ASSESSMENT OF THE HEAT TRANSFER MODEL AND TURBULENT WALL FUNCTIONS FOR TWO FLUID CFD SIMULATIONS OF SUBCOOLED AND SATURATED BOILING

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1 ASSESSMENT OF THE HEAT TRANSFER MODEL AND TURBULENT WALL FUNCTIONS FOR TWO FLUID CFD SIMULATIONS OF SUBCOOLED AND SATURATED BOILING D. Prabhudharwadar 1,, M. Lopez de Bertodano 1,, J. Buchanan Jr. 1 Purdue University, Schoo of Nucear Engineering, 400 Centra Drive, West Lafayette, IN-47906, USA Bechte Marine Propusion Corporation, West Miffin PA-151, USA Abstract This paper describes the detais of vaidation of the two-fuid mode for boiing simuations using the CFD code CFX. In particuar, since subcooed boiing occurs at a superheated wa paced in a subcooed iquid, most of the heat and mass transfer taes pace cose to the wa. Therefore, this study was focused on the assessment of two important aspects of near wa predictions, viz., the wa heat transfer mode and the turbuent wa functions. Vaidation was performed using the state-of-the-art mutidimensiona experimenta data avaiabe in the iterature. The first part is an assessment of the wa heat transfer mode which is based on spitting the wa heat fux into three components, viz., the singe phase convection, the evaporation at the bubbe surface and the quenching effect at the heated wa after the bubbe departure. The evaporation and quenching components use the foowing three cosure parameters: 1. Bubbe Nuceation Site Density,. Bubbe Departure Diameter, and 3. Bubbe Departure Frequency. Data sets where the bubbe diameter has been measured have been seected to eiminate one unnown so the assessment is imited to the nuceation site density and the departure frequency correations avaiabe in the open iterature. The second part of the paper is an assessment of the turbuent wa functions used to prescribe the wa boundary conditions for momentum and turbuence quantities and hence infuence the near wa veocity and turbuence distribution. The state-of-the-art two-fuid mode uses a two-phase -ε turbuence mode (Lopez de Bertodano et a., 1994) with the standard ogarithmic wa function (Launder and Spading, 1974). Athough the aw-of-the-wa is universa for singe phase fows, the same is not true when the boundary ayer contains a two-phase bubby mixture. Previous experiments (Marie et a., 1997) have shown that the veocity profie sti foows a ogarithmic profie but the sope and the intercept constant vary with the concentration of bubbes and the reative magnitude of the shear and buoyant forces. A modification of the aw of the wa coefficients has been suggested by Marie et a. (1997) based on the experimenta data obtained from adiabatic air-water bubby fow over a vertica fat pate. The wa aw coefficients of CFX were modified using the two-phase wa function mode and significant improvements were noticed in the turbuent parameter predictions. NOMENCLATURE Interfacia Area Concentration Drag Coefficient C VM Coefficient of Virtua Mass (0.5) D b Bubbe Diameter D dep Bubbe departure diameter e Phase enthapy f dep Bubbe departure frequency g Gravitationa Acceeration h g Latent heat of evaporation j L Superficia Veocity of Liquid j G Superficia Veocity of Gas Turbuence Kinetic Energy M i Interfacia Momentum Source Mass Source a i C D m nsite Nuceation site density q" Heat Fux υ Veocity (vector) υ Reative Veocity (vector) R T Temperature Gree Symbos Voume Fraction of th phase α α Void Fraction ε Dissipation of Turbuence Kinetic Energy Density ρ τ Re Turbuent (Reynods) Stress µ Dynamic viscosity 1

