Influences of Interfacial Shear in Turbulent Film Boiling on a Horizontal Tube with External Flowing Liquid

Transcription

1 Engineeing, 05, 7, Publihe Online Novembe 05 in SciRe. Influence of Intefacial Shea in Tubulent Film Boiling on a Hoizontal Tube with Extenal Flowing Liqui Hai-Ping Hu Depatment of Maine Engineeing, National Taiwan Ocean Univeity, Taiwan Receive Octobe 05; accepte Novembe 05; publihe 6 Novembe 05 Copyight 05 by autho an Scientific Reeach Publihing Inc. Thi wok i licene une the Ceative Common Attibution Intenational Licene (CC BY). Abtact The pape peent a theoetical tuy about tubulent film boiling on a hoizontal tube with extenal flowing liqui. The high velocity flowing liqui i etemine by potential flow theoy. By uing Colbun analogy, the peent pape uccefully aee a new moel to peict the vapoliqui intefacial hea, applie the intefacial hea into the foce balance equation an then combine the foce balance equation, the enegy equation an themal enegy balance equation. At lat, both the film thickne an Nuelt numbe can be obtaine. Beie, the peent analyi alo inclue aiation effect, tempeatue atio an ey iffuivity. Finally, a compaion between the eult of the peent tuy an thoe epote in peviou theoetical an expeimental tuie i povie. Keywo Ey Diffuivity, Potential Flow, Colbun Analogy. Intouction The eeach of film boiling on a hoizontal tube wa conucte by the pioneeing invetigato, Bomley []. Afte Bomley eeach, many elate eeache ha been epote. In 966, Nihikawa et al. [] analyze two-phae bounay-laye teatment of fee-convection film boiling. The theoetical tuy ha been mae of film boiling fom an iothemal vetical plate an a hoizontal cyline without conieing aiative effect. Joan [] invetigate the lamina film boiling an tanition boiling, an the epaate egion ha alo been icue. Sakuai et al. [4] peente the pool film boiling on a hoizontal cyline with theoetical olution. The analytical heat tanfe moel i bae on lamina bounay theoy incluing aiation effect. Fo the How to cite thi pape: Hu, H.-P. (05) Influence of Intefacial Shea in Tubulent Film Boiling on a Hoizontal Tube with Extenal Flowing Liqui. Engineeing, 7,

2 foce convection film boiling, Huang et al. [5] invetigate the flow film boiling aco a hoizontal cyline with unifom heat flux. The numeical eult agee with thoe expeimental ata whee the wall tempeatue i not vay a lot aoun the heate at high heat fluxe, within ±8%. Lamina film boiling on hoizontal tube ha been wiely icue in publihe liteatue, an thee i alo ome evelopment on the eeache of tubulent film boiling. Fo example, Sama et al. [6] peente tubulent film boiling une unifom heat flux conition on a hoizontal cyline. In the eeach, the aumption of equal hea conition both at the wall an the vapo-liqui inteface i eaonable. About the tubulent film boiling on a hoizontal iothemal cicula cyline, Sama et al. [7] peente ome theoetical eult. The analyi compae the theoetical eult with peviou expeimental eult, an foun that thei eult wee in a goo ageement with the expeimental ata. In aition, Hu [8] invetigate the fee convection in tubulent film boiling une the quiecent liqui on a phee with vaiable wall tempeatue. Even though thee wee many eeache about lamina film boiling an tubulent film boiling on hoizontal tube, thee wa little publication about the tubulent film boiling with high velocity liqui. Peicting intefacial hea in a tubulent film boiling ytem une high velocity liqui i not eay. Howeve, the peent pape uccefully peict the vapo-liqui intefacial hea by uing Colbun analogy. The peent tuy applie the intefacial hea into the foce balance equation, an then combine the foce balance equation with the enegy equation an themal enegy balance equation. At lat, both the film thickne an Nuelt numbe ae obtaine. Beie, the peent analyi alo inclue ey iffuivity, aiation effect an tempeatue atio. Finally, a compaion between the eult of the peent tuy an thoe epote in peviou theoetical an expeimental tuie i povie. It i foun that a goo ageement exit between the two et of eult.. Fomulation Conie a hoizontal tube immee in flowing liqui with the high velocity u an atuate tempeatue T. The wall tempeatue T w i aume high enough to occu tubulent film boiling on the uface of the tube, an then a continuou film of vapo un upwa ove the tube. The phyical moel an the cooinate ytem aopte in the peent tuy ae hown in Figue. Boiling une the foce convection, the vicoity component an the buoyancy effect ae aume moe ignificant than the inetia foce. Then the foce balance equation fo the vapo film can be expee a: τ τ = gδ ( ρ ρ )inθ () w δ l It i aume the thickne of vapo film i much thinne than the aiu of the tube ( δ R). An it futhe aume the tubulent conuction tem aco the vapo laye i moe ignificant than the convective tem, an hence the convective tem can be neglecte. The enegy equation can be expee a: ε m T P = 0 y ν y () Figue. Phyical moel an cooinate ytem. 755

