Taylor flow heat transfer in microchannels—Unification of liquid–liquid and gas–liquid results

Chemical Engineering Science 138 (2015) 140–152

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Taylor flow heat transfer in microchannels—Unification of liquid–liquid and gas–liquid results Zhenhui Dai, Zhenyi Guo, David F. Fletcher n, Brian S. Haynes School of Chemical and Biomolecular Engineering, University of Sydney, Sydney, NSW 2006, Australia

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T


Heat transfer data on liquid–liquid Taylor flow in a 2 mm diameter tube.
High sensitivity of liquid–liquid heat transfer rate to flow conditions.
Validation of CFD model of liquid– liquid Taylor flow heat transfer.
Developed a generalised model to interpret and predict Taylor-flow heat transfer.
Unification of liquid–liquid and gas– liquid results using a single correlation.

art ic l e i nf o

a b s t r a c t

Article history: Received 12 June 2015 Received in revised form 7 August 2015 Accepted 8 August 2015 Available online 18 August 2015

The flow and heat transfer behaviour of liquid–liquid Taylor flow is examined by performing both experiments and CFD simulations for 1 and 2 mm vertical tubes with constant wall heat flux boundary conditions. Water and hexadecane are used as the disperse and continuous phases, respectively. The measured heat transfer coefficients are extremely sensitive to experimental uncertainties but, are in overall good agreement with the simulations. The simulations confirm the strong dependence on the flow conditions seen in the experiments. A generalised model of heat transfer in gas–liquid and liquid–liquid Taylor flows is developed from a combination of resistances for wall-to-film, film-to-slug and film-to-bubble or droplet. Good estimates for these individual resistances are described and validated. The overall heat transfer coefficient obtained by a rigorous weighting of the individual resistances correlates the entire set of CFD (liquid–liquid) and experimental (gas–liquid) data with 20% relative standard deviation. The model captures the complex parametric dependencies and sensitivities in the data. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Microchannels Heat transfer Enhancement Correlation

1. Introduction Driven by the need for more effective heat transfer devices for product and process miniaturization, significant research efforts have focused on liquid cooling in microchannels. The ability of gas–liquid non-boiling Taylor flow to increase the heat transfer rate several-fold relative to liquid-only values has been demonstrated (Asadolahi et al., 2011, 2012; Betz and Attinger, 2010;

n

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

http://dx.doi.org/10.1016/j.ces.2015.08.012 0009-2509/& 2015 Elsevier Ltd. All rights reserved.

Fukagata et al., 2007; Gupta et al., 2010; He et al., 2010; Horvath et al., 1973; Lakehal et al., 2008; Leung et al., 2012, 2010; Oliver and Wright, 1964; Walsh et al., 2010). Internal recirculations within the liquid slugs have been shown to explain the heat transfer enhancement. However, in gas–liquid Taylor flow, the contribution of the gas phase to the heat transfer is negligible, as the thermal capacity of the gas phase is much smaller than that of the liquid phase. Replacing the gas bubbles with immiscible liquid

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droplets to form liquid–liquid Taylor flow in microchannels has drawn increasing attention recently. This is because the disperse liquid phase has a thermal capacity of the same order as that of the continuous liquid phase and hence significant heat transfer enhancement can be expected. In addition, Taylor droplet flow in microchannels provides rapid internal mixing, as well as precise control of the reaction time and chemical composition in each droplet, making it very attractive to engineering applications. Nitration of benzene to toluene (Song et al., 2006), microseparation (Ookawara et al., 2007), polymerase chain reaction (Urbant et al., 2008) and electronics cooling (Asthana et al., 2011; Fischer et al., 2010; Urbant et al., 2008) are some examples of novel applications of liquid–liquid Taylor flow. There has been considerable research on the hydrodynamics and heat transfer of two-phase flow both numerically and experimentally. However, most of the studies have been carried out for gas–liquid two-phase flow with few on liquid–liquid flow. In recent years, there have been a few investigations of hydrodynamics of liquid–liquid Taylor flow in microchannels, including flow patterns (Foroughi and Kawaji, 2011; Kashid and KiwiMinsker, 2011; Kashid et al., 2007), liquid film thickness (Grimes et al., 2007; Mac Giolla Eain et al., 2013), pressure drop (Jovanović et al., 2011; Kashid and Agar, 2007; Mac Giolla Eain et al., 2015a, 2013) and mass transfer enhancement (Kashid et al., 2007). The flow characteristics of liquid–liquid two-phase flow are now well understood, as described in recent reviews (Bandara et al., 2015; Gupta et al., 2013). Most recently, with the rapid development of modern imaging and optical diagnostic instrumentation, a number of techniques have emerged to investigate and characterise fluid flow in microscale systems quantitatively. Several authors investigated velocity distributions in microscopic liquid–liquid two-phase flow using micro-particle image velocimetry (micro-PIV). Kashid et al. (2005), (2008) carried out experiments to visualize the internal circulations in the slug and compared the PIV velocity fields qualitatively with those obtained from CFD simulations. With high-speed confocal scanning microscopy, three-dimensional and complex circulating flow patterns were identified inside the droplet by Kinoshita et al. (2007). Miessner et al. (2008) investigated the 3D velocity distributions in disperse oil/water two-phase flows in a rectangular channel (100 μm  100 μm) by scanning the volume and using micro-PIV. The velocity fields in both phases were obtained simultaneously and secondary vortices were observed inside and outside the droplet. Ghaini et al. (2011) adopted laser induced fluorescence (LIF) to visualize the wall film for a variety of aqueous-organic two-phase systems in 1 mm capillaries. Internal circulation patterns within the liquid slugs were identified by observing the variations in the local fluorescent dye concentrations. These techniques permit the observation of hydrodynamics and mixing effects with a high spatial resolution and provide useful insights into the physical mechanisms governing heat transfer in Taylor flow. While the flow pattern and mass transfer enhancement in liquid–liquid segmented flows in microchannels have been demonstrated in both simulations and experiments, studies of heat transfer in this flow regime have mostly been restricted to computational approaches. Urbant et al. (2008) calculated the heat transfer characteristics of the flow of water droplets in continuous oil phase in capillaries with diameters of 100 and 1000 μm. Their work focused on short droplets, o1d, but a single run for a droplet length of 1.65d in the larger channel was also conducted to illustrate the heat transfer performance of flow with longer water droplets. Significant heat transfer enhancement over that for single-phase flow was reported due to the recirculating flow inside the water droplets and the oil slugs. The heat transfer rate was found to increase with the length of the water droplets. They

