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Flow pattern based correlations of two-phase pressure drop in rectangular microchannels
International Journal of Heat and Fluid Flow 32 (2011) 1199–1207
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Flow pattern based correlations of two-phase pressure drop in rectangular microchannels Chiwoong Choi ⇑, Moohwan Kim Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang, San 31, Hyoja Dong 790-784, Republic of Korea
a r t i c l e
i n f o
Article history: Received 10 January 2011 Received in revised form 1 August 2011 Accepted 30 August 2011 Available online 6 October 2011 Keywords: Rectangular microchannel Pressure drop Flow pattern Correlation
a b s t r a c t Numerous pressure drop correlations for microchannels have been proposed; most of them can be classified as either a homogeneous flow model (HFM) or a separated flow model (SFM). However, the predictions of these correlations have not been compared directly because they were developed in experiments conducted under a range of conditions, including channel shape, the number of channels, channel material and the working fluid. In this study, single rectangular microchannels with different aspect ratios and hydraulic diameters were fabricated in a photosensitive glass. Adiabatic water-liquid and Nitrogen-gas two-phase flow experiments were conducted using liquid superficial velocities of 0.06–1.0 m/s, gas superficial velocities of 0.06–72 m/s and hydraulic diameters of 141, 143, 304, 322 and 490 lm. A pressure drop in microchannels was directly measured through embedded ports. The flow pattern was visualized using a high-speed camera and a long-distance microscope. A two-phase pressure drop in the microchannel was highly related to the flow pattern. Data were used to assess seven different HFM viscosity models and ten SFM correlations, and new correlations based on flow patterns were proposed for both HFMs and SFMs. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Two-phase flow pressure drop correlations can be classified as either homogeneous flow models (HFMs) or separated flow models (SFMs). HFMs treat a two-phase flow as a single phase flow with mixed properties. In HFMs, two-phase density and viscosity should be defined. SFMs are based on a two-phase frictional multiplier (u), which is defined as the ratio of the pressure gradient of the liquid phase to that of the gas phase. A pressure drop of two-phase flow in a microchannel cannot be predicted using conventional correlations. Hence, many studies of the two-phase frictional pressure drop in microchannels have been conducted; most used circular tubes and rectangular multichannels, because of difficulties in fabricating circular microchannels and controlling small amounts of fluid and heat. For a heat sink application, the quantification of a pressure drops in rectangular microchannels is needed. In the present study, rectangular glass microchannels were fabricated to allow a visualization of two-phase flow patterns. Experiments were conducted in these microchannels using water liquid and Nitrogen gas flow for liquid superficial velocities of 0.06– 1.0 m/s, gas superficial velocities of 0.06–72 m/s, and microchannel hydraulic diameters of 141, 143, 304, 322 and 490 lm. Flow ⇑ Corresponding author. E-mail address: [email protected] (C.W. Choi). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.08.002
patterns were visualized using a high-speed camera and a longdistance microscope. Data were used to assess seven different HFM viscosity models and ten SFM correlations, In addition, new correlations based on flow patterns were proposed for both HFMs and SFMs. There are modeling works for two-phase pressure drop in microchannel (Garimella et al., 2003; Shiferaw et al., 2011). For example, Garimella et al. (2003) developed pressure drop model, which based on a unit cell concept, for intermittent flow regime for non-circular microchannels. In development of model, the number of unit cell was regressed with a hydraulic diameter and slug Reynolds number. However, their model shows highly underestimated the number of unit cell comparing with present data (Choi et al., 2010b). Shiferaw et al. (2011) also developed semimechanistic pressure drop model for specific flow pattern, which is based on a three zone model (Thome et al., 2004). Both models are considered the specific flow pattern, the periodic elongated bubble regime, thus two models are exempted from our assessment correlations.
2. Experimental setup 2.1. Test section The test section (Fig. 1; the unit is mm) was fabricated in a sheet of a photosensitive glass using UV rays. Details can be found in Choi and Kim (2008). The test section has two inlets (i.e., one for
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Fig. 1. A microchannel test section (top view).
each phase flow), one outlet and three pressure ports. Total channel length was 60 mm and the distance between pressure ports was 15 mm. The test section was held horizontally with a support frame to connect inlets and outlet tubing. The interconnections were achieved using fitting and ferrule (Upchurch Scientific), and a fitting holder. Pressure was measured directly through the three embedded pressure ports; this overcame the uncertainties in a measurement that occurred in most previous studies, in which pressure was measured outside of the channel. Moreover, the fully developed condition was checked with comparing pressure differences among three pressure ports (Fig. 1). Flow patterns were visualized directly through the transparent glass using a high-speed camera and a long-distance microscope. Cross sections of five different fabricated rectangular microchannels were measured using a 3D-profiler (Veeco-Wyko DMEMS NT1100). To verify this measuring method, the cross-section of a test sample was measured using a microscope (Table 1).
rates of the liquid were measured using flow-meters. The pressure drop was measured directly through embedded pressure ports in the microchannel using four different pressure transducers (Druck LPM9000, Setra 230-5, 10, and 25 PSID). Data were collected using a data acquisition system (Agilent 34970) and were saved on a personal computer. The details of the experimental setup were described in Choi et al. (2010a). Before the main experiment, water was pumped into the inlet of the test section to extract non-degassed water from the tubes, and tubes connected to pressure ports were fully filled with water to offset the effect of a static head, which was significant under low pressure conditions. Then the gas flow was applied to the other inlet to create a two-phase flow. When the temperature and the pressure upstream of each phase flow were steady, the pressure and the flow rate were measured every 2 s. Then, the gas flow rate was changed while the liquid flow rate was held a constant (Table 1). All experimental uncertainties are summarized in Table 2. 3. Results and discussion
2.2. Experimental facilities and procedure 3.1. Single phase pressure drop In the experimental setup (Fig. 2), the liquid and gas flows were controlled using a pneumatic pumping system, which consisted of electric regulators and pressurized gases. Nitrogen gas was used to pressurize the gas phase loop and Helium gas was used to pressurize the liquid phase loop. The two-phase flow was mixed in a T-junction. The effect of the mixing method is not well understood and still hot issues in this research area (Kawaji, 2008). Therefore, in this study, we used T-junction configuration, which is well proven mixing method in various fields. The volume and mass flow
Before conducting the two-phase flow experiment, experiments for a single phase pressure drop in the rectangular microchannels were conducted to verify the experimental apparatus and to evaluate the friction factor of the rectangular microchannel. The value of fRe is 16 for circular tubes, and Hartnett and Kostic (1989) proposed a correlation of fRe for ducts of different ARs (Eqs. (1)–(3))
f Re ¼ 24ð1 1:3553AR þ 1:9467AR2 1:7012AR3 þ 0:9564AR4 0:2537AR5 Þ;
Table 1 Experimental conditions.
where
Variables
Ranges
Diameters of microchannels, Dh (WCh HCh) (lm)
490(510 470) 322(501 237) 143 (503 83) 304 (332 280) 141 (201 109) 66–1000 0.075–80 32–477 2–2134 0.06–1 0.06–72
Liquid mass flux, GL (kg/m2s) Vapor mass flux, Gg (kg/m2s) Liquid Reynolds number, ReL Vapor Reynolds number, Reg Liquid superficial velocity, jL (m/s) Vapor superficial velocity, Jg (m/s)
ð1Þ
Re ¼
quch Dh ; l
ð2Þ
and
f ¼
Dh DP : 2qu2ch Dz SP
ð3Þ
The predicted values from Hartnett and Kostic (1989)’s correlation were 19.71, 15.74 and 14.25 for 143, 322 and 490 lm diameters, respectively; measured fRe values were agreed well with predicted values. Results of the single phase frictional pressure
C.W. Choi, M.H. Kim / International Journal of Heat and Fluid Flow 32 (2011) 1199–1207
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Fig. 2. Experimental facilities for the adiabatic two-phase flow in microchannnel.
Table 2 Experimental uncertainties. Variables
Uncertainty
Diameter (lm) Area (lm2) Pressure (kPa) Temperature (°C) Mass flux, G (kg/m2s) Friction factor Superficial velocities (m/s)
±17 (4.5%) ±6311 (3.8%) ±6% (±0.01, ±0.005, ±0.034, ±0.085) ±0.1–0.5 ±0.1–2% ±9% ±10%
drop in our test section for water liquid indicate that the conventional theory of pressure drops in a single phase-laminar flow is acceptable and that our experimental setup is appropriate.