2 λ Therma Conductivity Superscript D Drag TD Turbuent Diffusion L Lift W Wa Subscript Phase identifier 1,, f, L Continuous (Liquid) Phase, g, G Dispersed (Gas) Phase dep Departure t Turbuent 1. INTRODUCTION The prediction of boiing fows depends on accurate representation of the heat and mass transfer processes at the wa where the vapor is formed. The accuracy of the mode prediction depends on how we the foowing three independent parameters are estimated: 1. Bubbe Nuceation Site Density,. Bubbe Departure Diameter, and 3. Bubbe Departure Frequency. These parameters depend on compex physica phenomena that are not competey understood. Different cosure reations exist in the iterature for each of the above parameters which are discussed in detai in the next section. One of the goas of the present study is to perform an independent assessment of each of these parameters from the perspective of CFD impementation. However, it is difficut to test three separate uncertainties simutaneousy and hence ony those data sets have been seected where the bubbe diameter has been measured. A review of the departure frequency modes was done to seect the one best suited for the present data sets. Then, using the nown departure bubbe size and the chosen departure frequency mode, two we nown modes for the nuceation site density were impemented and the resuts were compared. The heat transfer at the wa aso depends on the turbuence intensity near the wa and this shoud be correcty predicted as we. The wa function approach that is usuay used with the Standard - epsion mode to obtain the near wa turbuence quantities using coarser meshes needs a correction that accounts for the effect of presence of a two-phase mixture. Such an approach has been impemented successfuy for adiabatic fows (Marie et a., 1997) and this has been extended for boiing fows here. The two-fuid mode used here was previousy tested for adiabatic fows where the interfacia momentum transfer terms were vaidated (Prabhudharwadar et a., 009). The energy equation aong with the interfacia heat and mass transfer correations were incorporated into the mode. The next section describes in detai the two-fuid mode used in this study.. TWO-FLUID MODEL WITH HEAT AND MASS TRANSFER For diabatic fows, the ensembe averaged two-fuid mode equations (Ishii and Hibii, 006) governing the motion of each phase has the foowing form: Mass Conservation: α ρ + α ρυ = m (1) t Momentum conservation: α ρυ + α ρ υ υ Re = α p + α ( ) τ + τ + α ρ g + M + i m, jυ () j t j Energy conservation: α ρ e + Re α ρ υ e = [ α ( )] t q + q p α + ( αυ ) + m H i + q (3) t i Re ρ ν t q = λ T, q = e (4) Prt In the present cacuations of subcooed boiing, the vapor generated was assumed to be at saturated temperature and hence was treated as isotherma phase..1 Interfacia momentum transfer The interfacia momentum transfer force comprises of force terms due to drag, turbuent diffusion, ift and a wa force:

3 i D i L i TD i W i M = M + M + M + M (5) There are other forces ie the virtua mass force and Basset force which are important under transient conditions but they are found to be negigibe for the current cass of probems (steady state bubby). The drag force of the bubbes is given by, M 3 C D D = αρ υ R υ (6) R 4 Db where the drag coefficient, C D, is given by the Ishii-Zuber correation (1979). D b is the Sauter mean diameter of the bubbe fied and the reative veocity is given by υ R = υ υ1. The wa force mode accounts for the effect that eeps the centers of the bubbes no coser than approximatey one bubbe radius from the wa. This force is important when the ift force is present. It is given by (Anta et a., 1991): W M = Cwaρ1α υ υ1 n, cw 1 c w Cwa = min 0, + Db ywa where n and y wa are the norma vector and the distance from the wa respectivey. The CFX vaues for the coefficients are: c w1 = and c w = In a previous study (Prabhudharwadar et a., 009) it was found that the Anta s CFX Defaut coefficients wor we for adiabatic bubby fows through vertica pipe. The turbuent diffusion force is cosed using the Lopez de Bertodano Mode (Lopez de Bertodano, 1991): TD M = CTD ρ11 α (8) The turbuent diffusion coefficient used for the present cacuations is C TD = 0.5. The ift force is given by (Auton (1987)): L M = C L ρ 1α ( υ υ 1 ) ( υ 1 ) (9) The ift coefficient used for the cacuations is C L = 0.1 (Lopez de Bertodano, 1991). This vaue is of the same order as Tomiyama s experimenta vaue (00) for a bubbe in Couette fow, C L = Interfacia heat transfer The interface to iquid heat transfer (=) is expressed as, qi = hiai ( Tsat T ), (10) where, h i is the heat transfer coefficient between the iquid and the interface, cosed as foows, h i λ = Nu (11) d s In the above equation, λ is the iquid therma conductivity and d s is a ength scae which is assumed equa to the bubbe diameter. The Nusset number (Nu) in the above cosure equation is given by foowing expression (Ranz and Marsha, 195): Nu = + (1) Reb Pr Reynods number (Re b ) used in the above cosures is defined as, D υ b R Re b = (13) ν The interfacia mass transfer is given as, ( ) ( ) hi Tsat T ai hi Tsat T ai & &, (14) m cond, L = max, 0, m = evap, L min, 0 Hg H g (7) 3

