7.3 Filmwise Condensation Regimes of Filmwise Condensation

Transcription

1 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe 7.3 Fimwise Condensation Regimes of Fimwise Condensation Re 30 Figure 7.14 Fow regimes of fim condensate on a ertica wa. Re 1800 Mutiphase Systems with 1

2 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe 7.3. Generaized Goerning Equations for Laminar Fim Condensation incuding Binary Vapor In order to use boundary ayer theory to describe the condensation probem, the foowing assumptions must be made: 1. T w, T I, T, ω, u I, u I and u are independent of x. 1. The condensate fim and binary apor boundary ayers both deeop from the eading edge of the ertica surface, x=0.. Condensation takes pace ony at the apor-iquid interface. In other words, no condensation takes pace within the binary apor boundary ayer in the form of a mist or fog. 3. Both temperature and eocity are continuous at the apor-iquid interface. 4. The condensate is miscibe. 5. The physica properties of the system are assumed to be constant with respect to concentration and temperature except in the case of buoyancy terms. 6. The density of the condensate iquid is assumed to be much greater than that of the binary apor. 7. The apor mixture can be treated as an idea gas. Mutiphase Systems with

3 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Figure 7.15 Physica mode and coordinate system for condensation of a binary apor mixture. Mutiphase Systems with 3

4 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The goerning equations for the aminar fim condensation of a binary apor mixture can be gien by taking the aboe assumptions into account and using boundary ayer anaysis, i.e., For the condensate fim: u + = 0 x y u u u 1 dp u + = ν + g x y y ρ dx (7.33) (7.34) u T T T α + = x y y (7.35) Mutiphase Systems with 4

5 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe For the apor boundary ayer: u + = 0 x y u u u ρ u + = ν + g 1 x y y ρ T T T ω 1 T u + = α + Dc p1 x y y y y ω ω ω u D x y y = (7.36) (7.37) (7.38) (7.39) p1 Isobaric specific c heat difference of the binary p1ω 1 + cpω cp apor (7.40) Mutiphase Systems with c c c c c = = p1 p p1 p 5

6 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Definitions of the terms in eqs. ( ) ρ 1 ω 1 = ρ ρ ω = ρ where = ρ 1 ρ ρ + ω 1 + ω = 1 (7.41) (7.4) (7.43) p 1 M1ω = 1 Partia pressures of the system + p are Mdetermined by ω 1 p p = + M M ω ω (7.44) (7.45) Mutiphase Systems with (7.46) 6

7 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Boundary conditions at the surface of the cod wa u = 0, y = 0 = 0, y = 0 T = T, y = 0 Boundary conditions at ocations far from the cod wa w u = u, y T = T, y ω 1 = ω 1, y (7.47) (7.48) (7.49) (7.50) (7.51) Mutiphase Systems with (7.5) 7

8 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Boundary conditions that exist at the iquid-apor interface u = u = u u u µ = µ y y d d ρ u = ρ u = m & = m & 1 + m & dx dx T = T = T T T k = h m& + k y y ω = ω 1 1 The mass fuxes of the apor in the binary apor system are ω 1 m& ( ) 1 = ρ D1 + ω 1 m & 1 + m& y ω m& = ρ D1 + ω ( m & 1 + m& ) y (7.53) (7.54) (7.55) (7.56) (7.57) (7.58) (7.59) (7.60) Mutiphase Systems with 8

9 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The moar fuxes of the apor in the binary apor system are c1 n& 1 = D1 + x1n& T y c n& = D1 + xn& T y Equations and are often expressed in terms of partia pressure, i.e., D p n& = + n& T RuT y p D p n& = + n& Mass fraction of component 1 in the condensate fim m& 1x ω 1 = m& + m& 1x x p p T RuT y p (7.61) (7.6) (7.63) (7.64) (7.65) Mutiphase Systems with 9