3.2. Flow pattern and two-phase pressure drop In our experimental region, the observed flow patterns were bubbly, slug bubble, elongated bubble, liquid ring, and transition flow (Fig. 3). Definitions are the same as in Choi et al. (2010a). As gas superficial velocity (jG) increased, the bubble length increased. The bubble nose and the tail collapsed, and formed a liquid ring; this flow pattern has been observed and given various names by various researchers (Kawahara et al., 2002; Chung and Kawaji, 2004 and Saisorn and Wongwises, 2008). A transition region occurred between the elongated bubble flow and the liquid ring flow regimes; in the transition regime, the elongated bubble flow and the liquid ring flow occurred periodically. Thus, we defined this flow regime as ‘‘transition flow patterns’’. Before this transition point, the bubble flows were stable and formed a perfect bubble train. As liquid superficial velocity (jL) increased, the bubble length decreased. Then, the elongated bubble flow changed to the single bubbly flow at higher jL. In this study, main flow regimes were grouped into (1) a bubble flow including bubbly, slug bubble and elongated bubble, (2) liquid ring flow, and (3) a transition between elongated bubble flow and ring flow.
Fig. 3. A classification of the flow patterns in the rectangular microchannel for Dh = 490 lm.
Pressure drop increased as jL and jG increased. At the constant jL, the pressure drop increased as jG increased. However, in the transition region, the pressure drop decreased as jG increased. The merged bubbles caused the bubble to elongate. The dominant component of pressure drops in a bubble occurs in the nose and tail, rather than in the bubble body. However, Choi et al. (2010b) reported the collapsed bubble reduces the number of parts that
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ReTP ¼
Fig. 4. Two-phase pressure drop in rectangular microchannel with Dh = 322 lm: Typical trend of the pressure drop in a rectangular microchannel ((I) bubbly or liquid slug flow, (II) elongated bubble flow, (III) transition flow regime (transition), (IV) liquid ring flow).
contribute to the pressure drop. The typical trend of pressure drops in the microchannel (Fig. 4) is composed of three regions. Region I: A bubble flow regime including bubbly, slug bubble and elongated bubble flow patterns. Pressure drops increased as gas superficial velocity increased. Region II: Transition flow patterns. Pressure drop decreased as gas superficial velocity decreased. Region III: Liquid ring flow. Pressure drop increased as gas superficial velocity increased. This typical trend indicates that the pressure drop in the rectangular microchannel is highly related to the flow pattern. 4. Assessment of two-phase pressure drop models
GDh
lTP
ð8Þ
;
where lTP is a two-phase viscosity. For a laminar flow in a circular tube, n = 1 and N = 16; in a rectangular channel N is a function of AR (Eq. (1)). Numerous two-phase viscosity models have been proposed. Recently, Costa-Patry et al. (2011) conducted flow boiling experiments in 85 lm wide multi-microchannels. They showed that Cioncolini et al. (2009)’s correlation is well agree with their pressure drop results. The Cioncolini et al.’s correlation is developed using a dimensionless analysis for an annular flow pattern. Costa-Patry et al. also mentioned the Cioncolini et al.’s correlation had a good accuracy for the annular flow pattern. Therefore, the Cioncolini et al.’s correlation is exempted from our assessment correlations. In this study, our experimental results were used to assess seven viscosity models: Owens (1961), MacAdams et al. (1942), Cicchitti et al. (1960), Dukler et al. (1964), Beattie and Whalley (1982), Lin et al. (1991) and Awad and Muzychka (2008).
Owen ð1961Þ : lTP ¼ lL ;
ð9Þ
McAdams etal: ð1942Þ : lTP ¼
x
lG
þ
1x
1
lL
;
ð10Þ
Cicchitti etal: ð1960Þ : lTP ¼ xlG þ ð1 xÞlL ;
ð11Þ
Dukler etal: ð1964Þ : lTP ¼ blG þ ð1 bÞlL ;
ð12Þ
Beattie and Whalley ð1982Þ : lTP ¼ blG þ ð1 bÞð1 þ 2:5bÞlL ; Lin etal: ð1991Þ : lTP ¼
4.1. Homogeneous flow model (HFM)
lG lL ; lG þ x1:4 ðlL lG Þ
ð13Þ
ð14Þ
2lG þ lL 2ðlG lL Þð1 xÞ : 2lG þ lL þ ðlG lL Þð1 xÞ
An HFM is the simplest two-phase flow model; it treats twophase flow as a single phase flow with mixture properties. A two-phase frictional pressure drop consists of frictional, accelerational, and gravitational terms (Eq. (4)) (see Carey, 1992).
Awad and Myuztchka ð2008Þ : lTP ¼ lG
dP dP dP dP ¼ þ þ : dz TP dz Friction dz Acceleration dz Gravitation
Pressure drop based on the HFM and viscosity models were compared to experimental data. Accuracy of model predictions was assessed using the mean absolute error (MAE), expressed as a percentage:
ð4Þ
In this study, the gravitational term was neglected, because a flow was horizontal, and the accelerational term was neglected because the flow was adiabatic. Therefore, the total measured pressure drop is the frictional term in Eq. (4), which can be defined using the HFM (Eq. (5)).
dP dP 2f G2 ¼ ¼ TP ; dz TP dz Friction qTP Dh
ð5Þ
where fTP is a two-phase friction factor, G is mass flux and qTP is two-phase density (Eq. (6))
qTP ¼
x
qG
þ
1x
qL
1 :
ð6Þ
The two-phase friction factor can be expressed as an exponential function of the two-phase Reynolds number (Eq. (7))
fTP ¼ NRen TP ; with
ð7Þ
ð15Þ
MAE ¼
1 X jDPTP;pred DPTP;exp j 100 : M DPTP;exp
ð16Þ
HFMs were assessed for microchannels having different diameters (Table 3). The most accurate viscosity model is the Beattie and Whalley (1982)’s model, which is based on the volumetric quality. Experimental pressure drop data and the viscosity models, were used to calculate f and Re (Fig. 5), and an original HFM for Dh = 322 lm with n = 1 and N = 15.75 (Eq. (7)). The two-phase Re was over-predicted by Dukler et al. (1964)’s model and under-predicted by other models. The two-phase viscosity predicted by the Beattie and Whalley (1982)’s model agreed marginally well with the original HFM. However, the other models except that of Dukler et al. (1964)’s model overestimated two-phase viscosity (Fig. 6). Beattie and Whalley (1982)’s two-phase viscosity model was developed for the bubble flow and annular flow patterns; this concept agreed well with our experimental result (Fig. 7), but it was deviated from measurements by approximately ±50%. With other microchannels, a similar trend was observed.
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C.W. Choi, M.H. Kim / International Journal of Heat and Fluid Flow 32 (2011) 1199–1207 Table 3 Comparison of the two-phase pressure drop based on the homogeneous flow model for two-phase viscosity models with the experimental results. Models
MAE (%)
Owens (1961) MacAdams et al. (1942) Cicchitti et al. (1960) Dukler et al. (1964) Beattie and Whalley (1982) Lin et al. (1991) Awad and Muzychka (2008)
Dh = 490
Dh = 322
Dh = 143
Dh = 304
Dh = 14 l
Average
352.10 40.81 300.76 66.84 30.69 111.99 107.87
647.37 101.35 554.13 51.76 22.72 225.10 204.17
1247.28 236.46 1076.53 23.25 75.38 464.32 426.47
801.18 123.84 715.23 63.29 24.41 309.52 145.01
709.07 169.87 650.12 42.10 43.50 346.56 298.29
761.98 134.46 659.35 49.44 39.34 291.49 236.36
Fig. 5. Relation of friction factor and Reynolds number based on different twophase viscosity models for Dh = 322 lm and jL = 0.4 m/s.
Fig. 7. Two-phase friction factor and Reynolds number for Dh = 322 lm using Beattie and Whalley’s two-phase viscosity model.
X vv ¼
Fig. 6. Two-phase viscosity for water liquid and Nitrogen gas for different twophase viscosity models.
4.2. Separated flow model (SFM) SFMs are based on a two-phase multiplier (u), which is defined
0:5 0:5 0:5 0:5 lL Dp Dp 1x qG ¼ : x Dz L Dz G qL lG
ð19Þ
In the right-hand side of Eq. (18), one represents liquid-only pressure drop, C/Xvv represents mixed pressure drop, and 1=X 2vv represents gas phase-only pressure drop. Therefore, C represents the interactional effect of the two-phase flow. Previously proposed correlations for frictional pressure drop in a microchannel have been developed by modifying the C-value in the SFM. In this study, our experimental data were used to assess ten correlations for the SFM: Lockhart and Martinelli (1949), Chisholm (1967) and Muller-Steinhagen and Heck (1986) for macro-scale correlations; Zhang et al. (2007), Tran et al. (2000) and Lee and Lee (2001) for mini-scale correlations, and Moriyama and Inoue (1992), Qu and Mudawar (2003), Lee and Mudawar (2005) and Li and Wu (2010) for micro-scale correlations. Lockhart and Martinelli (1949) proposed the C value for different flow structures. For example, C = 5 for laminar flow of both liquid and gas phases. Chisholm (1967) proposed a modified two-phase multiplier (Eq. (20)):
h 2s i 2s /2LO ¼ 1 þ ðY 2 1Þ Bx 2 ð1 xÞ 2 þ x2s ;
ð20Þ
as
/2L ¼
Dp Dp Dz TP Dz L
or /2LO ¼
Dp Dp : Dz TP Dz LO
ð17Þ
Lockhart and Martinelli (1949) suggested that /2L is a function of a Martinelli parameter (Eq. (18))
/2L ¼ 1 þ
C 1 þ ; X vv X 2vv
ð18Þ
where Xvv is a parameter of a laminar liquid–laminar gas flow (Eq. (19)).
