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Drag reduction of slug flows in microchannels by modifying the size of T-junctions
International Journal of Multiphase Flow 62 (2014) 67–72
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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Drag reduction of slug flows in microchannels by modifying the size of T-junctions Ken Yamamoto ⇑, Satoshi Ogata Department of Mechanical Engineering, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
a r t i c l e
i n f o
Article history: Received 19 September 2013 Received in revised form 20 January 2014 Accepted 14 February 2014 Available online 12 March 2014 Keywords: Drag reduction Slug flow Microchannel Bretherton law T-junction
a b s t r a c t Pressure drop measurements of air–water slug flows in three circular microchannels (d = 0.486 mm) with various T-junctions (0.136, 0.194 and 0.252 mm) were performed. The measured pressure drops were compared with common predictive models, and it was found that there were deviations between the models and measurements. The tendencies of the deviation between the models and measurements were different for different channels, implying the existence of a T-junction size effect. For confirmation of this effect, a scaling analysis based on the Laplace pressure and Bretherton law was performed. The analysis results showed that the inner pressure of a bubble increased when the T-junction size became small, resulting in drag reduction of the slug flow in the microchannel. Consequently, a modified predictive model for the pressure drop of slug flows was proposed. However, the drag reduction effect decreased when the T-junction was smaller than d/2 because the bubbles cannot maintain their shape without the depressurization due to the geometrical restriction. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Two-phase flows are one of the most interesting flows in microfluidic devices (Thorsen et al., 2001; Kreutzer et al., 2005a,b; Stone et al., 2004; Agostini et al., 2008; Revellin et al., 2008; Cubaud and Ho, 2004), and the performance improvement of these devices is an important topic. Especially, an accurate estimation of the pressure drop in these flows is necessitated because of the limitation of a pumping power in these devices (Garimella and Singhal, 2004). For this reason, numerous researchers contributed for proposing accurate predictive models (Kreutzer et al., 2005a,b; Walsh et al., 2009; Warnier et al., 2010; Garimella et al., 2002). The pressure drop of liquid–gas slug flows can be divided into two parts, namely frictional loss in liquid slugs (frictional loss term) and pressure loss involving gas bubbles (Laplace pressure term) (Kreutzer et al., 2005a,b; Walsh et al., 2009; Warnier et al., 2010). While the frictional loss term is dominated by the liquid laminar flow theory, the Laplace pressure term is dominated by the Bretherton law (Bretherton, 1961), which is a theoretical model to predict the pressure drop across a flowing bubble. In this model, Bretherton (1961) employed the lubrication approximation and the assumption that the pressure difference between the liquid phase and gas bubble at the bubble cap is 2r/r, and he finally ⇑ Corresponding author. Tel.: +81 42 677 2710; fax: +81 42 677 2701. E-mail address: [email protected] (K. Yamamoto). http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.02.011 0301-9322/Ó 2014 Elsevier Ltd. All rights reserved.
obtained the pressure drop across a flowing bubble by calculating the pressure drop at the front and rear bubble caps:
DPbubble ffi 3:58ð3CaÞ2=3 ¼ 4:52ð3CaÞ2=3
r n
ro
r
r
r r
0:930ð3CaÞ2=3
;
ð1Þ
where r and r denote the interfacial tension and radius of the channel cross section, respectively, and Ca denotes the capillary number, which is calculated from the bubble velocity, liquid viscosity, and interfacial tension between the phases. Note that the first term in the middle side of (1) expresses the pressure drop at the front cap and the second term expresses the pressure drop at the rear cap, respectively. This equation is only valid when the inertia is negligible and the thickness h of the liquid film that exists between the bubble and the wall, is much smaller than r. Based on the Bretherton law, Kreutzer et al. (2005a) derived a theoretical correlation for predicting the total pressure drop (per unit length):
DP 1 4 ¼ eL fapp qU 2TP þ qg ; L 2 d
ð2Þ
where q, U, g, and eL denote liquid density, velocity, gravitational acceleration, and liquid hold-up expressed as eL = UL/UTP, respectively, and the subscripts L and TP denote the liquid phase and two-phase, respectively. Furthermore, fapp denotes the apparent friction factor that comprises a frictional loss term in the liquid phase and Laplace pressure term. With the slug length Lslug and
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the two-phase Reynolds number ReTP, which is calculated from the liquid properties and the velocity of the two-phase flow, fapp can be expressed as (3) (for negligible inertia: ReTP < 100) and (4) (for nonnegligible inertia: ReTP > 100).
fapp
" 16 d ¼ 1þ ReTP Lslug 16 d 1þ ¼ ReTP Lslug
fapp ¼
7:16 32=3
#
32 Ca1=3 0:465 ðReTP < 100Þ; 1=3 Ca
" 1=3 # 16 d ReTP 1þa ðReTP > 100Þ: ReTP Lslug Ca
ð3Þ
ð4Þ
Note that the coefficient 7.16 32/3 in (3) was derived from the Bretherton law. However, this coefficient only accounts for the pressure drop across the front bubble cap (Bretherton, 1961) as it is stated in (1), and hence it should be replaced by 9.04 32/3. Similarly, the coefficient 0.465 in the final form should be 0.588. Note also that while (3) is a theoretical correlation, (4) is a semiempirical one with a fitting parameter a. The value of the fitting parameter was chosen as a = 0.17 from their experiment. However, their numerical results showed the appropriate value for a was 0.07. They explained this discrepancy as a result of impurities, which were mixed into the test fluids during the experiments. Additionally, experimental data by Walsh et al. (2009) showed that a = 0.12 was the best fit. A sufficient explanation for this contradiction in the fitting parameter has not been provided until now. While the correlation proposed by Kreutzer et al. (2005a) is based on a negligible liquid film thickness, a predictive model with non-negligible liquid film thickness (Aussillous and Quéré, 2000) was derived by Warnier et al. (2010).