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explained that the presence of the thin film of the carrier fluid between the wall and the elongated water droplet could lead to considerable augmentation of the heat transport in the radial direction. However, Janes et al. (2010) who performed oil–water segmented flow heat transfer experiments in a meandering channel with 901 bends and a spiral channel of square crosssection drew a different conclusion concerning the effect of the water droplets on heat transfer. They highlighted that the water droplets only play a minor role in overall heat transfer due to the low thermal penetration through the thick oil film. Fischer et al. (2010) performed two-dimensional, axisymmetric simulations to study the heat transfer for liquid–liquid slug flow in a microchannel using 5 cS silicone oil, water and polyalphaolefine as the working fluids. For some simulations, the droplet phase was loaded with 3 vol% Al2O3 nanoparticles. Up to a four-fold increase in the Nusselt number relative to the single liquid flow was reported. The effect of Marangoni flow induced by the temperature gradient at the liquid interfaces was included in their modelling. They found that the Marangoni effect decreased the heat transfer as it opposed the recirculating flow in the slug. They also found that the nanoparticles inside the droplets only provided a small additional heat transfer enhancement (3–5%). Che et al. (2015) carried out 3D numerical simulations of the heat transfer process to study the droplet heat transfer in microchannel heat sinks. The effects of the length of the droplets, the aspect ratio of the channel cross-section, and the Péclet number were analysed. Nusselt numbers in the range 10–15 were found for the dropletbased heat transfer in their rectangular microchannels with a constant wall temperature boundary condition. Talimi et al. (2013) performed single phase CFD simulations inside the slug region for Taylor flow in square-section channels and predicted heat transfer rates approximately four times greater than the experimental data used in the comparison. To date, only two experimental studies that examine the heat transfer performance of liquid–liquid flow are available in the literature. Asthana et al. (2011) reported experimental measurements of the flow and heat transfer of mineral oil droplets dispersed in a continuous water phase in a serpentine channel of square cross-section (100 μm  100 μm). Laser induced fluorescence (LIF) and micro-Particle Image Velocimetry (micro-PIV) techniques were employed to measure the temperature and velocity fields, respectively. The recirculation zones inside the water slug and the resulting enhanced radial heat transfer were clearly demonstrated. A heat transfer enhancement of up to fourfold in slug flow compared with that for water alone was observed. Most recently, Mac Giolla Eain et al. (2015b) carried out heat transfer experiments on liquid–liquid Taylor flows with a constant wall heat flux boundary condition. Local wall temperature measurements were made in both developing and fully-developed regions using a high resolution infrared thermography system. They examined the influence of slug length and carrier phase properties on the local Nusselt numbers. Enhancements up to six times over single phase flow were observed by introducing a second immiscible liquid phase. They noted that reductions in carrier slug length and increases in the droplet length result in improved thermal performance. Some studies have led to the development of correlations to model the thermal behaviour of two-phase Taylor flow, but mostly only for gas–liquid flows (Kreutzer et al., 2001; Leung et al., 2012; Oliver and Wright, 1964; Walsh et al., 2010). Mac Giolla Eain et al. (2015b) proposed a correlation to predict the Nusselt number in liquid–liquid Taylor flows with constant wall heat flux boundary condition. The correlation consists of separate expressions for the thermal entrance and fully-developed regions. The deviation between the correlation predictions and the experimental measurements ranged from 710% to 730%. However, only the slug

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length was taken into account in their model. A more accurate model based on flow parameters, such as channel diameter, slug and droplet lengths and film thickness that can describe the underlying physics of the phenomena is therefore needed. Although the experimental work carried out by different research groups has all clearly shown a significant heat transfer enhancement in liquid–liquid two-phase flow compared with that for single phase flow, there appears to be considerable variation in the experimentally-determined heat transfer coefficients obtained by various researchers even for identical conditions. Bandara et al. (2015) provided a detailed review of both numerical and experimental studies of the hydrodynamics and heat transfer of twophase flow in small channels, addressing specifically the challenges and difficulties in this area. They compared the Nusselt numbers for liquid–liquid two-phase flow calculated using different correlations from the literature and pointed out that there were discrepancies of over 500%. There was no comparison between the experimental data and the corresponding computational or theoretical solutions to validate the results. The differences in the heat transfer rate measurements may be related to differences in boundary conditions used in each study. In addition, the measurement accuracy for various parameters, especially the temperatures, may be an important source of discrepancy. Until recently, most temperature measurements in microfluidic systems were limited to measurements of bulk fluid temperature at the inlet and outlet of microfluidic sections or measurements of the substrate temperature. The direct local measurements of the bulk temperature of the fluid in microchannels and the direct measurements of temperature at the interface between the fluid and solid are very difficult to implement using conventional methods, such as thermocouples or resistance temperature detectors (RTDs). Although laser induced fluorescence (LIF) offers an opportunity to collect temperature data nonintrusively at micro-scale resolution and with high accuracy (Natrajan and Christensen, 2011; Ross et al., 2001; Sutton et al., 2008), such measurements in two-phase liquid–liquid systems are still very limited. The local temperature field of water within the microchannels was resolved by a one-dye LIF technique in Asthana et al., (2011). While they achieved a RMS temperature precision o0.9 1C, the error bars in their results indicated that the error in the calculation of average Nusselt number was as high as 35%. This result highlights the difficulty of obtaining precise results in liquid–liquid flows—the temperature difference between the wall and the bulk fluid is generally much smaller than that in the equivalent single phase flow or gas–liquid flow for a similar level of heat flux (Leung et al., 2012; Mac Giolla Eain et al., 2015b) and temperature measurement uncertainties can induce significant errors in Nusselt number calculations. In this paper, we use both computational and experimental results to understand heat transfer in liquid–liquid flows. The computational approach, which we have validated comprehensively in a series of papers on gas– and liquid–liquid Taylor flows (Asadolahi et al., 2012; Gupta et al., 2013), is not suitable for generating a large number of data points but it does provide clear mechanistic insights. At the same time, the computations create a basis for the assessment of the overall validity of the experimental results which can be obtained more readily over a wide range of flow parameters. The computational data are then used in the development of a mechanistic model that unifies gas–liquid and liquid–liquid heat transfer rates in a single correlation.

2. Experimental set-up The same experimental facility as that described in Gupta et al. (2013) was used to determine the hydrodynamic and heat transfer

characteristics of liquid–liquid vertically upward Taylor flow in a circular channel, as shown in Fig. 1. Hexadecane was used as the continuous phase, while water was used as the disperse phase. The liquids were introduced into the system using two HPLC pumps (Shimadzu, LC-20AT). A Swagelok T-junction of 2.2770.08 mm inner diameter with head-on flows of the immiscible liquids was used to generate the segmented flow. The heating test section was made of a 300 mm long hard-drawn stainless-steel circular channel of 2.0070.02 mm internal diameter with a wall thickness of 0.588 mm. A transparent fluorinated ethylene propylene (FEP) tube of similar diameter (d¼2.04 mm) was connected immediately after the heating test section to allow visualization of the flow structures. 2.1. Visualization measurements Backlit images of the liquid–liquid Taylor flow were recorded using a high-speed CMOS camera (Fastcam1024PCI, Photron) at a rate of 1000 frames per second with a resolution of 1024  1024 pixels with 4  magnification to give a 4.35  4.35 mm2 area of view. A flat-faced refractive index matching (RIM) system filled with water (RI¼ 1.33) was employed to minimize the optical distortion due to the combined effects of the mismatch of the refractive indices of air (RI¼ 1.00) and the FEP tube (RI¼ 1.33) and the outer curved surface of the tube. The droplet velocity and the lengths of the droplets and slugs were determined by image analysis using the ImageJ software (Version 1.48, NIH, Bethesda, MD) following the same procedure as for our earlier gas–liquid Taylor flow work (Leung et al., 2012, 2010). 2.2. Heat transfer measurements The inlet temperatures of both liquids were measured with tipsensitive K-type thermocouples. The room temperature was measured using a 1/16 inch K-Type thermocouple probe (Merlin Marlox series). Six miniature flexible platinum RTD sensors (R-MINI 29223, RdF Corporation, Hudson, NH, USA) were mounted on the wall surface along the heating test section using hightemperature tape (at 30, 80, 114, 154, 194, and 234 mm from the entrance). The heating section was wrapped in foam insulation to minimize ambient heat losses and the effect of environmental changes. A constant wall heat flux boundary condition in the heating section was achieved by Joule heating using a high current DC power supply (Xantrex, XFR 38–40), with a total heating length of 260 mm. A series of calibration experiments were conducted for cases without fluid flow under different power levels. The overall heat loss conductance, C, was found to increase slightly with increasing temperature difference between the channel and the environment. By fitting the data points with a second order polynomial, the heat loss conductance for the present setup was obtained, as expressed by Eq. (1).
2
 C ¼  5  10  6 T W  T a þ0:0005 T W  T a þ 0:0253 ð1Þ where T W and T a are the average wall and ambient temperatures, respectively. The local bulk mean temperature was determined using an energy balance between the heat added and the enthalpy of the fluid.
 Q_ in ¼ Q_ elec  C T w  T a ð2Þ
 x _ c^ p T b;x  T in ¼ Q_ in ð3Þ m L where Q_ elec and Q_ in are the rates of electrical heat input and heat _ is the total mass flow rate; c^ p is input to the fluid, respectively; m the average specific heat capacity; T b;x and T in are the local bulk