where s = 0.25, x is a quality and Y is a sort of Martinelli parameter (Eq. (21))
Dp
Y 2 ¼ DDzpGO ¼ Dz LO
1 X 2vv
;
ð21Þ
and B is a function of mass flux and Y. Muller-Steinhagen and Heck (1986) proposed a simple and convenient correlation for a two-phase frictional pressure drop using data points of 9300 various fluids and flow conditions, including channel diameters from 4 mm to 392 mm (Eqs. (22) and (23))
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Dp Dp ¼ Fð1 xÞ1=3 þ x3 ; Dz TP Dz GO F¼
ð22Þ
Dp Dp Dp x: þ2 Dz LO Dz GO Dz LO
ð23Þ
The correlation of Muller-Steinhagen and Heck has a form that is similar to Chisholm’s correlation without the B-coefficient. Mishima and Hibiki group (1993, 1996) investigated the effect of a mini-sized diameter on two-phase frictional pressure in experiments on an air–water adiabatic two-phase flow in minichannels with diameters of 1–4 mm. They proposed that the C-value of the L–M correlation is a function of the diameter. They defined the Cvalue as
C ¼ 21ð1 e333D Þ;
ð24Þ
for circular tubes and as
C ¼ 21ð1 e319Dh Þ;
ð25Þ
for rectangular tubes. Zhang et al. (2007) modified Mishima and Hibiki’s correlation to extend to the micro-scale and verified it using different experimental results. The hydraulic diameter was replaced by a Laplace constant:
C ¼ 21ð1 e358=Nconf Þ;
ð26Þ
where Nconf is a confinement number (Eq. (27))
Nconf
lc lc ½r=ðqL qV Þg1=2 ¼ ¼ ¼ ; D Dh Dh
where lc is the Laplace constant, and the unit of D or Dh is a meter. The applicable range was limited to 0.014 mm
ð28Þ
Lee and Lee (2001) conducted air–water adiabatic two-phase flow experiments for different rectangular minichannels with low aspect ratios. They proposed a new C-value as a function of three dimensionless numbers for laminar and turbulent regions: q
C ¼ Ak
Carj Resj
l ¼A qL rDh 2 L
p q jl L qL jDh r
r
lL
;
/2L ¼ 1 þ
K X 2vv
ð30Þ
:
This correlation lacks the second term in the original L–M correlation (Eq. (19)), i.e., the C-value that is an interactional effect in the two-phase pressure drop. They multiplied the third term (the gas phase portion in the two-phase pressure drop) by a factor K, which they postulated to be a function of liquid Reynolds number. This means that the influential parameter in the two-phase pressure drop in a microchannel is the gas phase portion rather than the interactional portion. Recently, a similar correlation was proposed by Saisorn and Wongwises (2008). Qu and Mudawar (2003) investigated the frictional two-phase pressure drop in water flow boiling experiments using 21 parallel microchannels of size 231 lm 713 lm. They proposed a modified Mishima et al. (1993)’s correlation, which considers the effect of mass flux (Eq. (31))
C ¼ 21ð1 e319Dh Þð0:00418G þ 0:0613Þ: ð27Þ
h i 2s 2s /2LO ¼ 1 þ ð4:3Y 2 1Þ N conf x 2 ð1 xÞ 2 þ x2s :
where k is a fluid property, Caj and Rej are capillary number and Reynolds number based on total superficial velocity (j), respectively, and A, p, q and r are empirically-determined coefficients. For laminar–liquid and laminar–gas flow, A = 6.833 108, p = 1.317, q = 0.719 and r = 0.557. Moriyama and Inoue (1992) studied a two-phase pressure drop in 35–110 lm gaps using R113 under boiling conditions, and Nitrogen and R113 under adiabatic conditions. They proposed a modified L–M correlation (Eq. (30))
ð29Þ
ð31Þ
Lee and Mudawar (2005) conducted flow boiling experiments with R134a using Qu and Mudawar (2003)’s experimental facilities. They proposed a new correlation composed of dimensionless numbers (Eq. (32))
C ¼ c1 RecLO2 WecLO3 ;
ð32Þ
where c1, c2 and c3 are experimentally-determined coefficients; for laminar–liquid and laminar–gas flow, c1 = 2.16, c2 = 0.047 and c3 = 0.60. Li and Wu (2010) studied dominant parameters using database of 769 points, which include 12 different working fluids, diameter range from 0.148 to 3.25 mm, and a channel configuration of circular, rectangular cross section and multi-channel. They found that influential parameters are Bond number (Bo) and liquid Reynolds number (ReL). Finally, they proposed a new correlation based on the Bo and ReL as follows:
Bo 6 1:5 :
C ¼ 11:9Bo0:45 ;
1:5 < Bo 6 11 :
ð33Þ
0:56 : C ¼ 109:4ðBoRe0:5 L Þ
ð34Þ
The ten correlations based on the SFM were assessed for channels with different diameters (Table 4). The most accurate correlation is that of Qu and Mudawar (2003), which considered the
Table 4 Comparison of the two-phase pressure drop correlations based on the separated flow model with the experimental results. Reference
Lockhart and Martinelli (1949) Chisholm (1967) Muller-Steinhagen and Heck (1986) Zhang et al. (2007) Tran et al. (2000) Lee and Lee (2001) Moriyama and Inoue (1992) Qu and Mudawar (2003) Lee and Mudawar (2005) Li and Wu (2010)
MAE (%) Dh = 490
Dh = 322
Dh = l43
Dh = 3O4
Dh = HI
Average
27.50 71.13 56.96 45.32 73.00 51.80 48.55 25.65 43.30 45.00
29.24 60.18 39.39 30.52 92.99 41.07 31.93 19.75 23.70 29.37
100.84 42.91 24.88 12.40 283.17 18.88 30.20 23.70 19.42 11.12
22.07 74.06 48.92 41.91 89.88 51.15 38.57 23.53 38.63 44.94
52.79 54.14 31.25 29.75 108.41 39.31 26.39 20.36 24.82 30.53
46.49 60.48 40.28 31.98 129.49 4044 35.13 22.60 29.97 32.19
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effects of confinement and mass flux. The second most accurate correlation is that of Lee and Mudawar (2005), which had been validated previously using that of Qu and Mudawar (2003). Although Qu and Mudawar (2003)’s experiments were conducted in multimicrochannels under boiling conditions, their correlation is wellpredictable with a deviation of approximately ±50%. Awad and Muzychka (2006) compared correlations of the twophase frictional pressure drop in microchannels, and reported that a modified C-value with L–M correlation predicts a pressure drop well. They proposed bounds on the two-phase frictional pressure drop in the microchannel: the upper bound was the L–M correlation with C = 5 (i.e., the original correlation for laminar–liquid and laminar–gas); the lower bound was Ali et al. (1993)’s correlation (i.e., the original correlation with the C = 0). Awad and Muzychka (2006)’s upper bound was slightly underestimated, though the lower bound matched the data well (Fig. 8). As hydraulic diameter increased, the two-phase multiplier decreased; this trend is similar to that of Zhang et al.’s correlation. The correlation suggested by Zhang et al. uses C values of 1.31, 0.87, 0.39, 0.82 and 0.38 for hydraulic diameters of 490, 322, 143, 304 and 141 lm, respectively. However, Zhang et al. (2007)’s correlation under-predicted pressure drop for larger hydraulic diameters. Originally, the L–M correlation represented the two-phase multiplier monotonically increased as the Martinelli parameter decreased. This means that the pressure drop always increased with increasing gas superficial velocity. The reason is that the Martinelli parameter is inversely proportional to this quantity in Eq. (19). Moreover, the
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C-values do not have a constant value for different flow conditions and well matched flow patterns (Fig. 9). Therefore, the flow pattern should be considered when the pressure drop analyzed in a microchannel.
5. New flow pattern based correlation 5.1. New flow pattern based HFM In the basic HFM, n in Eq. (7) indicates flow structure. In this study, three flow regimes were defined: (1) the bubble regime including bubbly, slug and elongated bubble flows, (2) the liquid ring regime, (3) the transition regime between bubble and liquid ring flows. The relation between f and Re was clearly distinct for the flow regimes (Fig. 7), which means that different flow regimes have different values of n. Therefore, a flow regime based on the HFM was developed using Beattie and Whalley (1982)’s viscosity model, which was assessed as the best two-phase viscosity model. The exponent n and the coefficient N in Eq. (7) were evaluated for flow regimes using a regression method. New correlations for different flow regimes (Fig. 10) show no dependency on AR and Dh in the bubble and transition regimes, but an obvious effect of hydraulic diameters in the liquid ring regime. As AR or Dh increased, a coefficient N in Eq. (7) increased. We proposed a new correlation of a function of AR, because the AR shows higher correlation (Eqs. (35)–(37)).
