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Two-phase frictional pressure drop in small rectangular channels
Experimental Thermal and Fluid Science 32 (2007) 60–66 www.elsevier.com/locate/etfs
Two-phase frictional pressure drop in small rectangular channels Ing Youn Chen a
a,*
, Yi-Min Chen a, Jane-Sunn Liaw b, Chi-Chuan Wang
b
Mechanical Engineering Department, National Yunlin University of Science and Technology, Yunlin 640, Taiwan b Energy and Environment Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan Received 26 July 2006; received in revised form 4 January 2007; accepted 4 January 2007
Abstract This study presents single-phase and two-phase frictional pressure drop data in three rectangular channels with channel width of 3, 6 and 9 mm, and a fixed gap of 3 mm. Adiabatic flows of air–water mixtures are tested at mass flux of 100–700 kg/m2 s and gas quality ranging from 0.001 to 0.8. The single-phase friction factors agree satisfactorily with the predictions of laminar and turbulent flows. The two-phase frictional pressure drop data are compared to predictions of homogeneous and Chisholm method, Wambsganss and Ide– Fukano correlations. It is found that none of the existing methods or correlations can predict the two-phase pressure gradient in rectangular channels satisfactorily. A modified C factor of Chisholm method considering the effect of aspect ratio is proposed from the empirical fit with the data sets of Wambsganss et al., Ide–Fukano, and this study. The corresponding mean deviations of the proposed correlation against the datasets are 24.99%, 10.83% and 10.73%, respectively. This correlation is applicable in wide rages of mass flux (50 < G < 700 kg/m2 s), gas quality (0.001 < x < 0.95), Martinelli parameter (0.05 < X < 20) and aspect ratio (0.1 < A < 1.0). 2007 Elsevier Inc. All rights reserved. Keywords: Two-phase frictional pressure drop; Rectangular channel; Two-phase friction multiplier; C factor of Chisholm type correlation
1. Introduction Recently, the advance of the high performance electronic chips and the microelectronic devices results in a great demand in developing efficient heat removal techniques for dissipating increasingly large heat flux from compact systems. One of the simplest arrangements for the heat removal is using the liquid flow and convection heat transfer with or without phase change in small rectangular channels. In particular, plate-fin compact heat exchanger involves rectangular channels with hydraulic diameter less than 5 mm and mass flux less than 350 kg/ m2 s [1]. Also, flow in small rectangular channels is an integral part of the design of compact heat exchangers in boiling or condensation in space, aircraft, and other applications [2]. Since two-phase heat transfer is intimately interrelated to pressure drop, the higher pressure drop *
Corresponding author. Tel.: +886 5 5342601x4137; fax: +886 5 5312062. E-mail address: [email protected] (I.Y. Chen). 0894-1777/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2007.01.005
would increase the pumping power and also affect the performance of cooling systems. As a consequence, the twophase pressure drop in small rectangular channels is very important for the design of compact heat exchanger. In predicting the frictional two-phase pressure drop in any channel, it is necessary to have a reasonable method to estimate the friction factor of sing-phase flow in the same channel. For estimation the frictional performance of a non-circular tube, it is a common practice to adopt the well-known hydraulic diameter concept by substituting the diameter of the circular tubes with the hydraulic diameter (Dh) in the non-circular tubes. Recently, several investigations, Wambsganss et al. [1], Mishima et al. [3] and Lee and Lee [4], reported that the turbulent friction factor for rectangular channels agreed well with the predictions by the Blassius equation: dP =dz ¼ 2f qu2 =Dh ;
f ¼ 0:079Re0:25
ð1Þ
f is the Fanning friction factor. The Fanning friction factor for the laminar flow through the rectangular channels proposed by Harnett and Kostic [5] is given as:
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61
Nomenclature A dP/dz C D Dh f g G j P DP Re ReL ReG ReLO u
aspect ratio, gap (S)/width (W), 0 < A < 1 frictional pressure gradient (Pa/m) Chisholm’s factor internal diameter of circular tube (m) hydraulic diameter (m) friction factor gravity (m/s2) total mass flux (kg/m2 s) superficial velocity (m/s) pressure (Pa) pressure drop across the straight test section (Pa) Reynolds number (quDh/l) Reynolds number based on liquid single-phase flow (qLjLDh/lL) Reynolds number based on gas single-phase flow (qGjGDh/lG) liquid Reynolds number based on the total mass flux (GDh/lL) mean axial velocity (m/s)
f ¼ 24ð1 1:3553A þ 1:946A2 1:7012A3 þ 0:9564A4 0:2537A5 Þ=Re
ð2Þ
This simplified polynomial equation fits the exact analytical solution with an accuracy of ±0.05%. Where A = S/W = channel gap (or height)/channel width represents the aspect ratio of the rectangular section, 0 < A < 1.0. The product of f and Re is seen to decrease with the rise of A. Concerning the frictional pressure loss for two-phase flow, Lockhart and Martinelli [6] proposed the first predictive method for estimation of the frictional two-phase pressure gradient. Lockhart and Martinelli put forward a Martinelli parameter, X, which is based on the ratio of the single-phase frictional pressure drops calculated via assuming liquid-phase flow alone to gas-phase flow alone, i.e., X2 ¼
dP L =dz dP G =dz
ð3Þ
dP 2-phase =dz ; dP G =dz
UG q
rectangular channel gap or height (mm) gas quality rectangular channel width (mm) Martinelli parameter axial flow direction (m) void fraction liquid holdup viscosity (N s/m2) two-phase frictional multiplier based on liquid flow alone two-phase frictional multiplier based on gas flow alone density (kg/m3)
Subscripts cal calculation or prediction exp experimental measurement G gas-phase L liquid-phase 2-phase two-phase flow had approximated the relationships of U2G and U2L versus X by the expressions: U2L ¼ 1 þ C=X þ 1=X 2 ;
U2L ¼
dP 2-phase =dz dP L =dz
ð4Þ
where dP2-phase/dz is the frictional two-phase flow pressure gradient. Using the two-phase pressure drop data in adiabatic horizontal tubes, they proposed plots of UG and UL versus X. Their data indicated that the multipliers are a function of the Martinelli parameter alone. Chisholm [7]
U2G ¼ 1 þ CX þ X 2
ð5Þ
