Prediction models of void fraction and pressure drop for gas-liquid slug flow in microchannels

Accepted Manuscript Prediction models of void fraction and pressure drop for gas-liquid slug flow in microchannels Ryo Kurimoto, Kento Nakazawa, Hisato Minagawa, Takahiro Yasuda PII: DOI: Reference:

S0894-1777(17)30154-1 http://dx.doi.org/10.1016/j.expthermflusci.2017.05.014 ETF 9107

To appear in:

Experimental Thermal and Fluid Science

Received Date: Revised Date: Accepted Date:

23 January 2017 12 April 2017 18 May 2017

Please cite this article as: R. Kurimoto, K. Nakazawa, H. Minagawa, T. Yasuda, Prediction models of void fraction and pressure drop for gas-liquid slug flow in microchannels, Experimental Thermal and Fluid Science (2017), doi: http://dx.doi.org/10.1016/j.expthermflusci.2017.05.014

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Prediction models of void fraction and pressure drop for gas-liquid slug flow in microchannels Ryo Kurimoto1*, Kento Nakazawa1, Hisato Minagawa1, Takahiro Yasuda1 1 Department of Mechanical Systems Engineering, The University of Shiga Prefecture 2500, Hassaka, Hikone, Shiga, Japan. *Corresponding author: [email protected], Tel: +81-749-28-8383 Highlight (1) Experiments of gas-liquid slug flow in a circular microchannel were carried out using four circular microchannels and three liquid properties. (2) Measured pressure drop was predicted using the model proposed by Warnier et al. (2010) with the liquid film thickness model proposed by Han and Shikazono (2009). (3) A distribution parameter model developed in this study showed good agreement with the measured data. Abstract The void fraction and pressure drop of gas-liquid slug flow in a circular microchannel were measured using a high-speed video camera and image processing method. Water and two types of glycerol-water solutions were used as the liquid phase, and N2 gas was used as the gas phase. Four microchannels with different diameters (200, 320, 500, and 700 m) were used. The measured data were compared with the existing and newly developed models. The following conclusions were obtained: (1) the pressure drop was predicted using the model proposed by Warnier et al. (2010) with the liquid film thickness model proposed by Han and Shikazono (2009) within + 20%, (2) a distribution parameter model developed in this study can predict the distribution parameter within + 10% and is more reliable than the other models. Keywords: slug flow, microchannel, void fraction, pressure drop, drift flux model 1. Introduction Gas-liquid two-phase flows in microchannels are prevalent in microdevices, e.g., heat exchangers, fuel cells, and microreactors. For the design and development of microdevices, it is essential to have the knowledge of two-phase flows in microchannels and many studies have already been carried out. Various flow patterns, e.g., bubbly flow, slug flow, and annular flow, are appeared in

gas-liquid two-phase flow in microchannels due to the effects of channel diameter, liquid properties, liquid and gas volumetric flux, shape of gas-liquid mixture, and so on. Several researchers have reported flow pattern maps obtained by their experiments (Triplett et al., 1999; Kawahara et al., 2002; Serizawa et al., 2002; Chung and Kawaji, 2004; Saisorn and Wongwises, 2008; Zhang et al., 2011; Sur and Liu, 2012). Regardless of the difference of their experimental conditions, the slug flow occupies a large region in their maps. In other words, the slug flow appears often in all microchannels. Hence, accurate prediction models of void fraction and pressure drop for slug flow in microchannels are very useful in the design and development of various microdevices. The drift-flux model proposed by Zuber and Findlay (1968) has been widely used for the prediction of void fraction in various multiphase flow systems: uG  c0 jT  vGj

(1)

where uG is the gas slug velocity, c0 the distribution parameter, jT (= jL + jG) the total volumetric flux (the sum of gas volumetric flux jG and liquid volumetric flux jL), and vGj the drift velocity. The drift velocity is regarded as zero in horizontal microchannels. It is well-known that the above equation can be written as



1  c0

(2)

where  (= JG/uG) is the void fraction and  (= jG/jT) the gas volumetric flow ratio. Since the distribution parameter depends on the type of multiphase flow system, various models of the distribution parameter have been proposed (Ishii and Hibiki, 2006). Mishima and Hibiki (1996) carried out experiments on air-water two-phase flow in vertical tubes with diameter 1.05-3.90 mm. They proposed the following distribution parameter model based on their experimental data and experimental data of Kariyasaki et al. (1992):

c0  1.2  0.510 exp(0.691D)

(3)

where D is the channel diameter. This model agrees with the experimental data with a standard deviation of 0.025. Since the model, however, is a function of only D, i.e., does not take into account the effects of liquid properties, it cannot be applied to other liquid

properties. Liu et al. (2005) proposed the following model based on the experimental data of gas-liquid slug flow in vertical channels using three liquids (water, ethanol and oil mixture), three circular channels (D = 0.91, 2.00, and 3.02 mm), and two square channels (D = 0.99 and 2.89 mm):

c0 

1 1  0.61CaT0.33

(4)

where CaT is the capillary number based on the gas slug velocity uG. CaT is defined by CaT 

 L jT 

(5)

where L is the liquid viscosity and  the surface tension. Kawahara et al. (2009) carried out experiments on gas-liquid two-phase flow in a circular microchannel of 250 m diameter using four liquids (distilled water, 4.8, 49, and 100 wt% ethanol-water solutions), and proposed the following distribution parameter model:

c0  aBo 0.19ReL0.01WeG0.01

(6)

where a is constant determined by the flow contraction at the channel inlet, Bo the Bond number, ReL the Reynolds number based on the liquid volumetric flux jL and WeG the Weber number based on the gas volumetric flux jG, respectively. Bo, ReL and WeG are defined by

( L  G ) gD 2 Bo  

(7)

 L jL D L

(8)

ReL 

WeG 

 L jG2 D 

(9)

where L is the liquid density, G the gas density and g the magnitude of the acceleration of gravity. Minagawa et al. (2013) compared eq. (6) with their experimental data on air-water slug flow in microchannels of D = 75, 100 and 200 m. Since the model did not give agree with the data, the following model was proposed:

c0  1.88CaT0.12

(10)