4 where, H g is the atent heat of phase change..3 Wa heat transfer Wa heat transfer is the most important aspect of subcooed boiing as it provides the sources for energy and phase mass baance equations. Most wa heat transfer modes are based on partitioning the wa heat fux into three components. These were impemented in the mutidimensiona CFD cacuations by Kuru and Podowsi (1990). The three components of the wa heat fux are a) Evaporation heat fux (q e ) The part of heat fux utiized in formation of vapor at the wa, b) Forced convection to the iquid (q c ), c) Wa quenching by iquid phase (transient conduction) (q q ) This accounts for the heat transfer to the subcooed iquid that repaces the detached bubbe at the wa. The wa surface is assumed to be spit into two parts (A 1, A ) each under the infuence of one phase. Fraction A is infuenced by the vapour bubbes formed on the wa and participates in the evaporation and quenching heat transfer. Fraction A 1 is the remaining part of the wa surface, (A 1 =1 A ) and participates in the convective heat transfer to the iquid. A 1 and A are reated to the nuceation site density per unit wa area (n site ) and to the infuence area of a singe bubbe forming at the wa nuceation site. The Kuru-Podowsi mode assumes, that the diameter of the bubbe infuence zone is twice as big as the bubbe departure diameter (a = ) ( a Ddep ) nsite, A1 = A A = π 4 1 (15) Evaporation Heat Fux: The evaporation heat fux is obtained from the rate of vapor generation at the wa which is given as a product of mass of a bubbe at detachment, the detachment (departure) frequency and the nuceation site density, 3 π Ddep m& e = ρg f dep n (16) site 6 qe = m& eh g (17) The specification of evaporation heat fux using above equation requires cosure for bubbe departure diameter, bubbe departure frequency and the site density. The defaut avaiabe mode in CFX uses foowing cosure reations: The bubbe departure diameter is given by the empirica correation of Toubinsy and Kostanchu (1970), the Nuceation Site Density is obtained using the correations of Lemmart and Chawa (1977) and the departure frequency is cosed using the simpest avaiabe expression (Coe, 1960), which uses a characteristic bubbe veocity (termina veocity of bubbe rise) and a characteristic bubbe size (departure diameter). The frequency of bubbe departure obtained by Coe (1960) was derived under the assumption of bubbe moving with termina veocity once detached from the surface. This is a fair assumption for poo boiing scenario at ow heat fuxes where bubbe detachment is hydrodynamicay governed, but may not be accurate for fow boiing at high heat fuxes where the thermodynamics governs the bubbe growth and detachment. The CFX defaut cosure reations are given beow: D = D exp T T T, D = 0.6 mm, T = 45 K (18) n dep ref ( ( ) ) sat iq ref site = nref (( Twa Tsat ) / Tref ) nref = m, Tref = 10 K f dep dep iq ref ref (19) 4g ρ = (0) 3D ρ Quenching Heat Fux: As mentioned previousy, the quenching heat fux is the component of wa heat fux utiized to heat the cod iquid repacing the detached bubbe adjacent to the heated wa. In order to evauate this component, Miic and Rohsenow (1969) used an anaytica approach starting with transient conduction in semi-infinite medium with heated wa being the ony boundary where temperature is specified. 4

5 T ( x, t) T( x, t) = α ; T( x,0) = T, T (0, t) = Tw, T(, t) = T (1) t x Kuru and Podowsi (1991) made an assumption that quenching occurs between detachment of one bubbe and appearance of next bubbe (nuceation), and this time was assumed to be 80% of the detachment period. The fina form of quenching heat transfer cosure of Kuru-Podowsi mode is as foows: t w 0.8 hq = λ f ; t w = () πα f qq = A hq ( Tw T ) (3) Singe Phase Convection Heat Fux: This is evauated under the standard assumption of a ogarithmic temperature profie across the turbuent boundary ayer (Kader (1981), ANSYS CFX Sover Theory Guide (006)). ρ C p, u qw ph = ( Tw Tiq ) (4),1 + T T + = Pr β = f (Pr ), Γ = u = C y e Γ + (.1 n( y ) + β ) f ( y ), µ, y = ρ u y µ e 1 Γ, (5) where, y is the distance of the wa adjacent node from the wa, is the iquid turbuence inetic energy. In the above equation β is a function of iquid Prandt number and Γ is a function of y (ANSYS CFX Sover Theory Guide (006)). The convective heat fux component through the iquid area fraction is thus given as, ρ Cp, u q = A h T T h = (6) c ( w iq ) c + 1 f c, T It shoud be noted here that a the wa heat fux mode correations above use a iquid temperature (T iq ). The singe phase heat transfer correation above uses the wa adjacent node temperature. However, in the quenching heat transfer correation which was based on a one-dimensiona mode, the iquid temperature refers to the bu mean temperature. As a good estimate, CFX approximates this temperature with the temperature at a fixed y (50). This is done in order to have a mesh size independent evauation of wa heat fux partitions. Review of the Bubbe Departure Frequency and Nuceation Site Density modes: Situ et a. (008) recenty reviewed a the avaiabe departure frequency modes and proposed a correation for bubbe departure frequency based on experimenta data of subcooed boiing fow for wide ranges of pressure and heat fux (Basu et a. (005), Thorncroft et a. (1998) and Situ (004)). The departure frequency was correated to the boiing heat fux as foows: N =.7N (7) fd 10 qnb N fd is a non-dimensiona representation of frequency: fdepddep N fd =, (8) α Liq where, α Liq is the iquid therma diffusivity. N qnb is the dimensioness nuceate boiing heat fux obtained from Chen s correation. " qnb Ddep NqNB = (9) α ρ h q h " NB NB = h = S NB Liq g fg ( T T ) w sat λ 0.79 ( ) p Liq ( 0.001) ( T T ) p σ 0.5 C µ 0.45 h ρ 0.49 ρ fg 0.4 g w sat (30) (31) 5