where C denotes Chisholm’s factor and its value depends on whether the liquid and gas flows are laminar or turbulent. For example, when both phases are laminar or turbulent, C = 5 or 20, respectively. However, the Chisholm’s C factor had been empirically obtained that it is not a constant by the liquid and gas flow conditions. C factor was reported as a function of Martinelli parameter, total mass flux, tube diameter, and other physical properties. Mishima et al. [3] conducted two-phase flow in narrow rectangular ducts with width of 40 mm and gaps of 1.0, 2.4 and 5.0 mm (A = 0.025–0.125). The two-phase friction loss was based on the Chisholm’s equation [7] by modifying the Chisholm’s constant C to account for the effect of hydraulic diameter Dh. C ¼ 21ð1 e0:27Dh Þ
dPL/dz and dPG/dz are the liquid and gas frictional pressure gradients if each phase flows alone in the same pipe with its mass flow rate, respectively. They also defined the two-phase multipliers, U2G and U2L : U2G ¼
S x W X z a g l UL
ð6Þ
Similar tests were conducted by Mishima and Hibiki [8] for circular tubes with D = 1–4 mm. Based on Eq. (6), Mishima and Hibiki [8] then proposed a new C factor equation for vertical and round tubes, as well as rectangular channels. C ¼ 21ð1 e0:319Dh Þ
ð7Þ
Ide and Matsumura [9] experimentally investigated the effects of aspect ratio, hydraulic diameter and inclination angle on the two-phase frictional pressure drop for 10 rectangular channels with aspect ratio from 0.025 to 1.0. For horizontal arrangement, UL was found to be higher in the transverse case (gap < width) than in the longitudinal
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case (gap > width) at the same value of Martinelli parameter, X. This fact becomes more obvious at smaller X in the channel having the larger aspect ratio. The gas and liquid flows in the longitudinal case with larger aspect ratio could easily lead to more separate flow pattern by the gravity than that of the transverse case. For vertical arrangement, UL is higher for the channel with the largest aspect ratio over the entire range of X. The two-phase frictional pressure drop data were compared with the predictions of Chisholm method [7] with C = 21 and Akagawa’s correlation [10], i.e. U2L ¼ ð1 aÞb
ð8Þ
where a is the void fraction and b depends on the inclination angle and the tube surface roughness, taking b = 1.4 and 1.5 for the horizontal and vertical cases, respectively. However, neither the Chisholm’s method [7] nor Akagawa’s correlation [10] could accurately predict their data. Based on a separated flow model, Ide and Matsumura [9] proposed a two-phase frictional pressure drop correlation by taking into account the effects of the inclination angle of the channel and liquid Reynolds number. 0:625
UL ¼ BX ½ðA þ 1ÞðA þ aÞ=ðA þ 2 2aÞ
1:5
½a=ð1 aÞ
ð9Þ where B is 0.03 in horizontal channels and 0.045 in vertical channels. This correlation gave an accuracy about ±30% against their data. Wambsganss et al. [1] measured two-phase frictional pressure gradients in a small horizontal channel with a cross section of 19.05 · 3.18 mm (A = 0.167) for application to plate-fin compact heat exchangers. The data were compared with Chisholm method [7] and Fridel correlation [11]. However, both predictions did not adequately correlate the data. A modified equation for the Chisholm’s factor C was proposed for qualities (x) being larger than those occurred at the transition from bubbly to slug flow. The transition at x 0.002 was visually observed in their study. The proposed C equation is given as: C ¼ functionðX ; ReLO Þ ¼ aX b
ð10Þ
where a = 2.44 + 0.00939ReLO, b = 0.938 + 0.000432 ReLO, and ReLO = GDh/lL. This C correlation is applicable for ReLO < 2200 and 0.05 < X < 1.0. The correlation gave an average error of less than ±19% for 0.05 < x < 0.95 and 100 < G < 400 kg/m2 s. For G > 400 kg/m2 s, the Chisholm correlation [7] with C = 21 is recommended. Since this correlation was developed from air–water flowing in a single narrow channel (A = 0.167), orientated at horizontal transverse with the wide side horizontal (HT) and horizontal longitudinal with the wide side vertical (HL) positions, more data with smaller cross section areas, different aspect ratios and different fluids are needed to check the applicability of this correlation. More recently, Ide and Fukano [12] measured the mean liquid holdup (g) and frictional pressure drop for air–water
two-phase flow in a capillary rectangular channel (9.9 · 1.1 mm, A = 0.11). The test ranges of air and water superficial velocities are 1.5–8 m/s and 0.1–0.8 m/s, respectively. They described laminar flow (V) or turbulent flow (T) of two-phase intermittent flow in the small rectangular tube by using the liquid and gas Reynolds numbers in single-phase water flow (ReL = qLjLDh/lL) and single-phase gas flow (ReG = qGjGDh/lG), respectively, where jG andjL are the superficial gas and liquid velocities. The transition Reynolds number from laminar flow to turbulent flow was taken as 2400 experimentally. By considering the effect of aspect ratio, new correlations of two-phase frictional pressure drop for the capillary rectangular channel were proposed. For the laminar region: UL ¼ 0:2485A0:355 fjL =ðgDh Þ0:5 g0:233 ReL
3=8
ð11Þ
The above equation was only used for few data included in the laminar region, while most of the data were located in the turbulent region. For the turbulent region: 3=8
UL ¼ 0:848A0:145 fðjG þ jL Þ=jL g0:425 ReL
ð12Þ
The comparison of Ide and Fukano experimental data [12] with the calculated values UL of Eq. (12) shows that all the data are within ±15% of the predictions. From the above review, almost all the proposed correlations for the two-phase frictional pressure drop in rectangular channels were based on the data with very small aspect ratio (A < 0.2). It is unclear whether the previous correlations can be extended outside their applicable range. In this regard, it is the purpose of this study to extend the available database pertaining to rectangular cross section with higher aspect ratios. In the present study, the experiments of the frictional air and water two-phase flow in three rectangular channels are conducted with channel width, W = 3, 6 and 9 mm, at a fixed gap (S) of 3 mm (A = S/W = 1, 0.666 and 0.333). The applicability of previous correlations is verified. The effects of aspect ratio, gas quality and mass flow rate on the related parameters are investigated. 2. Experiments Three rectangular test sections made of transparent acrylic resin have cross sections of 3.0 · 3.0 mm, 3.0 · 6.0 mm and 3.0 · 9.0 mm having hydraulic diameters of 3.0, 4.0 and 4.5 mm, respectively. The inlet and outlet of the test rectangular sections are connected with a 3.0 mm internal diameter circular tube. A differential pressure transducer is used to measure the pressure drop across a 300 mm length of the rectangular channel as shown in Fig. 1. The upstream and downstream lengths of the test section are 150 mm and 90 mm, respectively. The total air and water mass flux (G) is ranged from 100 s to 700 kg/ m2 s and the quality (x) is changed from 0.001 to 0.8.
