Frictional resistance of adiabatic two-phase flow in narrow rectangular duct under rolling conditions

Annals of Nuclear Energy 53 (2013) 109–119

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Frictional resistance of adiabatic two-phase flow in narrow rectangular duct under rolling conditions Dianchuan Xing ⇑, Changqi Yan, Licheng Sun, Guangyuan Jin, Sichao Tan National Defense Key Subject Laboratory for Nuclear Safety and Simulation Technology, Harbin Engineering University, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 8 May 2012 Received in revised form 28 September 2012 Accepted 28 September 2012 Available online 28 November 2012 Keywords: Rolling motion Two-phase flow Frictional pressure drop Narrow rectangular duct Correlations evaluation New correlation

a b s t r a c t Frictional resistance of air-water two-phase flow in a narrow rectangular duct subjected to rolling motion was investigated experimentally. Time-averaged and transient frictional pressure drop under rolling condition were compared with conventional correlation in laminar flow region (Rel < 800), transition flow region (800 6 Rel 6 1400) and turbulent flow region (Rel > 1400) respectively. The result shows that, despite no influence on time-averaged frictional resistance, rolling motion does induce periodical fluctuation of the pressure drop in laminar and transition flow regions. Transient frictional pressure drop fluctuates synchronously with the rolling motion both in laminar and in transition flow region, while it is nearly invariable in turbulent flow region. The fluctuation amplitude of the Relative frictional pressure gradient decreases with the increasing of the superficial velocities. Lee and Lee (2002) correlation and Chisholm (1967) correlation could satisfactorily predict time-averaged frictional pressure drop under rolling conditions, whereas poorly predict the transient frictional pressure drop when it fluctuates periodically. A new correlation with better accuracy for predicting the transient frictional pressure drop in rolling motion is achieved by modifying the Chisholm (1967) correlation on the basis of analyzing the present experimental results with a great number of data points. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction With the extensive application of nuclear power system in marine transportation, effects of ocean condition (rolling, heaving, pitching, inclination etc.) on a flow and heat transfer system have attracted growing interests in recent years. From a fluid mechanics point of view, the main difference between land-based and bargemounted equipment is that the latter is inevitable from the effect of sea wave shocks and winds (Ishida et al., 1995). The thermal hydraulic behavior of shipborne equipment is influenced by rolling, heaving and pitching motions, leading to the occurrence of unsteady flow as mentioned by Pendyala et al. (2008) and Tan et al. (2009a). A number of previous studies regarding single-phase flow behaviors under ocean condition have been performed in recent years. Studies of Gao et al. (1997), Ishida and Yoritsune (2002), Murata et al. (2002), Tan et al. (2009a,b) and Yan and Yu (2009) indicated that the flow rate of a natural circulation system will oscillate sinusoidally in rolling motion, whereas almost keeps constant for a forced circulation loop. Rolling parameters, flow rates and the component layout in the experimental loop have strong effects on the thermal hydraulic behavior of a natural

⇑ Corresponding author. Tel./fax: +86 451 82569655.

circulation system. Cao et al. (2006), Xing et al. (2012) and Zhang et al. (2009) performed a series of experiments to investigate the effect of rolling parameters, flow rates and tube radius on singlephase forced circulation in pipes. Their results indicated that the frictional pressure drop of single-phase flow oscillates periodically in rolling motion. New empirical correlations for calculating the single-phase friction factor in rolling pipes were achieved from their experimental data. Studies of Pendyala et al. (2008) indicated that heaving movement can lead to the fluctuation of a forced single-phase flow. The mean friction factor in heaving motion was considered to be greater than that under immobile condition. Yan et al. (2010, 2011) gave the velocity distribution of singlephase flow in tubes under rolling condition, and showed that rolling motion influences only the velocity distribution near the channel wall but not influence its mean frictional resistance. From afore-mentioned work, it is clear that the single-phase fluid flow in an oscillating pipe is rather different from that in a pipe at rest. However, few related studies deal with two-phase flow characteristics in rolling motion, and the summarizations are listed as follow. Cao et al. (2006) studied the frictional resistance of single-phase and two-phase flow in pipes under rolling condition, and demonstrated that the predicted friction factor by traditional correlations deviates dramatically from the experimental results. They also proposed a new correlation according to homogeneous flow model.