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Fig. 1. Schematic diagram of the experimental setup to study hydrodynamics and heat transfer of liquid–liquid Taylor flow.

mean and inlet temperatures, respectively; x is the axial location and L is the total heating length. In determining the two-phase heat transfer rates, a homogeneous mixture of the two immiscible liquid phases was assumed and a two-phase specific heat capacity was used to obtain the bulk mean temperature. c^ p;TP ¼ xD c^ p;D þ ð1  xD Þc^ p;C

ð4Þ

where c^ p;D and c^ p;C are the specific heat capacities of the disperse and continuous phases, respectively; and xD is the mass fraction of the disperse phase. The local Nusselt number ðNux Þ was determined as follows: Nux ¼

hx d qav d d Q_ 
in  ¼
¼ k k k T w;x  T b;x π dL T w;x T b;x

ð5Þ

where hx is the local heat transfer coefficient; k is the thermal conductivity of the fluid; qav is the average wall heat flux; T w;x is the local inner wall temperature. The thermal conductivity of the continuous phase, in this study hexadecane, was used to determine the two-phase Nusselt number (NuTP ) as only the continuous phase is in contact with the wall surface at any time. Since only the outer wall temperature, T 0w;x , was measured, T w;x was obtained from the measured outer wall temperature using the steady one-dimensional conduction equation, " #   R2o Ro 1 Q_ in 0  ln T w;x ¼ T w;x  ð6Þ 2 2π Lkss ðR2o  R2i Þ Ri where Ri and Ro are the inner and outer tube radii, respectively and kss is the thermal conductivity of the stainless steel tube. Single phase heat transfer data for water flow were collected and compared with the theoretical solution for simultaneously developing laminar flow for constant wall heat flux boundary conditions (Shah and London, 1978). Excellent agreement was observed, with most of the experimental Nusselt numbers falling within 5% of the theoretical values. In order to evaluate the experimental method on flows with higher Nusselt numbers, tests using single phase water were also carried out under turbulent flow conditions ð3000 oRe o 8000Þ with the results being

Fig. 2. Comparison of local experimental Nusselt number in turbulent ð3000 o Re o 8000Þ water flow with predictions of the Gnielinski (1976) correlation.

compared with predictions following Gnielinski (1976). As shown in Fig. 2, good agreement was achieved, with most of the experimental Nusselt numbers falling within 10% of the prediction values. The increase in the difference between experiment and prediction at the highest Nusselt numbers may derive from increasing errors as the difference between the wall temperature and the fluid temperature diminishes but, overall, the experiment has been shown to be capable of returning accurate results for heat transfer coefficients up to at least 20 kW m  2 K  1. 2.3. Working conditions and uncertainty analysis In the water-hexadecane (W-HX) system, since hexadecane has contact angles with the stainless steel ( o241) and FEP (  541) surfaces smaller than those of water (  721 on stainless steel and  1101 on FEP), the wall is expected to be wetted by hexadecane at all times—this was confirmed by visualizing the flow at the entrance to the FEP tube. The physical and thermal properties of

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Table 1 Properties of the working fluids at 30 1C and atmospheric pressure. Fluid

Water (W)

Hexadecane (HX)

Density, ρ (kg m  3) Viscosity, m (kg m  1 s  1) Surface tension, (N m  1) Refractive index Specific heat capacity, cp (J kg  1 K  1) Thermal conductivity, k (W m  1 K  1) Prandtl number, Pr

996 7.97  10–4 σW/HX ¼0.051 1.33 4180 0.616 5.4

767 2.67  10  3 – 1.42 2227 0.148 40.2

Table 2 The range of experimental parameters studied and uncertainties of individual measurements. Parameter

the water and hexadecane at 30 1C and atmospheric pressure were determined from the steam tables (IAPWS-IF97 database) and Prak et al. (2013), respectively (Table 1), with the surface tensions for the liquid–liquid system estimated based on the theory of Fowkes (1962). qffiffiffiffiffiffiffiffiffiffiffiffi σ D=C ¼ σ D þ σ C  2 σ dD σ dC ð7Þ where σ D and σ C are the surface tensions of the disperse and continuous phases, respectively; σ dD is the dispersion component of the surface tension. All experiments were conducted with the wall heat flux controlled to ensure the working fluid temperature was within 20–50 1C. While liquid properties at the calculated local temperatures were used for data analysis, the values at the mid-range temperature (30 1C) are used for the purposes of reporting characteristic conditions for each heat transfer condition. Table 2 shows the ranges of conditions studied and the related dimensionless numbers, together with the estimates of the uncertainties for individual measurements used in the evaluation of the two-phase Nusselt number (NuTP ). The ability of the experimental setup to generate well-characterized liquid–liquid Taylor flow was confirmed by the small standard deviations of the droplet and slug lengths (o10%), which were obtained by averaging over 20–70 unit cells, each comprising a liquid droplet and an adjacent liquid slug. The small fluctuations in the droplet and slug lengths and droplet velocity arose from small instantaneous variations of the flow rates of the individual fluids ( o4%). Each temperature sensor was calibrated using an ice water bath and a boiling water bath at atmospheric pressure and accuracies of 70.3 1C and 70.15 1C were found for the thermocouples and RTD sensors, respectively, in the temperature calibration experiments. Due to the relatively small temperature difference between the wall and the bulk fluid, the NuTP calculation for the liquid–liquid system was found to be very sensitive to the errors in the temperature measurements. Since the bulk temperatures were calculated from the heat balance based on the inlet fluid temperature, the error in the inlet temperature measurement will accumulate along the channel. A difference of 0.1–0.3 1C in the inlet temperature was found to give rise to up to 20% difference in NuTP . Therefore, it is very important to control the inlet temperature of both phases and reduce the measurement uncertainty. Additionally, the locations of measurement points need to be carefully determined for the bulk mean temperature calculation. A 2 mm error (comparable size to the sensor size) in the location determination was generally negligible in single-phase flow and our previous gas–liquid flow heat transfer. However, it gives rise to more than a 10% error in the liquid–liquid Nusselt numbers. It should be noted that uncertainty in the wall temperature measured by the RTDs could arise from poor contact of the RTD sensors with the tube surface and that great care was taken to ensure intimate contact. Another source of error in the liquid–liquid two-phase flow heat transfer experiments is the estimation of the heat loss. This