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
63
and ranges from 3.2% to 11.5% for U2L . The highest uncertainties were associated with the lowest Reynolds number. 3. Results and discussion
The tests are first conducted with single-phase water flow to obtain the friction factor data in the rectangular test section. The friction factor data for the single-phase flow will be used to check the applicability of the instrumentations and the reliability of the friction factor correlations, Eqs. (1) and (2). Adiabatic flows of air–water mixtures at 25 C and near atmospheric pressure are tested in three rectangular channels at horizontal and longitudinal (HL) orientation. Pressure fluctuations depending on the flow pattern are inherent in two-phase flow. The pressure drops are calculated from the time-average mean differential-pressure measurements. The test rig is designed to conduct tests with air–water mixtures as shown in Fig. 1. Air is supplied from an aircompressor and then stored in a compressed-air storage tank. Airflow through a pressure reducer and, depending on the mass flux range, is measured by three Aalborg mass flow meters for different ranges of flow rate. The water flow loop consists of a variable speed gear pump that delivers water. The mixer is designed to provide better uniformity of the air–water mixture stream. At the upstream of the test section, there is a set of quick closing valves for measuring the average volumetric void fraction of the two-phase flow. A very accurate Yokogawa magnetic flow meter is installed at the downstream of the gear pump. The accuracy of the air and water flow meters is within ±0.2% of the test span. The pressure drops of the air–water mixtures are measured by a Yokogawa EJ110 differential pressure transducer having an adjustable span of 1300– 13,000 Pa. Resolution of this pressure differential transducer is 0.3% of the measurements. Leaving the test section, the air–water mixture is separated by an open water tank in which the air is vented and the water is recirculated. The air and water temperatures are measured by resistance temperature device (Pt 100 X) having a calibrated accuracy of 0.1 K (calibrated by Hewlett–Packard quartz thermometer probe with quartz thermometer, model 18111A and 2804A). Uncertainties in the reported values of the friction factor f and UL are estimated by the method suggested by Moffat [13]. The uncertainties of f are from 1.3% to 6.4%
f = 14.22/Re f = 15.55/Re f = 17.09/Re f = 0.0791/Re0.25
-1
10
f
Fig. 1. Schematic illustration of experimental apparatus.
For single-phase tests, water is used as the working fluid for measuring the friction factors. The test results of friction factor data are plotted in Fig. 2. The base lines are the Fanning friction factor (f = 0.0791Re0.25) for turbulent flow and the predictions for laminar flow with the channel dimensions of 3.0 · 3.0 mm, 3.0 · 6.0 mm and 3.0 · 9.0 mm (A = 1.0, 0.66 and 0.33) with f = 14.22/Re, 15.55/Re and 17.09/Re, respectively. As seen, f is decreased with the rise of A. The f data agree favorably with the base lines. The good agreement shown in Fig. 2 substantiates the accuracy of the instrumentation and the experimental apparatus. For predicting the two-phase frictional pressure drops in small rectangular channels, the present study was intended to use Ide and Matsumura’s correlation [9], Eq. (9), and the existed void fraction correlations to calculate UL. For verifying the applicability of the available a correlations, predictions of homogeneous method and correlations of Premoli [14], Baroczy [15], Yashr et al. [16] and Smith [17] are made to compare the void fraction data obtained from two quick-closing solenoid valves as shown in Fig. 1. The measured void fraction data versus predictions are quite scattering and do not have a consistent trend with the mass flux. The homogeneous method is greatly over predicted all the data. For x < 0.01, the correlations of Premoli [14] and Smith [17] have a fair agreement with the data whereas the predictions of Baroczy [15] and Yashar et al. [16] are more close to the data when x > 0.01. Since the predictions can not give a good agreement to the scattering void fraction data, Eq. (9) is not able to utilize in this study. More accurate method for measuring void fraction is needed.
-2
10
Width=3mm,Gap=3mm Width=6mm,Gap=3mm Width=9mm,Gap=3mm -3
10
2
10
3
10 Re
Fig. 2. Friction factor in small rectangular channels.
4
10
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
Fig. 3 is the comparison between the present data and the predictive lines of Chisholm method [7] for C = 5, 12 and 20. The constants of C = 5, 10, 12 and 20 represents combinations of flow pattern in laminar and turbulent flow conditions for gas and liquid-phases. Notice that the turbulent and laminar friction factors are calculated by Eqs. (1) and (2), respectively, for evaluating the frictional multiplier, UL and the Martinelli parameter X. Range of X for this study spans from 0.15 to 10. Despite the predicted values of UL are quite scattering, most of the calculated UL values are within the prediction lines of C = 5 and 20. However, a few points of the rectangular channel having a smaller aspect ratio (9 mm · 3 mm) are greater than the prediction line of C = 20 and some points of the square channel (3 mm · 3 mm) fall below the prediction line of C = 5. Figs. 4 and 5 show the comparisons among the twophase frictional pressure gradient data with the predictions by homogeneous model [18] and by the Wambsganss [1] 100
ΦL
Width: 9mm x Gap: 3mm Width: 6mm x Gap: 3mm Width: 3mm x Gap:3mm C=5 C=12 C=20
10
1 0.1
1
10
X Fig. 3. Predictions of Chisholm method [7] versus present data.
100 0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
%
dP/dz)cal (kPa/m)
-50
10
1 Mean Deviation = 37.67%
0.1 0.1
1
10
100
dP/dz)exp (kPa/m)
Fig. 4. Predictions of homogeneous method [18] versus present data.
100
0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
dP/dz)cal (kPa/m)
64
%
-50
10
1 Mean Deviation = 54.63%
0.1 0.1
1
10
100
dP/dz)exp (kPa/m)
Fig. 5. Predictions of Wambsganss et al. [1] correlation versus present data.
correlation, respectively. It is found that the mean deviations for the comparison are 37.67%, and 54.63%, respectively. Note that the mean PN deviation is calculated by N1 jDP DP j=DP pred exp exp 100%. Apparently, the 1 predictive ability of the homogeneous model is better than that of the Wambsganss correlation. It is not surprising that the homogeneous model gives better predictions in small and micro channels due to the effect of surface tension [19]. As shown in Fig. 5, the data groups for different aspect ratio of the present study (A = 0.33, 0.66 and 1.0) are apparently separated. The over predictions for the channel with smaller aspect ratio (9 mm · 3 mm) and under predictions for the square channel (3 mm · 3 mm) are observed in Fig. 5. Ide and Fukano [12] recently proposed a correlation for calculating UL in a capillary rectangular channel (9.9 mm · 1.1 mm, A = 0.11). This correlation, Eq. (12), is utilized for comparing with the data sets from Wambsganss et al. [1], Ide and Fukano [12], and this study. As shown in Fig. 6, although this correlation gives a very good agreement for their own data with a mean deviation of 9.95%, it shows relatively high mean deviations of 26.75% against the present data and of 37.41% with the data of Wambsganss et al. [1]. Fig. 7 also shows the comparison of the present results and experimental data from [1,12] in terms of UL with the predictions of Wambsganss et al.’s correlation [1]. The predictions of this correlation, Eq. (10), give fair agreements with the present data (25.95%) and with the data from (14.31%) [12], however, it over predicts most of their own data and gives a high mean deviation of 40.41%. This is because the original development of this correlation had excluded the lower mass flux data (G = 50 kg/m2 s) and higher mass flux data (G > 400 kg/m2 s) [1]. Notice that the correlations of Wambsganss et al. [1] and Ide and Fukano [12] were developed based only on a single rectangular channel with corresponding aspect ratio
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66 100
100
0% +5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
% -50
Mean Deviation for Wambsganss etal. [1]
%
Φ
Lcal
= 37.41% Mean Deviation for Ide and Fukano [12] = 9.95%
1
-50
10
1
Mean Deviation for Present Data = 26.79%
0.1 0.1
1
0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
dP/dz)cal (kPa/m)
10
65
Mean Deviation = 20.8%
10
Φ
100
Lexp
Fig. 6. Comparison of the present results and experimental data [1,12] of UL versus predictions of Ide and Fukano correlation [12].
