Gas-liquid two-phase flow across a bank of micropillars

Transcription

1 PHYSICS OF FLUIDS 19, Gas-liquid two-phase flow across a bank of micropillars Santosh Krishnamurthy and Yoav Peles Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York Received 13 October 2006; accepted 8 March 2007; published online 30 April 2007 Adiabatic nitrogen-water two-phase flow across a bank of staggered circular micropillars, 100 m long with a diameter of 100 m and a pitch-to-diameter ratio of 1.5, was investigated experimentally for Reynolds number ranging from 5 to 50. Flow patterns, void fraction, and pressure drop were obtained, discussed, and compared to large scale as well as microchannel results. Two-phase flow patterns were determined by flow visualization, and a flow map was constructed as a function of gas and liquid superficial velocities. Significant deviations from conventional scale systems, with respect to flow patterns and trend lines, were observed. A unique flow pattern, driven by surface tension, was observed and termed bridge flow. The applicability of conventional scale models to predict the void fraction and two-phase frictional pressure drop was also assessed. Comparison with a conventional scale void fraction model revealed good agreement, but was found to be in a physically wrong form. Thus, a modified physically based model for void fraction was developed. A two-phase frictional multiplier was found to be a strong function of mass flux, unlike in previous microchannel studies. It was observed that models from conventional scale systems did not adequately predict the two-phase frictional multiplier at the microscale, thus, a modified model accounting for mass flux was developed American Institute of Physics. DOI: / I. INTRODUCTION Gas-liquid two-phase flow in microdomains has been a research topic of increasing interest in the past decade across diverse engineering disciplines. Microchemical reactors, 1 microrockets, 2,3 microbiological systems, 4 and microheat exchangers, 5 to name a few, utilized two-phase flow as the prime physical phenomenon governing the performance of these systems. In conventional scale, two types of rudimentary configurations are typically studied: in-tube flow and flow across a tube bundle. A survey of the literature reveals myriad reports dedicated to unveiling the hydrodynamic as well as the thermal and chemical characteristics of adiabatic and diabatic two-phase flows In microdomains, the phenomenon is much less explored, and the fundamental processes governing adiabatic flow were only studied for the in-tube channel Both deviations and similarities to conventional scale channels were reported and discussed. Kawahara et al. 14 reported a new flow pattern in microchannels termed ring flow, while Chung and Kawaji 15 and Kawaji 16 showed that the extensively used Martinelli pressure drop model can be employed in very small length scales only when proper scaling laws are introduced. However, Cubaud and Ho 17 were not able to successfully use available large-scale models to predict their microchannel results and, therefore, they developed a new frictional pressure drop model. Chung and Kawaji 15 also showed that unlike large-scale systems, the two-phase frictional multiplier l 2 has no dependence on mass velocity as the archetype system size diminishes a result that was not observed in conventional scales. With the maturity of the microfluidics field, new flow configurations are being introduced. Micropillars entrenched inside microchannels are of particular interest as they provide increasing surface area and can promote mixing. However, much is unknown about the fundamental processes governing the hydrodynamic characteristics of gas-liquid two-phase flow in these very small systems. Similarly, the pressure drop, void fraction, and flow morphology are not very well understood. This paper reports a study of adiabatic nitrogen-water two-phase flow across a staggered array of 100 m diameter by 100 m deep micropillars entrenched inside a microchannel. Section II presents the essential background of the pertinent physical phenomena, including pressure drop, void fraction, and flow morphologies. Section III provides a detailed overview of the device geometry and presents the essential microfabrication process flow used. Additionally, an overview of the experimental setup and procedure is presented. Section IV presents the methodology used for the data reduction. Section V includes discussions on flow patterns, void fraction, and two-phase pressure drop. The validity of conventional and microchannel models is also assessed. Finally, the main conclusions of this study are presented in Sec. VI. II. BACKGROUND The study of hydrodynamic gas-liquid two-phase flow in channels and tube bundles often concern pressure drop, void fraction, and flow morphology /2007/19 4 /043302/14/$ , American Institute of Physics

