Validation of a CFD model of Taylor flow hydrodynamics and heat transfer

Chemical Engineering Science 69 (2012) 541–552

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Validation of a CFD model of Taylor flow hydrodynamics and heat transfer Azadeh N. Asadolahi, Raghvendra Gupta, Sharon S.Y. Leung, David F. Fletcher n, Brian S. Haynes School of Chemical and Biomolecular Engineering, The University of Sydney, NSW, Australia

a r t i c l e i n f o

abstract

Article history: Received 5 September 2011 Received in revised form 2 November 2011 Accepted 10 November 2011 Available online 19 November 2011

The growing importance of the Taylor flow regime in microchannel flow and heat transfer has led to the need for validated CFD models. Experimental data for Taylor flow of water/nitrogen and ethylene glycol/nitrogen in a 2 mm diameter channel having Reynolds numbers in the range 22–1189 and Capillary number of 0.003–0.160 are used to validate a CFD model developed in ANSYS Fluent. The model simulates two-dimensional, periodic flow and heat transfer in a unit cell (comprising a single bubble and its adjacent half slugs) in a frame of reference moving with the bubble. The simulation results show excellent agreement with the bubble shape, film thickness, bubble velocity and homogeneous void fraction. The above data and the pressure drop data are compared with the available correlations and are shown to agree well, linking the experimental data, CFD results and established correlations in one validation exercise. Heat transfer simulations also reproduce the data well with a maximum difference of 15% except for high Reynolds number cases (ReTP Z 951). It is shown that in this case the assumption of two-dimensional, axisymmetric flow is no longer valid. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Taylor bubble Microchannels Multiphase flow Slug flow Fully-developed H2 thermal boundary condition

1. Introduction The Taylor flow regime, characterised by the regular and periodic flow of capsular gas bubbles of the size of the channel in a continuous liquid stream, that occurs in gas–liquid flow in microchannels is an area of enormous interest due to its many and varied applications in micro-technology and micro-structured chemical plant. There are an ever increasing number of studies of the hydrodynamics, heat transfer and mass transfer in such flows using a range of experimental and computational techniques. Recent reviews include those of Angeli and Gavriilidis (2008) and Gupta et al. (2010a). Our group has carried out experimental studies of Taylor flow in microchannels using a bubble generation mechanism that allows us to produce gas bubbles and liquid slugs of controlled and highly repeatable lengths. We have used this methodology to perform detailed investigations of the hydrodynamics and heat transfer for water/nitrogen, 50 wt% ethylene glycol/water mixture/nitrogen and ethylene glycol/nitrogen systems allowing us to study a wide range of Reynolds and Capillary numbers (Leung et al., 2010, in press). In parallel with this experimental work we have developed CFD modelling techniques, based around the commercial software ANSYS Fluent, to study both the hydrodynamics and heat transfer behaviour of this system in a two-dimensional, axisymmetric

n

Corresponding author. E-mail address: david.fl[email protected] (D.F. Fletcher).

0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.11.017

computational domain. Initial work used simulations in a stationary frame of reference and a long tube (Gupta et al., 2009, 2010b) but this approach was found to be computationally very expensive. An alternative approach, using a single unit cell consisting of a gas bubble and a liquid slug and a moving mesh, has proved to be both reliable and computationally efficient to study fully-developed Taylor flow (Asadolahi et al., 2011). In this paper we unite the experimental and computational approaches to perform a detailed validation study and to determine the range of validity of the two-dimensional modelling approach developed. The remainder of the paper is laid out as follows: Section 2 describes the necessary details of the experimental system; Section 3 describes the CFD model; Sections 4 and 5 present the detailed comparison between the CFD and experimental data for the hydrodynamics and heat transfer, respectively; and Section 6 gives the conclusions of the study.

2. Summary of the experimental setup and data 2.1. Experimental setup The experimental facility was designed to determine the mean heat transfer rates and the hydrodynamic characteristics of vertical upward two-phase Taylor flow in a 2.00 mm diameter circular channel. Three Swagelok T-junctions of different internal diameters (2.27 mm, 3.10 mm and 4.81 mm) were used to generate bubbles and slugs of different lengths for the same mixture velocity and homogeneous void fraction. The data obtained from

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the smallest T-mixer were used in this comparison. Visualisation studies were conducted in a 300 mm long silica tube of 2.00 mm internal diameter using a high-speed camera under isothermal condition (21 1C). Backlit images of the two-phase flow were recorded at rates of 250–1000 frames per second (fps), adjusted according to the bubble and slug lengths. The bubble velocity, U B , bubble length, LB , slug length, LS , and bubble frequency, F B were determined by performing a frame-by-frame analysis, as described in Leung et al. (2010). The difference between the refractive indices of air (RI ¼1.000) and the silica tube (RI¼1.459) and that of water (RI¼1.333) and silica tube (when water was used as the liquid phase) gave rise to optical distortion from the outer and inner curved surfaces of the tube, respectively. The effect of refraction at the outer surface was corrected using a square refractive index matching (RIM) system filled with ethylene glycol (RI ¼1.432) for all fluid systems. The choice of ethylene glycol as a test liquid was in part motivated by the fact that the use of ethylene glycol also eliminates the refractive index mismatch at the inner surface and as will be seen later allows accurate bubble shapes to be determined. The heat transfer rate was determined using a multi-block heating system consisting of a copper tube (again 2.00 mm internal diameter) to which ten 25 mm long cylindrical copper blocks of 25 mm in diameter were soldered. Each block was in turn surrounded by a separate resistance band heater, and the blocks and heaters were isolated from their neighbours by a fibre sheet of 2 mm thickness. This system has been used to study twophase heat transfer (Bao et al., 2000b) and boiling (Baird et al., 2000; Bao et al., 2000a) and is described in detail in these references. As discussed in Leung et al. (2010), this experimental arrangement does not correspond strictly to any standard thermal boundary condition even though the heat flux in each heating zone is approximately the same. Although the apparatus returns pseudo-local heat transfer coefficients for steady, developing single-phase flows in agreement with analytic solutions for a constant heat flux thermal boundary condition (see Bao et al., 2000b), the long thermal time constant associated with the heating blocks (  200 s) means that its response to a pulsatile flow with a frequency of a few Hz represents an average for the flow. In the numerical simulations presented in the following sections, a constant wall heat flux boundary condition has been employed to compare with the experimental data. 2.2. Working fluids and experimental conditions Nitrogen was used as the gas phase and three fluids water, 50 wt% ethylene glycol/water mixture (EG/W) and pure ethylene glycol (EG), were employed as the liquid phases. Using these three fluids a wide range of Capillary numbers (0.001oCao0.180) for a similar range of mixture velocities (0.10oUTP o0.53 m s  1) were achieved. Here we consider only the water and ethylene glycol experiments, as these provide data that cover a sufficiently wide range of conditions. While ethylene glycol gives high Capillary numbers (0.037oCao0.177) and negligible inertia (10oReTP o65), water gives low Capillary numbers (0.001oCao0.007) and high inertia (210oReTP o1100). 2.3. Uncertainty analysis Through careful design of the mixing section, stable and reproducible Taylor flow patterns were achieved. For the experiments using ethylene glycol the pressure drop along the test section was large enough (inlet pressures of 170 kPa as compared with 105 kPa in the case of water and an exit pressure of 101 kPa) to give rise to significant changes in gas volume and mixture