0.1 0.1
1
dP/dz)exp (kPa/m)
10
100
Fig. 8. Comparison of the present two-phase frictional pressure gradient data versus predictions of the proposed correlation, Eq. (13).
100
MeanDeviation for Wambsganss etal. [1]
Φ
Lcal
= 40.41% Mean Deviation for Ide and Fukano [12] = 14.31%
1
Mean Deviation for Present Data = 25.95% 0.1 0.1
1
10
100
Φ
Lexp
Fig. 7. Comparison of the present results and experimental data [1,12] of UL versus predictions of Wambsganss et al. correlation [1].
of 0.167 and 0.11, respectively. Hence the Wambsganss et al. [1] correlation did not include the effect of aspect ratio. In the meantime, despite the Ide and Fukano [12] correlation includes the aspect ratio, one should be aware that it was only correlated from a narrow single rectangular channel and extension of their correlation outside their database is questionable. In that regard, efforts are made in this study to include the effect of aspect ratio. Based on the Wambsganss correlation [1], we have extended the applicability of this C factor correlation by including the influences of ReLO and X, as well as the aspect ratio, A. A new C factor correlation is proposed from the empirical fit with the data sets of Wambsganss et al. [1] (111 data points), Ide and Fukano [12] (69 data points), and the present study (157 data points). Note that we have included all the data from Wambsganss et al. [1]. The proposed modified C factor in the Chisholm correlation for the prediction of UL is given as:
where a = 5.55 0.7555 · A0.805 + 0.00439 · ReLO and b = 0.1001 + 0.0005 · A0.895. Fig. 8 gives the comparison between the measured twophase frictional pressure gradient data (52, 52 and 53 points for 3.0 · 9.0 mm, 3.0 · 6.0 and 3.0 · 3.0 test sections, respectively) and the predictions by Eq. (13) having a mean deviation of 20.81%. For assessing this proposed correlation, the published UL data of Wambsganss et al. [1], Ide and Fukano [12], and the present study are compared with the calculated values of UL as shown in Fig. 9, the associated mean deviations are 24.99%, 10.83% and 10.73%, respectively. In summary, the proposed correlation shows a very good accuracy against the existing data and is capable of handling the effect of aspect ratio. The proposed correlation is valid for wide rages of G, 100 0%
+5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
10
%
-50
Mean Deviation for Wambsganss et al. [1] = 24.99%
Lcal
% -50
ð13Þ
Φ
10
C ¼ functionðA; X ; ReLO Þ ¼ aX b
0%
+5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
Mean Deviation for Ide and Fukano [12] =10.83%
1
Mean Deviation for Present Data =10.73%
0.1 0.1 0.
1
ΦLexp
10
100
Fig. 9. Comparison of the predictions of the proposed correlation, Eq. (13) versus UL data [1,12] and present data.
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I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
x, X and A, i.e., 50 < G < 700 kg/m2 s, 0.001 < x < 0.95, 0.05 < X < 20, and 0.1 < A < 1.0. This proposed correlation provides a simple tool for the engineering designer with an assessment of prediction for the two-phase frictional pressure drop in small rectangular channels. 4. Conclusions In this study, measurements of frictional pressure drops for water single-phase flow and two-phase air–water flow in three small rectangular channels are presented. Based on the foregoing discussions, the following summaries are concluded: 1. The measured single-phase friction factor data matched well with the base line predictions in laminar and turbulent flows. 2. The two-phase frictional pressure drop data are compared with the predictions from homogeneous and Chisholm methods, Wambsganss and Ide–Fukano correlations. It is found none of them can satisfactorily predict all the data set. 3. A modified C factor of Chisholm method considering the effect of aspect ratio is proposed. The proposed correlation contains the data sets of Wambsganss et al., Ide–Fukano, and the present study. The mean deviations representing the predictive ability of the proposed correlation against these three data sets are 24.99%, 10.83% and 10.73%, respectively. This correlation is valid in wide rages of mass flux (50 < G < 700 kg/ m2 s), gas quality (0.001 < x < 0.95), Martinelli parameter (0.05 < X < 20) and aspect ratio (0.1 < A < 1.0). Acknowledgements The authors acknowledge the financial supports provided by the Energy Commission of the Ministry of Economic Affairs and National Science Committee (NSC 95-2212-E-224-079) of Taiwan. References [1] M.W. Wambsganss, J.A. Jendrzejezyk, D.M. France, N.T. Obot, Frictional pressure gradients in two-phase flow in a small horizontal rectangular channel, J. Exp. Therm. Fluid Sci. 5 (1992) 40–56.
[2] T. Willmarth, M. Ishii, Interfical area concentration and void fraction of two-phase flow in narrow rectangular vertical channels, Trans. ASME 119 (1997) 916–922. [3] K. Mishima, T. Hibiki, H. Nishihara, Some characteristics of gas– liquid flow in narrow rectangular duct, Int. J. Multiphase Flow 9 (1) (1993) 15–124. [4] H.J. Lee, S.Y. Lee, Pressure drop correlations for two-phase flow within horizontal rectangular channels with small heights, Int. J. Multiphase Flow 27 (2001) 783–796. [5] J.P. Harnett, M. Kostic, Heat transfer to Newtonian and nonNewtonian fluids in rectangular ducts, Adv. Heat Transf. 19 (1989) 247–356. [6] R.W. Lockhart, R.G. Martinelli, Proposed correlations for isothermal two-phase two-component flow in pipes, Chem. Eng. Prog. 45 (1) (1949) 39–48. [7] D. Chisholm, A theoretical basis for the Lockhart–Martinelli correlation for two-phase flow, Int. J. Heat Mass Transf. 10 (1967) 1767–1778. [8] K. Mishima, T. Hibiki, Some characteristics of air–water two-phase flow in small diameter vertical tubes, Int. J. Multiphase Flow 22 (1996) 703–712. [9] H. Ide, H. Matsumura, Frictional pressure drops of two-phase gas– liquid flow in rectangular channels, Exp. Therm. Fluid Sci. 3 (1990) 362–373. [10] K. Akagawa, The flow of mixture of air and water: the friction losses in horizontal, inclined and vertical tubes, Trans. JSME 23 (128) (1957) 292. [11] L. Friedel, Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow, in: European Two-phase Group Meeting, Ispra, Italy, Paper E2, 1979. [12] H. Ide, T. Fukano, Experimental research on the correlations of holdup (g) and frictional pressure drop in air–water two-phase flow in a capillary rectangular channel, Exp. Therm. Fluid Sci. 2 (2005) 833– 841. [13] R.J. Moffat, Describing the uncertainties in experimental results, Exp. Therm. Fluid Sci. 1 (1988) 3–17. [14] A.D. Premoli, D. Francessco, A. Prina, A dimensional correlation for evaluating the density of two-phase mixture, Lo Termotecnica 25 (1971) 17–26. [15] C.J. Baroczy, A systematic for two-phase pressure drop, Chem. Eng. Prog. Symp. Series 62 (1966) 232–249. [16] D.A. Yashar, M.J. Wilson, H.R. Kopke, D.M. Graham, J.C. Chato, T.A. Newell, An investigation of refrigerant void fraction in horizontal microfin tubes, Int. J. HVAC&R Res. 107 (part 2) (2001) 173–188. [17] S.L. Smith, Void fractions in two-phase flow: a correlation based upon an equal velocity and head model, Heat Fluid Flow 1 (1971) 22– 39. [18] J.G. Collier, J.R. Thome, Convective Boiling and Condensation, third ed. Oxford, New York, 1994. p. 44–45. [19] K.A. Tripplet, S.M. Ghiasiaan, S.L. Abdel-Khlik, A. LeMouel, B.N. Mc Cord, Gas–liquid two-phase flow in microchannels, Part II: void fraction and pressure drop, Int. J. Multiphase 25 (3) (1999) 395– 410.