E-mail address: [email protected] (D. Xing). 0306-4549/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.anucene.2012.09.024

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Nomenclature General symbols f rolling frequency (Hz) T rolling period (s) t time (s) hm rolling amplitude (rad) DPt total pressure drop (kPa) DPf frictional pressure drop (kPa) DPg gravitational pressure drop (kPa) DPadd additional pressure drop (kPa) Dq density difference between phases (kg/m3) j time-averaged superficial velocity (m/s) g gravity acceleration (m/s2) L length between pressure taps (m) h height of the duct (m) w width of the duct (m) j superficial velocity (m/s) dPf/dz two-phase frictional pressure gradient (kPa/m) DP pressure drop (kPa) d(Pf)g/dz gas frictional pressure gradient (kPa/m) d(Pf)g/dz liquid frictional pressure gradient (kPa/m) /21 frictional multiplier factor, Eq. (11) x mass quality X Martinelli parameter, Eq. (13) U0 rolling velocity, Eq. (20) (m/s) l the distance between the test section and the rolling shaft (m) Re Reynolds number (Re = jde/c) de hydraulic diameter of the test section (m)

The time-averaged frictional resistance of bubbly flow in rolling motion was predicted by their correlation with an accuracy of ±25%. The effects of rolling motion on air-water two-phase flow pattern transition were investigated experimentally by Luan et al. (2007) and Zhang et al. (2007). Their results showed that the rolling period, rolling amplitude and channel size affect the transitions between flow patterns, especially from the bubble to slug flow and churn to annular flow. Yan et al. (2008) measured the volume-averaged void fraction at a certain rolling angle with the help of quick-closing valves method. The result showed that rolling motion results in the decrease of the void fraction, but no correlation for such a condition was achieved. Tan et al. (2009c) experimentally studied the two-phase flow instability of natural circulation under rolling condition, and their result indicated that rolling motion causes the early occurrence of two-phase flow instability. The above reviewed researches regarding two-phase flow behavior in rolling motion were all performed for circular tube, whereas frictional resistance in narrow rectangular duct under rolling condition has not been studied in detail so far. In addition, none of the above work gives the transient frictional pressure drop. With demands for higher heat transfer efficiency and less space requirement in practical applications, rectangular duct is one of the choices as the heat transfer tube in a compact heat exchanger. Therefore, researches on thermal hydraulic characteristics of twophase flow in narrow rectangular ducts have been received increasing attention over the last few decades (Lee and Lee, 2002; Ma et al., 2010; Mishima et al., 1993; Sadatomi et al., 1982; Wang et al., 2011a; Zhou and Wang, 2011). Most studies on two-phase flow resistance in narrow ducts are concerning motionless condition, few can be found in rolling motion. Wang et al. (2011b) investigated two-phase flow patterns under rolling conditions and obtained the flow pattern map for a narrow rectan-

Greek letters h rolling angle (°) x angular velocity (rad/s) b angular acceleration (rad/s2) q fluid density (kg/m3) a void fraction e ratio of duct height to width (e = h/w) c kinematic viscosity (m2/s) l dynamic viscous (Pa s) r surface tension (N/m) k single-phase friction factor Subscripts roll under rolling condition g gas phase flows alone through the same pipe with its mass flow rate l liquid phase flows alone through the same pipe with its mass flow rate l0 liquid phase flows only through the same pipe with total mass flow rate 1, 2 start and end points pred prediction exp experiment Superscript 0 relative coordinate

