Evaluation analysis of correlations for predicting the void fraction and slug velocity of slug flow in an inclined narrow rectangular duct

Nuclear Engineering and Design 273 (2014) 155–164

Contents lists available at ScienceDirect

Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Evaluation analysis of correlations for predicting the void fraction and slug velocity of slug flow in an inclined narrow rectangular duct Chaoxing Yan a , Changqi Yan a,∗ , Yunhai Shen b , Licheng Sun a , Yang Wang a a b

Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001, China Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu 610041, China

h i g h l i g h t s • • • • •

46 void fraction correlations are evaluated on void fraction. Evaluation of void fraction correlations on slug velocity is studied. Effect of void fraction correlations on separated frictional pressure drop is studied. Drift-flux type correlation shows best agreement with experimental data. Evaluation is investigated in different flow regions.

a r t i c l e

i n f o

Article history: Received 24 November 2013 Received in revised form 17 March 2014 Accepted 21 March 2014

a b s t r a c t A visualized investigation was conducted on inclined upward air–water slug flow in a narrow rectangular duct with the cross section of 43 mm × 3.25 mm. The slug velocity and void fraction were obtained through image processing. 46 correlations for predicting void fraction, covering the types of slip ratio, Kˇ, Lockhart and Martinelli, drift-flux and general were evaluated against the experimental data. In the experiment, four inclined conditions including 0◦ , 10◦ , 20◦ and 30◦ were investigated and the ranges of gas and liquid superficial velocity were 0.16–2.63 m/s and 0.12–3.59 m/s, respectively. The results indicate that the inclination has no significant influence on prediction error for a given correlation and the drift-flux type correlations are more competitive than the others in the prediction of slug velocity and void fraction. In addition, most of drift-flux type correlations are quite accurate in turbulent flow region, while they provide relative poor predictions in laminar flow region. As for the frictional pressure drop separated from the measured total pressure drop, the deviation arising from the calculation of the void fraction by different correlations is significant in laminar flow region, whereas is negligible in turbulent flow region. © 2014 Elsevier B.V. All rights reserved.

1. Introduction With the extensive applications of narrow rectangular channel in energy and process systems including cooling systems of high performance microelectronics, cutting-edge electronic chips, compact heat exchangers and bioengineering devices, much attention has been paid to the investigation of two-phase flow in narrow rectangular channels recently. In actual application, the channels are often disposed at an inclination, for instance, power systems of a sailing vessel are often under inclined conditions due to the varying sea surface and the effect of ocean wave shocks. The coolant

∗ Corresponding author. Tel.: +86 0451 82569655; fax: +86 0451 82569655. E-mail addresses: [email protected] (C. Yan), Changqi [email protected] (C. Yan). http://dx.doi.org/10.1016/j.nucengdes.2014.03.019 0029-5493/© 2014 Elsevier B.V. All rights reserved.

flow channels may also incline when the avionics and space shuttle run in the landing area. It is anticipated that the characteristics of two-phase flow in such a narrow gap differ from those in other channel geometries, for example, slug flow covers a wide range of flow conditions due to the significant restriction. Void fraction, defined as the ratio of cross-section area occupied by the gas to the area of flow channel, plays a fundamental role in two-phase flow and is always involved in determining the flow pattern transition and pressure drop. Therefore, a comprehensive understanding of the pressure drop and void fraction of slug flow in inclined narrow rectangular channels is of considerable practical importance in the design and performance evaluation of related equipments. In view of the great importance of void fraction, plenty of correlations have been proposed. A summary of void fraction correlations is presented in Table 1. Vijayan et al. (2012) have classified the available void fraction correlations into four groups:

156

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

Nomenclature General symbols distribution parameter C0 D diameter (m) Dh hydraulic diameter (m) f camera frame rate (fps) Froude number Fr Ft Froude rate parameter g gravity acceleration (m/s2 ) mass flow rate (kg m−2 s−1 ) G h height of the test section (mm) j superficial velocity (m/s) K the factor of Kˇ correlations L length between two pressure pores (m) lengths (Fig. 2) l1 , l2 n number of data points n1 , n2 frame Re Reynolds number all liquid Reynolds number, Relo = GDh /l Relo S slip ratio between the phases Us slug velocity (m/s) Vgj drift velocity (m/s) w width of the test section (mm) width of the test section in image (pixel) wi x mass quality x1 , x2 heights of a slug above the image bottom (pixel) X Martinelli parameter P system pressure (MPa) critical pressure (MPa) Pc z axial position of test section (mm) Pf frictional pressure drop (kPa) gravitational pressure drop (kPa) Pg Pt total pressure drop (kPa) displacement of slug nose in image (pixel) x Greek letters void fraction ˛ ˇ volumetric fraction  scale factor  inclination angle (◦ ) dynamic viscous (Pa s)   fluid density (kg/m3 )  density difference between phases (kg/m3 )  surface tension (N/m) Subscripts gas phase g l liquid phase exp experiment prediction pred sepa separation

slip ratio correlations, Kˇ correlations, drift-flux correlations and general correlations. Five categories of void fraction correlations have been divided by Dalkilic et al. (2008): slip ratio correlations, Kˇ correlations, Lockhart and Martinelli parameter based correlations, flow regime based correlations as well as general correlations. In present study, 46 different void fraction correlations (Table 1) were selected covering the following five categories: the slip ratio correlations (1–13), Kˇ correlations (14–20), Lockhart and Martinelli parameter based correlations (21–27), drift-flux correlations (28–42) and general correlations (43–46).

