Computational heat transfer and two-phase flow topology in miniature ...

Microfluid Nanofluid DOI 10.1007/s10404-007-0176-1

RESEARCH PAPER

Computational heat transfer and two-phase flow topology in miniature tubes Djamel Lakehal Æ Guillaume Larrignon Æ Chidambaram Narayanan

Received: 9 November 2006 / Accepted: 16 April 2007 Springer-Verlag 2007

Abstract Detailed computational multi-fluid dynamics simulations have been performed to study the effect of twophase flow regime on heat transfer in small diameter pipes. Overall the heat removal rate in two-phase flow is higher than in single phase. Subtle differences in thermal removal rates are revealed when the flow-regime transitions from bubbly to slug and slug-train configurations. It is found that the wall thermal layer is affected by two separate mechanisms: an early-stage compression due to gas-jet fragmentation into slugs or bubbles, and a background inclusion-induced flow superimposed on the equivalent single-phase fully developed flow far downstream. The first mechanism resembles a confinement or blockage effect, and is shown to directly influence radial temperature gradients. The downstream mechanism is a cell-based developed flow (rather than fully developed), and is shown here to increase the wall shear in the vicinity of the cell, leading to higher heat transfer rates. The mean Nusselt number distribution shows that the bubbly, slug and slug-train patterns transport as much as three to four times more heat from the tube wall to the bulk flow than pure water flow. A mechanistic heat transfer model is proposed, based on frequency and length scale of inclusions.

1 Introduction The physics of two-phase flow heat transfer in miniature tubes is multifaceted, featuring significant differences with D. Lakehal (&) G. Larrignon C. Narayanan ASCOMP GmbH, Technoparkstrasse 1, 8005 Zurich, Switzerland e-mail: [email protected]

macroscale transport phenomena. The particularities concern pressure drop and friction coefficient, velocity profile and heat removal rate. More subtle differences are rooted into the imbalance between surface forces, which dominate with decreasing tube size, inertia and viscous forces and gravity. The high ratio of total surface area to volume characterising microchannels is useful in facilitating the removal of a large amount of heat from the tube confines vis-a`-vis heat capacity. Also, the use of convective boiling is particularly desirable for increasing heat removal efficiency, as the latent heat of vaporization is appreciably higher than sensible heat changes for a set of temperature operating ranges. There has been a growing interest in conducting experimental research focusing on thermal flow in miniature pipes, with two main objectives: (1) to establish a flow regime map and inherent transition mechanisms, and (2) to characterize heat transfer, mainly with phase change. The confinement effect has been identified as an important factor affecting heat transfer in small tubes (Cornwell and Kew 1997; Brauner and Moalem-Maron 1992; Akbar et al. 2003). Further, the surface condition of the inner wall was identified as another important parameter influencing thermal flow patterns by Serizawa et al. (2002), who studied air–water and steam-water flow in circular tubes (20–100 mm). A wide-spectrum pattern map was identified in this campaign, and special features were described. Chen et al. (2002) used visual observation from a high-speed camera to identify flow patterns in 1–3 mm vertical and horizontal pipes; they identified bubbly flow, slug flow, bubbly-train, churn and annular flow. In a recent paper, Chen et al. (2006) studied the effect of tube diameter on vertical two-phase flow regimes in small pipes. Their studies led to establishing flow regime maps showing the effect of tube diameter on the transition boundaries of slugchurn and churn-annular regimes.