3 The bounay conition of enegy equation une iothemal conition ae a follow: T = T w at y = 0 T = T at y = δ fo 0 θ π () Fo a pue ubtance, the themal enegy balance equation of the vapo film can be expee a: δ w ρvuy R θ 0 fg y= ( w T ) k T εσ RT = (4) h y h Une the high velocity liqui, the equation, which ecibe heat tanfe in the flow aoun the tube, can be expee a following equation: l n l fg Nu = CRe P (5) whee C i a contant in flow configuation. The contant value C i lite in Table. Accoing to Colbun analogy, the fiction facto can be expee a the following equation: f = St P (6) The mean fiction coefficient in the team wie iection may then be witten a: Futhemoe, the local fiction can be obtaine a: π f = f θ θ π (7) 0 f CπRe l n θ = inθ (8) The tubulent bounay laye exet a fiction foce on the liqui-vapo bounay. The hea te i etimate by conieing the extenal flowing liqui aco the uface of the tube when thee i no vapo film on the uface. The local fiction coefficient i efine a: f θ = τδ ρu l l Accoing to potential flow theoy, when the unifom liqui flow of velocity u paing a tube, the liqui velocity at the ege of the bounay i a follow: Combining Equation (8-0), the local hea te can be expee a: u = u inθ (0) e τ = πcρu Re in θ () δ n l l Incopoating the intefacial vapo hea te τ δ given by Equation () into the elemental foce balance equation enable Equation () to be ewitten in the following fom: ( ) τ = c ρu Re θ gδ ρ ρ θ () n π w l l in l in (9) Table. Value of C an n in Equation (5) ( ata ae ue in the peent pape). Re l C n , , ,

4 The foce balance equation Equation () yiel the following imenionle equation: n R = R ϕf in θ δ inθ () It futhe auming the peue aco the bounay laye i contant an the enity vaiation aco the bounay laye i given by the following equation: ρt = ρ T = ρ T (4) w w The enegy equation Equation () yiel the following imenionle enegy equation: y ε m P T = 0 ν µ ( T )( T ( T ) ) y The imenionle bounay conition of Equation (5) ae: ( ) ( ) at y = 0; T = at y = δ ; T = 0 whee the abolute vicoity equation µ in imenionle enegy equation Equation (5) i expee a the nitogen at the atuation tempeatue coeponing to a ytem peue une atm. i. e. µ T T µ = = µ T T Beie, the themal enegy balance equation Equation (4) can be ewitten in imenionle fom a follow: δ n u G T 4 y TS ( T ) R R F ( ) ( ) ϕ in θ kw NR T θ = (8) 0 T T y y = 0 whee the abolute conuctivity equation k in themal enegy balance equation Equation (8) i expee a the nitogen the atuation tempeatue coeponing to a ytem peue une atm. : 4 k T T T T k = = k T T T T Futhemoe, the imenionle themal enegy balance equation Equation (8) equie the velocity pofile u in the vapo film. An u can be obtaine by following equation [8]: The bounay conition i: u y y = δ T ε m µ ( T ) T ν µ T T T ( )( ( ) ) (5) (6) (7) (9) (0) u = 0, y δ = 0 at y = 0 () The ey iffuivity itibution peente by Kato et al. [9] i expee a: ε m = 0.4y exp ν ( y ) The heat tanfe of tubulent film boiling can be given by the following equation: T 4 4 ht ( T) = k σε T T w w w y y= 0 () () 757