gular duct having cross section of 40 mm  1.6 mm. Recently, Hong et al. (2012a,b) and Wei et al. (2011) studied the onset of nucleate boiling and the bubble behaviors in subcooled flow boiling under ocean condition. According to the authors’ knowledge, the frictional resistance of two-phase flow in rectangular duct under rolling condition has not been studied carefully. To better understand the effect of rolling motion on two-phase flow resistance, a series of experiments was performed by using a narrow rectangular duct having cross section of 43 mm  1.41 mm. The effects of rolling motion on time-averaged and transient frictional pressure drop were investigated in different flow regions. 2. Experimental apparatus 2.1. Description of the rolling platform The rolling movement of a ship was simulated by a simple harmonic motion. The rolling platform, a 2.5 m  3.5 m rectangular plane, rotates with the central shaft (O–O) as shown in Fig. 1. Rolling movement with required rolling period and amplitude is controlled by an automatic system (Wang et al., 2011b; Xing et al., 2012). Consequently, the rolling amplitude could be expressed as follow:

h ¼ hm sinð2pftÞ

ð1Þ

Clockwise direction is defined as the positive direction of the rolling movement as shown in Fig. 1. Accordingly, the angular velocity and angular acceleration are deduced as follow:

x ¼ 2pf hm cosð2pftÞ

ð2Þ

b ¼ 4p2 f 2 hm sinð2pftÞ

ð3Þ

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Fig. 1. Schematic diagram of the experimental facility.

where f and T denote rolling frequency and period respectively (f = 1/T). To evaluate the effects of rolling parameters, five following rolling conditions are included (hm is the rolling amplitude and T is the rolling period): hm10°T8s; hm10°T12s; hm10°T16s; hm15°T16s; hm30°T16s. 2.2. Experimental loop and instruments The schematic diagram of the experimental loop is also shown in Fig. 1, in which the solid and broken line denotes the water

Fig. 2. Location of pressure taps and the test section.

and air flow pipeline respectively. Purified water and air are supplied and introduced into a mixing chamber by centrifugal pump and air-compressor respectively. The mixture flows upward through the test section in which the pressure drop is obtained. The pressure at the inlet of the test section is maintained at gauge pressure of 0.2 MPa by a pressure regulator. The test section locates vertically on the rolling platform and rotates around the rolling shaft. The two pressure taps, located on center line of the wide wall as shown in Fig. 2, are spaced 1.5 m apart and the lower of which (p1) is 0.3 m from the entrance. The outlet of the test section is vented and therein air and water is separated. The wide wall of the test section is normal to the angular velocity in present work as shown in Fig. 2. Stainless steel flexible pipe is used to link the pipeline on rolling platform to that of immobile part. The test section is fabricated with the nominal gap of 1.5 mm and its cross-section is illustrated in Fig. 3. It mainly consists of two parts, a substrate on which a channel is machined and a cover

Fig. 3. Structure of the test section.

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Fig. 4. Pressure drop component in rolling motion: (a) x = 0.042; (b) x = 0.139.

Table 1 Time-averaged additional and gravitational pressure drop. Rolling amplitude (°)

Rolling period (s)

Time-averaged additional pressure drop (kPa)

Time-averaged gravitational pressure drop (kPa)

10 10 10 15 30

8 12 16 16 16

0.0178 0.0079 0.0044 0.0099 0.0397

14.587 14.588 14.588 14.449 13.708

plate, which are made of 10.0 mm thick apparent plexiglas. The width and the length of the duct are 43 mm and 2000 mm, respectively. After finishing all experiments, the test section was breached to measure the gap size with a clearance gauge. 11 measurement locations along flow direction and three locations in transverse direction were set uniformly as shown in Fig. 3. The gap size is 1.41 mm with an uncertainty of ±0.01 mm. Mass flow meters (Promass 83) are used to measure the gas and liquid flow rate. They have adjustable spans in 0–4000 kg/h with the uncertainty of 0.1%. Two pressure transducers are used to measure the local pressure (PR35X), with the accuracies of 0.2% and measurement range of 0–250 kPa (P1) and 0–100 kPa (P2), respectively. The temperature measurement is performed at the outlet of the test section by standard thermometer with a measured error of ±0.1 °C. All the test signals are recorded by NI data acquisition system (sampling frequency is 256 Hz and the uncertainty is ±0.1%) except for the water and air temperatures. The range of the present experiment is as follows.

