A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle

Annals of Nuclear Energy xxx (xxxx) xxx

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A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle Yang Liu a, Wei Liu b, Jianqiang Shan a,⇑, Bo Zhang a, L.K.H. Leung c a

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi Province, China Nuclear Power Institute of China, State Key Laboratory of Reactor System Design Technology, Chengdu 610213, Sichuan Province, China c Canadian Nuclear Laboratories, Chalk River, Ontario K0J 1J0, Canada b

a r t i c l e

i n f o

Article history: Received 14 April 2019 Received in revised form 17 September 2019 Accepted 25 September 2019 Available online xxxx Keywords: CHF Mechanistic model Bubble crowding Bundle

a b s t r a c t A mechanistic model has been developed for predicting critical heat flux (CHF) in subchannels of a bundle. It is based on the bubble crowding phenomena and takes into the account of effects of velocity distributions, turbulent mixing intensity and Prandtl mixing length on CHF in subchannels within the bundle. Several factors are introduced to represent these effects. The model has been applied together with the ATHAS subchannel code for assessing against CHF data obtained with high-pressure water through a 5
5 rod bundle having a uniform axial-power profile. Good agreement has been observed between predictions using this mechanistic model and experimental CHF values at operating conditions of Pressurized Water Reactors (PWRs). The average error between predicted and experimental CHF values is 0.76% (an average absolute error of 9.18%) with a standard deviation error of 11.86% for 417 data points. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Accurate prediction of critical heat flux (CHF) is essential to ensure safe operation of flow-boiling equipment, such as evaporators, boilers and heat exchangers. It is particularly of concern to the operation of water-cooled nuclear reactors, where safe operating power envelop and margin must be established to maintain fuel, cladding and core integrities. CHF has been extensively studied in the past 6 decades. Many empirical methods and mechanistic models were developed for predicting CHF in tubes (Biasi et al., 1967; Bowring, 1972; Weisman and Pei, 1983; Galloway and Mudawar, 1993; Lee and Mudawar, 1988). The look-up table approach (Groeneveld et al., 2007) has been widely adopted due to its accuracy and simplification to apply. Prediction of CHF for rod bundles, however, is more complex than that for tubes. The complex geometry, flow and enthalpy imbalances within the bundle, and spacing devices challenge the development of prediction methods. Nevertheless, several empirical methods were proposed for predicting CHF in bundles. These methods can be separated into two categories: 1) Tube-data-based methods: These methods were based on tube data and applied corrections for bundle applications. The W-3 correlation proposed by Tong (1986) includes two ⇑ Corresponding author. E-mail address: [email protected] (J. Shan).

corrections for the cold wall and grid effects in bundles. The look-up table (Groeneveld et al., 2007), on the other hand, includes several factors to account for changes in hydraulic diameter, heated length, spacer grid and axial power profile on CHF. A bundle specific correction factor has also been introduced but is applicable for a specific geometry. Another method applied the subchannel code in establishing the local flow conditions for developing a subchannel-data-based method, which is similar to the tube-data-based approach (Chun. et al., 1998). 2) Bundle-data-based methods: These methods were based directly on experimental data of specific bundles and hence provided in general the best prediction accuracy (Motley et al., 1976; Guo-han Chai et. al., 2003; Leung et al., 1998). However, each method is applicable to a specific bundle geometry and its grid (or spacer) configuration. Therefore, its application is rather limited. In view of limitations of tube- and bundle-data-based methods, a mechanistic CHF model, by taking into the account of physical CHF mechanisms, would provide the flexibility in predicting CHF for new bundle concepts or configurations, such as those proposed for small modular reactors (SMRs). Having said that, the prediction accuracy of the bundle-specific correlation is anticipated to be better than the model within the range of conditions of the bundle configuration for its database. However, the mechanistic model would provide appropriate physical trends for extension, if

https://doi.org/10.1016/j.anucene.2019.107085 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Nomenclature Symbol a D Dp F G G3

Units coefficient hydraulic diameter of tube or bundle m average bubble diameter m fraction of total heat flux for evaporation total axial mass velocity kg/m2s lateral mass velocity from core to bubbly layer due to turbulence kg/m2s hf saturated liquid enthalpy kJ/kg hfg latent heat of evaporation kJ/kg enthalpy of liquid kJ/kg hl hld enthalpy at point of bubble detachment kJ/kg ib turbulent intensity k coefficient L Heated Length m le Prandtl mixing length m Mea. CHF Experimental CHF value MW/m2 Pre. CHF Predicted CHF value MW/m2 qCHF CHF value kW/m2 qexp Experimental CHF value kW/m2

necessary, and is applicable to coolant other than water (Bruder et al., 2015). Validation of the model is required to improve the confident for these applications. CHF occurs generally through the departure from nucleate boiling (DNB) or liquid-film dry out (or simply dryout) mechanisms. DNB is encountered mainly at subcooled and low-quality conditions (corresponding to those of interest to pressurized water reactors or PWRs) while dryout at high-quality conditions (corresponding to those of interest to pressurized heavy water reactors or PHWRs). The physical mechanism for dryout is relatively well understood and hence reliable mechanistic models are available (Weisman, 1991). While the basic mechanism for DNB is known, details remain uncertain, even for simple tubes (Bricard et al., 1997). Several analytical models have been developed for DNB-type CHF in tubes, based on the bubble-crowding mechanism (Weisman and Pei, 1983; Weisman and Ying, 1985; Ying and Weisman, 1986; Lim and Weisman, 1990), the liquidsublayer-dryout mechanism (Lee and Mudawar, 1988; Celata et al., 1994; Katto, 1990; Katto, 1990); or the interfacial lift-off mechanism (Galloway and Mudawar, 1993). The bubblecrowding mechanism (Weisman and Pei, 1983) assumes the CHF occurrence when bubbles on the heating wall obstruct the liquid in the bulk flow from wetting the heated surface effectively. The liquid-sublayer-dryout mechanism (Lee and Mudawar, 1988) assumes the CHF occurrence when the bulk flow could not replenish the evaporating sublayer under the vapor bubble. The interfacial lift-off mechanism (Galloway and Mudawar, 1993) postulates the instability of the bubble interface prevents the continuous vapor layer from wetting the heated surface. The bubble crowding mechanism is more relevant than others for PWR applications and has been the focus of many studies. Weisman and Pei (1983) initiated the development of a CHF model for tubes. Ying and Weisman (1986) modified the model to accommodate the non-uniform void profile in a tube and extended the CHF prediction for void fractions up to 0.8. In addition, they revised the calculation for bubble diameters and included the slip ratio in the bubble layer to improve the CHF prediction at low mass-flow rates. Ying and Weisman (1986) revised the model for annuli with one unheated wall. Weisman et al. (1994) modified the model for tubes containing twisted tapes. Weisman and Illeslamlou (1988), on the other hand, extended the model to high-subcooling

qpre r0 RMS Us S v’ 1=2 ðv 0 Þ2

v0

x x1 x2 xin y

a2 d w

ql qg q


Predicted CHF value kW/m2 outer radius of rod bundle or inner radius of tube m root-mean-square error frictional velocity m/s sensitivity index radial fluctuating velocity m/s root mean square value of v’ mean value of v’ m/s average quality (across entire flow area) average quality in core layer average quality in bubble layer Inlet quality the distance from the wall m void fraction in bubble layer at CHF thickness of bubble layer m miscellaneous function liquid density at bulk temperature kg/m3 vapor density kg/m3 average density (across entire flow area) kg/m3