Although numerous of void fraction correlations have been achieved, only a few attentions have been paid to the comparison among different correlations. Woldesemayat and Ghajar (2007) have made the comparison among 68 void fraction correlations based on unbiased data set including 2845 data points for horizontal, inclined and vertical tubes. They confirmed that the drift-flux analysis method is a powerful tool in developing correlations for predicting void fraction as well as dealing with void fraction data. Dalkilic et al. (2008) evaluated 35 void fraction models and correlations based on the R134a condensation in vertical downward flow in a smooth tube. Their results indicated that void fraction correlations had no significant influence on separating frictional pressure drop. However, the drift-flux correlations were not mentioned in their work. Xing et al. (2013) conducted experimental study on effects of void fraction correlations on pressure gradient separation of two-phase flow in vertical rectangular ducts under room temperature and pressure. They selected some typical driftflux correlations for comparison with the Jones and Zuber (1979) correlation in different flow patterns, and recommended the latter for the separation of the component pressure drop from the total pressure drop in vertical mini rectangular ducts. It should be noted that Dalkilic et al. (2008) and Xing et al. (2013) just evaluated the void fraction correlation indirectly. To date, studies on evaluation of void fraction correlations for slug flow in inclined narrow rectangular duct are scarcely reported. In present study, comparisons were made among various void fraction correlations based on the experimental data involving void fraction and slug velocity in an inclined narrow rectangular duct. Meanwhile, effect of void fraction correlations on frictional pressure drop separation was also investigated. 2. Experimental apparatus and data processing 2.1. Experimental loop and procedures As shown in Fig. 1, the experimental facility mainly includes the rolling platform and two-phase flow experimental loop. The rolling platform was a rectangular plane that was operated by a hydraulic driving device, and it could make a simple harmonic motion around its central shaft or be fixed at an inclined angle. The two-phase flow experimental loop consisted of a test section, an air–water mixer, a centrifugal water pump, a water tank, compressed air lines, mass flow meters and various piping components. The test section was a transparent rectangular duct that was made of acrylite with a cross section of 43 mm × 3.25 mm × 2000 mm (width × height × length) and was installed vertically on the platform. It should be pointed out that the broad face of the rectangular duct always lies in a vertical plane under inclined conditions. Air and purified water were used as working fluids. The water was circulated in the loop by a centrifugal pump, and the air supplied by a gasholder was first introduced into the mixing chamber. The air and purified water were mixed in the mixing chamber before entering the test section. After the mixture flow upwards through the test section, the air was released into the atmosphere through the separator while the water was drained into a water tank. In order to identify the flow pattern and to record its characteristics, a high-speed camera (FASTCAM SA5) with the image resolution of 512 × 1021 pixels and the sampling speed from 1000 to 4000 frame/s was used. For each flow condition, images were taken at a fixed position (z/Dh = 163) to ensure the flow was fully developed. The present experimental conditions include gas flow rate, water flow rate and inclination. The flow rates of both the air and water were measured by mass flow meters with the uncertainties of ±1% and ±0.1%, respectively. Two pressure transducers located at the axial positions of z/Dh = 83 (0–100 kPa) and z/Dh = 248

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

157

Table 1 Correlations for predicting void fraction. Number 1

Author(s)

Void fraction correlations

Homogeneous flow model

˛=

1  1−x   g

1+

2

Lockhart and Martinelli (1949)

l

x

Fauske (1961) Thom (1964)

˛=

Zivi (1964)

˛=

Turner and Wallis (1965)

˛=

D = 25.4 mm, horizontal, steam–water

1  1−x  0.67 g

Large tube, steam–water, annular flow

Baroczy (1966)

˛=

Smith (1969)

l

x

 1−x 0.72  1g 0.4  l 0.08

Separate-cylinders model

 1−x 0.74  1g 0.65  l 0.13

–

g

l

x

1+

8

g

l

x

1+

7

g

l

x

1  1−x   g S l x   l 

˛=

Steam–water

l

 1−x  g 10.89  l 0.18

1+

6

1+

g

S = 0.4 + 0.6 Chisholm (1973)

˛=

Spedding and Chen (1984) Chen (1986) Hamersma and Hart (1987)

˛= ˛=

Petalaz and Aziz (1997)