10 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Fimwise Condensation in a Stagnant Pure Vapor Reseroir Laminar Fow Regime The fow is aminar. Constant fuid properties are assumed. Subcooing of the iquid is negigibe in the energy baance, i.e., a condensation occurs at the saturation temperature corresponding to the pressure in the iquid fim near the wa. Inertia and conection effects are negigibe in the boundary ayer momentum and energy equations, respectiey. The apor is assumed stagnant and therefore shear stress is considered to be negigibe at the iquid-apor interface. The iquid-apor interface is smooth, i.e., condensate fim is aminar and not in the way or turbuent stages. Mutiphase Systems with 10

11 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Figure 7.16 Oeriew of the contro oume under consideration in the Nusset anaysis. Mutiphase Systems with 11

12 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Pressure in the iquid fim dp dx = ρ g (7.66) u u u ρ u + = µ + g( ρ ρ ) x y y Substituting eq. (7.66) into eq. (7.34) (7.67) u g = ( ρ ρ ) y µ Negecting the inertia term, eq. (7.67) becomes Integrating twice and appying boundary conditions ( ρ ρ ) (, ) g u x y y = y µ ( ) g udy ρ ρ Γ = ρ ρ = 0 3µ 3 (7.68) (7.69) Mutiphase Systems with 1

13 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Heat fux across the fim thickness k( Tsat Tw ) q = k T dq = Heat transfer rate per unit width for the contro oume T = T T sat w dx (7.71) (7.7) where. dq = h d Γ Latent heat effects of condensation dominate the process (7.73) dγ is found by differentiating the expression for mass fow rate per unit surface eq. (7.70) Substituting into eq. (7.74) and (7.7) into (7.73): Mutiphase Systems with ρ ( ρ ρ ) g d Γ = µ 3 d dx = k µ T ρ ( ρ ρ ) gh d (7. 74) (7.75) 13

14 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe can be found be integrating eq. (8.10) and appying boundary 1/ 4 conditions 4k µ x T = ρ ( ρ ρ ) gh (7.76) Loca heat transfer coefficient 3/ 4 ρ ( ρ ρ ) gh hx = k 4µ x T 3 hx x ρ ( ρ ρ ) gh x Loca Nusset number Nux = = k 4k µ T Mean heat transfer coefficient 3 hll ρ ( ρ ρ ) gh L Nu L = = k µ k T Substituting eq. (7.77) into eq. (7.79) and integrating Mutiphase Systems with 1 L hl = h ( ) 0 x x dx L 1/ 4 1/ 4 1/ 4 (7.77) (7.78) (7.79) (7.80) 14

15 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Reynod s number 4Γ Re = µ (7.81) Substituting eq. (7.70) into eq. (7.81), Reynod s number for aminar 3 fim condensation becomes4 ρ ( ρ ρ ) g Re = 3µ (7.8) h x Substituting eqs. (7.81) and (7.76) into eqs. (7.78) and (7.80) k h k µ ρ ( ρ ρ ) g µ ρ ( ρ ρ ) g 1/ 3 1/ 3 = = 1.1Re 1/ Re 1/ 3 (7.83) dq k T d d Γ = = h + ρ c ( ) 0 pu Tsat T dy dx dx dx (7.84) Energy baance at the interface that takes subcooing of the iquid into account Mutiphase Systems with 15

16 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Substituting the eocity profie eq. (7.69) and using a inear temperature profie Tsat T y = 1 (7.86) T T sat w To eauate eq. (7.85) an energy baance where p sat w Rohsenow hincuded = h conection and iquid h subcooing effects to deeop Mutiphase Systems with k T d Γ = h dx 3 cp( Tsat Tw ) h = h h c ( T T ) (7.87) (7.88) (7.89) 16

17 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Way Condensate Regime Reynods number for onset of waes is reated to Archimedes number by (7.90) Re Archimedes number is defined as Ar = 9.3Ar 1/ 5 > 3/ ρ σ 1/ 3/ g ( ) µ ρ ρ (7.91) Kutateadze gae the foowing correation for the mean Nusset number of kfim condensation Re h = on a, ertica 30 pate Re where 1800 wae effects are present 1/3 1. ( ν / g) Re 5. (7.9) Mutiphase Systems with 17