6 ( T ) p( ) p = p (3) w T sat S is a suppression factor which is described in detai in the next section on site density (equation (41)). However, when the frequency data of Thorncroft et a. (1998), Basu et a. (005) and Situ (004) was compared with the prediction of Situ et a. s correation and Coe s mode, the accuracy of the modes was found comparabe as shown in the figure beow (Fig. 1 (a) and (b)). Situ et a. (008) Correation Prediction (Reproduced from the origina Paper) 1.E+06 Coe's Mode Prediction 1.E+05 1.E+04 Thorncroft et a. (1998) Basu et a. (005) Situ (004) 1.E+05 1.E+04 Thorncroft et a (1998) Basu et a. (005) Situ (004) N fd(predicted) 1.E+03 1.E+0 1.E+01 1.E+00 N fd (predicted) 1.E+03 1.E+0 1.E+01 1.E+00 1.E-01 1.E-0 (a) 1.E-01 1.E-0 (b) 1.E-03 1.E-03 1.E-0 1.E-01 1.E+00 1.E+01 1.E+0 1.E+03 N fd(experiment) 1.E+04 1.E+05 1.E-03 1.E-03 1.E-0 1.E-01 1.E+00 1.E+01 1.E+0 1.E+03 N fd (experiment) 1.E+04 1.E+05 1.E+06 Fig. 1: Comparison of the bubbe departure frequency modes The data scatter has been contained by the dotted ines in the figures above which are spaced by an order of magnitude from the correation. The data scatter about the prediction is comparabe and hence seection of the Coe s mode (defaut CFX mode) was reasonabe for the present probem. A correation for the nuceation site density has been proposed by Hibii and Ishii (003) which is vaid over bar and for most practica combinations of fuid and surface materia (e.g., Water- Stainess stee, Water-Copper, Water-Zr-4, R113-Nichrome etc.). This correation is essentiay an improvement of the previousy estabished correation of Kocamustaffaoguari and Ishii (1983) vaidated against water data for 0-50 bar. The Kocamustaffaoguari-Ishii (K-I) correation reates the site density to the wa superheat as foows: 4.4 c ( ρ ) nsite = R f, (33) where, n = n D (34) site site dep R = R /( D / ) (35) c C dep R c is the critica radius of the surface cavity which represents a minimum cavity size which can be activated at a given wa superheat. R c is given as, σ ( 1+ ( ρ g ρ f ) P R =. C ( exp ( hfg Tsat, e / ( RTgTsat )) 1) (36) Under the foowing conditions: ρ ρ, h ( T T ) /( RT T ) << 1, (37) g << fg g sat g sat R c can be simpified as, = σ T / ρ h T. (38) ( ) R C sat g fg sat, e The density dependent parameter in equation (33) is given as, f ρ = ρ ρ, ρ = ρ ρ ρ (39) ( ) ( ) ( g ) g The effective wa superheat in equation (36) is usuay ess than the actua wa superheat. The reason for this is as foows. A bubbe nuceated at the wa grows through a iquid fim adjacent to the wa 6

7 where a consideraby high temperature gradient exists. So, in reaity, it experiences a ower mean superheat than the wa superheat. In case of poo boiing, the difference is not significant and the superheat based on the wa temperature can be taen as the effective superheat. In case of forced convective boiing, there exists a steeper temperature gradient in the near wa fied due to therma boundary ayer. Hence, the effective superheat experienced by the bubbe is smaer than the actua wa superheat and it is given as, Tsat e = S T where,, sat Tsat = Tw T (40) sat The mutipier S in the above equation is the superheat suppression factor and is given as (Chen, 1966), ( 5 S = Re TP ) (41) The two-phase Reynods number is given as, 1. 5 ReTP = [ G ( 1 x) d h / µ ] F (4) F = 1.0, for X tt 10 (43) =.35( / X tt ), for X tt < 10 The Martinei parameter can be approximated as, x ρ g µ f X = (44) tt x ρ µ g Thus, equations (33)-(44) represent the compete mode of Kocamustafaoguari-Ishii (1983), which has been incorporated in CFX-1 for the present study..4 Turbuence Transport The cosure for the Reynods stresses in equation () is based on the two-phase -ε mode deveoped by Lopez de Bertodano et a. (1994). This mode assumes that the shear induced and bubbe induced turbuent stresses are added together: ( ) ( ) Re Re Re τ = τ + τ (45) SI BI The resuting expression for the diffusivity of momentum (effective viscosity) of the iquid phase is: ν µ + ε t1 = C CDB α Db υ R (46) where, the first term on the RHS corresponds to the -ε mode for the shear induced turbuence viscosity and the second term corresponds to Sato s mode (1981) for the bubbe induced turbuence viscosity. The coefficient C µ = 0.09 is the standard vaue according to the -ε mode. A vaue of 0.6 is recommended for C DB. It is important to note that the -ε mode transport equations for the iquid phase are soved together with the continuity and momentum equations (Eqs. (1) and ()). The standard coefficients of the -ε mode are eft untouched. Standard Wa Functions (Launder and Spading (1974)) are used to mode turbuence quantities near the wa which avoids need to resove the compete boundary ayer. Turbuence inetic energy dissipation rate (epsion) is specified using an agebraic cosure assuming turbuence production due to shear is baanced by dissipation. Shear stress at the wa is connected to the veocity in the fuy turbuent boundary ayer (computationa node next to the wa) using a ogarithmic aw of wa which is described in detai ater. This competes the description of the two-fuid mode and a the cosure reations used in the present study. The resuts of the comparison of two previousy mentioned nuceation site density mode are discussed in the subsequent section. The resuts reported here are restricted to the prediction of vapor void fraction distribution which of primary interest in this study. 7