" # 16 7:16 32=3 d F b 1 ; ¼ 1þ ReTP 32 U L ðCa1=3 þ 3:34CaÞ
ð5Þ
" # DP 32lL U L 7:16 32=3 d F b 1 ; ¼ 1 þ 2 L 32 U L ðCa1=3 þ 3:34CaÞ d
ð6Þ
fapp
Fb ¼
Ub ; Lbubble þ Lslug
explain this contradiction is the difference in the bubble/droplet generation method and/or mixer geometries. If this hypothesis is true, the inner pressure of the bubbles may also be changed by these factors (because detachment of the bubbles is often caused by the pressure difference between the two phases (Garstecki et al., 2006), which results in a change of the pressure drop characteristics. Therefore, we conducted pressure drop measurements of air–water two-phase flows in circular microchannels for investigating the effects of the size of the mixer. We chose three different sizes of T-junctions for the mixer and compared the measured pressure drop with the models presented above. Furthermore, we performed a scaling analysis for investigating the T-junction size effects. 2. Experimental setup and procedure In this study, we conducted pressure drop measurements and visualization of two-phase flow. The experimental setup (Fig. 1a) was similar to those used in our previous work (Yamamoto and Ogata, 2013); the only difference between them was the pressure ports for this study. Three test channels were fabricated by connecting two glass capillaries (d = 486 lm) with some distances win, which were used as air injectors, in transparent connecting chambers (Fig. 1b). The glass capillaries were prepared by cutting up single capillary into 50 mm long and grinding their edges; i.e. an error in the capillary diameter is negligible. For detecting the effects of the mixer, we prepared three different T-junction widths: win = 136, 194, and 252 lm for Channels #1, #2, and #3, respectively. We chose pure water and air for the continuous phase and dispersed phase, respectively. Both fluids were flowed at a constant flow rate by syringe pumps (NEXUS 6000, Chemyx Inc., and PHD 2000, HARVARD Apparatus), and mixed at T-junctions in the test channels after removing impurities with 0.2 lm filters (PURADISC™ 25 PP, Whatman Inc.). In order to decrease the effects of the pressure fluctuations in the gas phase and velocity fluctuations
ð7Þ
where UL, Ub, lL, Fb, and Lbubble denote the liquid superficial velocity, gas bubble velocity, liquid viscosity, frequency of gas bubbles, and bubble length, respectively. Note that the coefficient 7.16 32/3 should be replaced by 9.04 32/3 as mentioned above. However, several authors (Walsh et al., 2009; Sur and Liu, 2012) experimentally showed that neither semi-empirical models (see, for example, (Warnier et al., 2010; Garimella et al., 2002) nor a theoretical model (Kreutzer et al., 2005a) could predict the pressure drop accurately. From these facts, it can be said that the formulation of the predictive model for two-phase flow still remains an open issue. In addition to pressure drop, flow pattern prediction is also an important issue, and several researchers have contributed to make flow pattern maps (e.g. Sur and Liu, 2012; Chung and Kawaji, 2004; Shao et al., 2009; Serizawa et al., 2002; Nishisako et al., 2002; Xu et al., 2006). However, it is quite difficult to create a universal map because the flow pattern changes in different experiments and/or with different researchers. Several researchers (Sur and Liu, 2012; Chung and Kawaji, 2004; Shao et al., 2009) concluded that this is due to the difference of the channel size, and they have also reported that bubbly flow disappeared in smaller channels (mainly less than 100 lm). However, other authors (Serizawa et al., 2002; Nishisako et al., 2002) have observed bubbly flows in channels smaller than 100 lm. One possible reason that can
Fig. 1. Schematic of the experimental setup. (a) The test channel. Both the fluids were flowed at constant flow rates with syringe pumps. The fluids were flowed through the 0.2-lm filter and mixed at the T-junction fabricated in the connection chamber. (b) The inner structure of the connecting chamber. The T-junction was fabricated by aligning the ends of two glass capillaries with a gap of win. The chamber was filled with water.
