FORCED CONVECTION FILM CONDENSTION OF DOWNWARD-FLOWING VAPOR ON HORIZONTAL TUBE WITH WALL SUCTION EFFECT

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1 Journal of Marine Science and Tecnology DOI: /JMST Tis article as been peer reiewed and accepted for publication in JMST but as not yet been copyediting, typesetting, pagination and proofreading process. Please note tat te publication ersion of tis article may be different from tis ersion. Tis article now can be cited as doi: /JMST FORCED CONVECTION FILM CONDENSTION OF DOWNWARD-FLOWING VAPOR ON HORIZONTAL TUBE WITH WALL SUCTION EFFECT Fu-Jen Wang 1, Tong-Bou Cang, and Ru-Li Lin Key words: film condensation, flowing apor, potential flow, wall suction. ABSTRACT Te eat transfer performance of te condensate layer formed by a downward-flowing, dry saturated apor flowing oer an isotermal orizontal tube wit wall suction effects is examined. Under te effects of wall suction, te graity force, te potential flow pressure gradient and te interfacial sear stress, te local / mean sselt numbers of te condensate film are obtained as a function of te Jaob number Ja, te Prandtl number Pr, te yleig number, te two-pase mean Reynolds number Re, te dimensionless pressure gradient parameter P, and te suction parameter Sw. Te results sow tat te mean sselt number increases wit increasing Re and Sw. Furtermore, te dimensionless pressure gradient parameter as a negligible effect on te sselt number for 6 Re < 1, but as a negatie effect on te sselt number 6 for Re 1. I. INTRODUCTION Te problem of forced conection film condensation occurs in a ariety of industrial applications and as attracted many researcers interests. Te condensate layer flows oer te contact surface under te effects of graity, te interfacial apor sear stress, and te pressure gradient, and may lead to an accumulation of liquid in certain regions depending on te profile of te surface. In 1916, sselt [1] introduced te concept of local balance of iscous forces and te weigt of te condensate film and sowed tat te eat transfer in condensation depends on local film ticness. Te results sowed tat te eat transfer in a condensate layer is directly related to te Paper submitted /16/11; accepted 1/5/11. Autor for correspondence: Tong-Bou Cang ( tbcang@mail.stut.edu.tw). 1 Department of Refrigeration and Air Conditioning Engineering, National Cin-Yi Uniersity of Tecnology, Taicung, Taiwan, R.O.C. Department of Mecanical Engineering, Soutern Taiwan Uniersity, Tainan, Taiwan, R.O.C. local film ticness. Since tat time, many oter researcers ae also examined te laminar film c ondensation of quiescent apors. For example, Rosenow [], Sparrow and Gregg [3], Cen [4], Ko [5], Ko et al. [6], Denny and Mills [7], and Merte [8] improed te accuracy of sselt s original laminar film condensation solutions by remoing te oerly-restrictie assumptions, suc as te interfacial sear stress, te conectie and inertial effects were negligible, and te temperature witin te condensate layer aried linearly wit te film ticness. Condensation on orizontal tubes as many termal engineering applications, ranging from eat excange systems to cemical engineering processes, air-conditioning equipment, and so on. Te problem of laminar condensation on a orizontal tube was first analyzed by Sparrow and Gregg [9] in Gaddis [1] used a series expansion metod to sole te coupled boundary layer equations for laminar film condensation on a orizontal cylinder. Neglecting te inertial and conectie effects in te condensate film, Honda and Fujii [11] formulated te problem of forced flow condensation on a orizontal cylinder as a conjugate eat transfer problem. Yang and Cen [1] inestigated te role of surface tension and ellipticity in laminar film condensation on a orizontal elliptical tube. Te results sowed tat te mean eat transfer coefficient obtained using an elliptical tube wit its major axis orientated in te ertical direction was greater tan tat obtained from a simple circular tube. Wen a apor traels oer a orizontal tube at ig elocity, te eat