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Microfluid Nanofluid

Boiling heat transfer in small diameter tubes has been experimentally investigated by Huo et al. (2006) using R134a as the working fluid. The correlation between nucleate boiling dominance and vapour quality was determined based on pipe diameter. Revellin et al. (2006) used the same working fluid to characterize flow regime map, but under diabatic conditions. The two-phase flow pattern transitions observed did not compare well neither to the leading macroscale flow map for refrigerants nor to microscale map for air–water flow. In terms of heat transfer enhancement by two-phase flow, we note the experimental work of Monde and Mitsutake (1995), who measured the wall temperature fluctuations are different streamwise locations, and noted indeed a substantial enhancement in heat transfer due to bubble passage. From the computational front, the ability to predict the physics of microfluidics and associated heat transfer in miniature tubes can greatly benefit to various emerging technologies, including ink-jet, electro-thermal systems, MEMS design, computer chip cooling and medical diagnostics devices, e.g. bio-chips. In practical applications, the flow involves phenomena acting at different time and length scales, and this requires tailoring properly the prediction techniques. For bubbly and slug flow in 1 mm diameter pipes, typical time scales are of the order of 0.001 s. At each level of the scale cascade, the physics of the flow is amenable to numerical prediction by scalespecific strategies. The simulation approach should typically be capable of predicting flow motion and topology; inter-phase transfer mechanisms; capillary forces; and other thermal effects. Only with such capabilities could the physics of microfluidics be accurately predicted. CMFD studies are rare in this context as compared to experimental investigations, except maybe the contribution of Ua-Arayaporn et al. (2005) who conducted a similar investigation as the present one, though in a periodic small box rather than in an elongated tube. Their data for heat transfer in particular suffered in turn from domain size effects, and did not allow drawing a clear conclusion as to the real impact of two-phase flow on heat transfer. In this paper we report on the way this class of flow is tackled by use of the Level Set approach (Sussman et al. 1994), in which we have incorporated convective heat transfer, and surface tension modelling capabilities (Lakehal et al. 2002). The focus here is on the role played by flow regime patterns and associated blockage/confinement effects (without changing pipe dimensions) in controlling heat transfer. The 2D axisymmetric simulations were performed in a 1 mm diameter tube heated at the surface, in which air and water were injected as co-flowing streams. The computational strategy combines the unsteady Navier–Stokes equations for the flow and Level Sets (LS) for interface

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dynamics. The measurements used as reference are taken from (Chen et al. 2002).

2 Simulation framework 2.1 TransAT microfluidics code The CMFD code TransAT developed at ASCOMP is a multi-physics, finite-volume code based on solving multifluid Navier–Stokes equations. The code uses structured meshes, though allowing for multiple blocks to be set together. MPI parallel based algorithm is used in connection with multi-blocking. The grid arrangement is collocated and can thus handle more easily curvilinear skewed grids. The solver is pressure based (projection type), corrected using the Karki–Patankar technique for compressible flows (up to transonic flows). High-order time marching and convection schemes can be employed; up to third-order Monotone schemes in space. Multiphase flows are tackled using interface tracking techniques for both laminar and turbulent flows. The one-fluid formulation context on which TransAT is built is such that the flow is supposed to involve in one fluid having variable material properties, which vary according to the color function as it is advected by the flow, identifying gas flow regions from the liquid phase. Specifically, both the Level-Set and the Volume-ofFluid Interface Tracking Methods (ITM) (Lakehal et al. 2002) can be employed in the code to track evolving interfaces. 2.2 Interface tracking context When the exact shape of the interfaces separating two fluids is not known, or not relevant, one may resort to the averaged two-fluid approach, where separate conservation equations are required for each phase with appropriate interfacial exchange forces. ITMs may be invoked when the identification of interfaces needs to be precise, as in the breakup of large bubbles, droplets or liquid jets. The key to the methods is the use of a single-fluid set of conservation equations with variable material properties and surface forces. The concept is attractive, since it offers the prospect of a more accurate strategy than that offered by the two-fluid formalism, while minimizing modeling assumptions. 2.2.1 Transport equations The incompressible heat and fluid flow equations expressed within the single-fluid formalism take the following form