Range

Channel diameter, d (mm) 2.00 Mixture velocity, UTP (m s  1) 0.026–0.082 Homogeneous volume fraction, β 0.17–0.83 Normalized droplet length, LD/d 1.1–5.0 Normalized slug length, LS/d 0.4–3.6 Droplet velocity, uD (m s  1) 0.027–0.088 Heat flux, qav (kW m  2) 8.17–15.95 Wall to bulk mean temperature difference (1C) 1–4 Reynolds number, ReTP 15–48 Capillary number, Ca 0.0013–0.0043 Two-phase Nusselt number, NuTP

Uncertainty (%) 71 74 74 74 74 74 75 7 30 to 10 74 74 7 20 (typical)

was shown to be a minor effect as the heat loss itself was typically less than 5% in our study. In addition, the calculated NuTP is sensitive to the fluctuations of the droplet velocity and film thickness that arose from the flow rates unsteadiness. The details will be discussed in Section 4.2.1. Within the range and accuracy of the experimental data presented herein, the uncertainty of NuTP for the liquid–liquid system is expected to be up to 30% in the most challenging cases (wall to bulk-mean temperature difference  1 1C) and generally around 20%, which is somewhat greater than that for the single phase flow (o10%) and gas–liquid flow (o 15%) (Leung et al., 2012, 2014).

3. CFD methodology Our previously developed approach to model Taylor flow in a single unit cell using the VOF method is applied here. All fluids are assumed incompressible and the bulk properties of the fluids are calculated as volume fraction-weighted values. A full description of the governing equations was presented in the work of Asadolahi et al. (2011). The commercial software ANSYS Fluent 15.0.7 was used to perform the simulations. Steady, fully-developed liquid–liquid Taylor flow is periodic so it can be modelled as steady flows in a periodic unit cell. A moving domain method with periodic boundary conditions was used to model a unit cell of Taylor flow which consists of a droplet and two halves of adjacent slugs. In this method the flow is steady, fullydeveloped and periodic in a reference frame moving with the droplet velocity. First, transient simulations were performed to solve the continuity, momentum and volume fraction equations to obtain a steady hydrodynamic solution. Then the energy equation was solved using this “frozen” flow field to obtain a steady solution of the temperature field. A two-dimensional axisymmetric domain was used to perform the simulations. This method has been successfully validated using gas–liquid Taylor flow experimental data (Asadolahi et al., 2012) and for the hydrodynamics of liquid–liquid Taylor flow (Gupta et al., 2013). 3.1. Boundary and initial conditions As the flow is assumed steady and fully-developed, periodic boundaries were applied to the inlet and outlet of the computational domain. The velocity components and the temperatures at the inlet and outlet were made periodic following the procedures described in Asadolahi et al. (2011). The usual no slip condition and a specified heat flux were set at the wall. As the water droplet remains in the unit cell throughout the simulation and both fluids are assumed incompressible, the droplet volume is constant during the simulation period. Therefore, in order

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to simulate a specific experiment, the initialised droplet volume must be equal to the droplet volume in the experiment. There are two approaches to obtain the droplet volume from the experimental data. First, the droplet shape can be captured from the images taken in the experiments and therefore the volume of the droplet can be calculated. However, in the current experiments, the film thicknesses are very thin which introduces difficulties in the accurate identification of the droplet boundaries. Therefore, the droplet void fraction (ε) in the current numerical analyses was obtained from a simple volume balance (Suo and Griffith, 1964) as shown in Eq. (8):

ε U TP ¼ β uD

ð8Þ

where U TP and uD are the mixture and droplet velocities, respectively; β is the homogeneous void fraction which was obtained from the experimental flow rates of the two phases. The volume of the droplet was calculated as the product of the unit cell volume and the droplet void fraction. Once the droplet volume was obtained, the droplet phase was initialised as a rectangular zone, equal in length to the experimentally-determined unit cell length, in the two-dimensional axisymmetric computational domain. A fully-developed single-phase parabolic velocity profile with the average velocity equal to the mixture velocity (U TP ) was used to initialise the velocity field over the entire computational domain. For the energy equation, a constant initial temperature of 300 K was used.

3.2. Solver options The non-iterative fractional step scheme available in ANSYS Fluent was used for the time advancement and pressure-velocity coupling. The QUICK scheme was used to discretise the convective terms in the momentum and energy equations. The body-forceweighted scheme was chosen for pressure interpolation. The “implicit body force” option was enabled. The piecewise-linear interface calculation (PLIC) scheme was used for the discretisation of the volume fraction equation. A first order implicit scheme was used for the temporal discretisation of the transient terms. The gradients of scalars were calculated as cell centroid values from the centroid values of faces surrounding the cell using the Green–Gauss nodebased method. A time-step of typically 1  10  6 s was applied to the mass and momentum equations whereas a fixed time-step of 0.001 s was used for the energy equation. The meshes used here were made with reference to the detailed mesh resolution study performed by Gupta et al. (2009) and the original paper on the wrapping method (Asadolahi et al., 2011). The main improvement over the early work is that all code used to wrap boundary conditions was parallelised so the simulation could be run in parallel.

3.3. Material parameters It is noteworthy that the fluid temperature changes along the channel in the experiments, while constant properties at the midrange of the experimental temperature (30 1C) were used in the simulations (Table 1). Some representative simulations were repeated for both hydrodynamics and heat transfer using the properties at two extreme temperatures (20 1C and 60 1C). The differences in the hydrodynamics (indicated by film thickness) and heat transfer rate for the two cases are 24% and 11%, respectively. In practice, the experimental temperature range is much less than given by these extremes and the simulations are taken as providing heat transfer coefficients accurate to within  5%.