2 S. Krishnamurthy and Y. Peles Phys. Fluids 19, A. Pressure drop Two-phase pressure drop is composed of hydrostatic gravitational, acceleration, and frictional components. In microdomains, gravitational forces diminish, and thus, can be neglected. Acceleration pressure drop results from the change in the momentum of the two-phase mixture and is strongly dependent on the void fraction. Two basic modeling approaches have been widely used to predict the two-phase frictional pressure drop the homogeneous flow model and the separated flow model. When the velocities of both phases are similar i.e., in the absence of any slip between the phases, the homogeneous flow model is found to be quite accurate in predicting the two-phase frictional pressure drop, by assuming a single-phase flow type friction factor. For two-phase flow across a tube bundle, the homogeneous friction pressure drop is predicted in terms of the average mixture density eff according to P f tp = f tpng 2, 1 2 eff where G is the total mass flux, N is the number of rows, and eff is given by 1 = x + 1 x, 2 eff g l g is the density of the gas, l is the density of the liquid, and x is the quality of the two-phase flow. f tp is the two-phase Darcy friction factor, f tp = a Re b d,tp, 3 where a and b are constants derived from experiments, and Re d,tp is the two-phase Reynolds number based on the tube diameter, Re d,tp = Gd. 4 eff Several relations for the effective viscosity, eff, were proposed by Cicchitti et al., 18 Dukler et al., 19 Beattie and Whalley, 20 and McAdams, 21 respectively, eff = x g + 1 x l, eff = h g + 1 h l, eff = l 1 h h + g h, 7 eff = x + 1 x 1 g l, 8 where g is the gas viscosity, l is the liquid viscosity, and h is the homogeneous void fraction. One of the most widely used separated flow models for the two-phase frictional pressure drop is the Lockhart- Martinelli model, 22 which relates the two-phase frictional pressure drop to the single-phase frictional pressure drop using the gas-liquid frictional multiplier l 2. The two-phase frictional multiplier is often related to the Martinelli parameter X using the following equations: where + l 2 = P f tp =1+ C 1 2, 9 P f l X X X = P 1/2 f/ Z l P f / Z g, 10 P f tp is the two-phase frictional pressure drop, P f / Z l is the frictional pressure gradient across the channel that would result if the liquid flowed alone through the channel at a mass flow rate equal to G 1 x A, and P f / Z g is the frictional pressure gradient across the channel that would result if the gas flowed alone through the channel at a mass flow rate of GxA. The constant C is an empirically defined factor, which depends on the flow regimes laminar, turbulent, or a combination of the two, flow morphology e.g., bubbly, intermittent, etc. among other parameters. The Martinelli parameter, X, defined in Eq. 10 also depends on the individual two-phase flow regimes. If both phases have superficial velocities corresponding to laminar flow, the frictional pressure drop terms in Eq. 10 are defined based on laminar correlations and are termed the viscous-viscous Martinelli parameter X vv. B. Void fraction Void fraction, the ratio of the area occupied by the gas to the total cross-sectional area, is an important physical parameter, which is a key factor dictating the frictional and acceleration pressure drops. Knowledge of the void fraction can also provide important insight into the hydrodynamic mechanism and the means to obtain the slip velocity. In conventional scale systems, techniques such as the quick-closing valve 11 and gamma densitometry 8 10 have been successfully used to measure the void fraction, while in previous microchannel studies 14,15,17 image processing was used. These studies compared the measured void fraction to the homogeneous one, h = x g 1 x l + x g. 11 It has been widely observed that in many two-phase systems the homogeneous void fraction overpredicts the measured value even at low gas flow rates. C. Flow pattern Several distinct flow patterns can develop in internal gas-liquid two-phase flow, and transition from one flow pattern to another is accompanied by a significant change of the gas-liquid interfacial area. Some of the most commonly reported flow patterns for horizontal flow in pipes include bubbly, slug, spray, and stratified flows. 24 All these flow patterns, though particular to a horizontal in-tube configuration, form a subset of the flow patterns found in many different configurations including vertical in-tube flow and flow across tube bundles horizontal and vertical.

3 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, FIG. 1. a Device overview and geometry. b Inlet region of the device. c Mixer. III. DEVICE OVERVIEW AND FABRICATION A. Device overview A computer-aided design CAD schematic of the device consisting of a 1500 m wide and 1 cm long microchannel of depth 100 m is shown in Fig. 1. The microchannel encompasses 68 rows of 9 and 10 in tandem staggered circular 100 m diameter pillars Fig. 1 b. A Pyrex cover seals the device from the top and allows flow visualization. Pressure taps are placed at the inlet, exit, and in three different locations, indicated in Fig. 1 a, along the device to enable pressure measurements. Well-mixed gas and liquid twophase flow is obtained by passing the two phases through a mixer, which is located upstream of the main pillar array. Upstream the main pillar arrays, the gas and liquid enter the mixer and form a well mixed two-phase flow Fig. 1 c. Unlike other studies at the microscale, 14,15 where the two phases were mixed externally before entering the test section, in the current study the mixer is fabricated as a part of the device. The mixer has two inlets, one for the water and one for the nitrogen, and a series of closely spaced 50 m diameter circular pillars with a pitch-to-diameter ratio of 1.3. The design of the mixer is similar to that found in some conventional scale studies e.g., Grant et al. 6. B. Microfabrication process flow A double side polished, n-type 100 single crystal, silicon wafer is processed on both sides to create the microelectromechanical system MEMS device, which consists of a microchannel enclosing the array of pillars. In the fabrication process, the top side and bottom side masks are designed and fabricated. A 1 m thick oxide is deposited on both sides of the silicon wafer to protect the bare wafer surface. Next, the microchannel with the micropillars is formed on the top side of the wafer. For this, the wafer is taken through a photoli-