Howard and Walsh (2013) theoretically derived the following distribution parameter model considering the velocity profiles in a gas slug and a liquid film on the assumption that the gas slug shape is a long cylinder of radius RGS:

c0 

R2 2 RGS

(11)

where R is the channel radius. RGS can be calculated from R and liquid film thickness  as RGS = R  . They measured the liquid film thickness and the gas slug velocity of slug flow in a small circular channel of D = 1.58 mm using PD5 oil, ethylene glycol, and AS100 silicone oil, and confirmed that the liquid film thickness can be predicted using the following model proposed by Han and Shikazono (2009) and that the above equation gives good agreement with the measured data:

 0.670Ca 2 / 3  D 1  3.13Ca 2 / 3  0.504Ca 0.672Re 0.589  0.352We0.629

(12)

where Ca is the capillary number, Re the Reynolds number, and We the Weber number defined, respectively, by Ca 

 L uG 

(13)

Re 

 L uG D L

(14)

We 

 LuG2 D 

(15)

The characteristic velocities of these dimensionless numbers are the gas slug velocity uG. Pressure drop models based on a unit cell model have been developed to predict the pressure drop of slug flow in a microchannel. A single unit cell is composed of a single gas slug and a single liquid slug. Since single unit cells flow in turn in a microchannel, it can be assumed that the whole pressure drop of slug flow in microchannel is equal to the sum of the pressure drops of single unit cells. The pressure drop model based on a unit cell model is represented as follows:

PUC  PL  PG

(16)

where PUC is the pressure drop in a unit cell, PL the frictional pressure drop in a liquid slug and PG the pressure drop caused by the presence of a gas slug. Warnier et al. (2010) proposed a pressure drop model based on the unit cell model. In their model, PL is represented by

64 LL  L jT2 PL  ReT D 2

(17)

where ReT is the Reynolds number based on the total volumetric flux jT, and LL the liquid slug length on the assumption that the gas slug shape is a cylinder (Fig. 1). ReT is defined by

ReT 

 L jT D L

PG is represented by

(18)

PG  7.16

 f lf D

(19)

where flf is a function representing the contribution of the liquid film thickness to the pressure drop. They used the following model for flf:

flf 

(3Ca) 2 / 3

(20)

1  3.34Ca 2 / 3

The above equation is based on the following liquid film thickness model proposed by Aussillous and Quéré (2000)

 0.670Ca 2 / 3  D 1  3.34Ca 2 / 3

(21)

Sur and Liu (2012) and Minagawa et al. (2013) reported that Warnier et al.’s model agrees with their experimental data in gas-liquid systems. Eain et al. (2015) carried out experiments on slug flow in gas-liquid and liquid-liquid systems. In order to predict the pressure drop of slug flow in both gas-liquid and liquid-liquid systems, they extended Warnier et al.’s model and the extended model shows good agreement with their measured data. Although eq. (20) is used for flf in eq. (19), Han and Shikazono (2009) reported that eq. (21) can predict liquid film thickness only in low Ca region, in contrast to eq. (12) proposed for a wide range of Ca. Hence, the following flf model will be used for a wide range of Ca instead of eq. (20):

flf 

(3Ca) 2 / 3 1  3.13Ca 2 / 3  0.504Ca 0.672Re 0.589  0.352We0.629

(22)

Although the distribution parameter and pressure drop models have already been developed as described above, validation of the models for a wide range of dimensionless numbers has not been carried out yet. In this study, the void fraction and pressure drop of

gas-liquid slug flow in circular microchannels were measured using a high-speed video camera and image processing method. To carry out the experiment for a wide range of dimensionless numbers, four circular microchannels of channel diameter D = 200, 320, 500, and 700 m and three liquids (water and two glycerol-water solutions) were used. The measured data were compared with the pressure drop and distribution parameter models to investigate the models’ applicability. Hayashi et al. (2010), Hayashi et al. (2011), and Kurimoto et al. (2013) have developed Froude number correlation for Taylor bubbles and drops rising through a vertical pipe. Its functional form was deduced by making use of a scaling analysis based on local instantaneous field equations and the jump conditions at the interface, and some remaining constants after the scaling analysis were determined from experimental and numerical data. By referring to their process, we developed a distribution parameter model to accurately predict the void fraction for gas-liquid slug flow in circular microchannels. Cylindrical assumption

Slug unit Unit velocity: jT Gas

Gas

D

LL L PUC = PL + PG

Fig. 1 Slug flow in a microchannel 2. Derivation of functional form of the distribution parameter model The momentum equation for an incompressible Newtonian fluid in two-phase flow based on one-fluid formulation at the steady state is given by (V  V)  P      nI

(23)

where  is the density, V the velocity, P the pressure,  the viscous stress tensor, n the unit normal to the interface,  the curvature, and I the delta function, which is non-zero only at the interface. Although gas slugs gradually expand by decreasing the pressure from upstream to downstream in the channel, the expansion is neglected not to complicate the problem. Figure 1 shows the schematic of slug flow in a circular microchannel. A single slug unit is driven by the pressure difference between two points separated by the unit slug length L. The pressure difference can be represented by the pressure drop model. The

model has already included the effect of viscous and surface tension force on the pressure drop as mentioned above (Bretherton, 1961; Warnier et al., 2010). Hence, the viscous and surface tension force term in eq. (23) can be neglected by adopting the pressure drop model as the pressure term in eq. (23). The channel diameter D and total volumetric flux jT is adopted as the characteristic length and velocity, respectively. It is postulated that G << L for a gas slug in a liquid. Hence, by neglecting the liquid film,  can be represented as



LL L L

(24)

Substituting the above equation, pressure difference in the unit cell PUC, and the characteristic scales into eq. (23) yields

2 P L j  L L T  C1 UC L D L PL  PG  C1 L







(25)

C 2 LL  L jT2 ReT D  C3 ( / D) f lf L

where Ci (i = 1, 2, 3) are constants. The above equation can be written as follows:  C  L LL 1  2  ReT