18 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Mass fow rate of the condensate q hl L( Tsat Tw ) Γ = = h h Substituting eq. (7.93) into 4 h eq. ( (7.81) LL Tsat Tw ) Re = h ν Rearranging eq. (7.94) L h = Re h ν 4 L( T T ) sat w (7.93) (7.94) (7.95) Combining eq. (7.9) and (7.95) sat yieds w Re the Reynods = number for fim condensation µ h with waes ν Mutiphase Systems with 3.7 Lk ( T T ) g 1/3 0.8 (7.96) 18

19 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Turbuent Fim Regime Labuntso (1957) recommended the foowing empirica correation: 1/3 h x µ = 0.03Re Pr, Pr 10 (7.97) k ρ ( ρ ρ ) g Loca heat transfer coefficient for condensation: h x k ν 1/ = Re Pr, Re > 5800 Pr g (7.98) Mutiphase Systems with 19

20 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe h L Butterworth (1983) obtained the aerage heat transfer coefficient for fim condensation that coers aminar, way aminar, and turbuent fow by combining eqs. (7.84), (7.9) and (7.97) as foows k Re =, 1 < Re 700 ( ν / ) Pr (Re 53) 1/ g (7.99) Re The Reynods number,, is needed in order to use eq. (7.99) to determine the heat transfer coefficient for turbuent fim condensation. Mutiphase Systems with 0

21 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Equation (7.95) was obtained by energy baance and it is aid for a fim condensation regimes. Combining eqs. (7.99) and (7.95), the Reynods number for turbuent fow is obtained: 0.5 1/ Lk Pr ( Tsat Tw ) g 0.5 Re = 151Pr 53 + µ h ν 4/ 3 (7.100) Mutiphase Systems with 1

22 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Exampe 7. Saturated steam at 1 atm condenses on a ertica wa with a height of L= 1 m and width of b= 1.5 m. The surface temperature of the ertica wa is 80 C. What are the aerage heat transfer and condensation rates? Mutiphase Systems with

23 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Soution: The saturation temperature of steam at 1 atm is T sat =100 C. The apor density at this 3 temperature is ρ = , kg/m and the atent heat of aporization is h =. 51.kJ/kg The iquid properties eauated at = 90 C are ρ = kg/m, T = ( T + T ) / f sat w c = 4.06 kj/kg-k, 3 µ = kg/m-s, p 6 k = W/m-K, ν = µ / ρ = m /s. Mutiphase Systems with 3

24 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The reised atent heat of aporization is ( ) ' cp Tsat Tw h = h h Assuming the fim condensation is aminar (as wi be erified ater), the heat transfer coefficient can be obtained from eq. (7.80), i.e., Mutiphase Systems with 4.06 (100 80) = = kj/kg ρ ( ρ ρ ) g k h h = µ T L 1/ ( ) = (100 80) 1 = 5340.W/m -K 1/4 4

25 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The heat transfer rate is then q = hlb( T T ) = (100 80) = W sat The condensation rate is 5 q m& = = = kg/s h w The assumption of aminar fim condensation is now checked by obtaining the Reynods number defined in eq. (7.81), i.e., Re 4Γ 4m& = = = = 3 µ µ b which is greater than 30 and beow This means that the assumption of aminar fim condensation is inaid and it is necessary to consider the effect of waes on the fim condensation. 5 Mutiphase Systems with 5

26 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe For fim condensation with way effects, the Reynods number shoud be obtained from eq. (7.96), i.e., 1/ / ( ) (100 80) 9.8 Lk Tsat Tw g µ ν h Re = = ( ) = which confirms that the fim condensation is in the way regime. The heat transfer coefficient is obtained from eq. (7.9) h = k Re 1/3 1. g ( ν / ) Re = = 6 1/3 1. [( ) / 9.8] W/m -K Mutiphase Systems with 6

27 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The heat transfer rate is then q = hlb( T T ) = (100 80) = W sat The condensation rate is w q m& = = = h kg/s which is much higher than the condensation rate obtained by assuming aminar fim condensation. 5 Mutiphase Systems with 7