8 3. RESULTS 3.1 Vaidation of the Heat and Mass Transfer Modes Subcooed boiing of R1 at 15-7 bar (More et a (003)) More et a. (003) performed subcooed boiing experiments in a vertica pipe of 19. mm interna diameter and a ength of 5 m. The pipe had a heated section of 3.5 m preceded by and foowed by unheated engths of 1 m and 0.5 m respectivey. The input parameters for the four tests reported were as summarized in Tabe 1. The iquid-vapor density ratio in these experiments corresponds to that for water-steam at bar. Dua sensor fiber optics probes were used to measure void fraction. An experimenta error of ±0.0 was reported in the void fraction measurements. Tabe 1: Simuation input parameters for tests tp1 and tp6 Parameter Test1 (Deb5) Test (Deb6) Test3 (Deb13) Test4 (Deb10) Mass Fux (g/m ) Pressure (bar) Inet Subcooing ( o C) Heat fux (W/m ) A wedge of the pipe is used as the domain. The mesh had 0 radia and 50 axia uniformy spaced eements. Fig. shows the domain and cross-section of the mesh used. Fig. : Domain and Radia Mesh for R1 boiing simuation case A the data sets incude radia profies of bubbe diameter. Note that with Freon at these test pressures, the surface tension is of the order of N/m which resuts into very sma bubbe sizes (i.e., 0.5 mm). For the CFD cacuations the mean of the radia profie vaue is used for the bubbe diameter in the bu, whereas, the vaue at the wa is used as the departure bubbe size (Fig. 3). An experimenta error of ±1% was reported in the bubbe diameter measurements. It was found during a the cases that a better match with experimenta data is obtained with the coefficient of the turbuent diffusion force C TD = 0.5 instead of the vaue obtained for adiabatic air water fows, C TD = 0.5. This change may be attributed to bubbe diameters in these cases which are one order of magnitude smaer than that in atmospheric air-water systems (Prabhudharwadar et a., 009). Therefore there are more interactions of the bubbes with smaer turbuent eddies. Fig. 4 shows the mesh sensitivity resuts for Test 1 with three meshes incuding haf and doube the mesh size of the mesh size stated above. Simuations were performed using the Kocamustafaoguari- Ishii mode for site density (termed as K-I Mode hereafter). The mesh size dependent uncertainty in 8

9 the numerica soution was found to be esser than the uncertainty in the measured data at the chosen mesh with 0x50 eements. Fig.3: Bubbe diameter approximation for the simuations 0.6 P = 6.15 bar, q" = W/m, G = 1986 g/m s, dt sub = 18.1 o C, D b = 0.45 mm, D dep = 0.35 mm Experiment (More et a. (003)) Void Fraction x 15 Eements 0 x 50 Eements 40 x 500 Eements r (m) Fig. 4: Comparison of the mesh size dependent uncertainty in soution with the experimenta data 9

10 Fig. 5 shows the prediction of radia variation of void fraction obtained using two different site density modes. The Lemmart-Chawa (1977) correation which is used in the Kuru-Podowsi mode was found to under-estimate the void fraction in a the cases. K-I mode predicted the data we in a the cases. The prediction however deteriorated at ow pressure data (P = bar) and this needs further investigation. Fig. 5: Void Fraction prediction using the two Nuceation Site Density modes Fig. 6 shows the prediction of radia variation of interfacia area concentration (IAC) obtained using two different site density modes. It can be seen that the IAC prediction is better with K-I site density mode. However, the IAC prediction near the wa is not as accurate as that towards the pipe centre because of assumption of constant bubbe size in the entire pipe. The rapid change in IAC near the wa is due to coaescence of smaer nuceated bubbes (Fig. 3) and it can be predicted using a more sophisticated approach ie the one group interfacia area transport equation with a bubbe coaescence mode and this research is currenty underway. Fig. 7 shows the prediction of radia variation of iquid temperature obtained using two different site density modes for the two test cases for which iquid temperature is reported in the reference. Lemmart-Chawa mode over predicts iquid temperature near the wa and the wa superheat whereas the K-I mode predicts it we near the wa with an under-prediction in wa superheat but coser prediction than the Lemmart-Chawa mode. Overa, the K-I site density mode produces better resuts than Lemmart-Chawa mode. 10