velocity from the inlet to the exit. The magnitude of this effect was estimated from the expected fraction of the overall system pressure drop that occurred between the inlet pressure measurement (in the liquid, upstream of the mixer) and the observation point, being typically about 35% of the total pressure drop. For the worst case (ethylene glycol, UTP ¼0.42 m s  1, b ¼0.42, LS/d  2.3), the bubble volume was calculated to increase by 13% for the isothermal condition, however this case was excluded from the comparison. This calculation was confirmed experimentally by comparing the bubble lengths at the entrance and exit of the visualisation test section at room temperature. The variability in the mixture velocity and bubble length due to pressure drop effects was therefore o13% in the worst case, and considerably less than that in general. In the heat transfer experiments, the wall heat flux was controlled to ensure that the temperature rise was never more than 20 1C (and, typically, o10 1C). Therefore, the volume change due to thermal expansion of the gas was never more than a few %. In correlating the results from individual blocks, local estimates of the temperature and pressure were used; however, for the purposes of reporting characteristic conditions (such as velocity and homogeneous void fraction) for each heat transfer condition, the temperature and pressure at the middle of the test section were used. Measurements of local single-phase heat transfer coefficients in thermally developing laminar flow agreed with theoretical predictions to within 10%. An additional source of error in the two-phase flow heat transfer experiments is the estimation of the energy going to evaporation, but this was shown to be a minor effect ( o2%) because of the low working temperature (20–40 1C). The overall uncertainty in NuTP was assessed to be no more than 15% of the measured value.

3. Computational model The Volume-of-Fluid (VOF) method was used to model vertical upward Taylor flow in a microchannel. In the VOF method, a single set of mass, momentum and energy conservation equations are solved together with an additional advection equation to calculate the volume fraction of one of the phases. The bulk properties used in the conservation equations are the volume fraction-weighted average of the properties of the two fluids. The surface tension force which plays an important role in two-phase gas liquid flow in microchannels is approximated as a body force in the vicinity of the gas–liquid interface using the continuum surface force (CSF) model (Brackbill et al., 1992). The detailed equations can be found in our previous works (Gupta et al., 2009; Asadolahi et al., 2011). ANSYS Fluent, a commercial CFD software package, was used to solve the governing equations. The flow of gas bubbles and liquid slugs is regular and periodic in Taylor flow and therefore the hydrodynamic and thermal characteristics can be studied by modelling the flow in a representative unit cell consisting of a gas bubble and two halves of adjacent liquid slugs. A moving domain methodology (Asadolahi et al., 2011) in which the computational domain is moved with the bubble velocity (and thus the location of the bubble is kept fixed in the domain) to model the periodic Taylor flow in the unit cell was used to model fully-developed Taylor flow. The bubble velocity is not known a priori and is calculated at each time step as explained in Asadolahi et al. (2011). First, transient simulations for flow are performed in a two-dimensional, axisymmetric computational domain until a steady flow field and bubble shape are achieved. Once the steady solution was obtained, the flow field (continuity, momentum and VOF equations) was ‘frozen’ and only the energy equation was solved until a steady non-dimensional temperature

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field was obtained. However, this decoupling of flow and energy equations was not possible for high Reynolds number cases as the flow field was unsteady. The flow was assumed to be axisymmetric, laminar and incompressible. The physical properties of the working fluids used in the simulations were specified as being constant to allow periodic flow and heat transfer simulations to be performed. As the flow is modelled only in a unit cell, it is reasonable to neglect the gas density variation because of the small change in temperature and pressure over a unit cell (and therefore to assume the flow to be incompressible). It is noteworthy that the experimental hydrodynamics results were collected at a temperature of 21 1C while the temperature changes during the experimental study of heat transfer. Therefore, the constant properties used for the modelling of hydrodynamics were specified at a temperature of 21 1C for the ethylene glycol/nitrogen system. The temperature used for the constant properties for the water/nitrogen system was selected approximately in the middle range of the experimental temperatures for the heat transfer cases (25 1C). However, some representative simulations for the water/nitrogen system were repeated for the hydrodynamics using the properties evaluated at a temperature of 21 1C. These simulations did not show any appreciable change in the hydrodynamics or heat transfer confirming the validity of the assumption. The properties of water and ethylene glycol for temperatures ranging from 20 1C to 40 1C were determined from the steam tables (IAPWS-IF97 database) and Sun and Teja (2003), respectively and the values used are given in Table 1. The diameter of the channel was 2.00 mm, the same as used in the experiments and the length of the domain was set equal to the experimentally determined unit cell length in each case.