Two-phase frictional pressure drop in small rectangular channels Ing Youn Chen a
a,*
, Yi-Min Chen a, Jane-Sunn Liaw b, Chi-Chuan Wang
b
Mechanical Engineering Department, National Yunlin University of Science and Technology, Yunlin 640, Taiwan b Energy and Environment Laboratories, Industrial Technology Research Institute, Hsinchu 310, Taiwan Received 26 July 2006; received in revised form 4 January 2007; accepted 4 January 2007
Abstract This study presents single-phase and two-phase frictional pressure drop data in three rectangular channels with channel width of 3, 6 and 9 mm, and a fixed gap of 3 mm. Adiabatic flows of air–water mixtures are tested at mass flux of 100–700 kg/m2 s and gas quality ranging from 0.001 to 0.8. The single-phase friction factors agree satisfactorily with the predictions of laminar and turbulent flows. The two-phase frictional pressure drop data are compared to predictions of homogeneous and Chisholm method, Wambsganss and Ide– Fukano correlations. It is found that none of the existing methods or correlations can predict the two-phase pressure gradient in rectangular channels satisfactorily. A modified C factor of Chisholm method considering the effect of aspect ratio is proposed from the empirical fit with the data sets of Wambsganss et al., Ide–Fukano, and this study. The corresponding mean deviations of the proposed correlation against the datasets are 24.99%, 10.83% and 10.73%, respectively. This correlation is applicable in wide rages of mass flux (50 < G < 700 kg/m2 s), gas quality (0.001 < x < 0.95), Martinelli parameter (0.05 < X < 20) and aspect ratio (0.1 < A < 1.0). 2007 Elsevier Inc. All rights reserved. Keywords: Two-phase frictional pressure drop; Rectangular channel; Two-phase friction multiplier; C factor of Chisholm type correlation
1. Introduction Recently, the advance of the high performance electronic chips and the microelectronic devices results in a great demand in developing efficient heat removal techniques for dissipating increasingly large heat flux from compact systems. One of the simplest arrangements for the heat removal is using the liquid flow and convection heat transfer with or without phase change in small rectangular channels. In particular, plate-fin compact heat exchanger involves rectangular channels with hydraulic diameter less than 5 mm and mass flux less than 350 kg/ m2 s [1]. Also, flow in small rectangular channels is an integral part of the design of compact heat exchangers in boiling or condensation in space, aircraft, and other applications [2]. Since two-phase heat transfer is intimately interrelated to pressure drop, the higher pressure drop *
Corresponding author. Tel.: +886 5 5342601x4137; fax: +886 5 5312062. E-mail address: [email protected] (I.Y. Chen). 0894-1777/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2007.01.005
would increase the pumping power and also affect the performance of cooling systems. As a consequence, the twophase pressure drop in small rectangular channels is very important for the design of compact heat exchanger. In predicting the frictional two-phase pressure drop in any channel, it is necessary to have a reasonable method to estimate the friction factor of sing-phase flow in the same channel. For estimation the frictional performance of a non-circular tube, it is a common practice to adopt the well-known hydraulic diameter concept by substituting the diameter of the circular tubes with the hydraulic diameter (Dh) in the non-circular tubes. Recently, several investigations, Wambsganss et al. [1], Mishima et al. [3] and Lee and Lee [4], reported that the turbulent friction factor for rectangular channels agreed well with the predictions by the Blassius equation: dP =dz ¼ 2f qu2 =Dh ;
f ¼ 0:079Re0:25
ð1Þ
f is the Fanning friction factor. The Fanning friction factor for the laminar flow through the rectangular channels proposed by Harnett and Kostic [5] is given as:
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
61
Nomenclature A dP/dz C D Dh f g G j P DP Re ReL ReG ReLO u
aspect ratio, gap (S)/width (W), 0 < A < 1 frictional pressure gradient (Pa/m) Chisholm’s factor internal diameter of circular tube (m) hydraulic diameter (m) friction factor gravity (m/s2) total mass flux (kg/m2 s) superficial velocity (m/s) pressure (Pa) pressure drop across the straight test section (Pa) Reynolds number (quDh/l) Reynolds number based on liquid single-phase flow (qLjLDh/lL) Reynolds number based on gas single-phase flow (qGjGDh/lG) liquid Reynolds number based on the total mass flux (GDh/lL) mean axial velocity (m/s)
f ¼ 24ð1 1:3553A þ 1:946A2 1:7012A3 þ 0:9564A4 0:2537A5 Þ=Re
ð2Þ
This simplified polynomial equation fits the exact analytical solution with an accuracy of ±0.05%. Where A = S/W = channel gap (or height)/channel width represents the aspect ratio of the rectangular section, 0 < A < 1.0. The product of f and Re is seen to decrease with the rise of A. Concerning the frictional pressure loss for two-phase flow, Lockhart and Martinelli [6] proposed the first predictive method for estimation of the frictional two-phase pressure gradient. Lockhart and Martinelli put forward a Martinelli parameter, X, which is based on the ratio of the single-phase frictional pressure drops calculated via assuming liquid-phase flow alone to gas-phase flow alone, i.e., X2 ¼
dP L =dz dP G =dz
ð3Þ
dP 2-phase =dz ; dP G =dz
UG q
rectangular channel gap or height (mm) gas quality rectangular channel width (mm) Martinelli parameter axial flow direction (m) void fraction liquid holdup viscosity (N s/m2) two-phase frictional multiplier based on liquid flow alone two-phase frictional multiplier based on gas flow alone density (kg/m3)
Subscripts cal calculation or prediction exp experimental measurement G gas-phase L liquid-phase 2-phase two-phase flow had approximated the relationships of U2G and U2L versus X by the expressions: U2L ¼ 1 þ C=X þ 1=X 2 ;
U2L ¼
dP 2-phase =dz dP L =dz
ð4Þ
where dP2-phase/dz is the frictional two-phase flow pressure gradient. Using the two-phase pressure drop data in adiabatic horizontal tubes, they proposed plots of UG and UL versus X. Their data indicated that the multipliers are a function of the Martinelli parameter alone. Chisholm [7]
U2G ¼ 1 þ CX þ X 2
ð5Þ
where C denotes Chisholm’s factor and its value depends on whether the liquid and gas flows are laminar or turbulent. For example, when both phases are laminar or turbulent, C = 5 or 20, respectively. However, the Chisholm’s C factor had been empirically obtained that it is not a constant by the liquid and gas flow conditions. C factor was reported as a function of Martinelli parameter, total mass flux, tube diameter, and other physical properties. Mishima et al. [3] conducted two-phase flow in narrow rectangular ducts with width of 40 mm and gaps of 1.0, 2.4 and 5.0 mm (A = 0.025–0.125). The two-phase friction loss was based on the Chisholm’s equation [7] by modifying the Chisholm’s constant C to account for the effect of hydraulic diameter Dh. C ¼ 21ð1 e0:27Dh Þ
