Modelling of Subcooled Nucleate Boiling in a Vertical

Workshop on Modelling and Measurements of Two-Phase Flow and Heat Transfer in Nuclear Fuel Assemblies Stockholm, Sweden, 10-11 October 2006

Modelling of Subcooled Nucleate Boiling in a Vertical Channel with Coupling of Bubble-Tracking and Two-Fluid Models Ivo Kljenak, Boštjan Končar, Borut Mavko Reactor Engineering Division Jozef Stefan Institute Jamova 39, Ljubljana, Slovenia E-mail: [email protected], [email protected], [email protected] ABSTRACT A model of subcooled nucleate boiling flow in a vertical channel at low-pressure conditions is proposed, with specific application to the case of boiling in an annulus with a central heating rod. The model consists of a three-dimensional bubble-tracking model and a two-dimensional two-fluid model, which are coupled off-line. In the bubble-tracking model, vapour is distributed in the liquid in the form of individually tracked bubbles. The overall behaviour of the liquid-vapour system results from motion, interaction, coalescence and boiling mechanisms prescribed mostly at the level of bubbles. In the twofluid model, the so-called “interpenetrating continua” approach is used, where each phase is treated as a continuum that fills up the entire control volume and is described by its own system of averaged equations for mass, momentum and energy. The two-fluid model used in the present work consists of a generic two-fluid model, which is implemented within the Computational Fluid Dynamics code CFX, and additional closure relations introduced by the authors. The proposed model was applied to experiments on subcooled boiling that were carried out at Purdue University (USA) by Bartel (1999) and Situ et al. (2004). A promising agreement between measured and calculated void fraction profiles at different axial locations of a heated vertical annulus was obtained. INTRODUCTION In subcooled flow boiling in channels, liquid that flows near a heated channel wall evaporates, although the cross-sectional average flow temperature is lower than the local saturation temperature. The liquid temperature near the heated surface exceeds the saturation temperature and then gradually decreases as the distance from the surface increases. The proposed work deals with modelling of nucleate subcooled boiling in a vertical channel, where liquid evaporates on the heated wall in the form of vapour bubbles. Various approaches have been proposed in the literature to predict the non-homogeneous distribution of vapour over the channel cross-section in subcooled nucleate boiling. In the bubbletracking approach, the dynamic and thermal behaviour of each bubble in the surrounding liquid are simulated individually. The main difficulty in the strict bubble-tracking approach is the modelling of the liquid flow field, including the effects of liquid turbulence that affects bubble interaction. Namely, a model which would take into account the actual shape of the liquid domain at every instant should be based on local instantaneous description, which is computationally too demanding for a boiling system with several thousand bubbles. In the two-fluid approach, boiling flow consists of a dispersed phase (vapour bubbles) and a continuous phase (liquid flow), and is described with two sets of averaged transport equations. At averaging, the so-called “interpenetrating continua” approach is used: each phase is treated as a continuum that fills up the entire volume and is described by its own system of averaged equations. These equations are coupled with additional closure relations describing the exchange of mass, momentum and energy at the interface as well as the turbulence within each phase. A shortcoming of the two-fluid approach is that bubble coalescence is not modelled explicitly, so that the varying bubble size, which affects the liquid flow field and the non-homogeneous distribution of vapour over the channel cross-section, has to be calculated from some additional model or closure relation. To somewhat reduce these shortcomings, an approach based on coupling of bubble-tracking and two-fluid models is proposed. Both models were developed earlier (Kljenak et al. 2002, 2004a; 1/12