Bubble regime :
f TP ¼ 6:48Re0:85 ; TP
Transition regime : Liquid ring regime :
f TP ¼ 5:43Re0:85 ; TP f TP ¼ ðAR þ 0:46ÞRe0:6 TP :
ð35Þ ð36Þ ð37Þ
The new correlation based on the flow pattern more accurately predicted the two-phase pressure drop with MAE (%) of 10.48, 7.47, 21.5, 10.26 and 16.34 for Dh = 490, 322, 143, 304 and 141 lm, respectively. In the most of data, the error is reduced to approximately ±35% (Fig. 11). 5.2. New flow pattern based SFM
Fig. 8. Two-phase multiplier and Martinelli parameter for different diameter microchannels.
In the SFM, the hydraulic diameter was related to a two-phase multiplier (Fig. 8). As diameter decreased, C-value decreased in the same manner as in Zhang et al. (2007)’s correlation. The flow pattern was highly related to the pressure drop (Fig. 9). The C-value was higher in the bubble regime than in the liquid ring regime, which means that the interaction between the two phases is higher in the bubble regime than in the liquid regime. The C-value is linearly related to the mass flux in each flow pattern (Fig. 12); this relationship is the clearest in the liquid ring flow regime. Therefore, the flow pattern based correlation was developed by modifying Qu and Mudawar (2003)’s correlation, which was considered to be the best correlation. The correlation of Zhang et al. (2007) underestimated this experimental result, so the coefficient 0.358 in Eq. (19) was changed to 1.612, which is modified Zhang’s C-value (CM). Finally, the flow pattern based correlations were obtained using regression (Eqs. (38)–(40)):
Bubble regime :
C ¼ C M ð0:0012G þ 1:473Þ;
Transition regime : Liquid ring regime : Fig. 9. A C-value of Lockhart and Martinelli’s correlation in different flow conditions for Dh = 322 lm.
C ¼ C M ð0:0008G þ 0:95Þ;
ð38Þ ð39Þ
C ¼ C M ðaG þ bÞ;
ða ¼ 0:658 AR þ 0:13;
b ¼ 0:0016 AR þ 0:0003Þ:
ð40Þ
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Fig. 10. Two-phase friction factor and Reynolds number for different flow regimes: (a) bubble regime, (b) transition regime and (c) liquid ring regime.
Fig. 11. Comparison of new flow pattern based homogeneous flow model with all experimental data.
Fig. 13. Comparison of a new flow pattern based separated flow model with all experimental data.
The coefficient a and b had higher correlations with AR than with Dh. Finally coefficients were correlated (Eq. (40)). The new correlation based on the flow pattern was more accurately predicted a two-phase pressure drop with MAE (%) of 4.61, 8.08, 11.03, 20 and 11.93 for Dh = 490, 322, 143, 404 and 141 lm, respectively. For most of the data, an error was reduced to approximately ± 30% (Fig. 13). 6. Conclusions
Fig. 12. C-value and mass flux according to flow regimes for Dh = 490 lm.
A two-phase pressure drop in a microchannel has been widely studied. However, few studies have reported a pressure drop in single rectangular microchannels. Although the flow pattern is a critical parameter in two-phase flow phenomena, visualization results are rare due to the difficulties of fabricating microchannels. In this study, single glass rectangular microchannels with different diameters were fabricated. The adiabatic two-phase flow in the rectangular microchannels was achieved using water-liquid and Nitrogen-gas. A frictional
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pressure drop was directly measured using embedded pressure ports in the microchannels. The transparent channel walls allowed visualization of the flow pattern using a high-speed camera and a long–distance microscope. Experimental data were used to assess seven two-phase HFM viscosity models and ten correlations based on SFMs. Finally, an HFM and an SFM based on flow patterns were proposed. From the present study, the following main conclusions can be drawn. (1) Major flow patterns in the rectangular microchannels are (1) a bubble flow including a bubbly, slug bubble and elongated bubble, (2) liquid ring flow, and (3) transition between elongated bubble flow and liquid ring flow. Two-phase frictional pressure drops in rectangular microchannels are in accord with the flow patterns. This means that the flow pattern was highly related to the frictional pressure drop mechanism. (2) The experimental data were used to assess seven two-phase viscosity models. The most accurate two-phase viscosity model is that of Beattie and Whalley (1982). This model was developed considering the bubble and annular flow patterns. However, the minimum deviation is approximately ±50%. (3) The experimental data were used to assess ten correlations based on the SFM. The most accurate correlation is that of Lee and Mudawar (2005), which considers the effects of diameter and mass flux. However, the minimum deviation is approximately ±50%. (4) The relationship between two-phase friction and Reynolds number in the HFMs, and the C-value in the SFMs indicates that the flow pattern is strongly related to the two-phase frictional pressure drop in the rectangular microchannels. In addition, the flow patterns in the rectangular microchannel are different from existing patterns. Therefore, correlations based on the flow pattern were proposed for both HFM and SFM. The new HFM based on flow pattern is developed from Beattie and Whalley (1982)’s viscosity model, and is improved to its MAE of 13.21%. The new SFM based on the flow pattern was developed by modifying Qu and Mudawar (2003)’s correlation, and decreased its MAE of 11.13%. In both new correlations, a dependency on a channel diameter was observed only in the liquid ring flow regime. This could be due to the different ARs of the rectangular cross-sections. To achieve more accurate correlations for rectangular microchannels, studies of the effects of both diameter and AR on the pressure drop in microchannels are necessary.
References Ali, M., Sadatomi, M., Kawaji, M., 1993. Adiabatic two-phase flow in narrow channels between two Flat plates. Can. J. Chem. Eng. 71, 657. Awad, M.M., Muzychka, Y.S. 2006. Bounds on two-phase frictional pressure drop gradient in minichannels and microchannels. In: International Conference on Nanochannels, Microchannels and Minichannels, Limerick, Ireland. Awad, M.M., Muzychka, Y.S., 2008. Effective property models for homogeneous two-phase flows. Exp. Thermal Fluid Sci. 33, 106–113. Beattie, D.R.H., Whalley, P.B., 1982. A simple two-phase flow frictional pressure drop calculation method. Int. J. Multiph. Flow 8, 83–87. Carey, V.C., 1992. Liquid–Vapor Phase-Change Phenomena. Taylor and Francis, United States (Chapter 10). Chisholm, D., 1967. A theoretical basis for the Lockhart–Martinelli correlation for two-phase flow. Int. J. Heat Mass Transfer 10, 1767–1778.