11 P = 6.15 bar, q" = W/m, G = 1986 g/m s, dtsub = 18.1 o C, Db = 0.45 mm, Ddep = 0.35 mm P = 6.15 bar, q" = W/m, G = g/m s, dtsub = o C, Db = mm, Ddep = 0.5 mm Experiment (More et a (003)) Experiment (More et a (003)) Site Density: K-I Mode (1983) Site Density: K-I (1983) 8000 Site Density: Lemmart-Chawa (1977) 8000 Site Density: Lemmart-Chawa (1977)) IAC (1/m) 6000 IAC (1/m) r (m) r (m) P = 6.17 bar, q" = W/m, G = g/m s, dtsub = o C, Db = mm, Ddep = mm P = bar, q" = 76.4 W/m, G = 07 g/m s, dtsub = 3.4 o C, Db = mm, Ddep = 0.37 mm Experiment (More et a (003)) Site Density: K-I Mode (1983) Site Density: Lemmart-Chawa (1977) 6000 Experiment (More et a (003)) Site Density: K-I Mode (1983) Site Density: Lemmart-Chawa Mode (1977) IAC (1/m) 8000 IAC (1/m) r (m) r (m) Fig. 6: Interfacia area concentration prediction using the two Nuceation Site Density modes Liquid Temperature (K) P = 6.15 bar, q" = W/m, G = 1986 g/m s, dtsub = 18.1 o C, Db = 0.45 mm, D dep = 0.35 mm Experiment (More et a (003)) Site Density: K-I Mode (1983) Site Density:Lemmart-Chawa (1977) Tsat Liquid Temperature (K) P = 6.15 bar, q" = W/m, G = g/m s, dtsub = o C, Db = mm, Ddep = 0.5 mm Experiment (More et a. (003)) Site Density: K-I Mode (1983) Site Density: Lemmart-Chawa (1977)) Tsat r (m) r (m) Fig. 7: Liquid temperature prediction using the two Nuceation Site Density modes 11

12 3.1. Subcooed Boiing of R113 at.69 bar in an Annuus (Roy et a (00)) Roy et a. (00) performed subcooed boiing experiments in a vertica annuus of mm interna diameter and 38.0 mm outer diameter. The tota ength of the channe was 3.66 m of which the initia 0.91 m was unheated (adiabatic). The pipe had a heated section of.75 m. Six experiments were reported (designated as tp1-tp6) and two of them (having minimum and maximum mass fux, tp1 and tp6) have been seected for simuation purpose here. Radia Profies of Void Fraction were measured with a dua-sensor fiber-optic probe. An experimenta error of ±% was reported in the void fraction measurements. The input data for the two representative cases is stated in Tabe : Tabe : Simuation input parameters for tests tp1 and tp6 Parameter tp1 tp6 Mass Fux (g/m ) Pressure (bar) Inet Subcooing ( o C) Inner wa heat fux (W/m ) Simiar to previous probem, a section of the annuus is used as the domain. The mesh had 5 radia and 66 axia uniformy spaced eements. Fig. 8 shows the domain and cross-section of the mesh used. Fig. 8: Mesh cross-section for R-113 simuations These experiments had a characteristic near wa boundary ayer of vapor (boiing ayer) and the bu of the pipe (more than 50%) had ony iquid. Bubbe diameter was reported for the test tp6. As the pressure was same in a the experiments and the range of other parameters was aso narrow, same vaues were used for the other simuated case (tp1). The bubbe diameter profie is shown beow in Fig. 9. Fig. 9: Bubbe diameter for case tp6 For the simuations, the bu bubbe diameter was assumed to be 1 mm and the departure bubbe diameter 0.6 mm arrived at from the most probabe diameter vaues reported in the experiments (Fig. 9). Simuations were aso done with the Sauter mean diameter vaues in the bu and at the wa; however the predictions with most probabe vaues matched the data we. As stated earier, the 1