Holdup data, upward/downward inclined

l

 1−x 0.6 1g 0.33  l 0.07

Annular flow

 1−x 10.67  g 0.33

Horizontal tubes, liquid holdup

g

l

x

1+0.26

13

g

x

1+0.18

12

Pipes, steam–water, air–water

 1−x 10.65  g 0.65

˛=

1+2.22

11

x

1    g 1+ 1−x S l x   l 0.5

S = 1−x+x 10

6 < D < 38 mm, steam–water, air–water, horizontal/vertical

  0.5 +0.4 1−x  1−x x

1+0.4

9

D = 1.5–25.8 mm, adiabatic, several air–liquid combinations, 110.3–358.4 kPa

g

l

x

x

1+

5

S

1  1−x  0.5 g

˛=

1+

4

Model

 1−x 0.64 1g 0.36  l 0.07

˛=

1+0.28

3

Remarks

, S=1

l

x

 2 j2 0.074 1

˛=

l g 2

1+0.735

1−x x

−0.2  g −0.126

Pipes, mechanistic model, adiabatic

l

14

Armand (1946)

˛ = 0.833ˇ

15

Armand (1946), Chisholm (1983)

˛=

16

Bankoff (1960)

˛ = (0.71 + 0.0145P)ˇ, where P in MPa

Pipes, air–water, adiabatic

ˇ ˇ+(1−ˇ)

Pipes, air–water, adiabatic

0.5

˛ = ˇ − 0.7(1 − ˇ)

 −0.045

0.5

17

Kowalczewskia

18

Guzhov et al. (1967)

Fr 1− √ ˛ = 0.81(1 − exp(−2.2 Fr))ˇ

19

Armand and Massinaa

˛ = (0.833 + 0.167x)ˇ

20

Czop et al. (1994)

˛ = 1.097 −



0.285 ˇ

−0.378



,X =

22

Domanski and Didion (1983)

˛ = 0.823 − 0.157 ln(X)

23

Tandon et al. (1985)

50 < Relo < 1125, ˛ = 1 − 1.928 Relo >1125, ˛ = 1 − 0.38



1 Ft

+X

1 X

+

−0.321

Coiled tube, SF6-water, adiabatic, 102–1377 kPa

 1−x 0.9  g 0.5  l 0.1

˛ = (1 − X 0.8 )

, F(X) =

– –

Wallis (1969)

GDh l

High pressure

Pc

ˇ

21

Relo =

Pipes, steam–water, vertical/horizontal, bubbly flow

 P

x

l

Re−0.088 l F(X)

2.85 X 0.476



, Ft =

Re−0.315 l

F(X)

+

D = 1.5–25.8 mm, adiabatic, several air–liquid combinations, 110.3–358.4 kPa D = 1.5–25.8 mm, adiabatic, several air–liquid combinations, 110.3–358.4 kPa

g

+ 0.9293

Re−0.63 l

Analytical model for annular flow, steam–water

F(X)2

Re−0.176 0.0361 l 2 F(X)

x3 G 2 g2 gDh (1−x)



24

Yashar et al. (1998)

˛= 1+

25

Graham et al. (1998)

26

Harms et al. (2003)

Ft < 0.01032, ˛ = 0 2 Ft > 0.01032,˛ = 1 − exp{−1 − 0.3 ln(Ft) − 0.0328[ln(Ft)] } ˛ =

2

Microfin tubes, condensation



1 − 10.06Rel−0.875 (1.74 + 0.104Rel0.5 ) × 1.376 +

27

Wilson et al. (2003)



˛= 1+

a Ft

X+ X+

1 Ft 1 Ft

jg C0 j+Vgj

< 2, > 2,

+ bX

n

a = 1.84, b = 3.11, n = −0.21 a = 0.5, b = 1.2, n = −0.35



28

Nicklin et al. (1962)

˛=

, C0 =1.2, Vgj = 0.35

29 30

Hughmark (1965) Gregory and Scott (1969)

C0 =1.2, Vgi =0 C0 =1.19, Vgi =0

31

Bonnecaze et al. (1971)

C0 =1.2, Vgj = 0.35

32

Mattar and Gregory (1974)

C0 =1.3, Vgi =0.7





gDh 1 −

gDh

 g l

7.242 X 1.665

−0.5 2

Axially grooved tube, R134a, condensation Large tubes, horizontal flow, annular flow

Microfinned copper tubes, horizontal, R134a and R410A, condensation

Tubes, vertical Horizontal, slug flow, non-adiabatic Horizontal, air–water, slug flow Inclined pipes, gas–oil, slug flow Inclined pipe, air–oil, slug flow

158

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

Table 1 (Continued) Number

Author(s)

Void fraction correlations

33

Ishii (1977)

C0 = 1.35 − 0.35

34

Jones and Zuber (1979)

C0 = 1.35 − 0.35

g

, Vgj = 0.35

gl 

35 36

Sun et al. (1980) Anklam and Miller (1982)

C0 = 0.82 +



−1 P

C0 = 0.82 + 0.18 Pc

Kokal and Stanislav (1989)

C0 = 1.2, Vgj = 0.345

38

Qazi et al. (1994)

C0 = 0.82 + 0.18 Pc

39

Toshibab

C0 = 1.08, Vgj = 0.45

40 41

−1 P



C0 = jg

b

Dix

1+

43

45 46 a b

 , Vgj = 1.53

jg

b

Bestion

Huq and Loth (1992)

g





Steiner (1993)