4 S. Krishnamurthy and Y. Peles Phys. Fluids 19, FIG. 2. Schematic of experimental setup. thography step and a reactive ion etching RIE oxide removal process to mask certain areas on the wafer, which are not to be etched during the deep reactive ion etching DRIE process. The wafer is consequently etched in a DRIE process, and silicon is removed from places not protected by the photoresist/oxide mask. The DRIE process forms deep vertical trenches on the silicon wafer with a characteristic scalloped sidewall possessing a peak-to-peak roughness of 0.3 m. A profilometer and a scanning electron microscope SEM are employed to measure and record various dimensions of the device. The wafer is flipped and the backside is then processed, so that an inlet, exit, and pressure port taps for the transducers are formed. A photolithography followed by a buffered oxide etch BOE 6:1 oxide removal process is carried out to create a pattern mask. The wafer is then etched through in a DRIE process to create the fluidic ports. Finally, the processed wafer is stripped of any remaining resist or oxide layers and anodically bonded to a 1 mm thick polished Pyrex glass wafer to form a sealed device. After the completion of the bonding process, the wafer is visually inspected for any fringes or voids that are indicative of an ineffective bonding. In case any fringes are found on the wafer, the wafer stack is again taken through a bonding process in an attempt to remove the defects. After successful completion of the bonding process, the processed stack is die-sawed to separate the devices from the parent wafer. C. Experimental apparatus and procedure The schematic diagram of the experimental setup is shown in Fig. 2. The test section consists of the device with pressure ports and a packaging module to facilitate the transit of fluids to the microdevice and also to record the pressure drop across the channels. The MEMS device is placed in the package by means of o-rings, which act as hermetic seals and connect the pressure ports and inlet/outlet of the channel to the flow loop. The deionized water is delivered to the channel by means of a calibrated flow meter and a 0.2 m filter is used to avoid clogging of the device. The water flow meter Omega Engineering, Inc, FL-110 series is capable of reading flow rates from 0 to 1.24 ml/min. Nitrogen N 2 at desired pressure and flow rate is delivered to the channel through a mass flow meter Sierra Instruments, Inc., 820-S series and 0.5 m filter. The N 2 mass flow meter is capable of measuring volumetric flow rates ranging from 0 to 50 ml/ min. Before passing the two-phase mixture through the channel, the entire channel is vacuumed to 50 kpa and then flooded with water, including the pressure ports, to remove any gas bubbles trapped in the loop. After flushing the loop, nitrogen is allowed to enter the test section. Introducing the gas into the loop reduces the water flow rate due to an increase in the hydraulic resistance. The valves of the nitrogen and water loop are iteratively controlled to obtain the desired gas and liquid flow rates. The pressure ports are connected to pressure transducers Omega Engineering, Inc., each capable of measuring pressures of up to 50 psia via the packaging module. Data from the pressure transducer are delivered to the PC-based LabVIEW program and stored in a file for further analysis. The images from the device are taken by means of highspeed CMOS camera capable of capturing images up to frames per second and a maximum resolution of FIG. 3. Image before and after application of threshold.

5 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, pixels. Images from five different locations along the channel are captured for processing void fraction measurements and identifying flow patterns. In order to distinguish the gas from the liquid, an intensity threshold is applied to each image Fig. 3. This is followed by calculating the corresponding area of the gas and liquid phases using Image-pro plus software. It can also be observed that the intensity of the liquid and the pillars are of the same magnitude. Thus, for each image, the areas of the pillars are subtracted from the area of the liquid. The void fraction is then calculated as the ratio of the area of the gas to the total flow area. Any overlap between the liquid and gas intensities is accounted for by manually calculating the overlapped region and applying a suitable correction to the measured areas. Figure 4 shows the standard deviation of the void fraction measurements as a function of the number of frames. It appears that the standard deviation of the measurements does not vary significantly beyond 20 frames. Hence, all void fraction data are obtained by averaging 20 frames in the following form: 20 Number of pixels corresponding to gas in frame i g =. 12 i=1 Total number of pixels in frame i The comparison of the experimental data with existing models is done through the mean absolute error MAE, M MAE = 1 M i=1 exp pred pred 100, 13 where is the measured physical quantity and the subscripts exp and pred refer to experimental and predicted values, respectively. IV. DATA REDUCTION A. Single-phase flow The single-phase friction factor is obtained according to f = 2 P i P o l NG 2, 14 B. Two-phase flow The measured two-phase pressure drop consists of two components, namely frictional and acceleration terms, p measured = p f + p acc. 18 The frictional pressure drop is calculated once the acceleration term is determined by the following equation: 14,24 p acc = G2 x 2 G2 x 2 g + G2 1 x 2 l g + G2 1 x 2 l outlet, 19 inlet where is the time-averaged experimental void fraction. The Martinelli parameter defined in Eq. 10 is calculated using the following equations: where P i and P o are the inlet and outlet pressures and G is defined as G = Q l A min, 15 Q is the volumetric flow rate, and A min is the minimum cross-sectional flow area, which for the current staggered pillar device with a transverse pitch, S T, and longitudinal pitch, S D, is given by A min = S T D S T wh for S T + D S D Pressure drop data for different flow rates are recorded and the corresponding friction factor is elucidated for each single-phase Reynolds number defined as Re d =Gd/ l.to accurately represent the friction factor, a Blasius-type model is used, f = a Re d n. 17 FIG. 4. Standard deviation of void fraction measurements as a function of number of frames.

