Frictional pressure drop correlation for two-phase flows in mini and micro single-channels

International Journal of Multiphase Flow 90 (2017) 29–45

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Frictional pressure drop correlation for two-phase flows in mini and micro single-channels Xuejiao Li a,b, Takashi Hibiki a,∗ a

School of Nuclear Engineering, Purdue University, 400 Central Drive, West Lafayette 47907-2017, USA Beijing Key Laboratory of Micro-scale Flow and Phase Change Heat Transfer, Beijing Jiaotong University, Shangyuan Cun 3, Haidian District, Beiijing, 100004, PR China

b

a r t i c l e

i n f o

Article history: Received 14 August 2016 Revised 5 December 2016 Accepted 9 December 2016 Available online 13 December 2016 Keywords: Two-phase flow Frictional pressure drop Single channel Micro-channel Mini-channel

a b s t r a c t 1521 frictional pressure drop data points were collected from 12 literatures. The database included adiabatic and diabatic systems composed of 10 working fluids. The diameter range was from 0.1 to 3 mm, and the pressure drop ranged from 1.26 kPa/m to 2 MPa/m. 17 existing single channel two-phase pressure drop correlations including homogenous flow model and separated flow model were evaluated with the database. The comparison results showed that the correlations of Pamitran et al., Hwang and Kim, Mishima and Hibiki and Zhang et al. gave better predictions than the others. However, the performance evaluation results also showed the mean absolute errors higher than 40%. Based on the relatively large prediction error by the existing correlations, a new correlation was proposed by classifying the flow conditions into four regimes (namely, 1: gas laminar-liquid laminar, 2: gas laminar-liquid turbulent, 3: gas turbulent-liquid laminar, 4: gas turbulent-liquid turbulent). The newly developed correlation is expressed by a function of the two-phase Reynolds number, Retp , the two-phase viscosity number, Nμt p , and the gas quality, x, and was able to predict the two-phase frictional pressure drop with the mean absolute percentage error of 17.4%. The correlation demonstrated an excellent performance for predicting the two-phase frictional pressure drop in mini/micro single channels. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction In the development of the electronic devices, the power density increases rapidly. The thermal management of the high power electronic devices is the bottleneck problem for the development sustainably. There are two restrictions for the electronic equipment evolving: how to cool down devices efficiently and how to cool down devices in compact spaces. In comparison with conventional air-convective-flow cooling with fins, compact heat exchangers show excellent performances on solving these two problems. Compact heat exchangers are composed of mini/micro channels. Researchers have proposed the classifications of micro-channel and mini-channel. Mehendale et al. (20 0 0) defined that the microchannel and meso-channel ranges were from 1 to 100 μm and 100 μm to 1 mm, respectively. The range of the conventional channel diameter was defined as above 6 mm. In view of the singlephase and two-phase flow applications in compact heat exchangers, Kandlikar (2003) recommended that the ranges of micro-

∗

Corresponding author. E-mail address: [email protected] (T. Hibiki).

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2016.12.003 0301-9322/© 2016 Elsevier Ltd. All rights reserved.

channel and mini-channel diameters were from 10 to 200 μm and 200 μm to 3 mm, respectively. This definition was proposed based on the smallest channel dimension in the channels. This simple criterion is used in the present study. Since the channel dimension has a significant effect on the bubble motion in two-phase flow, Cornwell and Kew (1993) proposed a classification method based on the confinement number defined by:

 Co =

0.5

σ

g(ρl −ρg )

D

(1)

where D, σ , ρ l and ρ g are, respectively, the hydraulic diameter of the channel, the surface tension, the liquid density and the gas density. Observed heat transfer and flow characteristics suggest that the flow characteristics changed around Co = 0.5. Hence, they proposed Co = 0.5 as the threshold value between the microchannel and the conventional channel. Heat transfer coefficient, pressure drop, flow distributions and other parameters are employed for evaluating the capabilities of the compact heat exchanger. Numerous researchers have studied on the heat transfer characteristics of the compact heat exchanger. For micro-channel and mini-channel heat exchanger, they con-

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X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

Nomenclature Bo Bl C Co D e Fr f G g La MAPE ME MPE Nconf Nμ P PE Re RMSE RMSPE We x X z

Bond number Boiling number Chisholm parameter Confinement number Hydraulic diameter Error between experimental value and predicted value Froude number Friction factor Mass flow rate Gravity force Laplace number Mean absolute percentage error Mean error Mean percentage error Confinement number Viscosity number Pressure Percentage error Reynolds number Root mean square error Root mean square percentage error Weber number Vapor quality Martinelli parameter length

Subscripts crit Critical condition exp Experimental f Fluid g Gas l Liquid pre Predicted tp Two-phase flow Greek alphabet β Homogeneous void fraction ε Roughness μ Viscosity ρ Density σ Surface tension coefficient ∅2 Two-phase multiplier

ducted a lot of experiments, and developed numerous correlations for predicting the heat transfer coefficient in mini/micro channel heat exchangers. Qu and Mudawar (20 03a, 20 03b) studied the saturated flow boiling in micro-channel heat sink, and predicted the heat transfer coefficient. In their study, the heat transfer coefficient was predicted based on the annular flow pattern model. Their model captured the decreasing trend of the heat transfer coefficient at the low vapor quality range. Zhang et al. (2002) found that the region of the liquid-laminar and gas-turbulent was the general two-phase flow condition in the mini-channels. By considering the flow conditions with the factor, F, and the single-phase heat transfer coefficient, hsp , they developed a new correlation which showed a good agreement with their database. Lee and Mudawar (2005) investigated the heat transfer characteristics of R134a flow boiling in the micro-channel heat sink. They found that the heat transfer alteration was associated with the flow mechanisms at different vapor quality ranges. A heat transfer correlation was developed based on their experimental results.