3.1. Boundary and initial conditions At the upstream and downstream boundaries of the unit cell, periodic boundary conditions for the velocity components and scaled temperature were applied. The details of the implementation of the periodic boundary condition can be found in Asadolahi et al. (2011). At the channel wall, a no-slip boundary condition and a constant wall heat flux boundary condition were applied for the momentum and energy equations, respectively. As the gas bubble always remains inside the computational domain, the initial bubble volume must be set equal to the value obtained in the experiments. The initial bubble shape was specified to be cylindrical having the same volume as that of the gas bubble obtained from the experiments. For ethylene glycol, the bubble shape could be captured clearly in the experiments and therefore the volume of the gas bubble was determined directly from the images. In the water experiments, accurate measurement of the bubble shape was not possible. Therefore, the value of the gas hold-up (eG ) was obtained using

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Eq. (1), which can be derived from a simple volume balance

eG b

¼

U TP UB

ð1Þ

where the mixture velocity (UTP) and the homogeneous void fraction (b) are already known and bubble velocity (UB) is measured in the experiments. The volume of the gas bubble was obtained by multiplying the gas void fraction (eG ) by the unit cell volume. 3.2. Numerical methods The convective terms in the momentum and energy equations were discretised using the QUICK scheme. An explicit geometric reconstruction scheme, based on the piecewise linear interface calculation (PLIC) method (Youngs, 1982), was applied to keep the gas–liquid interface sharp. A maximum Courant number of 0.25 was used for the volume fraction equation. A first order noniterative time-marching scheme having a variable time step based upon a Courant number of 0.25 (giving a time step of the order of 1 ms) was used for the integration of the transient term while solving the flow field (flow and volume fraction equations) whereas a time step of typically 0.001 s was used for timemarching of the energy equation (when the flow-field was frozen). The fractional step co-located scheme, in which pressure and velocity were stored at the cell centres, was used for the pressure–velocity coupling and a body-force-weighted interpolation scheme was chosen to compute the face pressure. The ‘‘implicit body force’’ treatment was used. In our previous work, we carried out a detailed mesh resolution study (Gupta et al., 2009) and validated the numerical methods used by comparing with another interface capturing technique, level-set method (Gupta et al., 2010b). The mesh resolution and numerical methods used here reflect these best practices.

4. Hydrodynamics comparison In the simulations, the mixture velocity (UTP) and void fraction (eG) were specified as input parameters and the bubble velocity (UB) was obtained as a result of the simulations. The homogeneous void fraction (b) was then obtained using Eq. (1). In this section, the hydrodynamic characteristics of the flow, including the bubble shape, film thickness, bubble and slug lengths, bubble velocity and homogeneous void fraction obtained from the CFD simulations are compared with those obtained from our experiments. In addition to this, the pressure drop over a unit cell obtained from the simulations has been compared with the correlations available in the literature. Our experimental results showed an asymmetric bubble shape (as shown in Fig. 1) and the presence of unsteady ripples at the bubble tail for high mixture velocities of 0.42 m s  1 and 0.53 m s  1 for the water/nitrogen system, invalidating the assumption of

Table 1 Properties of the gas and the liquids used in the simulations. Fluid

Density (kg m  3)

Dynamic viscosity (kg m  1 s  1)

Thermal conductivity (W m  1 K  1)

Specific heat capacity (J kg  1 K  1)

Surface tension (N m  1)

Water at 25 1C Nitrogen at 25 1C

997.0 1.145

8.90  10  4 2.10  10  5

0.6000 0.0242

4182 1040

0.072

Ethylene glycol at 21 1C Nitrogen at 21 1C

1114 1.161

2.04  10  2 2.08  10  5

0.2510 0.0242

2627 1040

0.048

Ethylene glycol at 30 1C Nitrogen at 30 1C

1107 1.126

1.40  10  2 2.12  10  5

0.2530 0.0242

2707 1040

0.048

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Table 3 Flow conditions for the simulations of the water/nitrogen system. Case no.

Re at 25 1C

Ca at 25 1C

UTP (m s  1)

LUC/d

LS/d

7.68 7.82 8.38 10.08 8.40 6.97 6.90 7.29 7.93

4.28 3.57 3.05 2.74 6.43 4.57 3.86 3.37 2.87

Unsteady cases for the experiments, but steady for the simulations 10 951 0.0052 0.42 0.3 0.26 8.32 11 951 0.0052 0.42 0.4 0.35 6.83 12 951 0.0052 0.42 0.5 0.44 6.97 13 951 0.0052 0.42 0.6 0.53 7.32 14 951 0.0052 0.42 0.7 0.61 8.29

5.53 3.77 3.21 2.69 2.23

Unsteady cases for the experiments and simulations 15 1189 0.0066 0.53 0.4 0.34 16 1189 0.0066 0.53 0.5 0.43

5.31 3.52

Exp. b

Sim. input eG

Steady cases for the experiments and simulations 1 475 0.0026 0.21 0.4 0.36 2 475 0.0026 0.21 0.5 0.44 3 475 0.0026 0.21 0.6 0.53 4 475 0.0026 0.21 0.7 0.62 5 713 0.0040 0.32 0.2 0.18 6 713 0.0040 0.32 0.3 0.27 7 713 0.0040 0.32 0.4 0.36 8 713 0.0040 0.32 0.5 0.45 9 713 0.0040 0.32 0.6 0.53

9.37 7.73

Fig. 1. (a) Experimental images of bubble heads and tails at four different mixture velocities with b ¼ 0.5 for the water/nitrogen system. Fig. 1 (b) and (c) show the unsteady bubble tail for mixture velocities of 0.42 and 0.53 m s  1, respectively.

Table 2 Flow conditions for the simulations of the ethylene glycol/nitrogen system. Case no.

Re at 21 1C

Ca at 21 1C

UTP (m s  1)

Exp. b

Sim. LUC/d input eG

LS/d

1 2 3 4 5 6 7 8 9 10 11 12 13

22 22 22 22 22 22 32 32 31 40 40 40 40

0.0850 0.0850 0.0850 0.0850 0.0850 0.0850 0.1233 0.1233 0.1190 0.1573 0.1573 0.1573 0.1573

0.20 0.20 0.20 0.20 0.20 0.20 0.29 0.29 0.28 0.37 0.37 0.37 0.37

0.17 0.26 0.36 0.46 0.57 0.68 0.34 0.45 0.55 0.32 0.42 0.54 0.66

0.11 0.18 0.26 0.31 0.39 0.46 0.24 0.31 0.36 0.19 0.25 0.34 0.39

10.09 6.99 5.27 4.32 3.37 2.79 4.66 3.78 3.10 3.67 3.23 2.57 2.22

12.47 9.91 8.74 8.57 8.57 9.49 7.64 7.46 7.73 5.87 5.87 6.35 7.28

fully-developed flow and axisymmetric bubble shape and flow for these cases. However, we have performed two-dimensional, axisymmetric simulations for these cases to determine how well a two-dimensional approach performs. Unsteady bubble tail shapes were not observed for the ethylene glycol/nitrogen system in either the simulations or the experiments because of the low Reynolds numbers (22oReTP o40) studied. The flow conditions, relevant non-dimensional numbers and non-dimensional slug and unit cell lengths for the cases simulated for the ethylene glycol/nitrogen and water/nitrogen systems are given in Tables 2 and 3, respectively.