5 Obviouly, the local Nuelt numbe can be expee a: R T T = 4 Nu kw NR G y y = 0 T The mean Nuelt numbe fo the entie uface of the tube can be witten a:. Numeical Metho (4) π 4 R T T Num = kw NR π θ 0 G y (5) y = 0 T The imenionle govening Equation (), (5)-() an (4)-(5) ubject to the elevant bounay conition given can be ue to etimate δ, R an Nu fo the vapo film by mean of the following poceue by uing C : ) Suitable imenionle paamete, uch a T, S, NR, F an G ae pecifie. ) The bounay conition of velocity an tempeatue ae a follow: at y = 0; T =, u = 0, µ = T.48T k = T 045.4T 596.T 7.T at y = δ ; T = 0 4 ) At the bottom of the tube, θ = 0, i = 0, the imenionle film thickne δ i alo zeo δu δ0 δ = = = 0. At the next noe, i.e. i = i, the value of θ i given by θi = θi θ, whee ν ν θ = π 70. ( ) 4) Gue an initial value of δ ; ubtitute Equation (7), () into Equation (5) an (6) an then get the value of T y y = 0 5) Subtitute. T n y y = 0, Equation (9), (0) an () into Equation (8), an get the value of R R ϕf in θ. Afte calculating, the value of R can be gotten, an then ubtitute the value of into Equation (). 6) The citeion fo the accuacy of δ i aee by Equation (), an the equation can be ewitten in the following fom: fom: n δ inθ = R R ϕf in θ. The equation can be witten moe clealy a the following n R R ϕf in θ δ = 0 inθ 6 Futhemoe, the eo of the numeical calculation i le than 0, an the peciion i goo enough. It can be expee a the following unequal equation: n R R ϕf in θ 6 δ 0 inθ If the calculation i a convegence, poce the film thickne of next angula poition. If the calculation i not a convegence, gue a new thickne an epeat pocee (4)-(6). 7) The poce above i epeate at the next noe poition, i.e. θi = θi θ, an then ubequently at all (6) R 758

6 noe within the ange 0 θ π. 8) The local Nuelt numbe an mean Nuelt numbe ae then calculate. 9) The flow chat fo calculating the vapo thickne i expee in Figue. 4. Reult an Dicuion Figue peent imenionle local vapo velocity. The imenionle velocity of the tube wall i zeo une the no-lip conition. Then, the velocity inceae along the y-iection, an becaue of the hea te of vapo-liqui intefacial, it become lage an lage. Beie, the velocity will inceae along with the inceae of the angle. An the velocity will each the maximum value at the top of the tube. Figue 4 peent the imenionle tempeatue itibution fo entie angula poition along with the vapo film thickne iection y. Becaue auming the tagnation point i at the bottom of the tube, the figue how a linea tempeatue itibution at the bottom of the tube. Beie, the peent pape conie the intefacial Figue. The flow chat fo calculating the vapo thickne an Nuelt numbe. 759

7 Figue. Local velocity itibution in vapo film. Figue 4. Local tempeatue itibution in vapo film. hea with high velocity liqui une the effect of tubulence; thu, when the angula poition inceae, the non-linea tempeatue itibution of the pofile i becoming moe an moe ignificant. Figue 5 peent the vaiation of the imenionle vapo thickne of the tube. Specifically, the film thick- θ = 0 a the value of θ in- ne inceae continuouly fom a minimum value at the bottom of the tube ( ) ceae. It can be een that the film thickne eache it maximum value at the top of the tube ( θ =80 ). Futhemoe, the inceae of the Foue numbe, F will bing out the inceae of the intefacial hea an lea to the inceae of the evapoative ate. An the inceae of the evapoative ate alo lea to an inceae of the vapo film thickne. Figue 6 peent the elationhip between the mean Nuelt numbe an the Foue numbe fo five value of Gahof numbe. The figue how the eult of the foce convection film boiling. A highe Foue numbe will bing an inceae in the mean Nuelt. Beie, the Gahof numbe G i alo one of the ominant facto, an theefoe inceaing the Gahof numbe will bing out an inceae in the mean Nuelt. Figue 7 povie a compaion between the peent eult fo the mean Nuelt numbe an the eult gen- 7 8 eate in a peviou tuy [0]. The ange of Ra in Pomeantz eeach wa.5 0 Ra 0. Ra in the peent eeach i efine a moifie Rayleigh numbe, Ra = G P 0.5 S ( T ). An the ange of Ra 7 of the peent tuy i 0 Ra 0. The figue how that the mean Nuelt numbe calculate in the peent 7 tuy i boaly imila to the eult geneate by Sama et al. [8] fo the ange of Ra of 0 Ra 0. When F = 0, u = 0, thee i a goo ageement between Sama et al. eult which i une the conition of quiecent liqui an the peent ata. Beie, the figue alo compae the peent eult with the expeimental eult [9] obtaine in an ealie tuy fo lamina film boiling. It can be een that thee i a imila ten between the two et of eult at both 760

8 Figue 5. Dimenionle film thickne on tube uface. Figue 6. Effect of the F on mean Nuelt numbe. low an mi Ra numbe. Beie, the inceae of the Ra value lea to an inceae of the mean Nuelt numbe. The inceae in the Foue numbe will alo bing out an inceae in the mean Nuelt numbe. 5. Concluion The following concluion can be awn fom the eult of the peent theoetical tuy: 76