Xiong and Chung (2007) Saisorn and Wongwises (2010)

g

0.25 Rod bundle, high pressure

0.25 Heated rod bundle, 3.9–8.1 MPa, low flow conditions

2 l

Inclined pipes, air–oil, 230–350 kPa

 , Vgj = 2.9

  l g

g

Heated rod bundle, air–water, stagnation condition 4 × 4 rod bundle

0.25

For analysis of BWRs

2 l

,

Horizontal, swell conditions, 0.2–4 MPa

−1

 gDh  0.5

C0 = 1, Vgj = 0.188 ˛=1−

2 l

, Vgj = 0.35(gDh )0.5



 l

g

l

 jl (g /l )0.1 

For thermal hydraulic code CATHARE

g

 l 0.5

2(1−x)2

1−2x+ 1+4x(1−x)

44



, Vgj = 1.41

C0 = 1 + 0.796 exp −0.061

Jowittb

Vgj = 0.034 42

gw/l

 gDh  0.5

37



Rectangular duct, air–water, slug flow, adiabatic



−1

0.18 PPc

Circular tube, air–water, adiabatic

l

,

l

Vgj = (0.23 + 0.13h/w)



Remarks

 gDh  0.5

˛ = x g

[1 + 0.12(1 − x)]

˛=

Cˇ0.5 1−(1−C)ˇ0.5

˛=

0.036ˇ0.5

,C =

Derivation of an analytical formula

g −1

x g

+

1−x l



+

1.18(1−x)(g)0.25

−1

Boiling saturated flow

G0.5 l

0.266 1+13.8 exp(−0.688Dh )

Rectangular microchannels, nitrogen–water Circular microchannel, air–water

1−0.945ˇ0.5

Woldesemayat and Ghajar (2007). Coddington and Macian (2002).

(0–30 kPa) from the inlet, with the same accuracy of 0.2%. The experiments were conducted under ambient temperature and the water temperature was kept at approximately 20 ◦ C. A computer that was instrumented with a data acquisition board (NI SCXI-1338) was used to record the instantaneous output signals from the flow

meters and pressure transducers. The sampling rate and sampling time were set to 256 Hz and 20 s, respectively. In present study, four inclined conditions including 0◦ (vertical condition), 10◦ , 20◦ , 30◦ were investigated, and the ranges of gas and liquid superficial velocity were 0.16–2.63 m/s and 0.12–3.59 m/s, respectively. The

Fig. 1. Schematic diagram of the experimental facility.

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

159

Table 2 Uncertainties of the measure parameters. Parameters

Estimated uncertainty (%)

Gas mass flow rate (kg/s) Liquid mass flow rate (kg/s) Pressure drop (kPa) Slug velocity (m/s)

4.59 2.76 1.17 6.63 (laminar flow region) 5.26 (turbulent flow region)

Fig. 4. Distribution of experimental data under inclined condition.

Fig. 2. Image processing.

uncertainty analysis has been carried out according to the method proposed by Moffat (1988). The estimated uncertainties of measured parameter are listed in Table 2. 2.2. Image processing A scale factor  is acquired before image processing, which is defined as w = (1) wi where w and wi represent the width of test section and that in images, respectively. Fig. 2(a) and (b) shows two typical frames of n1 and n2 in time sequence extracted from a small period of high-speed video film under inclined condition. Thus, the slug velocity Us can be easily obtained Us =

f (x2 − x1 ) n2 − n1

(2)

where x1 and x2 denote the heights of a slug above the image bottom in frames n1 and n2 ; f is the camera frame rate. In order to reduce

the subjective error in the measurement, 15 slugs were taken an average under the same flow condition. For the case of slug flow in a narrow rectangular duct, gas is mainly concentrated in the gas slug, the small bubbles contained in the liquid slug and liquid film could be neglected. In other words, the slug velocity could be assumed to equal the gas phase velocity. Therefore, the void fraction can be given by (Sadatomi et al., 1982; Sowinski et al., 2009): ˛=

jg Us

(3)

In the present case of air–water two-phase flow under adiabatic condition, the quality remains unchanged and the acceleration pressure drop (Pa ) is negligible. The total pressure drop (Pt ) in inclined state consists of the frictional pressure drop (Pf ) and the gravitational pressure drop (Pg ): Pt = Pf + Pg

(4)

In which the gravitational pressure drop could be calculated from: Pg = [g ˛ + l (1 − ˛)]g cos L

(5)

where L represents the distance between two pressure pores; g is the gravity acceleration.

Fig. 3. Comparison of calculated slug velocity with experimental data.

160

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

Fig. 5. Evaluation of drift-flux type correlations on slug velocity under inclined conditions (a) vertical; (b) inclination 10◦ ; (c) inclination 20◦ ; (d) inclination 30◦ .