Superficial water velocity, jl Superficial air velocity, jg Liquid Reynolds number, Rel Gas Reynolds number, Reg

0.16–3.73 m/s 0.58–31.44 m/s 543–13,250 221–8310

3. Data processing The total pressure drop in rolling motion (DPt) for adiabatic two-phase flow consists of frictional pressure drop (DPf), gravitational pressure drop (DPg) and additional pressure drop (DPadd):

Dpt ¼ Dpg þ Dpf þ Dpadd

ð4Þ

In which the gravitational pressure drop could be calculated from:

Dpg ¼ qgLcosh

ð5Þ

where g is the gravity acceleration; q is the fluid density, which can be expressed as:

q ¼ qg a þ ql ð1  aÞ

ð6Þ

a denotes the area-averaged void fraction for which Jones and Zuber (1979) proposed drift flux correlation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jg =a ¼ C 0 j þ ð0:23 þ 0:13h=wÞ Dqgw=ql

ð7Þ

where the distribution parameter C0 could be calculated by the correlation of Ishii (1977):

qffiffiffiffiffiffiffiffiffiffiffiffi C 0 ¼ 1:35  0:35 qg =ql

ð8Þ

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narrow rectangular duct. Shah and London (1978) gave the following correlation with the Reynolds number below 2500:

kRe ¼ 96ð1  1:3553e þ 1:9467e2  1:7012e3 þ 0:9564e4  0:2537e5 Þ ð14Þ

Fig. 5. Experimental results of two-phase flow in the immobile pipe.

For the bubbly flow, Eq. (7) is not valid for rectangular duct. So the drift velocity is calculated by Ishii (1977):

jg = a ¼ C 0 j þ

pffiffiffirg Dq0:25 2 ð1  aÞ1:75 2

ql

ð9Þ

In Eqs. (7)–(9), j is the superficial velocity; h and w denote the height and width of the test section, respectively as shown in Figs. 2 and 3; qg, ql and Dq represent the densities of gas and liquid as well as their difference; r is the surface tension. Gao et al. (1997) analyzed the forces acting on coolant under rolling condition, proved the existence of additional pressure drop and gave its theoretical expression: 2

DPadd ¼ q

0 2 2 ðz02 2p 4p2 hm 2p 2  z1 Þ 4p hm  cos2 sin t t þ qy01 ðz02  z01 Þ  2 2 T T T T2 ð10Þ

z02 , z01 and y01 denote the coordinates in the relative coordinate system (o0 x0 y0 z0 ) fixed on the rolling platform as shown in Fig. 2. The first and the second term on the right-hand side of Eq. (10) represent the additional pressure drop caused by centrifugal inertial force and tangential inertial force respectively (Xing et al., 2012). Lockhart and Martinelli (1949) had put first forward a correlation for calculating the two-phase frictional pressure gradient by introducing a Martinelli parameter X to the two-phase multiplier /2l :

/2l ¼

dPf dz

 dðPf Þl dz

ð11Þ

Chisholm (1967) gave the following correlation to calculate /2l :

/2l ¼ 1 þ

C 1 þ X X2

ð12Þ

The parameter C depends on the flow regimes of the liquid and gas phases, ranging from 5 to 20, and the Martinelli parameter X is defined by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  dðPf Þl dðPf Þg X¼ dz dz

ð13Þ

where d(Pf)g/dz and d(Pf)l/dz represent the gas and liquid frictional pressure gradient as each phase flows alone through the same pipe with its mass flow rate, respectively. In present experiments, the gas and water flow rate fluctuate slightly, so traditional correlations are applied to calculate the single-phase flow friction factor in

Fig. 6. Time-averaged frictional resistance under rolling condition: (a) laminar flow region; (b) transition flow region; (c) turbulent flow region.