13 simuations were performed with the K-I site density mode ony since this data set is in the pressure range for which the K-I mode is aready found to be best suited through previous benchmar probems. The void fraction prediction for the two cases is shown beow in Fig. 10. The width of the boiing ayer was we predicted with the most probabe bubbe sizes; however the near wa void fraction was under-predicted in both the cases. Fig. 10: Void fraction distribution prediction for tp1 and tp6 with most probabe and mean diameter vaues 3. Two-phase wa function impementation The veocity profie in the turbuent boundary ayer next to the wa foows a ogarithmic profie as given beow: U = og( y ) + C (47) κ + where, U + uτ y τ w U =, y =, uτ = (48) uτ ν ρ κ is the von Karman constant (= 0.41) and C is a og-ayer constant (= 5.). U is the veocity component tangentia to the wa and y is the distance norma to the wa. The above definitions are in context of singe phase fow. Marie et a. (1997) performed experiments where a fat pate was paced parae to bubby air-water fow. Void fractions and veocity profies were measured in the boundary ayer next to the wa for different inet concentrations of the air. It was found that the veocity profie sti foows a ogarithmic aw-of-the-wa but with a modified sope and og ayer constant. When the streamwise momentum equation for iquid phase was integrated through the fuy deveoped two-phase boundary ayer, foowing equation was obtained: U L ( 1 α ) ν L u L υ L = u L g( α α ) y BL (49) τ, y where, u υ is the turbuent shear stress (Reynods stress), α is the average void fraction in the L L boundary ayer and α is the average void fraction in the free stream. y BL is the boundary ayer thicness. The second term on the right side of the above equation represents the buoyancy due to presence of dispersed phase. Negecting the viscous stresses, the above equation can be simpified for diute dispersed fows as, u υ u g α α y u (50) ( ) L L τ, L BL = τ The two-phase og ayer equation proposed by Marie et a. (1997) uses 13 uτ as the veocity scae instead of uτ in equations (47-48). When this veocity scae was used for the aw of the wa it was found that

14 the og ayer profies for a the inet void fraction were parae and the og ayer coefficient C increased with the inet void fraction. Fig. 11 beow shows the resuts obtained by Marie et a (1997) for different inet void fractions after using a modified veocity scae. Fig.11: Log ayer veocity profies measured by Marie et a. (1997) for different air void fractions The og ayer constant for two-phase mixture, C, was reated to the singe phase C with the foowing reation. 0+ C = C + y (1/ β 1) (1/ κ ) og β (51) u β = (5) u τ τ 0+ y represents the edge of the aminar sub-ayer which is usuay about 11. Based on the above mode, the og ayer coefficients were modified in the CFD code CFX and turbuence quantities were compared with singe phase and two phase aw of the wa. The above mentioned experimenta benchmar of Roy et a (00) incuded measurements for turbuence inetic energy and Reynods stress. A singe phase case was first tried to vaidate the singe phase wa function incuded in the CFD code CFX. Fig. 1 shows turbuence inetic energy and Fig. 13 shows the turbuent shear stress profie for a singe phase experiment reported by Roy et a (test sp1: Mass Fux = 568 g/m, Pressure =.69 bar, Inet Temperature = K, Wa Heat Fux = 16 W/m ). The resuts were satisfactory confirming adequacy of the mesh size and mode accuracy for the prediction of near wa turbuence quantities Experiment: Roy et a. (00) CFX Resuts Experiment: (Roy et a. (00))(sp1) CFX Resuts ) /s (m L <uv> (m /s ) (r-ri)/(ro-ri) Fig. 1: Turbuence inetic energy prediction for singe phase case (r-ri)/(ro-ri) Fig. 13: Reynods stress prediction for singe phase case 14

15 The two phase og aw coefficients were then impemented in CFX to compare the effective turbuence inetic energy and turbuent stress. The effective turbuence inetic energy superposes the shear induced and bubbe induced components as foows: 1 = + = α υ (53) L, eff L, SI L, BI ; BI R 4 Significant improvement was noticed in the prediction of the turbuence inetic energy (Fig. 14) for both the previousy reported cases especiay near the boiing boundary ayer edge where a singe phase wa function showed a characteristic dip. The Reynods stress (obtained using equations (45-46)) was found to be more accurate with the modified wa aw coefficients (Fig. 15). Fig. 14: Turbuence inetic energy prediction for wa-boiing cases tp1 and tp6 of Roy et a. (00) Fig. 15: Reynods Stress prediction for wa-boiing cases tp1 and tp6 of Roy et a. (00) 15