3. Results and discussion 3.1. Evaluation of void fraction correlations on slug velocity The void fraction could be obtained by various correlations as listed in Table 1, thus we could achieve the slug velocity by using Eq. (3). In other words, the void fraction correlations could be used to evaluate the slug velocity obtained by imaging processing. Fig. 3 presents the comparison of the calculated slug velocity with the experimental data. The number on the abscissa denotes the different void fraction correlations, and the ordinate represents the mean relative error (MRE) of these correlations against the experimental data. The error band of 10% is also plotted by the dotted line in Fig. 3. The MRE is defined as: 1 MRE = n

   Vpred − Vexp  × 100 Vexp

(6)

where n is the number of data points; V denotes the slug velocity and void fraction in present study; subscripts pred and exp are the predicted values and experimental data, respectively. The results indicate that the deviations of Zivi (1964), Turner and Wallis (1965), Czop et al. (1994), Tandon et al. (1985), Bestion, Xiong and Chung (2007) and Saisorn and Wongwises (2010) correlation are larger than 100%. It should be noted that the actual deviations larger than 100% are truncated in Fig. 3, and similar treatment could be found in the following discussion. For the slip ratio type, correlations of Lockhart and Martinelli (1949), Smith (1969), Chisholm (1973), Spedding and Chen (1984), Chen (1986) and Hamersma and Hart (1987) predict the slug velocity well, while the rest give the poor predictions. For the case of Kˇ type correlations, Kowalczewski and Czop et al. (1994) correlations perform poorly while other five

correlations show good agreement with experimental data. As to L&M type correlations, they have good performance on calculating slug velocity except for Tandon et al. (1985) correlation. It is evident that the drift-flux type correlations provide the best predictions, among which 11 correlations’ MREs are lower than 20%. Among the general correlations, the correlations of Huq and Loth (1992) and Steiner (1993) perform much better than the other two ones. Yan et al. (2014) investigated the slug behavior in a narrow rectangular duct and divided slug flow region into laminar flow (Rel < 3000) and turbulent flow (Rel ≥ 3000). They reported that the buoyancy played an important role in laminar flow region while the inertial force showed a dominant influence on slug behavior in turbulent flow region. The inclination shows a significant influence on slug behaviors in laminar flow region while it has little effect in turbulent flow region. According to the flow region suggested by Yan et al. (2014), the distribution of experimental data is presented in Fig. 4, showing the majority of experimental data lay in turbulent flow region. As previously discussed, the drift-flux type correlations provide the best predictions among all the categories. Therefore, the drift-flux type correlations are evaluated separately in different flow regions under inclined condition as shown in Fig. 5. It can be concluded from Fig. 5 that the drift-flux type correlations show larger deviations with slug velocity in laminar flow region than that in turbulent flow region except for Dix correlation. For laminar flow region case, only Toshiba correlation show relatively fair predictions under inclined condition, with the mean absolute errors around 5%. But it does not work as well as the correlations of Jones and Zuber (1979), Sun et al. (1980) and Anklam and Miller (1982) in vertical duct. In addition, 9 of the 15 correlations give poor predictions in laminar flow region, with MREs

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

161

larger than 20%. As to turbulent flow region, 11 of the 15 correlations show good agreement with experimental slug velocity, with the mean relative errors lower than 10%. Among all the drift-flux correlations, the MREs of Ishii (1977), Jones and Zuber (1979), Sun et al. (1980), Anklam and Miller (1982) and Toshiba are lower that 10%. The above correlations are evaluated on slug velocity as presented in Fig. 6(a) and (b). The results indicate that all the selected correlations underestimate the slug velocity in laminar flow region while they predict experimental data quite accurately in turbulent flow region. As mentioned in the study of Yan et al. (2014), the slug velocity increases with increasing the inclination in laminar flow region, while it is nearly invariable under the same flow conditions in turbulent flow region. Since the five correlations include no inclined parameters and underestimate the vertical slug velocity, their deviations should be even larger under inclined conditions, as evidenced in Fig. 6(a). 3.2. Evaluation of void fraction correlations on void fraction The experimental void fraction could be obtained by Eq. (3) and the comparison of void fraction calculated by 46 correlations with the experimental void fraction is presented in Fig. 7. The predicted void fraction by Kowalczewski correlation is negative, which is meaningless. Therefore, it will be omitted in the following discussion. Unlike the evaluated result on slug velocity, only Petalaz and Aziz (1997) correlation shows the deviation larger than 100%. Among slip ratio type correlations, Lockhart and Martinelli (1949), Smith (1969), Spedding and Chen (1984) and Hamersma and Hart (1987) correlations give reasonable predictions, and the others show large deviations. The maximum of mass quality is 0.005 in present study, the slip ratio type correlations greatly depend on mass quality may not work well at such low mass quality. Among L&M type correlations, only those of Domanski and Didion (1983), Yashar et al. (1998) and Wilson et al. (2003) show relatively fair predictions for all the inclined conditions, with MREs lay between 10% and 15%. While for general type correlations, correlations of Huq and Loth (1992) and Steiner (1993) work better than the others. Woldesemayat and Ghajar (2007) pointed out that the driftflux model could correlate experimental void fraction well and believed that it was a powerful tool to predict void fraction in circular tube. Fig. 3 indicates that correlations of Ishii (1977), Jones and Zuber (1979), Sun et al. (1980), Anklam and Miller (1982) and Toshiba give the favorable predictions. Zuber and Findlay (1965) first proposed the drift-flux model. Later Ishii (1977) developed a comprehensive set of drift-flux correlations for circular tubes. Jones

Fig. 6. Evaluation of selected drift-flux type correlations on slug velocity (a) laminar flow region; (b) turbulent flow region.

and Zuber (1979) developed a drift velocity correlation for slug flow in rectangular duct. So correlations of Ishii (1977) and Jones and Zuber (1979) could well correlate the experimental data, and the Jones and Zuber (1979) correlation shows a little better than Ishii

Fig. 7. Comparison of predicted void fraction with experimental data.