In practical applications, micro-channel heat exchanger is usually used with mini pumps. Pressure drop in the micro-channel may also cause hydrodynamic instabilities which would deteriorate the heat exchanger performance. Therefore, it is important to predict the pressure drop in compact heat exchangers. In order to predict the frictional pressure drop in the compact heat exchangers, the frictional pressure drop prediction in single channels is the fundamental procedure. In the present study, existing frictional pressure drop correlations of two-phase flow in single channels are first reviewed. These correlations include the homogenous flow model and the separated flow model. A frictional pressure drop database of two-phase flow in mini/micro single channels is established with 1521 data points. The data points are collected from 12 literatures, including the adiabatic and diabatic systems. The working fluids include R22, R134a, R410A, R290, R744, ammonia, nitrogen, water, R245fa and propane. The diameter range is from 0.1 to 3 mm, and pressure drop ranges from 1.26 kPa/m to 2 MPa/m. 17 frictional pressure drop correlations for single channels including the homogenous flow model and the separated flow model are evaluated with the database. Comparing the correlations with the database, the existing two-phase frictional pressure drop correlations do not show satisfactory performances. A new correlation is developed in the present study. The newly developed correlation is expressed by a function of the two-phase Reynolds number, Retp , the two-phase viscosity number, Nμt p , and the gas quality, x, and was able to predict the two-phase frictional pressure drop with the mean absolute percentage error of 17.4%. The correlation demonstrates an excellent performance for predicting frictional pressure drop in mini/micro single channels.

2. Existing correlations 2.1. Homogenous flow model The homogeneous flow model assumes the gas and liquid phases as two-phase mixture with no velocity difference (slip ratio is 1). The properties of the two-phase mixture are calculated with the gas quality and the properties of liquid and gas fluids. The two-phase frictional pressure drop based on the homogenous flow model is expressed by:



dp dz

 = ft p tp

2G2 Dρt p

(2)

G is the mass flow rate, D is the hydraulic diameter, ρ tp is the density of the two-phase mixture, and ftp is the two-phase frictional factor. In the present study, the two-phase frictional pressure drop is calculated by the Churchill (1977) correlation with the two-phase flow Reynolds number as:

 ft p = 8

8 Ret p

1/12

12 +

1

(3)

(A + B)3/2

where

A=

⎧ ⎪ ⎨

⎢

2.457ln⎣ 

⎪ ⎩ 

B=

⎡

37530 Retp

7 Ret p

⎤⎫16 ⎪ ⎬ 1 ⎥ ⎦ 0 . 9 ⎪ + 0.27 ε ⎭

(4)

D

16 (5)

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

31

Table 1 Summary of two-phase mixture viscosity, μtp , used in homogeneous flow models for single channel. Reference Dukler et al. (1964) Beattie and Whalley (1982) Lin et al. (1991) Fourar and Bories (1995) Yan and Lin (1998)

Viscosity μtp

Conditions

μt p = ρt p [x( μρgg ) + (1 − x)( μρll )] xρl (1.5μl +μg ) 2 l μt p = μl − 2.5μl [ xρl +(x1ρ−x )ρg ] + [ xρl +(1−x )ρg ] μt p = [μg +x1μ.4l(μμgl −μg )]   √ μt p = ρt p (xvg + (1 − x)vl + 2 x(1 − x)vg vl ) = ρt ( xvg + (1 − x)vl )2 .1 ft p = 0.11Re−0 eq

Ret p = GμDi [(1 − xm ) + xm ( ρρgl ) l

0.5

1

=

x

μg

+

1−x

μl

(6)

R-12, Single tube, 0.66 mm,1.17 mm Insulated Air-water, single tube, Insulated R134a, Single tube, 2 mm, Heated

2.2. Separated flow model The separated flow model considers the velocity difference between gas and liquid phases in calculating the two-phase frictional pressure drop. An example of widely used separated flow models is the Lockhart and Martinelli (1949) correlation given by:



Cicchitti et al. (1959) carried out experiments in the thermal power station and defined the two-phase viscosity μtp as:

μt p = xμg + (1 − x )μl

Water-steam, Single tube, Insulated

]

In order to calculate the two-phase Reynolds number, Retp , twophase viscosity μtp , should be properly given. McAdams et al. (1942) defined μtp as:

μt

Conventional single tube, Insulated

(7)

In the prediction of the two-phase frictional pressure drop in a conventional single channel, several researchers have used different definitions of the two-phase viscosity, μtp . Dukler et al. (1964) collected approximately 15,0 0 0 data points, and tested 25 correlations, and then recommended the viscosity used in the twophase flow. Beattie and Whalley (1982) derived the two-phase flow viscosity based on the flow patterns of the bubbly flow and the annular flow. They defined the viscosities of the bubbly flow and the annular flow differently with the homogeneous void fraction, and then proposed the viscosities with a hybrid viscosity used in the homogenous flow model. By comparing the viscosity model with the database, their viscosity was more suitable for gravity dominated flows such as bubbly flow in the conventional horizontal or vertical tubes. In comparison with the frictional pressure drop in conventional channels, the two-phase frictional pressure drop in mini/micro channels shows some different characteristics. Fourar and Bories (1995) carried out experiments using an air-water flow in narrow gaps with the width of 1 mm, 0.54 mm, 0.4 mm and 0.18 mm. They defined the mixture friction factor and the mixture Reynolds number. Their viscosity was similar to the Dukler’s viscosity (Dukler et al., 1964), and could predict the two-phase frictional pressure drop well when Retp is below 40 0 0. Lin et al. (1991) used R12 as a working fluid, and measured the two-phase frictional pressure drop in single tubes with the diameters of 0.66 mm and 1.7 mm. In their experiments, they observed that a low velocity ratio occurred between the vapor flow and the liquid flow. Hence, they introduced the viscosity with the vapor quality, x, with an exponent, n, into the correlation. By fitting with the experimental data, the exponent to the vapor quality was determined to be 1.4. Yan and Lin (1998) investigated R134a flow boiling in a horizontal small circular pipe with the diameter of 2 mm. Based on their experimental data, a new two-phase flow friction factor correlation was proposed. The equivalent Reynolds number defined by Akers et al. (1958) was employed in their friction factor. Since the friction factor based on the assumption such as the single-phase flow like two-phase homogeneously mixed flow, their model may be categorized as the homogeneous flow model. Table 1 summarizes the two-phase mixture viscosity used in the homogeneous flow models for the conventional and mini/micro single channels discussed above.