6 S. Krishnamurthy and Y. Peles Phys. Fluids 19, TABLE I. Uncertainty in variables. Uncertainty Error % Flow rate, Q for each reading 1 Inlet and exit pressures 0.25 Tube hydraulic diameter, D 1 Channel width, w 1 Channel height, H 0.67 Density of fluid, f 0.5 Void fraction, 3.1 Mass flux, G 3.8 P f l = Nf G 1 x 2 2 l, 20 P f g = Nf Gx g The uncertainty of the measured experimental values are listed in Table I and are derived from the manufacturers specification sheet while the uncertainties of the derived quantities are determined using the propagation of uncertainty method. V. RESULTS AND DISCUSSION A. Flow patterns Four different flow patterns were observed, namely bubbly slug, gas-slug, bridge, and annular flows. In the bubbly slug flow, bubbles and gas slugs coexist in the liquid Fig. 5. Here we define bubbles as dispersed gas in the liquid whose characteristic size is less than or equal to the spacing between the micropillars i.e., 50 m. Bubbles considerably smaller than 50 m were not detected, and the bubbles that were observed typically had nonspherical shapes as they were pressed against the pillar walls. Some of these bubbles coalesced and formed polydispersed bubbles in the liquid phase. The gas slug is defined as a continuous gas phase, which encompasses four or more micropillars. Occasionally, these slugs coalesced with smaller bubbles and formed an elongated slug just to be broken by the shear imposed by the downstream pillars. Such a flow pattern was detected for gas velocities less than 1 m/s. With increasing gas velocity, the flow pattern transitioned to purely slug flow. Here the gas traverses through the center of the channel, while the liquid FIG. 6. Image of gas slug flow with elongated gas slugs j g =3.86 m/s, j l =0.24 m/s. moved along the channel walls and pillar surfaces. Because of the staggered pillars arrangement, the gas slug tended to form finger-like structures, which extended both in the longitudinal and the traverse directions Fig. 6. With further increase in velocity, the gas occupied increasingly larger volume and the liquid slug size decreased. The gas stretched along the channel, resulting in the formation of elongated gas slugs. At even higher gas velocities, these gas slugs merged and the liquid traversed along the channel in the form of bridges Fig. 7. This flow pattern has only been previously reported before by Xu et al. 13 However, their study on flow patterns in vertical up and down flow across a tube bundle in conventional scale systems was governed by a completely different hydrodynamic mechanism. At low liquid and gas flow rates, they observed a falling film pattern where the liquid flows along the tube walls forming bridge-like structures between adjacent cylinders. Gravitational forces were a key factor in controlling such structures. In the current experiments where the characteristic length was O 100 m, FIG. 5. Images of bubbly gas slug flow j l =0.41 m/s; j g =0.822 m/s. FIG. 7. Image of liquid bridge flow, j g =14 m/s, j l =0.031 m/s.

7 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, FIG. 8. Sequence of images showing the thickening of bridges j l =0.09 m/s, j g =12.1 m/s. surface tension forces dictated such flow pattern. At some point when the drag forces exerted by the gas exceeded a certain threshold value, the liquid bridges broke and the liquid flowed downstream before forming another bridge between two adjacent pillars. While the surface tension forces tended to maintain the liquid in place, the drag force exerted by the gas tended to force the liquid bridges downstream. It should be noted that when the bridges were stationary, two adverse behaviors were observed and the thickness of the bridges was found to be unsteady. Figure 8 depicts how an initially narrow bridge gradually grew as a result of liquid supply from an adjacent bridge, and as the neighboring bridge broke, its excess liquid flowed into the bridge. The surface tension forces tended to pull the liquid surrounding the pillar and perhaps on the top and on the bottom of the walls into the bridge. As the thickness of the liquid layer around the pillar decreased, the shear forces exerted on the liquid layer by the pillars elevated and as a result hindered the liquid supply. For very thin liquid layer, other effects might begin to gradually dominate the process e.g., disjoining pressure. However, when this occurs, the mass encompassed in the liquid reservoir around the pillars is drained, and therefore hardly affects the liquid supply to the bridge. When a neighboring bridge failed to break, the liquid flow between adjacent bridges oscillated Fig. 9. The pumping of liquid between two adjacent pillars in this case was governed by the unsteady drag exerted on the bridge by the gas as a result of eddies shedding. When Bridge 1 Fig. 9 became very thin, it either broke to form a thin film around a cylinder, or it grew by the supply of liquid from the adjacent pillars. At high gas velocity, an annular flow pattern was established Fig. 10, where the flow consists of gas core with thin liquid film flowing along the pillar walls. Although the liquid bridges were still occasionally present, they tended to break owing to the increasing drag forces. Eventually, at sufficiently high gas velocity, the thin films become wavy and detached from the pillar surface. The above-mentioned flow patterns were mapped as a function of liquid and gas superficial velocities j l =Q l /A min ; j g =Q g /A min and are shown in Fig. 11 a. The flow map was also compared to the one constructed by Grant et al. 6 and Kawahara et al. 14 for conventional scale tube bundle systems FIG. 9. Sequence of images showing the thickening and thinning of bridges j l =0.09 m/s, j g =12.01 m/s. Fig. 11 b and for microchannels Fig. 11 c, respectively. Significant deviations with respect to flow patterns were found in both maps i.e., Grant et al. 6 and Kawahara et al. 14. However, some similarities in the slug-flow pattern were observed with Kawahara et al. s 14 map. In the microchannel study of Kawahara et al. 14 and Chung and Kawaji, 15 no bubbly flow was reported, while as discussed earlier in the micropillars, bubbles were observed at low gas velocities although with the existence of slugs. Bubbles form in the presence of high liquid velocities, where the liquid inertia and shear are sufficient to overcome surface tension and break the gas-liquid interface. However, with diminishing length scale low Reynolds number the inertial forces are typically not significant enough to form small bubbles, but shear forces can assist in breaking large bubbles. It is expected that this shear force is enhanced by the presence of FIG. 10. Annular flow showing the thinning of liquid bridges and their breakage.