Končar et al. 2004) and successfully validated by using experimental results from the literature. The bubble-tracking model is three-dimensional and implemented with an own computer code, whereas the two-fluid model is two-dimensional and implemented using the Computational Fluid Dynamics code CFX (AEA Technology, 2001). Similar approaches have also been used in the modelling of adiabatic bubbly flows (Tomiyama et al., 1997; Pokharna et al., 1997; Okawa et al., 2002). The proposed approach was first tested with experimental results from Bartel (1999) (Kljenak et al., 2004b), and later with experimental results from Situ et al. (2004) (Kljenak et al., 2006). Both experiments were performed at Purdue University (USA), using an annular channel with a heated central rod. The evolution of void fraction profiles along the channel, simulated with the proposed approach, was compared with those experimental runs, in which the local void fraction did not exceed 0.3. In the present work, a description of the coupled model and a synthesis of both comparisons are presented. As the model was applied for flow in an annular channel, it could potentially be extended to the modelling of flow in a nuclear fuel assembly. BUBBLE-TRACKING MODEL Bubble motion and interaction In the proposed model, bubbles assume a rigid ellipsoidal shape and move in three dimensions with their symmetry axis always vertical. The velocity of a bubble is calculated by first adding the bubble relative velocity to the local hypothetical “undisturbed” liquid velocity (a twodimensional axis-symmetric 1/7th power law profile is assumed). Then, if some other (leading) nearby bubble is found to be present ahead of the bubble whose velocity is being calculated, an increase due to wake drift is added. A necessary condition for a bubble to be influenced by a leading bubble through wake drift and eventually collide with it is that bubbles overlap laterally by more than a certain critical fraction, called minimum relative overlapping. A similar approach was already proposed by Mortensen and Trapp (1992). If, following axial collision, bubbles overlap in space, the upper bubble is displaced laterally (sideways) for a fraction of its width if there is no other bubble nearby to prevent the movement. Otherwise, the upper bubble is displaced upwards so that bubbles barely stick. If bubbles still stick after collision (that is, if they were not separated due to a lengthy lateral displacement of the upper bubble), they remain sticking, move along together with the upper bubble's velocity and eventually merge if they do not separate earlier due to either turbulent dispersion, subsequent movements of either bubble or coalescence of either bubble with some other bubble. If separate bubbles do not overlap laterally more than the critical fraction, the motion of the trailing bubble is not affected by the leading bubble and bubbles behave as if they would not overlap at all. This rule was prescribed to approximate the influence of bubble agitation, which occurs in real bubbly flow and allows tightly packed bubbles to overtake one another. The drawback of this approach is that bubbles may briefly overlap in space, which is not physically realistic. Bubble sliding, detachment and radial motion In various experiments on subcooled nucleate boiling of water at low-pressure conditions, it has been observed that bubbles nucleated on a heated wall first slide along the wall and then tend to detach and migrate towards the tube region away from the heated surface (Bibeau and Salcudean, 1994; Zeitoun and Shoukri, 1996; Prodanovic et al., 2002). To the authors’ knowledge, no consensus concerning bubble sliding distance has been reached yet. Prodanovic et al. observed that bubbles usually slide a couple of diameters before detaching. In the proposed model, the following empirical approach was adopted, based on above-cited experiments: after nucleation, bubbles slide along the heated surface for some distance before attempting detachment. During sliding, the ellipsoidal bubble vertical axis is longer than the bubble horizontal axis to approximate the bubble inclination, which was observed in experiments. Detachment, which may occur if no other bubble obstructs the radial motion away from the heated surface, is modelled probabilistically in the same way as bubble radial migration (see below). After detaching from the heated surface, bubbles tend to migrate towards the lowertemperature region (again, if other bubbles do not obstruct their motion), where they eventually condense. After detachment, the bubble shape changes and the horizontal axis is longer than the vertical axis. In bubbly flow in general (that is, in boiling as well as in adiabatic flow), bubbles are distributed over the channel cross-section, supposedly as the result of interaction of different phenomena: liquid turbulent flow, transverse lift force, wall lubrication force and bubble interaction (Okawa et al., 2002; 2/12

Ohnuki and Akimoto, 1998). Bubble transverse motion over the channel cross-section is presumably partly random, due to the interaction of bubbles with turbulent eddies. A probabilistic approach was thus implemented to model bubble radial motion (migration) towards the tube outer wall: the motion consists of finite steps that are equal to a fraction of bubble width, each displacement occurring with a certain probability. At present, the proposed model was developed for boiling systems in which all bubbles are located between the heated inner annulus wall and the middle of the annular gap. The lift force, which is related to the liquid velocity gradient, is assumed to represent a restraining force to bubble radial motion away from the heated wall. Thus, the probability of migration Pm was modelled to increase with decreasing velocity gradient over the channel cross-section:

 ∂w  Pm = 1.0 − C r  l∞   ∂r 

1/2

(1)

where Cr is an empirical coefficient. As bubble lateral motion may be affected by turbulent eddies of a comparable size as the bubble, bubble migration is attempted every time a bubble moves axially a distance equal to its maximum vertical chord length. Turbulent dispersion The relative motion between bubbles is mainly influenced by eddy motion of the length scale of bubble size (Prince and Blanch, 1990). The influence of turbulent eddies, which may affect wake drift or sticking bubbles, is modelled as a succession of random binary events. Thus, each event may have two possible outcomes: at a given instant, bubble motion is or is not disturbed by turbulent dispersion. Higher turbulence intensity is simulated by prescribing a higher probability of dispersion. Disruption of wake drift and of sticking bubbles by turbulent eddies is related to the turbulence length scale and is simulated every time a bubble has moved in the axial direction a distance equal to 1/20 of the channel hydraulic diameter. Bubble coalescence Following axial collision and sideways or upwards displacement of the upper bubble, bubbles which still overlap more than the minimum relative overlapping stick together. Bubbles eventually merge after sticking together for a certain time interval (so-called "rest time") if they are not dispersed by liquid turbulence or do not move apart due to axial collisions with other bubbles. Bubble coalescence occurs instantly only if the leading bubble is sliding on the heated surface whereas the trailing bubble is not, as the impact between bubbles is presumably stronger due to larger velocity differences. In the present work, it was assumed that the impact following bubble lateral collision is not strong enough to cause rupture of the vapour-liquid interface, so that coalescence following bubble lateral collision was not modelled. Liquid temperature In the proposed bubble-tracking model, the liquid temperature Tl depends on the distance from the inner heated wall r and obeys the following law (Sekoguchi et al., 1980):

 Ti − Tl  r =  Ti − To  R o − Ri 

(2)

1/m

Liquid temperature profiles at different axial locations along the channel are obtained using steadystate values of the average cross-sectional enthalpy . The liquid temperature profile must be such that the liquid specific enthalpy il fulfils the condition:

i

∫ ((1 − α ) ρ A

l

+ αρ g ) dA = ∫ (1 − α ) ρ l i l dA + ∫ αρ g i g dA A

(3)

A

where α denotes the local void fraction and integrals are calculated over the channel cross-section. Steady-state values of at different axial locations are obtained from thermal energy balances. The gas phase is assumed to be at saturation conditions. 3/12

The temperature of the heated wall is assumed to increase until it reaches a value determined from a correlation by Shah (as cited by Kandlikar, 1998):

(

q'' = 230(Gi lg ) − 0.5 h1Φ (T w − T sat )

)

(4)

2

where the single-phase heat transfer coefficient h1Φ has to be calculated from the Dittus-Boelter correlation (Collier, 1981) and ilg indicates the difference between vapour and liquid specific enthalpies at saturation conditions. Before reaching that value, the wall temperature is calculated from the relation:

q ' ' = h1Φ (T w − < Tl > )

(5)

where denotes the liquid temperature, averaged over the channel cross-section. Partitioning of wall heat flux In the proposed model, the wall heat flux is partitioned as follows:

q'' = C l h1Φ (Tw − < Tl > ) + q'' nucl + q'' slid

(6)

The first term on the r.h.s. of Eq. (6) represents heat transfer due to single-phase forced convection. The factor Cl accounts for the portion of the heated surface not covered by bubbles. The term q''nucl denotes the heat flux consumed for bubble nucleation. The term q''slid, which denotes the heat flux consumed for growing of bubbles that slide on the heated surface, is determined as in the work of Tsung-Chang and Bankoff (1990):

q ' ' slid =

2k l (T w − T sat )