16 4. CONCLUSIONS 1. The heat fux spitting mode for wa-boiing fows was vaidated using data sets where bubbe diameters were nown.. The nuceation site density mode by Kocamustaffaoguari and Ishii (1983) in combination with the Bubbe Departure Frequency mode of Coe (1960) resuts in the best prediction of the data. 3. Predictions were found to deteriorate at ower pressures of the R1 data; however the resuts were sti satisfactory. 4. Use of a modified og aw coefficients based on the approach of Marie et a. (1997) improved prediction of the turbuence quantities near the wa. REFERENCES Angart, H, Nyund, O., Kuru, N., Podowsi, M.Z. CFD prediction of fow and phase distribution in fue assembies with spacer, Nucear Engineering and Design, Vo. 177, 15-8 (1997) ANSYS CFX Sover Theory Guide, ANSYS CFX Reease 11.0, (006) Anta, S.P., Lahey, R.T. Jr., Faherty, J.E. Anaysis of phase distribution in fuy deveoped aminar bubby two-phase fow, Internationa Journa of Mutiphase Fow, Vo. 17, No. 5, (1991) Auton T. R., Hunt J. C. R. and Prud homme M. The force exterted on a body in inviscid unsteady non-uniform rotationa fow, Journa of Fuid Mechanics, Vo. 197, (1987) Basu, N., Warrier G. R., Dhir, V.K. Wa heat fux partitioning during subcooed fow boiing: Part II- Mode vaidation, Journa of Heat Transfer, Vo. 17, (005) Chen, J.C. Correation for boiing heat transfer to saturated fuids in convective fow, I& EC Proc. Des. Dev., Vo. 5, 3 39 (1966). Coe, R. A Photographic Study of Poo Boiing in the Region of The Critica Heat Fux, AIChE Journa, Vo. 6, (1960). Hibii, T. & Ishii, M. Active nuceation site density in boiing systems, Internationa Journa of Heat and Mass Transfer, Vo. 46, (003). Ishii, M. & Hibii, T., Thermo-fuid dynamics of two-phase fow, Springer (006) Ishii, M. & Zuber N., Drag coefficient and reative veocity in bubby, dropet or particuate fows, AIChE Journa, Vo. 5, No.5, (1979). Kader, B.A. Temperature and concentration profies in fuy turbuent boundary ayers, Internationa Journa of Heat and Mass Transfer, Vo. 4, Issue 9, (1981). Kocamustaffaoguari G. & Ishii M. Interfacia area and Nuceation Site Density in boiing systems, Internationa Journa of Heat and Mass Transfer, Vo. 6, (1983). Kuru, N. & Podowsi, M. Z., Mutidimensionna effects in forced convection subcooed boiing, Internationa. Heat Transfer Conference, Jerusaem, Vo. 1, 1-6 (1990). Launder B.E., and Spading D.B., The numerica computation of turbuent fows, Computer Methods in Appied Mechanics and Engineering, Vo.3, pp , Lemmert, M., Chawa, J.M. Infuence of fow veocity on surface boiing heat transfer coefficient, Heat Transfer in Boiing, Academic Press and Hemisphere, (1977) Lopez de Bertodano, M. Turbuent Bubby Fow in a Trianguar Duct, Ph. D. Thesis, Rensseaer Poytechnic Institute, Troy New Yor, (1991). Lopez de Bertodano, M. A., Lahey, R. T., Jr., and Jones, O. C., Jr. Deveopment of a -ε mode for bubby two-phase fow, Journa of Fuids Engineering, Vo. 116, (1994). 16

17 Marie, J.L., Moursai, E., Tran-Cong, S., Simiarity aw and turbuence intensity profies in a bubby boundary ayer at ow void fractions Internationa Journa of Mutiphase Fow, Vo. 3, No., pp 7-47 (1997). Miic, B.B. & Rohsenow, W.M. A new correation of poo-boiing data incuding the fact of heating surface characteristics, Journa of Heat Transfer, 91, 45 50, (1969). More C., Yao W. & Bestion D. Three dimensiona modeing of boiing fow for the NEPTUNE code, Proceedings of the 10 th Internationa Topica Meeting on Nucear Reactor Therma Hydrauics, Korea (003). Prabhudharwadar, D.M., Baiey, C.A., Lopez de Bertodano, M.A. & Buchanan Jr., J.R. Two-Fuid CFD mode of adiabatic air-water upward bubby fow through a vertica pipe with a One-Group Interfacia Area Transport Equation, Paper FEDSM , Proceedings of the ASME 009 Fuids Engineering Division Summer Meeting, Vai, Coorado, USA (009). Ranz, W.E. & Marsha, W.R. Evaporation from drops: Parts I&II, Chem. Eng. Prog., Vo. 48, Issue 3, (195) Roy, R.P., Kang, S., Zarate, J.A., Laporta, A., Turbuent subcooed boiing fow: Experiments and Simuations, Journa of Heat Transfer, Vo. 14, (00). Sato, Y., Sadatomi, M. & Seoguchi, K. Momentum and heat Transfer in Two-phase Bubbe Fow, Int. J. Mutiphase Fow, Vo. 7, (1981). Situ, R., Experimenta and theoretica investigation of adiabatic bubby fow and subcooed boing fow in an annuus, Ph.D. Thesis, Purdue University, USA (004). Situ R., Ishii M., Hibii, T., Tu, J.Y., Yeoh, G.H., Mori, M. Bubbe departure frequency in forced convective subcooed boiing fow, Internationa J. Heat and Mass Transfer, Vo. 51, (008). Thorncroft, G.E., Kausner, J.F., Mei, R. An experimenta investigation of bubbe growth and detachment in vertica upfow and downfow boiing, Internationa Journa of Heat and Mass Transfer, Vo. 41, (1998). Toubinsy, V.I. & Kostanchu, D.M. Vapour bubbes groth rate and heat transfer intensity at subcooed water boiing, Proceedings of the 4th Internationa Heat Transfer Conference, Vo. 5, Paris (Paper No. B-.8), (1970). Tomiyama, A., Tamai, H., Zun, I., Hosoawa, S. Transverse migration of singe bubbes in simpe shear fows. Chemica Engineering Science, Vo. 57, (00) 17