 2  jT  C3f lf 

(26)

Substituting eq. (1) into the above equation yields

c02 

 L LLuG2  c1  c2 ReT  f lf

(27)

where ci (i = 1,2) are constants. By adopting eq. (15), the above equation can be written as

WeLL D c1  c2 ReT  f lf

c0 

(28)

Since flf is mainly a function of Ca2/3 (Bretherton, 1961; Aussillous and Quéré, 2000; Han and Shikazono; 2009), liquid film thickness decreases with decreasing bubble velocity on the condition that the liquid does not change. Equation (11) demonstrates that c0 takes a value more than unity or unity by vanishing of the liquid film, i.e., uG > jT. Taking the limit uG → 0 and inevitably jT → 0 yields WeLL D c1  c2 ReT  f lf uG 0

c0  lim

jT 0

WeLL D c1  c2 ReT  uG 0 Ca 2 / 3

 lim

jT 0

L ReCa1 / 3  L u G 0  D

 lim

jT 0

(29)

c    c1  2  ReT  

0

The following equation, therefore, is obtained in the functional form:

c0  1 

WeLL D c1  c2 ReT  f lf

(30)

c1LL/D and c2LL/D are determined from measured data in Section 4.3. 3. Experimental method The experimental setup is shown in Fig. 2. The circular microchannels of 200, 320, 500 and 700 m diameter were used. The microchannels were made of silica glass (D = 200, 320 m; GL Sciences Inc.) or Pyrex (D = 500, 700 m; SUN-YELL Corp.). The tolerances of the diameters were + 6 m (D = 200, 320 m) and + 15 m (D = 500, 700 m), respectively. The lengths of the microchannels Lm were 100 mm (D = 200 m), 200 mm (D = 320 m) and 250 mm (D = 500, 700 m), respectively. Experiments were carried out at atmospheric pressure and room temperature. The temperature of the liquid

was kept at 298 + 1 K. Deionized water (Wako Pure Chemical Industries, Ltd.) or a glycerol-water solution (52 or 64 wt%) was used as the liquid phase, and nitrogen gas was used as the gas phase. The kinematic viscosity of the 52 and 64 wt% glycerol-water solutions was, respectively, five and ten times that of water. The glycerol-water solutions were made of deionized water and glycerol (Wako Pure Chemical Industries, Ltd.). The density and surface tension were measured at least five times using a densimeter and capillary tubes (glass tube, 700 m i.d.), respectively. Uncertainties estimated at 95% confidence level in the measured density and surface tension were 0.18% and 0.79%, respectively. The measured value of density agreed well with literature data (Ishikawa, 1968). The viscosity given in literature (Ishikawa, 1968) and the measured density and surface tension were used for the calculation of dimensionless numbers. The liquid was kept in a tank and was driven in the channel by compressed air produced by a compressor. Nitrogen gas was supplied from a gas cylinder. The flow rate of the gas was controlled by a mass flow controller (FUJIKIN Inc., FCST 1005(M)L). The liquid and nitrogen gas were mixed at a Tee junction made of transparent acrylic resin. The size of the Tee junction is shown in Fig. 3. The microchannel were connected to the end of the Tee junction horizontally. The inner diameter of the Tee junction was 0.5 mm for the microchannels of D = 200 and 320 m, and 1.0 mm for the microchannels of D = 500 and 700 m. The mixed liquid and gas phases flowed in the microchannel as slug flow. Images of slug flow were taken using a high-speed video camera (PHOTRON Ltd., FASTCAM Mini UX50, frame rate = 500-5000 frame/s, spatial resolution ~ 0.0014-0.0065 mm/pixel) from the top of the microchannel. The distances between the test section and the end of the Tee junction Lt were 50 mm (D = 200 mm) and 100 mm (D = 320, 500 and 700 m), respectively. The microchannel was placed between two parallel glass plates; space between the two plates was filled by the same liquid that flowed in the microchannel to reduce the optical distortion in slug flow images. A high lumen LED light source (Integrated Design Tools, Inc. Constellation 120 5600K) was used for back illumination to clearly identify the gas-liquid interface. The images were taken after gas slugs reached a constant length. The liquid flow rate was measured by an electric balance located at the end of the microchannel. For the estimation of uncertainties, the gas and liquid flow rate were measured in advance of the experiment. Uncertainties estimated at 95% confidence level in the measured gas and liquid flow rate were 2.5% and 1.9%, respectively. The void fraction in a single unit cell can be calculated from the gas slug volume and slug length. The gas slug volume was calculated from the slug flow images using image processing. An example of image processing is shown in Fig. 4. Figure 4(a) is the original

image of a single gas slug. The gas slug image with the background subtracted was transformed into binary images using an appropriate threshold level (Hosokawa and Tomiyama, 2003) as shown in Fig. 4(b). Since gas slugs were axisymmetric, all the cross-sections of a gas slug were circles of radius Ri, where the index i denotes the pixel number in the perpendicular direction to the cross-section. The height of a circular disk in the images was one pixel and its physical length was z. The resultant circular disks were stacked in the perpendicular direction to reconstruct a three-dimensional gas slug as shown in Fig. 4(c). Hence the gas slug volume is given by VG   Ri2 z where N is N i 1