transfer problem becomes a forced conection film condensation problem, and te analysis must tae account of te interfacial apor sear force. Seriladze and Gomelauri [13] sowed tat te surface sear stress depends on te momentum transferred to te condensate film by te suction mass. Fujii et al. [14] deried two-pase boundary layer equations for laminar film condensation on a orizontal cylinder. Hsu and Yang [15] proposed a combined forced and natural conection film condensation model to examine te effects of te pressure gradient and te wall temperature on te condensate layer produced by a apor stream flowing in a downward direction oer te surface of a orizontal tube. It was sown tat te mean sselt number decreased wit an increasing wall temperature ariation in te forced conection region, but was insensitie to te wall temperature in te natural conection region. More recently, Hu and Cen [16] analyzed te problem

2 of turbulent film condensation on an inclined elliptical tube, and sowed tat for lower apor elocities, a iger eccentricity increases te eat transfer oer te upper alf of te tube, but reduces te eat transfer oer te lower alf of te tube. In addition, it was sown tat te eat transfer performance of te condensate film improed wit an increasing apor elocity. Many preious studies [17-] ae reported tat te condensate eat transfer performance can be improed by applying a wall suction effect. For te problem of condensation on te out surface of a orizontal tube, an inner suction tube is needed to carry out te liquid condensate wic sucs from te outer surface of te tube. Accordingly, te present study examines te forced conection eat transfer performance of a condensate layer flowing in te downward direction oer te external surface of a orizontal tube wit suction force effects acting at te tube surface. Te analyses tae account of te effects of bot te potential flow pressure gradient and te interfacial sear stress. In general, it is sown tat te ydrodynamic caracteristics of te condensate film are significantly dependent on bot te wall suction effect and te interfacial sear stress effect. u dp = μ + ( ρ ρ ) gsin. () Rd Energy equation T. (3) = α Te boundary conditions are gien by: (1)At y =, u=, T=T w (4) ()At y =, u τ =, T=T sat (5) μ U, T sat Wall suction w R y Graity x Liquid-apor interface II. THE TRANSCEIVER STRUCTURE Consider a orizontal, clean, permeable tube wit a radius Wall temperature R and a constant temperature T w immersed in a downward T w flowing pure apor. Assume tat te apor is at its saturation temperature T sat and as a uniform elocity U. If T w is lower tan T sat, a tin condensate layer is formed on te surface of te Fig. 1. Pysical model and coordinate system. tube and runs downward oer te tube in te periperal direction under te combined effects of te graity force, te pressure gradient and te interfacial sear stress. Under steady-state Assuming tat te condensate film ticness is negligible compared to te tube radius, te pressure gradient term in Eq.() conditions, te ticness of te liquid film,, as a minimum can be deried by applying te Bernoulli equation. In oter alue at te top of te tube and increases gradually as te liquid words, Eq.() can be rewritten as flows in te downward direction. u due Te pysical model and coordinate system considered in μ = ( ρ ρ ) gsin ρ u. ( 6 ) e Rd te present study are sown in Fig. 1, in wic te curilinear According to potential flow teory, for a uniform apor coordinates (x, y)are aligned along te surface of te tube and te surface normal, respectiely. In analyzing te eat transfer flow traeling at a elocity U oer a circular cylinder, te caracteristics of te condensate film, te same set of assumptions as tose used by Rosenow [] are applied, namely (1).te gien by tangential apor elocity at te edge of te boundary layer is condensate film flow is steady and laminar, and tus te effects u e = U sin. ( 7 ) of inertia and conection are negligible and can be ignored(i.e. Te pressure gradient can ten be obtained as a creeping film flow is assumed); ().te wall temperature, due 4ρU sin apor temperature and properties of te dry apor and condensate, respectiely, are constant, and (3).te condensate film ρue =. ( 8 ) Rd D Te boundary condition for te interfacial sear stress as negligible inetic energy. Consequently, te continuity, (Eq.