28 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Effects of Vapor Motion Laminar Condensate Fow The boundary ayer momentum equation u u dp u ρ u + = + µ + ρ g x y dx y Another pressure gradient exists aong with the hydrostatic pressure gradient dp dx = ρ g + dx m (7.101) (7.10) The superimposed pressure gradient can be combined into a fictitious density * 1 dp (7.103) ρ dp Substituting eq. (7.103) into eqs. (7.10) and (7.101) = ρ + g dx m ρ µ ρ ρ x y y u u u * u + = + g( ) (7.104) Mutiphase Systems with 8

29 Mutiphase No Systems subcooing, with atent heat efffects of condensation dominate, thus 9 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Negecting inertia eq. (7.104) becomes u g * = ( ρ ) ρ y µ (7.105) * Integrating eq. (7.105) ( ρ twice ρ ) (, ) g and appying y τ u x y = y y boundary conditions µ + µ (7.106) ρ ( ρ ρ ) g τ ρ Γ = ρ udy Mass fow rate per = + unit 0 width of surface 3µ µ (7.107) k ( Tsat Tw ) q = Heat fux across the fim thickness can be obtained by Fourier s Law k T dq = * 3 dx (7.108) Heat fow dq = h d Γ (7.109)

30 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Differentiating eq. (7.110) * ρ ( ρ ρ ) d g τ ρ Γ = + µ µ d (7.111) Substituting into d the mass and kenergy µ T equation = dx * 3 ρ ρ ρ gh + τ ρ h ( ) (7.11) 3 4 4τ I 4k µ x T + = Integrating eq. (7.11) ( and * ) appying ( boundary * 3 ρ ρ ) conditions g ρ ρ ρ gh (7.113) * = Eq. (7.113) can be non-dimensionaized by 4 * x cp T x = LF Pr h L F (7.114) Mutiphase Systems with (7.115) 30

31 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe where Eq. (7.113) can be rewritten 4 as (7.116) (7.117) (7.118) Nusset and Reynods number for 3 aminar fow 1/3 * * * with finite apor h shear 4 ( ) ( ) τ L µ Mutiphase Systems with x L τ F * = = L F µ * ( ) ρ ρ ρ τ * ( ρ ρ ) g g ( ) 3 1/ 3 * = ( * ) 4 + * τ * Nu = = + k ρ ( ρ ρ ) g 3 x x 3 * * * 4Γ 4 Re ( ) ( ) = = + τ µ 3 * 3 * * (7.119) (7.10) 31

32 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe For the case that graitationa force is negigibe compared with the interfacia shear force imposed by the co-current apor fow, Butterworth (1981) recommended the foowing correation for oca heat transfer coefficient: * 1/ + 1/ Nux = 1.41Re ( τ ) (7.11) Nu x 1/3 h x µ = k ρ ( ρ ρ ) g (7.1) Mutiphase Systems with 3

33 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe τ + = ρ τ ( ) ρ ρ ρ µ g /3 (7.13) h = ( h + h ) 1/ shear gra (7.14) where h gra is heat transfer coefficient for graity-dominated fim condensation determined with eqs. (7.78) or (7.83) and h shear is heat transfer coefficient for sheardominated fim condensation, eq. (7.11). Mutiphase Systems with 33

34 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Turbuent Condensate Fow u = u + u (7.15) = + (7.16) p = p + p (7.17) The boundary-ayer equation for forced turbuent fow aong a panar surface is u u 1 p 1 u u + = + µ ρ u x y ρ x ρ y y (7.18) Eddy shear stress ρ u = ρ ε u y (7.19) u Apparent shear stress τ app = for µ turbuent ρ u fow y Mutiphase Systems with (7.130) 34

35 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Substituting eq. (7.19) into eq. (7.130) u u u τ app = µ + ρ ε = ρ ( ν + ε ) y y y ε + = 1+ ε ν (7.131) (7.13) Mutiphase Systems with 35