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Choi, C.W., Kim, M.H. 2008 The fabrication of a single glass microchannel to study the hydrophobicity effect on two-phase flow boiling of water. J. Micromech. Microeng., doi: 18.105016. Choi, C.W., Yu, D.I., Kim, M.H., 2010a. Adiabatic two-phase flow in rectangular microchannels with different aspect ratios: Part I – Flow pattern, pressure drop and void fraction. Int. J. Heat Mass Transfer. doi:10.1016/ j.ijheatmasstransfer.2010.07.067. Choi, C.W., Yu, D.I., Kim, M.H., 2010b. Adiabatic two-phase flow in rectangular microchannels with different aspect ratios: Part II – Bubble behaviors and pressure drop in single bubble. Int. J. Heat Mass Transfer. doi:10.1016/ j.ijheatmasstransfer.2010.07.035. Chung, P.M.-Y., Kawaji, M., 2004. The effect of channel diameter on adiabatic twophase flow characteristics in microchannels. Int. J. Multiph. Flow 30, 735–761. Cicchitti, A., Lombardi, C., Silvestri, M., Solddaini, G., Zavalluilli, R., 1960. Two-phase cooling experiments-pressure drop, heat transfer, and burnout measurement. Energia Nucl. 7, 407–425. Cioncolini, A., Thome, J.R., Lombardi, C., 2009. Unified macro-to-micro method to predict two-phase frictional pressure drops of annular flows. Int. J. Multiph. Flow 35, 1138–1148. Costa-Patry, E., Olivier, J., Nichita, B.A., Michel, B., Thome, J.R., 2011. Two-phase flow of refrigerants in 85 lm-wide multi-microchannels: Part I – Pressure drop. Int. J. Heat Fluid Flow 32, 451–463. Dukler, A.E., Wicks, M., Cleveland, 1964. Pressure drop and hold-up in two-phase flow: Part A – A comparison of existing correlations and part B-an approach through similarity analysis. AIChE J. 10, 38–51. Garimella, S., Killion, J.D., Coleman, J.W., 2003. An experimentally validated model for two-phase pressure drop in the intermittent flow regime for non-circular microchannels. J. Fluid. Eng. 125, 887–894. Hartnett, J.P., Kostic, M., 1989. Heat transfer to newtonian and non-newtonian fluids in rectangular ducts. Adv. Heat Transfer 19, 247–356. Kawahara, A., Chung, P.M.-Y., Kawaji, M., 2002. Investigation of two-phase flow pattern, void fraction and pressure drop in a microchannel. Int. J. Multiph. Flow 28, 1411–1435. Kawaji, M. 2008. Unique aspects of adiabatic two-phase flow in microchannels. In: International Conference on Heat Transfer and Fluid Flow in Microscale, Whistler, Canada. Lee, H.J., Lee, S.Y., 2001. Pressure drop correlations for two-phase flow within horizontal rectangular channels with small heights. Int. J. Multiph. Flow 27, 783–796. Lee, J., Mudawar, I., 2005. Two-phase flow in high-heat-flux microchannel heat sink for refrigeration cooling applications: Part I – Pressure drop characterisitics. Int. J. Heat Mass Transfer 48, 928–940. Li, W., Wu, Z., 2010. A general correlation for adiabatic two-phase pressure drop in micro/mini-channels. Int. J. Heat Mass Transfer 53, 2732–2739. Lin, S., Kwok, C.C.K., Li, R.Y., Chen, Z.H., Chen, Z.Y., 1991. Local frictional pressure drop during vaporization for R-12 through capillary tubes. Int. J. Multiph. Flow 17, 95–102. 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. MacAdams, W.H., Woods, W.K., Bryan, R.L., 1942. Vaporization inside horizontal tubes in benzene-oil mixture. Trans. ASME 64, 193. Mishima, K., Hibiki, T., Nishihara, H., 1993. Some characterisitics of gas-liquid flow in narrow rectangular ducts. Int. J. Multiph. Flow 19, 115–124. Mishima, K., Hibiki, T., 1996. Some characterisitics of air-water two-phase flow in small diameter vertical tubes. Int. J. Multiph. Flow 22, 703–712. Moriyama, K., Inoue, A., 1992. The thermal-hydraulic characteristics of two-phase flow in extremely narrow channels (the frictional pressure drop and heat transfer of boiling two-phase flow, analytical model). Heat Transfer-Jap. Res. 21, 838–856. Muller-Steinhagen, H., Heck, K., 1986. A simple friction pressure drop correlation for two-phase flow pipes. Chem. Eng. Process. 20, 297–308. Owens, W.L., 1961. Two-phase pressure gradient. In: International Development in Heat Transfer, Part II. ASME, New York, United States. Qu, W., Mudawar, I., 2003. Measurement and prediction of pressure drop in twophase micro-channel heat sinks. Int. J. Heat Mass Transfer 46, 2737–2753. Saisorn, S., Wongwises, S., 2008. Flow pattern, void fraction and pressure drop of two-phase air-water flow in horizontal circular micro-channel. Exp. Thermal Fluid Sci. 32, 748–760. Shiferaw, D., Mahmoud, M., Karayiannis, T.G., Kenning, D.B.R., 2011. Onedimensional semimechanistic model for flow boiling pressure drop in small to miro passages. Heat Transfer Eng. 32, 1150–1159. Thome, J.R., Dupont, V., Jacobi, A.M., 2004. Heat transfer model for evaporation in microchannels. Int. J. Heat Mass Transfer 47, 3375–3385. Tran, T.N., Chyu, M.-C., Wambsganss, M.W., France, D.M., 2000. Two-phase pressure drop of refrigerants during flow boiling in small channels: an experimental investigation and correlation development. Int. J. Multiph. Flow 26, 1739–1754. Zhang, W., Hibiki, T., Mishima, K. 2007. Two-phase frictional pressure drop in minichannel. In: 6th International Conference on Multiphase Flow. International Conference Multiphase Flow S7_Wed_C_38.
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Flow pattern based correlations of two-phase pressure drop in rectangular microchannels Chiwoong Choi ⇑, Moohwan Kim Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang, San 31, Hyoja Dong 790-784, Republic of Korea
a r t i c l e
i n f o
Article history: Received 10 January 2011 Received in revised form 1 August 2011 Accepted 30 August 2011 Available online 6 October 2011 Keywords: Rectangular microchannel Pressure drop Flow pattern Correlation
a b s t r a c t Numerous pressure drop correlations for microchannels have been proposed; most of them can be classified as either a homogeneous flow model (HFM) or a separated flow model (SFM). However, the predictions of these correlations have not been compared directly because they were developed in experiments conducted under a range of conditions, including channel shape, the number of channels, channel material and the working fluid. In this study, single rectangular microchannels with different aspect ratios and hydraulic diameters were fabricated in a photosensitive glass. Adiabatic water-liquid and Nitrogen-gas two-phase flow experiments were conducted using liquid superficial velocities of 0.06–1.0 m/s, gas superficial velocities of 0.06–72 m/s and hydraulic diameters of 141, 143, 304, 322 and 490 lm. A pressure drop in microchannels was directly measured through embedded ports. The flow pattern was visualized using a high-speed camera and a long-distance microscope. A two-phase pressure drop in the microchannel was highly related to the flow pattern. Data were used to assess seven different HFM viscosity models and ten SFM correlations, and new correlations based on flow patterns were proposed for both HFMs and SFMs. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Two-phase flow pressure drop correlations can be classified as either homogeneous flow models (HFMs) or separated flow models (SFMs). HFMs treat a two-phase flow as a single phase flow with mixed properties. In HFMs, two-phase density and viscosity should be defined. SFMs are based on a two-phase frictional multiplier (u), which is defined as the ratio of the pressure gradient of the liquid phase to that of the gas phase. A pressure drop of two-phase flow in a microchannel cannot be predicted using conventional correlations. Hence, many studies of the two-phase frictional pressure drop in microchannels have been conducted; most used circular tubes and rectangular multichannels, because of difficulties in fabricating circular microchannels and controlling small amounts of fluid and heat. For a heat sink application, the quantification of a pressure drops in rectangular microchannels is needed. In the present study, rectangular glass microchannels were fabricated to allow a visualization of two-phase flow patterns. Experiments were conducted in these microchannels using water liquid and Nitrogen gas flow for liquid superficial velocities of 0.06– 1.0 m/s, gas superficial velocities of 0.06–72 m/s, and microchannel hydraulic diameters of 141, 143, 304, 322 and 490 lm. Flow ⇑ Corresponding author. E-mail address: [email protected] (C.W. Choi). 0142-727X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2011.08.002
patterns were visualized using a high-speed camera and a longdistance microscope. Data were used to assess seven different HFM viscosity models and ten SFM correlations, In addition, new correlations based on flow patterns were proposed for both HFMs and SFMs. There are modeling works for two-phase pressure drop in microchannel (Garimella et al., 2003; Shiferaw et al., 2011). For example, Garimella et al. (2003) developed pressure drop model, which based on a unit cell concept, for intermittent flow regime for non-circular microchannels. In development of model, the number of unit cell was regressed with a hydraulic diameter and slug Reynolds number. However, their model shows highly underestimated the number of unit cell comparing with present data (Choi et al., 2010b). Shiferaw et al. (2011) also developed semimechanistic pressure drop model for specific flow pattern, which is based on a three zone model (Thome et al., 2004). Both models are considered the specific flow pattern, the periodic elongated bubble regime, thus two models are exempted from our assessment correlations.
2. Experimental setup 2.1. Test section The test section (Fig. 1; the unit is mm) was fabricated in a sheet of a photosensitive glass using UV rays. Details can be found in Choi and Kim (2008). The test section has two inlets (i.e., one for
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Fig. 1. A microchannel test section (top view).
each phase flow), one outlet and three pressure ports. Total channel length was 60 mm and the distance between pressure ports was 15 mm. The test section was held horizontally with a support frame to connect inlets and outlet tubing. The interconnections were achieved using fitting and ferrule (Upchurch Scientific), and a fitting holder. Pressure was measured directly through the three embedded pressure ports; this overcame the uncertainties in a measurement that occurred in most previous studies, in which pressure was measured outside of the channel. Moreover, the fully developed condition was checked with comparing pressure differences among three pressure ports (Fig. 1). Flow patterns were visualized directly through the transparent glass using a high-speed camera and a long-distance microscope. Cross sections of five different fabricated rectangular microchannels were measured using a 3D-profiler (Veeco-Wyko DMEMS NT1100). To verify this measuring method, the cross-section of a test sample was measured using a microscope (Table 1).