8 S. Krishnamurthy and Y. Peles Phys. Fluids 19, FIG. 12. Void fraction profile along the channel length for two different conditions j g =0.6 m/s; j l =0.41 m/s; j g =1.16 m/s; j l =0.3 m/s. force tends to stretch the gas cavity and form elongated slugs. Another noticeable deviation between the current flow map and the one developed by Grant et al. 6 was the absence of stratified flow pattern. Since stratified flow occurs due to gravitational forces, it is not expected to prevail in the current microscale system. FIG. 11. a Flow map for the current system, b comparison of the current flow map with the map of Grant et al. Ref. 6 developed for horizontal flow across a tube bundle arranged in equilateral configuration with a pitch to diameter ratio of 1.25, c comparing the flow map with the map of Kawahara et al. Ref. 14 developed for a 100 m plain microchannel. micropillars. This in turn results in the fission of gas and the formation of bubbles as small as the system characteristic length scale 50 m. The bubbles formed more readily at low gas velocities, but at elevated velocities, the inertial B. Void fraction Figure 12 shows the void fraction distribution along the micropillars for two representative gas/liquid flow velocities. The distribution changed little along the channel suggesting that the gas and liquid were well mixed and the two-phase flow developed rapidly over a small number of rows. The short developing length is expected since the Reynolds number was very low. The experimental result of the void fraction as a function of mass quality is shown in Fig. 13. Similar to conventional scale results, 8 12 the data are well below the homogeneous void fraction h, especially at low qualities, indicating significant slip velocities. Even at low gas velocities and qualities, considerable slip between the two phases was observed. For example, at a quality of the void fraction was 0.3, which resulted in a slip ratio as high as This high slip ratio suggests a weak momentum coupling between the gas and the liquid phases. The results were also compared with model developed by Kawahara et al. 14 for microchannels, given by = h h As clearly shown in Fig. 13, the above model predicts the data much better than the homogeneous model. Equation 22 underpredicts the results at low qualities, but provides increasingly better predictions at high qualities. Nevertheless, there still exists a considerable discrepancy of more than 20% between the data and the model under some conditions Fig. 14 a. Schrage et al. 11 also modeled the void

9 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, FIG. 13. Comparison of experimental void fraction data with the homogeneous model and the model of Kawahara et al. Ref. 14 for a microchannel., Experimental data; -----, the model of Kawahara et al. Ref. 14 ;, the homogeneous model. fraction in terms of the Froude number Fr and mass quality, = h Fr ln x, 23 where the Froude number Fr is defined as Fr = G l gd. 24 Surprisingly, their model adequately predicted the experimental results Fig. 14 b. Although the model signifies the dominance of the gravitational forces as implied by the use of the Froude number, 90% of the predicted values fall within ±20% of the experimental data. It is clear that the Froude number has no physical significance in microdomains, and the accurate prediction of the experimental results by Eq. 23 requires further examination. Since the experiments were conducted at low Reynolds number, viscous force is a key factor in controlling the flow patterns and void fraction. The agreement shows that the above model, despite its accurate prediction of the experimental data in both microscale and conventional scale systems, is not in its physically correct form. The gravitational acceleration, g, and the densities were kept constant in both the study of Schrage et al. 11 and in the current study, and only the ratio G/ D was apparently varied. A nondimensional term, which has a similar dependence on mass flux and characteristic length scale, is the friction factor, f, defined as f = 2 l dp/dx D G l Since friction forces are important in both the abovementioned scales, it is perhaps more appropriate to substitute the Froude number by the friction factor in 23. Thus, the above model is modified according to FIG. 14. Comparison of experimental data for void fraction with predicted values of models from a Kawahara et al. Ref. 14 and b Schrage et al. Ref. 11. h =1+A 1 f A 2ln x, 26 where A 1 and A 2 are empirically best fitted constants, which are found to be and 0.34, respectively, and provided MAE of 20.56%. In order to accurately predict the results at the microscale, the above model, though in physically correct form, needs to include the effect of surface tension and wettability. These parameters are important to determine the flow patterns and, hence, the void fraction. Therefore, to study the effects of surface tension and wettability on the void fraction, investigation using fluids with different surface tensions or with surface/fluid chemistry modifications is necessary.