36 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Re,tr Transition points Re Figure 7.17 Variation of the mean fim condensation heat transfer coefficient with Reynods number and as predicted by Rohsenow et a. (1956). Mutiphase Systems with 36

37 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Eq. (7.131) simpifies to τ = ρ ε ν app + (7.133) Reynods number at the transition point 1/ 3 in Fig Re = ρ τ ρ ( τ ) * *, tr I I ρ ρ u y 3 (7.134) Aerage heat transfer coefficient beyond 1/ 3 the transition point 1/ 1/ into * turbuent g fow h L ( τ ) I = 0.065Pr k (7.135) Mutiphase Systems with 37

38 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Turbuent Fim Condensation in a Tube with Vapor Fow Figure 7.18 Physica mode of the condensation phenomena in contact with fowing apor. Mutiphase Systems with 38

39 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe For countercurrent fow, a force baance resuts in dp p π r + ρ gv + ρ gv = p + x π r + τ π r x dx ( ) ( ) (7.136) Where the oume of apor and iquid are ( ) V = R π x ( ) π V = R y x V (7.137) (7.138) Substituting eq. R (7.137) y and dp (7.138) into eq. ( R (7.136) ) τ = g ( ) and ρ ρ ρ g diiding by Δx dx R y R dp τ I = ρ g dx The shear stress at the iquid-apor interface Mutiphase Systems with ( ) (7.139) (7.140) 39

40 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Substituting eq. (7.140) into eq. (7.139) R y R( y) + y τ = τ ( ) (7.141) + ρ ρ g R ( R y) The fim thickness is much Rsmaer than the radius of the tube, dp eq (7.140) reduces toτ = ρ g If ρ >> ρ τ = ± τ + ρ g (7.143) y Written in generaized form that incudes concurrent fow (7.14) (7.144) The shear stress at dthe wa is du ( m ) g 0 dy + ε dy + = (7.145) Veocity profie when a axia terms and curature are negected 40 Mutiphase Systems with dx ( y ) τ = τ + ρ g ( ) τ w = ± τ + ρ g

41 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Boundary conditions Veocity profie in the iquid fim The iquid Reynods number u + f * ν = ν g Nondimensiona ariabes u xu + + f u = x = u Mutiphase Systems with f u = u = 0, y = 0 u µ = ± τ, y = y 0 y [ g( y) ± τ / ρ ] m 4Γ ReL = = 4ρ = + ε udy µ 0 µ 1/3 y + = ε + = 1 + m yu f ν ε m dy * τ ( ν g) τ = ρ D + = Du f ν /3 (7.147) (7.148) (7.149) (7.150) (7.151) 41

42 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Where u f is the fractiona eocity defined as 1/ τ w u f = ρ (7.15) Appying these nondimensiona ariabes eqs. (7.145), 3 + /3 + (7.149) and (7.150), u m τ their u ( nondimensiona g) ( g) = 0forms are f u + f ( gν y / u ) f y + + = dy 0 + ε m + + Re = 4 u dy 0 (7.153) (7.154) ε m dt d( / µ ) ρ c p q µ h Γ + = = Pr Prt dy dx Energy baance for constant heat fux at the wa Mutiphase Systems with (7.155) (7.156) 4

43 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Eq. (7.156) can be nondimensionaized to 1 + d( Γ / µ ) Pr Prt + + = NT 1 ε 0 m dy + dx + Prt Pr ( T ) sat Tw cp where NT = h Pr 1 (7.157) (7.158) q ν ε m h x = = ρ c Loca heat transfer ( coefficient 0 p + dy T ) found from Pr eq. Pr (7.156) sat T w t 1 hx + + Pr Prt + + Nux = = 1 ε 0 m dy k + Prt Pr Nondimensionaizing eq. (7.159) as a Nusset number Mutiphase Systems with h 1/3 ( ν ) + x f Nux = = Nux g + k g u 1/ (7.159) (7.160) 43