the total number of pixels in the perpendicular direction. The error in the number of pixels in binarization was estimated as + 1 pixel. The smallest calculated gas slug volume was 0.0796 mm3 in the channel of D = 200 m. For a spatial resolution of 0.0014 mm/pixel, the relative error in the measured gas slug volume was estimated as less than + 3.0%. The errors can be decreased with increasing gas slug length and channel diameter. The gas slug velocity was calculated from the front position of a gas slug and time interval. The front position of a gas slug was observed in four or five consecutive images. The fluctuation of the gas velocity between the images was less than 1.7%. Hence, the gas slug velocity can be regarded as constant. The averaged values for five gas slugs were used as the measured value of the gas slug volume and velocity. The variabilities of the gas slug volume and velocity were less than 1.3% and 1.2%, respectively. Figure 5 shows the unit length calculation for a long slug unit. The unit length of a single slug unit often exceeds the image length, i.e., L > Li, where Li is the length of the image region. The gas slug front position outside of the image region can be predicted using the gas slug velocity obtained from the images in which the front position was observed. Hence, L can be calculated from the distance between the predicted front position and the next gas slug front position that appeared in the image region. When the gas slug length LG is larger than Li, the gas slug volume is calculated by summing the volumes gas slug regions. In Fig. 5, the front region of the gas slug is observed in the image region at t = 0, and the volume of the region, V1, can be calculated. The middle region is observed at t = t and its volume, V2, can be calculated by predicting the front position of the middle region using the gas slug velocity. The volume of the rear region, V3, can be also calculated by predicting the front position of the rear region at t = 2t. A pressure transducer often have been used to measure the pressure difference in a microchannel in other studies (Kawahara et al., 2002; Kawahara et al., 2009; Sur and Liu, 2012). Since the pressure transducer cannot be connected to the microchannel directly, an upstream channel with different channel diameter is required. Hence, the measured

pressure difference includes a pressure drop due to the expansion or contraction of the channel. However, the pressure drop cannot be estimated accurately in two-phase flow. Therefore, Boyle’s law was adopted to calculate the pressure difference between the test section and the end of the microchannel in the present study, i.e., pajGa = pjG, where pa is the atmospheric pressure, jGa the gas volumetric flux at the end of the microchannel, p the pressure at the test section, and jG the volumetric flux at the test section. jGa can be calculated from the gas flow rate controlled by the mass flow controller, and jG at the test section can be calculated from  and uG obtained by gas slug images. Hence, the pressure at the test section can be calculated from the Boyle’s law relationship, and the pressure difference was calculated by the difference between p and pa. Uncertainties estimated at 95% confidence level in the measured pressure difference was + 2.3%. High-speed video camera

Liquid Tank

Regulator Compressor

Tee junction

Microchannel N2 gas cylinder

LED light Electric balance

Mass flow controller

Regulator

Fig. 2 Experimental setup Lm Lt

Gas-liquid slug flow Gravity

Fig. 3 Schematic of Tee junction and microchannel

15 mm 15 mm

Test section

0.5 or 1.0 mm

Microchannel

25 mm

Liquid flow

N2 gas flow

(a) Original image

(b) Binary image

Ri z

(c) Reconstructed shape Fig. 4 Image processing Image region V1

t=0

t = t

t = 2t

V1

V1

V2

V2

V3 Li

LG L

Fig. 5 Unit length calculation for a long slug unit

Table 1 Experimental conditions Liquid

Deionized water

Glycerol-water solution (52 wt%)

Glycerol-water solution (64 wt%)

Gas

N2 997 1.19

N2 1129 1.19

N2 1164 1.19

L [kg/m3] G [kg/m3] L [mPa·s] G [mPa·s]  [mm2/s]  [mN/m]

0.890

5.62 2

1.80 × 10 0.893 71.9

1.80 × 10 4.98 68.7

11.6 2

1.80 × 102 9.97 67.3

D [m] JL [m/s] JG [m/s]

200, 320, 500, 700 0.184-1.52 0.0702-0.877

200, 320, 500, 700 0.105-0.431 0.0323-1.15

320, 500, 700 0.0916-0.263 0.0277-0.748

ReT We

139-941 0.834-26.2

19.5-84.3 0.802-22.5

6.03-34.3 0.341-10.3

Ca

0.00537-0.0310

0.0207-0.169

0.0425-0.235

4. Result and discussion 4.1 Comparisons of pressure gradient between models and measured data Since the distribution parameter model developed in the present study is based on the Warnier et al.’s pressure drop model (Eqs. (16), (17) and (19)), the applicability of the

model is validated in advance. Figure 6 shows the comparisons of pressure gradient dp/dz between Warnier et al.’s model and measured data. Equations (20) and (22) are used to calculate flf in the left and right panels of Fig. 6, respectively, to investigate the effects of the liquid film thickness models on the predicted dp/dz. Warnier et al.’s model agrees with the measured data within + 20% regardless of the liquid film thickness model. Figure 6 does not show a clear difference in dp/dz due to the difference the liquid film thickness models. The differences of dp/dz with eq. (20) and (22) are evaluated by the standard deviation calculated by



Nd

i 1

( xcal  xexp ) 2 ( N d  1) where Nd is the number of data, xexp

the each experimental data and xcal the each calculated data, respectively. The standard deviations of dp/dz with eq. (20) and (22) are 0.0767 and 0.0762, respectively. This is because the average of the ratio of PG to PUC is 10% in the experiment. Hence, the difference in the predicted liquid film thicknesses does not strongly affect dp/dz. However, this result indicates that Warnier et al.’s model can predict dp/dz more accurately with eq. (22) than with eq. (20) for slug flow in microchannels.

2

2 +20% Predicted dp/dz [MPa/m]

Predicted dp/dz [MPa/m]

+20%

20% 1 D = 200 m Water G-W 52 wt% D = 320 m Water G-W 52 wt% G-W 64 wt%

0

1 Measured dp/dz [MPa/m]

2

(a-1) Eqs. (16), (17), (19), and (20)

20% 1 D = 200 m Water G-W 52 wt% D = 320 m Water G-W 52 wt% G-W 64 wt%

0

1 Measured dp/dz [MPa/m]

2

(a-2) Eqs. (16), (17), (19), and (22)

(a) D = 200 and 320 m 0.3

0.3

+20%

0.2 20%

0.1

0

D = 500 m Water G-W 52 wt% G-W 64 wt% D = 700 m Water G-W 52 wt% G-W 64 wt%

0.1 0.2 Measured dp/dz [MPa/m]

(b-1) Eqs. (16), (17), (19), and (20)

0.3

Predicted dp/dz [MPa/m]

Predicted dp/dz [MPa/m]