(5)) can be simplified using te Seriladze and Gomelauri momentum and energy conseration equations for te liquid [13] model as film are as gien as follows: τ Continuity equation = m u e = m U sin, ( 9 ) u were m is te condensate mass flux. Neglecting te sensible eat of + =. (1) subcooling, te energy equation for steady laminar flow is x written as Momentum equation in x-direction T ΔT m = =, ( 1 )

3 were Δ T = T sat T. w Ja d 3 d 3 Re sin cos ( Re P ) Substituting Eqs.(9) and (1) into Eq.(5), te interfacial d sin + sin d. sear stress is obtained as 4 Ja u ΔTU sin = at y = + ( cos + 4 Re. (11) Pcos ) + 3Sw = 3 μ ( 17 ) Te numerical solution procedure commences by substituting te boundary condition gien in Eq.(16), i.e. Integrating te momentum equation gien in Eq. (6) wit Eq.(8), and applying te boundary conditions gien in Eqs. (4) and (11), te elocity distribution equation can be deried as d =, into Eq. (17). Te following polynomial equation 4ρU sin ( ρ ρ ) g sin + ΔTU sin y d 1 u = + D. = y y μ μ wit respect to te dimensionless condensate film ticness at (1) =,, can ten be obtained: According to sselt s classical analysis [1], an energy 4 Ja Ja balance exists on eac element of te liquid film wit eigt ( 1+ 4 Re P) + 3 Re + 3Sw = 3. (18) and widt dx, i.e. Obiously, te dimensionless liquid film ticness cannot d ΔT { ρu( + Cp( Tsat T )) dy} dx + ρ( + CpΔT ) wdx = dx dx ae a negatie alue, and tus in soling Eq. (18), te film ticness is constrained by. Note tat te exact alue of. (13) Substituting Eq. (1) into Eq. (13), introducing dimensionless parameters Ja, Pr,, Re, Re w, P, Sw, and te dituting te deried alue of into Eq.(17), te ariation of can be deried using te bisection metod [1(a)]. Substi- mensionless liquid film ticness parameter, = /R, ten in te direction can be calculated using te forward difference sooting metod [1(b)]. In accordance wit sselt s applying te relation dx=rd yields te following expression for te condensate film ticness: teory [1], te dimensionless local eat transfer coefficient (i.e. d Ja 3 Ja 3 Re sin + ( sin + Re Psin ) + 3 Sw = 3, te local sselt number) can be calculated as d D (14) =, (19) were R is te tube radius, Ja is te Jaob number, Pr is te were Prandtl number, is te yleig number, Re is te two-pase mean Reynolds number, Re w is te suction Reynolds =. number, P is te dimensionless pressure gradient parameter, Substituting and Sw is te suction parameter. Ja, Pr,, Re, Re w, P and Sw into Eq.(19), te local sselt number can be rewritten as are defined as follows: =. () CpΔT μcp 3, Ja = Pr =, ρ( ρ - ρ ) ( ) gprr =, 3 Meanwile, te mean sselt number is gien by μ + CpΔT 1 π 8 = d. (1) π ρur Re =, ρwr Rew μ = ρ Pr, P =, μ ρ Sw= 5 Pr 1 + Ja Re. (15) III. RESULTS & DISCUSSIONS w 8 In simulating practical engineering problems, te pysical Te corresponding boundary condition is gien as parameters used in Eq. (17) (i.e. R, Ja, Pr,, Re, P, and Sw) d must be assigned reasonable alues. Te present analyses assume te woring liquid to be water-apor and use te dimen- = at =. ( 16 ) d Te first term inside te differentiation bracets in Eq.(14) sional and dimensionless parameter alues presented in Table 1 results from te interfacial sear stress, wile te term inoling P describes te effect of te pressure gradient produced by (reproduced from []). Table 1 Pysical parameters used in present analyses te potential flow. Meanwile, te term inoling Sw represents te effect of te wall suction. Note tat wen tese tree Symbol Typical alue terms are omitted, Eq.(14) reduces to te pure natural conection film condensation problem (i.e. sselt type condensation R.1 m problem). Ja. For computational conenience, Eq.(14) can be rewritten Pr 1.76 as