9 Figue 7. Effect of the Ra on mean Nuelt numbe. ) With the help of Colbun analogy, the peent eeach uccefully peict the hea te of the vapoliqui inteface in a film boiling ytem une high velocity of liqui. ) The inceae in the Gah of numbe will lea to an inceae in the mean Nuelt numbe. Beie, tubulent film boiling une the high velocity of extenal flowing liqui, the inceae in the Foue numbe will bing out an inceae in the mean Nuelt numbe. ) It how a goo ageement between the peent eult which i une the conition of F = 0 an the peviou tuie with fee convection. Acknowlegement The autho gatefully acknowlege the uppot povie to thi poject by the Minity of Science an Technology of Taiwan une Contact Numbe MOST 04--E Refeence [] Bomley, L.A. (950) Heat Tanfe in Stable Film Boiling. Chemical Engineeing Poge, 46, -7. [] Nihikawa, K. an Ito, T. (966) Two-Phae Bounay-Laye Teatment of Fee Convection Film Boiling. Intenational Jounal of Heat an Ma Tanfe, 9, /007-90(66)905-6 [] Joan, D.P. (968) Film an Tanition Boiling. Avance in Heat Tanfe, 5, [4] Sakuai, A., Shiotu, M. an Hata, K.A. (990) Geneal Coelation fo Pool Film Boiling Heat Tanfe fom a Hoizontal Cyline to Subcoole Liqui: Pat A Theoetical Pool Film Boiling Heat Tanfe Moel Incluing Raiation Contibution an It Analytical Solution. Jounal of Heat Tanfe,, [5] Huang, L. an Witte, L.C. (995) Foce Convection Film Boiling Heat Tanfe Aoun Hoizontal Cyline in Highly Subcoole. ASME/JSME Themal Engineeing Confeence,, 5-. [6] Sama, P.K., Rao, V.D. an Begle, A.E. (997) Tubulent Film Boiling on a Hoizontal Cyline Effect of Tempeatue Depenent. Enegy Conveion an Management, 8, [7] Sama, P.K., Subahmanyam, T., Rao, V.D. an Begle, A.E. (00) Tubulent Film Boiling on a Hoizontal Cyline. Intenational Jounal of Heat an Ma Tanfe, 44,

10 [8] Hu, H.P. (008) An Analyi of Tubulent Film Boiling on a Sphee with Vaiable Wall Tempeatue. IMechE, Pat C: Jounal of Mechanical Engineeing Science,, [9] Kato, H., Shiwaki, N.N. an Hiota, M. (968) On the Tubulent Heat Tanfe by Fee Convection Fom a Vetical Plate. Intenational Jounal of Heat an Ma Tanfe,, [0] Pomeantz, M.L. (964) Film Boiling on a Hoizontal Tube in Inceae Gavity Fiel. Jounal of Heat Tanfe, 86, Nomenclatue a acceleation ue to gaviton foce (m/ec ) C p pecific heat capacity (J/kg K) D iamete of tube, R R hea Reynol, Ru ν R wall hea paamete, R G u F Foue numbe, gr gd ρl ρ G moifie Gahof numbe, ν ρ g acceleation ue to gavity (m/ec ) h heat tanfe coefficient, w/(m k) h fg latent heat (J/kg) k themal conuctivity (W/m k) k imenionle themal conuctivity, k( T) k( T ) k k( T) k( T ) w y = 0 Nu Local Nuelt numbe, hr k Nu m mean Nuelt numbe NR aiation paamete, εσ TRk P Pantl numbe, Cpµ k R Raiu of tube (m) Ra moifie Rayleigh numbe, G P 0.5 S ( T ) Re Reynol numbe of vapo, u D v ν ν S heat capacity paamete, C pt ( hfg P ) St Stanton numbe, Nu ( ReP ) T tempeatue (K) T tempeatue atio, Tw T T imenionle tempeatue, ( T T ) ( T T ) u vapo velocity in x-iection (m/) u hea velocity, τw ρ w 76

11 u imenionle velocity, uu v velocity nomal to the iection of flow (m/) x peipheal cooinate (m) y cooinate meaue itance nomal to tube uface (m) y imenionle itance, yu ν Geek Symbol δ vapo film thickne (m) δ imenionle film thickne, δu ν µ abolute vicoity (kg/m ) µ imenionle abolute vicoity, µ µ ν kinematic vicoity (m /) ρ enity (kg/m ) τ hea te (N/m ) θ angle meaue fom bottom of tube ϕ intefacial hea paamete, ε m ey iffuivity fo momentum ε emiivity σ Stefan-Boltzmann contant, wm Subcipt l liqui vapo at atuation tempeatue v vapo w tube wall x x-iection δ vapo-liqui inteface n n n n ν ρl ρ 6 πc νl ρ k 4 G 764