dp dz





= φl2 tp

dp dz



(8) l

( ddzp )l is the frictional pressure drop per unit length of the singlephase flow and φl2 is the two-phase multiplier. The two-phase multiplier is given as (Chisholm, 1967):

φl2 = 1 +

C 1 + 2 X X

(9)

X is the Martinelli parameter defined by:

 dp  2

X =

dz l  dp  , dz g



dp dz

 = fl l

2G2 (1 − x )2 and D ρl



dp dz

 = fg g

2G2 x2 D ρg (10)

where fl and fg are the liquid friction factor and the gas friction factor. The friction factors are calculated by Churchill (1977) correlation in the present study, because it applies to laminar, transition and turbulent flow regimes. G, ρ , D and x are the mass flow rate, the density, the channel hydraulic diameter and the gas quality, respectively. The Chisholm’s parameter, C, in Eq. (10) is given depending on gas and liquid Reynolds numbers. In the single channel two-phase flow, various correlations of the Chisholm’s parameter, C, have been proposed to predict the twophase frictional pressure drops for different fluids, channel diameters, gas qualities and so on. Mishima and Hibiki (1996) studied the characteristics of an air-water flow in capillary tubes with the diameters from 1 to 4 mm. By comparing the relations between the Lockhart–Martinelli parameter, X, and the two-phase multiplier, φl2 , for different diameters, they found the Chisolm’s parameter, C, decreased with decreasing the tube diameter. They gave the Chisholm’s parameter as a function of the tube inner diameter. Lee and Lee (2001) carried out experiments with working fluids of an air-water. The test section gap was in the range from 0.4 to 4 mm. Based on their 305 experimental data points, they correlated the Chisholm’s parameter, C, as a function of the gap size and the gas and liquid flow rates. The correlation could predict the two-phase frictional pressure drop in narrow channels where the surface tension effect was significant. Hwang and Kim (2006) used R134a to investigate the two-phase frictional pressure drop characteristics in circular tubes with the inner diameters of 0.244, 0.430, and 0.792 mm. In their study, they found the Chisholm’s parameter, C, changed with the variation of the liquid Reynolds number, Rel , and the Martinelli parameter, X. Since the surface tension was important in the mini/micro channels, they correlated the Chisholm’s parameter, C, as a function of the liquid Reynolds number, Rel , the

32

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45 Table 2 Summary of the existing separated flow models for single channels formulated with Chisholm’s parameter C. Reference Mishima and Hibiki (1996)

Correlations C = 21(1 − e

λ=

Lee and Lee (2001)

Conditions

− 0.319D

)

Air-water, Vertical single tube, 1 mm-4 mm, Adiabatic Air-water, Gap (0.4 mm, 1 mm, 2 mm and 4 mm), Adiabatic

64 Re

C = Aλq ψ r Resl ,

λ = ρμl σl D , ψ = μσl j 2

Hwang and Kim (2006)

C = 0.227Re0l .452 X −0.32 Ncon f −0.82

Pamitran et al. (2009)

.433 Ret1p.23 C = 3 × 10−3 × Wet−0 p

Zhang et al. (2010)

C = 21(1 − e− La =

Field and Hrnjak (2011)

0.358 La

( g(ρ σ−ρg ) )0.5

R134a, Single tube, 0.244 mm, 0.43 mm, 0.792 mm Heated R22, R134a, R410A, R290, R744 Single tube. 0.5, 1.5, 3 mm Heated Single tube, Adiabatic

)

l

D

β

C = β1 Reg 2 ψ β3

ψ=

R134a, R410A, R290, R717 Single tube, 0.07 mm - 0.305 mm, Adiabatic

μl j σ

Flow regime

β1

β2

β3

Surface tension-dominated Wevs ≤ 11.0Wels 0.14 Inertial-dominated Wevs > 11.0Wels 0.14

1.008e−5 0.0146

1.4591 0.4794

−0.6428 −0.6888

Martinelli parameter, X, and the confinement number, Co. Pamitran et al. (2010) studied the characteristics of the flow pattern transitions and the pressure drop for R22, R134a, R410A, R290 and R744 systems in tubes with the inner diameters of 0.5, 1.5 and 3 mm. They experimentally observed that the frictional pressure drop was expressed as a function of the mass flux, the tube diameter, the surface tension, the density and the viscosity. Based on the regression analysis using 812 data points, they correlated the Chisholm’s parameter, C, as a function of the two-phase Weber number, Wetp , and the two-phase Reynolds number, Retp . Zhang et al. (2010) developed an extensive database based on 13 literatures and classified them into three groups. They are (1) adiabatic liquid-gas flow, (2) adiabatic liquid-vapor flow, and (3) boiling flow. They introduced a neural network technique to identify major parameters for correlating the Chisholm’s parameter, C, as a function of the non-dimensional Laplace length, La. The correlation showed good performances for the collected database. For liquid turbulent and gas turbulent flow, the important parameter is considered to be Reynolds number. Field and Hrnjak (2007) measured the twophase frictional pressure drop of R134a, R410A, R290 and R714 in a rectangular channel with hydraulic diameter of 148 μm. They introduced the parameter, ψ , which was the product of liquid viscous and liquid slug velocity divided by the surface tension. Weber number was used to give a critical condition to distinguish the surface-tension dominated flow from the inertia-dominated flow. Table 2 summarizes the existing separated flow models for single channels formulated based on Chisholm’s parameter, C. Some separated flow model-based-correlations do not use the Chisholm’s parameter, C. Chisholm (1973) proposed a pressure drop correlation for evaporating flow in channels by recasting Eq. (10). In his correlation, a coefficient, B, represented the parameter, C. The parameter was a function of the mass flow rate, G, and the physical property coefficient, . Friedel (1979) proposed that the two-phase multiplier should be a function of Froude number, Fr, liquid Weber number, function of gas quality, liquid and gas densities. Müller-Steinhagen and Heck (1986) found that the frictional pressure drop increased with increasing gas quality, while the gas quality reached to 0.85, the two-phase frictional pressure drop became similar to the single-phase frictional pressure drop. Based on the observation, they developed correlations as a function of the gas quality, x, the liquid-phase frictional pressure drop and the gas-phase frictional pressure drop together. The comparison of the correlation with 14 existing correlations and 9300 data points