4 1 11 Re P Sw 1-1 Figures a and b sow te distributions of te dimensionless film ticness and local sselt number along te surface of te tube as a function of te two-pase Reynold number Re gien constant wall suction parameters of Sw= and 1-1, respectiely. (Note tat te remaining parameters are assigned te caracteristic alues sown in Table 1, i.e. Ja=., = 1 11 and P= ). As sown, te dimensionless film ticness as a minimum alue at te top of te tube and increases wit increasing. Tis is to be expected since te current analyses consider te case of falling film condensation, and tus te effects of graity minimize te film ticness on te upper surface of te orizontal tube, but cause te film ticness to increase toward an infinite alue at te lower surface of te tube. Moreoer, it can be seen tat te dimensionless liquid film ticness decreases, wile te local sselt number increases, wit an increasing alue of Re. Te pysical reason for tis is tat a iger alue of Re implies a stronger interfacial sear force, wit te result tat enance te liquid condensate falls along te tube circumference. As a result, te ticness of te liquid film is reduced and a steeper temperature gradient is formed. Finally, comparing Figs. a and b, it can be seen tat te sselt number increases wit an increasing wall suction parameter Sw due to te corresponding reduction in te condensate layer ticness. Figures 3a and 3b sow te effects of te dimensionless pressure gradient parameter, P, on te ariation of te dimensionless film ticness and local sselt number along te surface of te tube for wall suction parameters of Sw= and 1-1, respectiely. (Note tat Ja=., = 1 11 and Re = in bot cases). It is seen tat for bot alues of Sw, te ticness of te liquid film on te upper surface of te tube decreases wit increasing P. In oter words, te liquid film ticness at te upper surface of te tube decreases wen te effects of te potential flow pressure gradient are taen into account. Conersely, on te lower surface of te tube, te liquid film ticness increases wit a iger P. In oter words, te liquid film ticness at te lower surface of te tube increases wen te effects of te potential flow pressure gradient are taen into account. Tese findings are reasonable since te potential flow pressure gradient as a positie alue on te upper surface of te tube, but as a negatie alue on te lower surface of te tube. From Eq.(14), it can be seen tat for a positie alue of te pressure gradient (i.e. P sin >), te liquid film is pused down oer te side surfaces of te tube, and tus a reduction in te film ticness occurs. Conersely, for a negatie alue of te pressure gradient (i.e. P sin <), te liquid film is pused in te upward direction, causing an accumulation of te condensate and a corresponding increase in te ticness of te condensate layer. Figure 4 illustrates te ariation of te mean sselt number,, wit te two-pase Reynolds number, Re as a function of te dimensionless pressure gradient parameter, P, and te wall suction parameter, Sw. It can be seen tat te mean sselt number increases wit increasing Re and increasing Sw. In addition, it is obsered tat for lower alues of te two-pase Reynolds number (i.e. Re < 1 6 ), te dimensionless pressure gradient parameter, P, as a negligible effect on te mean sselt number. Howeer, decreases wit increasing P at iger alues of Re (i.e. Re 1 6 ). Te greater effect of P on at iger alues of Re is to be expected since a iger alue of Re corresponds to a reduced liquid film ticness on te upper surface of te tube, and ence a corresponding increase in te positie effects of te dimensionless pressure gradient parameter P..4x1-3.x1-3.x x x x1-3 1.x1-3 1.x1-3 8.x1-4 6.x1-4 4.x1-4 Ja=., = 1 11, P= , Sw=. : Re =1 6 : Re =1 5 : Re = x x1 3 3x1 3.5x1 3 x x1 3 1x1 3 5.x1 x1 Fig. a Variation of and wit as function of two-pase mean Reynolds number Re for constant suction parameter Sw=..4x1-3.x1-3.x x x x1-3 1.x1-3 1.x1-3 8.x1-4 6.x1-4 4.x1-4 : Re =1 6 : Re =1 5 : Re =. Ja=., = 1 11, P= , Sw= x x1 3 3x1 3.5x1 3 x x1 3 1x1 3 5.x1 x1 Fig. b Variation of and wit as function of two-pase mean Reynolds number Re for constant suction parameter Sw=1-1.