+20%

0.2 20%

0.1

0

D = 500 m Water G-W 52 wt% G-W 64 wt% D = 700 m Water G-W 52 wt% G-W 64 wt%

0.1 0.2 Measured dp/dz [MPa/m]

0.3

(b-2) Eqs. (16), (17), (19), and (22)

(b) D = 500 and 700 m Fig. 6 Comparison of pressure difference between Warnier et al.’s model with Eqs. (20) or (22) and measured data 4.2 Measured void fraction and prediction using models The relationship between  and  is shown in Fig. 7. All the measured values of  do not exceed the homogeneous line, i.e.,  <  and c0 is larger than unity. Regardless of the channel diameter and liquid properties,  increases with . The inclination of each measured data line slightly decreases in the high  region, i.e., the inclination is not linear and c0 increases with  In particular, the tendency is clearly observed in N2-glycerol-water solution systems. N2-water systems are the closest to the homogeneous line, and the inclination of the line decreases with increase in the weight percent

concentration of glycerol, i.e., with increase in c0. As shown in Table1, the viscosity increases with the weight percent concentration of glycerol. Hence, the main cause of the inclination decrease is the increase in the viscosity of the liquid phase. The inclination decreases with increasing channel diameter and this tendency is clearly observed in N2-water systems. Figure 8(a) shows the comparison of c0 between Liu et al.’s model, eq. (4), and the measured data. The model underestimates c0 with most of the measured data and only 72.0% of the data is predicted within + 10%. The standard deviation of the model is 0.120. Minagawa et al.’s model, eq. (10), also underestimate c0 as shown in Fig. 8(b). The model has a standard deviation of 0.129. These results indicate that the distribution parameter model is not a function of the capillary number alone. Figure 8(c) shows the comparison of c0 between Howard and Walsh’s model, eq. (11), and the measured data. This model agrees with 92.8% of the measured data within + 10%, with a standard deviation of 0.0818. As listed in Table 1, the experiment in the present study is carried out for a wide range of the Reynolds, Weber, and capillary numbers. The dimensionless numbers have been included in the model by using eq. (12) to calculate the liquid film thickness. Hence, this model is the most reliable model among the existing distribution parameter models. However, the dispersion is not so small in the range of + 10% and some of the data are outside the range.

1 0.8

1

Homogeneous flow Water G-W 52 wt%

0.8

0.6

Homogeneous flow Water G-W 52 wt% G-W 64 wt%





0.6

0.4

0.4

0.2

0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4



(a) D = 200 m

0.8

1

0.8

1

(b) D = 320 m

1 0.8

0.6 

1 Homogeneous flow Water G-W 52 wt% G-W 64 wt%

0.8



0.6



0.6

Homogeneous flow Water G-W 52 wt% G-W 64 wt%

0.4

0.4

0.2

0.2

0

0.2

0.4

0.6 

0.8

1

0

0.2

0.4

0.6 

(c) D = 500 m (d) D = 700 m Fig. 7 Relationship between  and 

2 +10%

c0 (Eq. (4))

1.8 10%

1.6 1.4 1.2 1 1

1.2

1.4 1.6 Measured c0

1.8

2

(a) Eq. (4) (Liu et al., 2005)

2 +10%

c0 (Eq. (10))

1.8 10%

1.6 1.4 1.2 1 1

1.2

1.4 1.6 Measured c 0

1.8

2

(b) Eq. (10) (Minagawa et al., 2013)

2 +10%

c0 (Eq. (11))

1.8 10%

1.6 1.4 1.2 1 1

1.2

1.4 1.6 Measured c 0

1.8

2

(c) Eq. (11) (Howard and Walsh, 2013) Fig. 8 Comparisons of distribution parameter between measured data and models

4.3 Development of the distribution parameter model LLc1/D and LLc2/D in eq. (30) are determined using measured data. At high Reynolds numbers, eq. (30) can be rewritten as

c0  1

c1 LL D We f lf

(31)

The relationship between 9. c0 increases linearly with

We / flf

and the measured c0 for ReT > 100 is shown in Fig.

We / flf

for ReT > 100. Hence, the following value can be

obtained: L  c1 L   0.000718  D 

(32)

1.5 0.5

1.4

0.0268(We/flf) +1

c0

1.3 1.2 1.1 1 0.9 0

5

10

15

0.5

(We/flf)

Fig. 9 Relationship between

We / flf

and c0 for ReT > 100

Substituting the above equation into (30) yields

c0  1 

We0.000718  LL D c2 ReT  f lf

(33)

Next, the remaining constant c2LL/D is determined from all measured data. This equation can be represented as follows  L  We f lf (c0  1) 2  0.000718We   c2 L  D  ReT 

(34)

c2LL/D can be determined by substituting the measured data into this equation, the obtained value of the constant c2LL/D is 0.327. The distribution parameter model using the obtained constant agrees with most of the measured data within + 10%. However, the accuracy is not that high for a high ReT range because the effect of (c2LL)/(DReT) on c0 does not disappear completely in that range. This shortcoming can be overcome by taking into account the nonlinear effect. Hence, the above equation is rewritten as

 L  We c f lf (c0  1) 2  0.000718 We   c2 L  (1  c4 ReT5 ) c D 3   Re T

(35)

where ci (i = 3, 4, 5) are constants. By substituting the measured data into this equation, the constants are determined by the least squared method as c2LL/D = 2.91, c3 = 0.875, c4 = 0.845, and c5 = 0.0259. Figure 10 shows the relationship between We(1  0.845ReT0.0259)/ReT0.875 and the LHS of eq. (35). The LHS of eq. (35) linearly increases with We(1  0.845ReT0.0259)/ReT0.875.

flf(c01)20.000718We

0.15 2.91We(10.845ReT0.0259)/ReT0.875

0.1

0.05

0 -0.01

0 0.01 0.02 0.03 0.04 0.0259 0.875 We(10.845Re T )/ReT

0.05

Fig. 10 Relationship between We(1  0.845ReT0.0259)/ReT0.875 and LHS of eq. (35)