44 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The aerage modified Nusset number is found from x Nu = Nu d (7.16) 0 L + The dimensioness shear * / stress f E ρ at the + interface + d( can Γ / µ be ) τ ( g) u ( written as = ν f u + u, ) ( u + u, ) + + ρ dx (7.163) f + f [ ( Mg 5.9)] Re 75 = 0. + f [ Re ( Mg 5.9)] Re > 75 Friction factor for apor fow E where Mg Mutiphase Systems with + ν ρ τ = ν ρ τ 0.6 I 0.78 Re Re 75 ν ρ τ c 1/ 1/ 0.7 I 0.50 Re Re > 75 ν ρ τ c (7.164) (7.165) 44

45 ε Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe τ 1 τ w The characteristic stress = + τ c 3 3 τ (7.166) τ y τ τ = y 1 exp + exp τ w A τ w τ w (7.167) + 3 τ / τ where A + = 5.1 and w = 1 y ( gν ) / u. f This profie represents the eddy diffusiity in the inner ayer cosest to the 0 y 0.6 wa ( + + ), where the infuence of the wa is important. In the outer ayer ( y + + ) the eddy iscosity is assumed to be constant, with a continuous transition to the inner ayer exp( y / A ) Prt = + 1 exp ( y Pr / B+ ) (7.168) + + m B + = 5 i= 1 c i 1 ( og 10 Pr ) i 1 (7.169) And c 1 = 34.96; c = 8.79; c 3 = 33.95; c 4 = 6.3; c 5 = Mutiphase Systems with 45

46 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Other Fimwise Condensation Configurations Nusset anaysis for a ertica pate Nu x ( ) g µ ( ) 3 hxx ρ ρ ρ cosθ h x = = k k T For aminar fim condensation on a horizonta cyinder Nu D ( ρ ρ ) 3 hdd D h g = = 0.79 k kν T For aminar fim condensation on a sphere, the aerage heat transfer coefficient can be obtained by Nusset anaysis Nu D ( ρ ρ ) 3 hdd D h g = = k kν T 1/ 4 1/ 4 1/ 4 (7.170) (7.171) (7.17) Mutiphase Systems with 46

47 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Nusset anaysis for a ertica array of horizonta tubes Nu D ( ρ ρ ) 3 hdd D h g = = 0.79 k nkν T (7.173) To find the aerage heat transfer h Dcoefficient for a singe h D, n = tube array 1/ 4 (7.174) For condensation oer a ong horizonta strip with a width 3 1/ 5 of L, the aerage heat hl transfer ρ ( can ρ ρbe ) gh obtained L Nu = = by k µ k ( T T ) Nu D n sat w ( ) ( ) 1/ 4 3 hd ρ ρ ρ gh D = = k µ k Tsat Tw 1/ 5 (7.175) Mutiphase Systems with (7.176) 47

48 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe For condensation on the outside of a horizonta tube in crossfow, the heat transfer coefficient is affected by both free-steam eocity of apor and graitationa force so that hd 1/ gh µ D Nu 0.64 Re = = D k + u k Tsat T ( ) (7.177) For aminar fim condensation on a horizonta fat pate in a parae stream of saturated apor, the aerage heat transfer coefficient is Nu hl / = = 0.87 ReL + 3/ k (1 + Ja / Pr ) Ja ρ µ Pr w ρ µ 1/ 1/ 1/ 1/ 3 (7.178) Mutiphase Systems with 48

49 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Exampe 7.3 Saturated acetone at T sat = 60 C condenses on the outside of a copper tube with a diameter of D = 3.0 cm. The outer surface temperature of the copper tube is T w = 40 C. The apor properties are eauated at saturation temperature 3 T sat = 60 ρ C. The apor density is =.37 kg/m, and the h atent heat of aporization is = 517 kj/kg 3. The ρ = kg/m, c iquid properties p = 55 J/kg-K 3 µ = are k = 0.17 W/m-K ν =,, and Find the heat transfer coefficient and the rate of condensation per unit ength of the tube. Mutiphase Systems with m /s. 49

