Development of a three-field mechanistic model for dryout prediction in annular flow

Annals of Nuclear Energy 135 (2020) 106978

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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Development of a three-field mechanistic model for dryout prediction in annular flow Minyang Gui, Wenxi Tian ⇑, Di Wu, Ronghua Chen, G.H. Su, Suizheng Qiu School of Nuclear Science and Technology, Shanxi Key Laboratory of Advanced Nuclear Energy and Technology, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 15 January 2019 Received in revised form 31 July 2019 Accepted 2 August 2019

Keywords: Critical heat flux (CHF) Annular flow Subchannel analysis

a b s t r a c t Critical heat flux (CHF) is one of the most important thermal criteria for nuclear power plants. It has traditionally been evaluated using look-up tables or empirical correlations. In this paper an annular film dryout (AFD) mechanistic model has been developed based on the interaction of three fields in annular flow region, i.e. the liquid film, entrained droplets and vapor core. The model describes the mass and momentum conservation equations of three fields together with a series of constitutive relations. The effect of some constitutive correlations (the entrainment and deposition of droplets and the onset of annular flow) on the prediction accuracy of the model is studied. The results are compared with CHF experimental data in flow boiling. Fairly good agreements are observed for the CHF in circular tubes with uniform and axially non-uniform heating, as well as rectangular channels with uniform heating. Through coupling with the subchannel analysis method, the model is used to predict the dryout-type CHF in the rod bundles with a good precision. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Critical heat flux (CHF) is one of the main factors affecting the heat transfer capacity of boiling heat exchanger such as boilers, reactors, and steam generators. On the one hand, it is hoped that these heat exchangers can be operated at as high heat flux as possible in order to pursue the highest possible heat transfer efficiency. On the other hand, when the heat flux increases to a critical value (i.e. CHF), the wall surface cannot be sufficiently wetted due to exposure to the vapor phase, and the surface temperature will rise suddenly due to decrease of heat transfer coefficient, and then boiling crisis will occur. Especially for the nuclear reactor system, the cladding temperature in the fuel assembly must not exceed a specific value, beyond which the cladding material can lose its integrity and fission products will be released into the coolant and threaten the safety of the reactor. The prediction of CHF is of significant importance to the safety operation of nuclear power plants. In general, CHF mechanisms can be roughly divided into two categories based on different flow regimes (Collier and Thome, 1994): one occurs in the bubbly flow regime at subcooled or low-quality flow, and is often referred to as departure from

⇑ Corresponding author. E-mail address: [email protected] (W. Tian). https://doi.org/10.1016/j.anucene.2019.106978 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.

nucleate boiling (DNB); the other occurs in the annular flow regime at high quality flow and is called annular film dryout (AFD), which is the main subject of the present paper. So far, many scholars have conducted a large number of experiments on CHF (Becker, 1965; Bowring, 1972; Griffith et al., 1977; Katto, 1979; Sudo and Kaminaga, 1989, 1993; Kureta and Akimoto, 2002; Lu et al., 2004), and some reliable experimental correlations have been proposed. Moreover, Groeneveld et al. (1996, 2007) developed the look-up tables to predict CHF based mostly on the data bank for flow in circular tubes which seem to be more effective. The advantages of the CHF look-up tables are that they have a wider range of application with satisfactory accuracy, and the database is easy to update. For annular-dispersed two-phase flow, the typical characteristic is that the part of the liquid flows along the wall as a liquid film and part as droplets in the gas core flow. Due to the fluctuation at the film-gas interface, the gas core continuously entrains droplets from the liquid film and at the same time the droplets are also continuously deposited on the liquid film surface due to diffusion effect (Hewitt and Govan, 1990; Sugawara, 1990; Kataoka et al., 2000; Okawa and Kataoka, 2005). Currently some scholars have begun to develop so-called mechanistic models for predicting CHF based on the entrainment and deposition process of droplets (Saito et al., 1978; Utsuno and Kaminaga, 1998; Okawa et al., 2003; Jayanti and Valette, 2004; Du et al., 2012; Chandraker

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Nomenclature A C DeðDhÞ De EnðqÞ Ev f g G hfg J M p PrwðqÞ q u W x z

Channel cross-sectional area, (m2) Droplet mass concentration, (kg
m3) Channel equivalent (hydraulic) diameter, (m) Deposition mass flux, (kg
m2
s1) Entrainment mass flux, (kg
m2
s1) Vaporization mass flux, (kg
m2
s1) Friction coefficient Gravitational constant, (m
s2) Mass flux, (kg
m2
s1) Latent heat of vaporization, (J
kg1) Superficial velocity, (m
s1) Interface friction force, (N
m3) Pressure, (Pa) Wetted (heated) perimeter, (m) Channel surface heat flux, (kW
m2) Velocity, (m
s1) Mass flow, (kg
s1) Vapor quality Channel axial coordinate, (m)

et al. 2012). Saito et al. (1978) carried out a detailed analysis of the annular two-phase flow based on general mathematical three-field formulation, Okawa et al. (2003) proposed a liquid film analysis model to predict the dryout in the circular tube, which focuses on the axial variation of film flow rate. Du et al. (2012) conducted a mechanistic study on the characteristics of CHF in vertical narrow rectangular channels. More recently, to remedy the existing disadvantages of the one-dimensional model such as the rough film-gas interface analysis and inherent inability to account for the threedimensional effects, the mechanistic film models have been implemented into Computational Fluid Dynamics (CFD) codes (Li and Anglart, 2016, 2014; Baglietto et al., 2019; Anglart et al., 2018). Anglart et al. (2018) presented a new mechanistic model for the diabatic annular two-phase flow to predict dryout and postdryout heat transfer in various channels. The model employed OpenFOAM to solve the governing equations of two-phase mixture flow together with additional closure laws. Baglietto et al. (2019) have developed the second-generation of multiphase-CFD (MCFD) mechanistic model for CHF prediction, based on new experimental observations of mesoscales resulting from advancements in measurement and visualization technologies. Although the current model is mainly used to deal with low-quality condition, some efforts are being devoted to extend the approach to higher void fraction regimes (i.e. AFD). Furthermore, there have been numerous efforts to predict the AFD in rod bundles also using the mechanistic approach. Codes like FIDAS (Sugawara, 1990), ASSERT-PV (Hammouda et al., 2016) and COBRA-TF (Thurgood et al., 1983) etc. have been developed and compared with the experimental data achieving good accuracy. Especially for COBRA-TF, which is the most famous subchannel code based on three-field model, it has been applicable for both Pressurized Water Reactors (PWRs) and Boiling Water Reactors (BWRs). For AFD mechanistic model, the simplest approach is to only consider mass conservation equation for the liquid film and the liquid film mass flow rate is calculated from a first-order nonlinear differential equation. In comparison, the three-field model is more elaborate, i.e. the three fields (liquid film, vapor core, droplet entrained in vapor core) are considered separately. To be solved, some additional closure models are required, such as droplet deposition and entrainment, onset of the annular flow, Initial Entrained Fraction (IEF) and dryout criterion etc. There are at least five widely