5 IV. CONCLUSION Tis study as inestigated te forced conection eat transfer performance of a condensate layer flowing in te downward direction oer a orizontal tube wit wall suction effects. Te results ae sown tat te graity effect causes te condensate layer ticness to ae a minimum alue on te upper surface of te tube but to increase toward an infinite alue on te lower surface of te tube. In addition, it as been sown tat te interfacial sear stress causes a reduction in te ticness of te liquid film, and terefore increases te eat transfer performance. Moreoer, te potential flow pressure gradient enances te eat transfer coefficient on te upper surface of te tube, but reduces te eat transfer coefficient on te lower surface of te tube. Finally, te results ae sown tat te effects of te potential pressure gradient on te mean sselt number must be taen into account wen te two-pase Reynolds number as a alue greater tan Re = x x x x1 3 3.x1 3 3.x1 3.8x1 3.6x1 3.4x1 3.x1 3.x1 3 : P= : P=1-14.x1 5.x1 5 1.x x1 6.x1 6.5x1 6 3.x x1 6 4.x x1 6 5.x1 6 Re Sw=1-1 Sw=. Ja=., = 1 11 Fig. 4 Variation of wit Re as function of P and Sw. 3.x1-3.75x1-3.5x1-3.5x1-3.x x x x1-3 1.x x1-4 5.x1-4 : P=1-1 : P= : P=. Ja=., = 1 11, Re= , Sw= x1 3.5x1 3 x x1 3 1x1 3 5.x1 x1 Fig. 3a Variation of and wit as function of dimensionless pressure gradient parameter P for constant suction parameter Sw=. 3.x1-3.75x1-3.5x1-3.5x1-3.x x x x1-3 1.x x1-4 5.x1-4 Ja=., = 1 11, Re= , Sw=1-1 : P=1-1 : P= : P= x x1 3 3x1 3.5x1 3 x x1 3 1x1 3 5.x1 x1 Fig. 3b Variation of and wit as function of dimensionless pressure gradient parameter P for constant suction parameter Sw=1-1 V. ACKNOWLEDGMENTS Tis study was supported by te National Science Council of Taiwan (NSC 99 1 E 18 13). REFERENCES 1. sselt, W., Die oberfläcen Kondensation des Wasserdampfes, Zeitscrift des Vereines Deutscer Ingenieure,, Vol.6, No., pp (1916).. Rosenow, W. M., Heat Transfer and Temperature Distribution in Laminar Film Condensation, Trans. ASME, J. Heat Transfer, Vol.78, pp (1956). 3. Sparrow, E. M. and Gregg, J. L., Laminar Condensation Heat Transfer on a Horizontal Cylinder, Trans. ASME, J. Heat Transfer, Vol.81, pp (1959). 4. Cen, M. M., An Analytical Study of Laminar Film Condensation: Part 1-Flat Plates, J. Heat Transfer, Vol.83, pp.48-54(1961). 5. Ko, J. C. Y., On Integral of Treatment of Two Pase Boundary Layer in Flim Condensation, J. Heat Transfer, Vol.83, pp (1961). 6. Ko, J. C. Y., Sparrow, E. M. and Hartnett, J. P., Te Two Pase Boundary Layer in Laminar Film Condensation, Int. J. Heat and Mass Transfer, Vol., pp.69-8(1961). 7. Denny, V. E. and Mills, A. F., Nonsimilar solutions for laminar film condensation on a ertical surface, Int. J. Heat and Mass Transfer, Vol.1, pp (1969). 8. Merte, Jr. H., Condensation eat transfer, Adances in Heat Transfer, Vol.9, pp.181(1973). 9. Sparrow, E. M. and Gregg, J. L., Laminar Condensation Heat Transfer on a Horizontal Cylinder, Trans. ASME J. Heat Transfer, Vol.81, pp (1959). 1. Gaddis, E. S., Solution of te two pase boundary layer equations for laminar film condensation of apour flowing perpendicular to a orizontal cylinder, Int. J Heat Mass Transfer, Vol., pp.371 8(1979). 