Development of void fraction-quality correlation for ...

Progress in Nuclear Energy 97 (2017) 38e52

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Development of void fraction-quality correlation for two-phase flow in horizontal and vertical tube bundles Takashi Hibiki a, *, Keyou Mao a, Tetsuhiro Ozaki b a b

School of Nuclear Engineering, Purdue University, 400 Central Drive, West Lafayette, IN 47907-2017, USA Tokai Works, Nuclear Fuel Industries, Ltd., 3135-41, Muramatsu, Tokai, Naka, Ibaraki 319-1196, Japan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 September 2016 Received in revised form 8 December 2016 Accepted 3 January 2017

A steam generator thermal-hydraulic code based on homogeneous flow model has been useful based on its numerical stability and simpler formulation. One of key parameters for a steam generator thermalhydraulic analysis is void fraction which determines two-phase mixture density and affects two-phase mixture velocity. These parameters are important for a heat transfer tube vibration analysis. A void fraction-quality correlation is very important to accurately convert the quality into the void fraction. The void fraction-quality correlation should preferably be applicable to parallel and cross flows in rod or tube bundles since two-phase flow in the steam generator encounters flow configuration change from the parallel flow along the tube bundle in the riser section of the steam generator to the cross flow in the Ubend section of the steam generator. A set of correlations depending on flow configuration such as parallel and cross flows, rod or tube array pattern and mass flux is developed based on legacy Smith correlation. The correlation agrees with the parallel and cross flow data with the mean absolute error (or bias) of 0.117% and the standard deviation (random error) of 2.26% and with the mean absolute error (or bias) of 0.760% and the standard deviation (random error) of 6.21%, respectively. The correlations are further simplified to a single correlation applicable for parallel and cross flow in rod or tube bundles. The Smith correlation with a modified constant entrainment parameter e being 0.5 is recommended for predicting void fraction in the steam generators. The Smith correlation with e ¼ 0.5 is expected to be applicable for parallel and cross flows with various rod or tube array patterns including normal square, parallel triangular and normal triangular arrays. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Void fraction Cross flow Horizontal bundle Rod bundle U tube Steam generator

1. Introduction A steam generator is a type of heat exchanger. Water that has passed through a nuclear reactor core (“the primary fluid”) is carried through the steam generator within thousands of metal tubes, known as “heat transfer tubes.” Some of the heat contained in the primary fluid is conveyed through the walls of the heat transfer tubes to water flowing outside of the tubes (“the secondary fluid”). The secondary fluid is water at the steam generator inlet, but the water boils into a two-phase mixture (steam/water) as heat transfers from the primary fluid to the secondary fluid, so that a good portion of the secondary fluid has become steam as it reaches the steam generator outlet. After leaving the steam generator, the steam is the driving force that rotates a turbine to generate

* Corresponding author. E-mail address: [email protected] (T. Hibiki). http://dx.doi.org/10.1016/j.pnucene.2017.01.003 0149-1970/© 2017 Elsevier Ltd. All rights reserved.

electricity. Some of steam generators have experienced some problems such as tube support corrosion, tube-sheet corrosion, tubing corrosion, fretting fatigue cracking and impingement, which have led to unplanned outages (Green and Hetsroni, 1995). To avoid these problems, a steam generator is designed with an input based on detailed three-dimensional local thermal-hydraulic conditions computed by steam generator thermal-hydraulic codes. A porous media approach is usually utilized in the steam generator thermalhydraulic codes. A control volume in the porous media approach includes volumes of structures and flow channels. The porosity is defined by the ratio of volume of flow channels to total volume. Various reliable steam generator thermal-hydraulic codes have been developed based on different two-phase flow porous media formulations. CAFCA code developed by EDF (Electricite de France in France), THIRST code developed by AECL (Atomic Energy of Canada Limited) and FIT-III code developed by MHI (Mitsubishi Heavy Industries,

T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

Nomenclature

NG;P

a B1 B2 C0 C1 c1 c2 D DC e

NLa NRe NRi NWe Nmf P p pcrit q_ s S sd t Vgj Vgj;B

eV f G GH GP g hfg hm j jg jþ g j*g jg;crit K0 L La md mrel mrel;ab N NCa NFr NG NG;H

gap between tubes parameter parameter distribution parameter parameter parameter parameter tube (or rod) diameter flow path diameter entrainment factor defined as ratio of mass of liquid droplets entrained in gas core to total mass of liquid volume porosity friction force per unit volume exerted on secondaryside fluid by embedded solids mass flux mass flux based on hydraulic equivalent diameter mass flux based on minimum pitch between rods (or minimum gap between rods) gravitational acceleration latent heat mixture enthalpy of secondary fluid mixture volumetric flux superficial gas velocity non-dimensional superficial gas velocity non-dimensional superficial gas velocity critical superficial gas velocity parameter parameter Laplace length scale mean absolute error mean relative deviation mean absolute relative deviation number of sample Capillary number Froude number non-dimensional mass flux non-dimensional mass flux based on hydraulic equivalent diameter