50 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Soution The reised atent heat of aporization is The heat transfer coefficient can be obtained from eq. (7.171): Transport Phenomena = W/m in -K Mutiphase Systems with cp ( Tsat Tw ) h = h h.55 (60 40) = = kj/kg ( ρ ρ ) g k h h = 0.79 ν TD 1/4 3 3 ( ) = (60 40) /4 50

51 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The heat transfer rate per unit width is then q = hπ D( T T ) = π 0.03 (60 40) = W sat w The condensation rate per unit width is q m& = = = h kg/s-m Mutiphase Systems with 51

52 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Effects of Noncondensabe Gas Fimwise condensation for both stagnant apor and forced conection has been anayzed using boundary ayer treatment. If fim condensation takes pace in an atmosphere where a noncondensabe gas exists, the condensing apor must diffuse through the noncondensabe gas to the iquid-apor interface (Stephan, 199). Therefore, a partia pressure gradient must exist in the apor-gas atmosphere. The partia pressure of the condensabe gas, p c, decreases from a constant aue p c,res in the apor-gas reseroir to the aue p c, at the phase interface where the apor is condensing to iquid. The partia pressure of the noncondensabe gas, p ncg, on the other hand, increases from its reseroir aue, p ncg,res, to the aue p ncg, at the iquidapor interface. At any point in space and time the summation of the partia pressures of this binary system must equa the constant tota pressure. Mutiphase Systems with 5

53 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe p + p = p g (7.179) The partia pressure of the condensing gas decreases as it approaches the phase interface and its saturation temperature T sat (p c ) aso fas. Depending on the noncondensabe gas content, the temperature at the interface can be much ower than if no such gas were present. The temperature difference across the interface woud aso be ower as a resut, which woud ead to a ower oera heat transfer coefficient. This ceary demonstrates the benefit of remoing as much noncondensabe gas from the system as possibe. Howeer, systematic purity cannot aways be achieed and the noncondensabe gas content must be taken into account. Mutiphase Systems with 53

54 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe y Vapor iquid Tsat p, T ω m &, p, T ω,, p, ( ) Figure 7.19 Mass transfer in the equiaent aminar fim. x Mutiphase Systems with 54

55 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe the moar fux of the condensabe apor at any point within the equiaent aminar ayer can be obtained by c n& = Dg + n& x y D (7.180) g c which can be rearranged to obtain n& = 1 c / ct y (7.181) + & = Integrating eq. (7.181) oer the equiaent aminar ayer, one obtains g Equation can be rearranged to yied h m, G = D 1, n dy D c dc c, g T ct c c D c c T g,, n T c c, n T n& = = ct hm G ct c, ct c, c (7.18) (7.183) where is the mass transfer coefficient. Mutiphase Systems with 55

56 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The mass transfer coefficient can be approximatey reated to the heat transfer coefficient, h G, through the Lewis equation, i.e., 1 hg hm, G = ρ g cp, g (7.185) If the condensabe apor and noncondensabe gas can be treated as idea gas, the moar fux in eq. (7.183) can aso be expressed in p p, terms of mass fux and partia m& pressure of the condensabe apor in = ρ ghm, G n the mixture p p, (7.186) The energy baance across h ( T Ta w) differentia = m& h + hg ( contro Tg T ) oume at the iquidapor interface, as shown in Fig. 7.0, is hg = ξ hg (7.187) G, Substituting T eqs. (7.185) and (7.186) into the energy baance Tw = n + ξ Tg T equation (7.187) h and cpusing, g p p, res, the foowing is obtained Mutiphase Systems with h h p p ( ) (7.188) 56

57 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Condensate iquid Vapor & m h h ( T T w ) h ( T T ) G g Figure 7.0 Energy baance at the iquidapor interface for fim condensation on a ertica pate incuding the effects of noncondensabe gases. Interface Mutiphase Systems with 57

58 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe T g Figure 7.1 Faing fim condensation of steam with non-condensabe gas (Coier and Thome, 1994; Reprinted with permission from Oxford Uniersity Press). Mass fraction of noncondensabe gas Mutiphase Systems with 58