ru ¼ 0

ð1Þ

ot ðquÞ þ rðquu rÞ ¼ Fg þ Fs ot ðqCp TÞ þ r qCp Tu q00 ¼ 0

ð2Þ ð3Þ

where the RHS terms in the momentum equation (Eq. 2) represent respectively the body forces, the surface tension force expressed by Eq. 5 below. In Eqs. 2 and 3 where phase change is not accounted for, r is the Newtonian stress tensor, T is the temperature, Cp is the heat capacity of the fluid, and q† is the heat flux. 2.2.2 The level set method In the LS method employed here, the interface between immiscible fluids is represented by a continuous function /, representing the distance to the interface that is set to zero on the interface, is positive on one side and negative on the other. This way, both fluids are identified, such that the location of the physical interface is associated with the zero level. The LS evolution equation is given by (Sussman et al. 1994) ot / þ ur/ ¼ 0

ð4Þ

Material properties such as the density, the viscosity, the heat capacity and the thermal conductivity are updated locally based on /, and smoothed across the interface using a smooth Heaviside function. Further, the fact that / is a continuous function across the interface helps determine the normal vector n to the interface, and thereby the interface curvature k required for the determination of the surface tension force, Fs ¼ cjnds

ð5Þ

where c is the surface tension coefficient of the fluid. In practice, the level set function ceases to be the signed distance from the interface after a single advection step of Eq. 4. To restore its correct distribution near the interface, a re-distancing problem has to be solved, in which the equation below has to be integrated to steady state (Sussman et al. 1994): os d sgn ðd0 Þ ð1 jrdjÞ ¼ 0;

d0 ðx; s ¼ 0Þ ¼ /ðx; tÞ ð6Þ

Equation 6 is solved after each advection step of Eq. 4, using the non-oscillatory third-order WENO scheme. Details about alleviating the mass conservation issue in the approach can be found in Takahira et al. (2004). The Navier–Stokes transport equations and the level set advection function are solved using the third-order Runge–Kutta

explicit scheme for time integration. The convective fluxes are discretized using the third-order Quick scheme bounded using a TVD limiter (Leonard 1979). The diffusive fluxes are differenced using the second-order central scheme.

3 Thermal flow in miniature tubes 3.1 The experiment and CMFD set-ups The data used for reference in this work are taken from the Chen et al.’s (2002) experiments, which were conducted without heat transfer. In that campaign, air–water flow was pumped at various flow rates in a closed loop into a 1 mm diameter pipe. Five flow regimes were investigated in horizontal pipes: bubbly, slug, bubbly-train slug, churn and annular. The measurements provide the exact void fractions for specific inlet mass flow rates, which help set the corresponding flow conditions in computations. The simulations were conducted under axisymmetric conditions in circular horizontal tubes for single and twophase flow, without gravity. Only the first three flow patterns were investigated. To study the flow-pattern effects on the heat transfer, we have set the pipe wall to a constant temperature (Tw = 340 K), and the inflow to Tin = 300 K. The inlet flow conditions were extracted from the experiment, which provided the exact void fraction distribution for each set of superficial liquid and gas velocities. No-slip conditions were applied at the wall. The temperature was taken as a passive scalar. Preliminary grid and domain sensitivity studies have revealed that a domain extension of at least 40 diameters is necessary for the multiphase flows to establish a sort of steady state; 70D were necessary though for the single-phase flow. Various grid resolutions were first tested to assure that the velocity and wall thermal layers are well resolved. Although the intermediate grid of 30 · 600 nodes was sufficient to resolve the macro flow topology and transition (half pipe), grid-dependence study required to increase the resolution in the streamwise direction up to 900. The liquid and gas inflow velocities and corresponding void fractions a are listed in Table 1; the void fractions were set by adjusting the inlet area. Note though that the bubbly flow simulation did not match exactly the experiment, in which the pipe diameter was higher 1.5 mm, but the void fraction and mass flow rate were corrected accordingly. 3.2 Flow topology and transition Figure 1 depicts the simulated flow patterns in the bubbly flow regime. The iso-contours refer to the pressure field;

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Microfluid Nanofluid Table 1 Inflow velocity and void fraction conditions Case study

a

UG (m/s)

UL (m/s)