Ltd.) adopt homogeneous flow model composed of three transport equations such as mass, momentum and energy conservation equations (Boivin et al., 1987; Carver et al., 1981; Hirao et al., 1993). The velocity slip is considered through a void fraction-quality correlation. ATHOS code developed by EPRI (Electric Power Research Institute) utilizes algebraic slip model composed of three transport equations such as mass, momentum and energy conservation equations (Singhal et al., 1982). The velocity slip is considered through the momentum equation with a drift-flux type correlation. PORTHOS code developed by EPRI uses two-fluid model composed of six transport equations such as mass, momentum and energy conservation equations for gas and liquid phases (Chan et al., 1986). The outputs of the steam generator thermal-hydraulic codes are utilized for improving the steam generator design and stability analysis of fluid-elastic vibration. In order to enhance the code prediction capability, reliable constitutive equations are indispensable. Among the constitutive equations, void fraction correlation is very important, because void fraction affects two-phase mixture density directly and two-phase mixture velocity. Void fraction constitutive correlations are often

Vgj;P Vgm vf vg vm x

39

non-dimensional mass flux based on minimum pitch between rods (or minimum gap between rods) non-dimensional Laplace length scale Reynolds number Richardson number Weber number viscous number tube (or rod) pitch pressure critical pressure heat source (rate of heat transfer) per unit volume slip ratio standard deviation time drift velocity drift velocity computed by Ishii's bubbly flow correlation drift velocity computed by Kataoka-Ishii's correlation relative velocity of vapor with reference to mixture velocity liquid velocity gas velocity mixture velocity quality

Greek symbols void fraction parameter density difference absolute viscosity of liquid phase gas density liquid density mixture density surface tension

a g Dr mf rg rf rm s

Subscripts cal. calculated value exp. experimental value

given for each flow regime, each channel geometry, and each channel orientation, but it is preferred to use a single void fraction constitutive correlation in the code. However, since the two-phase flow structure changes from parallel flow along tube bundles in a vertically straight section to cross flow in a U-bend tube section, it is challenging to develop a single void fraction constitutive correlation which is applicable for all void fraction range in the steam generator. From this point of view, this study aims to develop a void fraction-quality correlation for maintaining the prediction accuracy of steam generator thermal-hydraulic codes developed based on homogeneous flow model. First, basic two-phase flow porous media formulations are reviewed to highlight the importance of void fraction correlations. A brief literature survey on existing void fraction correlations and tube (or rod) bundle data for parallel and cross flows follows. Then new void fraction correlation is developed for parallel and cross flows encountered in a steam generator.

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T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

2. Basic two-phase flow porous media formulations A porous media two-phase flow formulation has been utilized to simulate thermal-hydraulic parameters in a steam generator (Carver et al., 1981; Singhal et al., 1982; Boivin et al., 1987; Hirao et al., 1993). A homogeneous equilibrium flow model in Cartesian coordinate is given by (Singhal et al., 1982). Mass conservation equation

  v
 v
V  v
V v
V e rm þ e rm vm;x þ e rm vm;y þ eV rm vm;z vt vx vy vz ¼0 (1)

Momentum conservation equation

where t, eV , rm , vm , p, f , g, hm and q_ s are, respectively, the time, volume porosity, mixture density, mixture velocity, pressure, friction force per unit volume exerted on secondary-side fluid by embedded solids, gravitational acceleration, mixture enthalpy of secondary fluid and heat source (rate of heat transfer) per unit volume. Subscripts of x, y and z are, respectively, the components of x, y and z directions. A constitutive equation relating void fraction, a, and quality, x, plays an important role in the homogeneous equilibrium model. The homogeneous equilibrium model has been utilized in several steam generator thermal-hydraulic analysis codes such as FIT-III code (Hirao et al., 1993) developed by Mitsubishi Heavy Industries, Ltd. in Japan, CAFCA code (Boivin et al., 1987) developed by Electricite de France in France, THIRST code (Carver et al., 1981) developed by Atomic Energy of Canada Limited in Canada. In the homogeneous equilibrium model, the velocity slip between the two phases are considered through the two-phase mixture density computed by a void fraction-quality correlation. An algebraic slip model considers the velocity slip between the two phases through the drift velocity in the momentum equation. The algebraic slip model in Cartesian coordinate is given by (Singhal