conditions based on the energy balance at the outer edge of the bubble layer in round tubes. Chang and Lee (1989) revised the calculation of the lateral mass velocity from the core to the bubble layer at low qualities in uniformly heated tubes in their CHF model. Kwon and Chang (1999) introduced a drag force due to the roughness of wall-attached bubbles in the momentum balance, which determines the limiting transverse interchange of mass flux crossing the interface of the wall bubbly layer and the core. Furthermore, the bubbly layer was assumed to be a single layer of wall-attached bubbles, which acts as an equivalent surface roughness. The critical void fraction at the bubble layer was represented with an exponential function related to the quality, which was determined from CHF data for tubes. Kodama and Kataoka (2002) expressed the critical void fraction at the bubble layer in terms of the channel-average void fraction, which was also determined from CHF data for tubes. Kinoshita et al. (2001) considered spherical bubbles in the bubble layer at high velocity and subcooling conditions. These bubbles were assumed contacting each other in a cubical configuration. Interference of bubbles was taken at the ratio of bubble diameter to bubble distance of 0.5. The critical void fraction at the bubble layer was determined to be p/12. Recently, Xie and Yang (2019) improved the model for rifled tubes and proposed a correlation for the critical void fraction at the bubble layer based on their experimental data. Pan and Li (2016) adopted the approach of Kwon and Chang (1999) in developing a new correlation for bubble departure diameter, which accounted for the effect of buoyancy on CHF in tubes at high pressures and low flow rates. Ahmad (2012) modified the Weisman and Pei model for low mass fluxes through the optimization of the coefficient ‘‘a” in the turbulent intensity, ib, with tube data. Rui and Xiaojin (2016) extended the model to inclined tubes by introducing a correction factor, based on the inclination angle, in the calculation of the bubble velocity and bubble layer thickness. Among the three types of the mechanistic model (i.e., bubble crowding, liquid-sublayer dryout, and interfacial lift-off), the bubble crowding model is the most appropriate for predicting CHF at high pressures and low subcoolings (Chang and Baek, 2003; Chun et al., 2000). Bloch et al. (2016) showed that, with increasing void fraction, the local flow pattern near the heated wall corresponds to the bubble crowding phenomena. Their probe measurements

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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disputed the possibility of the interfacial lift-off phenomena. There were no direct visualization measurements to support the sublayer-liquid-dryout phenomena at high pressures and low subcoolings. Therefore, the bubble crowding model has been selected as the basis for developing a mechanistic model for bundles. Despite of the experimental support, challenges (such as establishing the critical void fraction at the heated surface for the CHF occurrence) remain in advancing the model. Bundle-data-based correlations are currently applied in PWR analyses. These correlations are applicable to specific bundle and grid configurations for the range of experimental conditions. Extensions to other configurations or outside the experimental range are not recommended or justifiable. Mechanistic models, on the other hand, are theoretically based and provide phenomenologically correct parametric and asymptotic trends. Therefore, these models are more applicable to various bundle and grid configurations as well as a wide range of flow conditions than correlations. In particular, these models are ideal for supporting fuel assembly designs for advanced nuclear reactors (such as SMRs) or fuel assembly optimization for current generations of nuclear reactors. The objective of this study is to develop a generalized DNB-type CHF model for subchannels in rod bundles, based on the bubblecrowding concept (Weisman and Pei, 1983). This current development is more relevant to PWR assembly analyses than those for tubes, annuli, rifled, inclined or twisted tube. It takes into the account of the influence of rod bundle geometry on the bubble crowding model, which has not been applied in previous models. Analytical models have been introduced for separate effects of the radial velocity, turbulence intensity and heat-transfer enhancement due to grids in subchannels. Furthermore, the new model has been validated against experimental data for a bundle obtained under the PWR operating conditions. Applying the model based on local conditions evaluated with a subchannel code would facilitate CHF prediction in a bundle.

Bubble layer

Bulk flow

Bubble crowding CHF(+)

Radial liquid heated wall

CHF(-) Radial liquid

flow Fig. 1. Typical flow regime patterns in vertical flow (Weisman and Pei, 1983).

Pei, 1983). The lateral mass velocity from the core to the bubble layer due to turbulence was expressed in terms of the total axial mass velocity, G, the turbulent intensity at the interface between the bubbly layer and the core, ib, and a miscellaneous function, w:

G3 ¼ Gib w 2. Bubble-crowding CHF model for tubes The bubble-crowding model separates the flow area into two regions: the bubble layer and the bulk flow, as illustrated in Fig. 1. It takes into the account of the influence of turbulent intensity and wall-forming bubbles on the mass exchange of liquid phase between these two regions. Bubbles generated at the wall are considered hindering the bulk-liquid phase from pulsating into the bubble layer, where phase change occurs, to replenish the liquid. As the bubble population near the wall increases with increasing heat load, it becomes difficult for turbulent pulsation to transport sufficient liquid from the bulk flow to the wall for cooling, eventually leading to the occurrence of CHF. Based on the mass conservation at the CHF point, Weisman and Pei (1983) expressed the CHF as

qCHF ¼ hfg G3

ðx2  x1 Þ F

ð1Þ

where hfg is the latent heat of evaporation, G3 is the lateral mass velocity from the core to the bubble layer due to turbulence, x2 is the average quality at the bubble layer and x1 is the average quality at the core layer. The fraction of total heat flux for evaporation, F, was calculated with (Lahey and Moody, 1977):

F¼

hl  hld hf  hld

ð2Þ

The average quality at the bubble layer was calculated using the critical void fraction, a2, which was assumed as 0.82 (Weisman and

ð3Þ

Weisman and Pei (1983) applied the turbulent radial-velocity fluctuations in a tube (Laufer, 1953) and considered the radial turbulent velocity distribution independent of the Reynolds number (see Eq. (5)), as concluded by Lee and Durst (1980). They expressed the turbulent intensity at the interface between the bubbly layer and the core as

ib ¼ ðv 0 Þ2

1=2


 q
G

ð4Þ

The root-mean-square value of m’, (m’)2, is defined as

½ðv 0 Þ2

1=2

=U s =ðle =r0 Þ ¼ 2:9


0:4 r0 y

ð5Þ

where Us is the frictional velocity, le is the Prandtl mixing length, r0 is the inner radius of the tube, and y is the radial distance from the wall (i.e., r0-r). Weisman and Pei (1983) defined the Prandtl mixing length as

le ¼ Ky ¼ 0:4y

ð6Þ

where K is the empirical coefficient and is 0.4 for tubes or flat plates. Weisman and Pei (1983) assumed that the ratio of two phase to single-phase turbulent intensity is independent of the radial position and introduced a two-phase factor, F1, in Eq. (5):

½ðv 0 Þ2

1=2

=U s  ¼ 2:9F 1


0:4 r0 ðle =r0 Þ y

ð7Þ

The frictional velocity, Us, is expressed in terms of the friction factor, f’, which is a function of the Reynolds number in the turbulent region (Weisman and Pei, 1983):

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Us ¼


0 0:5 f G ¼ 2 q


0:046Re0:2 2

!0:5

G G ¼ ð0:023Þ0:5 Re0:1 q
q


ð8Þ

Applying Eqs. (6) and (8), Eq. (7) can be rewritten as

ðv 0 Þ2 G

1=2

¼


0:6 0:176 y F 1 Re0:1 r0 q


ð9Þ

Weisman and Pei (1983) indicated evaluating the turbulent intensity at a distance from the wall:

F 2 le ¼ kDp

ð10Þ

where Dp is the average bubble diameter, F2 is a two-phase factor that accounts for the effect of bubble motion on the fluctuation velocity, and k is coefficient. The distance from the wall for the bubbly layer-core interface, yc, is determined by applying Eq. (6) in Eq. (10), and is expressed as



yc ¼

 kDp 0:4F 2

ð11Þ

Replacing ‘‘y” in Eq. (9) by ‘‘yc”, the turbulent intensity at the interface between the bubbly layer and the core is expressed as:

ib ¼ ðv 0 Þ2

1=2

#
0:6
0:6 "
 q
k Dp F1 ¼ 0:176Re0:1 0:4 G r0 F 0:6 2

ð12Þ

Both F1 and F2 were introduced to take into the account of the two-phase effect, Weisman and Pei (1983) related these two factors in the form:

"

F1

F 0:6 2

#

"

ql  qg ¼ 1þa qg

ib ¼ ðv

1=2

ð13Þ

#
0:6
0:6 "
 ql  qg q
k Dp 0:1 1þa ¼ 0:176Re 0:4 G r0 qg ð14Þ

where a and k are coefficients derived from CHF data, Dp is the average bubble diameter, ql is the liquid density at bulk-fluid temperature, and qg is the vapor density. Weisman and Pei (1983) assessed their model against experimental CHF data obtained with water in a uniformly heated tube. The assessment resulted in about a 10% standard deviation of the ratio between experimental and predicted CHF values for about 1500 data points with maximum void fractions up to 0.6. 3. Bubble-crowding-based CHF model for subchannels in a bundle The bubble-crowding-based CHF model for subchannels in a bundle is based on the Weisman-Pei model for tubes, but takes into the account of the complex flow, enthalpy and geometric configurations in subchannels. Key parameters in the Weisman and Pei (1983), which are related to the geometric changes from a tube to a subchannel, are established through a sensitivity analysis. Bricard et al. (1997) introduced a sensitivity index, S, to assess the sensitivity of an intermediate parameter, X, to CHF. The sensitivity index is defined as:

X 0 @qc S, qc ðX 0 Þ @X

in assessing the impact of various parameters. Table 1 lists average sensitivity indexes for four key parameters of the model. The prediction of CHF is more sensitive to the critical void fraction in the bubbly layer, a2, and the radial turbulent intensity factor at the interface, ib, than the average bubble diameter, Dp. Therefore, the analysis focuses on developing models for the critical void fraction in the bubbly layer, a2, and the radial turbulent intensity factor at the interface, ib, for subchannels in the bundle. The bubble-crowding-based CHF model is applicable to the DNB-type of CHF mechanism encountered in bubbly, slug and churn flows. Lahey and Moody (1993) indicated the occurrence of the slug/churn flow transition over the range of global void fractions from 0.65 to 0.80. The proposed model is considered valid for conditions up to the slug/churn flow transition at the maximum global (subchannel cross-sectional average) void fraction of 0.7. Extension of the tube-based bubble crowding model to rod bundles takes into the account of the geometry effect on various modelling parameters. Details on the development are presented in sections below. Overall, the sensitivity analysis of the tube-based model (see Table 1) indicated that the CHF prediction depends strongly on the turbulence intensity factor, ib, among key parameters. Therefore, the model development has been focusing on the derivation of this factor for bundles by considering the impact of axial and radial velocity distributions as well as the CHF enhancement effect of the grid.

!#

Eq. (12) is rewritten into 0 Þ2

X, would result in a CHF variation of 10%). In view of the dependency of the sensitivity index on experimental conditions, Bricard   et al. (1997) applied the average value of the sensitivity index, S,

ð15Þ

where X0 is the value of X for the proposed closure relation at experimental conditions. An absolute value of the sensitivity index at unity (i.e., |S| = 1) signifies the direct impact of the uncertainty in the parameter on CHF (i.e., an uncertainty of 10% on the parameter,

3.1. Turbulent intensity at the interface between the bubbly layer and the core The turbulent intensity at the interface between the bubbly layer and the core depends strongly on the radial and axial velocity distributions (see Eq. (4)), which includes both the radial and axial variations. In addition, the enhancement in turbulent intensity due to a localized structure, such as a twisted tape (Weisman et al., 1994) or a spacer grid in fuel assemblies, must be considered. Furthermore, parameters in Eq. (4) were established mainly from characteristics of single-phase flow. The impact of boiling on these parameters must be considered. 3.1.1. Radial velocity distribution Weisman and Pei (1983) applied the radial velocity distribution in a tube, which was established by Lee and Durst (1980) using the experimental data of Laufer (1953) (see Eq. (4)). The complex variation of subchannel configurations in a bundle has a significant impact on the radial velocity distribution. Krauss and Meyer (1996) investigated the turbulent air flow in a wall subchannel of a heated 37-rod bundle (pitch to diameter ratio, P/D, of 1.12 and width-to-diameter ratio, W/D, of 1.06). Measurements were obtained with a hot-wire probe with x-wires and a temperature wire. The radial fluctuating velocity was measured in a wall channel. Krauss and Meyer (1996) observed the decay in radial fluctuating velocity from 0.9 at the near-wall region to 0.6 at the

Table 1 CHF sensitivity to uncertainty in closure relationships (Bricard et al., 1997). X Thickness of the bubble layer, d Turbulent intensity at interface between bubbly layer and core, ib Critical void fraction, a2 Average bubble diameter, Dp




jSj 0.56 0.92 1.9 0.56

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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centerline of the tube, which differs from measurements in a tube (Laufer, 1953). Trupp and Azad (1973) measured velocity distributions for fully developed turbulent flow of Reynolds numbers from 12,000 to 84,000 through triangular-array-rod bundles of three different rod spacings (i.e., P/Ds of 1.20, 1.35 and 1.50). They observed higher radial fluctuating velocity in the bundle than that in a tube at the near-wall region, but approaches that in the tube at the centerline. This difference at the near-wall region signifies a change in mass exchange of the bubble layer with the bulk flow region between a subchannel and a tube impacting the CHF. Measurements for the P/D of 1.35 (which is close to that of the PWR fuel assembly) of Trupp and Azad (1973) are applied in establishing the radial turbulence fluctuation in the subchannel. Fig. 2 compares measurements of Trupp and Azad (1973) and those of a tube (Lee and Durst, 1980). In addition, parabolic fits of all and near-wall (up to radius ratios, r/r0, of 0.5) measurements of Trupp and Azad (1973) are also shown. In view of the CHF occurrence at the wall, the near-wall velocity fluctuation is of the most interest and is expressed as

qffiffiffiffiffiffiffiffiffiffiffi

0:538 ðv 0 Þ2 le r0 ¼ 2:318 = Us r0 y

ð16Þ

Assuming the ratio of two-phase to single-phase turbulent intensity independent of the radial position, the near-wall velocity fluctuation for a subchannel is rewritten as

qffiffiffiffiffiffiffiffiffiffiffi
0:538
 ðv 0 Þ2 r0 le ¼ 2:318F 1 y r0 Us

ð17Þ

where F1 is the two-phase factor. 3.1.2. Axial velocity distribution Weisman and Pei (1983) applied the Karman three-layer approach for the velocity distribution, as defined in Eq. (18). The coefficient ‘‘C” was established from measurements of tubes or flat plates. Therefore, both coefficients ‘‘K” and ‘‘C” in the model of Weisman and Pei (1983) are not applicable to subchannels in a bundle.