demonstrated that the correlation had a good agreement with the data. Tran et al. (20 0 0) modified B and introduced by Chisholm (1973). Considering the channel size, the fluid physical properties, the mass flux, and the gas quality, Confinement number, Co, was replaced with B. The comparison of the correlation with their experimental data demonstrated its excellent performance. Yu et al. (2002) studied the frictional pressure drop of the boiling water in 2.98 mm tube. In their experiments, they found the two-phase frictional pressure drop had the stronger correlation with 1/X2 term than the other two terms in the Chisholm correlation. They replaced the three terms in the original two-phase multiplier with X − 1.9 . The correlation predicted the two-phase frictional pressure drop within the mean absolute error of 7% for their experimental data including laminar liquid and turbulent gas flow condition. Table 3 summarizes the existing separated flow model-basedcorrelations which do not use the Chisholm’s parameter, C.

3. Existing databases 1521 data points of the frictional pressure drop in mini/micro single channels were collected from literatures and summarized in Table 4. The database includes adiabatic and diabatic conditions. The ranges of the frictional pressure drop are from 4 kPa/m to 2 MPa/m. The diabatic database is composed of 10 fluid systems including refrigerants, water and carbon dioxide. The channel diameter ranges from 0.1 to 2.98 mm. The heat flux range is from 5 to 500 kW/m2 . The adiabatic two-phase flow database is composed of 5 fluid systems with diameters ranging from 0.108 to 0. 792 mm. The channel classification such as “mini channel” or “micro channel” is based on the definition proposed by Kandlikar (2003). All data are collected from the two-phase flow experiments in the mini/micro single channels. Comparing with the two-phase frictional pressure drop characteristics in multi channels, the characteristics of the two-phase frictional pressure drop in single channels show some difference. The design of parallel channels may cause pressure instabilities between the neighboring channels. The present study focus on the two-phase frictional pressure drop characteristics of two-phase flow in mini/micro single channels. Pamitran et al. (2010) observed that the two-phase frictional pressure drop increased with increasing the mass and heat flux and decreased with increasing the channel diameter. The twophase frictional pressure drop was affected by the fluid physical properties significantly. Yan and Lin (1998) also observed that the

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

(e)

(f)

33

Fig. 1. Comparison between existing correlations and collected database. (a) Dukler et al. (1964); (b) Friedel (1979); (c) Müller-Steinhagen and Heck (1986); (d) Lin et al. (1991); (e) Mishima and Hibiki (1996); (f) Yu et al. (2002); (g) Hwang and Kim (2006); (h) Pamitran et al. (2009); (i) Zhang et al., (2010).

34

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(g)

(h)

(i) Fig. 1. Continued

higher heat flux resulted in higher two-phase frictional pressure drop. Maqbool and Khodabandeh (2012) measured the ammonia two-phase frictional pressure drops in two tubes. The effects of the mass flux, saturation temperature and inner diameters on the two-phase frictional pressure drop were studied. The two-phase frictional pressure drop at the higher saturation temperature was lower than that at the low saturation temperature. They explained that this result was due to the decreased liquid viscosity at the higher temperature. Qi et al. (2007) observed the oscillation of the liquid nitrogen two-phase frictional pressure drop. They discussed that the major reason for the two-phase frictional pressure drop oscillations was the coalesced vapor at the outlet and the flash evaporation in the channel. Hwang and Kim (2007) pointed out that the geometries of the channels also have some effects on twophase frictional pressure drop due to the liquid film thickness and the distributions were different in rectangular or circular channels. Revellin and Thome (2007) classified their data into three groups in terms of two-phase Reynolds number. They neglected the laminar and transition flows, since the characteristics of the two-phase frictional pressure drop in these two flow regimes didn’t show any significant dependence on the Reynolds number.

According to the existing experiments carried out in single channels, flow velocity, heat flux, channel diameter and fluid properties have significant effects on the two-phase frictional pressure drop. The above brief review of the existing experimental data for mini/micro single channels suggests that all influential parameters should be properly considered for developing the two-phase frictional pressure drop correlation applying for two-phase flow in mini/micro single channels.

4. Performance evaluation of existing correlations using existing databases 4.1. Statistical parameters for evaluating existing correlations To evaluate the agreement between the existing correlations and the existing database, 5 statistical parameters including the scale (or value) dependent measures and the scale (or value) independent measures are adopted.