K. Yamamoto, S. Ogata / International Journal of Multiphase Flow 62 (2014) 67–72
at the outlet due to the Laplace pressure jump, we flowed the air phase through a 4-m-long polytetrafluoroethylene (PTFE) tube (ID: 0.8 mm) before entering the T-junctions, and the channel outlet was directly connected to the outlet reservoir. The water level in the outlet reservoir was kept constant and higher than the channel outlet. The pressure measurement procedure was as follows. First, we measured the single-phase pressure drop (including the inlet and entrance regions) by connecting a pressure transducer (DP15, 5.5 kPa, ±0.25% FS, Validyne) to the inlet chamber and the T-junction, which were filled with water and kept unconnected to the syringe pump for air phase. Second, we reconnected the pressure transducer to the channel inlet and outlet reservoir for the overall pressure drop measurement. Finally, the two-phase pressure drop was calculated by subtracting the single-phase pressure drop from the overall pressure drop. The uncertainty of the calculated two-phase pressure drop was estimated as less than ±3.6%. For the visualization, we used a high-speed camera (FASTCAMNEO, Photron) operating at 2000–8000 fps. By using two resolutions [high resolution (9.71 lm/pixel), low resolution (19.42 lm/ pixel)], we could measure all the slug and bubble lengths at directly downstream of the T-junctions. Note that all the bubble lengths were 1.6 times longer than the capillary inner diameter and that even though we could not measure the thickness of the liquid film (theoretically it is on the order of 100–101 lm, which is comparable to the pixel size), we estimated from the obtained images and the fact that the liquid film exists even when the bubble is stationary (Chen, 1986) that a film exists between the bubble and the wall. After more than 15-min run of the system, we conducted the overall pressure drop measurements and visualization. Every liquid flow was fully developed ahead of the T-junctions, and all the experiments were conducted at ambient pressure and room temperature. We also recorded the room and water temperatures and reflected those values in the post-processing of the data. The tested flow rate ratio Qair/Qwater was 0.15–2.0, the two-phase Reynolds number ReTP was 40–180, and the capillary number Ca was <5 103. We determined the two-phase Reynolds number based on the liquid properties and mean interfacial velocity. 3. Results and discussion 3.1. Comparison with the predictive models We compared the deviation of three predictive models from the measured pressure drop. The theoretical model with negligible inertia (Kreutzer et al., 2005a) and the two semi-empirical models with negligible inertia (Warnier et al., 2010) and non-negligible inertia (Kreutzer et al., 2005a) were calculated from a combination of (2) and (3), from a combination of (5)–(7), and from a combination of (2) and (4), respectively. Note that we replaced coefficients 7.16 32/3 in (3), (5), and (6) by 9.04 32/3 for the reason mentioned earlier. The deviations were calculated by
Pressure drop deviation ½% ¼
ðDP=LÞmodel ðDP=LÞexp 100; ðDP=LÞexp
ð8Þ
where (DP/L)model and (DP/L)exp denote the calculated pressure drop by predictive models and the experimentally obtained pressure drop, respectively. In addition, we observed quasi-stable flows when Qair/Qwater ranged from 0.2 to 0.7 in the experiment with Channel #1, which contains the smallest air injector (Yamamoto and Ogata, 2013). The pressure drop became larger in these cases and, although the bubble and slug lengths were constant, the generated bubbles were longer than those observed in other channels at the same operating condition and their pinch-off motions from the T-junction
69
were different from that observed by Garstecki et al. (2006); they first expanded toward both upstream and downstream of the T-junction and then flowed downstream (15–35 ms in Fig. 2). Fig. 3 shows the relationship between the deviation of the theoretical model (Kreutzer et al., 2005a) and the Reynolds number. We can clearly see in Fig. 3 that for ReTP < 100, which is a valid range for the theoretical model (Kreutzer et al., 2005a), the model overestimates by approximately 40% and 60%, on average, for Channels #2 and #3, respectively. However, for the case of Channel #1, although the scattering of the experimental data becomes large, the average of the deviation becomes small. In a comparison with the theoretical model (Kreutzer et al., 2005a), the semi-empirical model by Warnier et al. (2010) (Fig. 4) shows a better agreement with the experimental data. The averaged deviations in ReTP < 100 are approximately 20% for Channel #1, 25% for Channel #2, and 45% for Channel #3. The relationship between the semi-empirical model by Kreutzer et al. (2005a) and the Reynolds number is shown in Fig. 5. We found the best fit to the experimental data with the fitting parameter of a = 0.13 for Channel #1, and a = 0.03 for Channels #2 and #3. Although the value of the fitting parameter is different, Fig. 5 shows that the model predicts the pressure drop well for ReTP > 120. However, in 100 < ReTP < 120, where it is experimentally defined as the non-negligible inertia region (Kreutzer et al., 2005a), the deviation becomes large. Moreover, the value of the fitting parameter is much smaller than the experiments by both Kreutzer et al. (2005a) and Walsh et al. (2009) and the numerical calculation by Kreutzer et al. (2005a). Since the fitting parameter a involves with the Laplace pressure term and the fact that the only difference between the channels is size of the T-junction, let us consider the effects caused by the difference in generated bubbles. 3.2. Scaling analysis of the Laplace pressure term As mentioned earlier, the Bretherton law assumes that the Laplace pressure difference at the bubble cap DPLaplace,cap (Laplace pressure difference between the bubble cap and liquid phase in slug flow) is 2r/(d/2). However in this experiment, the air phase must first exceed the Laplace pressure difference at the T-junction DPLaplace,ent ffi 2r/(win/2), for entering the main channel. Therefore, with an assumption that the air-phase (bubble) pressure is constant, we assume the Laplace pressure between the liquid phase and flowing bubble as:
DPLaplace;cap DPLaplace;ent 2r=ðwin =2Þ 2r=Ar;
ð9Þ
where A = win/d and r is a radius of the channel. Note that this value may be somewhat overestimated because the pressure in the water phase fluctuates and becomes minimum when the air phase starts to enter the main channel (Garstecki et al., 2006; Yamamoto and Ogata, 2013). By employing this modified assumption, the balance of the Laplace pressure at dynamic and static menisci (Aussillous and Quéré, 2000), which are connected smoothly because the pressure should be equal at their boundary, for bubbles longer than d (i.e. bubbles having front and rear cap and cylindrical part) can be expressed by
r rh r
k2
2r ; Ar
ð10Þ
where h and k denote the thickness of the liquid thin film and the length scale of the dynamic region, respectively. However, the relationship between the viscous force and pressure gradient along the dynamic meniscus is given by
lU b h
2
1r 2 1 : k r A
ð11Þ
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Fig. 2. Bubble pinch-off motion at Qair/Qwater = 0.54 in Channel #1 (left) and Channel #2 (right).
By solving (10) and (11), we now obtain the liquid film thickness as follows:
h 0:643 ð3CaÞ2=3 ; r ð2=A 1Þ
ð12Þ
where the coefficient 0.643 32/3 was derived by Bretherton. Because the radius of the actual bubble caps, r0 , can be expressed as r0 ffi r bh, where b is a coefficient, the total curvature of the bubble caps, j, can be expressed as follows:
2 r
j¼ 0¼ Fig. 3. Deviation of the theoretical (negligible inertia) model by Kreutzer et al. (2005a) from the experimental data.