10 S. Krishnamurthy and Y. Peles Phys. Fluids 19, FIG. 15. Comparison of single-phase friction factor data with the model., Experimental data; -----, the model of Koşar et al. Ref. 25 ; experimental data Koşar et al., Ref. 25. C. Pressure drop 1. Single-phase flow Single-phase pressure drop as a function of the Reynolds number, is shown in Fig. 15. A Blasius-type friction factor model was used to fit the liquid single-phase flow according to f = Re d Re d The above equation predicted the experimental data with a MAE of ±4.4%. A similar pillar configuration was previously studied by Koşar et al. 25 for single-phase flow. It is apparent that the model suggested by Koşar et al. 25 and Eq. 27 agree well and, therefore, Eq. 27 was used for subsequent analysis of the two-phase pressure drop. 2. Two-phase flow Figure 16 shows the variation of the pressure along the length of the channel. The linear dependency of the pressure drop on the longitudinal position coupled with the uniform distribution of the void fraction Fig. 12 further indicate that the two-phase flow is fully developed and well mixed. The homogeneous pressure drop model was compared to the experimental results with four viscosity models. As discussed earlier, the gas-liquid slip velocity was significantly large, and therefore it was not expected to predict the twophase frictional pressure drop adequately, as indeed was the case Fig. 17. Cubaud and Ho 17 modeled their microchannel two-phase frictional pressure drop as a function of the homogeneous void fraction and the single-phase liquid pressure drop by using the following piecewise flow pattern based equations: P tp = P l 1 h 1/2 for gas slug, annular, and dry flow, 28 FIG. 16. Pressure profile along the length of the channel for different patterns. P tp = P l 1 h 1 for bubbly flow. 29 Since purely bubbly flow has not been observed in the current study, the two-phase pressure drop predicted from Eq. 28 has been compared with the experimental data Fig. 18. Although the model provided a much better prediction for the micropillar results, both in absolute values and trend, it underpredicts the pressure drop by as much as 50% with MAE of 30%. The frictional multiplier, when plotted as a function of the viscous-viscous Martinelli parameter Fig. 19, revealed a clear dependence on the mass flux, where the frictional multiplier increased with increasing mass velocity. The dependence is more apparent at low X vv higher gas velocities than at high X vv lower gas velocities. Dependence of the frictional multiplier on mass flux has previously been reported also by Dowlati et al. 10 and Schrage et al. 11 However, in these studies the frictional multiplier was found to increase or decrease with mass flux with increasing values of the turbulent-turbulent Martinelli parameter, X tt. With bubbly flow, the frictional multiplier was found to decrease with increasing mass flux Schrage et al. 11 since the flow tended to better follow the homogeneous flow model assumption i.e., no gas/liquid slip. Such a trend has not been observed here, perhaps because the flow never assumes completely bubbly flow. A similar dependence of the frictional multiplier on the mass flux was also reported by Chung and Kawaji 15 in microchannels. However, it was argued that this dependence diminishes with decreasing channel diameter. For channel hydraulic diameter less than 50 m, the frictional multiplier was independent of the mass flux. Although the micropillars have similar characteristic length scale i.e., 50 m, the mass flux dependence in the micropillar device was significant. The characteristic of the frictional multiplier with respect to mass flux dependence is perhaps more sensitive to flow geometry than to the characteristic length scale. It should be noted that the physical process controlling the dependence of the frictional multiplier on the mass velocity is

11 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, FIG. 17. Comparison of experimental data with the predicted frictional pressure drop from homogeneous models proposed by a Cicchhitti et al. Ref. 18, b McAdams Ref. 21, c Dukler et al. Ref. 19, and d Beattie and Whalley Ref. 20. still not very well understood even at a conventional scale. Table II lists the different models used to compare the current data along with their mean average error. Figure 19 compares the experimental data with the predictions from these models when both the gas and liquid flows are assumed to be laminar. Chisholm s model equation 9 for laminar flow, with the C parameter obtained by fitting the experimental data, is also shown in the figure C= The disagreement between the predicted and experimental data suggests that the model needs to account for the mass flux dependence. Since none of these models accounts for the mass flux dependence, all resulted in appreciable errors. However, it is apparent that the models developed for microchannels predicted the micropillar results much better than conventional tube bundle models. This suggests that the characteristic length scale is far more dominant in dictating the hydrodynamic characteristics than the detailed flow configurations. FIG. 18. Comparison between the predicted two-phase pressure drop from the model of Cubaud and Ho Ref. 17 and experimental data.