Using these values and relations, the following distribution parameter model is obtained:

c0  1 









We 0.000718  2.91 ReT0.875 1  0.845ReT0.0259

(36)

f lf

As illustrated in Fig. 11, this equation shows good agreement with experimental data. This model agrees with the measured data within + 10% except for only one measured data point and the standard deviation is 0.0543. This distribution parameter model, therefore, is more reliable than the other models for the prediction of void fraction for gas-liquid slug flow in a microchannel. 2 +10%

c0 (Eq. (36))

1.8 10%

1.6 1.4 1.2 1 1

1.2

1.4 1.6 Measured c 0

1.8

2

Fig. 11 Comparison of distribution parameter between measured data and eq. (36) 5. Conclusion The void fraction and pressure drop of gas-liquid slug flow in circular microchannels were measured to examine the applicability of various models. The experiment was carried out using three liquids and N2 gas for the liquid and gas phases, respectively, and four circular microchannels of channel diameter 200, 320, 500, and 700 m. The void fraction and pressure difference were measured using a high-speed video camera and image processing method. For accurate prediction of the void fraction, a distribution parameter model was newly developed in this study. The functional form of the model

was derived using a pressure drop model and a scaling analysis of the momentum equation for incompressible Newtonian fluid in two-phase flow based on a one-field formulation. The remaining constants were determined from the measured data. The following conclusions were obtained: (1) The pressure drop can be predicted using the model proposed by Warnier et al. (2010) with the liquid film thickness model proposed by Han and Shikazono (2009) within + 20%. (2) The distribution parameter model developed in this study can predict the measured data within + 10% and is more reliable than the other models. Acknowledgements We would like to thank Mr. Tomoya Shiraki for his contribution to the experiments and image processing. Appendix A All measured data of uG, jG, jL, and dp/dz are summarized in Table A1. The measured gas slug shapes are shown in Fig. A1. Since the length of all gas slugs was larger than that of the images in the cases of D = 200 m, water; D = 200 m, 52 wt% glycerol-water solution; and D = 500 m, 64 wt% glycerol-water solution, the measured bubble shapes were not inserted in the figure.

Fig. A1 Measured gas slug shapes (j) (a) (k) (b) (c)

(l)

(m)

(d) (e)

(n)

(f) (o) (g) (h) (i)

(p)

Table A1 Measured data Fig.

uG

jG

jL

dp /dz

Fig.

uG

jG

jL

dp /dz

Fig.

uG

jG

jL

dp /dz

A1

[m/s]

[m/s]

[m/s]

[MP a/m]

A1

[m/s]

[m/s]

[m/s]

[MP a/m]

A1

[m/s]

[m/s]

[m/s]

[MP a/m]

D = 200 mm, Water 1.12 0.429 0.518 1.60 0.877 0.484 1.17 0.381 0.610 1.69 0.833 0.578 1.22 0.359 0.719 1.77 0.788 0.679 1.37 0.354 0.838 1.85 0.741 0.784 1.47 0.338 0.909 1.85 0.679 0.851 1.59 0.332 1.00 1.91 0.637 0.962 1.64 0.319 1.10 1.99 0.614 1.04 1.81 0.306 1.17 2.09 0.593 1.13 1.92 0.294 1.27 2.16 0.587 1.23 1.96 0.279 1.33 2.24 0.562 1.29 2.08 0.271 1.43 2.41 0.579 1.39 2.27 0.264 1.52 2.51 0.528 1.45 1.83 0.342 1.20 2.26 0.678 1.15 1.91 0.327 1.32 2.34 0.663 1.22 2.03 0.320 1.37 2.33 0.638 1.30 2.20 0.311 1.40 2.46 0.622 1.42 D = 200 mm, G-W soln. (52 0.783 0.325 0.295 1.18 0.612 0.296 0.897 0.265 0.413 1.24 0.542 0.405 0.898 0.466 0.231 0.922 0.451 0.266 1.23 0.649 0.258 0.998 0.486 0.287 1.24 0.653 0.280 1.02 0.470 0.315 1.27 0.639 0.314 1.04 0.456 0.332 1.28 0.624 0.328 1.02 0.413 0.362 1.28 0.602 0.364 1.05 0.427 0.396 1.30 0.596 0.367 1.05 0.410 0.403 1.30 0.576 0.392 1.09 0.388 0.431 1.30 0.550 0.426 D = 320 mm, Water 0.909 0.171 0.581 0.897 0.172 0.588 0.876 0.171 0.579 0.880 0.170 0.586 0.434 0.205 0.184 0.566 0.198 0.304 (a) 0.693 0.179 0.421 0.790 0.183 0.492 0.988 0.171 0.642 1.06 0.160 0.705 0.709 0.395 0.196 0.777 0.363 0.297 0.860 0.359 0.361 0.933 0.341 0.444 1.01 0.324 0.521 1.09 0.315 0.599 1.17 0.305 0.672 (b) 1.24 0.292 0.759 0.937 0.595 0.200 0.995 0.562 0.302 1.21 0.740 0.261 1.08 0.545 0.355 1.26 0.717 0.325 D = 320 mm, G-W soln. (52 0.453 0.145 0.200 0.657 0.282 0.204 0.899 0.425 0.201

0.380 0.349 0.560 0.421 0.659 0.506 0.684 0.606 0.764 0.758 0.797 0.876 0.874 0.950 0.958 1.02 1.04 1.04 1.15 1.13 1.21 1.07 1.27 1.28 0.851 0.868 0.945 0.917 0.995 1.00 1.06 1.06 wt%) 1.06 1.21 1.60 1.54 1.10 1.18 1.22 0.999 1.21 1.08 1.26 1.15 1.32 1.40 1.42 1.31 1.44 1.42 1.53 1.57 1.67 0.165 0.164 0.168 0.170 0.056 0.076 0.135 0.122 0.168 0.211 0.078 0.128 0.135 0.169 0.204 0.224 0.248 0.280 0.075 0.108 0.116 0.127 0.136 wt%) 0.458 0.491 0.488