59 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe In the case of sma inert gas content, the aboe equation wi reduce to h G h p p, T Tw = n h cp, g p p, (7.189) No noncondensibe gas in the k apor, the heat fux across the iquid fim q = T T ( ) (7.190) If noncondensibe gas is present, the heat fux across the iquid fim q g TI Tw = 1 Ratio of heat fuxes obtained q Tby eqs. T (7.191) and (7.190) sat q = k T T w ( ) g I w sat w (7.191) q g hg h p p = n q ( T T ) hc p p Substituting eq. (7.189) into eq. (7.19) Mutiphase Systems with, sat w g,, (7.19) (7.193) 59

60 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe m& c or substituting from eq. (7.186) q g m& h = q ( T T ) h sat (7.194) It can be seen from the aboe expression m& that for arge (T sat T w ), the eocity or mass fow rate c, must be made sufficienty arge to acquire a arge heat transfer coefficient for the heat transfer from the apor-gas mixture to the iquid-apor interface. q g This must be done in order to make not too sma and therefore remoing the undesirabe effects of the noncondensabe gas as much as possibe. w Mutiphase Systems with 60

61 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe T g Mass fraction of noncondensabe gas Figure 7.1 Faing fim condensation of steam with non-condensabe gas (Coier and Thome, 1994; Reprinted with permission from Oxford Uniersity Press). Mutiphase Systems with 61

62 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Exampe 7.4 A mixture of 0% steam and 80% air at 100 C and 1 atm fows across a horizonta cyinder. The diameter of the cyinder is 0.1 m and the eocity of the mixture is 30 m/s. The condensation rate on the cyinder is 0.0 kg/m -s. m The & = properties of the mixture are, ρ g = kg/m µ g, and = N-s/m D, g respectiey. = What is the temperature at the iquid-apor interface? If the temperature of the tube is 80 C, what is the percentage of heat transfer reduction due to the existence of noncondensabe gas? m /s Mutiphase Systems with 6

63 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe Soution The mass fraction of the steam is ω, = 0.9. The moecuar mass of water and air are M = 18.0 kg/kmo and M g = 8.96 kg/kmo. The partia pressure of the steam can be obtained from eq. (7.45), i.e., p, = p 1 + M (1 ω, ) Mgω, (1 0.9) 5 = = kpa Mutiphase Systems with 63

64 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The Reynods number of the mixture is ρ gu D ReD = = = µ g The Schmidt number of the mixture is µ 6 g Sc = = = 0.39 ρ D g g The empirica correation for forced conectie heat transfer across a cyinder is (see Tabe 1.9) 1/ 1/3 0.6 Re Pr D ReD Nu D = /3 1/4 [1 (0.4 / Pr) ] Anaogy between mass and heat transfer gies us 1/ 1/3 0.6 Re Sc D ReD ShD = /3 1/4 [1 (0.4 / Sc) ] / 1/3 5/ = /3 1/4 = [1 + (0.4 / 0.39) ] /8 5/8 4/5 4/5 4/5 Mutiphase Systems with 64

65 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The mass transfer coefficient is therefore h m, G The partia pressure of the apor at the iquid-apor interface, p,, can be obtained from eq. (7.186) m& p, = p ( p p, )exp ρ gh m, G = ( ) exp = Pa Sh 5 DD g = = = D m/s Mutiphase Systems with 65

66 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe The interfacia temperature, T, is the saturation temperature corresponding to the aboe partia pressure. Assuming that the apor behaiors ike idea gas, the Capeyron-Causius equation (.168) can be used to obtain p, h 1 1 = p Rg T T n sat (7.195) Mutiphase Systems with 66

67 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe where h = 51. kj/kg and R g = kj/kg-k for water at 1 atm. Equation (7.195) can be rearranged to obtain T 1 Rg p = n Tsat h p, o = n = K = C The ratio of heat fuxes with and without noncondensabe gas can be obtained from eq. (7.19) qg T Tw = = = q T T sat w Mutiphase Systems with 67

68 Adanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howe In other words, the heat transfer is decreased by 16.75% due to the presence of noncondensabe gas. Mutiphase Systems with 68