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4. Results and discussion Heat transfer and visualization experiments were performed in the circular channel and CFD simulations were carried out for the same flow conditions as those used in the experiments. The flow conditions, relevant non-dimensional numbers and droplet and slug lengths for 19 different cases are listed in Table 3. The Reynolds and Capillary numbers shown in the table are based on the properties of the continuous phase liquid and the mixture velocity (U TP ). The flow structures for all reported conditions (0.026 oU TP o0.082 m s  1, 0.17 o β o0.83) were confirmed to be in the Taylor flow regime, for which periodic water droplets and hexadecane slugs of almost constant lengths were observed for a period of time. 4.1. Hydrodynamics In the experiments, the droplet velocity ðuD Þ was determined by performing a frame-by-frame analysis of the recorded images as explained in Leung et al. (2010). However, the film thickness is very hard to measure accurately for the cases studied here due to the very low Capillary number. The film thickness is typically on the order of tens of microns or less (4–8 pixels under the 4  objective in our present system), which are dimensions that start to push the limits of standard measurement techniques in circular channels. The liquid film thickness was captured in numerical simulations using high grid resolution near the wall and is given in Table 3. The droplet and slug lengths, droplet velocity and homogeneous void fraction obtained from the CFD simulations are compared with those obtained from our experiments. The ratios of the simulation values to the experimental results are very close to 1, indicating that the hydrodynamics obtained from the simulations are in good agreement with those obtained from the experiments (the average difference being less than 1%). The comparison of the droplet shapes extracted from experimental images and those obtained from simulation also showed very good agreement. A detailed comparison of our experimental data and simulation results on the hydrodynamics has been undertaken in Gupta et al. (2013). The film thicknesses obtained from our simulations and experiments were found to be close to those predicted by the Bretherton and Aussillous and Quéré correlations for pure gas–liquid flow (Aussillous and Quéré, 2000; Bretherton, 1961). 4.2. Heat transfer measurements Although the flow is pulsatile, the measured wall temperatures are essentially steady, typically showing a standard deviation o 0:1 1C when sampled with frequency 22 Hz (significantly greater than the droplet frequencies of 5 to 14 Hz). This steadiness arises from the heat capacity of the wall—the CFD results, which do not include conjugate wall effects, show larger but still relatively small variations (e.g., o1 1C when qW ¼ 5000 W m  2 ) over a unit cell. Typical results for the time-averaged wall temperature, bulk fluid temperature and the local Nusselt numbers are shown in Fig. 3. It can be seen from Fig. 3 (a) that the temperatures increase linearly with the axial location, as expected. The temperature difference is about 2 1C, which is much smaller than that observed in the single phase flow with the same Reynolds number and heat flux. Fig. 3(b) shows that the local Nusselt number is rather uniform throughout the heating zone except at the first location, indicating that thermally fully-developed flow is achieved after the first monitor point. In the following analysis, we consider only the fully-developed two-phase Nusselt numbers, NuTP , determined by averaging the local Nusselt number in the downstream region where relatively constant values were observed (standard deviation o5%). In order to minimize the entrance effects and the uncertainties arising from the end of the

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Table 3 Flow conditions for water-hexadecane studied experimentally and via simulations. Tube diameter ¼ 2.00 mm. Experiments

CFD

Case no.

ReTP at 30 1C

Ca at 30 1C

U TP;e (m s  1)

uD;e (m s  1)

β

LD; e (mm)

LS; e (mm)

ε

NuTP:e

δF

NuTP:s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

15 18 18 24 24 27 30 30 30 30 36 36 36 36 36 42 42 45 48

0.0013 0.0016 0.0016 0.0021 0.0021 0.0024 0.0027 0.0027 0.0027 0.0027 0.0032 0.0032 0.0032 0.0032 0.0032 0.0038 0.0038 0.0040 0.0043

0.026 0.031 0.031 0.041 0.041 0.046 0.051 0.051 0.051 0.051 0.061 0.061 0.061 0.061 0.061 0.072 0.072 0.077 0.082

0.027 0.031 0.032 0.044 0.043 0.049 0.055 0.054 0.054 0.054 0.064 0.065 0.065 0.067 0.067 0.077 0.079 0.083 0.088

0.60 0.50 0.67 0.50 0.75 0.33 0.20 0.40 0.60 0.80 0.17 0.33 0.50 0.67 0.83 0.57 0.71 0.67 0.50

3.99 3.72 5.00 3.64 7.40 2.89 2.43 3.01 4.44 9.96 2.27 2.92 3.86 5.40 9.65 4.23 6.26 5.46 3.86

1.40 2.18 1.32 2.07 1.30 3.57 6.11 2.56 1.65 1.33 7.01 3.60 2.32 1.37 0.82 1.72 1.21 1.32 2.16

0.57 0.49 0.65 0.47 0.72 0.32 0.19 0.38 0.57 0.76 0.16 0.32 0.48 0.61 0.76 0.53 0.65 0.62 0.46

20.97 28.79 29.59 26.73 27.90 18.83 20.99 20.58 16.24 32.24 15.03 20.57 25.66 18.34 20.41 18.43 14.48 22.40 22.30

0.007 0.009 0.011 0.012 0.011 0.012 0.013 0.013 0.013 0.013 0.014 0.015 0.015 0.014 0.014 0.016 0.016 0.017 0.017

24.40 26.92 28.54 27.24 29.19 23.90 22.02 26.33 29.25 30.17 21.71 24.21 27.40 30.01 26.73 27.08 26.33 26.34 24.82

d

Fig. 4. Comparison of the average Nusselt number between the experimental data and the simulation results. Experimental error bars represent total uncertainties in the NuTP calculations.

Fig. 3. (a) Typical mean wall temperature and bulk fluid temperature profiles, and (b) local Nusselt numbers along the tube at U TP ¼ 0.041 m s  1 and β ¼0.5.

channel, the data for the first and last measurement points are not used.

4.2.1. Comparison between experimental data and CFD results The experimental Nusselt numbers and those calculated from the CFD simulations are compared in Fig. 4. The error bars on the

experimental data represent the total uncertainties in the NuTP calculations associated with related individual measurement uncertainties. It can be seen that the experimental data are generally in reasonable agreement with the simulations (relative standard deviation ¼22.9%). However, there are some cases where the experimental values are up to 45% lower than the predictions, a discrepancy which cannot be explained purely by uncertainties in the evaluation of NuTP from the temperature measurements. Considering the parametric sensitivities in the comparison of experimental and simulated results, we suggest that the unaccounted discrepancies in Fig. 4 may arise from errors in the measurements of droplet velocity. Such an error feeds into the simulations to give a film thickness that differs from that pertaining in the experiments. As described above, uD and U TP are critical input parameters for the simulation of specific experimental conditions. This is illustrated by comparing the flow characteristics and Nusselt numbers obtained for Case 18 (U TP ¼0.077 m s  1, β ¼0.67) when the value of uD is changed slightly, keeping other

Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

147

Table 4 Flow parameters and Nusselt numbers for simulations for Case 18 (U TP ¼ 0.077 m s  1, β ¼ 0.67).

Case 18.1 Case 18.2 Difference (%)

U TP (m s  1)

uD (m s  1)

β

0.00769 0.00769 –

0.0836 0.0817  2.3

0.67 5.26 0.64 5.02  4.5  4.6

LD (mm)

LS (mm)

δF (μm)

NuTP

1.50 1.74 þ15.7

35.2 33.1  6.0

26.4 29.1 þ10.2

simulation input parameters the same. As shown in Table 4, a reduction in uD by 2.3% leads to apparently small differences in flow pattern, with the droplet and slug lengths changing in accord with the volume balance in the cell. The film becomes thinner by 2.1 μm or 6.0% and the Nusselt number rises by 10%. These results highlight the fact that the liquid–liquid flow heat transfer is highly sensitive to the flow conditions. The same feature was reported by Mac Giolla Eain et al. (2015b) who found that for carrier slugs of approximately equal length, an increase in film thickness of 2.1% corresponds to a reduction of 17% in NuTP . In addition to the 74% uncertainty estimate for the experimental value of uD (Asadolahi et al., 2012; Gupta et al., 2013), in the experiments, the unsteadiness in the flow rate or surface roughness in the wall may also change the flow pattern or film thickness close to the wall. The high sensitivity of the Nusselt number calculation to small differences in the droplet velocity appears to account for the large uncertainty and difficulty for liquid–liquid flow heat transfer measurements. As these parameters do not enter directly into the experimental calculation of the Nusselt number, their uncertainties are not included in the error analysis shown in Fig. 4. 4.3. Parametric effects The uncertainties in the experimental results tend to obscure the details of the effect of parameter variations. Therefore, for the purposes of this discussion, we consider only the simulated results. 4.3.1. Effect of slug and droplet lengths Fig. 5 shows the variation of the normalized Nusselt number with normalized slug (LS =d), and droplet lengths (LD =d) for different mixture velocities with 0:17 o β o 0:83. The normalized Nusselt number is defined as Nun ¼