1.09 0.557 0.201 0.509 1.35 0.717 0.197 0.472 1.60 0.862 0.194 0.469 0.692 0.263 0.249 0.587 0.934 0.415 0.245 0.518 0.545 0.136 0.292 0.540 (c) 0.731 0.256 0.286 0.626 1.83 1.01 0.175 0.477 2.01 1.12 0.160 0.524 1.09 0.568 0.207 0.504 1.30 0.707 0.184 0.510 1.59 0.874 0.172 0.470 2.07 1.15 0.162 0.490 0.891 0.422 0.204 0.517 1.09 0.554 0.193 0.538 1.34 0.698 0.200 0.526 0.391 0.0691 0.234 0.542 (d) 0.404 0.0670 0.259 0.584 D = 320 mm, G-W soln. (64 wt%) 0.341 0.153 0.0947 0.392 0.588 0.304 0.0961 0.400 0.891 0.485 0.0937 0.329 1.16 0.639 0.0916 0.342 1.36 0.748 0.0937 0.419 0.346 0.149 0.104 0.422 0.589 0.300 0.102 0.416 0.842 0.443 0.0984 0.435 1.13 0.608 0.0993 0.400 1.33 0.732 0.0983 0.446 0.342 0.138 0.107 0.517 0.557 0.279 0.108 0.510 (e) 0.249 0.0711 0.117 0.482 0.362 0.148 0.112 0.433 0.388 0.121 0.153 0.678 0.625 0.263 0.148 0.565 0.812 0.369 0.152 0.656 0.407 0.121 0.164 0.678 0.600 0.243 0.164 0.673 0.801 0.355 0.160 0.712 0.422 0.121 0.175 0.676 0.598 0.232 0.171 0.744 0.822 0.358 0.172 0.699 (f) 0.424 0.116 0.187 0.747 0.623 0.235 0.181 0.725 0.782 0.330 0.177 0.824 D = 500 mm, Water 0.602 0.0860 0.433 0.0379 0.680 0.170 0.404 0.0443 0.770 0.262 0.387 0.0308 0.836 0.344 0.370 0.0379 0.996 0.423 0.393 0.0458 (g) 1.15 0.0788 0.837 0.0820 1.18 0.148 0.793 0.118 1.27 0.233 0.777 0.0899 1.30 0.311 0.740 0.0897 1.80 0.0702 1.29 0.147 1.86 0.141 1.24 0.144 1.95 0.207 1.20 0.158 (h) 1.93 0.280 1.17 0.147 1.07 0.526 0.333 0.0288 1.10 0.593 0.303 0.0451 1.19 0.679 0.288 0.0447 D = 500 mm, G-W soln. (52 wt%) 0.740 0.378 0.178 0.103 0.837 0.443 0.179 0.115 0.896 0.487 0.179 0.148 (i) 0.373 0.0704 0.222 0.142 0.591 0.198 0.255 0.180 0.678 0.261 0.254 0.187 0.774 0.326 0.263 0.187 0.634 0.188 0.292 0.213 0.708 0.244 0.291 0.231 0.788 0.304 0.283 0.233 0.360 0.145 0.129 0.126 0.564 0.283 0.127 0.138 0.853 0.470 0.125 0.0837 0.426 0.132 0.188 0.182 0.518 0.195 0.188 0.188 0.627 0.269 0.186 0.169 0.737 0.328 0.189 0.183 0.839 0.390 0.186 0.189 (j) 0.371 0.0616 0.222 0.224

0.465 0.125 0.219 0.215 0.423 0.0588 0.251 0.256 0.498 0.118 0.247 0.254 D = 500 mm, G-W soln. (64 wt%) 0.247 0.0637 0.119 0.212 0.337 0.126 0.119 0.218 0.459 0.198 0.117 0.188 0.563 0.258 0.115 0.203 0.630 0.301 0.115 0.249 0.744 0.377 0.115 0.220 0.270 0.0612 0.135 0.239 0.353 0.121 0.133 0.250 0.460 0.186 0.132 0.231 0.583 0.253 0.131 0.217 0.666 0.305 0.129 0.241 0.746 0.350 0.131 0.272 0.295 0.0575 0.151 0.284 0.373 0.116 0.148 0.275 0.305 0.0549 0.164 0.318 0.389 0.109 0.168 0.324 D = 700 mm, Water 1.20 0.0783 0.874 0.0441 1.36 0.109 0.882 0.0794 1.26 0.152 0.825 0.0577 1.30 0.192 0.790 0.0525 1.28 0.230 0.744 0.0551 1.28 0.268 0.706 0.0548 1.25 0.310 0.666 0.0489 1.40 0.112 0.974 0.0665 1.43 0.144 0.951 0.0843 1.47 0.183 0.926 0.0757 1.47 0.218 0.890 0.0799 1.64 0.257 0.939 0.0752 1.64 0.296 0.904 0.0706 1.30 0.342 0.643 0.0577 1.35 0.388 0.618 0.0481 1.01 0.328 0.458 0.0237 1.11 0.401 0.470 0.0333 (k) 1.04 0.401 0.426 0.0330 (l) 0.991 0.396 0.347 0.0386 D = 700 mm, G-W soln. (52 wt%) 0.656 0.179 0.303 0.132 0.709 0.214 0.302 0.133 0.773 0.248 0.308 0.138 0.745 0.169 0.370 0.162 0.793 0.204 0.364 0.161 0.830 0.238 0.354 0.160 0.604 0.279 0.182 0.0775 0.685 0.323 0.187 0.0722 0.741 0.359 0.180 0.0778 0.749 0.381 0.180 0.100 0.597 0.309 0.111 0.0510 0.684 0.342 0.109 0.0583 0.740 0.395 0.105 0.0403 0.253 0.0388 0.151 0.0489 0.294 0.0760 0.144 0.0582 (m) 0.349 0.115 0.139 0.0543 0.398 0.151 0.135 0.0609 0.452 0.185 0.142 0.0706 0.534 0.228 0.143 0.057 (n) 0.488 0.0323 0.315 0.142 0.518 0.0638 0.291 0.149 0.557 0.0957 0.288 0.149 0.620 0.132 0.281 0.131 D = 700 mm, G-W soln. (64 wt%) 0.283 0.0324 0.164 0.140 0.327 0.0648 0.157 0.140 0.372 0.0970 0.154 0.142 (o) 0.428 0.131 0.153 0.136 0.474 0.163 0.152 0.138 0.532 0.199 0.149 0.127 0.598 0.235 0.148 0.122 0.643 0.263 0.151 0.132 0.695 0.294 0.149 0.136 0.780 0.341 0.149 0.113 0.310 0.0322 0.182 0.143 0.358 0.0646 0.177 0.142 (p) 0.398 0.100 0.174 0.126 0.363 0.0300 0.215 0.186 0.407 0.0574 0.213 0.212 0.445 0.0277 0.263 0.236 0.481 0.0552 0.253 0.237