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

35

Table 3 Summary of the existing separated flow model for single channels formulated without Chisholm’s parameter, C. Reference

Correlations

Chisholm (1973)

Pt p

Pl

Conditions

= 1 + ( X − 1 )[ B x 2

0.875

ρ

)2−n ρg ( μμgl )n X 2 = ( 1−q q

0.875

(1 − x )

+x

1.75

]

Conventional single tube, Insulated

l

( dp ) = ( dp ) ∅2 , dz t p dz l lo

Friedel (1979)

∅2l = E +

Conventional single tube, Insulated

3.24F X F r 0.045 We0l .035

G2 gDρt2p 0.78

Fr =

F = x (1 − x)0.224 2 W el = Gσ ρD l

ρ f

E = (1 − x )2 + x2 ρlg fg l

μ μ X = ( ρρgl )0.91 ( μg )0.19 (1 − μg )0.7 l l

Müller-Steinhagen and Heck (1986)

( dp ) t p = E (1 − x )1/c + Bxc dL E = A + 2(B − A)G 2 2 ( dp ) l = fl 2ρGD = A, ( dp ) g = fg ρ2Gg D = B dz dz l

C=3 Tran et al. (20 0 0)

Conventional single and multi channels, 9300 data points

p

p 2 (

L )t p = ( L )l ∅l 0.875 ∅2l = 1 + (4.3X 2 − 1)[Cox0.875 (1 − x ) + x1.75 ]

R134a, R12, R113, Single tube, 2.46 mm, 2.92 mm, Insulated

[ g(ρ σ−ρ ) ] g l D (dp/dz )g (dp/dz )l

0.5

Co = X = 2

Yu et al. (2002)

Re0.1

X = 18.65( ρρGL )0.5 ( 1−x ) ReG0.5 x

 =X 2 FL

−1.9

Single tube, 2.98 mm, Heated

L

Table 4 Collected database of two-phase frictional pressure drop in mini/micro single channels. Reference

Geometry

Working fluid and operating condition

Data points

Yan and Lin (1998)

2 mm

160

Yu et al. (2002)

2.98 mm

Shuai et al. (2003)

2.67 mm, 0.8 mm

Hwang and Kim (2006)

0.244 mm 0.430 mm 0.792 mm 0.531 mm 0.834 mm 1.042 mm 1.931 mm 0.509 mm 0.790 mm

R134a Heat flux range: 5–20 kW/m2 Mass flux range: 50,100,150,200 kg/m2 s Saturation temperature range:5,15,31 ◦ C Water Heat flux range: 50–200 kW/m2 Mass flux range: 50–151 kg/m2 s Saturation temperature range:80 ◦ C Water Heat flux range: 10–120 kW/m2 Mass flux range: 10 0–70 0 kg/m2 s Saturation temperature range: R134a Adiabatic Mass flux range: 480–950 kg/m2 s Nitrogen heat flux range: 5.09–21.39 W/cm2 Mass flux range: 797.7–1743.1 kg/m2 s Saturated temperature range: 78.2–79.8 ◦ C R134a, R245fa Adiabatic Mass flux range: 350–20 0 0 kg/m2 s Saturation temperature range: 26, 30 and 35 ◦ C R134a, R410A, propane and ammonia adiabatic Mass flux range: 290,330,450,440 kg/m2 s Saturation temperature range: 22–25 ◦ C Water Heat flux range: 20 0–50 0 kW/m2 Mass flux range: 90,169,267 kg/m2 s R22, R134a, R410A, R290 and R744 Heat flux range: 5–40 kW/m2 Mass flux range: 50–600 kg/m2 s Saturation temperature range:0–15 ◦ C R245fa Heat flux range: 0–55 kW/m2 Mass flux range: 10 0–70 0 kg/m2 s Saturation temperature range: 31–41 ◦ C CO2 Heat flux range: 7.5–29.8 kW/m2 Mass flux range: 30 0–60 0 kg/m2 s Saturation temperature range:–40–0 ◦ C Ammonia Heat flux range: 15–355 kW/m2 Mass flux range: 10 0–50 0 kg/m2 s Saturation temperature range:23,33,43 ◦ C

Qi et al. (2007)

Revellin and Thome (2007)

Field and Hrnjak (2007)

0.148 mm

Huh and Kim (2007)

0.1 mm

Pamittran et al. (2010)

0.5 mm 1.5 mm 3 mm

Tibiriçá and Ribatski (2011)

2.32 mm

Wu et al., (2011)

1.42 mm

Maqbool et al., (2012)

1.7 mm 1.224 mm

230

150

122

30

150

57

39

74

48

280

180

36

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

Fig. 2. Relationship between Chisholm’s parameter, C, and two-phase Reynolds number, Retp . (a) liquid-turbulent and gas-turbulent flow condition; (b) liquid-turbulent and gas-laminar flow condition; (c) liquid-laminar and gas-turbulent flow condition; (d) liquid-laminar and gas-laminar flow condition.

The scale dependent measures involve the Mean Error (ME) and the Root Mean Square Error (RMSE). The definition of the error between the corresponding predicted value and the experimental value is expressed as ei :

ei = P r ei − Ex pi

(11)

The Mean Error (ME) and the Root Mean Square Error (RMSE) are defined as Eqs. (12) and (13), respectively: n 1 ME = ei n i=1

 RMSE =

n 1 2 ei n

(12)

0 . 5

Mean Absolute Percentage Error (MAPE) are, respectively, defined by Eqs. (15), (16), and (17). Percentage error (PE):

PEi =

ei Ex pi

(14)

Mean percentage error (MPE):

MPE =

n 1 P Ei n

(15)

i=1

Root mean square percentage error (RMSPE):



(13)

i=1

The scale independent measures are the Mean Percentage Error (MPE), the Root Mean Square Percentage Error (RMSPE), and the Mean Absolute Percentage Error (MAPE). With the definition of the Percentage Error (PE) in Eq. (14), the Mean Percentage Error (MPE), the Root Mean Square Percentage Error (RMSPE) and the

RMSPE =

n 1 2 PE n

0 . 5 (16)

i=1

Mean absolute percentage error (MAPE):

MAPE =

n 1 |PE | n i=1

(17)

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

37

Fig. 3. Relationship between common logarithm of (Cexp /Retp n ) and the gas quality, x. (a) liquid-turbulent and gas-turbulent flow condition; (b) liquid-turbulent and gaslaminar flow condition; (c) liquid-laminar and gas-turbulent flow condition; (d) liquid-laminar and gas-laminar flow condition.