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Table 2 Evaluations of some conventional correlations with the time-averaged frictional pressure drop. Correlations

Conditions Data number

Vertical 171

hm10°T8s 163

hm10°T12s 153

hm10°T16s 148

hm15°T16s 151

hm30°T16s 122

Lee and Lee (2002)

MAE (%) Proportion(%) MAE (%) Proportion(%) MAE (%) Proportion(%)

16.97 90.06 18.74 80.12 24.14 70.19

17.11 87.73 12.49 95.71 19.29 80.98

15.90 97.39 13.07 92.81 16.58 83.01

18.11 88.51 12.86 91.89 17.36 81.08

17.66 86.09 14.56 90.07 19.41 76.16

16.41 86.89 13.31 94.26 18.37 81.15

Chisholm (1967) Mishima et al. (1993)

where e is the ratio of duct height to duct width. For Re P 2500, correlation of Sadatomi et al. (1982) was used:

k ¼ 0:3164  ½ð0:0154C V =64  0:012Þ1=3 þ 0:85  Re0:25

ð15Þ

where CV = kRe is given by Eq. (14). The experimental uncertainties of /2l and X under non-rolling condition are ±14.5% and ±13.56% respectively based on Eqs. (11) and (13). 4. Results and discussion 4.1. Pressure drop component in rolling motion The transient frictional and additional pressure drop of twophase flow oscillates with the same period of rolling motion, whereas the gravitational pressure drop fluctuates with the half rolling period, as shown in Fig. 4. It is indicated that the fluctuation amplitude of additional pressure drop is 1–2 orders of magnitude smaller than that of gravitational pressure drop. As the mass quality increases, gravitational and additional pressure drop decreases as the result of decreasing two-phase density as shown in Eqs. (5) and (10). Meanwhile, much larger frictional loss is resulted as shown in Fig. 4. Therefore the proportion of gravitational and additional pressure drop to the total pressure drop decreases sharply as the mass quality increases. Table 1 summarizes time-averaged gravitational and additional pressure drop in rolling motion by integrating Eqs. (5) and (10) respectively, assuming that the density is 1000 kg/m3. It is shown from Table 1 that for the case of larger rolling amplitude and smaller rolling period, rolling motion gives rise to the increasing of time-averaged additional pressure drop. In addition, time-averaged gravitational pressure drop only increases as the rolling amplitude decreases, which is nearly independent of the rolling period. The time-averaged additional pressure drop is 2–3 orders of magnitude smaller than the gravitational pressure drop, much smaller than frictional pressure drop, so the effect of time-averaged rolling inertial force on flow resistance could be neglected. 4.2. Time-averaged frictional resistance in rolling motion The experimental /2l varying with X in vertical non-rolling condition is plotted in Fig. 5, in which three flow regions are exhibited apparently against the liquid Reynolds number (It is defined as the Reynolds number assuming the liquid phase flows alone in the same pipe with its mass flow rate, Rel = jlde/cl), namely, laminar flow region (Rel < 800), transition flow region(800 6 Rel 6 1400) and turbulent flow region (Rel > 1400). Wang et al. (2011a) and Zhou and Wang (2011) obtained the similar characteristics of frictional resistance in mini/micro rectangular ducts. Therefore, the method is used to classify the experimental data in present study. Chisholm parameter C strongly depends on the liquid and gas phase flow condition and the present experimental data in steady condition varies in the range of (C =5–20) as shown in Fig. 5.