(7)

(πυ l )1/2

Bubble nucleation According to many experimental results, the bubble departure diameter db varies along the heated wall. The experimental data of Unal (1967), Zeitoun and Shoukri (1996), Bartel (1999), Prodanovic et al. (2002) and Situ et al. (2004) showed that at low pressure (of about 1 bar), bubble sizes significantly increased with decreasing subcooling along heated channels. The generated bubbles of spherical or ellipsoidal shape were relatively large (they reach maximum diameters of a few millimetres). In the proposed bubble-tracking model, bubbles are nucleated at fixed nucleation sites randomly distributed over the heated surface, instantly reach departure size and assume an ellipsoidal shape. The bubble equivalent departure diameter dd is constant at each site. Bubble diameters over nucleation sites are distributed according to Gaussian distributions and are randomly generated in intervals [dd – 3s, dd + 3s], where dd denotes the local mean bubble equivalent diameter and s the local standard deviation. The mean bubble size was assumed to depend on local subcooling and was calculated in the same way as in the work of Končar et al. (2004), using Unal’s mechanistic model that describes the bubble departure diameter dd as a function of pressure, liquid subcooling, heat flux and liquid flow velocity. Since, in the present work, the heat flux in the considered experimental data is below the range of applicability of the correlation, the bubble diameter dd was multiplied by a coefficient Cbw to describe larger bubbles at low-pressure conditions. The frequency of bubble nucleation at individual sites is calculated from a correlation by Cole (1960, as cited by Ivey, 1967):  4 g ( ρl − ρ g )  f =   3d d ρ l 

(8)

1/ 2

4/12

Bubble evaporation and condensation In the proposed model, bubbles may further grow while part of them is still within the region near the heated wall where the temperature is higher than the saturation temperature. The interfacial heat transfer coefficient hint is calculated from a correlation already used by Mortensen and Trapp (1992):

hint =

(

kl 0.5 0.67 0.4 2 + (0.4Re b + 0.06Re b )Prl db

)

(9)

Bubbles that move laterally into the lower-temperature region collapse instantly if the liquid temperature at the bubble tip closest to the heated surface is lower than the saturation temperature. TWO-FLUID MODEL The two-fluid model of subcooled nucleate boiling flow consists of a dispersed phase (vapour bubbles) and a continuous phase (liquid flow) and is based on two sets of averaged transport equations. Each phase is treated as a continuum that fills up the entire control volume and is described by its own system of averaged equations for mass, momentum and energy. Averaged equations for both phases are coupled with additional closure relations describing the exchange of mass, momentum and energy at the interface as well as the turbulence within each phase. The twofluid model has been described extensively in many works (see, for instance, Ishii, 1975). In the present work, the general-purpose CFD (Computational Fluid Dynamics) code CFX-4.4 was used as a framework for solving the generic two-fluid model with additional relevant closure relations, introduced by the authors, which describe the mechanisms of phase change and lateral transport of mass, momentum and energy. The development of the CFX code was started at AEA Technology, and the code is currently being further developed by ANSYS, Inc. The discretisation of transport equations in the CFX4.4 code is based on a conservative finite-volume method. A nonstaggered grid arrangement is employed, where all variables (velocity components and scalars) are stored in the geometrical centres of control volumes (cells) that fill up the considered flow domain. The two-dimensional two-fluid model of convective boiling flow, which is used in the present work, is extensively described in another reference (Končar et al., 2004). As, in the model proposed in the present work, the bubble-tracking model uses solely the results of turbulence modelling from the two-fluid model, the description is limited to the features concerning modelling of dynamical phenomena. Turbulence modelling Due to the lower density of vapour, it is commonly assumed that, in nucleate boiling flow, the motion of the dispersed vapour phase follows the fluctuations in the continuous liquid phase (Kurul and Podowski, 1991). Accordingly, the turbulence stresses are modelled only for the liquid phase, whereas the vapour phase is assumed to be laminar. In the present work, the following option from the CFX4.4 code was applied: turbulence in the liquid phase was modelled using a k–ε model with an additional term describing the bubble-induced turbulence. Shear and bubble-induced turbulence are linearly superimposed, according to an assumption from Sato et al. (1981), where the effective eff viscosity µl of the continuous liquid phase is expressed as: (10)