rates of the liquid were measured using flow-meters. The pressure drop was measured directly through embedded pressure ports in the microchannel using four different pressure transducers (Druck LPM9000, Setra 230-5, 10, and 25 PSID). Data were collected using a data acquisition system (Agilent 34970) and were saved on a personal computer. The details of the experimental setup were described in Choi et al. (2010a). Before the main experiment, water was pumped into the inlet of the test section to extract non-degassed water from the tubes, and tubes connected to pressure ports were fully filled with water to offset the effect of a static head, which was significant under low pressure conditions. Then the gas flow was applied to the other inlet to create a two-phase flow. When the temperature and the pressure upstream of each phase flow were steady, the pressure and the flow rate were measured every 2 s. Then, the gas flow rate was changed while the liquid flow rate was held a constant (Table 1). All experimental uncertainties are summarized in Table 2. 3. Results and discussion
2.2. Experimental facilities and procedure 3.1. Single phase pressure drop In the experimental setup (Fig. 2), the liquid and gas flows were controlled using a pneumatic pumping system, which consisted of electric regulators and pressurized gases. Nitrogen gas was used to pressurize the gas phase loop and Helium gas was used to pressurize the liquid phase loop. The two-phase flow was mixed in a T-junction. The effect of the mixing method is not well understood and still hot issues in this research area (Kawaji, 2008). Therefore, in this study, we used T-junction configuration, which is well proven mixing method in various fields. The volume and mass flow
Before conducting the two-phase flow experiment, experiments for a single phase pressure drop in the rectangular microchannels were conducted to verify the experimental apparatus and to evaluate the friction factor of the rectangular microchannel. The value of fRe is 16 for circular tubes, and Hartnett and Kostic (1989) proposed a correlation of fRe for ducts of different ARs (Eqs. (1)–(3))
f Re ¼ 24ð1 1:3553AR þ 1:9467AR2 1:7012AR3 þ 0:9564AR4 0:2537AR5 Þ;
Table 1 Experimental conditions.
where
Variables
Ranges
Diameters of microchannels, Dh (WCh HCh) (lm)
490(510 470) 322(501 237) 143 (503 83) 304 (332 280) 141 (201 109) 66–1000 0.075–80 32–477 2–2134 0.06–1 0.06–72
Liquid mass flux, GL (kg/m2s) Vapor mass flux, Gg (kg/m2s) Liquid Reynolds number, ReL Vapor Reynolds number, Reg Liquid superficial velocity, jL (m/s) Vapor superficial velocity, Jg (m/s)
ð1Þ
Re ¼
quch Dh ; l
ð2Þ
and
f ¼
Dh DP : 2qu2ch Dz SP
ð3Þ
The predicted values from Hartnett and Kostic (1989)’s correlation were 19.71, 15.74 and 14.25 for 143, 322 and 490 lm diameters, respectively; measured fRe values were agreed well with predicted values. Results of the single phase frictional pressure
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Fig. 2. Experimental facilities for the adiabatic two-phase flow in microchannnel.
Table 2 Experimental uncertainties. Variables
Uncertainty
Diameter (lm) Area (lm2) Pressure (kPa) Temperature (°C) Mass flux, G (kg/m2s) Friction factor Superficial velocities (m/s)
±17 (4.5%) ±6311 (3.8%) ±6% (±0.01, ±0.005, ±0.034, ±0.085) ±0.1–0.5 ±0.1–2% ±9% ±10%
drop in our test section for water liquid indicate that the conventional theory of pressure drops in a single phase-laminar flow is acceptable and that our experimental setup is appropriate.
3.2. Flow pattern and two-phase pressure drop In our experimental region, the observed flow patterns were bubbly, slug bubble, elongated bubble, liquid ring, and transition flow (Fig. 3). Definitions are the same as in Choi et al. (2010a). As gas superficial velocity (jG) increased, the bubble length increased. The bubble nose and the tail collapsed, and formed a liquid ring; this flow pattern has been observed and given various names by various researchers (Kawahara et al., 2002; Chung and Kawaji, 2004 and Saisorn and Wongwises, 2008). A transition region occurred between the elongated bubble flow and the liquid ring flow regimes; in the transition regime, the elongated bubble flow and the liquid ring flow occurred periodically. Thus, we defined this flow regime as ‘‘transition flow patterns’’. Before this transition point, the bubble flows were stable and formed a perfect bubble train. As liquid superficial velocity (jL) increased, the bubble length decreased. Then, the elongated bubble flow changed to the single bubbly flow at higher jL. In this study, main flow regimes were grouped into (1) a bubble flow including bubbly, slug bubble and elongated bubble, (2) liquid ring flow, and (3) a transition between elongated bubble flow and ring flow.
Fig. 3. A classification of the flow patterns in the rectangular microchannel for Dh = 490 lm.
Pressure drop increased as jL and jG increased. At the constant jL, the pressure drop increased as jG increased. However, in the transition region, the pressure drop decreased as jG increased. The merged bubbles caused the bubble to elongate. The dominant component of pressure drops in a bubble occurs in the nose and tail, rather than in the bubble body. However, Choi et al. (2010b) reported the collapsed bubble reduces the number of parts that
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ReTP ¼
Fig. 4. Two-phase pressure drop in rectangular microchannel with Dh = 322 lm: Typical trend of the pressure drop in a rectangular microchannel ((I) bubbly or liquid slug flow, (II) elongated bubble flow, (III) transition flow regime (transition), (IV) liquid ring flow).
contribute to the pressure drop. The typical trend of pressure drops in the microchannel (Fig. 4) is composed of three regions. Region I: A bubble flow regime including bubbly, slug bubble and elongated bubble flow patterns. Pressure drops increased as gas superficial velocity increased. Region II: Transition flow patterns. Pressure drop decreased as gas superficial velocity decreased. Region III: Liquid ring flow. Pressure drop increased as gas superficial velocity increased. This typical trend indicates that the pressure drop in the rectangular microchannel is highly related to the flow pattern. 4. Assessment of two-phase pressure drop models
GDh
lTP
ð8Þ
;
where lTP is a two-phase viscosity. For a laminar flow in a circular tube, n = 1 and N = 16; in a rectangular channel N is a function of AR (Eq. (1)). Numerous two-phase viscosity models have been proposed. Recently, Costa-Patry et al. (2011) conducted flow boiling experiments in 85 lm wide multi-microchannels. They showed that Cioncolini et al. (2009)’s correlation is well agree with their pressure drop results. The Cioncolini et al.’s correlation is developed using a dimensionless analysis for an annular flow pattern. Costa-Patry et al. also mentioned the Cioncolini et al.’s correlation had a good accuracy for the annular flow pattern. Therefore, the Cioncolini et al.’s correlation is exempted from our assessment correlations. In this study, our experimental results were used to assess seven viscosity models: Owens (1961), MacAdams et al. (1942), Cicchitti et al. (1960), Dukler et al. (1964), Beattie and Whalley (1982), Lin et al. (1991) and Awad and Muzychka (2008).
Owen ð1961Þ : lTP ¼ lL ;
ð9Þ
McAdams etal: ð1942Þ : lTP ¼
x
lG
þ
1x
1
lL
;
ð10Þ
Cicchitti etal: ð1960Þ : lTP ¼ xlG þ ð1 xÞlL ;
ð11Þ
Dukler etal: ð1964Þ : lTP ¼ blG þ ð1 bÞlL ;
ð12Þ
Beattie and Whalley ð1982Þ : lTP ¼ blG þ ð1 bÞð1 þ 2:5bÞlL ; Lin etal: ð1991Þ : lTP ¼
4.1. Homogeneous flow model (HFM)
lG lL ; lG þ x1:4 ðlL lG Þ
ð13Þ
ð14Þ
2lG þ lL 2ðlG lL Þð1 xÞ : 2lG þ lL þ ðlG lL Þð1 xÞ
An HFM is the simplest two-phase flow model; it treats twophase flow as a single phase flow with mixture properties. A two-phase frictional pressure drop consists of frictional, accelerational, and gravitational terms (Eq. (4)) (see Carey, 1992).
Awad and Myuztchka ð2008Þ : lTP ¼ lG
dP dP dP dP ¼ þ þ : dz TP dz Friction dz Acceleration dz Gravitation
Pressure drop based on the HFM and viscosity models were compared to experimental data. Accuracy of model predictions was assessed using the mean absolute error (MAE), expressed as a percentage:
ð4Þ
In this study, the gravitational term was neglected, because a flow was horizontal, and the accelerational term was neglected because the flow was adiabatic. Therefore, the total measured pressure drop is the frictional term in Eq. (4), which can be defined using the HFM (Eq. (5)).
dP dP 2f G2 ¼ ¼ TP ; dz TP dz Friction qTP Dh
ð5Þ
where fTP is a two-phase friction factor, G is mass flux and qTP is two-phase density (Eq. (6))
qTP ¼
x
qG
þ
1x
qL
1 :
ð6Þ
The two-phase friction factor can be expressed as an exponential function of the two-phase Reynolds number (Eq. (7))
fTP ¼ NRen TP ; with
ð7Þ
ð15Þ
MAE ¼
1 X jDPTP;pred DPTP;exp j 100 : M DPTP;exp
ð16Þ
HFMs were assessed for microchannels having different diameters (Table 3). The most accurate viscosity model is the Beattie and Whalley (1982)’s model, which is based on the volumetric quality. Experimental pressure drop data and the viscosity models, were used to calculate f and Re (Fig. 5), and an original HFM for Dh = 322 lm with n = 1 and N = 15.75 (Eq. (7)). The two-phase Re was over-predicted by Dukler et al. (1964)’s model and under-predicted by other models. The two-phase viscosity predicted by the Beattie and Whalley (1982)’s model agreed marginally well with the original HFM. However, the other models except that of Dukler et al. (1964)’s model overestimated two-phase viscosity (Fig. 6). Beattie and Whalley (1982)’s two-phase viscosity model was developed for the bubble flow and annular flow patterns; this concept agreed well with our experimental result (Fig. 7), but it was deviated from measurements by approximately ±50%. With other microchannels, a similar trend was observed.