   v
 v
V v
V v
V vp e rm vm;x þ e rm vm;x vm;x þ e rm vm;y vm;x þ eV rm vm;z vm;x ¼ eV  fx vt vx vy vz vx    v
 v
V v
V v
V V V vp e rm vm;y þ e rm vm;x vm;y þ e rm vm;y vm;y þ e rm vm;z vm;y ¼ e  fy vt vx vy vz vy      v
 v
V v
V v
V vp e rm vm;z þ e rm vm;x vm;z þ e rm vm;y vm;z þ eV rm vm;z vm;z ¼ eV þ rm g  fz vt vx vy vz vz

(2)

et al., 1982). Mass conservation equation Energy conservation equation

  v
 v
V  v
V v
V e rm þ e rm vm;x þ e rm vm;y þ eV rm vm;z vt vx vy vz

  o vn V v
V v
V e ðrm hm  pÞ þ e rm vm;x hm þ e rm vm;y hm vt vx vy  v
V e rm vm;z hm þ vz ¼ q_ s

¼0 (4)

(3)

Momentum conservation equation

   v
 v
V v
V v
V vp e rm vm;x þ e rm vm;x vm;x þ e rm vm;y vm;x þ eV rm vm;z vm;x ¼ eV  fx vt vx vy vz vx    v
 v
V v
V v
V vp e rm vm;y þ e rm vm;x vm;y þ e rm vm;y vm;y þ eV rm vm;z vm;y ¼ eV  fy vt vx vy vz vy ) (  



    rm rg a 2 v V v v v v V V V V vp V e rm vm;z þ e rm vm;x vm;z þ e rm vm;y vm;z þ e rm vm;z vm;z ¼ e þ rm g  fz  V e vt vx vy vz vz vz rf ð1  aÞ gm;z

(5)

T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

41

A two-fluid model in Cartesian coordinate is given by Lee et al. (2009) and Ishii and Hibiki (2010). 3. Existing works

Energy conservation equation

3.1. Existing void fraction correlations

  o vn V v
V v
V e ðrm hm  pÞ þ e rm vm;x hm þ e rm vm;y hm vt vx vy  v
V e rm vm;z hm þ vz  v
V e rg aVgm;z hfg ¼ q_ s  vz

As indicated in Fig. 1, parallel and cross flows along tube bank appear in a steam generator. The parallel and cross flows may be dominant in the riser and U-bend sections of the steam generator, respectively. In what follows, existing void fraction correlations for parallel and cross flows are briefly reviewed.

(6) where rg , rf and hfg are, respectively, the gas density, liquid density and latent heat. The relative velocity of vapor with reference to mixture velocity in z direction, Vgm;z , is represented by

Vgm;z ¼

rf Vgj;z þ ðC0  1Þrf vm;z C0 arg þ ð1  C0 aÞrf

(7)

where C0 and Vgj are, respectively, the distribution parameter and drift velocity appeared in a drift-flux model (Zuber and Findlay, 1965) as

jg vg ≡ ¼ C0 j þ Vgj

a

(8)

where vg , jg and j are, respectively, the gas velocity, superficial gas velocity and mixture volumetric flux. It should be noted here that the velocity slip in horizontal directions in the momentum equation is neglected in the above algebraic slip model for simplicity but this assumption has not been validated. Constitutive equations predicting the distribution parameter and drift velocity play an important role in the algebraic slip model. The algebraic slip has been utilized in a steam generator thermal-hydraulic analysis code such as ATHOS code (Singhal et al., 1982) developed by Electric Power Research Institute in the USA.

3.1.1. Parallel flow in vertical rod bundle Existing void fraction correlations for parallel flows in vertical tube banks (or rod bundles) have been reviewed by Julia et al. (2009) and Ozaki et al. (2013). The performance of void fraction correlations for parallel flows in vertical rod bundles has been evaluated by Coddington and Macian (2002). Recently, Ozaki et al. (2013) and Ozaki and Hibiki (2015) developed the following driftflux type correlation using Nuclear Power Engineering Corporation (NUPEC) 8  8 vertical rod bundle data taken at elevated pressure conditions from 1.0 MPa to 7.2 MPa, which covered void fraction condition up to 0.87 under typical steam generator pressure conditions. Ozaki correlation (2013).

sffiffiffiffiffi C0 ¼ 1:1  0:1

rg rf

(9)

and

 n
o
þ : Vgj ¼ Vgj;B exp 1:39jþ g þ Vgj;P 1  exp 1:39jg

(10)

Vgj;B and Vgj;P are, respectively, the drift velocities computed by Ishii's bubbly flow correlation (1977) and Kataoka-Ishii correlation

Fig. 1. Schematic diagram of parallel flow and cross flows.