The time-averaged two-phase frictional multiplier changing with X in rolling motion is also divided into three regions as illustrated in Fig. 6. In each flow region, the rolling period and amplitude nearly have no influence on time-averaged frictional resistance. Section 4.1 have concluded the neglectable time-averaged rolling inertial force, additionally, the time-averaged flow rate and frictional pressure drop under rolling condition are very close to that under non-rolling condition. Therefore, we can conclude that the effect of rolling motion in present condition nearly has no influence on time-averaged frictional resistance of adiabatic two-phase flow. Similar results have been achieved concerning single-phase flow resistance in rolling motion as shown in recent literatures of Xing et al. (2012) and Yan et al. (2010, 2011). Lee and Lee (2002) presented a correlation to calculate air– water two-phase flow frictional pressure drop in narrow rectangular duct. The gap size ranges from 0.4 to 4 mm while the width being fixed to 20 mm. The corresponding aspect ratio ranges from 0.02 to 0.2, in which the present aspect ratio is included. The Chisholm parameter C in their correlation could be expressed as:

C¼A



l2l ql rde

q 



ll j r s Rel0 r

ð16Þ

The constant A and exponent (q, r and s) are determined by experimental data with the corresponding value referred to Lee and Lee (2002), Mishima et al. (1993) conducted experiments to study the flow regime, void fraction, bubble velocity and the pressure loss in narrow rectangular duct. Their results indicated that the Chisholm parameter C depends on the hydraulic diameter, decreasing from 21 to 0 as the hydraulic diameter decreases from 10 to 0.1 mm. The also proposed a correlation both for round tubes and rectangular ducts using an appropriate hydraulic diameter, which could be expressed as:

C ¼ 21½1  expð0:27de Þ

ð17Þ

Kandlika (2002) differentiated the conventional channel, minichannel and microchannel associated with the hydraulic diameter. The channels employing hydraulic diameter ranging from 200 lm to 3 mm are referred as minichannels. When the gap width of narrow channel is less than 3 mm, it has been shown that effect of surface tension on flow and heat transfer characteristics is very significant. Therefore the critical duct height differentiated narrow channel and conventional channel is considered to be 3mm in some literatures (Huang et al., 2009; Xu et al., 2008). In present study, the test section is thought to be narrow or mini rectangular duct from the above discussion. Table 2 shows the comparison of the time-averaged frictional pressure drop with predictions by Lee and Lee (2002) correlation, Chisholm (1967) correlation and Mishima et al. (1993) correlation. The Mean Absolute Error (MAE) is defined as:

MAE ¼

1 X jDP pred  DPexp j  100 n DPexp

where n is the number of the data points.

ð18Þ

D. Xing et al. / Annals of Nuclear Energy 53 (2013) 109–119

Fig. 7. Transient frictional pressure gradient in rolling motion: (a) laminar flow region; (b) transition flow region; (c) turbulent flow region.

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Fig. 8. Comparison of conventional correlations with the experimental results: (a) laminar flow region; (b) transition flow region; (c) turbulent flow region.

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DPpred and DPexp are the predicted frictional pressure drop and the experimental results, respectively. The experimental results under vertical flow condition agree favorably with the Lee and Lee (2002) correlation and Chisholm (1967) correlation, which imply the accuracy and reliability of our experimental data. It can be seen from Table 2 that Lee and Lee (2002) correlation and Chisholm (1967) correlation could calculate the time-averaged frictional resistance under rolling conditions with a rather good precision. Evaluating all the experimental data in rolling motion, the MAE of Lee and Lee (2002) correlation, Chisholm (1967) correlation and Mishima et al. (1993) correlation for time-averaged frictional pressure drop is 17.04%, 13.26% and 18.20%, respectively, and the Chisholm (1967) correlation gives the most accurate prediction.