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MAE (%)
Owens (1961) MacAdams et al. (1942) Cicchitti et al. (1960) Dukler et al. (1964) Beattie and Whalley (1982) Lin et al. (1991) Awad and Muzychka (2008)
Dh = 490
Dh = 322
Dh = 143
Dh = 304
Dh = 14 l
Average
352.10 40.81 300.76 66.84 30.69 111.99 107.87
647.37 101.35 554.13 51.76 22.72 225.10 204.17
1247.28 236.46 1076.53 23.25 75.38 464.32 426.47
801.18 123.84 715.23 63.29 24.41 309.52 145.01
709.07 169.87 650.12 42.10 43.50 346.56 298.29
761.98 134.46 659.35 49.44 39.34 291.49 236.36
Fig. 5. Relation of friction factor and Reynolds number based on different twophase viscosity models for Dh = 322 lm and jL = 0.4 m/s.
Fig. 7. Two-phase friction factor and Reynolds number for Dh = 322 lm using Beattie and Whalley’s two-phase viscosity model.
X vv ¼
Fig. 6. Two-phase viscosity for water liquid and Nitrogen gas for different twophase viscosity models.
4.2. Separated flow model (SFM) SFMs are based on a two-phase multiplier (u), which is defined
0:5 0:5 0:5 0:5 lL Dp Dp 1x qG ¼ : x Dz L Dz G qL lG
ð19Þ
In the right-hand side of Eq. (18), one represents liquid-only pressure drop, C/Xvv represents mixed pressure drop, and 1=X 2vv represents gas phase-only pressure drop. Therefore, C represents the interactional effect of the two-phase flow. Previously proposed correlations for frictional pressure drop in a microchannel have been developed by modifying the C-value in the SFM. In this study, our experimental data were used to assess ten correlations for the SFM: Lockhart and Martinelli (1949), Chisholm (1967) and Muller-Steinhagen and Heck (1986) for macro-scale correlations; Zhang et al. (2007), Tran et al. (2000) and Lee and Lee (2001) for mini-scale correlations, and Moriyama and Inoue (1992), Qu and Mudawar (2003), Lee and Mudawar (2005) and Li and Wu (2010) for micro-scale correlations. Lockhart and Martinelli (1949) proposed the C value for different flow structures. For example, C = 5 for laminar flow of both liquid and gas phases. Chisholm (1967) proposed a modified two-phase multiplier (Eq. (20)):
h 2s i 2s /2LO ¼ 1 þ ðY 2 1Þ Bx 2 ð1 xÞ 2 þ x2s ;
ð20Þ
as
/2L ¼
Dp Dp Dz TP Dz L
or /2LO ¼
Dp Dp : Dz TP Dz LO
ð17Þ
Lockhart and Martinelli (1949) suggested that /2L is a function of a Martinelli parameter (Eq. (18))
/2L ¼ 1 þ
C 1 þ ; X vv X 2vv
ð18Þ
where Xvv is a parameter of a laminar liquid–laminar gas flow (Eq. (19)).
where s = 0.25, x is a quality and Y is a sort of Martinelli parameter (Eq. (21))
Dp
Y 2 ¼ DDzpGO ¼ Dz LO
1 X 2vv
;
ð21Þ
and B is a function of mass flux and Y. Muller-Steinhagen and Heck (1986) proposed a simple and convenient correlation for a two-phase frictional pressure drop using data points of 9300 various fluids and flow conditions, including channel diameters from 4 mm to 392 mm (Eqs. (22) and (23))
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Dp Dp ¼ Fð1 xÞ1=3 þ x3 ; Dz TP Dz GO F¼
ð22Þ
Dp Dp Dp x: þ2 Dz LO Dz GO Dz LO
ð23Þ
The correlation of Muller-Steinhagen and Heck has a form that is similar to Chisholm’s correlation without the B-coefficient. Mishima and Hibiki group (1993, 1996) investigated the effect of a mini-sized diameter on two-phase frictional pressure in experiments on an air–water adiabatic two-phase flow in minichannels with diameters of 1–4 mm. They proposed that the C-value of the L–M correlation is a function of the diameter. They defined the Cvalue as
C ¼ 21ð1 e333D Þ;
ð24Þ
for circular tubes and as
C ¼ 21ð1 e319Dh Þ;
ð25Þ
for rectangular tubes. Zhang et al. (2007) modified Mishima and Hibiki’s correlation to extend to the micro-scale and verified it using different experimental results. The hydraulic diameter was replaced by a Laplace constant:
C ¼ 21ð1 e358=Nconf Þ;
ð26Þ
where Nconf is a confinement number (Eq. (27))
Nconf
lc lc ½r=ðqL qV Þg1=2 ¼ ¼ ¼ ; D Dh Dh
where lc is the Laplace constant, and the unit of D or Dh is a meter. The applicable range was limited to 0.014 mm
ð28Þ
Lee and Lee (2001) conducted air–water adiabatic two-phase flow experiments for different rectangular minichannels with low aspect ratios. They proposed a new C-value as a function of three dimensionless numbers for laminar and turbulent regions: q
C ¼ Ak
Carj Resj
l ¼A qL rDh 2 L
p q jl L qL jDh r
r
lL
;
/2L ¼ 1 þ
K X 2vv
ð30Þ
:
This correlation lacks the second term in the original L–M correlation (Eq. (19)), i.e., the C-value that is an interactional effect in the two-phase pressure drop. They multiplied the third term (the gas phase portion in the two-phase pressure drop) by a factor K, which they postulated to be a function of liquid Reynolds number. This means that the influential parameter in the two-phase pressure drop in a microchannel is the gas phase portion rather than the interactional portion. Recently, a similar correlation was proposed by Saisorn and Wongwises (2008). Qu and Mudawar (2003) investigated the frictional two-phase pressure drop in water flow boiling experiments using 21 parallel microchannels of size 231 lm 713 lm. They proposed a modified Mishima et al. (1993)’s correlation, which considers the effect of mass flux (Eq. (31))
C ¼ 21ð1 e319Dh Þð0:00418G þ 0:0613Þ: ð27Þ
h i 2s 2s /2LO ¼ 1 þ ð4:3Y 2 1Þ N conf x 2 ð1 xÞ 2 þ x2s :
where k is a fluid property, Caj and Rej are capillary number and Reynolds number based on total superficial velocity (j), respectively, and A, p, q and r are empirically-determined coefficients. For laminar–liquid and laminar–gas flow, A = 6.833 108, p = 1.317, q = 0.719 and r = 0.557. Moriyama and Inoue (1992) studied a two-phase pressure drop in 35–110 lm gaps using R113 under boiling conditions, and Nitrogen and R113 under adiabatic conditions. They proposed a modified L–M correlation (Eq. (30))
ð29Þ
ð31Þ
Lee and Mudawar (2005) conducted flow boiling experiments with R134a using Qu and Mudawar (2003)’s experimental facilities. They proposed a new correlation composed of dimensionless numbers (Eq. (32))
C ¼ c1 RecLO2 WecLO3 ;
ð32Þ
where c1, c2 and c3 are experimentally-determined coefficients; for laminar–liquid and laminar–gas flow, c1 = 2.16, c2 = 0.047 and c3 = 0.60. Li and Wu (2010) studied dominant parameters using database of 769 points, which include 12 different working fluids, diameter range from 0.148 to 3.25 mm, and a channel configuration of circular, rectangular cross section and multi-channel. They found that influential parameters are Bond number (Bo) and liquid Reynolds number (ReL). Finally, they proposed a new correlation based on the Bo and ReL as follows:
Bo 6 1:5 :
C ¼ 11:9Bo0:45 ;
1:5 < Bo 6 11 :
ð33Þ
0:56 : C ¼ 109:4ðBoRe0:5 L Þ
ð34Þ
The ten correlations based on the SFM were assessed for channels with different diameters (Table 4). The most accurate correlation is that of Qu and Mudawar (2003), which considered the
Table 4 Comparison of the two-phase pressure drop correlations based on the separated flow model with the experimental results. Reference
Lockhart and Martinelli (1949) Chisholm (1967) Muller-Steinhagen and Heck (1986) Zhang et al. (2007) Tran et al. (2000) Lee and Lee (2001) Moriyama and Inoue (1992) Qu and Mudawar (2003) Lee and Mudawar (2005) Li and Wu (2010)
MAE (%) Dh = 490
Dh = 322
Dh = l43
Dh = 3O4
Dh = HI
Average
27.50 71.13 56.96 45.32 73.00 51.80 48.55 25.65 43.30 45.00
29.24 60.18 39.39 30.52 92.99 41.07 31.93 19.75 23.70 29.37
100.84 42.91 24.88 12.40 283.17 18.88 30.20 23.70 19.42 11.12
22.07 74.06 48.92 41.91 89.88 51.15 38.57 23.53 38.63 44.94
52.79 54.14 31.25 29.75 108.41 39.31 26.39 20.36 24.82 30.53
46.49 60.48 40.28 31.98 129.49 4044 35.13 22.60 29.97 32.19
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effects of confinement and mass flux. The second most accurate correlation is that of Lee and Mudawar (2005), which had been validated previously using that of Qu and Mudawar (2003). Although Qu and Mudawar (2003)’s experiments were conducted in multimicrochannels under boiling conditions, their correlation is wellpredictable with a deviation of approximately ±50%. Awad and Muzychka (2006) compared correlations of the twophase frictional pressure drop in microchannels, and reported that a modified C-value with L–M correlation predicts a pressure drop well. They proposed bounds on the two-phase frictional pressure drop in the microchannel: the upper bound was the L–M correlation with C = 5 (i.e., the original correlation for laminar–liquid and laminar–gas); the lower bound was Ali et al. (1993)’s correlation (i.e., the original correlation with the C = 0). Awad and Muzychka (2006)’s upper bound was slightly underestimated, though the lower bound matched the data well (Fig. 8). As hydraulic diameter increased, the two-phase multiplier decreased; this trend is similar to that of Zhang et al.’s correlation. The correlation suggested by Zhang et al. uses C values of 1.31, 0.87, 0.39, 0.82 and 0.38 for hydraulic diameters of 490, 322, 143, 304 and 141 lm, respectively. However, Zhang et al. (2007)’s correlation under-predicted pressure drop for larger hydraulic diameters. Originally, the L–M correlation represented the two-phase multiplier monotonically increased as the Martinelli parameter decreased. This means that the pressure drop always increased with increasing gas superficial velocity. The reason is that the Martinelli parameter is inversely proportional to this quantity in Eq. (19). Moreover, the
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C-values do not have a constant value for different flow conditions and well matched flow patterns (Fig. 9). Therefore, the flow pattern should be considered when the pressure drop analyzed in a microchannel.