4.2. Performance of existing correlations Table 5 compares 9 correlations with the MAPE smaller than 80% among 17 correlations with the corrected data. Among the existing correlations, the correlations of Pamitran et al. (2009), Mishima and Hibiki (1996), and Zhang et al. (2010) predict the frictional pressure drop within the MAPE of 50%, and the correlations are developed based on the Chisholm’s parameter, C. The RMSPEs of these three correlations are the smallest among the 9 correlations. Fig. 1 shows the comparison between the 9 correlations and the collected database. The abscissa presents the experimental twophase frictional pressure drop and the ordinate presents the predicted two-phase frictional pressure drop by the existing correlations. The broken lines in these figures indicate the error band of 30%. Correlations of Hwang and Kim (2006), Müller-Steinhagen and Heck (1986) and Yu et al. (2002) do not predict the two-phase frictional pressure drop data with a proper tendency. The detailed discussion is given in Appendix A.

Among the nine correlations, homogenous correlations (Dukler et al., 1964; Lin et al., 1991) predict the two-phase frictional pressure drop with the MAPEs of 56.3% and 62.0%, respectively. The separated flow models without the Chisholm’s parameter, C, (Friedel, 1979; Müller-Steinhagen and Heck, 1986; Yu et al., 2002) predict the two-phase frictional pressure drop with the MAPEs of 63.8%, 76.0% and 70.5%, respectively, while the MAPEs for the correlations with the Chisholm’s parameter, C, are smaller than 50%. The above performance evaluation of the existing correlations suggests that the separated flow models with the Chisholm’s parameter, C, can predict the two-phase frictional pressure drop better than the homogeneous flow models. 5. New correlation development and its performance analysis 5.1. New correlations development The performance analysis of the existing correlations indicates a promising performance of the separated flow model-based-

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X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

Fig. 4. Relationship between common logarithm of (Cexp / Retp n xm ) and two-phase viscosity number, Nμt p . (a) liquid-turbulent and gas-turbulent flow condition; (b) liquidturbulent and gas-laminar flow condition; (c) liquid-laminar and gas-turbulent flow condition; (d) liquid-laminar and gas-laminar flow condition.

Table 5 Comparison between existing correlations and collected database. Correlations

ME [kPa]

RMSE [kPa]

MAPE [%]

MPE [%]

RMSPE [%]

Dukler et al. (1964) Friedel (1979) Müller-Steinhagen and Heck (1986) Lin et al. (1991) Mishima and Hibiki (1996) Yu et al. (2002) Hwang and Kim (2006) Pamitran et al. (2009) Zhang et al. (2010) Newly developed correlation

−2.80 −7.79 6.62 −9.86 −4.14 5.38 −6.46 −6.96 −4.52 −2.12

16.6 30.9 41.6 49.4 24.8 57.9 37.9 15.3 26.6 11.4

56.3 63.8 76.0 62.0 43.7 70.5 48.9 43.2 44.1 17.4

1.84 −9.69 −53.3 7.60 −2.18 −34.9 −6.39 26.6 −4.36 −5.52

126 80.7 93.5 183 65.6 89.3 60.5 53.5 59.8 23.6

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

39

2 2 Fig. 5. Comparison between predicted two-phase multiplier φ pre and experimental two-phase multiplier φexp .

correlations with the Chisholm’s parameter, C. In order to identify the dependence of key non-dimensional parameters used in the existing correlations on the Chisholm’s parameter, C, some sensitivity analyzes have been done at the first stage of the new correlation development. The sensitivity analyzes suggest that the correlation to be developed should properly consider the effects of channel diameter, viscosity and the surface tension on the twophase frictional pressure drop and the regimes of liquid and gas flow, and identify that the two-phase Reynolds number, Retp , the gas quality, x, and the two-phase viscosity number, Nμt p , are the key parameters to correlate the Chisholm’s parameter, C. The flow conditions are classified into (a) liquid-turbulent and gas-turbulent, (b) liquid-turbulent and gas-laminar, (c) liquid-laminar and gaslaminar, and (d) liquid-laminar and gas-laminar. The thresholds of liquid Reynolds number, Rel , and gas Reynolds number, Reg , between laminar and turbulent flows are proposed by Chisholm (1967) and are listed in Table 6. The similar criteria were also used

by other researchers such as Huh and Kim (2007), Kandlikar et al. (20 03), Li (20 03), Maynes and Webb (2002) and Sharp and Adrian (2004). A total of 1135 data is classified into four flow regimes based on the thresholds of gas and liquid Reynolds numbers, and a total of 386 data is located in the transition regions. As briefly discussed above, in order to find key non-dimensional parameters to correlate the Chisholm’s parameter, C, the dependence of several parameters on the Chisholm’s parameter is examined, and the analysis indicates that the two-phase Reynolds number, Retp , the gas quality, x, and the two-phase viscosity number, Nμt p , to be the key parameters for all flow conditions (a) – (d) . The two-phase Reynolds number, gas quality and two-phase viscous number represent the kinematic effect, thermal effect and physical property effect on the two-phase frictional pressure drop, respectively. Figs. 2, 3 and 4 compare the experimentally obtained Chisholm’s parameter with Retp , x, Nμt p , respectively. The two-

40

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45 Table 6 Thresholds of Reynolds number between laminar and turbulent flows. Flow regimes

Liquid Reynolds number

Vapor Reynolds number

Data number

Liquid Liquid Liquid Liquid

Rel > 20 0 0 Rel < 10 0 0 Rel > 20 0 0 Rel < 10 0 0

Reg > 20 0 0 Reg > 20 0 0 Reg < 10 0 0 Reg < 10 0 0

356 675 19 85

turbulent and gas turbulent laminar and gas turbulent turbulent and gas laminar laminar and gas laminar