2 h 1þb : r r
ð13Þ
Therefore, the Laplace pressure at the bubble caps is given by
DPLaplace;cap ¼ rj ¼
2r h 1þb : r r
ð14Þ
Here, the coefficient b takes on a different value for the front and rear caps, and so there is a difference between the Laplace pressures at the front and rear menisci. Hence, we obtain the pressure drop across the bubble as the difference in the Laplace pressure:
DPbubble ¼ ðbfront brear Þ
Fig. 4. Deviation of the semi-empirical (negligible inertia) model by Warnier et al. (2010) from the experimental data.
ð15Þ
where again, the coefficients bfront = 2.79 and brear = 0.723 were derived by Bretherton (1961), and (15) corresponds to (1) when A = 1. This equation suggests that the pressure drop across the bubble (i.e., the Laplace pressure term) will decrease for bubbles generated by the narrower T-junction, resulting in a change in the Laplace pressure term in (3) and (4) and drag reduction in the slug flow. This explains why the fitting parameter a in (4), which is a part of the Laplace pressure term, took such a low value in our experiment. Finally, we obtain the apparent friction factor, which is a modification of (3) and can predict the pressure drop of slug flows by combining with (2):
fapp Fig. 5. Deviation of the semi-empirical (non-negligible inertia) model by Kreutzer et al. (2005a) from the experimental data. The fitting parameters were chosen as a = 0.13 for Channel #1, and a = 0.03 for Channels #2 and #3.
h 2r 4:52 r ð3CaÞ2=3 ; r r ð2=A 1Þ r
" # 16 d 0:588 ðReTP < 100Þ: ¼ 1þ ReTP Lslug ð2=A 1ÞCa1=3
ð16Þ
However, whereas we assumed r0 ffi ar for the beginning of the analysis, (16) was derived from r0 ffi r bh. This contradiction limits the validity of (16) to A ffi 1. For smaller values of A, the
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et al., 2010) in Fig. 7. From Fig. 7, it can be said that the proposed model showed the best fit to our experimental data. Furthermore, we compared the pressure drop deviation of the proposed model with the bubble length deviation, which was calculated by (17), in Fig. 8.
Bubble length deviation ½% ¼
ðLbubble Þmodel ðLbubble Þexp ðLbubble Þexp 100;
ð17Þ
where (Lbubble)model denotes the bubble length calculated from a predictive model by van Steijn et al. (2007) (18): Fig. 6. Deviation of the proposed predictive model from the experimental data.
bubble caps cannot maintain their shape, and the bubbles may expand themselves in order to depressurize, which may decrease the effect of the overpressure. Despite this limitation, we compared the new predictive model [a combination of (16) and (2)] with the experimental results in Fig. 6. Clearly, we see a good agreement between the new model and the experimental data (95% of the data were within ±25% deviation) for the entire Reynolds number region (even in the non-negligible inertia region) for Channel #3 (A = 0.52). However, the model underestimates the pressure drop for Channels #1 and #2 (A = 0.28 and 0.40, respectively). This implies that the model is invalid for Channels #1 and #2 because of their small A value, and the lower threshold of (16) for practical use lies around A = 0.5. The reason why the lower threshold of A value is 0.5, instead of 1, may be due to the overestimation of DPLaplace,cap in (9). As the proposed predictive model showed a good agreement with the experimental data, we now compare the proposed model with negligible inertia models (Kreutzer et al., 2005a; Warnier
Fig. 7. Comparison of the deviation between the theoretical (Kreutzer et al., 2005a), semi-empirical (Warnier et al., 2010), and proposed model for Channel #3. The proposed model showed the best fit to the experimental data.
Q air : ðLbubble Þmodel ¼ win a1 þ a2 Q water
ð18Þ
Note that a1 and a2 are fitting parameters chosen as a1 = 3 and a2 = 3.5, respectively, for this experiment (Yamamoto and Ogata, 2013). We see in Fig. 8 a trend of increasing pressure drop deviation along with the bubble length deviation, and the deviations in the bubble length are not large (±10%) for Channels #2 and #3. This is because of the change of the bubble pinch-off mechanism (Fig. 2), and it implies that the quasi-stable state in Channel #1 was induced by depressurization of the bubble inner pressure, which results in a decrease of the drag reduction effect. 4. Conclusions We conducted pressure drop measurements and visualization of air–water two-phase slug flow in circular microchannels (d = 486 lm) to investigate the effects of the T-junction size. The measured pressure drop in all the test channels showed discrepancies from previous predictive models. To determine the reasons for such discrepancies, we examined the effects of generated bubbles by means of a scaling analysis of the pressure drop across the bubble. The analytical results showed that the pressure inside the bubble increased with a decrease in the T-junction size, and this overpressure resulted in a drag reduction in the pressure drop across the flowing bubble and the slug flow. Because this drag reduction effect changed the Laplace pressure term in the apparent friction factor, we modified the correlation derived by Kreutzer et al. (2005a). The proposed correlation showed a good agreement for A (=win/d) > 0.5. However, the modified correlation had a lower threshold of A ffi 0.5, and below this point the drag reduction effect gradually decreased with the T-junction size. Moreover, for ReTP > 120, it was found that we can apply the semi-empirical correlation derived by Kreutzer et al. (2005a), but the fitting parameter a should be changed for different T-junction sizes. The minimum a value may be found at A ffi 0.5. Finally, from the deviation between the measured bubble length and the predictive model, it was implied that the bubbles generated at narrow T-junctions depressurize themselves in the channel. Acknowledgements We would like to express our appreciation for the financial support by the Japan Society for the Promotion of Science through the Research Fellowship for Young Scientists (No. 23-283). References
Fig. 8. Relationship between the deviations of the proposed predictive model to the experimentally obtained pressure drop and deviations of the slug length.