162

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

Fig. 8. Evaluation of drift-flux type correlations on void fraction under inclined conditions (a) vertical; (b) inclination 10◦ ; (c) inclination 20◦ ; (d) inclination 30◦ .

(1977) correlation. Although correlations of Sun et al. (1980) and Anklam and Miller (1982) were obtained from experiments under high pressure and diabatic conditions, they showed exactly good agreement with present experimental data. For Toshiba correlation, constant values for C0 and Vgj were obtained from a least squares fit to their experimental data from a 4 × 4 rod bundle test section, while it is the best for present experimental data among all the correlations. The drift-flux type correlations are evaluated separately in different flow regions under inclined condition as shown in Fig. 8. It could be concluded that except the Dix correlation, the others have larger MRE in laminar flow region than that in turbulent flow region. For the cases in laminar flow region, only Toshiba correlation could give favorable predictions under inclined condition, with a MRE about 5%. Correlations of Jones and Zuber (1979), Sun et al. (1980) and Anklam and Miller (1982) could provide reasonable predictions with MREs lower than 10% in vertical duct, while show large deviations under inclined conditions. What is more, 11 correlations among the 15 drift flux give poor predictions in laminar flow region, evidenced by MREs larger than 20%. For the case of turbulent flow region, 11 of 15 correlations are compatible with each other, with the mean absolute errors lower than 10%. Although the predictions for void fraction in laminar flow region are remarkable worse than those in turbulent flow region, the MRE based on all experimental data are similar to each other for a given correlation due to most experimental data lay in turbulent flow region. Drift-flux type correlations could well correlate the void fraction, so several drift-flux type correlations should be further analyzed in different flow regions. Fig. 9(a) and (b) shows the experimental results of void fraction against the gas superficial velocity,

where the predicted values by Ishii (1977), Jones and Zuber (1979), Sun et al. (1980), Anklam and Miller (1982) and Toshiba are also plotted. In the laminar flow region, the void fraction decreases as the inclined angle increases. The Toshiba correlation shows the best predictions, whereas the Ishii (1977) model works relatively poorly. The other three correlations provide a little bit higher predictions of void fraction for inclined conditions, but for vertical condition they show relatively fair predictions. As for turbulent flow region, all the five correlations show good agreement with the experimental data, as evidenced in Fig. 9b. 3.3. Effect of void fraction correlations on predicted frictional pressure drop The frictional pressure drop could be obtained by subtracting gravitational pressure drop from the total pressure drop: Pf = Pt − Pg = Pt − [g ˛ + l (1 − ˛)]g cos L

(7)

Eq. (7) indicates that the predicted frictional pressure drop may be influenced by void fraction correlation. In the present study, the separated frictional pressure drop by applying different void fraction correlations to Eq. (7) are compared with the experimental results, in which the void fraction is calculated by Eq. (3). Section 3.2 indicates that drift-flux type correlations give the best predictions on void fraction. Therefore, the effect of drift-flux type correlations on the separation of the frictional pressure drop from the measured total pressure drop is investigated in this section. Fig. 10 compares the separated frictional pressure drop with experimental data, in which the error bands of 20% are also presented. The abscissas denote liquid Reynolds number, while

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

163

Fig. 9. Evaluation of selected drift-flux type correlations on void fraction (a) laminar flow region; (b) turbulent flow region.

Fig. 10. Comparisons between the predicted frictional pressure drop and the experimental data: (a) vertical; (b)  = 10◦ ; (c)  = 20◦ ; (d)  = 30◦ .

the ordinates represent the ratio of separated frictional pressure drop ((Pf )sepa ) to experimental one ((Pf )exp ). It can be seen from Fig. 10 that the predictions show large scatter in laminar flow region. For the case of turbulent flow region, the predictions show good agreement with the experimental frictional pressure drop, with almost all of the points lying in the ±20% error bands. That is to say, the effect of void fraction correlations on predicted pressure drop is significant in laminar flow region but negligible in turbulent flow region. Therefore, appropriate void fraction correlation should be adopted to acquire the accurate frictional pressure drop in laminar flow region.

