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2nd RCM on the IAEA CRP on Natural Circulation Phenomena, Modelling and Reliability of Passive Safety Systems that Utilize Natural Circulation Effect of noncondensable gas on steam condensation in a vertical tube N.K. Maheshwari, P.K. Vijayan and D. Saha Reactor Engineering Division, Bhabha Atomic Research Centre, Trombay, Mumbai, INDIA - 400 085 Corvallis, USA Oregon State University, August 29 to September 02, 2005 Effect of noncondensable gas on Steam Condensation in a vertical tube Objectives Development of a theoretical model for the calculation of the condensation heat transfer coefficient in presence of noncondensable gas flowing inside vertical tube Comparison of various models for condensate film heat transfer coefficient including the effect of film roughness Comparison of the heat transfer coefficients obtained from theoretical studies and experiments . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Cooling surface Condensate film Gaseous state Wnc,i Steam/Air Heat Noncondensable going Wnc,b Gas Concentration out Tb Saturation bulk temperature Ti Gas/ vapour boundary layer Interface Fig.1 Schematic Illustration of the Model . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Theoretical Model Heat transfer through gas /vapor boundry layer dQ dm cond H fg h g dA(Tb Ti ) (1) Heat transfer through condensate film dQ h f dATi Tw (2) Heat balance at boundary layer h (T - T ) m // H h (T - T ) (3) f i w cond fg g b i Condensation heat transfer is defined as h (T - T ) m // H (4) cond b i cond fg Condensation heat transfer is defined as h (T T ) h f i w T T h T T cond b i g b i 1 cond h g Tb Ti 1 1 h h = = (5) tot h h h (6) f cond g . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube A mass balance at the interface is done to yield the following equation W m // ρD v W m// cond y v, i tot i i As the condensate surface is impermeable to the noncondensables, Eq. can be simplified as, W // ρD v 1 W m cond v, i y i W W v, b v, i = h m 1 W v, i . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube where hm is the mass transfer coefficient. The above equation can be recast, in terms of Sherwood number (hmd/ρD) , as // m cond,x d Wnc,i,x (7) Sh x ρD Wnc,i,x Wnc,b,x Following modifications are carried out to account for the • Film Waviness/ripple effect on condensate film heat transfer coefficient • Condensate film roughness effect on condensation and convective heat transfer • Suction effect • Developing flow effect on heat and mass transfer . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Condensate film model Two models for calculating the film heat transfer coefficient are used In the first model the Nusselt equation for film heat transfer coefficient is modified (McAdams (1954)) as, kl hf β (8) δx Where, β accounts for the increase in heat transfer due to film waviness and rippling. The correction is used only if film Reynolds number is greater than 30. For estimating the film thickness following eq. is used (siddique (1992)) 2π gρ l (ρ l ρ v ) Rδ3 5δ 4 m cond (x) 3 24 (9) μl Knowing the local condensate flow rate the local film thickness can be calculated . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube The second model to calculate the heat transfer coefficient in the condensate film is due to Blangetti et al. [4]. The model provides a weighted correction to the laminar Nusselt film solution. The Nusselt theory neglects interfacial shear and convective effects leading to a linear temperature distribution in the condensate film. Blangetti et al. (1982) considers that the condensate film thickness depends on the local mass flow rate and the interfacial shear stress g, as gρl (ρl ρ g ) 3 ρl τ g 2 Γ δ δ (10) 3μ l 2μ l The Nusselt number can be estimated Nu 1 x, la δ for laminar condensate film Beyond laminar zone h L Nu f (Nu 4 Nu4 )1/4 x, la x, tu , where Nu aRe b Prc (1 eτ f ) k x, tu f g l 1/3 μ2 δ δ and L l (11) L ρ2g l δ is condensate film thickness . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Film roughness considerations Film roughness increases the heat transfer from the gas phase by influencing the turbulence pattern close to the interface and disrupting the gaseous laminar sublayer. Using the corrections suggested by Norris [20] for the roughness of the heat transfer surface, n n fr fr Nu = Nu Sh Sh or os f and or os f (12) s s Moody correlation is used for friction factor to account for film roughness 2ε 100 1/3 f 1.375 10 3 1 + 21.544 (13) r d Re Where ε = δ/2 (Siddique (1992)) and Re is the bulk mixture Reynolds number . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube To account for the film roughness an alternate model proposed by Wallis for interfacial friction in the vertical annular flow is used δ f f (1 300 ) δ is condensate film thickness (14) r s d Suction effect considerations Kays and Moffat experimentally obtained the following correlation for sucked boundary layer, St (1 B ) ln h St B o h Nu B h m cond /G St // Where, St and RePr G∞ is the local mixture mass flux in the tube St/Sto defines the ratio of Stanton number with suction to that without suction at the same Reynolds and Prandtl numbers . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube The equation is recast 1 1 // m Re Pr G Nu exp cond x - 1 m // (15) G Nu Re Pr x ox cond x Using the analogy between heat and mass transfer, the above Eq. can be written as 1 1 m // Re Sc G Sh exp cond x - 1 (16) x m // G Sh Re Sc cond x ox Gnielinski correlation is used for Nusselt number without suction (f /2)(Re - 1000)Pr Nu s Re is the local mixture Reynolds ox number in the bulk fluid 1 + 12.7(f /2)1/2 (Pr2/3 - 1) s (17) . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube The local condensation mass flux can be given as G Sh Re Sc D ρ1 ω // ox ln 1 x m Where, Sc=ρD/μ G d cond Re Sc x (18) Where, ω is the ratio of the noncondensable gas mass fraction in the bulk to that at the liquid/gas interface Developing flow considerations Reynolds et al. suggested the following correlations for the thermal entrance zone, 0.8 (1 + 7 104 Re- 3/2 ) Nu Nu 1 Where, Nu=hg d/K ot o x d (19) 0.8 (1 + 7 104 Re- 3/2 ) Sh Sh 1 Where, ot o x d Sh=hmd/ρD (20) x is the distance from the inlet of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Input data for the Computation Temperature, noncondensable gas mass fraction, steam mass flow rate at inlet, wall temperature (Tw) , total pressure (Pt) Solution procedure Step 1. Assuming ∆Tb, the bulk temperature at (j+1)th step is calculated as T (j 1) T (j) - ΔT b b b Step 2. The local noncondensable gas mass fraction is evaluated from the Gibbs-Dalton ideal gas mixture equation using the partial pressure of steam corresponding to the bulk temperature. Step 3. As the noncondensable gas flow rate is constant, the local steam flow rate is calculated as 1 W (j 1) b, nc m (j 1) m s nc, in W (j 1) (21) b, nc . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Step 4. The local condensate flow rate at (j+1)th location can be found as follows, m (j 1) m - m (j 1) (22) cond s, in s Step 5. The local condensate film thickness δ(x) can be found from the relationship containing the expression for mcond and δ The local film heat transfer coefficient, hf can be calculated by equation either (8) or (11) Step 6. By using a trial value of Ti and Tw the local heat flux is then calculated as follows, q ( j 1 ) h (T - T ) (23) w f i w Step 7. The total heat transfer for the jth step is calculated as Δ Q [ m ( j ) - m (j 1) ] H (T ) m (j 1) C (Δ T ) s s fg i s ps b m C (Δ T ) nc, in pnc b (24) . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Step 8. The step length ΔL is calculated using the following equation 2 ΔQ j ΔL(j) πd q // ( j ) + q // (j + 1) (24) w w Step 9. To get m// cond , hcond and hg, eqs. (18), (4) and (15) are used Step 10. The heat balance at the interface yields the following eq. h (T - T ) m // H h (T - T ) f i w cond fg g b i (25) If the heat balance is not satisfied then improve the value of Ti and return to step 6. Step 11. The value of Tw is then calculated from the stored wall temperature profile. If these values differ by more than 0.10C from the trial value, the guessed value of Tw is improved. The overall heat transfer coefficient htot is then calculated. . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Step 12 Repeat the calculations from step 5 to step 12 for other space steps till the end of the tube has been reached The theoretical formulation has been tested against In-house as well as literature data • Maheshwari (2003) • Siddique (1992) • Tanrikut (1998) . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Test section characteristics for the selected database Characteristics Maheshwari Siddique Tanrikut Test tube material SS SS SS Tube length ( m ) 1.7 2.54 2.15 Cooled segment of the 1.6 2.44 2.15 tube ( m ) Inner diameter ( m ) 0.04276 0.046 0.033 Thickness ( m ) 0.00277 0.0024 0.003 Measurement error 31% 18% NA . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Test inlet conditions Case no. Pressure Steam flow Non- Mixture Reference (Pa) rate (kg/s ) condensable Reynolds mass fraction number (%) Maheshwari E19701 266000.0 0.004 11.5 9755 E19704 266000.0 0.004 23 10549 Siddique RUN # 47 214238.6 0.008863 9.8 18873 RUN # 52 221140.6 0.008635 35.4 22733 Tanrikut RUN-6.4.1 390600.0 0.01526 52 45195 RUN-4.4.1 398200.0 0.02987 27.8 85898 . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 10 9 8 With McAdams modifier and Moody corr. Heat transfer coefficient (kW/m K) With Blangetti model and Moody corr. 2 With Blangetti model and Wallis corr. 7 experiment 6 Case No. E19701 flow rate of steam =0.004 kg/s Mass fraction of air =11.5% 5 Total pressure =0.266 MPa Reynolds number =9755 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Distance from inlet (m) Variation of total heat transfer coefficient along the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 10 McAdams modifier and Moody corr. Blangetti model and Moody corr. 8 Blangetti model and Wallis corr. Total heat transfer coefficient (kW/m K) 2 Experiment Case No. RUN 52 flow rate of steam =0.008635 kg/s 6 Mass fraction of air =35.4% Total pressure =0.221 MPa Reynolds number =22733 4 2 0 0.0 0.5 1.0 1.5 2.0 Distance from inlet (m) Variation of total heat transfer coefficient along the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 0 2 4 6 8 8 8 McAdams model and Moody corr. (rms error 44%) Blangetti model and Wallis corr. (rms error 64%) 0% Theoritical heat transfer coef. (kW/m K) +2 6 6 2 % 4 -20 4 2 2 0 0 0 2 4 6 8 2 Heat transfer coef. (measured) (kW/m K) Comparison between experimental and theoretical heat transfer coefficients (Maheshwari's data) . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 0.009 With McAdams modifier and Moody corr. With Blangetti model and Moody corr. 0.008 With Blangetti model and Wallis corr. Experiment Case No. RUN 52 0.007 flow rate of steam =0.008635 kg/s Mass fraction of air =11% Steam flow rate (kg/s) Total pressure =.485 MPa 0.006 0.005 0.004 0.003 0.002 0.001 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Distance from inlet (m) Variation of steam mass flow rate along the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 0.000 0.002 0.004 0.006 0.008 0.010 0.010 0.010 With McAdams model and Moody corr. (rms error 13.5%) With Blangeti model and Wallis corr. (rms error 16.7%) Mass flow of steam calculated theoretically (kg/s) 0% 0.008 0.008 +2 0.006 % 0.006 -20 0.004 0.004 0.002 0.002 0.000 0.000 0.000 0.002 0.004 0.006 0.008 0.010 Mass flow rate of steam calculated experimentally (kg/s) Comparison between experimental and theoretical steam flow rate insde the tube(Maheshwari's data) . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 80 McAdams modifier and Moody corr. hf hg 60 hcond Heat transfer coefficient (kW/m K) 2 Blangetti model and Wallis corr. hf hg hcond 40 Case No. E19701 flow rate of steam =0.004 kg/s Mass fraction of air =11% Total pressure =.266 MPa 20 Reynolds number =9755 0 0.0 0.5 1.0 1.5 Distance from inlet (m) Variation of heat transfer coefficient along with the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 60 McAdam modifier and Moody corr. 55 hf 50 hg hcond Heat transfer coefficient (W/m K) 45 Blangetti model and Wallis corr. 2 40 hf hg 35 hcond 30 25 Case No. RUN-6.4.1 flow rate of steam =0.01526 kg/s 20 Mass fraction of air =52% Total pressure =.3906 MPa 15 Reynolds number =45195 10 5 0 -5 0.0 0.5 1.0 1.5 2.0 Distance from inlet (m) Variation of heat transfer coefficient along with the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube 100 90 McAdam modifier and Moody corr. Heat transfer coefficient (kW/m K) hf hg 2 80 hcond 70 Blangetti model and Wallis corr. hf 60 hg hcond Case No. RUN-4.4.1 flow rate of steam =0.0298 kg/s 50 Mass fraction of air =27.8% Total pressure =0.398 MPa 40 Reynolds number =85898 30 20 10 0 0.0 0.5 1.0 1.5 2.0 Distance from inlet (m) Variation of heat transfer coefficient along with the length of the tube . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube Conclusions • The heat transfer coefficient decreases sharply at the initial length and then slowly as the mass fraction of noncondensable gas increases along the length • The prediction of total heat transfer coefficient by all models is close to the experimental data up to Reynolds number 90000. The Blangetti model and Wallis correlation predicts higher local heat transfer coefficient compared to those predicted with McAdams modifier and Moody correlation and with Blangetti model and Moody correlation • The decrease in mass flow of steam along the length of the tube with Blangetti model and Wallis correlation is more compared to the model with McAdams modifier and Moody correlation and with Blangetti model and Moody correlation. The predicted steam flow rate with McAdams modifier and Moody correlation gives the lowest deviation when compared with the Maheshwari’s experimental data. . . . RCM-2 Effect of noncondensable gas on Steam Condensation in a vertical tube • The thermal resistance offered by gas/vapor boundary layer to condensation is higher than that offered by condensing film for low inlet Reynolds number. But this phenomenon may get reversed at higher Reynolds number. As the inlet mixture Reynolds number increases, the condensation heat transfer coefficient increases due to the higher turbulence in the gas/vapor boundary layer. Thank you . . . RCM-2

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posted: | 10/12/2010 |

language: | English |

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