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Contents lists available at ScienceDirect
International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Drag reduction of slug flows in microchannels by modifying the size of T-junctions Ken Yamamoto ⇑, Satoshi Ogata Department of Mechanical Engineering, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
a r t i c l e
i n f o
Article history: Received 19 September 2013 Received in revised form 20 January 2014 Accepted 14 February 2014 Available online 12 March 2014 Keywords: Drag reduction Slug flow Microchannel Bretherton law T-junction
a b s t r a c t Pressure drop measurements of air–water slug flows in three circular microchannels (d = 0.486 mm) with various T-junctions (0.136, 0.194 and 0.252 mm) were performed. The measured pressure drops were compared with common predictive models, and it was found that there were deviations between the models and measurements. The tendencies of the deviation between the models and measurements were different for different channels, implying the existence of a T-junction size effect. For confirmation of this effect, a scaling analysis based on the Laplace pressure and Bretherton law was performed. The analysis results showed that the inner pressure of a bubble increased when the T-junction size became small, resulting in drag reduction of the slug flow in the microchannel. Consequently, a modified predictive model for the pressure drop of slug flows was proposed. However, the drag reduction effect decreased when the T-junction was smaller than d/2 because the bubbles cannot maintain their shape without the depressurization due to the geometrical restriction. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Two-phase flows are one of the most interesting flows in microfluidic devices (Thorsen et al., 2001; Kreutzer et al., 2005a,b; Stone et al., 2004; Agostini et al., 2008; Revellin et al., 2008; Cubaud and Ho, 2004), and the performance improvement of these devices is an important topic. Especially, an accurate estimation of the pressure drop in these flows is necessitated because of the limitation of a pumping power in these devices (Garimella and Singhal, 2004). For this reason, numerous researchers contributed for proposing accurate predictive models (Kreutzer et al., 2005a,b; Walsh et al., 2009; Warnier et al., 2010; Garimella et al., 2002). The pressure drop of liquid–gas slug flows can be divided into two parts, namely frictional loss in liquid slugs (frictional loss term) and pressure loss involving gas bubbles (Laplace pressure term) (Kreutzer et al., 2005a,b; Walsh et al., 2009; Warnier et al., 2010). While the frictional loss term is dominated by the liquid laminar flow theory, the Laplace pressure term is dominated by the Bretherton law (Bretherton, 1961), which is a theoretical model to predict the pressure drop across a flowing bubble. In this model, Bretherton (1961) employed the lubrication approximation and the assumption that the pressure difference between the liquid phase and gas bubble at the bubble cap is 2r/r, and he finally ⇑ Corresponding author. Tel.: +81 42 677 2710; fax: +81 42 677 2701. E-mail address: [email protected] (K. Yamamoto). http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.02.011 0301-9322/Ó 2014 Elsevier Ltd. All rights reserved.
obtained the pressure drop across a flowing bubble by calculating the pressure drop at the front and rear bubble caps:
DPbubble ffi 3:58ð3CaÞ2=3 ¼ 4:52ð3CaÞ2=3
r n
ro
r
r
r r
0:930ð3CaÞ2=3
;
ð1Þ
where r and r denote the interfacial tension and radius of the channel cross section, respectively, and Ca denotes the capillary number, which is calculated from the bubble velocity, liquid viscosity, and interfacial tension between the phases. Note that the first term in the middle side of (1) expresses the pressure drop at the front cap and the second term expresses the pressure drop at the rear cap, respectively. This equation is only valid when the inertia is negligible and the thickness h of the liquid film that exists between the bubble and the wall, is much smaller than r. Based on the Bretherton law, Kreutzer et al. (2005a) derived a theoretical correlation for predicting the total pressure drop (per unit length):
DP 1 4 ¼ eL fapp qU 2TP þ qg ; L 2 d
ð2Þ
where q, U, g, and eL denote liquid density, velocity, gravitational acceleration, and liquid hold-up expressed as eL = UL/UTP, respectively, and the subscripts L and TP denote the liquid phase and two-phase, respectively. Furthermore, fapp denotes the apparent friction factor that comprises a frictional loss term in the liquid phase and Laplace pressure term. With the slug length Lslug and
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the two-phase Reynolds number ReTP, which is calculated from the liquid properties and the velocity of the two-phase flow, fapp can be expressed as (3) (for negligible inertia: ReTP < 100) and (4) (for nonnegligible inertia: ReTP > 100).
fapp
" 16 d ¼ 1þ ReTP Lslug 16 d 1þ ¼ ReTP Lslug
fapp ¼
7:16 32=3
#
32 Ca1=3 0:465 ðReTP < 100Þ; 1=3 Ca
" 1=3 # 16 d ReTP 1þa ðReTP > 100Þ: ReTP Lslug Ca
ð3Þ
ð4Þ
Note that the coefficient 7.16 32/3 in (3) was derived from the Bretherton law. However, this coefficient only accounts for the pressure drop across the front bubble cap (Bretherton, 1961) as it is stated in (1), and hence it should be replaced by 9.04 32/3. Similarly, the coefficient 0.465 in the final form should be 0.588. Note also that while (3) is a theoretical correlation, (4) is a semiempirical one with a fitting parameter a. The value of the fitting parameter was chosen as a = 0.17 from their experiment. However, their numerical results showed the appropriate value for a was 0.07. They explained this discrepancy as a result of impurities, which were mixed into the test fluids during the experiments. Additionally, experimental data by Walsh et al. (2009) showed that a = 0.12 was the best fit. A sufficient explanation for this contradiction in the fitting parameter has not been provided until now. While the correlation proposed by Kreutzer et al. (2005a) is based on a negligible liquid film thickness, a predictive model with non-negligible liquid film thickness (Aussillous and Quéré, 2000) was derived by Warnier et al. (2010).