Greek symbols Volume fraction Time, (s) d Liquid film thickness, (m) l Dynamic viscosity, (kg
m1
s1) q Density, (kg
m3) r Surface tension, (N
m2) u Fraction of liquid flux flowing as droplets

a s

Subscripts ave Average ann Onset of annular flow cal Calculated value CHF Critical heat flux d Droplets exp Experimental value f Liquid film g Gas (vapor) l Liquid phase tran Slug-annular transition point

used droplet deposition and entrainment models, including WurtzSugawara correlations (Sugawara, 1990), Govan correlations (Govan et al., 1988), Okawa & Tsuyoshi correlations (Okawa et al., 2002), Kataoka & Ishii correlations (Kataoka et al., 2000) and Okawa & Kataoka correlations (Okawa and Kataoka, 2005), which were developed based on a large number of droplet entrainment and deposition experimental data, with good channel suitability. Another major challenge is choosing the appropriate slug-annular transition point, the common models are Wallis (1969) correlation and Mishima and Ishii (1984) correlation. Once the onset of annular flow has been determined, it is required to know the proportion of droplets in the liquid phase at the starting point, i.e. IEF. No agreement has been achieved yet regarding the value of the IEF, the reason is that there is a dearth of experimental data, if it exists, is obtained from adiabatic air-water experiments. Many scholars considered IEF as a tuning parameter to obtain a good match with experimental results. Notable scholars are Whalley (1977, 1978), Saito et al. (1978), Govan et al. (1988), Sugawara (1990), Azzopardi (1996), Naitoh et al (2002), Ahmad et al. (2013) who all have given a fixed value of the IEF in their models. For most mechanistic models, the dryout criterion is generally assumed that film thickness (or film flow rate) becomes zero, some studies (Ueda et al., 1981; Milashenko et al., 1989) showed that CHF occurs before the liquid film disappears absolutely, which can be ascribed to the breakup or the interfacial waves of the film. Overall, different scholars performed different model combinations and some AFD mechanistic models have been developed in the literature, but the comparative analysis of models is rare. Spirzewski et al. (2017), Spirzewski and Anglart (2018) deeply analyzed the effects of boiling entrainment and IEF on the CHF prediction, it was found that boiling entrainment do not contribute significantly at low heat fluxes but underestimate the dryout at higher heat fluxes. At the same time, the sensitivity analysis of the predicted CHF to IEF was performed and a new expression for the IEF was subsequently developed. In aggregate there are still many disputes about the impact of related constitutive relations on the prediction ability of mechanistic model. Thus, the objective of current study is to establish the conservation equations for three-field (i.e. liquid film, vapor core and entrained droplets) of annular flow region together with a set of closure relationships, further develop an AFD mechanistic model

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based on the entrainment and deposition behavior of droplets. Besides, the effect of some constitutive correlation (including the entrainment and deposition of droplets and the onset of annular flow) on the prediction accuracy of the model is studied. Finally, the model is used to predict the CHF in the rod bundles by coupling with the subchannel analysis code.

(4) The friction force between the liquid film and the droplets is ignored; (5) The liquid film, droplets and vapor in the annular flow region are all saturated, and the energy conservation equations can be omitted. 2.1. Conservation equations

2. Mathematical model and calculation procedure It is known that annular flow is an important flow pattern in the study of two-phase flow, which generally occurs after the churnturbulent flow with a higher vapor flow rate, as shown in Fig. 1 (a). In the areas with high vapor quality, liquid film flows along the flow path against the wall, and the vapor flows in the center of the flow path entraining the droplets (Fig. 1(b)). On the one hand, the liquid film adhering to the wall surface is continuously thinned by the entrainment of the central high-speed vapor stream (En) and the water evaporation on the heated surface (Ev). On the other hand, the droplets entrained in the vapor stream are continuously deposited on the surface of the liquid film (De), which is represented in Fig. 1(c). Finally, the vapor quality is continuously increased, and the liquid film is further thinned until it is completely evaporated to dryout, i.e. the thickness of liquid film (df) in Fig. 1(b) is reduced to zero. In order to describe the above process, a three-field model (i.e. liquid film, vapor core and entrained droplets) will be established based on the entrainment and deposition behavior of droplets. The major assumptions used in the derivation of the three-field balance equations are as follows, (1) The flow is incompressible and the gravity is the only body force; (2) The circumferential liquid film of the channel is uniform; (3) The pressure is uniform across the channel cross-sectional area;

2.1.1. The continuity equations In three-field model, the continuous liquid film (f), entrained droplets (d) and vapor (g) in the annular flow region of the channel are mainly controlled by film evaporation, droplet deposition and entrainment, thus the continuity equations are as follows:


  @ qf af uf @
qP rq Prw De  ðPrw Enh þ P rq Enq Þ ¼ q af þ þ @z @s f A Ahfg @ @ ðqd ad ud Þ P rw De  ðPrw Enh þ Prq Enq Þ ðq ad Þ þ ¼ @z @s d A
  @ qg ag ug @
qP rq q ag þ ¼ @s g Ahfg @z

ð1Þ

ð2Þ

ð3Þ

where qk, ak and uk represent the density, volume fraction and axial velocity at axial location z for each phase (k = f, d or g). Prq and Prw are the heated and wetted perimeters of the channel, respectively. If the channel is heated uniformly, then Prq = Prw. De is the droplet deposition rate per unit area. For droplet entrainment rate per unit area, it consists of two parts, i.e. the droplet entrainment occurs due to shearing effect of central high-speed gas flowing over the fluctuation surface of liquid film (Enh), and the droplet entrainment produced by bursting of bubbles (Enq). In general, dryout is the result of competition of droplet deposition, entrainment and film evaporation. Liquid is lost from the film due to droplet entrainment and water evaporation, and it is gained as a result of droplet deposition. 2.1.2. The momentum conservation equations For the momentum conservation equations, in addition to considering the gravity, the pressure and the friction force from the wall surface, the friction forces among of the phases interfaces should also be included. Combining the force analysis of each phase and the momentum exchange between phases, the momentum conservation equation of each phase can be obtained separately:


 @ qf af uf @s ¼ af

þ


 @ qf af u2f @z



De Prw qP rq Enh Prw þ Enq Prq uf þ ud þ uf A Ahfg A

@p þ af qf g  M wf þ M fg @z

  @ ðqd ad ud Þ @ qd ad u2d De Prw Enh Prw þ Enq Prq þ þ ud  uf @s @z A A @p ¼ ad þ ad qd g þ M gd @z
 @ ag qg ug @s ¼ ag

Fig. 1. (a) The schematic diagram of annular two-phase flow; (b) The schematic diagram of circumferential liquid film thickness of round tube; (c) The schematic diagram of droplet entrainment and deposition in annular flow region.