Fig. 2. Comparison of the bubble shapes obtained from the simulations and experiments for the ethylene glycol/nitrogen system with UTP ¼0.20 m s  1 and different values of homogenous void fractions (b). The simulation results are shown by dashed lines. The bubbles were aligned at the nose.

4.1. Bubble shape The gas–liquid interface locations in the experimental images obtained for the ethylene glycol/nitrogen system were extracted using the DigitizeIt software. In Fig. 2, the bubble shapes from the experiments and simulations are compared for the mixture velocity of 0.20 m s  1 at various homogenous void fractions (b). The experimental and simulation bubble lengths differ by only  1%. The bubble shapes obtained from simulations are very close to those obtained from experiments. The shape of the bubble from the nose to the constant thickness film region is similar for all four homogeneous void fractions. For the lowest homogeneous void fraction of 0.17, a constant film thickness region is not observed. A comparison of the bubble shapes by matching the tails together (not shown here) confirmed that the tail regions are also similar for all of the homogeneous void fractions. The effect of the mixture velocity (UTP) on the experimental and simulated bubble shapes is shown in Fig. 3 for the cases having experimental homogenous void fractions (b) in the range 0.54–0.57. It is worth pointing out that though the homogeneous void fractions are the same for the three cases, the unit cell

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lengths and hence unit cell volume and bubble volumes are different among the cases. As shown in the figure, the bubble shape for different mixture velocities has been captured very well

Fig. 5. Comparison of the shapes of bubble nose for ethylene glycol/nitrogen (UTP ¼ 0.2 m s  1, b ¼ 0.57) and water/nitrogen (UTP ¼ 0.21 m s  1, b ¼ 0.6) systems. The bubbles were aligned at the nose. Fig. 3. The effect of the mixture velocity (UTP) on the bubble shape for the experimental homogenous void fractions (b) of  0.55 for the ethylene glycol/ nitrogen system. The simulation results are shown by dashed lines. The bubbles were aligned at the nose.

Fig. 4. (a) Comparison of bubble shape obtained from CFD simulations (left in colour) and experiments (right, black and white) for water/nitrogen system at a homogeneous void fraction of 0.5. The red colour represents liquid and blue colour denotes gas phase. (b) Bubble shape as obtained from CFD simulations at UTP ¼0.53 m s  1 and b ¼ 0.4 for the water/nitrogen system at three different time instants in a periodic cycle of oscillation.

in the CFD simulations. With the increase in mixture velocity, the bubble nose becomes sharper and the length of the transition region between the bubble head and constant thickness film region increases. The bubble tail becomes flatter with an increase in the mixture velocity. Similar results for the bubble shape at the nose and tail have also been observed in the simulations of Giavedoni and Saita (1997, 1999) and Taha and Cui (2004). As is apparent from Fig. 1, for the water/nitrogen system it was not possible to extract the exact bubble shape because of the mismatch between the refractive indices of tube wall and water. Fig. 4(a) compares the bubble shapes obtained from the simulations for water/nitrogen system for a homogeneous void fraction of 0.5 for different mixture velocities with the bubble shape images captured in the experiments. The figure confirms that the shapes of the bubble nose and tail have also been captured very well for the water/nitrogen system. The unsteady behaviour observed at UTP ¼0.42 m s  1 for the water/nitrogen system in the experiments was not found in the simulations but it was present at 0.53 m s  1 and the bubble tail was observed to oscillate periodically. The bubble shape shown in Fig. 4(a) for a mixture velocity of 0.53 m s  1 is shown at a time instant when the tail shape was similar to that in the experimental image. As shown in Fig. 4(b) by plotting the bubble shape at three different time instants in a periodic cycle, the bubble tail shape fluctuates between convex and concave shapes in a cycle. In Fig. 5, the shapes of the bubble nose as obtained from CFD simulations for the ethylene glycol/nitrogen and water/nitrogen systems are compared at a similar mixture velocity (  0.2 m s  1) and homogeneous void fraction (  0.6). The transition length between the nose front and constant thickness film region for the ethylene glycol/nitrogen system is significantly longer than that of the water/nitrogen system.

4.1.1. Film thickness Several correlations have been proposed in the literature to calculate the film thickness in Taylor flow (Angeli and Gavriilidis, 2008; Gupta et al., 2010a). Amongst these the correlation proposed by Aussillous and Que´re´ (2000), given in Eq. (2), is widely used to estimate the film thickness in the case of negligible inertia (low Reynolds number) as is the case for the ethylene glycol/ nitrogen system (22oReTP o40) and for negligible gravity (not the case here).

dF R

¼

1:34Ca2=3 1 þ 2:5ð1:34Ca2=3 Þ

ð2Þ

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Fig. 6. The effect of the Capillary number (Ca) on the non-dimensional film thickness (dF/R) for the ethylene glycol/nitrogen system. The symbols (&) and (J) are used for the experimental and simulation results, respectively. Values from the Aussillous and Que´re´ (2000) correlation are shown by a dashed line.

The effect of the Capillary number on the film thickness for the ethylene glycol/nitrogen system, together with the results from the experiments, the simulations and the Aussillous and Que´re´ (2000) correlation are plotted in Fig. 6. The experimental and simulated film thicknesses are measured at the middle of the bubble for all the cases, as the film thickness varies significantly along the bubble for low homogeneous void fractions. In fact, this is the reason for the small differences in the film thickness at different homogeneous void fraction for a given velocity (and Capillary number). The direct comparison of the film thickness obtained from the experiments and simulations shows an average difference of 3% and a maximum difference of 7%. The thickness of the liquid film increases with increasing Ca in all cases. For the ethylene glycol/nitrogen system, the range of the simulation result deviations from the Aussillous and Que´re´ (2000) correlation, which was developed for negligible inertia and no gravity, is from 5% to 14%. Experimental film thicknesses for the water/nitrogen system could not be measured directly from photographic images, but it was calculated using Eq. (3), which was derived from volume conservation by Suo and Griffith (1964), assuming a stationary, constant thickness film. sffiffiffiffiffiffiffiffi! dF 1 U TP 1 ¼ ð3Þ 2 d UB In Fig. 7, the film thicknesses obtained from the simulations and the calculated values from Eq. (3) evaluated using the experimental data are plotted versus the Capillary number (Ca). The film thicknesses obtained from the simulations that used constant properties evaluated at 21 1C temperature are presented elsewhere (Asadolahi, 2011). The film thicknesses calculated from the Aussillous and Que´re´ (2000) correlation is also shown but we note that the assumption of negligible inertia is no longer valid. The film thickness is also calculated using Eq. (3) where the bubble velocity (UB) obtained from the simulations is used. The two methods employed to calculate the film thickness from the simulations are in agreement with each other and confirm the validity and accuracy of Eq. (3) to calculate the film thickness from the experimental data. The results derived from Eq. (3) using the experimental data show the liquid film thickness is approximately the same for the first two Capillary numbers (Ca ¼0.0029, 0.0043) and then it increases. However, the film thickness calculation using Eq. (3) is very sensitive to changes in bubble velocity. For example, a 5% difference between the simulated (0.224 m s  1) and experimentally obtained (0.235 m s  1) bubble velocities gave rise to a 42%