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T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

(1987) and non-dimensional superficial gas velocity is defined by




jg !1=4 :

jþ g≡

Dowlati correlation (1992).

(11)

Drgs r2f

a ¼ 1  1 þ c1 j*g þ c2 j*2 g

0:5

:

(16)

Non-dimensional superficial gas velocity, j*g , is defined by

where Dr and s are, respectively, the density difference between phases and surface tension. Lellouche and Zolotar (1982) developed the following drift-flux type correlation which has been utilized in the EPRI ATHOS code (THD-AESJ, 1995; Singhal et al., 1982).

xGP ffi: j*g ≡qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DrgDrg

(17)

Feenstra correlation (2000). Lellouche-Zolotar correlation (1982).

1  x rg 1þS x rf

a¼ L C0 ¼ K0 þ ð1  K0 Þag

(12)

where

rg K0 ¼ B1 þ ð1  B1 Þ rf ¼

!1=4

r

1  eC1 a ;L ¼ ;g ¼ 1  eC1

!

1 þ 1:57 rg f 1  B1

; C1

The slip ratio, S, is given by

S ¼ 1 þ 25:7ðNRi NCa Þ0:5

 1 P D

(19)

where P is the tube (or rod) pitch. Richardson number, NRi , and Capillary number, NCa , are defined by

NRi ≡

pðpcrit  pÞ

B1 ¼

(18)

Dr2 ga

4p2crit

(

!1

(20)

G2P

where a is the gap between tubes, and

minð0:80 ; B2 Þ minð0:71 ; B2 Þ

for round tube or rod bundle ; B2 for rectangular channel

mf NCa ≡ s

1   ¼ 1 þ exp  NRe 105

Vgj ¼ 1:41

Drgs r2f

!1=4

ð1  aÞ1=2 1þa

(13)

where NRe , p and pcrit are, respectively, the Reynolds number, pressure and critical pressure. Lellouche-Zolotar correlation is applicable at pressure higher than 1.36 MPa. 3.1.2. Cross flow in horizontal tube bundle Feenstra et al. (2000) reviewed existing void fraction correlations for cross flows in horizontal tube bundles. Those correlations include correlations developed by Dowlati et al. (1992) and Schrage et al. (1988). Feenstra et al. (2000) also developed a correlation based on their R-11 cross flow data. They evaluated the performance of void fraction correlations for cross flows in horizontal tube bundles.

xGP

! (21)

arg

where mf is the absolute viscosity of liquid phase. It should be noted here that Feenstra correlation (2000) is not explicit for void fraction, which means that an iteration calculation is necessary to compute void fraction. 3.1.3. Other correlations In addition to the correlations developed for parallel and cross flows in rod bundles, two important void fraction-quality correlations are briefly reviewed. They are Armand-Massena correlation (Armand, 1946; Massena, 1960) and Smith correlation (Smith, 1969) as given by Eqs. (22) and (23), respectively, Armand-Massena correlation (1960).

1  x rg a ¼ ð0:833 þ 0:167xÞ 1 þ x rf

!1 :

(22)

Schrage correlation (1988).

n

0:191 a ¼ 1 þ 0:123NFr lnðxÞ

o

1  x rg 1þ x rf

Smith correlation (1969).

!1 :

(14) 2

a ¼ 41 þ

The Froude number, NFr , is defined by

G Fr≡ pPffiffiffiffiffiffi rf gD

(15)

where GP and D are, respectively, the pitch mass flux defined by the mass flux based on the minimum pitch between rods (or minimum gap between rods) and tube (or rod) diameter.

rg rf



  8rf 9 3    e=1=2 1 þ 1x < r x g rg 1  x 1x 5   eþ ð1  eÞ : ; x x rf e 1 þ 1x x (23)

where e is the entrainment factor defined as the ratio of the mass of liquid droplets entrained in the gas core to the total mass of liquid. An alternative form of Smith correlation is

T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

(

)1



r 1x a¼ 1þS g x rf

:

(24)