" # 16 7:16 32=3 d F b 1 ; ¼ 1þ ReTP 32 U L ðCa1=3 þ 3:34CaÞ
ð5Þ
" # DP 32lL U L 7:16 32=3 d F b 1 ; ¼ 1 þ 2 L 32 U L ðCa1=3 þ 3:34CaÞ d
ð6Þ
fapp
Fb ¼
Ub ; Lbubble þ Lslug
explain this contradiction is the difference in the bubble/droplet generation method and/or mixer geometries. If this hypothesis is true, the inner pressure of the bubbles may also be changed by these factors (because detachment of the bubbles is often caused by the pressure difference between the two phases (Garstecki et al., 2006), which results in a change of the pressure drop characteristics. Therefore, we conducted pressure drop measurements of air–water two-phase flows in circular microchannels for investigating the effects of the size of the mixer. We chose three different sizes of T-junctions for the mixer and compared the measured pressure drop with the models presented above. Furthermore, we performed a scaling analysis for investigating the T-junction size effects. 2. Experimental setup and procedure In this study, we conducted pressure drop measurements and visualization of two-phase flow. The experimental setup (Fig. 1a) was similar to those used in our previous work (Yamamoto and Ogata, 2013); the only difference between them was the pressure ports for this study. Three test channels were fabricated by connecting two glass capillaries (d = 486 lm) with some distances win, which were used as air injectors, in transparent connecting chambers (Fig. 1b). The glass capillaries were prepared by cutting up single capillary into 50 mm long and grinding their edges; i.e. an error in the capillary diameter is negligible. For detecting the effects of the mixer, we prepared three different T-junction widths: win = 136, 194, and 252 lm for Channels #1, #2, and #3, respectively. We chose pure water and air for the continuous phase and dispersed phase, respectively. Both fluids were flowed at a constant flow rate by syringe pumps (NEXUS 6000, Chemyx Inc., and PHD 2000, HARVARD Apparatus), and mixed at T-junctions in the test channels after removing impurities with 0.2 lm filters (PURADISC™ 25 PP, Whatman Inc.). In order to decrease the effects of the pressure fluctuations in the gas phase and velocity fluctuations
ð7Þ
where UL, Ub, lL, Fb, and Lbubble denote the liquid superficial velocity, gas bubble velocity, liquid viscosity, frequency of gas bubbles, and bubble length, respectively. Note that the coefficient 7.16 32/3 should be replaced by 9.04 32/3 as mentioned above. However, several authors (Walsh et al., 2009; Sur and Liu, 2012) experimentally showed that neither semi-empirical models (see, for example, (Warnier et al., 2010; Garimella et al., 2002) nor a theoretical model (Kreutzer et al., 2005a) could predict the pressure drop accurately. From these facts, it can be said that the formulation of the predictive model for two-phase flow still remains an open issue. In addition to pressure drop, flow pattern prediction is also an important issue, and several researchers have contributed to make flow pattern maps (e.g. Sur and Liu, 2012; Chung and Kawaji, 2004; Shao et al., 2009; Serizawa et al., 2002; Nishisako et al., 2002; Xu et al., 2006). However, it is quite difficult to create a universal map because the flow pattern changes in different experiments and/or with different researchers. Several researchers (Sur and Liu, 2012; Chung and Kawaji, 2004; Shao et al., 2009) concluded that this is due to the difference of the channel size, and they have also reported that bubbly flow disappeared in smaller channels (mainly less than 100 lm). However, other authors (Serizawa et al., 2002; Nishisako et al., 2002) have observed bubbly flows in channels smaller than 100 lm. One possible reason that can
Fig. 1. Schematic of the experimental setup. (a) The test channel. Both the fluids were flowed at constant flow rates with syringe pumps. The fluids were flowed through the 0.2-lm filter and mixed at the T-junction fabricated in the connection chamber. (b) The inner structure of the connecting chamber. The T-junction was fabricated by aligning the ends of two glass capillaries with a gap of win. The chamber was filled with water.