Fig. 7. The effect of Capillary number (Ca) on the non-dimensional film thickness (dF/R) for the water/nitrogen system for an experimental homogenous void fraction (b) of 0.4. The symbols (&) and (W) are used for film thicknesses calculated from Eq. (3) using the experimental and simulated bubble velocities (UB), respectively. The symbol (J) is used for the results obtained directly from the simulations. The Aussillous and Que´re´ (2000) correlation is also shown by a dashed line.

Fig. 8. The effect of the mixture velocity (UTP) on the bubble velocity (UB) obtained from CFD simulations and the experiments. The correlation of Liu et al. (2005) is shown by large and small dashed lines for ethylene glycol/nitrogen and water/ nitrogen systems, respectively. The Armand correlation is shown by the dotted line and solid line shows UB ¼ UTP. For the ethylene glycol/nitrogen system, the CFD results and experimental data are represented by (J) and ( ), respectively. For the water/nitrogen system, the CFD results and experimental data are represented by (&) and (D), respectively.

difference in the film thickness. This high sensitivity of film thickness to small differences in the bubble velocity appears to account for the large differences (15–42%) between the film thicknesses obtained from experiments and CFD simulations. As seen in Fig. 4(b), for the water/nitrogen system at the high mixture velocity of 0.53 m s  1, the liquid film thickness remains unchanged even though the bubble tail shape becomes unsteady. Therefore, a constant value has been used for presenting the nondimensional film thickness for Ca ¼0.0072 in Fig. 7.

4.1.2. Bubble and slug lengths The bubble and slug lengths obtained from the modelling were in good agreement with those obtained from experiments with the average difference being 2% and the largest being 9%. The

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ripples observed at the bubble tails of the unsteady simulated cases for the water/nitrogen system cause a maximum variation of 1% for the bubble and slug lengths. 4.2. Bubble velocity In the experiments, the bubble velocity was determined by performing a frame-by-frame analysis of the recorded images as explained in Leung et al. (2010) and therefore is an important parameter to validate the computational results. The Armand correlation (Armand and Treschev, 1946; Armand, 1959), given by Eq. (4a) is often used to predict the bubble velocity or void fraction (in combination with Eq. (1)) in Taylor flow regime in microchannels. Liu et al. (2005) obtained a bubble velocity correlation for upward Taylor flow in vertical channels, given in Eq. (4b). This correlation is valid for Ca from 0.0002 to 0.39. UB ¼ 1:2 U TP UB 1 ¼ U TP 10:61Ca0:33

ð4aÞ

ð4bÞ

In Fig. 8 the variation of the bubble velocity (UB) with the mixture velocity (UTP) is shown for the ethylene glycol/nitrogen and water/nitrogen systems obtained from experiments, CFD and the Armand (Armand and Treschev, 1946; Armand, 1959) and Liu et al. (2005) correlations. The variation of the simulated bubble velocity for the unsteady cases with UTP ¼0.53 m s  1 in the water/ nitrogen system was negligible ( 0.3%). There is very good agreement between the bubble velocities obtained from the simulations and experiments for both systems, especially given that the standard deviations for the experimental values are less than 3% and 4% for the ethylene glycol/nitrogen and water/ nitrogen systems, respectively. As can be seen from the graph, the data from ethylene glycol/ nitrogen and water/nitrogen systems fall along separate lines so the Armand correlation cannot represent the relation between UB and UTP for all the fluid systems. The correlation of Liu et al. (2005) incorporates a Capillary number effect and is in better agreement with the bubble velocities obtained from CFD simulations and experiments. 4.3. Homogeneous void fraction In the experiments, the homogeneous void fraction was an input parameter and the void fraction was calculated using Eq. (1) employing the experimentally determined bubble velocity. On the other hand, in simulations the void fraction was set equal to that obtained from experiments and the homogeneous void fraction was calculated from Eq. (1). As mentioned in Section 3.1 earlier, the void fraction for ethylene glycol/nitrogen system was calculated directly from the images and the homogeneous void fraction was calculated from the CFD simulations using Eq. (1). Fig. 9 shows a comparison of b values thus calculated with those used in the experiments. The b values in the two cases were in good agreement, the largest difference between the two being 6%. For the water/nitrogen system the input value of the void fraction was calculated using the homogeneous void fraction and, therefore, the comparison of homogeneous void fraction is similar to the comparison of bubble velocities between the simulations and experiments. 4.4. Pressure drop In the experiments, a pressure sensor was connected close to the liquid inlet to monitor the system pressure and flow stability

Fig. 9. Comparison of homogeneous void fraction obtained from CFD and experiments for ethylene glycol/nitrogen system.

(Leung et al., 2010). This measurement gave the total pressure drop in the system including the pressure losses in the liquid inlet, mixing section, developing and developed Taylor flow and hence cannot be compared with the pressure drop over a fullydeveloped unit cell obtained from CFD simulations. Therefore, the pressure drops calculated from the CFD simulations were compared with those obtained from correlations developed for estimating the pressure drop in the fully-developed Taylor flow regime. A normalised pressure drop per unit length can be defined as the ratio of pressure drop per unit length over a unit cell and pressure drop per unit length for fully-developed, liquid-only flow having the average velocity the same as the mixture velocity, as given in Eq. (5) below.  n DP DP=L ¼ ð5Þ L ðDP=LÞLO Bretherton (1961) derived a correlation to estimate the pressure drop over the bubble caps for negligible inertia as given in Eq. (6) below.