The slip ratio is given as

8rf < e ¼ 0:0397NG;P for Normal Square Array 0:571 for Parallel Triangular Array e ¼ 0:0523NG;P > : 0:571 for Normal Triangular Array e ¼ 0:0637NG;P

(34)

It should be noted here that the mass flux in Eq. (34) is pitch mass flux, GP calculated using a minimum pitch between rods (or minimum gap between rods). In order to emphasize the mass flux based on the minimum pitch between rods, GP , used in Eq. (34), the subscript of P is added to the symbol of the non-dimensional mass flux as NG;P ð≡GP =rg jg;crit Þ. It should be also noted that the maximum e parameter should be physically unity. When non-dimensional mass flux increases, the value of e parameter reaches unity at a certain non-dimensional mass flux (i.e. critical non-dimensional mass flux). The value of e parameter should be fixed at unity at non-dimensional mass flux higher than the critical nondimensional mass flux.

e ¼ min½Eq: ð34Þ; 1

(35)

5. Results and discussion 5.1. Performance evaluation of existing void fraction correlations for parallel and cross flow data Some statistical parameters are introduced here to evaluate the performance of existing void fraction correlations for existing parallel and cross flow data. They are the mean absolute error, md , standard deviation, sd , mean relative deviation, mrel , and mean absolute relative deviation, mrel;ab .

md ¼

N


E  D 1 X aðiÞcal:  aðiÞexp: ; N i¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N n


E o2 u 1 X  D aðiÞcal:  aðiÞexp:  md ; sd ¼ t N1

(36)

(37)

i¼1

mrel

E

 D N aðiÞcal:  aðiÞexp: 1 X D E ; ¼ N i¼1 aðiÞ

(38)

exp:

mrel;ab



E  D N aðiÞ cal:  aðiÞexp: 1 X D E ; ¼ N i¼1 aðiÞ

(39)

exp:

Fig. 4. Dependence of e parameter on non-dimensional mass flux for cross flow in horizontal tube bundles.

where N, aexp: and acal: are, respectively, the number of sample, measured void fraction, and calculated void fraction. Table 3 shows the results of the performance evaluation of existing void fraction correlations for parallel flow data in vertical rod bundles. The Smith correlation with modified e parameter calculated by Eq. (32) agrees with the parallel flow Type-I data with the mean absolute error (or bias) of 0.117% and the standard deviation (or random error) of 2.26%. The original Smith correlation with e ¼ 0.4, Armand-Massena correlation and Ozaki correlation also show excellent agreements with the mean absolute errors of 0.586%, 1.36% and 0.498% and the standard deviations of 3.70%, 3.74% and 2.02%, respectively. Lellouche-Zolotar correlation tends to overestimate the void fraction by 3.05%. The Smith correlation with modified e parameter calculated by Eq. (32) agrees with the parallel flow Type-II data with the mean absolute error (or bias) of 0.216% and the standard deviation (random error) of 1.73%.

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T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

Table 3 Performance evaluation of existing void fraction correlations for parallel flow data in vertical rod bundles. Comparison of Correlations with NUPEC Type-I Bundle Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Smith Correlation with e ¼ 0.5 Armand-Massena Correlation Ozaki's Correlation Lellouche-Zolotar Correlation

0.117 0.586 2.21 1.36 0.498 3.05

1.22 0.277 5.16 3.73 1.35 5.82

3.67 5.47 6.29 5.87 3.17 5.83

2.26 3.70 3.54 3.74 2.02 1.46

Comparison of Correlations with NUPEC Type-II Bundle Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Smith Correlation with e ¼ 0.5 Armand-Massena Correlation Ozaki's Correlation Lellouche-Zolotar Correlation

0.216 2.19 0.634 0.354 1.30 1.44

0.0557 2.86 1.78 0.270 2.06 2.42

2.39 5.85 4.67 4.88 2.46 2.66

1.73 3.51 3.46 3.63 1.36 1.50

Table 4 Performance evaluation of existing void fraction correlations for cross flow data in horizontal tube bundles. Comparison of Correlations with All Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Armand-Massena Correlation Schrage Correlation Dowlati Correlation Feenstra Correlation Lellouche-Zolotar Correlation

0.760 9.84 11.1 16.0 6.07 1.93 22.1

13.5 38.0 40.3 45.8 19.5 3.66 58.7

20.5 39.5 41.9 47.6 27.6 12.9 58.7

6.21 8.81 9.77 11.6 10.3 4.89 6.60

Comparison of Correlations with Normal Square Pitch Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Armand-Massena Correlation Schrage Correlation Dowlati Correlation Feenstra Correlation Lellouche-Zolotar Correlation