4.3. Transient frictional pressure gradient in rolling motion The transient frictional pressure gradient fluctuates with the same period of rolling motion both in laminar and in transition flow region, whereas, it is nearly invariable in turbulent flow region as shown in Fig. 7. The Relative Frictional Pressure Gradient is defined as the ratio of the transient frictional pressure gradient to its time-averaged value. It can be seen from Fig. 7 that the fluctuation amplitude of the Relative Frictional Pressure Gradient decreases sharply as the superficial liquid velocity increases. As the superficial gas velocity increases, despite that the fluctuation amplitude of frictional pressure gradient becomes larger, the Relative Frictional Pressure Gradient still decreases due to the larger increasing of frictional loss. The superficial gas velocity has less

Table 3 Constants and exponents for empirical correlation. Ref < 800; Reg 6 2000

m a b d

800 6 Ref 6 1400; Reg 6 2000

800 6 Ref 6 1400; Reg > 2000

k0

k1

k2

Ref < 800; Reg > 2000 k0

k1

k2

k0

k1

k2

k0

k1

k2

4.517E1 4.781E2 1.146 0

2.200E1 8.702E2 1.603 0

9.376E9 7.123E1 1.000 2.159

1.549E2 1.642E2 4.414E1 4.306E1

2.897E4 2.793E2 9.396E1 1.235

2.679E6 7.246E-1 1.144 0

4.550E3 1.490E1 4.900E1 1.512

1.285E5 1.313E-1 9.689E-1 2.434

1.563E6 6.312E-1 1.316 0

6.266E3 1.653E-2 1.484 7.141E-1

3.864E5 0 1.777 2.100

8.511E3 1.649E1 7.869E1 0

Fig. 9. Comparison of the predicted C with the experimental results: (a) laminar flow, Reg 6 2000; (b) laminar flow, Reg > 2000; (c) transition flow, Reg 6 2000; (d) transition flow, Reg > 2000.

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influence on fluctuation amplitude of frictional pressure gradient than that of superficial liquid velocity. Fig. 7 also indicated that the phase lag between the rolling motion and transient frictional pressure gradient could be neglected. Fig. 8 presents the comparison of experimental transient frictional pressure gradient with Lee and Lee (2002) correlation and Chisholm (1967) correlation in rolling motion. It can be seen from Fig. 8 that the both correlations poorly predict the transient frictional pressure gradient under rolling condition in laminar and transition flow regions because in these regions frictional pressure gradient fluctuates strongly with the same period of rolling motion. The parameter C of Chisholm (1967) correlation is determined by the gas and liquid phase flow conditions, varying discretely in 5, 10, 12 and 20. If the gas and liquid phase flow with the Reynolds number fluctuating around 1000, the predicted transient frictional pressure gradient changes abruptly and deviates dramatically from the experiments, as shown in Fig. 8b. In turbulent flow region, the frictional pressure gradient oscillates randomly, so the Lee and Lee (2002) correlation and Chisholm (1967) correlation work better as shown in Fig. 8c. Lee and Lee (2002) correlation and Chisholm (1967) correlation are developed based on experimental data under non-rolling condition. It works for the time-averaged condition as shown in Table 2, but not for transient pressure gradient which oscillates periodically. 4.4. New correlation for transient frictional resistance in rolling motion Up to now, no correlation is suitable for calculating transient frictional pressure drop of two-phase flow in rolling motion, so new correlation is required for engineering application. Murata et al. (2002), Xing et al. (2012) and Zhang et al. (2009) took into account of the influence of rolling motion by adopted the rolling Reynolds number, which denoted a ratio between the inertial force caused by rolling motion and viscous force, and could be expressed as:

Reroll ¼ U h l=c

ð19Þ

where c represents the kinematic viscosity. The rolling velocity Uh is the scale of the rolling motion, and is described as:

U h ¼ 4hm l=T

ð20Þ

where l is the distance between the test section and the rolling shaft (l ¼ y01 ). Based on Chisholm (1967) correlation, we modified the parameter C, which is affected by many factors and could be expressed: 2 C ¼ f ðbl =j2 ; x2 lde =j2 ; Reroll ; Reg ; Rel ; X; xÞ

ð21Þ 2

2

2

2

where the dimensionless parameters of bl =j and x lde =j denote the ratio of tangential and the centrifugal inertial force caused by rolling motion to flow inertial force respectively. The first three groups represent the effect of rolling motion on frictional resistance. As discussed in study of Xing et al. (2012), the effect of centrifugal inertial force is much slighter than the tangential one, so the effect of x2 lde =j2 is neglected in the new correlation. By analysis of the experimental data, the parameter C could be expressed as: 2 C ¼ k0 þ k1 X þ k2 ðbl =j2 Þ