5. New flow pattern based correlation 5.1. New flow pattern based HFM In the basic HFM, n in Eq. (7) indicates flow structure. In this study, three flow regimes were defined: (1) the bubble regime including bubbly, slug and elongated bubble flows, (2) the liquid ring regime, (3) the transition regime between bubble and liquid ring flows. The relation between f and Re was clearly distinct for the flow regimes (Fig. 7), which means that different flow regimes have different values of n. Therefore, a flow regime based on the HFM was developed using Beattie and Whalley (1982)’s viscosity model, which was assessed as the best two-phase viscosity model. The exponent n and the coefficient N in Eq. (7) were evaluated for flow regimes using a regression method. New correlations for different flow regimes (Fig. 10) show no dependency on AR and Dh in the bubble and transition regimes, but an obvious effect of hydraulic diameters in the liquid ring regime. As AR or Dh increased, a coefficient N in Eq. (7) increased. We proposed a new correlation of a function of AR, because the AR shows higher correlation (Eqs. (35)–(37)).
Bubble regime :
f TP ¼ 6:48Re0:85 ; TP
Transition regime : Liquid ring regime :
f TP ¼ 5:43Re0:85 ; TP f TP ¼ ðAR þ 0:46ÞRe0:6 TP :
ð35Þ ð36Þ ð37Þ
The new correlation based on the flow pattern more accurately predicted the two-phase pressure drop with MAE (%) of 10.48, 7.47, 21.5, 10.26 and 16.34 for Dh = 490, 322, 143, 304 and 141 lm, respectively. In the most of data, the error is reduced to approximately ±35% (Fig. 11). 5.2. New flow pattern based SFM
Fig. 8. Two-phase multiplier and Martinelli parameter for different diameter microchannels.
In the SFM, the hydraulic diameter was related to a two-phase multiplier (Fig. 8). As diameter decreased, C-value decreased in the same manner as in Zhang et al. (2007)’s correlation. The flow pattern was highly related to the pressure drop (Fig. 9). The C-value was higher in the bubble regime than in the liquid ring regime, which means that the interaction between the two phases is higher in the bubble regime than in the liquid regime. The C-value is linearly related to the mass flux in each flow pattern (Fig. 12); this relationship is the clearest in the liquid ring flow regime. Therefore, the flow pattern based correlation was developed by modifying Qu and Mudawar (2003)’s correlation, which was considered to be the best correlation. The correlation of Zhang et al. (2007) underestimated this experimental result, so the coefficient 0.358 in Eq. (19) was changed to 1.612, which is modified Zhang’s C-value (CM). Finally, the flow pattern based correlations were obtained using regression (Eqs. (38)–(40)):
Bubble regime :
C ¼ C M ð0:0012G þ 1:473Þ;
Transition regime : Liquid ring regime : Fig. 9. A C-value of Lockhart and Martinelli’s correlation in different flow conditions for Dh = 322 lm.
C ¼ C M ð0:0008G þ 0:95Þ;
ð38Þ ð39Þ
C ¼ C M ðaG þ bÞ;
ða ¼ 0:658 AR þ 0:13;
b ¼ 0:0016 AR þ 0:0003Þ:
ð40Þ
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Fig. 10. Two-phase friction factor and Reynolds number for different flow regimes: (a) bubble regime, (b) transition regime and (c) liquid ring regime.
Fig. 11. Comparison of new flow pattern based homogeneous flow model with all experimental data.
Fig. 13. Comparison of a new flow pattern based separated flow model with all experimental data.
The coefficient a and b had higher correlations with AR than with Dh. Finally coefficients were correlated (Eq. (40)). The new correlation based on the flow pattern was more accurately predicted a two-phase pressure drop with MAE (%) of 4.61, 8.08, 11.03, 20 and 11.93 for Dh = 490, 322, 143, 404 and 141 lm, respectively. For most of the data, an error was reduced to approximately ± 30% (Fig. 13). 6. Conclusions
Fig. 12. C-value and mass flux according to flow regimes for Dh = 490 lm.
A two-phase pressure drop in a microchannel has been widely studied. However, few studies have reported a pressure drop in single rectangular microchannels. Although the flow pattern is a critical parameter in two-phase flow phenomena, visualization results are rare due to the difficulties of fabricating microchannels. In this study, single glass rectangular microchannels with different diameters were fabricated. The adiabatic two-phase flow in the rectangular microchannels was achieved using water-liquid and Nitrogen-gas. A frictional
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pressure drop was directly measured using embedded pressure ports in the microchannels. The transparent channel walls allowed visualization of the flow pattern using a high-speed camera and a long–distance microscope. Experimental data were used to assess seven two-phase HFM viscosity models and ten correlations based on SFMs. Finally, an HFM and an SFM based on flow patterns were proposed. From the present study, the following main conclusions can be drawn. (1) Major flow patterns in the rectangular microchannels are (1) a bubble flow including a bubbly, slug bubble and elongated bubble, (2) liquid ring flow, and (3) transition between elongated bubble flow and liquid ring flow. Two-phase frictional pressure drops in rectangular microchannels are in accord with the flow patterns. This means that the flow pattern was highly related to the frictional pressure drop mechanism. (2) The experimental data were used to assess seven two-phase viscosity models. The most accurate two-phase viscosity model is that of Beattie and Whalley (1982). This model was developed considering the bubble and annular flow patterns. However, the minimum deviation is approximately ±50%. (3) The experimental data were used to assess ten correlations based on the SFM. The most accurate correlation is that of Lee and Mudawar (2005), which considers the effects of diameter and mass flux. However, the minimum deviation is approximately ±50%. (4) The relationship between two-phase friction and Reynolds number in the HFMs, and the C-value in the SFMs indicates that the flow pattern is strongly related to the two-phase frictional pressure drop in the rectangular microchannels. In addition, the flow patterns in the rectangular microchannel are different from existing patterns. Therefore, correlations based on the flow pattern were proposed for both HFM and SFM. The new HFM based on flow pattern is developed from Beattie and Whalley (1982)’s viscosity model, and is improved to its MAE of 13.21%. The new SFM based on the flow pattern was developed by modifying Qu and Mudawar (2003)’s correlation, and decreased its MAE of 11.13%. In both new correlations, a dependency on a channel diameter was observed only in the liquid ring flow regime. This could be due to the different ARs of the rectangular cross-sections. To achieve more accurate correlations for rectangular microchannels, studies of the effects of both diameter and AR on the pressure drop in microchannels are necessary.
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