylinde and didn't attempt to account fo the coupling e ects of shea stess in tube axial diection. The pupose of pesent study is to simultaneously account fo gavity, vapo shea along axial diection and suface tension e ects on the condensate laye in hoizontal concentic annulus with cold small/mini-diamete inne tube and an adiabatic oute wall. Flow condensation outside along a taditional inne tube of lage diamete is also eckoned as fo compaison. T w of inne wall suface and adiabatic oute wall is consideed. The satuated vapo at tempeatue T s, which is geate than T w, ows in the annulus with inlet eynolds numbe e v and is condensed on the inne tube wall suface. The condensate lm is simultaneously dained by shea stess due to the vapo ow in axial diection, dz, and by eithe gavity o suface tension in cicumfeence diection, dy, espectively. The simplifying assumptions ae made as: (1) thee ae no intefacial waves between the condensate and vapo; (2) thee is no pesence of non-condensable gas; (3) neglect the convective tems in enegy equation and the inetia tems of momentum equation; (4) the tempeatue of the oute suface of the lm was the same 2. Theoetical analysis Taking the cylindical co-odinate system shown in Fig. 1, a hoizontal annulus with unifom tempeatue Fig. 1. Coodinating system and elementay condensate lm volume fo analysis.

3 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391± as the satuated vapo tempeatue; (5) the density, l, themal conductivity, l l, and viscosity, m l, of the condensate ae taken as constants, being independent of tempeatue. Then, the govening equations fo consevation of mass ow and momentum tansfe acoss the condensate lm will ˆ whee v is the condensate lm velocity along cicumfeence, u is the velocity along axial diection, and 1 if neglecting the fictional pessue dop and inetia pessue dop along the peiphey. Combined with the bounday conditions, integating Eq. (2), we obtain the velocity po le along cicumfeence as v ˆ A 2 2 m l 2 B m l ln whee d l d l ln d l 8 m 2 2 lg sin y ˆ with the coesponding bounday conditions: 2 A ˆ l v g sin y B ˆ and d l d l 9 v ˆ, fo ˆ ˆ, fo ˆ d l 3b whee is the outside adius of the inne tube, d l is the condensate lm thickness, o d l =f(, y ). In axial diection, we l ˆ with bounday conditions as: u ˆ, fo ˆ 5a ˆ t d, fo ˆ d l 5b t d is the shea stess on the vapo±liquid inteface due to vapo ow. The liquid lm thickness will vay along the axial and cicumfeence diection. An additional pessue gadient can be poduced by suface tension due to the vaiation of condensate lm cuvatue. That is, p l ˆ p v s d l d l d l we ˆ d l vg sin y d l The st tem of the ight-hand side of Eq. (8) esults fom the gavity, wheeas the second tem involving B e ects the e ect of suface tension on vapo±liquid inteface. The lm thickness in axial diection is faily unifom as compaed with the vaiation along cicumfeence, and theefoe, the capillay pessue dop along axial diection can be neglected. Hence, the pessue gadient of condensate lm espected to z comes mainly fom fictional pessue dop due to the vapo ow. A momentum balance fo the vapo egion yields ˆ 2p 2p t d d l dy pd i t w : 1 D i =2 d l Šdy The condensate lm thickness is compaatively so small, with espect to the hydaulic diamete of the annulus, (D i d o ), that, the pessue dop of liquid along axis can be obtained ˆ 4 t dd o t w D i D 2 : 11 i d 2 o No mass tansfe exists on the adiabatic oute wall of the annulus, the shea stess would be esulted fom the ow fiction, i.e. t w ˆ cf 2 vu 2 v : 12 But also, the intefacial shea stess, t d, due to the vapo condensed onto liquid suface should be included on the inne wall as t d ˆ cf 2 v u v u j ˆ dl 2 m u v u j ˆ dl 13 among which, the two-phase ow fiction coe cient,

4 1394 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391±1398 c f, was given by Henstock and Hamatty [12] fo cicula pipe ow with e v 2, and was late adopted by Faghi and Chow [13] to annula ow as c f =2 ˆ c f =2 1 85F whee c f =2 ˆ :85=e :25 v 14 15a F ˆ gm = :5 e:9 v 15b g ˆ 1:414e :5 l 2:5 :132e :9 l 2:5 Š :4 : 15c Consequently, the velocity po le in axial diection can be deduced fom integating Eq. (4) along with the bounday conditions, Eq. (5), and applying Eq. (11)± (13) as u ˆ A 2 C 1 ln C m l m l whee A ˆ 1 dp l 2 dz C 1 ˆ d l t d A d l Š 17a 17b C 2 ˆ A 2m l 2 C 1 m l ln 17c The heat conducted, though the lm to the incemental element dz dy in Fig. 1, causes a change of condensate ow as T s T w dz dy l l ln d l