1.62 10.5 11.4 15.8 2.60 1.13 20.6

21.5 50.9 51.8 53.1 9.66 1.06 60.6

28.8 52.5 53.8 57.3 17.5 14.7 60.6

7.23 10.1 10.9 11.6 6.24 4.14 7.04

Comparison of Correlations with Parallel Triangular Pitch Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Armand-Massena Correlation Schrage Correlation Dowlati Correlation Feenstra Correlation Lellouche-Zolotar Correlation

0.252 13.3 15.3 7.38 16.8 0.125 22.2

5.65 35.2 39.3 20.5 50.0 0.128 55.0

12.2 35.2 39.4 20.8 51.1 8.97 55.0

4.13 5.16 6.09 3.89 7.03 4.14 4.76

Comparison of Correlations with Normal Triangular Pitch Data

md [%]

mrel [%]

mrel.ab [%]

sd [%]

Smith Correlation with Entrainment Factor Computed by New Correlation Original Smith Correlation with e ¼ 0.4 Armand-Massena Correlation Schrage Correlation Dowlati Correlation Feenstra Correlation Lellouche-Zolotar Correlation

0.140 4.64 5.63 26.7 2.05 5.61 24.2

11.5 22.9 24.9 66.1 3.59 12.1 60.3

18.7 26.0 27.7 66.1 13.4 15.0 60.3

6.65 8.07 9.08 8.73 7.01 4.76 7.33

Armand-Massena correlation also shows excellent agreements with the mean absolute error of 0.354% and the standard deviation of 3.63%. The original Smith correlation with e ¼ 0.4 and Ozaki correlation tend to underestimate the void fraction by 2.19% and 1.30%, respectively, and Lellouche-Zolotar correlation tends to overestimate the void fraction by 1.44%. The above performance evaluation for parallel flow in vertical rod bundles is conducted for the conditions of pressure from 1.0 to 8.6 MPa, mass flux from 280 to 2000 kg/m2s, void fraction up to 0.87 and quality up to 0.25. Table 4 shows the results of the performance evaluation of existing void fraction correlations for cross flow data in horizontal tube bundles. The Smith correlation with modified e parameter calculated by Eq. (34) agrees with the cross flow data with the mean

absolute error (or bias) of 0.760% and the standard deviation (random error) of 6.21%. Feenstra correlation and Dowlati correlations also show good agreements with the mean absolute errors of 1.93% and 6.07% and the standard deviations of 4.89% and 10.3%, respectively. The prediction accuracies of the original Smith correlation with e ¼ 0.4, Armand-Massena correlation and Schrage correlation are deteriorated with the mean absolute errors of 9.84%, 11.1% and 16.0% and the standard deviations of 8.81%, 9.77% and 11.6%, respectively. Lellouche-Zolotar correlation tends to underestimate the void fraction by 22.1%. It should be noted that the applicable range of original Smith correlation, Armand-Massena correlation and Lellouche-Zolotar correlation does not cover cross flow. The above performance evaluation for cross flow in horizontal

T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

47

Fig. 5. Comparison between newly developed correlation and NUPEC data for Type-I bundle.

tube bundles is conducted for the conditions of density ratio from 0.00629 to 0.00220, mass flux from 90 to 1000 kg/m2s, void fraction up to 0.90 and quality up to 0.27.

5.2. Performance evaluation of modified Smith correlation for parallel and cross flow data The section 5.1 shows the overall performance of existing void fraction correlations for parallel and cross flow data. The newly developed correlation predicts the NUPEC parallel flow data with the mean absolute error of 0.117% and the standard deviation of 2.26% for the Type-1 bundle and with the mean absolute error of 0.216% and the standard deviation of 1.73% for the Type-II bundle. The correlation also predicts the existing cross flow data from 5 different sources including normal square, parallel triangular and normal triangular arrays with the mean absolute error of 0.760% and the standard deviation of 6.21%. Figs. 5 and 6 compare the void fraction computed by newly developed correlation and NUPEC data for Type-I and Type-II bundles, respectively. The upper left, upper right, lower left and lower right figures show the comparisons for the pressures of 1.0,