Table 7 Summary of newly developed correlation. Models and parameters

Definitions and correlations to be used

Separated flow model Two-phase multiplier

( dp ) = φl2 ( dp ) dz t p dz l φl2 = 1 + XC + X12

Martinelli parameter

X2 =

( ddzp )l ( ddzp )g

) and ( dp ( dp ) = fl 2G D(1ρ−x ) = fg 2DGρxg dz l dz g l 2

Chisholm’s parameter, C

2

2 2

Liquid-turbulent and gas-turbulent C = 6.28Nμ0tp.78 Re0tp.67 x0.32 (Rel > 20 0 0 and Reg > 20 0 0) Liquid-turbulent and gas-laminar C = 1.54Nμ0tp.14 Re0tp.52 x0.42 (Rel > 20 0 0 and Reg < 10 0 0)Liquid-laminar and gas-turbulent C = 245.5Nμ0tp.75 Re0tp.35 x0.54 (Rel 10 0 0 and Reg 20 0 0) Liquid-laminar and gas-laminar C = 41.7Nμ0tp.66 Re0tp.42 x0.21 (Rel < 10 0 0 and Reg < 10 0 0 )

Table 8 Statistical analysis for the prediction of the frictional pressure drop for each flow regime and transition flow regime. Flow regimes

ME [kPa]

RMSE [kPa]

MAPE [%]

MPE [%]

RMSPE [%]

Liquid turbulent and gas turbulent Liquid turbulent and gas laminar Liquid laminar and gas turbulent Liquid laminar and gas laminar Transition flow regions

2.80 −2.37 −5.21 1.31 −2.43

7.94 9.68 15.6 6.52 8.24

17.2 18.1 19.2 9.07 16.1

7.60 −8.31 −13.4 2.38 −5.43

19.8 22.4 29.7 18.7 17.5

phase Reynolds number, Retp , is defined as:

Ret p =

Gd

(18)

μt p

The two-phase viscosity, μtp , used in the two-phase Reynolds number is given by:

1

μt p

=

1−x

μl

+

x

μg

(19)

The common definition of the viscosity number (Ishii and Hibiki, 2010) is based on the single-phase viscosity and density, but in the present study, the two-phase viscosity number, Nμt p , is introduced as:

Nμt p =



ρt p σ

μt p  σ 0 . 5

(20)

g ρ

The two-phase viscosity in Eq. (20) is given by Eq. (19). The two-phase mixture density, ρ tp , is given by:

ρt p = xρg + (1 − x )ρl

(21)

Fig. 2 shows the relationship between the Chisholm’s parameter, C, and the two-phase Reynolds number, Retp . The abscissa is the common logarithm of two-phase Reynolds number. The ordinate is the common logarithm of experimentally determined Chisholm’s parameter, C. The slopes shown in Fig. 2 represent the exponent to the two-phase Reynolds number in the Chisholm’s parameter, C. The exponents to the two-phase Reynolds number for

the four flow conditions (a) – (d) are 0.67, 0.52, 0.35 and 0.42, respectively. Fig. 3 shows the relationship between (Cexp /Retp n ) and the gas quality x. The abscissa is the common logarithm of gas quality, x. The ordinate is the common logarithm of (Cexp / Retp n ) where n is the exponent to the two-phase Reynolds number shown in Fig. 2. The slopes shown in Fig. 3 represent the exponent to the gas quality in the Chisholm’s parameter, C. The exponents to gas quality for the four flow conditions (a) – (d) are 0.32, 0.42, 0.54 and 0.21, respectively. Fig. 4 shows the relationship between (Cexp / Retp n xm ) and twophase viscosity number, Nμt p . The abscissa is the common logarithm of the two-phase viscosity number, Nμt p . The ordinate is the common logarithm of (Cexp / Retp n xm ) where n and m are, respectively, the exponents to the two-phase Reynolds number and gas quality. The slopes shown in Fig. 4 represent the exponent to the two-phase viscosity number, Nμt p , in the Chisholm’s parameter, C. The exponents to the two-phase viscosity number for the four flow conditions (a) – (d) are 0.78, 0.14, 0.75 and 0.66, respectively. The coefficients in the Chisholm’s parameter, C, are calculated based on these exponents determined in Figs. 2–4 and collected data. Table 7 lists the correlations to calculate the Chisholm’s parameters for the four flow conditions (a) – (d). The newly developed correlation is applicable to 10 fluid systems: R22, R134a, R410A, R290, R744, ammonia, nitrogen, R245fa, Propane and water, pipe diameter ranging from 0.1 to 3 mm, twophase Reynolds number ranging from 100 to 35,0 0 0, gas quality

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

Fig. 6. Comparison between predicted two-phase frictional pressure drop and experimental two-phase frictional pressure drop.

ranging from 0.01 to 0.9 and two-phase viscosity number ranging from 0.0 0 063 to 0.01. In the transition liquid and gas Reynolds number regions, parameter, C, can be obtained by a simple interpolation scheme detailed in Appendix B.

41

or without Chisholm’s parameter, C. The performance evaluation of the correlations was performed by comparing them with the collected database. The evaluation results indicated that 9 out of 17 correlations had the MAPE within 80%. The correlations formulated with the Chisholm’s parameter, C, showed a promising performance. Some preliminary analyzes indicated that a correlation to be developed should consider tube dimension, fluid properties and flow velocity. A correlation to predict the two-phase frictional pressure drop in mini/micro single channels was developed based on the separate flow model with the Chisholm’s parameter. The correlation of the Chisholm’s parameter was developed for four different flow conditions. They are (a) liquid-turbulent and gas-turbulent, (b) liquidturbulent and gas-laminar, (c) liquid-laminar and gas-laminar, and (d) liquid-laminar and gas-laminar. 1135 data points in the four flow conditions were compared with the new correlations. The Chisholm’s parameter was well-correlated by two-phase Reynolds number, Retp , gas quality, x, and the two-phase viscosity number, Nμt p . The newly developed correlation showed excellent performance for predicting the two-phase frictional pressure drop. The MAPEs for the two-phase frictional pressure drop are 17.2%, 19.2%, 18.1% and 9.07% for turbulent liquid-turbulent gas, turbulent liquid-laminar gas, laminar liquid-turbulent gas and laminar liquidlaminar gas, respectively. 386 data points in the transition conditions were compared with the new correlations with the aid of a simple interpolation scheme. The interpolation scheme showed excellent performance for predicting the two-phase frictional pressure drop. The MAPE of the two-phase frictional pressure drop prediction for 386 data is 16.1%. The MAPE of the two-phase frictional pressure drop prediction for 1521 data including the data taken in the transition regions is 17.4%.