d l dl ˆ l h lv u d dy # dl v d dz hydaulic diamete of the annulus, we can detemine the aveage vapo velocity, u v, and vapo quality, x, at di eent axial position, z, fom the mass balance fo vapo±liquid system, i.e. _m v _m l u v ˆ 2p 2 l d l dy and x ˆ _m v _m l : 21 _m v Assuming a linea tempeatue po le in the thin condensate lm, we have the local lm thickness, d l,as l l a z, y ˆ ln d : 22 l The aveage heat tansfe coe cient along the peiphey can be theeby pedicted as a z ˆ 1 p p a z, y dy with mean Nusselt numbe, Nu, as 23 Nu ˆ a z d o l l : 24 This poposed analytical model can be solved numeically. The discetizations of govening equations wee taken along axis and peiphey, espectively. The step size in z diection was Dz =.1, while the step size taken in cicumfeence diection was Dy=18. Using the last calculated vapo velocity u v, shea stess t d, t w, and pescibed condensate lm thickness, which is assumed unifom in evey gid, the momentum conse- whee l l is the conductivity of condensate lm, h lv is the latent heat of condensation. Then, the mass ow ate of condensate, m. l, at axial position z can be acquied by _m l ˆ ll T s T w h lv z 2p dz ln 1 dy: d l d l 19 Fo the known inlet vapo mass ow ate, m. v, by the pescibed inlet vapo eynolds numbe, e v, and the Fig. 2. Schematic diagam of the test appaatus.

5 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391± vation equation was solved to detemine the value of liquid mass ux. If it is not satis ed with the mass ow ate calculated fom the enegy consevation Eq. (19), the value of lm thickness is changed slightly and the same iteative pocess is epeated until the elative discepancy is less than.1%. Afte all the values of d l, m l, x, u v, t d, t w and a ae thus calculated at a coss section, the iteation is stated at the next axial location. 3. Expeimental examination Expeiments wee conducted to examine the pedicted esults. The expeimental appaatus is shown in Fig. 2. Wate steam was used fo test uid. The test setup consists of two main loops, namely, the vapo±condensate and cooling wate loops. The cleaned wate stoed in the eceive is pumped into the evapoato. The by-pass line at the dischage side of the pump enables to egulate the liquid volume in the evapoato. The evapoato is an electic boile of which the powe can vay fom to 3 kw. The ow ate of supeheated vapo geneated by the evapoato is adjusted by anothe by-pass valve at the outlet of evapoato. The vapo passes though the adiabatic section to come out in a satuated/supeheated condition. It is then condensed in the test section to a satuated condition. Tanspaent tubes ae connected to both ends of the test section to visually inspect the vapo conditions. The outlet condensate ows though the sub-cooling section to each a subcooled condition and comes into the volumetic cylinde to ead the volumetic ow ate. Finally, it is collected to the liquid eceive. The cooling wate loop designed fo condensing the vapo contains a wate themostat bath with a pump and the adjustable tempeatue fom 4 to 18C. A by-pass valve is also povided to adjust the ow ate. Vaying the tempeatue and ow ate of cooling wate, we can expect di eent vapo quality at the outlet of test section fo a xed inlet vapo eynolds numbe. The ow ate of cooling wate is measued by weighing the cooling wate collected in a cetain time. The test section is eally a tube-in-tube condense with counte ow aangement, which is 3 mm in length. The oute tube of the condense is mm i.d. and is coveed by a laye of insulation outside. The inlet vapo though a mixe, ows into the annulus and is condensed along the outside suface of the inne tube, while the cooling wate ows into the test tube in the opposite diection, as shown in Fig. 3. Two types of annulus ae selected as test section with the same oute tube but di eent inne tube of 2.97 mm o.d. and 5.98 mm o.d., signed by Type A and Type B, espectively. At the both ends of the total test section, the tempeatues of vapo/condensate wee measued. The inlet and outlet tempeatues of cooling wate of the condense and the condensate tempeatue in volumetic cylinde wee also measued. All the tempeatues wee measued by the.1 mm T-type coppe± constantan themocouples calibated with an accuacy of.18c. The pessues wee measued by pessue gauge with an accuacy of 2.25%. The weighing accuacy of cooling wate is.5 g, while the accuacy of volumetic cylinde is.1 ml. Fig. 3. Details of test section. Fig. 4. Plot of pedicted Nu vs vapo quality, x.