ð22Þ

where j is the time-averaged superficial velocity, and the coefficient ki(i = 0, 1, 2) in Eq. (22) could be written as the function of timeaveraged Reynolds number for gas and liquid, rolling Reynolds number and mass quality:

ki ¼ mRearoll ðReg =Ref Þb xd

ð23Þ

Constant m and exponents a, b and d are determined by gas and liquid phase flow conditions, and given in Table 3 by multiple

regression against a large number of experimental data in laminar and transition flow region. The comparisons of the proposed correlation and the experimental results are plotted in Fig. 9, showing a good agreement. The proposed correlation predicts the periodical change of the transient frictional pressure drop much better than Lee and Lee (2002) correlation and Chisholm (1967) correlation, because that it takes into account of the rolling effect on frictional resistance by introducing dimensionless parameters (such as the rolling Rey2 nolds number and bl =j2 ). The modified correlation is validated over the flow and rolling parameter ranges: X varying from 0.24 to 2.98; rolling period ranging from 8 s to 16 s and rolling amplitude ranging from 10° to 30°. The correlation is developed from a single duct. More data sets are required to extend the application of the proposed correlation. 5. Conclusion In the present paper, frictional resistance of adiabatic twophase flow in a narrow rectangular duct under rolling condition is investigated experimentally. Comparisons are made between conventional correlations and the experimental data. Major conclusions of this study are summarized as follows: Frictional resistances of two-phase flow under rolling condition are divided into three regions according to liquid Reynolds number, similar with that in vertical non-rolling condition. The timeaveraged additional pressure drop is so small compared with frictional pressure drop that the effect of rolling motion on time-averaged frictional resistance is neglectable. Different from characteristics of time-averaged frictional resistance, transient frictional pressure drop fluctuates synchronously with rolling motion both in laminar and transition flow region. The fluctuation amplitude of the Relative Frictional Pressure Gradient decreases as the gas and liquid velocities increase. For the case of turbulent flow, transient frictional pressure drop does not oscillate periodically. Although Lee and Lee (2002) correlation and Chisholm (1967) correlation are suitable for time-averaged frictional pressure drop under rolling condition, they cannot predict the periodical change of the transient frictional pressure drop. Finally, a new correlation applied to calculate the transient frictional pressure drop in narrow rectangular duct under rolling condition is achieved by modifying the factor of C in Chisholm (1967) correlation. The proposed correlation takes into account the influence of rolling motion and flow conditions, expressed as dimensionless groups, and shows good agreement with the experimental data. Acknowledgement The authors are profoundly grateful to the financial supports of the National Natural Science Foundation of China (Grant Nos.: 51076034 and 11175050). References Cao, X.X., Yan, C.Q., Gao, P.Z., et al., 2006. Pressure drop correlations of single-phase and two-phase flow in rolling rubes. In: Proceedings of the 14th International Conference on Nuclear Engineering (ICONE 14-89237), Miami, Florida, USA. Chisholm, D., 1967. A theoretical basis for the Lockhart-Martinelli correlation for two-phase flow. Int. J. Heat Mass Transfer 10, 1767–1778. Gao, P.Z., Pang, F.G., Wang, Z.X., 1997. Mathematical model of primary coolant in nuclear power plant influenced by ocean condition. J. Harbin Eng. Univ. 18, 24– 27 (in Chinese). Hong, G., Yan, X., Yang, Y.H., et al., 2012a. Experimental study on onset of nucleate boiling in narrow rectangular channel under static and heaving conditions. Ann. Nucl. Energy 39, 26–34. Hong, G., Yan, X., Yang, Y.H., et al., 2012b. Experimental research of bubble characteristics in narrow rectangular channel under heaving motion. Int. J. Therm. Sci. 51, 42–50.

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