þ


 @ ag qg u2g @z



ð4Þ

ð5Þ

qP rq uf Ahfg

@p þ ag qg g  Mfg  Mgd @z

ð6Þ

where M is the friction force per unit volume. Assuming that the liquid film completely covers the wall surface of the flow channel, the friction forces between the wall surface and the vapor core as well as between the wall surface and the droplets are negligible. The first term on the right-hand side of Eqs. (4)–(6) represents the pressure

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force, the second term represents the body force due to gravity. The terms starting from the third one to the end on the left-hand side represent the momentum exchange due to mass exchange. Combining the continuity Eqs. (1)–(3), the above Eqs. (4)–(6) can be further transformed into the following forms:

qf af

@uf @uf @p þ af qf g  Mwf þ M fg þ qf af uf ¼ af @z @s @z  De Prw  þ ud  uf A

@u @u @p qd ad d þ qd ad ud d ¼ ad þ ad qd g þ Mgd @z @s @z  Enh Prw þ Enq Prq  þ uf  ud A

qg ag

@ug @ug @p þ ag qg g  M fg  Mgd þ qg ag ug ¼ ag @z @s @z   qP rq þ uf  ug Ahfg

ð7Þ

ð8Þ

ð9Þ

2.1.3. The energy conservation equations The liquid film, droplets and vapor in the annular flow region are assumed to be all saturated, therefore, the energy conservation equations can be omitted. Of course, this assumption is unreasonable for the annular flow in the non-equilibrium state.

uniformly along the circumference of the channel, Pr,fg is evaluated by Pr,fg = (4pAag)1/2, and the thickness of liquid film df is computed by df = De/2  (Aag/p)1/2. Mfg is proportional to the square of velocity difference between liquid film and vapor core. Owing to the radial velocity gradient of thin liquid film, it is necessary to determine the surface velocity of the liquid film, which is also important for droplet entrainment. In this paper, the velocity distribution of the liquid film is obtained by the von Karman formula (Arpaci and Larsen, 1984):

8 þ þ 0  yþ  5 > : þ u ¼ 2:5lnðyþ Þ þ 5:5 30  yþ

ð13Þ

where u+ and y+ are the dimensionless velocity and distance, respectively. Their expressions are u+ = u/Us, y+ = yUsqf/lf. The expressions of other parameters are as follows:

sw qf

Us ¼

!0:5 ;

sw ¼

fu2f qf 8

;

  1x 1 ; uf ¼ G 1  a qf

f ¼

0:3164 Re0:25 f ð14Þ

2.2.4. The friction force between vapor and entrained droplets The friction force between vapor and entrained droplets per unit volume (Mgd) is evaluated with the following equation:

   1 C d agd qg ug  ud  ug  ud 8

2.2. Constitutive equations

Mgd ¼

The majority of the constitutive equations are derived from those of Stevanovic and Studovic (1995) and Sugawara (1990), including the shear stresses and friction coefficients. Some models are optimized, such as the radial velocity gradient of thin liquid film. The main difference is in the constitutive relationships representing the mechanism of droplet entrainment and deposition, which will be analyzed in Section 3.

where agd is the interfacial area concentration, agd = 6ad/Ded, Ded is the diameter of the droplet, which is given by:

2.2.1. The volume fraction The three-field model established in this paper contains seven variables, i.e. pressure (p), volume fraction (af, ad, ag) and axial velocity (uf, ud, ug) of liquid film, droplets and vapor. In addition to the preceding six conservation equations, the relationship of volume fraction is also needed, it is:

af þ ad þ ag ¼ 1

ð10Þ

2.2.2. The friction force between liquid film and wall The wall friction force per unit volume i.e. Mwf in Eq. (4) can be calculated by following correlation:

M wf

  1 f w qf uf uf ¼ Prw A 2

ð11Þ

where fw is the wall friction coefficient proposed by Wallis (1969), fw = max (16/Ref, 0.005), Ref is the Reynolds number of liquid film, Ref = af qf uf De/lf. 2.2.3. The friction force between liquid film and vapor The friction force between liquid film and vapor per unit volume (Mfg) has the same form as Mwf, that is:

   1 f fg qg ug  uf  ug  uf M fg ¼ Pr;fg 2 A

ð12Þ

where ffg is the film-to-vapor interfacial friction coefficient proposed by Wallis (1969), ffg = 0.005 (1 + 300 df/De). Pr,fg is the perimeter of the interface. Assuming that the liquid film distributes

Ded ¼

8 < :

5r 2 qg ðug ud Þ

103 ; Ded < 103

ð15Þ

ð16Þ

The interfacial force between the droplets and the vapor is mainly reflected in the droplets drag coefficient Cd, in this paper the coefficient is derived based on the Clift et al.’s (1978) theory, it is:

Cd ¼

 24
0:42 1 þ 0:15Re0:687 þ d Red 1 þ 4:25  104 Re1:16 d

ð17Þ

where Red is the Reynolds number of droplets, Red = qg |ug  ud| Ded/

lg, and lg is the dynamic viscosity of vapor. 2.3. Calculation procedure In order to solve the partial differential equations effectively, the finite difference method is used in this paper. For the spatial discretization, a staggered grid is adopted, that is, the momentum control volume (including the variables uf, ud, ug) and the energy, mass control volume (including the variables p, af, ad, ag, qf, qd, qg) are utilized which are staggered from one another. For the time discretization, the semi-implicit difference scheme is applied. Except for the velocity of the convection term in the continuity equation, the pressure gradient and the interface-related velocity term in the momentum equation are in implicit scheme, all other items adopt the explicit scheme. Dividing the channel into a number of control volumes along the axial direction, and integrating the Eqs. (1)–(9) from the point where the equilibrium quality is 0 to the channel exit, the volume fraction of each phase at any axial point can be calculated, which is used to predict whether the channel exit has dried up. The solving procedure is described as follow:

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Beside the equations of void fraction and friction forces, some correlations such as the entrainment and deposition of droplets and the onset of annular flow also have an important impact on the prediction ability of the mechanistic model. In order to compare the different constitutive correlations of the entrainment and deposition of droplets and the onset of annular flow, the experimental data of Becker (1965) were used. The total number of the database is 269. The parameter ranges are inside tube diameter of 3.93–24.95 mm, system pressure of 1.1–10.1 MPa and mass flux of 144–4050 kg m2 s1. In order to compare the prediction results with the experimental CHF data, a parameter is defined as follows:

CHFR ¼ qcal =qexp

ð18Þ

where qcal and qexp are calculated CHF and experimental CHF, respectively. A perfect CHF model will give CHFR values of 1 for every test point. In addition, the three statistical parameters, Ravg (average value of CHFR), RMS (root mean square error), and STD (standard deviation), used in this paper are defined as: N 1X CHFRi N i¼1

"

N 1X ðCHFRi  1Þ2 RSM ¼ N i¼1

"

1.0

0.8

0.6 Kataoka Ravg = 0.965 RSM = 0.090 STD = 0.083 Okawa (2002) Ravg = 0.947 RSM = 0.098 STD = 0.082

0.4

Okawa (2005) Ravg = 0.822 RSM = 0.200 STD = 0.094

0

1

2

3

4

5

6

q (MW.m-2) Fig. 2. Comparison of different droplets deposition and entrainment models for CHF prediction.

3. Analysis of the models

Rav g ¼

1.2

CHFR

(1) An initial value is supposed for the heat flux of heated wall (q); (2) Solve the velocity (un+1) of the new time step with respect to the expression of pressure (pn+1); (3) Obtain the pressure matrix of the new time step and solve the pressure by direct elimination or Gauss–Seidel method; (4) Calculate the velocity (un+1) and volume fraction (an+1) of each phase in the new time step; (5) Determine whether the time step meets the limitation of Courant number rule, that is u Ds/Dz < 1, if not, decrease the time step and repeat steps (2)–(5); (6) Calculate the volume fraction of liquid film for the next time step. If the liquid film volume fraction at outlet is less than the set value, the calculation is completed and the corresponding wall heat flux (q) is the CHF value (qCHF). Otherwise, increase the supposed value of the heat flux q in step (1) and repeat steps (2)–(6).

ð19Þ #1=2

N  2 1 X STD ¼ CHFRi  Ravg N  1 i¼1

ð20Þ #1=2 ð21Þ

3.1. Droplet deposition and entrainment Fig. 2 compares the effects of three different correlations of the entrainment and deposition of droplets on CHF prediction, which are Okawa & Tsuyoshi correlations (Okawa et al., 2002), Kataoka & Ishii correlations (Kataoka et al., 2000) and Okawa & Kataoka correlations (Okawa and Kataoka, 2005), respectively. They were all developed based on a large number of droplet entrainment and deposition experimental data and applied to the mechanistic model of different scholars. As can be seen from the figure, for the same experimental data, the present model derived from Kataoka et al. (2000) and the model of Okawa et al. (2002) both can obtain better prediction results, the absolute value of the deviation between the average value of CHFR and 1 is below 6% and the

magnitude of the RMS error is below 10%. By comparison, the model of Okawa and Kataoka (2005) underestimates CHF, the error is beyond 40% bound at certain conditions. In this paper, the Kataoka & Ishii correlations (Kataoka et al., 2000) are chosen in the present model. For droplet deposition rate, it is generally considered to be proportional to the concentration of droplets in the vapor core C:

De ¼ kd C

ð22Þ

where kd is the droplet deposition coefficient. Paleev and Filippovich (1966) proposed that the droplet deposition coefficient is also related to the droplet concentration, on the basis of their work, the deposition rate is given by Kataoka et al. (2000), that is: 0:74 f Ref

De ¼ 0:022l

lg lf

!0:26

u0:74 =De

ð23Þ

where u is the fraction of liquid flux flowing as droplets, and defined by u = Wd/(Wf + Wd), Wd and Wf are the mass flow of droplets and liquid film, respectively. For droplet entrainment, it mainly occurs due to shearing effect of central high-speed vapor flowing over the fluctuation surface of liquid film, but for the cases of higher heat flux and shorter channel, bursting of bubbles will also produces entrained droplets which cannot be ignored (Ueda et al., 1981). Since very few experimental data are available, it is unclear what nonlinear relation is between the shearing effect and the boiling effect, Okawa et al. (2003) assumed the total entrainments are the linear superposition of both. In this paper the same treatment to droplet entrainment is adopted as Okawa et al., the shearing entrainment rate Enh is evaluated with the following equations proposed by Kataoka et al. (2000): 8 E De 0:25 9 1:75 nh >  ð1  u=u1 Þ2 > lf ¼ 0:72  10 Ref Weð1  u1 Þ > > > ðu=u1  1Þ
0:26 > > l 0:185 > < þ6:6  107 Re0:925 We0:925 g ð1  uÞ f

lf


0:26 > l 7 Enh De 0:925 > > We0:925 lg ð1  uÞ0:185 > lf ¼ 6:6  10 Ref > l > > > : Enh De ¼ 0 l f

ðu=u1  1Þ   Relf < Relfc ð24Þ

where, u1 is the equilibrium value of fraction entrained, u, which is given by:




u1 ¼ tanh 7:25  10 - 7 We1:25 Re0:25 f



ð25Þ

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Ref and Relf are Reynolds number of total liquid phase and film, respectively. Relfc is the minimum liquid film Reynolds number required to entrainment. The equations are as follows:

Ref ¼

Relf ¼

Gð1  xÞDe

ð26Þ

lf Gð1  xÞð1  uÞDe

ð27Þ

lf

!0:75 1:5 qf 10 ¼ 0:347 qg 

Relfc

lg lf

!1:5 ð28Þ

We is the Weber number and defined by:

We ¼

qg J2g De qf  qg r qg

!1=3 ð29Þ

For boiling entrainment rate Enq, Ueda et al. (1981) and Milashenko et al. (1989) carried out some experiments to study the effect of boiling on the entrainment, and proposed their experimental relations subsequently. In this paper the boiling entrainment rate Enq is evaluated with Ueda et al.’s (1981) equation, that is:

Enq

q ¼ 4:77  10 hfg

" 

2

2 !#0:75 q=hfg df

3.2. Onset of annular flow and the inception entrainment fraction Another major challenge in applying the mechanistic model is choosing the appropriate slug-annular transition point which may affect the dryout prediction. Mishima and Ishii (1984) proposed when the churn flow in a circular tube is transformed into annular flow, the superficial gas velocity needs to satisfy the following conditions:

!0:25

r
g
Dq Jg  q2g

xann ¼

ð31Þ

J g
qg G

ð32Þ

While Wallis (1969) correlation has also been extensively used for determining the slug-annular transition point (Okawa et al., 2003; Talebi and Kazeminejad, 2012), the vapor quality at the transition point can be written as:

xann

In fact, the boiling-induced entrainment is an uncertain factor. As indicated in Fig. 3, the CHF for different L/D ratios is compared when the boiling-induced entrainment is neglected. A clear increase in CHF prediction error in terms of the average CHFR, RMS and the standard deviation can be observed. It can be found that for the smaller L/D region (L/D < 220), the boiling entrainment has a significant effect on the prediction of CHF. But for larger L/D region, with considering boiling or without to predict results are almost identical. Trace its root, the extent to which boiling entrainment affects the dryout prediction depends on whether there is enough time in the channel to reach a droplets entrainment/deposition equilibrium before dryout occurs. Even for larger wall heat flux, when the channel has a larger L/D, the liquid film dryout might occur far from the churn-to-annular flow transition, so the effects of boiling become negligible.