12 S. Krishnamurthy and Y. Peles Phys. Fluids 19, FIG. 19. Mass flux dependence and comparison of various models with the experimental data. A model that does account for mass velocity and flow pattern was developed by Schrage et al. 11 according to L 2 =1+ C + C 5 2 X tt X, 30 tt where C is a constant defined as C = C 1 Fr C 2ln X tt + C 3 Fr C 4, 31 C 1, C 2, C 3, and C 4 are empirically derived coefficients that vary depending on the flow pattern. Figure 20 a compares the experimental data for the two-phase frictional multiplier FIG. 20. Comparison of experimental data with the model of Schrage et al. Ref. 11. with Eqs. 30 and 31. A large discrepancy between the predicted and experimental data is evident, which might be linked to the presence of the Froude number Fr and the use of the turbulent-turbulent Martinelli parameter X tt. Obviously, the Froude number cannot be used to predict the current microscale frictional multiplier, and the turbulentturbulent Martinelli parameter needs to be substituted by the viscous-viscous Martinelli parameter. 3. New model In an attempt to develop an improved model based on both the relevant scaling effects observed in this study and adoption from previous studies, a modified form of Eq. 9 has been developed to better predict the experimental result, TABLE II. C factors used in different experiments with the corresponding MAE. No. Reference Geometry Type of flow C factor MAE % 1 Chisholm et al. Ref. 23 Tube Laminar/Turbulent C=5 for laminar 170 C=10 for turbulent Kawahara et al. Ref. 14 Microchannel Laminar C= D h =100 m 3 Mishima and Hibiki Ref. 27 Tube Laminar/Turbulent C=21 1 exp 0.319D h 26 4 Dowlati et al. Ref. 10 Horizontal staggered tube bundle P/ D=1.3/ 1.75 Turbulent C= Dowlati et al. Ref. 9 Horizontal inline tube bundle P/ D=1.3/ Schrage et al. Ref. 11 Inline tube array P/ D=1.3 7 Chung and Kawaji Ref. 15 Microchannel D h =530 m, 250 m, 100 m, 50 m Turbulent C=8 320 Turbulent C=C1 Fr C2 ln X tt +C3 Fr C4 567 Laminar C=3.18 for D h =530 m C=1.74 for D h =250 m C=0.22 for D h =100 m C=0.15 for D h =50 m 23.88

13 Gas-liquid two-phase flow across a bank of micropillars Phys. Fluids 19, TABLE III. The values of the constant B for different flow patterns and their respective MAEs. Flow patterns B MAE % Bubbly gas/gas slug flow % Bridge flow % Annular flow % All the flow patterns % L 2 =1+ B Re d X. 32 vv X vv The constant B is an empirically defined constant, which is obtained by fitting the experimental data using the leastsquares method. 26 The values of the constant for the individual flow patterns and for the entire data, along with their mean average errors, are shown in Table III. A total of 90% of the data fall within ±20% of the model when using a flow-pattern-dependent model Fig. 21, while 90% of the data fall within ±25% of the predicted values while using all of the data Fig. 22. The above result indicates that the frictional multiplier is less sensitive to the flow pattern and strongly dependent on the liquid Reynolds number, contrary to some conventional scale models e.g., Schrage et al. 11. VI. CONCLUSION Two-phase flow across a bank of micropillars entrenched inside a microchannel was experimentally studied. Flow pattern, two-phase pressure drop, and void fractions were obtained, discussed, and compared to conventional and microscale models. Both similarities and deviations to conventional scale and other microchannels studies were FIG. 21. Comparison of the prediction from the new model dependent on the flow patterns with experimental data. FIG. 22. Comparison of the prediction from the new model independent of the flow patterns with experimental data. found. The main conclusions derived from this study are presented below: 1 A unique flow pattern driven by surface tension was observed in the current study and termed bridge flow. 2 Flow patterns were mapped and compared with conventional scale as well as microchannel flow maps. Significant deviations from conventional scale system were observed both in terms of flow patterns and transition lines. Microchannel flow patterns better predicted the trend observed in the current study. 3 The homogeneous void fraction was compared to the measured void fraction and was found to overpredict the data significantly, especially at low qualities. 4 Comparison of void fraction data with other conventional scale models revealed good agreement. However, the conventional scale model was found not to be in the correct physical form. A new model was proposed. 5 The two-phase frictional multiplier was modeled in terms of the Martinelli parameter and was found to be dependent on mass flux, but much less on flow pattern. Existing models for conventional scale systems did not predict well the two-phase frictional multiplier, perhaps because the mass velocity was not accounted for. 6 While most conventional scale results showed that the frictional multiplier strongly depend on the mass flux, the results for microchannels did not show such a dependence. This contradicting result for two microscale systems for the two-phase frictional multiplier suggests that both length scales and flow configuration strongly affect the two-phase pressure drop. 7 A modified form of the Chislom et al. 6 model was developed, which includes the mass velocity effects. The proposed model was able to predict the experimental data with an MAE of 20% and 25%, for flow-patterndependent and -independent models, respectively. The result suggests that the model is less sensitive to flow