4. Conclusions In present paper, evaluations of 46 void fraction correlations including slip ratio, Kˇ, Lockhart and Martinelli, drift-flux and general types on void fraction as well as slug velocity for slug flow in a narrow rectangular duct are investigated. Meanwhile, effect of void fraction correlations on predicted frictional pressure drop is also studied. The main conclusions were as follow: (1) For a given correlation, the mean relative errors for slug velocity under different inclinations are similar to each other. The

164

C. Yan et al. / Nuclear Engineering and Design 273 (2014) 155–164

drift-flux type correlations provide the best predictions, among which 11 correlations’ MREs are lower than 20%. Almost all the drift-flux type correlations show larger deviations with slug velocity in laminar flow region than that in turbulent flow region. In addition, most of drift-flux type correlations are quite accurate in turbulent flow region, while they provide relative poor predictions in laminar flow region. (2) The drift-flux type correlations show better agreement with experimental void fraction than the other four types and present good compatibility. The mean relative errors in laminar flow are significantly larger than those in turbulent flow. In laminar flow region, Toshiba correlation could give favorable predictions under inclined condition and correlations of Jones and Zuber (1979), Sun et al. (1980) and Anklam and Miller (1982) could provide reasonable predictions in vertical duct. For the case of turbulent flow region, 11 of drift-flux type correlations show good agreement with experimental data with MRE lower than 10%. (3) Influence on separated frictional pressure drop arising from different drift-flux type correlations is significant in laminar flow region, whereas is negligible in turbulent flow region. The appropriate void fraction correlation should be adopted to acquire the accurate frictional pressure drop in laminar flow region. Acknowledgements The authors are profoundly grateful to the financial supports of the National Natural Science Foundation of China (Grant Nos. 11175050 and 51376052) as well as by the Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University. References Anklam, T.M., Miller, R.F., 1982. Void fraction under high pressure, low flow conditions in rod bundle geometry. Nucl. Eng. Des. 75, 99–108. Armand, A.A., 1946. The resistance during the movement of the two-phase system in horizontal pipes. Izv. Vses. Teplotekh. Inst. 1, 16–23. Bankoff, S.G., 1960. A variable density single fluid model for two phase flow with particular reference to steam water flow. Trans. ASME, J. Heat Transf. 82, 265–272. Baroczy, C.J., 1966. A systematic correlation for two phase pressure drop. Chem. Eng. Prog. Symp. Ser. 62, 232–249. Bonnecaze, R.H., Erskine, W., Greskovich, E.J., 1971. Holdup and pressure drop for two phase slug flow in inclined pipes. AIChE J. 17, 1109–1113. Chen, J.J.J., 1986. A further examination of void-fraction in annular two-phase flow. Int. J. Heat Mass Transf. 29, 1760–1763. Chisholm, D., 1973. Void fraction during two-phase flow. J. Mech. Eng. Sci. 15, 235–236. Chisholm, D., 1983. Two phase flow in pipelines and heat exchangers, George Godwin in association with The Institution of Chemical Engineers, London. Coddington, P., Macian, R., 2002. A study of the performance of void fraction correlations used in the context of drift-flux two-phase flow models. Nucl. Eng. Des. 215, 199–216. Czop, V., Barbier, D., Dong, S., 1994. Pressure drop, void fraction and shear stress measurements in adiabatic two-phase flow in coiled tube. Nucl. Eng. Des. 149, 323–333. Dalkilic, A.S., Laohalertdecha, S., Wongwises, S., 2008. Effect of void fraction models on the two-phase friction factor of R134a during condensation in vertical downward flow in a smooth tube. Int. Commun. Heat Mass Transf. 35, 921–927. Domanski, P., Didion, D., 1983. Computer modeling of the vapor compression cycle with constant flow area expansion device. Building Science Series, National Bureau of Standards of USA. Fauske, H.,1961. Critical two-phase, steam–water flows. In: Proceedings of the 1961 Heat Transfer and Fluid Mechanics Institute. Stanford University Press, Stanford, CA, pp. 79–89. Graham, D., Chato, J.C., Newell, T.A., 1998. Heat transfer and pressure drop during condensation of refrigerant 134a in an axially grooved tube. Int. J. Heat Mass Transf. 42, 1935–1944. Gregory, G.A., Scott, D.S., 1969. Correlation of liquid slug velocity and frequency in horizontal co-current gas liquid slug flow. AIChE J. 15, 933–935. Guzhov, A.L., Mamayev, V.A., Odishariya, G.E., 1967. A study of transportation in gas liquid systems. In: 10th International Gas Union Conference, June Hamburg, Germany, pp. 6–10.