8 þ þ 0 6 yþ < 5 > < UL ¼ y ; þ þ U L ¼ 5lny  3:05; 5 6 yþ < 30 > : þ 1 U L ¼ K lnyþ þ C; yþ P 30

ð18Þ

Rehme (1978) measured velocity distributions at the side channel of four parallel bundles with a P/D of 1.071. The velocity distribution at the distance from the rod surface was smaller than the logarithmic region of the smooth tube. Rehme (1978) suggested reducing the coefficient ‘‘C” from 5.5 to 5.0. The velocity gradient

Fig. 2. Comparisons of radial turbulence fluctuations and fitted representations.

5

for the side channel remains the same as that for the smooth tube. Therefore, the coefficient ‘‘K” for the smooth tube is applicable for the subchannel. The analysis of experimental velocity distributions obtained in Trupp and Azad (1973) has led to a coefficient ‘‘K” of approximately 0.4 for the bundle with the P/D of 1.35 but greater than 0.4 for bundles with other pitch-to-diameter ratios. Renksizbulut and Hadaller (1986) conducted experiments with bundles of P/Ds varying from 1.15 to 1.217. They derived a coefficient ‘‘K” of about 0.37 from their measurements. Hooper and Wood (1984) measured the velocity distribution of a bundle with the P/D of 1.107. They proposed a coefficient ‘‘K” of 0.418. In view of the scatter among recommended values for the coefficient ‘‘K”, the recommendation of Trupp and Azad (1973) for the bundle with the P/D of 1.35 (which is close to that of the PWR fuel assembly) is applied in the current model (i.e., the coefficient ‘‘K” of 0.4). 3.1.3. Effect of grid on turbulence intensity One of the fixtures in a fuel assembly, that has not been included in the Weisman and Pei model (Weisman and Pei, 1983), is the presence of spacer grids with mixing vanes. These grids enhance the turbulence intensity in the flow stream and the heat transfer accordingly. This enhancement effect is captured in the turbulence intensity factor, ib, of the present model. Fig. 3 illustrates schematically the enhancement effect of grids on CHF in a bundle with axially uniform heating (Groeneveld et al., 2015). The local CHF is reduced monotonically along the bare bundle (i.e., without grids) due to the increase in flow quality and reaches a minimum at the end of the heated length. In the presence of grids, the local CHF is enhanced at the grid location but decays gradually with increasing distance downstream from the grid. The CHF decays until encountering the next grid, where the enhancement recovers, or back to that of the bundle without grids if the distance between neighboring grids is long. The enhancement effect of the grid on CHF is attributed to the heat-transfer enhancement and is proportional to the enhancement in turbulence intensity at locations downstream of the grid. Yao et al. (1982) observed the enhancement in heat transfer and the decay at locations downstream of straight-type grids. They expressed the heat-transfer characteristic as


 Nusp x ¼ 1 þ 5:55e2 e0:13D Nu0

ð19Þ

where Nusp is the Nusselt number for the bundle with straight-type grids, Nu0 is the reference Nusselt number for the bare bundle without grid, x is the axial distance downstream from the grid, D is the

Fig. 3. Enhancement effect of grids on CHF in a bundle with axially uniform heating (Groeneveld et al., 2015).

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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hydraulics-equivalent diameter and e is the blockage-area ratio of the grid. The heat-transfer enhancement in a bundle due to grids is considered proportional to the turbulence-intensity enhancement. Therefore, the exponential decay in heat transfer is representative to that in turbulence intensity. Yang and Chung (1998) and Takuji et al. (Nagayoshi and Nishida, 1998) applied a similar approach in analyzing changes to radial and axial turbulence velocity distributions due to grids. Takuji et al. (Nagayoshi and Nishida, 1998) studied the radial turbulent velocity distribution of the straight-type grid and observed a decay in radial turbulence intensity factor downstream of the grid. The normalized velocity fluctuation reaches a maximum value at the grid and decays with increasing distance from the grid. It approaches the fluctuation for the bare bundle after a distance of approximately 10 times the hydraulic-equivalent diameter from the grid. Takuji et al. (Nagayoshi and Nishida, 1998) expressed the change in radial turbulence intensity factor as

0qffiffiffiffiffiffiffiffiffiffiffi1 ðv 0 Þ2 @ A Us

0qffiffiffiffiffiffiffiffiffiffiffi1 ðv 0 Þ2 A ¼ 1 þ 6:5e2 e0:27Dx =@ Us

ð20Þ

(Kinoshita et al., 2001) assumed a cubical configuration of contacting bubbles and considered a mutual interference of bubbles at the diameter-to-distance ratio of 0.5. They established the critical void fraction as p/12. Kwon and Chang (Kwon and Chang, 1999) correlated the critical void fraction in terms of the quality at the outlet, which is expressed as

a2 ¼ 0:83  0:29expð4:71xout  1:89Þ

ð24Þ

On the other hand, Kodama and Kataoka (Kodama and Kataoka, 2002) expressed the critical void fraction as a function of the average void fraction; a, at CHF in the channel. They derived a correlation from the tube CHF data, which is expressed as

a2 ¼ 0:77a0:25

ð25Þ

Styikovich et al. (Styrikovich, et al., 1970) measured void fractions at CHF locations and observed a range of critical void fraction from 0.3 to 0.95. In view of the scatter in establishing the critical void fraction, the recommendation of Weisman and Pei (Weisman and Pei, 1983) (i.e., the value of 0.82 for the critical void fraction, a2) is adopted in the current model.

grid

The radial turbulence intensity factor, ib, is modified with a grid correction factor, Fgrid, to include the enhancement effect. It is expressed as



ibuse ¼ ib F grid x

F grid ¼ 1 þ a2 e2 eb2 D

ð21Þ

The coefficients a2 and b2 are grid dependent. Experimental data of Yao et al. (1982) indicated the decay in heat-transfer enhancement due to grid down to the reference value (i.e., a bundle without grids) after 25–30 times the hydraulics-equivalent diameter. Based on these data, the coefficient b2 has been established as 0.13. The coefficient a2 has been derived from data of Takuji et al. (Nagayoshi and Nishida, 1998) as 6.5. 3.1.4. Turbulence intensity due to boiling The turbulence intensity factor, ib, expressed in Eq. (14), is based primarily on single-phase parameters. Weisman and Pei (Weisman and Pei, 1983) introduced the coefficient ‘‘a” to account for the enhancement in turbulence intensity for two-phase flow, which is expressed as


a1 G a ¼ 0:135 Gcr

ð22Þ

where Gcr is the reference mass flow rate at 970,000 kg/m2/h. Lim (Lim, 1988) indicated a slight pressure dependency for the coefficient ‘‘a”, based on their experimental data. To capture the pressure effect, the coefficient ‘‘a” has been revised to


a1
b1 G P a ¼ 0:135 Gcr Pcr

ð23Þ

where Pcr is the reference pressure at 1450 kPa. 3.2. Critical void fraction in bubble layer The critical void fraction in the bubble layer, a2, corresponds to the volume fraction of steam in the bubbly layer for CHF to occur. It has mainly been established through assumptions due to a lack of relevant experimental data. Weisman and Pei (Weisman and Pei, 1983) proposed a value of 0.82 for the critical void fraction, which was based on the maximum packing density of ellipsoid with an axis ratio of three to one. The critical void fraction was applied to calculate the quality at the bubbly layer, x2. Kinishita et al.