NuTP Nu0

ð9Þ

where Nu0 is the Nusselt number for fully-developed laminar flow with constant heat flux boundary conditions, having a value of 4.364. It can be seen that flow with shorter slugs or longer droplets results in improved thermal performance. The heat transfer enhancement can be up to seven times relative to the liquidonly flow. As reported in many gas–liquid Taylor flow studies (Horvath et al., 1973; Leung et al., 2012, 2010; Walsh et al., 2010), the heat transfer rate increases with decreasing liquid slug length due to the higher frequency and greater strength of the internal recirculations—similar effects arise in liquid–liquid flows. In the gas–liquid Taylor flow, the disperse gas phase makes a negligible contribution thermally to the overall heat transfer. However, for the liquid–liquid flow system, the thermal capacity of the disperse phase is comparable with that of the continuous phase and a significant portion of the heat transferred from the wall is taken up by the liquid droplet. The quantity of heat carried by the disperse phase depends on its thermophysical properties and on the internal heat transfer rates within the unit cell which

Fig. 5. The effect of (a) normalized slug length (LS =d), and (b) normalized droplet length (LD =d) on the normalized Nusselt number obtained for different mixture velocities U TP with 0:17o β o 0:83. The dashed line shows the prediction
 U TP ¼ 0:051 m s  1 of the model developed in this paper.

change as the flow conditions are varied. We return to this topic later but it should be noted that, without internal recirculation, long slugs could be expected to approach the single-phase limit, Nun -1: this is clearly not occurring in Fig. 5(a), underlining the importance of internal recirculation even in these relatively long slugs. By the same token, long droplets approach core-annular flow, for which (neglecting any wall film resistance) we would expect Nun -4:2 (being the ratio of the thermal conductivities of water and hexadecane)—again, we remain well above this limit even for the longest droplets in Fig. 5(b).

4.3.2. Effect of homogenous void fraction (β ) and mixture velocities (U TP ) The variation of normalized Nusselt number (Nun ) with homogeneous void fraction (β) at different mixture velocities (U TP ) with 1:1 o LD =d o 5:0 and 0:4 o LS =d o 3:6 is shown in Fig. 6. It can be seen that the heat transfer enhancement increases from the limiting value of Nun ¼1 at β ¼0 (pure hexadecane) up to a maximum value (6.91), beyond which Nun declines, apparently towards the wateronly value (4.2). These results contrast with those for gas–liquid systems which show a negative impact of increasing β on the heat

148

Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

transfer enhancement due to the relative reduction in liquid flow rate (Asadolahi et al., 2012; Leung et al., 2010). In the liquid–liquid flows, the slug length decreases and the droplet length increases with an increase in β, resulting in better heat transfer performance as discussed in the preceding paragraphs. The effect of the mixture velocity (U TP ) on the normalized Nusselt number (Nun ) is shown in Fig. 7. The figure shows Nun to be nearly independent of the mixture velocity for different homogeneous void fractions. Our gas–liquid flow studies also suggested the Nusselt number to be almost independent or weakly dependent on mixture velocity for short slugs (LS =d  3) (Gupta et al., 2010). It must also be remembered here that at a given homogeneous void fraction and for different mixture velocities, the slug and droplet lengths are not always the same. Therefore, the effect of mixture velocity on Nun seen in Fig. 7 is a combined effect of different slug/droplet lengths and mixture velocities. 4.4. General correlation for Taylor flow heat transfer It is clear from the above discussion that the dependence of heat transfers coefficient on flow conditions in liquid–liquid Taylor flows is the result of complex interplays of the effects of diverse parameters. In this section we develop a simple model to understand and predict the Taylor flow heat transfer in both gas–liquid and liquid–liquid situations. In fully-developed heat transfer with constant wall heat flux, the temperatures of all the phases must increase at the same rate. The fraction of the wall heat flow that goes into the disperse phase is therefore _ D cpD m QD ¼ _ D cpD þ m _ C cpC m QW

V_ D ρD cpD m ¼ ¼ mþ1 V_ D ρD cpD þ V_ C ρC cpC

ð10Þ

within the system (one unit cell) that the redistribution occurs. Heat is stored in the (nearly stationary) film around the bubble, the excess energy subsequently being transferred into the next liquid slug coming by. The film temperature varies dramatically between the bubble and slug regions (Asadolahi et al., 2011). The same thermal constraints apply to liquid–liquid flows and the film region continues to act as a buffer that redistributes the uniform wall heat flux into the droplet and slug zones accordingly. It should therefore be possible to describe gas–liquid and liquid– liquid flows according to a common model, so long as the heat flow from the film into the droplet region in liquid–liquid flow is accounted for, the corresponding flow in gas–liquid systems (m-0) being insignificant. In order to understand the interactions in this system, the unit cell is divided into droplet (i.e. bubble for gas–liquid flows) and slug zones; within each zone a wall film region and a bulk region are identified. The identification of the film and bulk regions in the droplet zones is obvious, with the wall film of continuous phase fluid being physically distinct from the droplet of the disperse phase occupying the core region. In the slug zone, the zones are not physically distinct but one can logically identify the dividing streamline that separates the slug recirculation cell from the wall flow as the boundary between the wall film and the bulk, as shown schematically in Fig. 8. Analysis presented in Fig. 11 of Leung et al. (2012) shows this to be a good assumption for Cao0.1, which is the case for all of the data considered here. The two-phase overall heat transfer coefficient is defined as:
 Q W ¼ hTP AW T W  T b

where T b is the bulk mean temperature and can be calculated neglecting the contribution of the wall film:

where   V_ D ρD cpD β ρD cpD ¼ m¼ 1  β ρC cpC V_ C ρC cpC

ð11Þ

Similarly, QC 1 ¼ mþ1 QW

ð12Þ

For low-pressure gas–liquid flow, m-0, and practically all the energy goes into the continuous phase, essentially the slug. The energy flow into the system is uniformly distributed, so it is only

Fig. 6. The effect of the homogenous void fraction (β) on the normalized Nusselt number for different mixture velocities (U TP ) with 1:1o LD =d o 5:0 and
 0:4 o LS =d o 3:6. The dashed line shows the prediction U TP ¼ 0:051 m s  1 of the model developed in this paper.