dPL/dz and dPG/dz are the liquid and gas frictional pressure gradients if each phase flows alone in the same pipe with its mass flow rate, respectively. They also defined the two-phase multipliers, U2G and U2L : U2G ¼
S x W X z a g l UL
ð6Þ
Similar tests were conducted by Mishima and Hibiki [8] for circular tubes with D = 1–4 mm. Based on Eq. (6), Mishima and Hibiki [8] then proposed a new C factor equation for vertical and round tubes, as well as rectangular channels. C ¼ 21ð1 e0:319Dh Þ
ð7Þ
Ide and Matsumura [9] experimentally investigated the effects of aspect ratio, hydraulic diameter and inclination angle on the two-phase frictional pressure drop for 10 rectangular channels with aspect ratio from 0.025 to 1.0. For horizontal arrangement, UL was found to be higher in the transverse case (gap < width) than in the longitudinal
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case (gap > width) at the same value of Martinelli parameter, X. This fact becomes more obvious at smaller X in the channel having the larger aspect ratio. The gas and liquid flows in the longitudinal case with larger aspect ratio could easily lead to more separate flow pattern by the gravity than that of the transverse case. For vertical arrangement, UL is higher for the channel with the largest aspect ratio over the entire range of X. The two-phase frictional pressure drop data were compared with the predictions of Chisholm method [7] with C = 21 and Akagawa’s correlation [10], i.e. U2L ¼ ð1 aÞb
ð8Þ
where a is the void fraction and b depends on the inclination angle and the tube surface roughness, taking b = 1.4 and 1.5 for the horizontal and vertical cases, respectively. However, neither the Chisholm’s method [7] nor Akagawa’s correlation [10] could accurately predict their data. Based on a separated flow model, Ide and Matsumura [9] proposed a two-phase frictional pressure drop correlation by taking into account the effects of the inclination angle of the channel and liquid Reynolds number. 0:625
UL ¼ BX ½ðA þ 1ÞðA þ aÞ=ðA þ 2 2aÞ
1:5
½a=ð1 aÞ
ð9Þ where B is 0.03 in horizontal channels and 0.045 in vertical channels. This correlation gave an accuracy about ±30% against their data. Wambsganss et al. [1] measured two-phase frictional pressure gradients in a small horizontal channel with a cross section of 19.05 · 3.18 mm (A = 0.167) for application to plate-fin compact heat exchangers. The data were compared with Chisholm method [7] and Fridel correlation [11]. However, both predictions did not adequately correlate the data. A modified equation for the Chisholm’s factor C was proposed for qualities (x) being larger than those occurred at the transition from bubbly to slug flow. The transition at x 0.002 was visually observed in their study. The proposed C equation is given as: C ¼ functionðX ; ReLO Þ ¼ aX b
ð10Þ
where a = 2.44 + 0.00939ReLO, b = 0.938 + 0.000432 ReLO, and ReLO = GDh/lL. This C correlation is applicable for ReLO < 2200 and 0.05 < X < 1.0. The correlation gave an average error of less than ±19% for 0.05 < x < 0.95 and 100 < G < 400 kg/m2 s. For G > 400 kg/m2 s, the Chisholm correlation [7] with C = 21 is recommended. Since this correlation was developed from air–water flowing in a single narrow channel (A = 0.167), orientated at horizontal transverse with the wide side horizontal (HT) and horizontal longitudinal with the wide side vertical (HL) positions, more data with smaller cross section areas, different aspect ratios and different fluids are needed to check the applicability of this correlation. More recently, Ide and Fukano [12] measured the mean liquid holdup (g) and frictional pressure drop for air–water
two-phase flow in a capillary rectangular channel (9.9 · 1.1 mm, A = 0.11). The test ranges of air and water superficial velocities are 1.5–8 m/s and 0.1–0.8 m/s, respectively. They described laminar flow (V) or turbulent flow (T) of two-phase intermittent flow in the small rectangular tube by using the liquid and gas Reynolds numbers in single-phase water flow (ReL = qLjLDh/lL) and single-phase gas flow (ReG = qGjGDh/lG), respectively, where jG andjL are the superficial gas and liquid velocities. The transition Reynolds number from laminar flow to turbulent flow was taken as 2400 experimentally. By considering the effect of aspect ratio, new correlations of two-phase frictional pressure drop for the capillary rectangular channel were proposed. For the laminar region: UL ¼ 0:2485A0:355 fjL =ðgDh Þ0:5 g0:233 ReL
3=8
ð11Þ
The above equation was only used for few data included in the laminar region, while most of the data were located in the turbulent region. For the turbulent region: 3=8
UL ¼ 0:848A0:145 fðjG þ jL Þ=jL g0:425 ReL
ð12Þ
The comparison of Ide and Fukano experimental data [12] with the calculated values UL of Eq. (12) shows that all the data are within ±15% of the predictions. From the above review, almost all the proposed correlations for the two-phase frictional pressure drop in rectangular channels were based on the data with very small aspect ratio (A < 0.2). It is unclear whether the previous correlations can be extended outside their applicable range. In this regard, it is the purpose of this study to extend the available database pertaining to rectangular cross section with higher aspect ratios. In the present study, the experiments of the frictional air and water two-phase flow in three rectangular channels are conducted with channel width, W = 3, 6 and 9 mm, at a fixed gap (S) of 3 mm (A = S/W = 1, 0.666 and 0.333). The applicability of previous correlations is verified. The effects of aspect ratio, gas quality and mass flow rate on the related parameters are investigated. 2. Experiments Three rectangular test sections made of transparent acrylic resin have cross sections of 3.0 · 3.0 mm, 3.0 · 6.0 mm and 3.0 · 9.0 mm having hydraulic diameters of 3.0, 4.0 and 4.5 mm, respectively. The inlet and outlet of the test rectangular sections are connected with a 3.0 mm internal diameter circular tube. A differential pressure transducer is used to measure the pressure drop across a 300 mm length of the rectangular channel as shown in Fig. 1. The upstream and downstream lengths of the test section are 150 mm and 90 mm, respectively. The total air and water mass flux (G) is ranged from 100 s to 700 kg/ m2 s and the quality (x) is changed from 0.001 to 0.8.