4. Assessment of the new bundle CHF model The new mechanistic CHF model, described in Section 3, has been assessed against experimental data obtained with water flow through a 5
5 rod bundle with axially uniform heating at typical operating conditions of PWRs. Since it is developed for predicting CHF in subchannels of a bundle, the local flow conditions in subchannels of the test bundle are evaluated using the subchannel code, ATHAS (Liu, 2014). The final CHF prediction equation of the model in this paper is as follows (as shown in the first equation of Appendix A)

qpre ¼ hfg ib - use wGðx2 - x1 Þ


 hf  hld hl  hld

ð26Þ

The assessment is based on a comparison of predicted CHF value using the model against the experimental CHF value, in term of the CHF ratio (or referred to as the DNB ratio or DNBR) defined as


 hf  hld =qexp DNBR ¼ qpre =qexp ¼ hfg ib - use wGðx2  x1 Þ hl  hld

ð27Þ

Appendix A describes the calculation procedures for the model. While most parameters in the model are based on local flow conditions, local heat flux is required in the evaluation of the onset of significant void and is based on the experimental heat flux (i.e., qexp). 4.1. 5
5 Rod-bundle experiments The experiments were performed at the Nuclear Power Institute of China (NPIC) with water flow over a 5
5 rod bundle simulating the PWR fuel assembly with flat mixing vanes. Data covered mainly the low-quality DNB-type CHF, but a few points were obtained at subcooled CHF conditions. Three test bundles were constructed for the experiments. Two of these bundles simulated a PWR fuel assembly with nine hot rods at the central locations and 16 cold rods at the peripheral locations inside a square frame of 66.1 mm in width. Each rod had an outer diameter of 9.5 mm. Spacing between heated rods and between the cold rod and the frame was 3.1 mm (i.e., the rod pitch was 12.6 mm). One of these bundles was heated over a length of 2438 mm (or 8 ft) and the other of a length of 3657 mm (or 12 ft). Fig. 4 illustrates the rod configuration of the 5
5 rod bundle

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Fig. 4. Test bundle setup in the NPIC 5
5 rod-bundle experiment (rods setup).

Fig. 5. Test bundle setup in the NPIC 5
5 rod-bundle experiment (grids setup).

experiment. Eight grids were installed along the axial length of the short bundle (four of these grids were simple support grids without mixing vanes (SS) while four with mixing vanes installed in the grid (MVG)). Thirteen grids were installed along the axial length of the long bundle at locations shown in the figure (six SS grids and seven MVG grids). Fig. 5 illustrates the grid configurations of the long and short bundles. The span between grids with mixing vanes was 0.56 m. A set of thermocouples was installed at the location 56 mm upstream of the end of the heated length (10 mm upstream of the last grid with mixing vanes). The coolant travelled vertically upward from the bottom to the top. Another test bundle in the experiments was equipped with a guide tube of 12.45 mm in outer diameter, which replaced the hot rod at the central position (see Fig. 4). The spacing between the hot rod and the guide tube was 1.625 mm. Thirteen grids were installed over the heated length of 3657 mm. Other geometric configurations of this bundle are the same as those of the long bundle described above.

Six sets of data have been selected from the database to assess the prediction accuracy of the bubble-crowding-based mechanistic CHF model for subchannels in the test bundles. Table 2 lists specific geometric parameters and number of data points in each dataset. These data covered bundles of both short and long heated lengths with and without the guide tube (GT). Table 3 lists the overall flow

Table 2 Selected 5
5 rod bundle data groups for assessment. Group ID

Grid number

Heated length (mm)

Data

Run-1 Run-2 Run-3 Run-4 Run-5 (with GT) Run-6 All data number

8 13 13 13 13 13

2438 3657 3657 3657 3657 3657

71 38 104 73 68 63 417

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Table 3 Range of CHF data selected in the assessment. Pressure (MPa)

Mass Flux (t/m2/s)

Subchannel Quality ()

Cross-Sectional Average Void Fraction ()

Inlet Subcooling (°C)

12.4–16.7

0.95–4.05

0.3

0.7

13.3–235.3

3.5

conditions covered in these data, which are typical for PWR operations. Pre. CHF (MW/m2)

1.0

0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Mea. CHF (MW/m2) Fig. 6. Comparison of predicted and experimental CHF values by model.

Parametric trends of predicted MDNBR in the bundles are examined against the experimental conditions (i.e., pressure, mass flux and quality) in Figs. 7–9. There is no apparent trend of predicted MDNBRs with pressure (Fig. 7) and mass flux (Fig. 8). However, predicted MDNBRs are smaller than 1 at low- and highquality regions but larger than 1 at qualities of about 0.18 (Fig. 9). The variation is probably attributed to the neglect of the 1.6 1.4 1.2 1.0 0.8 0.6 0.4

0.0 10000

#1=2

12000

13000

14000

15000

16000

17000

18000

Fig. 7. Variation of Predicted MDNBRs with Pressure.

1.6

#1=2

where MDNBRi is the MDNBR for each data point and N is the total number of data points. Fig. 6 compares predicted and experimental CHF values for 417 data points. The average value of MDNBRs, Rav, is 0.9924 (which is equivalent to an average error of 0.76%, and an average absolute error of 9.18%) with a standard deviation, SD, of 11.86%, and a root-mean-square error, RMS, of 11.87%. Almost all data are predicted within the ± 30% error band. Applying the original model of Weisman and Pei (Weisman and Pei, 1983) has led to an average error of 12% with a standard deviation of 22% for the same set of data points. This demonstrates the improvement in prediction accuracy using the current model.

11000

P (KPa)

1.4 1.2

MDNBR

N 1X RMS ¼ ðMDNBRi  1Þ2 N i¼1

±30%

1.5

0.2

N 1 X Rav ¼ MDNBRi N i¼1

"

2.0

0.5

MDNBR

CHF is predicted using the bubble-crowding-based mechanistic CHF model with the local flow conditions at each subchannel along the axial distance of the bundle evaluated using the subchannel code ATHAS. Local flow conditions in the subchannel are directly applied in the model to evaluate the CHF. The prediction uncertainty of CHF using the model is assessed against the experimental CHF based on the experimental conditions. This is referred to as the Direct Substitution Method (DSM) to determine CHF, which differs from the Heat Balance Method (HBM) in establishing the CHF power (see Appendix B for the description). The DSM method apples the experimental conditions and power as input values to the subchannel code ATHAS and calculates the local conditions and heat flux in each subchannel at each axial position. The local subchannel conditions are then implemented into the mechanistic model to predict the CHF. The predicted CHF at the critical subchannel is then compared against the experimental CHF to assess the prediction uncertainty. This approach differs from the HBM approach, where the CHF power is predicted for the bundle at possibly different critical quality and subchannel from the experiment (see Appendix B). One of the empirical factors in the subchannel code is the turbulent mixing coefficient, which depends on the specific mixing-grid configuration. A separate experiment provided the value of 0.066 for the type of mixing grid installed in the tested bundles. The predicted CHF value is compared against the experimental heat flux to determine the DNBR in the subchannel. The location with the minimum DNBR (or MDNBR) is considered the initial CHF point. The prediction accuracy of the model is assessed from the average value of predicted MDNBRs, Rav, the standard deviation on MDNBR, SD, and the root-mean-square error on MDNBR, RMS, which are defined as

N 1 X SD ¼ ðMDNBRi  Rav Þ2 N i¼1

SD(MDNBR)=11.86% RMS(MDNBR)=11.87%

2.5

4.2. Results

"

ERROR BAND±30% Rav(MDNBR)=0.9924

3.0

1.0 0.8 0.6 0.4 0.2 0.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

G (t/m2/h) Fig. 8. Variation of Predicted MDNBRs with mass flux.