4.0, 7.2 and 8.6 MPa, respectively. Except for lowest mass flux cases such as GH ¼ 290 and 300 kg/m2s, all measured void fractions show insignificant effect of mass flux on void fraction-quality relationship regardless of the bundle type. The developed correlation agrees with the NUPEC data in the tested conditions and simulates the effect of mass flux on void fraction-quality relationship very well. As discussed above, the newly developed correlation has been validated by rod bundles with normal square pitch. It is expected that the newly developed correlation obtained for the parallel flow in the square pitch rod bundle geometry may be applicable for triangular pitch because the flow channel is very similar between square and triangular pitch rod bundles for parallel flow considering the fact that even significant channel geometry difference between square pitch rod bundle and round tube (e ¼ 0.4) does not cause significant e parameter difference. Thus, it is considered that the rod pitch array does not affect the void fraction of parallel flow in a vertical bundle significantly. Figs. 7e9 compare the void fraction computed by newly developed correlation with R-11 data taken by Feenstra et al. (1995), air-water data taken by Dowlati et al. (1992) and airwater data taken by Noghrehkar (1996), respectively. The

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T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

Fig. 6. Comparison between newly developed correlation and NUPEC data for Type-II bundle.

developed correlation agrees with the data and simulates the effect of mass flux on void fraction-quality relationship very well. Red lines in Figs. 7e9 show the approximate void fraction (¼0.8) at the boundary between the riser section and U-bend region of a steam generator. 5.3. Simplification of modified Smith correlation for parallel and cross flows in steam generators

Fig. 7. Comparison between newly developed correlation and Feenstra's data (1995) with parallel triangular array of P/D ¼ 1.44.

measured void fractions show significant effect of mass flux on void fraction-quality relationship regardless of the rod array type. The

It is validated in the previous section that the Smith correlation with the modified e parameter can predict void fraction very well. It is also indicated that the e parameter for parallel gas-liquid flows in vertical rod bundles, Eq. (32) is different from the e parameter for cross gas-liquid flows in horizontal tube bundles, Eq. (34). A twophase mixture in a steam generator first flows through a riser section (parallel flow along a vertical tube bundle) and passes through a U-bend region (mixture of parallel and cross flows along or over an inclined tube bundle). In order to use the Smith correlation for a steam generator thermal-hydraulic code, an interpolation scheme at the boundary between the riser section and the Ubend region should be introduced because the constitutive

T. Hibiki et al. / Progress in Nuclear Energy 97 (2017) 38e52

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Fig. 8. Comparison between newly developed correlation and Dowlati's data (1992) with (a) normal square array of P/D ¼ 1.3, (b) normal square array of P/D ¼ 1.75, (c) normal triangular array of P/D ¼ 1.3, and (d) normal triangular array of P/D ¼ 1.75.

Fig. 9. Comparison between newly developed correlation and Noghrehkar's (1996) with (a) normal square array of P/D ¼ 1.47 and (b) normal triangular array of P/D ¼ 1.47.

correlation of e parameter is different between the riser section and the U-bend region. An additional interpolation scheme for a complicated flow (mixture of parallel and cross flows in an inclined tube bundle) in the U-bend region should be also considered. However, if a constant e parameter can be approximately used for both parallel and cross flows, a single e parameter can be used for a

whole region of the steam generator. This approximate approach can solve the above problem. The void fraction at the boundary between the riser and U-bend regions is considered about 0.8. Thus, it is preferable to find the constant e parameter for parallel flow in a vertical rod bundle at void fraction lower than 0.8 to be equal to that for cross flow in a

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Fig. 10. Comparison of void fractions calculated by Smith correlation with e ¼ 0.4 and 0.5.

vertical rod bundles with the bias of 2.21%. As indicated in Fig. 8, the Smith correlation with e ¼ 0.5 also predicts the void fraction of cross flow in horizontal tube bundles with triangular array at the void fraction higher than 0.8 indicated by a red solid line fairly well but tends to overestimate the void fraction of cross flow in horizontal tube bundles with square array at void fraction higher than 0.8. However, the overestimated void fraction is conservative for a vibration analysis. The stability ratio is calculated by the ratio of effective two-phase mixture velocity and critical two-phase mixture velocity for fluid elastic vibration. Local two-phase mixture velocity is calculated by dividing local mass flux by local two-phase mixture density. Higher void fraction causes lower twophase mixture density or higher two-phase mixture velocity. Thus, higher void fraction is conservative for the vibration analysis. It should be also noted here that Smith found the most suitable value of e parameter to be between 0.3 and 0.5 (Smith, 1969). The e parameter of 0.5 is within the value originally recommended by

Fig. 11. Sample calculations at prototypic steam generator pressure condition.

horizontal rod bundle at void fraction higher than 0.8. It is recommended to use e ¼ 0.5 for simulating two-phase flow in a whole region of a steam generator. As indicated in Table 3, the Smith correlation with e ¼ 0.5 predicts the void fraction in parallel flow in

Smith for pipe flows. Fig. 10 compares the void fractions calculated by Smith correlation with e ¼ 0.4 and 0.5. A red line in Fig. 10 shows the approximate void fraction (¼0.8) at the boundary between the riser section and U-bend region of a steam generator.