6 1396 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391±1398 Fig. 5. Vaiation of condensate lm thickness along cicumfeence of inne tube, e v =

7 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391± Taking the aithmetic mean of inlet and outlet cooling wate tempeatues as its aveage value, t - cool, and the aithmetic mean of vapo tempeatue at both ends of test section as vapo tempeatue, t s, we can obtain the aveage condensation heat tansfe coe cient ā exp fo the test section as 1 a exp ˆ ts t cool q ˆ do 2l w ln d o d i d o d i 1 a cool 25 whee a cool is the convective heat tansfe coe cient of cooling wate in inne tube, calculated by Dittus±Boelte coelation. The aveage Nusselt numbe can be theeby acquied by Nu exp ˆ a exp d o =l l : 26 Befoe the tests stated, the oute wall of the test tube was teated with mechanical polishing and the inne wall of the tube was washed by acetone epeatedly to avoid oganic and dit contamination. At most test conditions, the eo of heat balance between sensible heat of cooling wate and latent heat of condensate is less than 5%. Fo the uncetainty analysis, the basic methodology of Kline and McClintock [14] was adopted. Fom the analysis, the estimated aveage uncetainty in the measuement of the pesent expeiments fo Nu was 225%. 4. esults and discussions Fo caying out a concete analysis, we consideed the case of ow condensation of steam with inlet vapo eynolds numbes, e v, fom 23 to 1 4. The analytical pedicting esults ae illustated in Figs. 4 and 5. The Nusselt numbe, Nu, fo ow condensation heat tansfe though annulus of both Type A and Type B with di eent inlet eynolds numbes, ae plotted as function of vapo quality, x, in Fig. 4. The esults with neglecting the e ect of suface tension on vapo±liquid inteface ae given simultaneously fo compaison. Accoding to Eq. (7), the e ect of suface tension is mainly embodied by the additional capillay pessue dop due to the lm thickness vaied along cicumfeence of tube. The degee of this e ect is detemined by the condensate lm cuvatue and vaiation of lm thickness. Fo annulus of Type A, in which vapo is condensed on a small-diamete tube, the condensate lm cuvatue become obvious as compaed with the tube diamete, especially in lowe vapo quality zone, x.45. Also fo small-diamete tube, the bending of condensate lm makes moe liquid dained by gavity along peiphey, which leads to the lm accumulating, and hence, the vaiation of lm thickness is enhanced Fig. 6. Compaison of pedicted value with expeimental data. too. Theefoe, an obvious deviation exists fo neglecting suface tension shown in Fig. 4(a). Howeve, fo annulus of Type B with lage outside diamete of inne tube, the vaiation of condensate lm cuvatue does not actually in uence the condensation pocess yet, so neglecting the capillay pessue dop due to suface tension almost has no e ect on Nu. As the e ect of shea stess on phase-change inteface enhances with inceasing e v, moe impotant ole of shea stess would be expected fo thinning the condensate lm thickness, and hence, Nu will incease. Howeve, the lm thickness inceases apidly with deceasing x due to condensate accumulation. This may lead to the case shown in Fig. 5(c), whee the themal esistance comes dominantly fom the condensate lm laye, and so, thee will be little in uence on Nu fo di eent e v. It is well known that, both the shea stess and the suface tension on vapo±liquid inteface can pomote the condensate distibuting moe unifomly along tube cicumfeence. Theefoe, fo ow condensation of vapo in hoizontal annulus, the inceasing shea stess on vapo±liquid inteface will impove accodingly the condensate lm thickness distibution moe even along the inne tube peiphey. This can be illustated, at di eent