DPcap ¼ 7:16ð3CaÞ2=3

s d

ð6Þ

Combining this with the pressure drop in the liquid slug assuming the liquid slug to have fully-developed flow of liquid, Kreutzer et al. (2005) proposed a correlation, given by Eq. (7), to predict the pressure drop in Taylor flow in case of negligible inertial effects. !     DP n LS d 7:16ð3CaÞ2=3 ¼ 1þ ð7Þ LUC LS LUC 32Ca For non-negligible inertia (100oReTP o1000), they proposed another correlation (given in Eq. (8)), which includes the effect of Reynolds number on the flow. It is interesting to note here that the important non-dimensional number in this correlation is (Re/ Ca) as identified by de Ryck (2002) to determine the thickness of the liquid film in the case of non-negligible inertia. It is also important to note here that the non-dimensional pressure drop is independent of the mixture velocity, meaning that the increase in the pressure drop due to the introduction of the gas bubble is independent of the mixture velocity. !       DP n LS d Re 0:33 ¼ 1þ a ð8Þ LUC LS Ca LUC

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Fig. 10. Variation of normalised pressure drop over a unit cell with non-dimensional time at a mixture velocity of 0.53 m s  1 and a homogeneous void fraction of 0.5.

Fig. 12. The effect of the mixture velocity (UTP) on the normalised pressure drop (J) for a homogenous void fraction of (a)  0.56 for the ethylene glycol/nitrogen system (b)  0.5 for the water/nitrogen system. The normalised pressure drops calculated from Eq. (7) are shown by the (W) symbol and those calculated from Eq. (8) using a¼ 0.07 and a ¼0.17 are shown by () and (&) symbols, respectively.

Fig. 11. The effect of the homogenous void fraction (b) on the calculated normalised pressure drop (J) at (a) UTP ¼ 0.20 m s  1 for the ethylene glycol/ nitrogen system (b) UTP ¼0.32 m s  1 for the water/nitrogen system. The normalised pressure drops predicted from Eq. (7) are shown by the (W) symbol and those from Eq. (8) using a¼ 0.07 and a ¼0.17 are shown by () and (&) symbols, respectively.

Kreutzer et al. (2005) found two different values of a, 0.07 and 0.17 from numerical simulations and experiments, respectively. In their experiments, performed over a wide range of Reynolds and Capillary numbers, Walsh et al. (2009) found the value of a to be the average of the two values suggested by Kreutzer et al. (2005). Kreutzer et al. (2005) attributed the experimental value of a (0.17) to Marangoni effects from impurities in his experiments. As pointed out by Bretherton (1961), dissolved impurities and surface charges can reduce the surface tension by a factor of 4 and thus the value of a ¼0.17 (0.07  42/3) is expected to give the maximum pressure drop that can occur because of impurities. We have compared pressure drop obtained from our simulations with the values obtained from Eq. (8) using both values of a, though we

expect simulation pressure drop values closer to those obtained using a ¼0.07 as our CFD model also does not consider Marangoni effects. In the CFD simulations large fluctuations of the pressure drop with time were observed in the case of the high mixture velocity (UTP ¼0.53 m s  1), when the bubble shape and flow field were not steady. Fig. 10 shows the variation of normalised pressure drop over a unit cell with time (non-dimensionalised by tUC, flow time of a unit cell) for the case of UTP ¼0.53 m s  1 and a homogeneous void fraction of 0.5. The normalised pressure drop fluctuates periodically over a wide range from 25% to more than 200% of the liquid only pressure drop having a time period of  0.06 tUC. The peak in the pressure drop corresponded to convex shape of the tail (longer bubble length, as shown in Fig. 4 b) while the troughs corresponded to the concave tail shape (shorter bubble length). While these large fluctuations in pressure drop suggest the flow and therefore temperature field to be highly transient, the accurate flow physics for such a case can only emerge in full in a three-dimensional simulation with a number of unit cells. The pressure drop results from the simulations for these high Reynolds number cases are not included in the following analysis. In Fig. 11(a) and (b) the variation of the normalised pressure drops with the homogenous void fraction are plotted for the ethylene glycol/nitrogen system for UTP ¼0.20 m s  1 and for the water/nitrogen system for UTP ¼0.32 m s  1, respectively. In these figures, predicted normalised pressure drops Kreutzer et al. (2005) correlations for negligible inertia (Eq. (7)) and non-negligible inertia (Eq. (8) using a¼ 0.07 and 0.17) are also shown. The normalised pressure drop decreases when the homogenous void fraction increases for both the ethylene glycol/nitrogen and water/nitrogen systems. The increase in b corresponds to a

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decrease in the volume fraction of the liquid in a unit cell, and therefore, a decrease in the pressure drop can be expected. For the ethylene glycol/nitrogen system, the pressure drops calculated using Eqs. (7) and (8) employing a ¼0.17 were almost the same for each case. When a ¼0.07 was employed in Eq. (8), the calculated pressure drop was lower than that calculated for the case of negligible inertia using Eq. (7). The pressure drops calculated from the CFD simulations was between the two values obtained from Eq. (8) using a ¼0.07 and 0.17 but closer to the values for a ¼0.07. For the water/nitrogen system, the pressure drops calculated from the CFD simulations were closer to those obtained from Eq. (8) employing a ¼0.07 and were significantly lower than the values obtained using a ¼0.17. At high values of homogeneous void fractions, the pressure drops were observed to be closer to the values obtained using Eq. (7). Fig. 12(a) and (b) shows the effect of the mixture velocity on the normalised pressure drop for ethylene glycol/nitrogen and water/nitrogen systems, respectively. The normalised pressure drop obtained from the CFD modelling increases with increasing mixture velocity for both systems. Although the simulated normalised pressure drop depends on the mixture velocity, it can be understood from Eq. (8) that the normalised pressure drop is independent of the mixture velocity and the small variations seen in the values calculated using Eq. (8) in Fig. 12(a) and (b) are because of variations in the slug and unit cell lengths upon increasing the mixture velocity. It is evident that employing a¼ 0.17 over predicts the pressure drop obtained from CFD simulations for the water/nitrogen system, which has significant inertia (as Marangoni effects are not considered in the CFD model) whereas a ¼0.07 results in pressure drop values lower than that obtained from the relation for negligible inertia. Our CFD simulations are overall in good agreement with Eq. (7) and with Eq. (8) using a ¼0.07.