ð13Þ

Tb ¼

V_ D ρD cpD T D þ V_ C ρC cpC T S mT D þ T S ¼ mþ1 V_ D ρ cpD þ V_ C ρ cpC D

ð14Þ

C

where T D and T S are the temperatures of the droplet and the slug, respectively. We neglect direct heat transfer between the core regions (slug and bubble/droplet) because the exchange area and the velocities at the interface are small. Therefore, defining heat transfer coefficients in terms of the respective wall areas and assuming negligible mass flow in the film, the droplet and slug

Fig. 7. The effect of the mixture velocity (U TP ) on the normalized Nusselt number for different homogenous void fractions (β) with 1:1o LD =d o 5:0 and 0:4 o LS =d o 3:6. The dashed line shows the prediction ðβ ¼ 0:5Þ of the model developed in this paper.

Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

149

Fig. 9. Comparison of gas–liquid data of Leung et al. (2012) with the film-to-slug heat transfer coefficient (NuFS ) evaluated in terms of Eq. (19). The solid line shows the best-fit Hausen-type developing flow correlation and the dashed line shows the mean Nusselt number (Num;H1 ) for thermally developing single-phase laminar flow with constant wall heat flux H1 boundary condition. Fig. 8. Schematic of the heat transfer system in a unit cell. D and S denote the droplet and slug zones, respectively. Heat transfer mechanisms in the system are indicated by the arrows: (1) heat transfer from the wall to the wall film surrounding the droplet and slug, (2) heat transfer from the wall film to the slug, (3) transfer of heat from the wall film to the droplet.

temperatures can be obtained by Eqs. (15a) and (15b):   1 LUC 1 1 þ T S ¼ T WS  qW hWS LS 1 þ m hFS and



T D ¼ T WD  qW

1

hWD

þ

LUC m 1 LD 1 þm hFD

ð15aÞ

 ð15bÞ

where T WS and T WD are the wall temperature in the slug and droplet zones, respectively. From the CFD analysis, the wall temperature in liquid–liquid flow is shown to be remarkably constant across a unit cell (T W 71 1C when qW ¼ 5000 W m  2 ); for gas–liquid flows, the value of the wall in the bubble region is immaterial. Therefore, the wall temperatures in both zones are assumed equal ðT WD ¼ T WS ¼ T W Þ. hWS and hWD are the heat transfer coefficients from the wall to the films in the slug and droplet zones, respectively, and again are assumed to be equal ðhWS ¼ hWD ¼ hW Þ. Substituting Eqs. (14) and (15a), (15b) into Eq. (13), the overall heat transfer correlation can be obtained upon rearrangement:  2 1 1 LUC m 1 LUC 1 1 ¼ þ þ ð16Þ hTP hW LD m þ 1 hFD LS ðm þ 1Þ2 hFS As the axial velocity in the liquid film is negligible compared with the slug and droplet velocities, the film heat transfer conductance can be approximated as being for conduction through a stationary film of thickness δF : hW ¼

kC

δF

ð17Þ

Gupta et al. (2013) have shown that the correlation proposed by Aussillous and Quéré (2000) agreed well with the film thicknesses obtained using CFD. This semi-empirical correlation is therefore used to calculate the film thickness, as given by Eq. (18):

δF

1:34Ca2=3 ¼ R 1 þ 3:35Ca2=3

ð18Þ

For the slug heat transfer, we note that this term is most clearly observed in gas liquid flows, for which m-0 and the droplet term is negligible. Leung et al. (2010) found a dimensionless Hausentype expression to predict the apparent slug Nusselt number for nitrogen-water flow in terms of an inverse Graetz number for the slug LnS ¼ LS =ðReTP  Pr C  dÞ: Nu LnS ¼ 4:364 þ

a1 n1=3

LnS þ a2 LS

ð19Þ

However, this approach failed when applied to more viscous fluids (Leung et al., 2012). Since it is precisely for these more viscous fluids (larger values of Ca) that the wall film resistance may become significant, we have re-analysed the earlier data in terms of Eq. (16) in which the droplet term is neglected, and Eqs. (18) and (19) are used for the relevant film and slug terms, respectively. This analysis reveals that the wall film resistance with the most viscous liquid (ethylene glycol) is up to 65% of the total whereas with water it is always o 25%. A least-squares fit yields values of a1 ¼ 0:171 and a2 ¼ 0:0663, as shown in Fig. 9. Clearly the fit to the entire data set is vastly improved, with average relative correlation error for Eq. (19) being 1.7%, with a standard deviation of 20%. The success of Eq. (19) confirms that the heat transfer in the slug region can be treated as a developing flow, in accord with earlier arguments. Also plotted is the mean Nusselt number for thermally developing flow alone for a constant wall heat flux H1 boundary condition, for which the tabulated data provided by Shah and London (1978) are fitted in the Hausen form (Eq. (19)) with a1 ¼0.0894 and a2 ¼ 0.0490. It is evident that, as expected, the recirculating flow in the slug enhances the heat transfer above that for developing laminar flow. These same arguments apply to liquid–liquid flows and we proceed now on the basis that Eq. (19) may be used without modification of the parameters. Considering the droplet zone, the apparent value of the film-todroplet heat transfer coefficient in liquid–liquid flows can be extracted from the data as:

hFD ¼

LUC LD 1  h1W hTP



2

m mþ1



LUC 1 1 LS ðm þ 1Þ2 hFS

ð20Þ

where hW and hFS are evaluated according to Eqs. (18) and (19) respectively.

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Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

Fig. 10. Comparison of liquid–liquid (water–hexadecane) data generated using CFD for the film-to-droplet heat transfer coefficient (NuFD ) with the mean Nusselt number (Num;H1 ) for thermally developing single-phase laminar flow with constant wall heat flux H1 boundary condition.

Fig. 10 shows the results extracted from a total of 55 runs, including all the conditions in Table 3 together with additional simulations with a wider range of conditions—overall, the parameter range covered d ¼1 and 2 mm; 2:28 o LUC o 11:29 mm; 0:16 o β o 0:83; and 0:026 o U TP o 0:246 m s  1. The results of this analysis are
presented interms of NuFD based on the droplet characteristics ¼ hFD  d=kD versus dimensionless droplet length, LnD ¼ LD =ðReD  Pr D  d), in keeping with laminar, developing characteristics of the flow and heat transfer in the droplet. Because they are determined by difference (Eq. (20)), these results include all the model mismatch error but a clear trend nevertheless emerges. Indeed, as shown by the solid line, the results overall are well represented by the solution for the mean heat transfer coefficient in developing laminar flow in a tube with constant wall heat flux ðNum;H1 Þ. Note that, for convenience, the Reynolds number used
in the definition of LnD is based on U TP and d  when use of uD and d 1  2δF is formally more correct. However, the differences are small for thin films ðCa ⪅0:1Þ and negligible in the overall evaluation. Interestingly, the Nusselt number for the droplet shows no enhancement over that for developing laminar flow in a tube, unlike for the slug where recirculation effects were significant. Ultimately, the robustness of the result that NuFD ¼ Num;H1 needs to be demonstrated across a wider range of conditions. However, it is a very satisfying result as it stands because it fits the present data without any manipulation and conforms with the expected limit for single-phase flow of the disperse fluid. Finally, Fig. 11 compares the Nusselt numbers from these simulations, and all liquid–liquid (CFD) and gas–liquid (experimental) data with the values predicted by Eq. (16). Overall a very good fit is obtained, with nearly all data points within 720% of the parity line. In Eq. (16), the overall heat transfer resistance is the sum of the individual resistances corresponding to the wall-film, film-droplet and film-slug transfer processes. Applying the model to the conditions listed in Table 3 yields a contribution of the wall-film resistance in the range 24 to 42%; the effective film-to-slug resistance is from 5 to 57%; and the effective film-to-droplet resistance is from 17 to 57%. When the gas–liquid data are included, the correlation range for relative resistances in Fig. 11 becomes 3 to 42%, 5 to 97%, and 0 to 57%, respectively. It is not surprising then that Eq. (16) not only provides accurate correlation over a wide range of conditions but also captures the individual

Fig. 11. Comparison of the predicted Nusselt number from Eq. (16) with the experimental (gas–liquid data of Leung et al., (2012), N2-HX, N2-W, N2-EG, N2-EG/ W) and CFD (liquid–liquid data, W-HX) results.