10:2

lf B C
@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA qf r r=ðg
DqÞ

where Dq is the difference between qf and qg. Hibiki and Mishima (2001) though that Eq. (31) is equally applicable to rectangular channels. Once the superficial gas velocity of the transition point is determined, the vapor quality at the starting point can be obtained by the following formula:

ð30Þ

rqg

0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 0:6 þ 0:4 qf qf  qg gDh=G qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:6 þ qf =qg

ð33Þ

Fig. 4 compares the effects of the two different annular flow starting point correlations on model validation. The result shows that the predicted values of CHF obtained by using these two correlations are basically the same, which indicates the present model is not sensitive to the onset of the annular flow. It seems that the sensitivity of boundary conditions depends on the constitutive model chosen. Tomiyama and Yokomizo (1988) examined two typical models developed by Whalley (1977, 1978) and Saito et al. (1978), and found that the Whalley’s model is not affected by the choice of onset of annular flow while Satio’s model have been affected obviously. Particularly, in present model, the calculation of the entrainment and deposition of droplets begins with the equilibrium quality of 0 instead of the onset of annular flow, which would lead to the current conclusion.

1.3 With boiling entrainment Ravg = 0.970 RSM = 0.085 STD = 0.080

2.4

1.2

Without boiling entrainment Ravg = 1.164 RSM = 0.285 STD = 0.234

2.0

1.1

CHFR

CHFR

1.0 1.6

0.9 0.8

1.2

Wallis Ravg = 0.969 RSM = 0.086 STD = 0.081

0.7 0.8

Mishima Ravg = 0.970 RSM = 0.093 STD = 0.089

0.6 0

50

100

150

200

250

300

350

400

450

L/D Fig. 3. Sensitivity analysis of L/D ratio and boiling entrainment model for CHF prediction.

0

1

2

3

4

5

6

q (MW.m-2) Fig. 4. Comparison of different correlations of slug-annular transition point for CHF prediction.

M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Eqs. (31) and (32) are adopted as criterion for onset of annular flow in current model. In the common AFD mechanistic models, it is necessary to first determine the IEF at the onset of annular flow, that is, the share of the liquid droplets in the liquid phase. However, little agreement has been achieved yet and many scholars considered IEF as a tuning parameter to obtain a good agreement with experimental results. In this paper, to avoid the unknown IEF, a simpler approach was adopted. The calculation of the entrainment and deposition of droplets begins with the point where the equilibrium quality is 0. Then, the expressions of entrainment rate and deposition rate are as follows:

En ¼

z En;z xxann

xz < xann

En;z

xz  xann

;

De ¼

z De;z xxann

xz < xann

De;z

xz  xann

ð34Þ

where En,z and De,z are the entrainment rate and deposition rate calculated by Eqs. (23), (24) and (30) at position z. A fixed value is used as the fraction of entrained droplets at the starting point (i.e.  = 0), that is, the volume fraction of droplets ad is 0.001. 3.3. Dryout criterion Due to the entrainment of the droplets and the continuous evaporation of the liquid film, the thickness of the liquid film will be gradually thin along the flow channel to dryout. Some scholars proposed the CHF is triggered by the breakup (Ueda et al., 1981; Milashenko et al., 1989) or the interfacial waves (Tu et al., 1998) of the film before the exit. However, Okawa et al.’s (2003) sensitivity analysis of dryout criterion found little impact on CHF prediction results. In present model it is considered that the dryout occurs when the volume fraction of liquid film is less than 109. The correlations adopted in the paper are summarized in Table 1. 4. Results and discussion To validate the proposed dryout model, the prediction results are compared with experimental data in uniformly heated vertical round tubes reported by Becker (1965), Thompson and Macbeth (1964) and Saito et al. (1978), non-uniformly heated round tubes reported by Judd et al. (1965), as well as uniformly heated vertical rectangular channels reported by Siman-Tov et al. (1995), Jacket et al. (1958) and Troy (1958). Table 2 shows the operating conditions of the experimental data, although the currently available database is relatively small, it contains a wide range of parameters. Besides, all data points are basically in the annular flow regions judged by Eqs. (31) and (32), as shown in Fig. 5, which guarantees the applicability of the dryout mechanistic model.

7

4.1. CHF prediction in uniformly heated vertical circular tubes Fig. 6 shows the comparisons between calculated and experimental CHF in uniformly heated vertical circular tubes. It can be found that almost all of these data (96.7% of data) are predicted within the error range of ±20%. Subsequently the dependence analysis of the present model on several thermal-hydraulic parameters is investigated, as shown in Fig. 7(a)–(f). The effects of inlet mass flux, vapor quality at outlet, pressure, inlet subcooling, tube diameter and length/diameter ratio on CHF prediction are compared. In general, there are the tendencies that CHFR increases with the increases of pressure and inlet subcooling and with the decreases of length/diameter ratio and vapor quality at outlet. For instance, the model underestimates CHF at the lower pressure but overestimates that at higher pressure. The major reasons of these tendencies would be attributed to the insufficiencies of the constitutive models. Fig. 7(b) clearly indicates the model overestimates CHF at a smaller outlet vapor quality, and a tendency is showed that the smaller the vapor quality is, the larger the prediction deviation is. The model proposed in this study is based mainly on the mechanism of droplet entrainment and deposition in typical annular flow region, eventually resulting to evaporation of the wall liquid film. However, at certain conditions (i.e. the vapor quality is small at the exit and the annular flow length is short enough), the annular flow pattern in the channel is not obvious. Although the critical value can also be calculated by the AFD model, the CHF may be triggered by other mechanisms instead of entrainment and deposition of droplets. Liu et al (2012) have found that the DNB-based liquid sublayer dryout mechanism can also be used to predict the CHF in a low-quality annular flow region. The effects of inlet subcooling and mass flow rate on CHF are demonstrated in Figs. 8 and 9. Two different experimental conditions are selected, i.e. the system pressure is 4 MPa and 7 MPa, respectively. As expected, the CHF increases almost linearly with inlet subcooling consistent with the experimental point, and the CHF increases with mass flow rate as well. An increase in inlet subcooling will result in a decrease in the length of the annular flow. The influence of the inlet subcooling on the CHF becomes smaller in a lower mass flux condition (i.e. a smaller linear slope). 4.2. CHF prediction in non-uniformly heated vertical circular tubes