14 S. Krishnamurthy and Y. Peles Phys. Fluids 19, pattern and more sensitive to the liquid Reynolds number, as opposed to some conventional scale studies. ACKNOWLEDGMENTS This work was supported by the Office of Naval Research through the Young Investigator Program, under Contract No. N Program Officer: Dr. Mark Spector. Graduate student support from Rensselaer Polytechnic Institute is also gratefully appreciated. The microfabrication was performed in part at the Cornell NanoScale Facility a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation under Grant No. ECS , its users, Cornell University, and industrial affiliates. 1 M. W. Losey, J. Jackman, S. L. Firebaugh, M. A. Schmidt, and K. Jensen, Design and fabrication of microfluidic devices for multiphase mixing and reaction, J. Microelectromech. Syst. 11, D. L. Hitt, C. M. Zakrzwski, and M. A. Thomas, MEMS-based satellite micropropulsion via catalyzed hydrogen peroxide decomposition, Smart Mater. Struct. 10, L. A. Christel, K. Peterson, W. McMillan, and M. A. Northrup, Rapid automated nucleic acid probe assays using silicon microstructures for nucleic acid concentration, J. Biomech. Eng. 121, A. London, Development and testing of a microfabricated bipropellant rocket engine, Ph.D. thesis, Massachusetts Institute of Technology, Boston A. Koşar and Y. Peles, Boiling heat transfer in a hydrofoil-based micro pin fin heat sink, Int. J. Heat Mass Transfer 50, I. D. R. Grant and D. Chislom, Two-phase flow on the shell-side of a segmentally baffled shell-and-tube heat exchanger, J. Heat Transfer 101, G. P. Xu, C. P. Tso, and K. W. Tou, Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle, Int. J. Multiphase Flow 24, R. Dowlati, M. Kawaji, and A. M. C. Chan, Void fraction and friction pressure drop in two-phase flow across a horizontal tube bundle, AIChE Symp. Ser. l84, R. Dowlati, M. Kawaji, and A. M. C. Chan, Pitch-to-diameter effect on two-phase flow across an in-line tube bundle, AIChE J. 36, R. Dowlati, A. M. C. Chan, and M. Kawaji, Hydrodynamics of twophase flow across horizontal in-line and staggered rod bundle, ASME Trans. J. Fluids Eng. 114, a. 11 D. S. Schrage, J-T. Hsu, and M. K. Jensen, Two-phase pressure drop in vertical cross-flow across a horizontal tube bundle, AIChE J. 34, G. P. Xu, K. W. Tou, and C. P. Tso, Two-phase void fraction and pressure drop in horizontal crossflow across a tube bundle, J. Fluids Eng. 120, J. Xu, Experimental study on gas-liquid two-phase flow regimes in rectangular channels with mini gaps, Int. J. Heat Fluid Flow 20, A. Kawahara, P. M.-Y. Chung, and M. Kawaji, Investigation of twophase flow pattern, void fraction and pressure drop in a microchannel, Int. J. Multiphase Flow 28, P. M.-Y. Chung and M. Kawaji, The effect of channel diameter on adiabatic two-phase flow characteristics in microchannels, Int. J. Multiphase Flow 30, M. Kawaji and P. M. Y. Chung, Adiabatic gas-liquid flow in microchannels, Microscale Thermophys. Eng. 8, T. Cubaud and C.-M. Ho, Transport of bubbles in square microchannels, Phys. Fluids 16, A. Cicchitti, C. Lombardi, M. Silvestri, G. Solddaini, and R. Zavalluilli, Two-phase cooling experiments Pressure drop, heat transfer and burnout measurement, Energ. Nucl. Milan 7, A. E. Dukler, M. Wicks III, and R. G. Cleveland, Pressure drop and hold-up in two-phase flow, AIChE J. 10, D. R. H. Beattie and P. B. Whalley, A simple two-phase flow frictional pressure drop calculation method, Int. J. Multiphase Flow 8, W. H. McAdams, Heat Transmission, 3rd ed. McGraw-Hill, New York, R. W. Lockhart and R. C. Martinelli, Proposed correlation of data for isothermal two-phase, two-component flow in pipes, Chem. Eng. Prog. 45, D. Chisholm and A. D. K. Laird, Two-phase flow in rough tubes, Trans. ASME 80, V. P. Carey, Liquid-Vapor Phase-Change Phenomena Taylor & Francis, London, A. Koşar, C. Mishra, and Y. Peles, Laminar flow across a bank of low aspect ratio micro pin fins, J. Fluids Eng. 127, J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering Prentice Hall, Englewood Cliffs, NJ, 1992, Vol K. Mishima and T. Hibiki, Some characteristics of air water two-phase flow in small diameter vertical tubes, Int. J. Multiphase Flow 22,