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at the outlet due to the Laplace pressure jump, we flowed the air phase through a 4-m-long polytetrafluoroethylene (PTFE) tube (ID: 0.8 mm) before entering the T-junctions, and the channel outlet was directly connected to the outlet reservoir. The water level in the outlet reservoir was kept constant and higher than the channel outlet. The pressure measurement procedure was as follows. First, we measured the single-phase pressure drop (including the inlet and entrance regions) by connecting a pressure transducer (DP15, 5.5 kPa, ±0.25% FS, Validyne) to the inlet chamber and the T-junction, which were filled with water and kept unconnected to the syringe pump for air phase. Second, we reconnected the pressure transducer to the channel inlet and outlet reservoir for the overall pressure drop measurement. Finally, the two-phase pressure drop was calculated by subtracting the single-phase pressure drop from the overall pressure drop. The uncertainty of the calculated two-phase pressure drop was estimated as less than ±3.6%. For the visualization, we used a high-speed camera (FASTCAMNEO, Photron) operating at 2000–8000 fps. By using two resolutions [high resolution (9.71 lm/pixel), low resolution (19.42 lm/ pixel)], we could measure all the slug and bubble lengths at directly downstream of the T-junctions. Note that all the bubble lengths were 1.6 times longer than the capillary inner diameter and that even though we could not measure the thickness of the liquid film (theoretically it is on the order of 100–101 lm, which is comparable to the pixel size), we estimated from the obtained images and the fact that the liquid film exists even when the bubble is stationary (Chen, 1986) that a film exists between the bubble and the wall. After more than 15-min run of the system, we conducted the overall pressure drop measurements and visualization. Every liquid flow was fully developed ahead of the T-junctions, and all the experiments were conducted at ambient pressure and room temperature. We also recorded the room and water temperatures and reflected those values in the post-processing of the data. The tested flow rate ratio Qair/Qwater was 0.15–2.0, the two-phase Reynolds number ReTP was 40–180, and the capillary number Ca was <5 103. We determined the two-phase Reynolds number based on the liquid properties and mean interfacial velocity. 3. Results and discussion 3.1. Comparison with the predictive models We compared the deviation of three predictive models from the measured pressure drop. The theoretical model with negligible inertia (Kreutzer et al., 2005a) and the two semi-empirical models with negligible inertia (Warnier et al., 2010) and non-negligible inertia (Kreutzer et al., 2005a) were calculated from a combination of (2) and (3), from a combination of (5)–(7), and from a combination of (2) and (4), respectively. Note that we replaced coefficients 7.16 32/3 in (3), (5), and (6) by 9.04 32/3 for the reason mentioned earlier. The deviations were calculated by
Pressure drop deviation ½% ¼
ðDP=LÞmodel ðDP=LÞexp 100; ðDP=LÞexp
ð8Þ
where (DP/L)model and (DP/L)exp denote the calculated pressure drop by predictive models and the experimentally obtained pressure drop, respectively. In addition, we observed quasi-stable flows when Qair/Qwater ranged from 0.2 to 0.7 in the experiment with Channel #1, which contains the smallest air injector (Yamamoto and Ogata, 2013). The pressure drop became larger in these cases and, although the bubble and slug lengths were constant, the generated bubbles were longer than those observed in other channels at the same operating condition and their pinch-off motions from the T-junction
69
were different from that observed by Garstecki et al. (2006); they first expanded toward both upstream and downstream of the T-junction and then flowed downstream (15–35 ms in Fig. 2). Fig. 3 shows the relationship between the deviation of the theoretical model (Kreutzer et al., 2005a) and the Reynolds number. We can clearly see in Fig. 3 that for ReTP < 100, which is a valid range for the theoretical model (Kreutzer et al., 2005a), the model overestimates by approximately 40% and 60%, on average, for Channels #2 and #3, respectively. However, for the case of Channel #1, although the scattering of the experimental data becomes large, the average of the deviation becomes small. In a comparison with the theoretical model (Kreutzer et al., 2005a), the semi-empirical model by Warnier et al. (2010) (Fig. 4) shows a better agreement with the experimental data. The averaged deviations in ReTP < 100 are approximately 20% for Channel #1, 25% for Channel #2, and 45% for Channel #3. The relationship between the semi-empirical model by Kreutzer et al. (2005a) and the Reynolds number is shown in Fig. 5. We found the best fit to the experimental data with the fitting parameter of a = 0.13 for Channel #1, and a = 0.03 for Channels #2 and #3. Although the value of the fitting parameter is different, Fig. 5 shows that the model predicts the pressure drop well for ReTP > 120. However, in 100 < ReTP < 120, where it is experimentally defined as the non-negligible inertia region (Kreutzer et al., 2005a), the deviation becomes large. Moreover, the value of the fitting parameter is much smaller than the experiments by both Kreutzer et al. (2005a) and Walsh et al. (2009) and the numerical calculation by Kreutzer et al. (2005a). Since the fitting parameter a involves with the Laplace pressure term and the fact that the only difference between the channels is size of the T-junction, let us consider the effects caused by the difference in generated bubbles. 3.2. Scaling analysis of the Laplace pressure term As mentioned earlier, the Bretherton law assumes that the Laplace pressure difference at the bubble cap DPLaplace,cap (Laplace pressure difference between the bubble cap and liquid phase in slug flow) is 2r/(d/2). However in this experiment, the air phase must first exceed the Laplace pressure difference at the T-junction DPLaplace,ent ffi 2r/(win/2), for entering the main channel. Therefore, with an assumption that the air-phase (bubble) pressure is constant, we assume the Laplace pressure between the liquid phase and flowing bubble as:
DPLaplace;cap DPLaplace;ent 2r=ðwin =2Þ 2r=Ar;
ð9Þ
where A = win/d and r is a radius of the channel. Note that this value may be somewhat overestimated because the pressure in the water phase fluctuates and becomes minimum when the air phase starts to enter the main channel (Garstecki et al., 2006; Yamamoto and Ogata, 2013). By employing this modified assumption, the balance of the Laplace pressure at dynamic and static menisci (Aussillous and Quéré, 2000), which are connected smoothly because the pressure should be equal at their boundary, for bubbles longer than d (i.e. bubbles having front and rear cap and cylindrical part) can be expressed by
r rh r
k2
2r ; Ar
ð10Þ
where h and k denote the thickness of the liquid thin film and the length scale of the dynamic region, respectively. However, the relationship between the viscous force and pressure gradient along the dynamic meniscus is given by
lU b h
2
1r 2 1 : k r A
ð11Þ
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Fig. 2. Bubble pinch-off motion at Qair/Qwater = 0.54 in Channel #1 (left) and Channel #2 (right).
By solving (10) and (11), we now obtain the liquid film thickness as follows:
h 0:643 ð3CaÞ2=3 ; r ð2=A 1Þ
ð12Þ
where the coefficient 0.643 32/3 was derived by Bretherton. Because the radius of the actual bubble caps, r0 , can be expressed as r0 ffi r bh, where b is a coefficient, the total curvature of the bubble caps, j, can be expressed as follows:
2 r
j¼ 0¼ Fig. 3. Deviation of the theoretical (negligible inertia) model by Kreutzer et al. (2005a) from the experimental data.