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Y. Liu et al. / Annals of Nuclear Energy xxx (xxxx) xxx 1.6

3.5

1.4

3.0 2.5

Pre. CHF (MW/m2)

MDNBR

1.2 1.0 0.8 0.6 0.4

1.5

0.0 -0.10

0.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

±50%

1.0 0.5

0.00

SD(MDNBR)=42.4% RMS(MDNBR)=45.8%

2.0

0.2

-0.05

ERROR BAND±50% Rav(MDNBR)=1.1734

W-3_correlation 0.0

0.5

x (-)

F g ¼ 1:0 þ 0:6144
102

G

106

0:35

C TD 0:019

ð28Þ

where G is the mass velocity and CTD is the turbulent mixing coefficient, which is 0.066 in the ATHAS code. Fig. 10 compares the measured and predicted CHF values using the Biasi correlation for the bundle data. Overall, the Biasi correlation predicts the CHF for this set of data with an average error of 15.73%, an average absolute error of 26.22% (i.e., MDNBRs, Rav, of 1.1573) and a standard deviation, SD, of 33.0%, RMS, of 36.5%. The scatter of the predictions is significantly larger than that of the current model. The majority of the CHF values are predicted within the ±50% range.

3.5 3.0

Pre. CHF (MW/m2)

2.5

ERROR BAND±50% Rav(MDNBR)=1.1573 SD(MDNBR)=33.0% RMS(MDNBR)=36.5%

2.0 1.5

±50%

1.0 0.5 0.0

Biasi_correlation 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fig. 11. Comparison of measured and predicted CHF values using the W-3 correlation (Tong, 1986).

quality effect on the constant coefficient ‘‘a” in the turbulence intensity factor, ib, which is based only on mass flux and pressure. Based on the study of Carlucci et al. (2004), turbulence under twophase conditions is closely related to the quality and hence the quality effect must be considered in the coefficient ‘‘a”. At this point, there is a lack of data to establish the quality dependency. Further study is recommended to model the turbulent intensity factor. The underprediction of MDNBRs at high qualities (with void fractions higher than 0.7) is probably attributed to the change in flow pattern from the churn to the annular flow, where a liquidfilm dryout model is more applicable than the bubble-crowding model. The prediction accuracy of the model has been compared against those of the Biasi correlation (Biasi et al., 1967), which was developed for vertical uniform-heated tubes, and the W-3 correlation (Tong, 1986), which was developed for tubes and bundles. The grid effect correction in the W-3 correlation, Fg, is expressed as




1.5

Mea. CHF (MW/m2)

Fig. 9. Variation of Predicted MDNBRs with quality.




1.0

2.0

2.5

3.0

3.5

Mea. CHF (MW/m2) Fig. 10. Comparison of measured and predicted CHF values using the Biasi correlation (Biasi et al., 1967).

Fig. 11 compares the measured and predicted CHF values using the W-3 correlation for the bundle data. Overall, the W-3 correlation predicts the CHF for this set of data with an average error of 17.34%, an average absolute error of 28.99% (i.e., MDNBRs, Rav, of 1.1734) and a standard deviation, SD, of 42.4%, a RMS, of 45.8%. The scatter of the predictions is significantly larger than that of the current model. Most of the CHF values are predicted within the ± 50% range. It is clear that both the Biasi and the W-3 correlations are not valid for predicting CHF in this bundle configuration. The phenomenological model, on the other hand, provides significant better prediction accuracy than these correlations. 5. Conclusions A new mechanistic CHF model has been developed for subchannels in a bundle. It is based on the bubble-crowding concept for tubes but considered effects of subchannel geometry on axial and radial velocity distributions, boiling characteristics and spacer grids on CHF. The turbulence intensity factor in the model has been revised to capture the change in velocity profile between a tube and a subchannel. It includes also the enhancement effect due to a grid. The mechanistic CHF model has been assessed against experimental data obtained with water flow through 5
5 rod bundles having two different uniform-heated lengths and rod configurations (i.e., with and without a guide tube). Local flow conditions in subchannels of the bundle were calculated using the subchannel code ‘‘ATHAS”. Experimental CHF values were predicted with an average error of 0.76%, an average absolute error of 9.18% and a standard deviation of 11.86%, a root-mean-square error of 11.87% for 417 data points at PWR conditions of interest. The predicted MDNBRs appear independent of pressure and mass flux, but vary slightly with quality. This is probably attributed to the neglect of the quality effect in the turbulent intensity factor and the change in flow pattern, where the bubble-crowding phenomena is no longer applicable. Comparing the prediction accuracy of the mechanistic CHF model against those of the Biasi and the W-3 correlations, the model has been shown to perform better than these correlations with less bias and scatter in CHF prediction. Acknowledgement This work was financially supported by the National Key R&D Program of China (No. 2018YFB1900402).

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Appendix A:. Evaluation procedure of the mechanistic bubblecrowding model

hf  hld hl  hld

qb ¼

The bubble departure enthalpy, hld, is the liquid enthalpy at which bubbles begin to break away from the heated surface. It is calculated using the Levy model (Levy, 1966):

hld ¼ hf  DT ld C pf 8 0 6 yþb 6 5:0; DT ld ¼ Hq  fq 1=2 Pryþb > > db > Gð8Þ > > 

< 5:0 6 yþb 6 30:0; DT ld ¼ Hq  5:0 fq 1=2 Pr þ In 1 þ Prðyþb =5:0  1:0Þ db Gð8Þ > >  > q q > þ þ > : yb P 30:0; DT ld ¼ Hdb  5:0 f 1=2 Pr þ Inð1:0 þ 5:0PrÞ þ 0:5Inðyb =30:0Þ Gð8Þ

where

yþb ¼

0:015

mf

The single-phase heat transfer coefficient is calculated using the Dittus-Boelter correlation:

where the Reynolds number and Prandtl number are defined as:

GD

gf

; Pr ¼

"

cpf gf kf

! # ibuse G G G G x rv 0 ¼
¼ ibuse þ  q ql qg ql av g

The parameters, x1 and x2, depend on the critical void fraction,

a2, which is 0.82 as recommended by Weisman and Pei (1983). The densities are calculated with:

q2 ¼ ð1  a2 Þqf þ a2 qg

Re0:1





Dp D

0:462 "

1þa

ql  qg qg

!#

where k has a value of 0.58 and ‘‘a” is defined as

G a ¼ 0:135 Gcr

a1


P Pcr

b1

;a1 ¼ 0:3; G > Gc ; a1 ¼ 0:6;G < Gc ; b1 ¼ 2:0

with Gcr as 970,000 kg/m2/h, and Pcr as 1450 kPa. The Levy model (Levy, 1966) model is applied to calculate the mean bubble diameter, Dp,