11. Honda, H. and Fujii, T., Condensation of flowing apor on a orizontal tube numerical analysis as a conjugate problem. Trans. ASME J. Heat Transfer Vol.16, pp (1984). 1. Yang, S. A. and Cen, C. K., Role of surface tension and ellipticity in laminar film condensation on a orizontal elliptical tube, Int. J. Heat Mass Transfer, Vol.36, pp (1993). 13. Seriladze, I.G. and Gomelauri, V.I., Te teoretical study of laminar film condensation of a flowing apor, Int. J. Heat Mass Transfer, Vol.9, pp (1966). 14. Fujii, T., Ueara, H. and Kurata, C., Laminar filmwise condensation of flowing apor on a orizontal condenser, Int. J. Heat Mass Transfer, Vol.15, pp.35 46(197). 15. Hsu, C.H. and Yang, S.A., Pressure gradient and ariable wall temperature effects during filmwise condensation from downward flowing apors

6 onto a orizontal tube, Int. J. of Heat Mass Transfer, Vol.4, pp (1999). 16. Hu, H. P. and Cen, C. K., Simplified approac of turbulent film condensation on an inclined elliptical tube, Int. J. of Heat Mass Transfer, Vol.49, pp (6). 17. Yang, J. W., Effect of Uniform Suction on Laminar film Condensation on a Porous Vertical Wall, Trans. ASME J. Heat Transfer, Vol.9, pp.5-56 (197). 18. Yang, S. A. and Cen, C. K., Laminar Film Condensation on a Finite-Size Horizontal Plate wit Suction at Te Wall, Appl. Mat. Modelling, Vol.16, pp.35-39(199). 19. Cang, T. B., Effects of Surface Tension on Laminar Filmwise Condensation on a Horizontal Plate in a Porous Medium wit Suction at te Wall, Cemical Engineering Communications, Vol.195, pp (8).. Cang, T. B., Laminar filmwise condensation on orizontal dis embedded in porous medium wit suction at wall, Trans. ASME J. Heat Transfer, Vol.13, paper No: 715(8). 1. James, M. L., Smit, G. M. and Wolford, J. C., Applied merical Metods for Digital computation, 3 rd ed., Happer & Row, New Yor, (a) pp.87-9, and (b) pp (1985).. Holman, J. P., Heat Transfer, 9 t ed., McGraw-Hill, New Yor, pp.66 (). NOMENCLATURE Cp specific eat at constant pressure D diameter of circular tube g acceleration of graity eat transfer coefficient eat of aporization Ja Jaob number defined in Eq. (15) termal conductiity K permeability of porous medium m condensate mass flux sselt number defined in Eq. (19) p static pressure of condensate P dimensionless pressure gradient parameter defined in Eq. (15) Pr Prandtl number defined in Eq. (15) R radius of circular tube yleig number defined in Eq. (15) Re two-pase mean Reynolds number defined in Eq. (15) Re w suction Reynolds number defined in Eq. (15) Sw suction parameter defined in Eq. (15) T temperature Δ T saturation temperature minus wall temperature u elocity component in x-direction u e tangential apor elocity at edge of boundary layer defined in Eq. (7) U apor elocity of free stream elocity component in y-direction Gree symbols condensate film ticness μ liquid iscosity liquid density ρ ρ α τ apor density termal diffusiity angle measured from top of tube interfacial sear stress Superscripts aerage quantity dimensionless ariable Subscripts sat w saturation property quantity at wall