Domain size

Water

0.0

0

1.11

1 · 70D

Bubbly

0.205

0.66

1.11

1 · 40D

Slug

0.376

0.66

1.11

1 · 40D

Slug-train

0.480

1.57

1.11

1 · 40D

Fig. 2 Slug flow formation and evolution. The panels are spaced by 10D

Fig. 1 Bubbly flow formation and evolution. The panels are spaced by 10D

the white line indicates the interface. The upper panel of the figure shows the evolution of the flow while the breaking of the initial air jet is taking place. The capsule is elongated by the action of interfacial shear, and breaks at x/D = 4 when surface tension effects exceed the inertia. Individual bubbles are then released periodically. The periodicity of the bubble passage is somewhat disrupted in the second panel, then restored later on at 30D downstream, as shown in the third panel. One can visually notice that the void fraction distribution along the tube axis is well reproduced by the CMFD. Figure 2 shows the slug detachment from the initial flow breakup, which takes place now further downstream as compared to the bubbly flow; i.e. at x/D = 7. Surface tension effects dominate the inertia at a later position as compared to the bubbly flow. Like in the previous scenario, the breakup is in essence a stability phenomenon, in which surface tension, inertia and shear are simultaneously in work. The simulations compares pretty well with the visualized result shown in the second panel of Fig. 3. The figure clearly reveals the strong interaction between the slug and the wall, a proximity shown later to be responsible for substantial heat transport down to the core-flow region. Note, too, that the front slug is some what more elongated than the upstream ones, because of its merging with the neighbouring slug. The scenario is in fact pretty similar to turbulent flows, where turbulent eddies of various size tend to randomly wash the wall-adjacent layer, transferring lowmomentum fluid to the bulk, and thus heat. In this case the mechanism is rather coherent and is controlled by the

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Fig. 3 Experimental flow regime patterns (bubbly, slug and slugtrain). Visualizations taken by permission from Chen et al. (2006)

frequency of successive slugs/bubbles passing through the pipe. The analogy with turbulent flows will be detailed later on. The above two figures are meant to show the flow structure from the initial breakup to the formation of single slugs and bubbles. 3.3 Heat transfer results Inspecting closely the flow-topology results discussed previously helps suggest that heat transfer is locally controlled by the interface topology along the pipe. The analysis reveals indeed the presence of two distinct heat transfer regions: one right at the initial breakup and one far downstream in the fully developed region. The mechanisms are explained below for the bubbly and slug flow only. As a clarification note though, in contrast to singlephase flow, where it straightforward to distinguish the dynamic and the thermal boundary layers, in two-phase flow of the class predicted herein the definition would not


be justifiable, since liquid boundary layer is often perturbed by the light-phase inclusion. It is therefore more judicious to refer to this region as ‘‘wall thermal layer’’ instead. 3.3.1 The breakup zone The fluid dynamics mechanisms affecting the wall thermal layer in this area of the bubbly flow can be explained by inspecting Fig. 4 below, depicting the pressure and temperature iso-contours obtained at an advanced time of simulation. The upper panel shows a strong radial pressure gradient building up at the breakup location. The wall thermal layer shown in the lower panel indicates that this has been compressed by the pressure gradient after fragmentation (x/D > 5), yielding a higher temperature gradient from the wall. Only far downstream (x/D = 27) when the flow is fully developed (Fig. 6) the heat penetrates deeply the core flow. The same mechanism is observed now in the slug flow regime, where again the breakup of the flow into slugs can

Fig. 6 Temperature iso-contours in the developed region of the bubbly flow

clearly be seen to be associated with a significant radial pressure gradient (Fig. 5). A strong compression of the thermal boundary layer occurs at the location washed out by the rear area of the second slug (indicated by an arrow). The intensity of the radial pressure gradient is somewhat weaker than in the bubbly flow, because the distance between slugs is larger than between bubbles. The immediate consequence of the sudden compression of the thermal layer against the wall is to promote locally the radial gradient of temperature. In some sense, this increase in the radial thermal gradient is the precursor for promoting heat transfer along the pipe. 3.3.2 The developed zone