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Fig. 11 shows sample calculations of void fraction and quality relationship for parallel and cross flows at prototypic steam generator pressure condition. The upper left, upper right, lower left and lower right figures are the calculations for parallel flow with normal square pitch, cross flow with normal square pitch, cross flow with parallel triangular pitch and cross flow with normal triangular pitch, respectively. A red line in Fig. 11 shows the approximate void fraction (¼0.8) at the boundary between the riser section and U-bend region of a steam generator. The mass flux in a steam generator is around 400e600 kg/m2s. The Smith correlation with e ¼ 0.5 overestimates the newly developed correlation for parallel flow up to about 5% at void fraction of 0.8. Considering the standard deviation of the newly developed correlation being 2.26%, the choice of e ¼ 0.5 is reasonable to secure sufficient conservatism. The Smith correlation with e ¼ 0.5 overestimates the newly developed correlation for cross flow up to about 7% at void fraction of 0.8. Considering the standard deviation of the newly developed correlation being 6.21%, the choice of e ¼ 0.5 is reasonable to secure sufficient conservatism. Thus, it is recommended to use e ¼ 0.5 for simulating two-phase flow in a whole region of a steam generator. 6. Conclusions To maintain the accuracy of void fraction predicted by a steam generator thermal-hydraulic code developed based on homogeneous flow model, it is indispensable to develop a reliable void fraction-quality correlation applicable to both parallel and cross flows in rod or tube bundles. From this point of view, intensive literature survey of existing correlations and data for parallel and cross flows in rod or tube bundles is performed. It is shown that the relationship between quality and void fraction is dependent on flow configuration such as parallel and cross flows, rod or tube array pattern and mass flux. The framework of Smith correlation with one empirical constant being entrainment factor or e parameter defined as the ratio of the mass of liquid droplets entrained in the gas core to the total mass of liquid is found to be useful for

51

developing a correlation. Based on this observation, the e parameter depending on the flow configuration and rod or tube array pattern is correlated with a dominant parameter such as nondimensional mass flux. The scaled non-dimensional parameter is obtained by considering a key velocity scale being the critical superficial gas velocity at annular flow transition. The correlations for parallel flow with normal square array rod or tube and cross flow with normal square, parallel triangular and normal triangular array rod or tube are obtained based on test data taken in various flow configurations and rod or tube array patterns. The Smith correlation with modified e parameter agrees with the parallel and cross flow data with the mean absolute error (or bias) of 0.117% and the standard deviation (random error) of 2.26% and with the mean absolute error (or bias) of 0.760% and the standard deviation (random error) of 6.21%, respectively. The developed correlations can be utilized for predicting void fraction in steam generators as well as common heat exchanges. For the purpose of developing a useful correlation for steam generators, the developed correlations are further simplified to a single correlation applicable to parallel and cross flow in rod or tube bundles. The Smith correlation with a modified constant e parameter being 0.5 is recommended for predicting void fraction in a whole region of a steam generator. The Smith correlation with e ¼ 0.5 is expected to be applicable to parallel and cross flows with various rod or tube array patterns including normal square, parallel triangular and normal triangular arrays.

Appendix. NUPEC Type-I and Type-II rod bundles In the NUPEC experiments, two types of test section simulating different types of fuel assembly were used. They are Type-I and Type-II rod bundle test sections. The schematic diagram of the cross-sectional views for each test sections are given in Fig. A1, and their detailed geometrical information is given in Table A1 (Ozaki et al., 2013).

Fig. A1. Schematic diagram of the cross-sectional views for (a) NUPEC Type-I bundle and (b) NUPEC Type-II bundles (Ozaki et al., 2013).

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Table A1 Detailed geometrical information of NUPEC Type-I and Type-II bundles (Ozaki et al., 2013). Assembly Type

Type-I

Type-II

Rod array Number of heater rods Diameter of heater rod Active heater length Number of simulated water rod (unheated) Diameter of simulated water rod Pitch of heater rod Width of flow channel

88 62 12.3 mm 3.7 m 2 15.0 mm 16.2 mm 132.5 mm

88 60 12.3 mm 3.7 m 1 34.0 mm 16.2 mm 132.5 mm

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