µ leff = µ l + µ lturb + µ lb b

The bubble-induced turbulence viscosity µl in the liquid phase depends on the vapour phase volume fraction α, the local bubble diameter db and the relative velocity between the phases:

r

r

(11)

µ lb = C µb ρ lαd b u g − u l

The liquid phase turbulence in the near wall region is described by so-called “wall functions” (as, for example, in the work of Lahey and Drew, 2001). The turbulent boundary layer near the wall consists of a very thin laminar sub-layer adjacent to the wall and a so-called “buffer region”, which describes the flow between the laminar sub-layer and the core of the turbulent flow (modelled with the k-ε model). + The thickness of the laminar sub-layer is defined by the non-dimensional distance from the wall y = 5/12

11.23 (Launder and Spalding, 1972). In the laminar sub-layer, the axial liquid velocity is a linear + function of the distance from the wall. In the buffer region (11.23 ≤ y ≤ 300), the velocity profile is described by the logarithmic wall function. Interphase momentum transfer In the CFX4.4 code, the interphase transfer of momentum in bubbly flows is modelled with the following interfacial forces: drag force, lift force, turbulent dispersion force and wall lubrication force (AEA Technology, 2001). The interphase drag force is flow-regime dependent and is modelled according to a correlation by Ishii and Zuber (1979). The lift force on the liquid phase can be calculated as:

v v v v FL = αC L ρ l (u g − ul )× ∇ × (ul )

(12)

where CL is the lift force coefficient. The effect of diffusion of the vapour phase, caused by liquid phase turbulence, is described with the turbulent dispersion force:

v FTD = −C TD ρ l k ∇α

(13)

where CTD is a turbulent dispersion coefficient. Since the drainage rate of liquid is restrained by the noslip condition at the wall, the wall lubrication force acts in the radial direction away from the wall and prevents the accumulation of bubbles near the wall. The wall lubrication force is modelled using a correlation from Antal et al. (1991):

v v v αρ l (u g − u l )2  v d ⋅ max  C1 + C 2 b ,0 n FW = db yw  

(14)

where yw denotes the distance from the wall. According to Eq. (14), the wall lubrication force strongly depends on the local bubble diameter db. Modelling of local bubble diameter Bubble size determines the interfacial momentum transfer (drag force, wall lubrication force) as well as interfacial heat and mass transfer (evaporation and condensation). In the original CFX4.4 code, the local bubble diameter is treated as an adjustable parameter. In the present work, the local bubble diameter (in both radial and axial directions) was tabulated from results of simulations with the bubble-tracking model. COUPLING OF BUBBLE-TRACKING AND TWO-FLUID MODELS The bubble-tracking and the two-fluid models were coupled “off-line”. For a given set of boundary conditions (subcooling or inlet temperature, wall heat flux, mass flux or average liquid inlet velocity), an initial simulation with the bubble-tracking model was performed first. The probability of turbulent dispersion was constant over the entire channel (both in the axial direction and over the channel cross-section). The space-dependent time-averaged bubble equivalent diameter determined by the simulation was then tabulated and prescribed in the two-fluid input model. An intermediate simulation of the flow was then performed with the two-fluid model (using the CFX code), where the ensemble-averaged turbulent kinetic energy field in the liquid phase was calculated. A final simulation with the bubble-tracking model was then carried out, where the non-homogeneous probability of turbulent dispersion over the flow domain was based on the turbulent kinetic energy calculated with the two-fluid model. RESULTS AND DISCUSSION Experimental conditions Among experimental results on subcooled boiling in channels at low-pressure conditions, not many authors have measured the non-homogeneous radial distributions of two-phase flow parameters, such as void fraction and bubble size. Recently, these kinds of experiments have been carried out at Purdue University (USA) by Bartel (1999) and Situ et al. (2004), and at Seoul National 6/12