Model

Equations in the paper

Related references

To further test the applicability of the model, the CHF data of Judd et al. (1965) with chopped cosine, peak towards inlet, and peak towards outlet heat flux distributions are analyzed. As shown in Fig. 10, the overall prediction results are within ±25%, and the model overestimates the CHF when the location of peak heat flux is shifted towards the outlet. Fig. 11 shows the variations of CHFR with the vapor quality at outlet. It can be found that the CHFR decreases with the increase of the vapor quality at outlet. Especially for the smaller outlet vapor quality, the model is poorly predicted, which is similar to that of uniform axial heat flux data.

Shear stresses and friction coefficients Droplet deposition Shear-induced entrainment Boiling entrainment Onset of annular flow Inception entrainment fraction Dryout criterion

Eqs. (11)–(17)

Stevanovic and Studovic (1995);Sugawara (1990)

4.3. CHF prediction in uniformly heated vertical rectangular channels

Eqs. (22) and (23) Eqs. (24)–(29)

Kataoka et al. (2000) Kataoka et al. (2000)

Eq. (30)

Ueda et al. (1981)

Eqs. (31) and (32)

Mishima and Ishii (1984)

Eq. (34)

/

The thickness or volume fraction of liquid film is zero

Adamsson and Corre (2011)

Table 1 Summary of correlations adopted in the paper.

In addition to the circular tubes, the present model is validated by experimental data of rectangular channels (Jacket et al., 1958; Troy, 1958; Siman-Tov et al., 1995). The experiments include higher (14 MPa) and lower (2MPa) pressure conditions, and for rectangular channels, the two wide sides are heated and the narrow sides are adiabatic. As shown in Fig. 12, most of these data (91.8% of data) are predicted within a ±25% error band, which shows the present model in this paper has better ability to predict dryout in rectangular channels. Fig. 13(a)–(d) show the dependence analysis of the model on several thermalhydraulic parameters, including heated length/gap ratio, inlet

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Table 2 Experimental data for single channel used in the paper. Reference

No. of data

Channel type

Siman-Tov et al. (1995)

269 109 20 10 10 7 25

Jacket et al. (1958)

44

Troy (1958)

53

Circular Circular Circular Circular Circular Circular Rectangular (1.27  12.7 mm) Rectangular (1.27  25.4 mm/ 2.46  25.4 mm) Rectangular (1.9  57.15 mm)

Becker (1965) Thompson and Macbeth (1964) Saito et al. (1978) Judd et al. (1965)

Parameters range D (mm)

L (mm)

p (MPa)

G (kg m2 s1)

DTin (K)

q (MW m2)

3.93–24.95 4.57–37.47 12.6 11.3 11.3 11.3 2.309

432–2727 238.7–3657.6 1828–5328 1819.3 1819.3 1819.3 507

1.1–10.1 0.99–7.0 7.0 6.9–10.4 6.9–13.8 6.9–10.4 1.7

144–4050 93.58–2023.53 393–5235 660.5–2061.5 655–700 664.6–2146.9 2.8–28.4 (m s1)

3.9–240.4 4.4–247.6 10.7–90 27.3–88.3 11.5–173.3 26.5–58 16.2–31.6

0.491–5.146 0.502–3.534 0.39–2.05 63.4–143.3 kW (CC) 62–119.2 kW (IP) 53.7–103.7 kW (OP) 0.7–16.8

2.419/4.486

304.8/685.8

13.79

212.93–4462.06

4.44–186.1

0.536–3.912

3.678

1828.8

4.14–13.79

508.59–5434.49

20–297.78

0.410–2.685

CC: chopped cosine, IP: peak towards inlet, OP: peak towards outlet.

5. Application to rod bundles 1.4

Experimental data

1.2 1.0 0.8

xout

0.6

Annular and subsequent flow regions Mishima's correlation

0.4 0.2

Bubbly to churn flow regions 0.0

xe=0

-0.2

Subcooled flow regions -0.4 0.0

0.1

0.2

0.3

0.4

0.5

xtran Fig. 5. Categories of the present experimental data for single channel.

1.4 Data of circular tubes Ravg = 0.949 RSM = 0.097 STD = 0.082

1.3

5.1. Combining the present model with subchannel analysis

1.2

CHFR

1.1 1.0 0.9 0.8 0.7 0

1

2

3

4

Since the AFD mechanistic model has been proved to be practicable in single channel, its application to rod bundles is a natural step forward. Whalley (1977, 1978) and Saito et al. (1978) are pioneers in this field, who have applied their respective mechanistic models to the CHF prediction in rod bundles. A typical feature of rod bundles is the cross flow between subchannels, the flow and quality distributions need to be determined by considering lateral conservation equations. In fact, any subchannel code based on subchannel analysis, which has been verified on its accuracy, could be used for the analysis. Therefore, recently many studies have been reported on coupling the subchannel analysis code with the AFD mechanistic model (Tomiyama and Yokomizo, 1988; Mitsutake et al., 1990; Naitoh et al., 2002; Adamsson and Corre, 2008, 2011, 2014; Chandraker et al., 2012; Ahmad et al., 2013; Dasgupta et al., 2015; Zhang et al., 2019). Especially, most of the widely used correlations such as the deposition and entrainment of droplets are derived from flows in vertical tubes by fitting to experimental data of air-water, and their application in rod geometries has some uncertainty. However, the mechanistic models developed by previous scholars have showed that a good prediction of CHF in rod bundles can be obtained by using the correlations for tubes.

5

6

q (MW. m-2) Fig. 6. Comparisons between calculated and experimental CHF in uniformly heated vertical circular tubes.

subcooling, inlet mass flux and pressure. The model underestimates CHF at the lower mass fluxes but overestimates that at higher mass fluxes.