5. Heat transfer comparison As discussed in Section 2.1, a constant heat flux boundary condition was applied on the channel wall. A value of 32 kW m  2 was set as the wall heat flux for all the cases. As the temperature changes during the heat transfer experiments, constant properties were specified at a temperature falling in the middle of the range of temperature variation in the experiments. These temperatures were 25 1C and 30 1C for the water/nitrogen and ethylene glycol/ nitrogen systems, respectively. The heat transfer comparison between the experimental data and CFD simulations was made by comparing the average Nusselt number. In the experiments, the time-averaged Nusselt number at the centre of each block (Nuj) was calculated using Eq. (9). Nuj ¼

qj T bj, lnððT wj T bj,

out T bj,

in

in Þ=ðT wj T bj,

out ÞÞ

d kL

ð9Þ

where qj and Twj are the wall heat flux and wall temperature, respectively, at the jth block and Tbj,in and Tbj,out are the bulk fluid temperatures at the upstream and downstream ends, respectively, of the jth block. These bulk fluid temperatures were calculated from the heat balance assuming the gas–liquid mixture to be homogeneous (Leung et al., 2010). The experiments showed that while the ethylene glycol/nitrogen system required 3 blocks to achieve thermally developed Taylor flow, for the water/nitrogen system the flow was thermally developed after the first heating block. Therefore, the average Nusselt number (Nuav) was obtained by averaging the local Nusselt numbers (Nuj) over blocks 4 to 9 for the ethylene glycol/nitrogen system and over blocks 2 to

549

Fig. 13. Normalised average Nusselt number (Nu*) versus the homogenous void fraction (b) for the ethylene glycol/nitrogen system for different mixture velocities of 0.20 m s  1 (J), 0.29 m s  1 (&), 0.37 m s  1 (W) and 0.53 m s  1 (). The experimental and simulated results are shown by solid and open symbols, respectively.

9 for the water/nitrogen system to eliminate entrance, development and end-block effects on the average value. In the CFD simulations, the average Nusselt number (Nuav) for a unit cell was calculated based on the following equation: Nuav ¼

qw,av d ðT w,av T b,av Þ kL

ð10Þ

where qw,av and T w,av are the average of the heat flux and temperature, respectively, over the wall surface and the average bulk temperature (T b,av ) is calculated using the following equation: R x þ LUC R R r9ux 9cp Tð2prÞdrdx0 T b,av ¼ Rx x þ L R0 R ð11Þ UC 0 x 0 r9ux 9cp ð2prÞdrdx A normalised Nusselt number is defined via Nun ¼

Nuav NuLO

ð12Þ

where NuLO is the Nusselt number for the liquid only, fullydeveloped flow for a constant wall heat flux boundary condition and takes a value of 4.364 (Shah and London, 1978). The comparison between the average Nusselt numbers obtained from the CFD simulations and experiments using ethylene glycol/ nitrogen and water/nitrogen systems are presented in Sections 5.1 and 5.2, respectively. A detailed comparison of our experimental heat transfer data with the existing correlations to predict heat transfer in the Taylor flow regime and with a new empirical correlation developed by us has been undertaken recently (Leung et al., in press) and is not repeated here. 5.1. Ethylene glycol/nitrogen system The normalised average Nusselt numbers obtained from the simulations and the experiments are shown in Fig. 13 for different mixture velocities for the ethylene glycol/nitrogen system. The error bars on the experimental data represent 72SD calculated from the scatter of the experimental Nusselt numbers among the individual copper blocks used to determine the mean Nusselt number for each case. Recall that the homogenous void fraction is a result of the simulation and is not set as an input parameter; therefore, small differences between the simulated and experimental homogenous void fractions are observed. Most of the

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simulated Nu* fall in the range of experimental uncertainty for the different mixture velocities and homogenous void fractions. The difference between the simulated and experimental results lies in the range of 1–22% with the average difference being 8%. In addition to the cases listed in Table 2, a few additional simulations were run at high mixture velocity (UTP) or high homogeneous void fraction (b) in order to cover a wider range of these parameters. The Nu* obtained from the simulations and the experiments decrease with increasing b for all of the mixture velocities, with the effect being more pronounced at the lower mixture velocities. As for the pressure drop over a unit cell, the decrease in Nusselt number with b can also be attributed to the decreasing amount of liquid and therefore thermal mass, in a unit cell. As the simulations were carried out at a temperature of 30 1C, an average of the middle range of all the cases in Table 2, case 6 (UTP ¼0.20 m s  1 and b ¼0.67), which had the largest difference with the experimental data was rerun using the fluid properties at the minimum (23.6 1C) and maximum (30 1C) temperature for this case. The value of normalised Nusselt numbers for the two cases (at the minimum and maximum temperatures) differed by only 4% (Asadolahi, 2011). The Nusselt numbers obtained from the experiments show a weak dependence on the mixture velocity with the Nu* increasing with increasing mixture velocity. However, the CFD simulations show the Nusselt numbers to be independent of mixture velocity for homogeneous void fractions less than 0.5. At high homogeneous void fractions, a dependence of Nu* on the mixture velocity is observed in the CFD simulations too. It must also be remembered here that at a given homogeneous void fraction and for different mixture velocities, the slug length is not always the same. Therefore the effect of mixture velocity on Nu* seen in Fig. 13 is a combined effect of different slug lengths and mixture velocities.

The differences between the simulated and experimental Nu* are small and within the range of uncertainty for the mixture velocities of 0.21 m s  1 and 0.32 m s  1 with the largest difference being 9%, but the deviation increases as the mixture velocity becomes larger for a specified homogenous void fraction (b). For a mixture velocity of 0.42 m s  1, the difference between the experimental data and CFD simulations is within 20%. For a mixture velocity of 0.53 m s  1, the differences of 32% and 29% values are significant at b 0.4 and 0.5, respectively (At the mixture velocity of 0.53 m s  1, the bubble shape and flow field were unsteady in the CFD simulations and therefore the flow and energy equations were solved simultaneously. However, very small ( o1%) fluctuations were observed in the value of the average Nusselt number). It is important to remember here that unsteady and asymmetric bubble shapes were observed for these higher mixture velocities of 0.42 and 0.53 m s  1 and two-dimensional, axisymmetric CFD simulations do not necessarily capture the correct physics for these cases. The values of the Nusselt numbers obtained from the CFD simulations at the same homogeneous void fraction but different mixture velocities (except at a mixture velocity of 0.53 m s  1) are similar suggesting the Nusselt number to be independent of mixture velocity. Our previous computational work (Gupta et al., 2010b) for the water/nitrogen system using a fixed domain approach has also shown independence of mixture velocity (a value of Nu* 2.5 over the range of mixture velocity of 0.3–1.0 m s  1 for b ¼0.51 and LS/d  1). The experimental results also show independence of Nusselt number for low mixture velocities (0.21 and 0.32 m s  1) but Nu* increases with the mixture velocity for two high mixture velocities of 0.42 m s  1 and 0.53 m s  1, the cases for which the bubble shape was found to be unsteady and asymmetric in the experiments.