Dp ¼ 0:015

rD sw ð1 þ

a1 ¼

f G2 8qavg

The grid correction factor Fgrid is defined as: x

F grid ¼ 1 þ a2 e2 eb2 D ibuse ¼ ib F grid where a2 is 6.5, b2 is 0.13, and e is the blockage-area ratio of the grid. The function,w, is the sum of all turbulent fluctuations larger than the mean vapour velocity ‘‘v 1l ”away from the wall:

"
2 #

1 1 v 1l 1 v 1l 1 pffiffiffiffiffiffiffi exp   erfc pffiffiffi 2 rv 0 2 rv 0 2p 2

(

w¼

ðr  sÞ2

ql  q1 ql  qg

a1
qg a2
qg ; x2 ¼ q1 q2

Based on these parameters, the CHF is predicted using the first equation in this appendix. Appendix B:. CHF evaluation methods A CHF predictor can be written in the inlet conditions form or the local conditions form. The inlet form directly using the experimental control parameters, such as the inlet G, P, xin etc. The local form using the parameters at the CHF location, such as G, P, xout etc. For uniform heated tube is the exit location. In fact, the two forms can transform each other for round tube by the heat balance equation as fellow:

xout  xin ¼ 4

0:1ðqf qg Þ sw D Þ

The wall shear stress,sw , is defined as

sw ¼

  2 r  2s s

s ¼ 5:5Dp

x1 ¼

The turbulence intensity factor, ib-use, is defined as




ðr  sÞ

q  q2 2 av g

Assuming a homogeneous flow, the parameters, x1 and x2, are calculated with:

f ¼ 0:0046Re0:2

0:462

r2

where ‘r’ is the channel radius and ‘‘s” is the bubbly layer thickness defined as:

The friction factor for turbulent flow is calculated with:

ib ¼ 0:2962
ðkÞ

rv 0 is defined as

The void fraction of the core region is given as

Hdb ¼ 0:023Re0:8 Pr0:4 kf =D

Re ¼

hl  hld q hf  hld exp

The turbulent RMS velocity,

q1 ¼

sffiffiffiffiffiffiffiffiffiffiffiffi rg c D

lf

qb

qg hfg

where qb is the portion of the total heat flux effective in generating bubbles

The CHF is predicted using

qpre ¼ Ghfg ðx2  x1 Þwibuse

v 1l ¼

v 1l rv 0

)

qc L Ghfg D

ð29Þ

Although for the open-type bundle fuel assembly, the Eq. (29) is no longer applicable due to the cross flow and the turbulent effect, there are still two ways to predict the bundle CHF. Groeneveld et al. (1996) have described two different methods of predicting CHF and CHF margins using local-conditions-type correlations: (1) the constant-inlet-conditions approach or the heat balance method(HBM), and (2) the so-called constant dryout quality approach or the direct substitution method (DSM). Groeneveld et al. (1996) have theoretically shown that the standard deviation obtained with HBM is significantly lower than the DSM, when comparing CHF data to the predicted value. More intuitively, from the Fig. 12 we can understand this difference. Fig. 12 illustrates the margin to CHF for a uniformly heated tube having fixed inlet conditions and operating at a given heat flux for the two above methods. We consider the experimental point A with CHF value q1.At fixed inlet quality Xin, CHF2 is the calculated CHF value corresponds to the intersection point B between the

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Fig. 12. CHF prediction with HBM or DSM.

HBM and the CHF curve, after a series of iteration calculations. However, for a local quality X1, the CHF1 corresponds to point C is the predicted CHF value. Thus, it is clear that CHF2 is closer to q1 than CHF1. From the above analysis, we can know that the HBM method is more accurate than DSM method, this difference is caused by the HBM or DSM method, not by the forecasting method or model itself. The magnitude of this difference is related to the slope of the CHF curve, that is, the interval in which
is located. In some cases, the error of the DSM method can be four to five times that of the HBM method (Groeneveld, 1996). In the early CHF prediction, the HBM method was widely used, but it could not effectively consider some separate effects, such as grid effect and axial flux distributions etc. The HBM method adjusts the input heat flux until the predicted CHF value is consistent with the adjusted heat flux value, and the adjusted heat flux value is the final predicted CHF value. Because DSM method uses actual experimental values as subchannel code input, its local parameters are accurate. For most CHF conditions, especially DNB type CHF, CHF is a local phenomenon and is insensitive to inlet conditions. Because of this, DSM method can account for separate effects, were found to be more reliable. The HBM method is mostly used for the critical power calculation of boiling water reactors (BWR). For PWR, the DSM method is now more commonly used. The work of this paper is also based on the DSM method. Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.anucene.2019.107085. References Ahmad, Masroor, 2012. Critical heat flux and associated phenomena in forced convective boiling in nuclear systems. Imperial College London, London. Biasi, L., Clerici, G.C., Garribba, S., et al. Studies on burnout. Part 3. a new correlation for round ducts and uniform heating and its comparison with world data[R]. ARS SpA, Milan. Univ., Milan, 1967. Bloch, Gregor, Bruder, Moritz, Sattelmayer, Thomas, 2016. A study on the mechanisms triggering the departure from nucleate boiling in subcooled vertical flow boiling using a complementary experimental approach. Int. J. Heat Mass Transf. 92, 403–413. Bowring, R.W. A simple but accurate round tube, uniform heat flux, dryout correlation over pressure range 0.7- 17 MN/m2 (100-2500 psia) [R]. Atomic Energy Establishment, Winfrith (England), 1972. Bricard, B., Peturaud, P., Delhaye, J.M., 1997. Understanding and modelling DNB in forced convective boiling: Modelling of a mechanism based on nucleation site dryout. Multiphase Sci. Technol. 9 (4), 329–379. Bruder, M., Bloch, G., Sattelmayer, T., 2015. Critical heat flux in flow boiling-review of the current understanding and experimental approaches. Heat Transfer Eng. 38, 347–360.