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
63
and ranges from 3.2% to 11.5% for U2L . The highest uncertainties were associated with the lowest Reynolds number. 3. Results and discussion
The tests are first conducted with single-phase water flow to obtain the friction factor data in the rectangular test section. The friction factor data for the single-phase flow will be used to check the applicability of the instrumentations and the reliability of the friction factor correlations, Eqs. (1) and (2). Adiabatic flows of air–water mixtures at 25 C and near atmospheric pressure are tested in three rectangular channels at horizontal and longitudinal (HL) orientation. Pressure fluctuations depending on the flow pattern are inherent in two-phase flow. The pressure drops are calculated from the time-average mean differential-pressure measurements. The test rig is designed to conduct tests with air–water mixtures as shown in Fig. 1. Air is supplied from an aircompressor and then stored in a compressed-air storage tank. Airflow through a pressure reducer and, depending on the mass flux range, is measured by three Aalborg mass flow meters for different ranges of flow rate. The water flow loop consists of a variable speed gear pump that delivers water. The mixer is designed to provide better uniformity of the air–water mixture stream. At the upstream of the test section, there is a set of quick closing valves for measuring the average volumetric void fraction of the two-phase flow. A very accurate Yokogawa magnetic flow meter is installed at the downstream of the gear pump. The accuracy of the air and water flow meters is within ±0.2% of the test span. The pressure drops of the air–water mixtures are measured by a Yokogawa EJ110 differential pressure transducer having an adjustable span of 1300– 13,000 Pa. Resolution of this pressure differential transducer is 0.3% of the measurements. Leaving the test section, the air–water mixture is separated by an open water tank in which the air is vented and the water is recirculated. The air and water temperatures are measured by resistance temperature device (Pt 100 X) having a calibrated accuracy of 0.1 K (calibrated by Hewlett–Packard quartz thermometer probe with quartz thermometer, model 18111A and 2804A). Uncertainties in the reported values of the friction factor f and UL are estimated by the method suggested by Moffat [13]. The uncertainties of f are from 1.3% to 6.4%
f = 14.22/Re f = 15.55/Re f = 17.09/Re f = 0.0791/Re0.25
-1
10
f
Fig. 1. Schematic illustration of experimental apparatus.
For single-phase tests, water is used as the working fluid for measuring the friction factors. The test results of friction factor data are plotted in Fig. 2. The base lines are the Fanning friction factor (f = 0.0791Re0.25) for turbulent flow and the predictions for laminar flow with the channel dimensions of 3.0 · 3.0 mm, 3.0 · 6.0 mm and 3.0 · 9.0 mm (A = 1.0, 0.66 and 0.33) with f = 14.22/Re, 15.55/Re and 17.09/Re, respectively. As seen, f is decreased with the rise of A. The f data agree favorably with the base lines. The good agreement shown in Fig. 2 substantiates the accuracy of the instrumentation and the experimental apparatus. For predicting the two-phase frictional pressure drops in small rectangular channels, the present study was intended to use Ide and Matsumura’s correlation [9], Eq. (9), and the existed void fraction correlations to calculate UL. For verifying the applicability of the available a correlations, predictions of homogeneous method and correlations of Premoli [14], Baroczy [15], Yashr et al. [16] and Smith [17] are made to compare the void fraction data obtained from two quick-closing solenoid valves as shown in Fig. 1. The measured void fraction data versus predictions are quite scattering and do not have a consistent trend with the mass flux. The homogeneous method is greatly over predicted all the data. For x < 0.01, the correlations of Premoli [14] and Smith [17] have a fair agreement with the data whereas the predictions of Baroczy [15] and Yashar et al. [16] are more close to the data when x > 0.01. Since the predictions can not give a good agreement to the scattering void fraction data, Eq. (9) is not able to utilize in this study. More accurate method for measuring void fraction is needed.
-2
10
Width=3mm,Gap=3mm Width=6mm,Gap=3mm Width=9mm,Gap=3mm -3
10
2
10
3
10 Re
Fig. 2. Friction factor in small rectangular channels.
4
10
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66
Fig. 3 is the comparison between the present data and the predictive lines of Chisholm method [7] for C = 5, 12 and 20. The constants of C = 5, 10, 12 and 20 represents combinations of flow pattern in laminar and turbulent flow conditions for gas and liquid-phases. Notice that the turbulent and laminar friction factors are calculated by Eqs. (1) and (2), respectively, for evaluating the frictional multiplier, UL and the Martinelli parameter X. Range of X for this study spans from 0.15 to 10. Despite the predicted values of UL are quite scattering, most of the calculated UL values are within the prediction lines of C = 5 and 20. However, a few points of the rectangular channel having a smaller aspect ratio (9 mm · 3 mm) are greater than the prediction line of C = 20 and some points of the square channel (3 mm · 3 mm) fall below the prediction line of C = 5. Figs. 4 and 5 show the comparisons among the twophase frictional pressure gradient data with the predictions by homogeneous model [18] and by the Wambsganss [1] 100
ΦL
Width: 9mm x Gap: 3mm Width: 6mm x Gap: 3mm Width: 3mm x Gap:3mm C=5 C=12 C=20
10
1 0.1
1
10
X Fig. 3. Predictions of Chisholm method [7] versus present data.
100 0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
%
dP/dz)cal (kPa/m)
-50
10
1 Mean Deviation = 37.67%
0.1 0.1
1
10
100
dP/dz)exp (kPa/m)
Fig. 4. Predictions of homogeneous method [18] versus present data.