Fig. 4 Pressure and temperature iso-contours at breakup into train of bubbles

Figure 6 presents a bubbly flow scenario taken where the center of the selected cell is located around x/D = 27, in the fully developed region. Although the wall thermal layer does not seem to be affected by the cell, or slightly at the early stage of the train, the results of the heat transfer rates discussed in the context of Table 2 indicate the contrary. Compared to the slug-flow regime results shown in the next figure, the heat penetration in the bubble core flow is less pronounced, which is well illustrated in the figures comparing temperature profiles. The difference between heat penetrations into the bulk represents about 20% of the total subcooling rate. The thickness of the wall thermal layer remains smaller than in the slug flow, as if it were permanently controlled by the fragmentation-induced pressure gradient occurring upstream. These mechanistic behaviors of the heat convection and fluid flow are corroborated with

Table 2 Heat transfer and bubble/slug frequency detachment data

Fig. 5 Pressure and temperature iso-contours at breakup into train of slugs

Frequency (Hz)

LGB/L

3.67

0.0

0.0

10.65

600

0.8

15.90

450

1.8

32

15.61

519

2.7

35

17.20

350

5.14

Case study

Numax

Water

–

Bubbly

17

Slug

25

Slug-train-1 Slug-train-2

Numean

123


the detailed plots of thermal flow profiles and Nusselt number distribution presented in the next section. Figure 7 displays now a post-fragmentation sequence of the slug flow taken in the fully developed region, where the center of the selected cell is at about x/D = 28. This zone has indeed reached fully-developed steady-state conditions. The heat diffuses first from the wall down starting from the front surface of the slug. As the slug grows in scale and expands closer to the wall, the heat is driven from the large-curvature back circumference. But the main heat removal mechanism is seen to be essentially a convective transport taking place at the back of the slug, where largecurvature surfaces approach the wall. A jet-like flow forms there and penetrates the cell, transporting heat into the core. The thermal field gradually penetrates the entire slug in this way. We can also notice that the slug interfacial area helps exchange a larger portion of heat between the fluids, as compared to the bubbly flow discussed earlier. 3.4 Thermal-flow profiles 3.4.1 Single-phase flow profiles In order to compare the convective heat transfer between the two-phase and single-phase flow, it is first important to define equivalent flow profiles. This is achieved by simulating three single-phase flow profiles: a water plug in which the inflow consists of a plug (constant profile), a bubbly profile where the velocity is split between the bubble region and the surrounding liquid area assuming equivalent superficial velocities, as if it were a two-phase flow, and a slug profile defined in the same way. The fully developed profiles are plotted in Fig. 8. The axisymmetric behavior of the plug profile is correctly reproduced by the simulation, with the maximum axial velocity tending towards the exact analytical value Umax = 2.2 m/s, for an inflow velocity of 1.11 m/s. The bubbly and slug profiles depict a defect by reference to the water plug, because the inflow liquid velocity is now composed of two portions (0.66 and 1.11 m/s). The heat transfer rates are discussed next.

Fig. 7 Temperature iso-contours in the developed region of the slug flow

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Fig. 8 Water flow fully developed profiles: plug, bubbly and slug

3.4.2 Two-phase flow profiles Turning now to the two-phase flow structure, we first compare in Fig. 9 the velocity profiles around individual bubble and slug cells. These profiles ahead and in the wakes are very much similar to single-phase axisymmetric profiles, whereas within the cells the flow is subject to substantial overturning. The profiles labeled in the figure by 1, 2, 3 and 4 are compared in Figs. 10 and 11 to the corresponding single-phase flow profiles; the figures include thermal profiles as well. In the bubbly flow case the analysis of the data indicates that the flow in stations 1 and 4 (in the wake and ahead) is not affected by the presence of the bubbles, unlike in the cell core (at stations 2 and 3) where a recirculation is clearly taking place in the front part. The slopes of the corresponding thermal field profiles show an interesting behaviour: at all the stations the slopes are clearly sharper than in the single-phase flow taken at the same location—in the water flow the thermal wall layer develops fully only at x/D = 70,—meaning that the heat