11

Carlucci, L.N., Hammouda, N., Rowe, D.S., 2004. Two-phase turbulent mixing and buoyancy drift in rod bundles. Nucl. Eng. Des. 227 (1), 65–84. Celata, G.P., Cumo, M., Mariani, A., Simoncini, M., Zummo, G., 1994. Rationalization of existing mechanistic models for the prediction of water subcooled flow boiling critical heat flux. Int. J. Heat Mass Transf. 37, 347–360. S.H. Chang, W.P. Baek, Understanding, predicting, and enhancing critical heat flux, in: 10th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-10), Seoul, Korea, October 5–9, 2003. Chang, S.H., Lee, K.W., 1989. A critical heat flux model based on mass, energy, and momentum balance for upflow boiling at low qualities. Nucl. Eng. Des. 113 (1), 35–50. Chun, T.H., Baek, W.P., Chang, S.H., 2000. An integral equation model for critical heat flux at subcooled and low quality flow boiling. Nucl. Eng. Des. 199 (1), 13–29. Chun., T.H., Hwang., D.H., Baek., W.P., Chang., S.H., 1998. Assessment of a tubebased bundle CHF prediction method using a subchannel code. Ann. Nucl. Energy 25, 1159–1168. Galloway, J., Mudawar, I., 1993. CHF mechanism in flow boiling from a short heated wall – II. Theoretical CHF model. Int. J. Heat Mass Transf. 36, 2527–2540. Galloway, J., Mudawar, I., 1993. CHF mechanism inflowboiling from a short heated wall. II. Theoretical CHF model. Int. J. Heat Mass Transfer 36, 2527–2540. Groeneveld, D.C., 1996. On the definition of critical heat flux margin. Nucl. Eng. Des. 163 (1), 245–247. Groeneveld, D.C., Shan, J.Q., Vasic´, A.Z., et al., 2007. The 2006 CHF look-up table. Nucl. Eng. Des. 237 (15), 1909–1922. Groeneveld, D.C., Leung, L.K.H., Park, J.H., 2015. Overview of methods to increase dryout power in CANDU fuel bundles. Nucl. Eng. Des. 287, 131–138. Guo-han CHAI etc, 2003. Review of correlation FC-2000 for critical heat flux calculation. Nucl. Power Eng. 2, 84–87 (In Chinese). Hooper, J.D., Wood, D.H., 1984. Fully developed rod bundle flow over a large range of Reynolds number. Nucl. Eng. Des. 83 (1), 31–46. Katto, Y., 1990. Prediction of critical heat flux of subcooled flow boiling in round tubes. Int. J. Heat Mass Transfer 33, 1921–1928. Katto, Y., 1990. A physical approach to critical heat flux of subcooled flow boiling in round tubes. Int. J. Heat Mass Transf. 33, 611–620. Kinoshita, H., Nariai, H., Inasaka, F., 2001. Study of critical heat flux mechanism in flow boiling using bubble crowding model. JSME Int. J. 44 (44), 81–89. Kodama, S., Kataoka, I., 2002. Study on analytical prediction of forced convective CHF in the wide range of quality. Int. Conf. Nucl. Eng. 62 (82), 147–154. Krauss, T., Meyer, L., 1996. Characteristics of turbulent velocity and temperature in a wall channel of a heated rod bundle. Exp. Therm Fluid Sci. 12 (1), 75–86. Kwon, Y.M., Chang, S.H., 1999. A mechanistic critical heat flux model for wide range of subcooled and low-quality flow boiling. Nucl. Eng. Des. 188 (1), 27–47. R.T. Lahey, F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, 3nd Edition, chapter 5 Two Phase Flow [M]. 1993. Lahey, R.T., Moody, F., 1977. The Thermal hydraulics of a Boiling water Reactor. American Nuclear Society, La Grange Park, Illinois, pp. 214–220. J. Laufer, The structure of turbulence in fully developed pipe flow, NACA Technical Note No. 2954, 1953. S.L. Lee, F. Durst, On the motions of particles in turbulent flow, U.S. Nuclear Regulatory Commission Report NUREG/CR-1556, 1980. Lee, C.H., Mudawar, I., 1988. A mechanistic critical heat flux model for subcooled flow boiling based on local bulk flow conditions. Int. J. Multiph. Flow 14, 711– 728. Leung, L.K.H., Groeneveld, D.C., Hotte, G., 1998. A Generalized Prediction Method for Critical Heat Flux in CANDU Fuel-Bundle Strings. In: Proc. 11th International Heat Transfer Conference, Kyongju, Korea, Aug. 23–28, pp. 15–20. Levy, S., 1966. Forced convection subcooled boiling – prediction of Vapor volumetric fraction. Int. J. Heat Mass Transf. 10, 91–965. Lim, J.C., Weisman, J., 1990. A phenomenologically based prediction of the critical heat flux in channels containing an unheated wall. Int. J. Heat Mass Transf. 33, 203–205. Lim. J.C., Improvements in the theoretical prediction of the departure from nucleate boiling, United States, 1988. Liu, W. et al., 2014. Study on reactor thermal-hydraulic sub-channel analysis code ATHAS. Nucl. Sci. Eng. 34 (1), 59–66. Motley, F.E., Hill, K.W., Cadek, F.F., et al. New Westinghouse correlation WRB-1 for predicting critical heat flux in rod bundles with mixing vane grids. Westinghouse Electric Corp., Pittsburgh, PA (USA). Nuclear Energy Systems Div., 1976. Nagayoshi, Takuji, Nishida, Koji, 1998. Spacer effect model for subchannel analysis. J. Nucl. Sci. Technol. 35 (6), 399–405. Pan, Jie, Li, Ran, et al., 2016. Critical heat flux prediction model for low quality flow boiling of water in vertical circular tube. Int. J. Heat Mass Transf. 99, 243–251. Rehme, K., 1978. The structure of turbulent flow through a wall subchannel of a rod bundle. Nucl. Eng. Des. 45 (2), 311–323. Renksizbulut, M., Hadaller, G.I., 1986. An experimental study of turbulent flow through a square-array rod bundle. Nucl. Eng. Des. 91 (1), 41–55. Rui, Guo, Xiaojin, Liu, et al., 2016. Theoretical Investigation on Critical Heat Flux at Inclined Heater Surface in Low Pressure. Nucl. Power Eng. 37 (4), 11–14 (In Chinese). Styrikovich, M.A. et al., Effect of Two Phase Flow Pattern on the Nature of Heat Transfer Crises in Boiling, Proc.4th Int. Heat Transfer Conf, Paris-Versailles, vol. 9, p. 360, 1970. Tong, L.S., 1986. An evaluation of the departure from nucleate boiling in bundles of reactor fuel rods. Nucl. Sci. Eng. 33 (1), 7–15.

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085

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Trupp, A.C., Azad, R.S., 1973. The structure of turbulent flow in triangular array rod bundles. Nucl. Eng. Des. 32 (1), 47–84. Weisman, J., 1991. The current status of theoretically based approaches to the prediction of the critical heat flux in flow boiling. Nucl. Technol. 9 (1), 1–21. Weisman, J., Illeslamlou, S., 1988. Phenomenological model for prediction of critical heat flux under highly subcooled conditions. Fusion Technol. 13 (4), 654–659. Weisman, J., Pei, B., 1983. Prediction of critical heat flux in flow boiling at low qualities. Int. J. Heat Mass Transf. 26 (10), 1463–1477. Weisman, J., Yang, J.Y., Usman, S., 1994. A phenomenological model for boiling heat transfer and the critical heat flux in tubes containing twisted tapes. Int. J. Heat Mass Transfer 37, 69–80.

Weisman, J., Ying, S., 1985. A theoretically based critical heat flux prediction for rod bundles at PWR conditions. Nucl. Eng. Des. 85 (2), 239–250. Xie, Haiyan, Yang, Dong, et al., 2019. An improved DNB-type critical heat flux prediction model for flow boiling at subcritical pressures in vertical rifled tube. Int. J. Heat Mass Transf. 132, 209–220. Yang, S.K., Chung, M.K., 1998. Turbulent flow through spacer grid in rod bundles. J. Fluids Eng. 120 (4), 786–791. Yao, S.C. et al., 1982. Heat-transfer augmentation in rod bundles near grid spacers. Int. J. Heat Mass Transf. 104 (1), 76–81. Ying, S.H., Weisman, J., 1986. Prediction of critical heat flux in flow boiling at intermediate qualities. Int. J. Heat Mass Transf. 29, 1639–1648.

Please cite this article as: Y. Liu, W. Liu, J. Shan et al., A mechanistic bubble crowding model for predicting critical heat flux in subchannels of a bundle, Annals of Nuclear Energy, https://doi.org/10.1016/j.anucene.2019.107085