100
0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
dP/dz)cal (kPa/m)
64
%
-50
10
1 Mean Deviation = 54.63%
0.1 0.1
1
10
100
dP/dz)exp (kPa/m)
Fig. 5. Predictions of Wambsganss et al. [1] correlation versus present data.
correlation, respectively. It is found that the mean deviations for the comparison are 37.67%, and 54.63%, respectively. Note that the mean PN deviation is calculated by N1 jDP DP j=DP pred exp exp 100%. Apparently, the 1 predictive ability of the homogeneous model is better than that of the Wambsganss correlation. It is not surprising that the homogeneous model gives better predictions in small and micro channels due to the effect of surface tension [19]. As shown in Fig. 5, the data groups for different aspect ratio of the present study (A = 0.33, 0.66 and 1.0) are apparently separated. The over predictions for the channel with smaller aspect ratio (9 mm · 3 mm) and under predictions for the square channel (3 mm · 3 mm) are observed in Fig. 5. Ide and Fukano [12] recently proposed a correlation for calculating UL in a capillary rectangular channel (9.9 mm · 1.1 mm, A = 0.11). This correlation, Eq. (12), is utilized for comparing with the data sets from Wambsganss et al. [1], Ide and Fukano [12], and this study. As shown in Fig. 6, although this correlation gives a very good agreement for their own data with a mean deviation of 9.95%, it shows relatively high mean deviations of 26.75% against the present data and of 37.41% with the data of Wambsganss et al. [1]. Fig. 7 also shows the comparison of the present results and experimental data from [1,12] in terms of UL with the predictions of Wambsganss et al.’s correlation [1]. The predictions of this correlation, Eq. (10), give fair agreements with the present data (25.95%) and with the data from (14.31%) [12], however, it over predicts most of their own data and gives a high mean deviation of 40.41%. This is because the original development of this correlation had excluded the lower mass flux data (G = 50 kg/m2 s) and higher mass flux data (G > 400 kg/m2 s) [1]. Notice that the correlations of Wambsganss et al. [1] and Ide and Fukano [12] were developed based only on a single rectangular channel with corresponding aspect ratio
I.Y. Chen et al. / Experimental Thermal and Fluid Science 32 (2007) 60–66 100
100
0% +5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
% -50
Mean Deviation for Wambsganss etal. [1]
%
Φ
Lcal
= 37.41% Mean Deviation for Ide and Fukano [12] = 9.95%
1
-50
10
1
Mean Deviation for Present Data = 26.79%
0.1 0.1
1
0%
+5
Width:9mm x Gap:3mm Width:6mm x Gap:3mm Width:3mm x Gap:3mm
dP/dz)cal (kPa/m)
10
65
Mean Deviation = 20.8%
10
Φ
100
Lexp
Fig. 6. Comparison of the present results and experimental data [1,12] of UL versus predictions of Ide and Fukano correlation [12].
0.1 0.1
1
dP/dz)exp (kPa/m)
10
100
Fig. 8. Comparison of the present two-phase frictional pressure gradient data versus predictions of the proposed correlation, Eq. (13).
100
MeanDeviation for Wambsganss etal. [1]
Φ
Lcal
= 40.41% Mean Deviation for Ide and Fukano [12] = 14.31%
1
Mean Deviation for Present Data = 25.95% 0.1 0.1
1
10
100
Φ
Lexp
Fig. 7. Comparison of the present results and experimental data [1,12] of UL versus predictions of Wambsganss et al. correlation [1].
of 0.167 and 0.11, respectively. Hence the Wambsganss et al. [1] correlation did not include the effect of aspect ratio. In the meantime, despite the Ide and Fukano [12] correlation includes the aspect ratio, one should be aware that it was only correlated from a narrow single rectangular channel and extension of their correlation outside their database is questionable. In that regard, efforts are made in this study to include the effect of aspect ratio. Based on the Wambsganss correlation [1], we have extended the applicability of this C factor correlation by including the influences of ReLO and X, as well as the aspect ratio, A. A new C factor correlation is proposed from the empirical fit with the data sets of Wambsganss et al. [1] (111 data points), Ide and Fukano [12] (69 data points), and the present study (157 data points). Note that we have included all the data from Wambsganss et al. [1]. The proposed modified C factor in the Chisholm correlation for the prediction of UL is given as:
where a = 5.55 0.7555 · A0.805 + 0.00439 · ReLO and b = 0.1001 + 0.0005 · A0.895. Fig. 8 gives the comparison between the measured twophase frictional pressure gradient data (52, 52 and 53 points for 3.0 · 9.0 mm, 3.0 · 6.0 and 3.0 · 3.0 test sections, respectively) and the predictions by Eq. (13) having a mean deviation of 20.81%. For assessing this proposed correlation, the published UL data of Wambsganss et al. [1], Ide and Fukano [12], and the present study are compared with the calculated values of UL as shown in Fig. 9, the associated mean deviations are 24.99%, 10.83% and 10.73%, respectively. In summary, the proposed correlation shows a very good accuracy against the existing data and is capable of handling the effect of aspect ratio. The proposed correlation is valid for wide rages of G, 100 0%
+5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
10
%
-50
Mean Deviation for Wambsganss et al. [1] = 24.99%
Lcal
% -50
ð13Þ
Φ
10
C ¼ functionðA; X ; ReLO Þ ¼ aX b
0%
+5
Wambsganss et al. [1] Ide and Fukano [12] Present Data
Mean Deviation for Ide and Fukano [12] =10.83%
1
Mean Deviation for Present Data =10.73%
0.1 0.1 0.
1
ΦLexp
10
100
Fig. 9. Comparison of the predictions of the proposed correlation, Eq. (13) versus UL data [1,12] and present data.
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x, X and A, i.e., 50 < G < 700 kg/m2 s, 0.001 < x < 0.95, 0.05 < X < 20, and 0.1 < A < 1.0. This proposed correlation provides a simple tool for the engineering designer with an assessment of prediction for the two-phase frictional pressure drop in small rectangular channels. 4. Conclusions In this study, measurements of frictional pressure drops for water single-phase flow and two-phase air–water flow in three small rectangular channels are presented. Based on the foregoing discussions, the following summaries are concluded: 1. The measured single-phase friction factor data matched well with the base line predictions in laminar and turbulent flows. 2. The two-phase frictional pressure drop data are compared with the predictions from homogeneous and Chisholm methods, Wambsganss and Ide–Fukano correlations. It is found none of them can satisfactorily predict all the data set. 3. A modified C factor of Chisholm method considering the effect of aspect ratio is proposed. The proposed correlation contains the data sets of Wambsganss et al., Ide–Fukano, and the present study. The mean deviations representing the predictive ability of the proposed correlation against these three data sets are 24.99%, 10.83% and 10.73%, respectively. This correlation is valid in wide rages of mass flux (50 < G < 700 kg/ m2 s), gas quality (0.001 < x < 0.95), Martinelli parameter (0.05 < X < 20) and aspect ratio (0.1 < A < 1.0). Acknowledgements The authors acknowledge the financial supports provided by the Energy Commission of the Ministry of Economic Affairs and National Science Committee (NSC 95-2212-E-224-079) of Taiwan. References [1] M.W. Wambsganss, J.A. Jendrzejezyk, D.M. France, N.T. Obot, Frictional pressure gradients in two-phase flow in a small horizontal rectangular channel, J. Exp. Therm. Fluid Sci. 5 (1992) 40–56.
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