Fig. 9 Flow velocity profiles within individual bubble and slug cells


Fig. 10 Velocity and temperature profiles around a bubble cell

Fig. 11 Velocity and temperature profiles around a slug cell

transfer should be larger. In contrast to the velocity field then, the temperature profiles at locations 1 and 4 deviate from the single phase, suggesting that it is the velocity profiles that are repeatedly affected by the cells (stations 2 and 3) that prevent the wall thermal layer to naturally develop along the pipe. It is also interesting to note that the scalar penetration from the wall is maximal within the cell core. The velocity and temperature profiles around the slug cell are compared in Fig. 11. The behaviour observed for the bubbly flow can be noticed here, too. The velocity profiles deviate from the corresponding single-phase results only slightly in the wake and ahead of the cell, whereas the structure inside the slug exhibits a reverse flow tending to trap heat longer within the cell. The backflow is more intense than in the bubbly regime (–3 m/s as compared to –2 m/s), which explains why, in average, scalar penetration in the bulk is larger than in the bubbly flow. The temperature profiles within the wall thermal layer are again shown to present sharper slopes than the single phase. The scalar penetration inside the core is larger than

in the bubbly flow, in particular at station 3 where the flow overturns. Figures 10 and 11 give a clear indication on the structure of the wall thermal layers in each case, which are somewhat equivalent in terms of thickness, but still larger than the single-phase flow. 3.4.3 Defect-flow analysis We have so far observed that in the bubbly and slug flows, the thermal wall layer is affected by the presence of the cells, preventing it from growing naturally with the dynamic boundary layer along the pipe. A defect flow field builds up indeed by reference to the equivalent singlephase flow. In order to scrutinize the mechanism in detail, the resulting flow has systematically been decomposed into the sum of the mean single-phase field and a perturbation field superimposed on top of it; the later being induced by the presence of individual cells, i.e. U = Usp + Up. The single-phase flow profiles designated by Usp are exactly those discussed in the context of Fig. 8. Results of this flow decomposition are presented below in Fig. 12 for bubbly

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Fig. 13 Nusselt number distribution along the axis Fig. 12 Mean and defect flow fields in bubbly and slug flows

Nux ¼ and slug flows; the snapshots are taken far downstream in the developed region (the ‘‘control cell’’ is selected when it compares perfectly with its neighbouring cells). The total flow field is shown to be affected by the presence of the cells, the flow recirculation in particular. Although less intense than the mean flow U, the defect flow field Up can be seen to establish around the cells with substantial deviations from the Usp field. Flow recirculation is visible in the wake and ahead of the cells. But the additional shear created by the defect flow field between the wall and the cell surfaces is precisely what promotes heat transfer in the two-phase flow in general. The defect-flow induced shear is the main feature promoting heat transfer and preventing the wall thermal layer to naturally develop with the dynamic boundary layer.

4 Nusselt number distribution Figure 13 compares the Nusselt number distributions along the axis for single and two-phase flow regimes. The local Nusselt number Nux and the wall heat flux q†w are defined by

123

q00w jx D ; ðTw jx Tb Þ k

q00w jx ¼ k

oT jw on

ð7Þ

Noting that both the density and heat capacity are space and time dependent, the bulk temperature Tb can be defined in the following way: RR Tb ¼