In this paper, the SACROM code developed by Cai et al. (2016) is used as subchannel analysis tool to obtain the distribution of thermal-hydraulic parameters. The basic equations of SACROM code are shown in Table 3. Owing to the fact that the SACROM code is developed based on coolant-centered approach, i.e. the open space between each set of four or three rods serves as control volume, as shown in Fig. 14, while the present model is established based on rod-centered, it is necessary to design a strategy for coupling calculation. In general, the strategy that each segment of rod (facing a specific subchannel) having different liquid-film flow has been widely adopted for subchannel codes. However, by such an approach, the non-uniform liquid film flow rates along circumferential direction for a rod would be obtained, while the film flow on the rod has a strong tendency to be uniform as mentioned by Butterworth (1968) and Whalley (1977). So Whalley (1978), Lim and Weisman (1988) and Talebi and Kazeminejad (2012) applied the rod-centered approach in their models. Based on the above reasons, this paper attempts to combine the AFD mechanistic model with the conventional subchannel code to make full use of their respective advantages. The subchannel code provides the neces-

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Data

1.4

+20%

1.2

+20% 1.2

CHFR

CHFR

Data

1.4

1.0

0.8

1.0

0.8

-20%

-20%

0.6

0.6

0

1000

2000

3000

4000

5000

0

50

100

G (kg.m-2 .s-1)

200

250

DTin (K)

(d) Inlet subcooling

(a) Inlet mass flux

Data

1.4

150

Data

1.4

1.2

+20%

1.2

CHFR

CHFR

+10% 1.0

0.8

0.8

-20%

-20%

0.6

0.6 0.0

0.2

0.4

0.6

0.8

1.0

0

50

100

150

200

250

300

350

xout

L/D

(b) Vapor quality at outlet

(e) Dimensionless tube length

Data

1.4

+20%

0.8

-20%

0.6

450

+20%

1.2

CHFR

1.0

400

Data

1.4

1.2

CHFR

1.0

1.0

0.8

-20% 0.6

0

2

4

6

p (MPa)

(c) System pressure

8

10

0

5

10

15

D (mm)

(f) Tube diameter

Fig. 7. Independence analysis of calculated CHF on relevant Thermal-Hydraulic parameters for circular tubes.

20

25

10

M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

1.5

Experimantal data Predicted value

3.5

3.0

1.3

+25%

1.2

2.5

CHFR

q (MW.m-2)

Chopped cosine Peak towards inlet Peak towards outlet

1.4

2.0

1.1 1.0

1.5

0.9 p=4.1MPa,

Tin=211.4K

L=2000mm, d=10.04mm

1.0 400

800

1200

1600

-20%

0.8 0.7

2000

0.1

G (kg. m-2 .s-1)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

xout

Fig. 8. The effect of mass flow rate on CHF.

Fig. 11. The effect of vapor quality at outlet on CHF.

1.6 Data of rectangular channels Ravg=1.024 RSM=0.144 STD=0.142

1.4

CHFR

1.2

1.0

0.8

0.6 0

2

4

6

8

10

12

14

16

18

q (MW. m-2) Fig. 12. Comparisons between calculated and experimental CHF in uniformly heated vertical rectangular channels.

Fig. 9. The effect of inlet subcooling on CHF.

sary boundary parameters for the mechanistic model and the latter calculates the mass flow rate of liquid film along the axial direction for each rod to judge the occurrence of CHF. In this paper the rod-centered approach is applied. As shown in Fig. 15, the rod bundle is divided into a series of coolant-centered subchannels based on geometric features, the thermal-hydraulic parameters in rod bundle are firstly calculated by the SACROM code. Then, for each heated rod, the fluid conditions in the surrounding area are obtained by weighted average of the respective control volumes. For example, the values of region (1, 1), (1, 2), (1, 3), and (1, 4) in Fig. 15 are averaged to get the vapor quality for volume (1), that is:

1.6

Ravg = 1.057 RSM = 0.151 STD = 0.143 1.4

CHFR

1.2

1.0

P4

0.8

0.6 40

xav g ð1Þ ¼

Chopped cosine Peak towards inlet Peak towards outlet 60

80

100

120

140

Q (kW) Fig. 10. Comparisons between calculated and experimental CHF in non-uniformly heated vertical circular tubes.

i¼1 xð1; iÞP ð1; iÞ P4 i¼1 P ð1; iÞ

ð35Þ

A similar equation is written to obtain mass fluxes G. Since the subchannel is divided into many segments along the axial direction, the AFD mechanistic model begins to work at the point where the equilibrium quality is 0. For an axial segment, the flow rates of liquid film, entrained droplets and vapor are first calculated, then, the entrained droplets and vapor in current volume is corrected at

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Data

1.6

1.6

Data

1.4

1.4

+25%

+25% 1.2

CHFR

CHFR

1.2

1.0

0.8

1.0

0.8

-20%

-20% 0.6

0.6 200

300

400

500

600

700

800

900

1000

0

50

100

150

(a) Heated length/gap ratio

300

(b) Inlet subcooling

Data

Data

1.6

1.4

1.4

+25%

+25%

1.2

1.2

CHFR

CHFR

250

DTin (K)

L/s

1.6

200

1.0

0.8

1.0

0.8

-20% 0.6

-20%

0.6

0

1000

2000

3000

4000

5000

6000

4

2

G (kg/m s)

6

8

10

12

14

p (MPa)

(c) Inlet mass flux

(d) Pressure

Fig. 13. Independence analysis of calculated CHF on relevant thermal-hydraulic parameters for rectangular channels.

dW g dðWxÞ ¼ dz dz

Table 3 Conservation equations in SACROM code. Mass continuity equation: P @ Ai @t qi þ @z@ mi þ j2i wij ¼ 0 Energy conservation equation:   P  P P s   @ qh Ai h@t ii þ @m@zi hi þ j2i hi  hj w0ij þ j2i h  wij ¼ qi  j2i lijij K T i  T j Axial momentum conservation equation:
 2  0 P P  K s mi 1 fi i þ @m@xi ui þ Ai @p j2i ui  uj wij  j2i u  wij  2 Di þ Dz Ai q @z ¼ g qi Ai cosh  f T

@mi @t

ð37Þ

where W and Wl are the total flow rate and flow rate of liquid phase calculated by SACROM code at every axial location, respectively. Once the liquid film vanishes on any of the rods (i.e. minimum film flowrate = 0), dryout in the rod bundle is considered to occur. Especially for unheated regions such as the channel wall, the vaporization term is absent in Eqs. (1)–(9).

i

Transverse momentum conservation equation:
 K w w @ ðuwij Þ @wij s G j ij j ij ¼ l ij pi  pj  2s @t þ @z l q ij

5.2. Assessment against rod bundles data

ij ij

each axial location before the next axial segment calculation to consider lateral flow. That is, ignoring the lateral flow of the liquid film, the liquid phase in the cross flow is considered to be all droplets, the flow rate of droplets at next axial location is given by:

dW d dW l dW f ¼  dz dz dz

ð36Þ

To assess the predictive ability of the present model, the prediction results are compared with experimental data for vertical rod bundles. All data come from the reports of Fighetti and Reddy (1982a,b), the experiments are mainly conducted in 4  4 rod bundles arranged in square pitch under Boiling Water Reactor (BWR) conditions, and contain the uniform and non-uniform radial heat flux distributions. Table 4 shows the operating conditions of the experimental data. An example of geometric cross-section of a rod bundle is shown in Fig. 16.