vapo quality, x, fo Type B compaed with that fo Type A, as shown in Fig. 5. Fo annulus of Type A with small-diamete of inne tube, the shea stess deceases with deceased quality, theeby, the suface tension will take moe obvious e ect in distibuting the lm thickness unifomly at low quality zone. Fig. 6 shows the compaison of the pedicted aveage Nu numbe fom analytical model with the expeimental esults. It indicates that most of the expeimental values ae within a ange of 3% to +2% of the pedicted values. As the estimated uncetainty of the pesent expeimental measuement of Nu exp is about 225%, the deviation of pedicted and

8 1398 B.-X. Wang, X.-Z. Du / Int. J. Heat Mass Tansfe 43 (2) 1391±1398 expeimental esults may indicate a little ove-pediction of the analytical model. 5. Conclusions An analytical model, taking into account the e ect of suface tension on the natue of ow condensation on the outside suface of a small/mini diamete tube is poposed. Fo taditional industial tube of which the diamete is lage than about 6 mm, the suface tension has actually negligible e ect on Nusselt numbe fo ow condensation and distibution of condensate lm along the tube suface. Howeve, fo small/mini tube of outside diamete less than 3 mm, the e ect of suface tension could not be neglected, especially in low vapo quality zone. In hoizontal annulus, both the shea stess and suface tension in uence the ow condensation by means of distibuting condensate lm unifomly along tube cicumfeence. Compaison with the expeimental data vei es the poposed analytical model to some extent. Acknowledgements The nancial suppot fo this eseach fom the National Natual Science Foundation of China (Gant No ) is geatly acknowledged. efeences [1] W. Nusselt, Die Obe aè chenkondensation des Wassedampfes, Ve Deut. Ing. 6 (1916) 541±546. [2] J.W. ose, E ect of pessue gadient in foce convection lm condensation on a hoizontal tube, Int. J. Heat Mass Tansfe 27 (1984) 39±47. [3] K. Suzuki, Y. Hagiwaa, H. Izumi, A numeical study of foced-convective lmwise condensation in a vetical tube, JSME Int. J., Se. II 33 (199) 134±14. [4] W.M. ohsenow, J.H. Webe, A.T. Ling, E ect of vapo velocity on lamine and tubulent- lm condensation, Tans. ASME 78 (1956) 1637±1643. [5] W.M. ohsenow, Film condensation, Applied Mechanics eviews 23 (197) 487±496. [6] S. Hiasawa, K. Hijikata, Y. Moi, W. Nakayama, E ect of suface tension on condensate motion in lamine lm condensation (study of liquid lm in a small tough), Int. J. Heat Mass Tansfe 23 (198) 413±418. [7] M. Yanadoi, K. Hijikata, Y. Moi, M. Uchida, Fundamental study of lamina lm condensation heat tansfe in a downwad hoizontal suface, Int. J. Heat Mass Tansfe 28 (1985) 1937±1944. [8] S.B. Memoy, V.H. Adams, P.J. Mato, Fee and foce convection lamina lm condensation on hoizontal elliptical tubes, Int. J. Heat Mass Tansfe 4 (1997) 3395±346. [9]. Kupiczka, E ect of suface tension on lamina lm condensation on a hoizontal cylinde, Chem. Eng. Pocess. 19 (1985) 199±23. [1] V.M. Buznik, V.A. Aleksandov, G.A. Sminov, The e ect of suface-tension foces on the heat-tansfe coef- cient in the condensation of vapos on inclined tubes, Int. Chem. Eng. 9 (1969) 216±219. [11] A.M. Jacobi, V.W. Goldschmidt, The e ect of suface tension vaiation on lmwise condensation and heat tansfe on a cylinde in coss ow, Int. J. Heat Mass Tans. 32 (1989) 1483±149. [12] W.H. Henstock, T.J. Hodgson, The intefacial dag and height of the wall laye in annula ows, AIChE J. 22 (1976) 99±1. [13] A. Faghi, L.C. Chow, Annula condensation heat tansfe in a mico-gavity envionment, Int. Comm. Heat Mass Tans. 18 (1991) 715±792. [14] S.J. Kline, F.A. McClintock, Descibing uncetainties in single-sample expeiments, Mech. Engng 75 (1953) 3±12.