University (Korea) by Lee et al. (2002). The experiments of Bartel and Situ et al. seem to have been carried out on the same experimental device, but the authors have not been able to ascertain this with certainty. The experiments include measurements at different axial locations along the flow, so that the axial evolution of radial distribution of flow parameters was observed. The experiments were carried out at near atmospheric pressure in a vertical annulus with a heated inner rod. The diameter of the rod was 19.1 mm, whereas the inner diameter of the outer tube was 38.1 mm. The length of the heated part of the annulus was 1.5 m. At the beginning of the heated section, the flow was single-phase. Local void fraction was measured using an electrical conductivity probe technique. The data were collected simultaneously at different axial locations. The modelling of bubbles as rigid ellipsoids in the bubble-tracking model limits the void fraction, at which the proposed approach may be applied. In the present work, experiments where the maximum measured void fraction did not exceed 0.3 were simulated. Experimental conditions for experimental runs from Bartel, which were simulated with the proposed model, are presented in Table 1. The inlet subcooling refers to conditions at atmospheric pressure. Experimental conditions for experimental runs from Situ et al., which were simulated with the proposed model, are presented in Table 2. Table 1. Experimental conditions form Bartel (1999)

∆Tsub [°C]

q’’ 2 [kW/m ]

G 2 [kg/m s]

1

8.9

105

470

2

6.1

128

701

3

4.8

128

701

4

5.2

145

700

Run

Table 2. Experimental conditions from Situ et al. (2004) Run

Tinlet [°C]

q’’ 2 [kW/m ]

inlet [m/s]

G (calculated) 2 [kg/m s]

1

95.0

98.7

0.661

636

2

95.0

149

0.982

3

95.0

150

1.22

944 1173

Model parameters The present work is a synthesis of earlier simulations of experiments from Bartel and Situ et al. Basically, the same model was used and most of the physical parameters and adjustable coefficients are either equal or consistent. However, all the details have not been unified yet. At each nucleation site, the bubble departure diameter was constant but generated as a random variable with a Gaussian distribution. The value of the multiplicative coefficient in Unal’s correlation to calculate the local mean bubble departure diameter dd was set to 1.5 for Bartel’s experiment and 1.25 for run 1, 1.50 for run 2 and 1.75 for run 3 of experiments from Situ et al. For Bartel’s experiments, it was assumed that the smallest bubble diameter (lower boundary of the interval: dd – 3s) was always 0.0005 m, from which the interval upper boundary was calculated. For experiments from Situ et al., the lower boundary of the distribution dd – 3s was set to 0.5dd, whereas the upper boundary dd + 3s was set to 1.5dd.. However, the lower boundary could never be below 0.0005 m and the upper boundary never above 0.0025 m. The ratio of horizontal to vertical bubble axis before bubble detachment from the wall was set to 0.8, to approximate the inclination of bubbles while they are sliding on the walls. After detachment, the ratio was set to 1.2. Bubble radial motion (including detachment from the heated wall) consisted of finite steps of 1/20 bubble width. The factor Cr in Eq. (1) was set to 0.03. 7/12