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M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

Fig. 16. The schematic diagram of cross-section of 4  4 rod bundle.

1.4

Fig. 14. The schematic of control volume of SACROM code.

Uniform radial power distributions Ravg = 1.032 RSM = 0.086 STD = 0.081

1.3

CHFR

1.2

1.1

1.0

0.9

0.8 0.6

0.9

1.2

1.5

1.8

2.1

2.4

q (MW. m-2) Fig. 17. Comparisons between calculated and experimental CHF in vertical rod bundles for uniform radial power distributions.

1.4 Fig. 15. Control volumes used by the driving subchannel code (SACROM) and the present dryout model.

1.3

Non-uniform radial power distributions Ravg = 0.986 RSM = 0.103 STD = 0.102

1.2

System pressure (MPa) Inlet mass flux (kg
m2
s1) Inlet subcooling (K) Radial heat flux distribution Axial heat flux distribution Structure of rod bundle Number of data

4.0–10.0 131.55–2442.61 5.0–156.0 uniform and non-uniform uniform 44 696

1.1

CHFR

Table 4 Experimental data for rod bundles used in the present work.

1.0 0.9 0.8 0.7 0.0

Figs. 17 and 18 show the comparisons between calculated and experimental CHF in rod bundles for the uniform and nonuniform radial power distributions, respectively. A total of 696 experimental data points are selected, including 113 data with uniform power distribution. It can be found that the vast majority of data (96.7% of data) are predicted within a ±20% error band. Then the dependence analysis of the present model on several thermal-hydraulic parameters is investigated, as shown in

0.5

1.0

1.5

2.0

2.5

3.0

q (MW. m-2) Fig. 18. Comparisons between calculated and experimental CHF in vertical rod bundles for non-uniform radial power distributions.

Fig. 19–21, including pressure, inlet mass flux and inlet subcooling. The results indicate that the calculated values of the model overestimate CHF for higher pressure and higher flow flux, and underes-

M. Gui et al. / Annals of Nuclear Energy 135 (2020) 106978

13

6. Conclusions

Data 1.4

+20%

CHFR

1.2

1.0

0.8

-20%

0.6 2

3

4

5

6

7

8

9

10

11

p (MPa) Fig. 19. CHF prediction accuracy against system pressure.

Data

1.4

+20%

CHFR

1.2

1.0

0.8

-20% 0.6 0

500

1000

1500

2000

2500

G (kg.m-2.s-1) Fig. 20. CHF prediction accuracy against inlet mass flux.

Data

1.4

In this paper, an annular film dryout (AFD) mechanistic model has been developed based on the mass and momentum conservation equations of three fields (the liquid film, entrained droplets and vapor core) together with a series of constitutive relations. The effect of some constitutive correlations (the entrainment and deposition of droplets and the onset of annular flow) on the prediction accuracy of the model is studied. It can be found: (1) the droplet entrainment and deposition correlations derived from Kataoka et al. (2000) and Okawa et al. (2002) are more suitable for current model; (2) the boiling entrainment has a significant effect on the prediction of CHF for the smaller L/D region (L/D < 220), but not for larger L/D region; (3) the present model has little dependence on different correlations of the onset of annular flow. To validate the proposed model, the prediction results are compared with experimental data both for uniformly heated and non-uniformly heated vertical circular tubes, as well as uniformly heated rectangular channels. The results indicates that almost all of the data are predicted within a ±25% error band. On the whole, the model underestimates CHF at the lower pressure but overestimates that at higher pressure, and for annular flow conditions of low vapor quality at outlet, the predicted deviation is relatively large. Further, through coupling with the subchannel analysis code (SACROM), the present model is used to predict the dryout-type CHF in the rod bundles. For uniform and non-uniform radial power distributions the present model predicted CHF well, within ±20%. The main shortcomings of the current work which need for further improvement are the following: (1) The liquid film, droplets and vapor in the annular flow region are assumed to be all saturated, which is unreasonable for the annular flow in the nonequilibrium state; (2) the entrainment and deposition correlations of droplets play a crucial role in establishment and prediction of the present model. However, current several of droplets entrainment and deposition correlations are developed based mainly on air-water adiabatic annular flow experiment at specific pressures, whose applicability is not guaranteed; (3) the current validation of the application of the mechanistic model to the rod bundles is not sufficient, especially the location of the dry-out spot for nonuniform axial power distribution; (4) the grid is the typical component in the rod bundles and it has been shown to have a great impact on CHF. Due to its complexity, the influence mechanism is not considered in the current model. Acknowledgements

+20%

CHFR

1.2

This work is supported by Science and Technology on Reactor System Design Technology Laboratory of NPIC (Nuclear Power Institute of China), to which the authors would like to express great gratitude. The authors also appreciate the support from Natural Science Foundation of China (Grant No. 11622541) and National Key R&D Program of China (Grant No. 2017YFE0302100).

1.0

0.8

-20% References 0.6 0

20

40

60

80

100

120

140

160

DTin (K) Fig. 21. CHF prediction accuracy against inlet subcooling.

timate CHF for lower pressure. Overall, after combining with subchannel analysis, the present model can also be used to predict the dryout-type CHF in the rod bundles with high precision.

Adamsson, C., Corre, J.M.L., 2008. MEFISTO: a mechanistic tool for the prediction of critical power in BWR fuel assemblies. In: 16th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, pp. 735–747. Adamsson, C., Corre, J.M.L., 2011. Modeling and validation of a mechanistic tool (MEFISTO) for the prediction of critical power in BWR fuel assemblies. Nucl. Eng. Des. 241 (8), 2843–2858. Adamsson, C., Corre, J.M.L., 2014. Transient dryout prediction using a computationally efficient method for sub-channel film-flow analysis. Nucl. Eng. Des. 280, 316–325. Ahmad, M., Chandraker, D.K., Hewitt, G.F., Vijayan, P.K., Walker, S.P., 2013. Phenomenological modeling of critical heat flux: the GRAMP code and its validation. Nucl. Eng. Des. 254 (702), 280–290. Anglart, H., Li, H., Niewinski, G., 2018. Mechanistic modelling of dryout and postdryout heat transfer. Energy, 352–360.

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