Fig. 12. Variation of the relative resistance contribution with homogeneous void fraction (β) at different mixture velocities (U TP ).

parametric influences, as shown in Figs. 5, 6 and 7 where the dashed lines represent the model predictions. Investigating the individual contributions to the overall heat transfer resistance in the application of Eq. (16), Fig. 12 shows the relative contribution of each of these resistances as a function of homogeneous void fraction (β) for two of the velocity conditions (U TP ), shown in Fig. 6. The quantity of heat carried by each zone changes as the flow condition is varied so the overall effect represents the influence of both m and the internal heat transfer coefficients. The film-droplet resistance contribution increases with increasing β , largely due to the increased proportion of disperse phase in the overall flow. Similarly, the contribution of the film-to-slug resistance decreases with increasing β due to the relative reduction in continuous phase flow rate. The contribution of the wall film resistance is always significant but changes relatively little over the range 0:17 o β o 0:83. The model is capable of capturing the complex parametric dependencies and sensitivities of the Taylor flow heat transfer. For the two cases listed in Table 4, a change of þ 9.8% ( þ10.2% from the CFD simulations) in the overall Nusselt number was obtained from the model prediction, with only 2.3% uncertainty in the droplet velocity and corresponding changes in other

Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

hydrodynamic parameters. The fact that relatively small differences in the bubble velocity (  0.002 m s  1), the slug and droplet lengths (  0.24 mm) and the film thickness (  2.1 mm) cause a significant change in the heat transfer explains that the difficulty of making accurate experimental measurements of these hydrodynamic parameters makes it very hard experimentally to analyse liquid–liquid Taylor flow heat transfer in microchannels.

5. Conclusions The heat transfer performance of liquid–liquid Taylor flow has been studied using a combination of experimental and CFD techniques in circular channels of 1 and 2 mm diameter using water and hexadecane as the disperse and continuous phases, respectively. Visualization and heat transfer experiments were conducted for 0.026oU TP o0.082 m s  1 and 0.17o β o0.83 and CFD modelling was carried out in a periodic computational domain for these and other flow conditions. A mechanistic model was developed to understand and predict the Taylor flow heat transfer in both gas– and liquid– liquid situations. Overall, the measured heat transfer coefficients are in good agreement with the simulations. However, the Nusselt number calculation is extremely sensitive to experimental uncertainties. The simulations confirm the strong dependence on the flow conditions seen in the experiments, explaining the large uncertainty and difficulty for liquid–liquid flow heat transfer measurements. A simple compartment model that includes heat transfer from the wall to a continuous film region, and from the film separately to the slug and droplet (or bubble in gas–liquid flows) can correlate our entire set of CFD (liquid–liquid) and experimental (gas–liquid) data with a relative standard deviation of 20%. In dimensionless terms, the correlation derived from the model is based on the summation of apparent resistances:  2 1 1 LUC m 1 kC LUC 1 1 ¼ þ þ NuTP NuW LD m þ1 NuFD kD LS ðm þ 1Þ2 NuFS

ð21Þ

where the inter-compartment heat-transfer coefficients are calculated as NuW ¼

d

δF


¼ 5 þ 1:5 Ca

−2=3

 assuming the correlation of Aussilous and Queˊ reˊ

NuFD ¼ Num;H1 ¼ 4:364 þ NuFS ¼ 4:364 þ

0:0894 n1=3

LnD þ 0:0490 LD

0:171 n1=3

LnS þ0:0663 LS

;

;

with LnS ¼

with LnD ¼

LD =d ReD Pr D

LS =d ReTP Pr C

The terms in m in Eq. (21) reflect the importance of the relative heat capacity rate of the two phases. For gas–liquid flows, m-0 and film-droplet heat transfer is negligible. However, especially for gas flow with viscous liquids (larger Ca), the wall-film term remains important. The film-droplet resistance term becomes important in liquid–liquid flows, and dominates when the droplets are relatively long (high volume fraction of the disperse phase). These trends account quantitatively for the observed heat transfer rates in gas–liquid and liquid–liquid Taylor flow over a very wide range of conditions. Strikingly, the model provides better agreement with CFD predictions than do our experiments in liquid– liquid flows (which are very sensitive to flow conditions, as captured in the model). The model developed here therefore provides a robust phenomenological basis for the interpretation and prediction of Taylor-flow heat transfer.

151

Nomenclature a1 ; a2 A C Ca cp d h k L Ln m _ m Nun Nu0 NuFS NuFD NuTP Pr qav Q_ elec Q_ in R ReD ReTP Ta Tb T in Tw T 0w u U U TP V_ x

constants area (m2) heat loss conductance (W K  1) capillary number (U TP μC =σ ) heat capacity (J kg  1 K  1) tube diameter (m) heat transfer coefficient (W m  2 K  1) thermal conductivity (W m  1 K  1) length (m) dimensionless length (L=ðRe

TP PrdÞ)
  _ p D = mc _ p C heat capacity rate ratio mc 1 mass flow rate (kg s ) normalized Nusselt number (NuTP =Nu0 ) fully-developed single-phase Nusselt number with constant heat flux boundary conditions (¼4.364) Nusselt number for film-to-slug heat transfer, based on continuous-phase properties (hFS d=kC ) Nusselt number for film-to-droplet heat transfer, based on disperse-phase properties (hFD d=kD ) two-phase Nusselt number (hTP d=kC ) Prandtl number (cp μ=k) average wall heat flux (W m  2) electrical heat input (W) heat input to the fluid (W) radius (m) Droplet Reynolds number, (U TP ρD d=μD ) two-phase Reynolds number (U TP ρC d=μC ) ambient temperature (K) bulk mean fluid temperature (K) inlet temperature (K) wall temperature (K) tube outer wall temperature (K) velocity (m s  1) superficial velocity (m s  1) two phase superficial velocity ( ¼ U D þ U C ) (m s  1) volumetric flow rate (m3 s  1) axial location (m)

Greek symbols

β δ ε μ ρ σ

homogeneous void fraction (U D =U TP ) film thickness (m) disperse phase void fraction in a unit cell dynamic viscosity (kg m  1 s  1) density (kg m  3) surface tension (N m  1)

Subscripts av b C D F FD FS i o S ss TP UC W

average bulk continuous phase disperse phase, droplet or bubble film film to droplet or bubble film to slug inner tube outer tube slug stainless steel two-phase unit cell wall

152

x

Z. Dai et al. / Chemical Engineering Science 138 (2015) 140–152

axial location

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