In the simulation of Bartel’s experiments, it was assumed that bubbles slide along the wall for 2 a distance equal to twice the ellipsoid vertical axis before attempting detachment for G=470 kg/m s, 2 whereas for G=700-701 kg/m s, it was assumed that the “sliding distance” is equal to one vertical axis. For experiments from Situ et al., it was assumed that bubbles slide along the wall for a distance of twice their vertical axis before attempting detachment in all runs. The rest time during which bubbles stick together before merging was set to 0.02 s for the simulation of Bartel’s experiment, whereas in the simulation of experiments from Situ et al. it was calculated as in the work of Prince and Blanch (1990). In the initial simulations with the bubble2 tracking model, the probabilities of turbulent dispersion were set to 0.09 for G=470 kg/m s and 0.13 2 for G=700-701 kg/m s for Bartel’s experiments, and to 0.11 for run 1, 0.18 for run 2 and 0.21 for run 3 of experiments from Situ et al. The ratios of these probabilities correspond roughly to the ratios of bulk Reynolds numbers. In the final simulations, the local probability of turbulent dispersion was obtained by multiplying the local turbulent kinetic energy, obtained from the two-fluid model, by 10.0 The factor m in Eq. (2) (liquid temperature profile) was set to 4.0. The coefficient Cµb in Eq. (11) was set to 0.6, as recommended by Sato et al. (1981). The lift force coefficient CL in Eq. (12) was set to 0.1, which is adequate for weakly turbulent bubbly flow (Lahey, 1995). The turbulent dispersion coefficient CTD in Eq. (13) was set to 0.1 (Kurul and Podowski, 1991). The coefficient values in Eq. (14) were set as C1= –0.04 and C2= 0.08 to obtain a good agreement between calculated and experimental void fraction radial profiles. With this set of coefficients, the wall lubrication force acts within the region of two bubble diameters away from the heated wall. Results Figures 1-4 show experimental and simulated time-averaged void fraction radial profiles at different experimental conditions and different axial locations along the channel for experiments from Bartel (1999), whereas figures 5-7 show the same parameters for experiments from Situ et al. (2004). The coordinate z=0 corresponds to the beginning of the heated section. As bubbles instantly assume the shape of ellipsoids, which have their axis always vertical and touch the wall only with their tip, void fraction assumes zero values at the heated wall. In actual boiling flow, the void fraction at the heated wall is not zero due to the presence of hemispherical growing bubbles with their bases attached to the wall. However, the modelling simplification of the bubble shape does not significantly affect the void fraction distribution over the channel cross-section. The proposed model simulates the gradual increase and widening of the void fraction profile along the channel. In general, the agreement between measured and simulated profiles is good at the highest observed location, whereas the agreement at the lowest and middle location is variable.

0.20 z/Dh exp. theory 6.9 44.6 65.6

0.10

0.05

0.00 0.0

z/Dh exp. theory 6.9 44.6 65.6

0.15 Void fraction

Void fraction

0.15

0.10 0.05 0.00

0.2

0.4 0.6 (r / (Ro - Ri )

0.8

1.0

Figure 1. Simulated and experimental void fraction profiles (exp. from Bartel, Run 1)

8/12

0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 2. Simulated and experimental void fraction profiles (exp. from Bartel, Run 2)

0.25

0.30 z/Dh exp. theory 6.9 44.6 65.6

0.15 0.10 0.05

0.15 0.10

0.00 0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 3. Simulated and experimental void fraction profiles (exp. from Bartel, Run 3)

0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 4. Simulated and experimental void fraction profiles (exp. from Bartel, Run 4) 0.4

0.3 z/Dh exp. theory 31.3 52.6 89.4

0.2

z/Dh exp. theory 31.3 52.6 89.4

0.3 Void fraction

Void fraction

0.20

0.05

0.00 0.0

z/Dh exp. theory 6.9 44.6 65.6

0.25 Void fraction

Void fraction

0.20

0.1

0.2

0.1

0.0 0.0

0.0

0.2

0.4 0.6 (r / (Ro - Ri )

0.8

0.0

1.0

Figure 5. Simulated and experimental void fraction profiles (exp. from Situ et al., Run 1)

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 6. Simulated and experimental void fraction profiles (exp. from Situ et al., Run 2) 0.25

0.3 z/Dh exp. theory 31.3 52.6 89.4

0.2

initial simul. final simul.