2 h 1þb : r r
ð13Þ
Therefore, the Laplace pressure at the bubble caps is given by
DPLaplace;cap ¼ rj ¼
2r h 1þb : r r
ð14Þ
Here, the coefficient b takes on a different value for the front and rear caps, and so there is a difference between the Laplace pressures at the front and rear menisci. Hence, we obtain the pressure drop across the bubble as the difference in the Laplace pressure:
DPbubble ¼ ðbfront brear Þ
Fig. 4. Deviation of the semi-empirical (negligible inertia) model by Warnier et al. (2010) from the experimental data.
ð15Þ
where again, the coefficients bfront = 2.79 and brear = 0.723 were derived by Bretherton (1961), and (15) corresponds to (1) when A = 1. This equation suggests that the pressure drop across the bubble (i.e., the Laplace pressure term) will decrease for bubbles generated by the narrower T-junction, resulting in a change in the Laplace pressure term in (3) and (4) and drag reduction in the slug flow. This explains why the fitting parameter a in (4), which is a part of the Laplace pressure term, took such a low value in our experiment. Finally, we obtain the apparent friction factor, which is a modification of (3) and can predict the pressure drop of slug flows by combining with (2):
fapp Fig. 5. Deviation of the semi-empirical (non-negligible inertia) model by Kreutzer et al. (2005a) from the experimental data. The fitting parameters were chosen as a = 0.13 for Channel #1, and a = 0.03 for Channels #2 and #3.
h 2r 4:52 r ð3CaÞ2=3 ; r r ð2=A 1Þ r
" # 16 d 0:588 ðReTP < 100Þ: ¼ 1þ ReTP Lslug ð2=A 1ÞCa1=3
ð16Þ
However, whereas we assumed r0 ffi ar for the beginning of the analysis, (16) was derived from r0 ffi r bh. This contradiction limits the validity of (16) to A ffi 1. For smaller values of A, the
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et al., 2010) in Fig. 7. From Fig. 7, it can be said that the proposed model showed the best fit to our experimental data. Furthermore, we compared the pressure drop deviation of the proposed model with the bubble length deviation, which was calculated by (17), in Fig. 8.
Bubble length deviation ½% ¼
ðLbubble Þmodel ðLbubble Þexp ðLbubble Þexp 100;
ð17Þ
where (Lbubble)model denotes the bubble length calculated from a predictive model by van Steijn et al. (2007) (18): Fig. 6. Deviation of the proposed predictive model from the experimental data.
bubble caps cannot maintain their shape, and the bubbles may expand themselves in order to depressurize, which may decrease the effect of the overpressure. Despite this limitation, we compared the new predictive model [a combination of (16) and (2)] with the experimental results in Fig. 6. Clearly, we see a good agreement between the new model and the experimental data (95% of the data were within ±25% deviation) for the entire Reynolds number region (even in the non-negligible inertia region) for Channel #3 (A = 0.52). However, the model underestimates the pressure drop for Channels #1 and #2 (A = 0.28 and 0.40, respectively). This implies that the model is invalid for Channels #1 and #2 because of their small A value, and the lower threshold of (16) for practical use lies around A = 0.5. The reason why the lower threshold of A value is 0.5, instead of 1, may be due to the overestimation of DPLaplace,cap in (9). As the proposed predictive model showed a good agreement with the experimental data, we now compare the proposed model with negligible inertia models (Kreutzer et al., 2005a; Warnier
Fig. 7. Comparison of the deviation between the theoretical (Kreutzer et al., 2005a), semi-empirical (Warnier et al., 2010), and proposed model for Channel #3. The proposed model showed the best fit to the experimental data.
Q air : ðLbubble Þmodel ¼ win a1 þ a2 Q water
ð18Þ
Note that a1 and a2 are fitting parameters chosen as a1 = 3 and a2 = 3.5, respectively, for this experiment (Yamamoto and Ogata, 2013). We see in Fig. 8 a trend of increasing pressure drop deviation along with the bubble length deviation, and the deviations in the bubble length are not large (±10%) for Channels #2 and #3. This is because of the change of the bubble pinch-off mechanism (Fig. 2), and it implies that the quasi-stable state in Channel #1 was induced by depressurization of the bubble inner pressure, which results in a decrease of the drag reduction effect. 4. Conclusions We conducted pressure drop measurements and visualization of air–water two-phase slug flow in circular microchannels (d = 486 lm) to investigate the effects of the T-junction size. The measured pressure drop in all the test channels showed discrepancies from previous predictive models. To determine the reasons for such discrepancies, we examined the effects of generated bubbles by means of a scaling analysis of the pressure drop across the bubble. The analytical results showed that the pressure inside the bubble increased with a decrease in the T-junction size, and this overpressure resulted in a drag reduction in the pressure drop across the flowing bubble and the slug flow. Because this drag reduction effect changed the Laplace pressure term in the apparent friction factor, we modified the correlation derived by Kreutzer et al. (2005a). The proposed correlation showed a good agreement for A (=win/d) > 0.5. However, the modified correlation had a lower threshold of A ffi 0.5, and below this point the drag reduction effect gradually decreased with the T-junction size. Moreover, for ReTP > 120, it was found that we can apply the semi-empirical correlation derived by Kreutzer et al. (2005a), but the fitting parameter a should be changed for different T-junction sizes. The minimum a value may be found at A ffi 0.5. Finally, from the deviation between the measured bubble length and the predictive model, it was implied that the bubbles generated at narrow T-junctions depressurize themselves in the channel. Acknowledgements We would like to express our appreciation for the financial support by the Japan Society for the Promotion of Science through the Research Fellowship for Young Scientists (No. 23-283). References
Fig. 8. Relationship between the deviations of the proposed predictive model to the experimentally obtained pressure drop and deviations of the slug length.
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