Hamersma, P.J., Hart, J., 1987. A pressure drop correlation for gas/liquid pipe flow with a small liquid holdup. Chem. Eng. Sci. 42, 1187–1196. Harms, T., Eckart, D., Groll, A., Braun, J., 2003. A void fraction model for annular flow in horizontal tubes. Int. J. Heat Mass Transf. 46, 4051–4057. Hughmark, G.A., 1965. Holdup and heat transfer in horizontal slug gas-liquid flow. Chem. Eng. Sci. 20, 1007–1010. Huq, R.H., Loth, J.L., 1992. Analytical two-phase flow void fraction prediction method. J. Therm. Phys. 6, 139–144. Ishii, M., 1977. One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two phase flow regimes, ANL-77-47, USA. Jones, O.C., Zuber, N., 1979. Slug-annular transition with particular reference to narrow rectangular ducts in two phase momentum. Heat Mass Transf. Chem. Process Energy Eng. Syst. 1, 345–355. Kokal, S.L., Stanislav, J.F., 1989. An experimental study of two phase flow in slightly inclined pipes. II. Liquid holdup and pressure drop. Chem. Eng. Sci. 44, 681–693. Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem. Eng. Prog. 45, 39–48. Mattar, L., Gregory, G.A., 1974. Air oil slug flow in an upward-inclined pipe. I. Slug velocity, holdup and pressure gradient. J. Can. Petrol. Technol. 13, 69–76. Moffat, R.J., 1988. Describing the uncertainties in experimental results. Exp. Therm. Fluid Sci. 1, 3–17. Nicklin, D.J., Wilkes, J.O., Davidson, J.F., 1962. Two-phase flow in vertical tubes. Trans. Inst. Chem. Eng. 40, 61–68. Petalaz, N., Aziz, K., 1997. A mechanistic model for stabilized multiphase flow in pipes. In: Technical Report for Members of the Reservoir Simulation Industrial Affiliates Program (SUPRI-B) and Horizontal Well Industrial Affiliates Program (SUPRI-HW). Stanford University, Palo Alto, CA. Qazi, M.K., Guido-Lavalle, G., Clausse, A., 1994. Void fraction along a vertical heated rod bundle under flow stagnation conditions. Nucl. Eng. Des. 152, 225–230. Sadatomi, M., Sato, Y., Saruwatari, S., 1982. Two-phase flow in vertical noncircular channels. Int. J. Multiphase Flow 8, 641–655. Saisorn, S., Wongwises, S., 2010. The effects of channel diameter on flow pattern, void fraction and pressure drop of two-phase air water flow in circular microchannels. Exp. Therm. Fluid Sci. 34, 454–462. Smith, S.L., 1969. Void fractions in two phase flow: a correlation based upon an equal velocity head model. Proc. Inst. Mech. Eng., London 184, 647–657, Part 1. Sowinski, J., Dziubinski, M., Fidos, H., 2009. Velocity and gas-void fraction in twophase liquid–gas flow in narrow mini-channels. Arch. Mech. 61, 29–40. Spedding, P.L., Chen, J.J.J., 1984. Holdup in two phase flow. Int. J. Multiphase Flow 10, 307–339. Steiner, D., (J.W. Fullarton, Trans.) 1993. In: Verein Deutscher Ingenieure, VDIGessellschaft Verfahrenstechnik, Chemie-ingenieurwesen (GCV) (Eds.), Heat transfer to boiling saturated liquids, VDI-Warmeatlas (VDI Heat Atlas). , Dusseldorf. Sun, K.H., Duffey, R.B., Peng, C.M., 1980. A thermal-hydraulic analysis of core uncovery. In: Proceedings of the 19th National Heat Transfer Conference, Experimental and Analytical Modeling of LWR Safety Experiments, July 27–30, Orlando, FL, pp. 1–10. Tandon, T.N., Varma, H.K., Gupta, C.P., 1985. A void fraction model for annular twophase flow. Int. J. Heat Mass Transf. 28, 191–198. Thom, J.R.S., 1964. Prediction of pressure drop during forced circulation boiling of water. Int. J. Heat Mass Transf. 7, 709–724. Turner, J.M., Wallis, G.B., 1965. The separate-cylinders model of two-phase flow. Paper No. NYO-3114-6. Thayer’s School Eng., Dartmouth College, Hanover, NH, USA. Vijayan, P.K., Patil, A.P., Pilkhawal, D.S., Saha, D., Venkat Raj, V., 2012. An assessment of pressure drop and void fraction correlations with data from two-phase natural circulation loops. Heat Mass Transf. 36, 541–548. Wallis, G.B., 1969. One Dimensional Two-Phase Flow. McGraw-Hill Inc., New York. Wilson, M.J., Newell, T.A., Chato, J.C., Infante, F.C.A., 2003. Refrigerant charge, pressure drop and condensation heat transfer in flattened tubes. Int. J. Refrig. 26, 442–451. Woldesemayat, M.A., Ghajar, A.J., 2007. Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. Int. J. Multiphase Flow 33, 347–370. Xing, D.C., Yan, C.Q., Ma, X.G., Sun, L.C., 2013. Effects of void fraction correlations on pressure gradient separation of air–water two-phase flow in vertical mini rectangular ducts. Prog. Nucl. Energy 70, 84–90. Xiong, R., Chung, J.N., 2007. An experimental study of the size effect on adiabatic gas-liquid two-phase flow patterns and void fraction in microchannels. Phys. Fluids 19, 0333301. Yan, C.X., Yan, C.Q., Sun, L.C., Wang, Y., Zhang, X.N., 2014. Slug behavior and pressure drop of adiabatic slug flow in a narrow rectangular duct under inclined conditions. Ann. Nucl. Energy 64, 21–31. Yashar, D.A., Newell, T.A., Chato, J.C., 1998. Experimental Investigation of Void Fraction During Horizontal Flow in Smaller Diameter Refrigeration Applications, ACRC TR141. Air Conditioning and Refrigeration Center, University of Illinois, Urbana-Champaign, IL. Zivi, S.M., 1964. Estimation of steady state steam void fraction by means of the principle of minimum entropy production. Trans. ASME, J. Heat Transfer 86, 247–252. Zuber, N., Findlay, J.A., 1965. Average volumetric concentration in two-phase flow systems. J. Heat Transf. 87, 435–468.