0.20 Void fraction

Void fraction

0.2

0.1

0.15 0.10 0.05 0.00

0.0 0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 7. Simulated and experimental void fraction profiles (exp. from Situ et al., Run 3)

0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

Figure 8. Initial and final simulated void fraction profiles with bubble-tracking model (exp. from Bartel, Run 2)

As an illustration of the influence of the two-fluid model, Fig. 8 shows the calculated void fraction radial profiles after initial and final simulations with the bubble-tracking model for Run 2 from Bartel’s experiments. The influence of the two-fluid model is not significant, as only the calculated turbulent kinetic energy is applied to the bubble-tracking model. In future developments, additional variables from the two-fluid model, in particular the liquid velocity and liquid temperature profiles, will be transferred to the bubble-tracking model. 9/12

1.0

As an example of intermediate results of the procedure, Fig. 9 shows the radial profiles of bubble equivalent diameter obtained at different axial locations with the initial simulation with the bubble-tracking model. These values are used as input for the intermediate simulation with the twofluid model. Figure 10 shows the radial void fraction profiles, obtained from the two-fluid model but not used in the further procedure. The relatively good agreement between measured and calculated values illustrates the consistency of the procedure. Figure 11 shows the profiles of the turbulent kinetic energy at the same axial locations as the void fraction profiles in Figs. 2 and 8. One may observe the increase of the energy due to rising void fraction along the flow.

0.25

0.004

z/Dh exp. theory 6.9 44.6 65.6

0.003

Void fraction

Bubble eq.diam. [m]

0.20

0.002

0.15 0.10 0.05

0.001 0.0

0.00

0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 9. Bubble eq. diameter at different axial locations from initial simul. with bubble-tracking model (exp. from Bartel, Run 2)

0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

Figure 10. Experimental and simulated void fraction profiles, obtained with two-fluid model (exp. from Bartel, Run 2)

z/Dh 6.9 44.6 65.6

2

2

Turb.kin.energy [m /s ]

0.015

0.010

0.005

0.000 0.0

0.2

0.4 0.6 r / (Ro - Ri )

0.8

1.0

Figure 11. Turbulent kinetic energy obtained with two-fluid model (exp. from Bartel, Run 2)

CONCLUSIONS A model of subcooled convective boiling was developed by coupling a three-dimensional bubble-tracking model and a two-dimensional two-fluid model. The model was used to simulate experiments performed with water in vertical annuli with a heated central rod. A good overall agreement between calculated and measured radial profiles of void fraction at different axial locations along the channel was obtained. Although the model contains a number of adjustable parameters, the comparison of simulated results with experimental data indicates that the proposed approach captures the basic mechanisms that govern the development and evolution of subcooled nucleate boiling at near atmospheric pressure and relatively low void fraction. In future developments of the coupled model, additional variables calculated with the two-fluid simulation will be used in the bubble-tracking simulation. The model may also be extended to boiling flow in more complex geometries, including nuclear fuel assemblies. 10/12

1.0

ACKNOWLEDGMENTS The present work was financed by the Ministry of Higher Education, Science and Technology of the Republic of Slovenia (Research programme P2-0026 “Reactor engineering”). Greek letters α void fraction [dimensionless] 3 ρ density [kg/m ] 2 υ thermal diffusivity [m /s] µ viscosity [Pa s] 2 ν kinematic viscosity [m /s]

NOMENCLATURE A Dh G P Ro Ri T d g h i k p q'' r s

2

w y + y z

channel cross-section [m ] channel hydraulic diameter [m] 2 mass flux [kg/m s] probability [-] radius of annulus outer wall [m] radius of annulus inner wall [m] temperature [K] bubble equivalent diameter [m] 2 gravitational acceleration [m/s ] 2 heat transfer coefficient [W/m K] specific enthalpy [J/kg] thermal conductivity [W/m K] pressure [Pa] 2 heat flux [W/m ] distance from annulus inner wall [m] std dev. of bubble diam. distribution [m] velocity velocity in z-direction [m/s] distance from wall [m] non-dimensional distance from wall [m] axial coordinate [m]

Prl Reb

liquid Prandtl number = νl/ υl bubble Reynols number = wbl · db / νl

v u

Subscripts b bubble bl bubble relative velocity d bubble departure equivalent diam. g gas i annulus inner wall l liquid l∞ undisturbed liquid velocity o annulus outer wall sat saturation conditions sub subcooling w heated wall Superscripts eff effective turb turbulent

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