5.2. Performance analysis of newly developed correlation 2 , with Fig. 5 compares the predicted two-phase multiplier, φ pre 2 the experimental two-phase multiplier, φexp . Fig. 5 demonstrates an excellent performance of the newly developed correlation for the four flow conditions. The MAPE of the correlation for all database is smaller than 20%. Fig. 6 compares the predicted two-phase frictional pressure drop and the experimental two-phase frictional pressure drop. Fig. 6 also demonstrates an excellent performance of the newly developed correlation for the four flow conditions. The MAPE, RMSPE and MPE are, respectively, 17.2%, 15.3% and 7.12%. Table 8 summarizes the ME, RMSE, MPE, RMSPE and MAPE for the prediction of the two-phase frictional pressure drop for each flow regime and transition flow regime. As shown in Table 5, the MAPE of the twophase frictional pressure drop prediction for 1521 data including the data taken in the transition regions is 17.4%. The prediction accuracy comparison between the correlations demonstrates that the newly developed correlation significantly improves the prediction accuracy of the two-phase frictional pressure drop.

6. Conclusions In order to study the two-phase frictional pressure drop in mini/micro single channels, 1521 data points were collected from 12 literatures. The database included adiabatic and diabatic systems composed of 10 different fluids. The diameter range in the database was from 0.1 to 3 mm, which covered the range of the mini/micro channels. The present study reviewed 17 existing correlations developed with two-phase frictional pressure drop data. The reviewed correlations included the homogenous flow model-based-correlations and the separated flow model-based-correlations formulated with

Acknowledgments This work was performed when one of the authors (Li) studied at Purdue University as a visiting student. Beijing Jiaotong University provided the financial support for her travel expense and living cost.

Appendix A. Additional explanation of correlation performance This Appendix will provide the additional explanation of the conclusion such that correlations of Hwang and Kim (2006), Müller-Steinhagen and Heck (1986) and Yu et al. (2002) do not predict the data with a proper tendency. The performance of a correlation is expressed by the ratio of predicted two-phase frictional pressure drop to measured two-phase frictional pressure drop, Ppre / Pexp . Although the correlation performance depends on various non-dimensional parameters, the performance ratio,

Ppre / Pexp , is plotted against the two-phase Reynolds number, Retp , as an example. If a correlation properly models the dependence of the two-phase frictional pressure drop on Retp , the performance ratio should randomly scatter around Ppre / Pexp = 1. However, as shown in Fig. A.1(c), (f) and (g), the performance ratios for the correlations of Hwang and Kim (2006), MüllerSteinhagen and Heck (1986) and Yu et al. (2002) have a clear negative slope against Retp . This demonstrates that the correlations of Hwang and Kim (2006), Müller-Steinhagen and Heck (1986) and Yu et al. (2002) do not predict the data with the proper tendency.

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X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(a)

(b)

(c)

(d)

(e)

(f)

Fig. A.1. Comparisons between Ppre / Pexp and Retp of existing correlations. (a) Dukler et al. (1964); (b) Friedel (1979); (c) Müller-Steinhagen and Heck (1986); (d) Lin et al. (1991); (e) Mishima and Hibiki (1996); (f) Yu et al. (2002); (g) Hwang and Kim (2006); (h) Pamitran et al. (2009); (i) Zhang et al. (2010).

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

(g)

(h)

(i) Fig. A.1. Continued

Appendix B. Interpolation scheme to calculate Chisholm parameter in transition regions This Appendix will provide a simple interpolation scheme to calculate Chisholm parameter in the transition region. Fig. B.1 shows the schematic diagram of the four flow regions where the newly developed correlations are available and five transition regions classified by Rel and Reg . Interpolation scheme for region (1):

T C=







Il − Il1 CLL + IlT − Il2 CT L IlT

(B.1)

where

Il1 = l og10 Rel − l og10 10 0 0

(B.2)

Il2 = log10 20 0 0 − log10 Rel

(B.3) Fig. B.1. Regions classified by Rel and Reg .

IlT

= log10 20 0 0 − log10 10 0 0

(B.4)

43

44

X. Li, T. Hibiki / International Journal of Multiphase Flow 90 (2017) 29–45

Fig. B.2. Comparison between predicted two-phase frictional pressure drop and experimental two-phase frictional pressure drop for the transition regions (386 data points).

Interpolation scheme for region (2):

T







Il − Il1 CLT + IlT − Il2 CT T

C=

IlT

(B.5)

Interpolation scheme for region (3):

T







Ig − Ig1 CLL + IgT − Ig2 CLT

C=

IgT

(B.6)

where

Ig1 = l og10 Reg − l og10 10 0 0

(B.7)

Ig2 = log10 20 0 0 − log10 Reg

(B.8)

Interpolation scheme for region (4):

T







Ig − Ig1 CT L + IgT − Ig2 CT T

C=

IgT

(B.9)

Interpolation scheme for region (5):

C=

(IlT − Il1 )C + (IlT − Il2 )C IlT

where

C = C =

T







Ig − Ig1 CLL + IgT − Ig2 CLT IgT

(IgT − Ig1 )CT L + (IgT − Ig2 )CT T IgT

(B.10)

(B.11)

(B.12)

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