Reference Aussillous, P. and Quéré, D., 2000. Quick deposition of a fluid on the wall of a tube. Physics of Fluids, 12, 2367-2371. Bretherton, F. P., 1961. The motion of long bubbles in tubes. Journal of Fluid Mechanics, 10, 166-188. Chung P. M. -Y. and Kawaji, M., 2004. The effect of channel diameter on adiabatic two-phase flow characteristics in microchannels. International Journal of Multiphase Flow, 30, 735-761. Eain, M. M. G., Egan, V., Howard, J., Walsh, P., Walsh, E. and Punch, J., 2015. Review and extension of pressure drop models applied to Taylor flow regimes. International Journal of Multiphase Flow, 68, 1-9. Han, Y., and Shikazono, N., 2009. Measurement of the liquid film thickness in micro tube slug flow, International Journal of Heat and Fluid Flow, 30, 842-853. Hayashi, K., Kurimoto, R. and Tomiyama, A., 2010. Dimensional analysis of terminal velocity of Taylor bubble in a vertical pipe. Multiphase Science and Technology, 22 (3), 197-210. Hayashi, K., Kurimoto, R. and Tomiyama, A., 2011. Terminal velocity of a Taylor drop in a vertical pipe. International Journal of Multiphase Flow, 37, 241-251. Hosokawa, S. and Tomiyama, A., 2003. Lateral force acting on a deformed single bubble due to presence of wall. Transaction of Japanese Society of Mechanical Engineers, Series B 69 (686), 2214-2220 (in Japanese). Howard, J. A. and Walsh, P. A., 2013. Review and extensions to film thickness and relative bubble drift velocity prediction methods in laminar Taylor or slug flows. International Journal of Multiphase Flow, 55, 32-44. Ishii, M. and Hibiki, T., 2006. Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York. Ishikawa, T., 1968. Kongou Ekinendo No Riron. Maruzen (in Japanese). Kariyasaki, A., Fukano, T., Ousaka, A. and Kagawa, M., 1992. Isothermal Air-Water Two-Phase Up-and Downward flows in a Vertical Capillary Tube (1st Report, Flow Pattern and Void Fraction). Transaction of the Japan Society of Mechanical Engineers Series B, 58 (553), 2684-2690. Kawahara, A., Chung, P. M. –Y. and Kawaji, M., 2002. Investigation of two-phase flow pattern, void fraction and pressure drop in a microchannel. International Journal of Multiphase Flow, 28, 1411-1435. Kawahara, A., Sadatomi, M., Nei, K. and Matsuo, H., 2009. Experimental study on bubble velocity, void fraction and pressure drop for gas-liquid two-phase flow in a

circular microchannel. International Journal of Heat and Fluid Flow, 30, 831-841. Kurimoto, R., Hayashi, K., and Tomiyama, A., 2013. Terminal velocities of clean and fully-contaminated drops in vertical pipes. International Journal of Multiphase Flow, 49, 8-23. Liu, H., Vandu, C. O. and Krishna, R., 2005. Hydrodynamics of Taylor Flow in Vertical Cappilaries: Flow Regimes, Bubble Rise Velocity, Liquid Slug Length, and Pressure Drop. Industrial & Engineering Chemistry Research, 44 (14), 4884-4897. Minagawa, H., Asama H. and Yasuda, T., 2013. Void Fraction and Frictional Pressure Drop of Gas-Liquid Slug Flow in a Microtube. Transaction of Japanese Society of Mechanical Engineers, Series B 79 (804), 1500-1513 (in Japanese). Mishima, K. and Hibiki, T., 1996. SOME CHARACTERISTICS OF AIR-WATER TWO-PHASE FLOW IN SMALL DIAMETER VERTICAL TUBES. International Journal of Multiphase Flow, 22 (4), 703-712. Saisorn S. and Wongwises, S., 2008. Flow pattern, void fraction and pressure drop of two-phase air-water flow in a horizontal circular micro-channel. Experimental Thermal and Fluid Science, 32, 748-760. Serizawa, A., Feng, Z. and Kawara, Z., 2002. Two-phase flow in microchannels. Experimental Thermal and Fluid Science, 26, 703-714. Sur, A. and Liu, D., 2012. Adiabatic air-water two-phase flow in circular microchannels. International Journal of Thermal Sciences, 53, 18-34. Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I. and Sadowski, D. L., 1999. Gas-liquid two-phase flow in microchannels Part I: two-phase flow patterns. International Journal of Multiphase Flow, 25, 377-394. Warnier, M. J. F., de Croon, M. H. J. M., Rebrov, E. V. and Schouten J. C., 2010. Pressure drop of gas-liquid Taylor flow in round micro-capillaries for low to intermediate Reynolds numbers. Microfluidics and Nanofluidics, 8 (1), 33-45. Zhang, T., Cao, B., Fan, Y., Gonthier, Y., Luo, L. and Wang, S., 2011. Gas-liquid flow in circular microchannel. Part I: Influence of liquid physical properties and channel diameter on flow patterns. Chemical Engineering Science, 66, 5791-5803. Zuber, N. and Findlay, J. A., 1968. Average volumetric concentration in two-phase flow system. Journal of Heat Transfer, 87, 453-468.