6. Conclusions 5.2. Water/nitrogen system Fig. 14 shows the simulated and experimental normalised average Nusselt numbers for different mixture velocities for the water/nitrogen system. As observed for ethylene glycol/nitrogen system, the Nusselt number also decreases with increasing b for the water/nitrogen system except for the case of mixture velocity of 0.42 m s  1 where it first increases and then decreases after reaching a maximum value at b ¼0.4 in the experiments.

Fig. 14. Normalised average Nusselt number (Nu*) versus the homogenous void fraction (b) for the water/nitrogen system for different mixture velocities of 0.21 m s  1 (J), 0.32 m s  1 (&), 0.42 m s  1 (W) and 0.53 m s  1 ( ). The experimental and simulation results are shown by solid and open symbols, respectively.

In this work, the results obtained from two-dimensional, axisymmetric CFD simulations in a frame of reference moving with the bubble carried out in a periodic computational domain for the hydrodynamics and heat transfer of fully-developed vertically upward Taylor flow in microchannels are compared with experimental data over a wide range of Reynolds and Capillary numbers using ethylene glycol/nitrogen (22oReTP o40; 0.085oCao0.160) and water/nitrogen (475oReTP o1189; 0.003oCao0.007) systems. The simulations were performed in a unit cell consisting of a gas bubble and two halves of adjacent liquid slugs having the same unit cell length, channel diameter and void fraction as obtained in the experiments. A comparison of bubble shapes obtained from the simulations with those obtained in the experiments showed them to be in good agreement having similar nose shapes, transition lengths from the nose to the constant film thickness region and tail shape except at high Reynolds numbers for which the bubble tail shape was observed to be asymmetric and unsteady in the experiments. For the ethylene glycol/nitrogen system the film thicknesses obtained from CFD were also found to be in very good agreement with those obtained from experiments (within 7%) and with the Aussillous and Que´re´ correlation (within 15%), a correlation developed for negligible inertia and zero gravity. For the water/ nitrogen system, large differences (15–42%) were observed in the film thickness between the CFD simulations and experimental data. This can be attributed to the high sensitivity of the film thickness calculation to the small error in bubble velocity measurement as the film thickness could not be measured directly for water/nitrogen system because of channel curvature and refractive index mismatch. The bubble velocity calculated from the

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simulations was found to be in excellent agreement (within 5%) with the experimental data. The pressure drop obtained from the CFD simulations was compared with the Bretherton (neglecting inertia) and Kreutzer (considers inertia) correlations for pressure drop. Overall good agreement was found between the CFD simulations, the Bretherton correlation and with the Kreutzer correlation employing a ¼0.07 (appropriate in the absence of Marangoni convection effects). To compare the heat transfer between the CFD simulations and experiments, a Nusselt number was defined, which was similar to the Nusselt number definition employed experimentally, which used time-averaged temperature data. The Nusselt numbers obtained from the simulations were in good agreement with the experimental data for both ethylene glycol/nitrogen (within 22%) and water/nitrogen (9%) systems except at high Reynolds number for the water/nitrogen system. The water/nitrogen system (maximum value  3 times that of single phase fully-developed laminar flow) offered very high heat transfer enhancement over fully-developed single phase flow as compared with that of ethylene glycol/nitrogen system (maximum value 1.5 times). Both the systems showed a decrease in Nusselt number with an increase in homogeneous void fraction. The Nusselt number was found to be almost independent or weakly dependent on mixture velocity (for ethylene glycol at b 40.5) except for high Reynolds number unsteady cases. Overall, the CFD simulations were in very good agreement with the experimental data except for the high Reynolds number cases for which the assumption of an axisymmetric bubble shape and flow was not valid. However, a good agreement was also observed for bubble shape and velocity even for these cases. At ReTP ¼951, while the experiments showed the bubble shape to be unsteady and asymmetric, the simulations showed the bubble shape to be steady suggesting the asymmetric disturbances causing the unsteadiness. At ReTP ¼1189, the bubble shape and flow were observed to be unsteady in both the experiments and simulations. While the liquid film thickness and bubble velocity were in good agreement even for these cases, the Nusselt number was significantly lower than those observed in the experiments.

eG m r s x

551

volume fraction of gas in a unit cell (also termed the void fraction and gas hold-up) dynamic viscosity (kg m  1 s  1) density (kg m  3) surface tension coefficient (N m  1) non-dimensional radius ( ¼r/R)

Subscripts av B b F G j L S UC w

average value for a unit cell bubble bulk mean film gas value for jth block liquid slug unit cell wall

Superscripts n

value normalised with that for single-phase fully developed flow at the same mixture velocity

Acknowledgements A.N. Asadolahi was supported through a University of Sydney Postgraduate Award. S. Leung was supported through the Endeavour International Postgraduate Research Scholarship funded by the Australian Government. The work was supported under Australian Research Council Discovery Grant DP0985453. The financial support of the Heatric Division of Meggitt (UK) is also gratefully acknowledged.

References Nomenclature a Ca cp d k L Nu q r R ReTP t T UB UTP ux x

constant in the pressure drop correlation capillary number (mL U TP =sÞ specific heat capacity (J kg  1 K  1) diameter of the channel (m) thermal conductivity (W m  1 K  1) length (m) Nusselt number heat flux (W m  2) radial coordinate (m) channel radius (m) two-phase (liquid-only) Reynolds number (UTPrLd/mL) time (s) temperature (K) bubble velocity (m s  1) mixture velocity (UG þUL) (m s  1) axial velocity (m s  1) axial coordinate (m)

Greek symbols

b DP dF

homogeneous void fraction (UG /UTP) pressure drop (Pa) film thickness (m)

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