0

qðrÞ Cp ðrÞ uðrÞ TðrÞ dr RR 0 q ðrÞCp ðrÞ uðrÞ dr

ð8Þ

The results obtained for the fine mesh resolution are detailed in Figs. 13 and 14 below. Panel (b) of Fig. 13 is a zoom of data taken in the fully developed region, around x/D = 24. The water flow results depicted in both the panels corroborate with the Hagen–Poiseuille flow solution, according to which, Numean asymptotes towards 3.67 for constant wall temperature. The water bubbly and water slug flow return slightly lower Nusselt values than the water plug (results not shown here). Turning now to the two-phase flow results, the comparison clearly reveals a substantial increase in the heat transfer rate with increasing void-inclusion length scale, denoted in Table 2 as LGB. The slug-train flow data show the largest fluctuations around the mean. The RMS of these fluctuations is comparable to the slug flow, but substantially larger than the bubbly


flow. The maximum value Nu = 32 is reached for the slugtrain flow regime, while for the slug it attains 25. These results are very close to those obtained by Ua-Arayaporn et al. (2005) who conducted a similar investigation as the present one, and confirm the experimental findings of Monde and Mitsutake (1995). The successive marked frequencies are the signature of the passage of individual slugs, or more precisely their wall-adjacent, rear circumferences, as discussed previously. The average values presented in Fig. 14 indicate that for the slug-train and slug flows, the data asymptote towards Numean = 17.2 and 15.95, respectively, which is four times larger than singlephase heat transfer. Lower but coherent fluctuations around the mean value Numean = 10.65 are observed in the bubbly flow signal, with marked peaks (Nu = 17) corresponding to the liquid zones squeezed between the bubbles and the wall. The data asymptote towards the average value rather quickly compared to the slug flow. Overall the average Nusselt number is three to four times higher in the two-phase flow regimes, but the slug and slug-train flows remove more heat due to the large blockage effect, and to the background fluctuating flow induced as discussed previously. Table 2 compiles typical heat transfer data for all simulations; the table includes bubble/slug detachment frequency and typical length scale data, too. The quantities reported were taken in the developed flow region, at the center of individual slugs and bubbles (as shown in Figs. 6, 7). The mean Nusselt number is statistically defined using local values in the fully developed region starting from x/D = 15, as indicated by the solid lines in Fig. 14. The water flow simulation provides a Nusselt number perfectly in line with the analytical data (Sparrow and Patankar 1977). The slug flow results are higher than the corresponding single-phase slug flow value (Sparrow and Patankar 1977), i.e. Numean = 5.78. The frequency of bubble/

slug detachment together with the inclusion characteristic length scale LGB reported in Table 2 helps make a better estimate of the flow characteristic timescale, other than simply the superficial inflow velocity and void fraction. Bubbles detach with the highest frequency as compared to the slug and slug-train, and this should be taken into account in the mechanistic modelling of heat transfer of this flow discussed next.

5 Mechanistic heat transfer modelling For practical applications it is useful to determine the heat transfer coefficient using a simple mechanistic model. According to the analogy we have previously made with turbulent flow, void inclusions such as bubbles, and slugs, or more precisely the wall-adjacent liquid layer drifted by their passage can be considered as the flow structures responsible for renewing the surface area. This is the spirit of surface renewal theory of Higbie (1935). From scaling analysis, the heat transfer rate defined by b¼

q00w J ¼ Nu ; L qCp ðTw T1 Þ

with J ¼

m Pr

ð9Þ

is proportional to the diffusivity of the media, i.e. b ¼ ðJ= sÞ1=2 ; at least for high Shmidt/Prandtl numbers Sc/ Pr (where m is the kinematic viscosity). If one allows for a random distribution of wall surface ages for the renewed inclusions, which is typical of what might be expected from a slug or bubbly flow, time-scale s may be thought of as the mean time between bubble or slug passage. Using the definitions introduced in Eq. 9, and making specifically reference to the Dittus–Boelter (Dittus and Boelter 1930) expression for heat transfer in turbulent pipe flows, the Nusselt number may be expressed as follows 4=5

Nu Nuw þ CPrL0:4 ReLS

ð10Þ

where Nuw is the value for fully developed single-phase flow (equal to 3.67 for constant wall temperature and 4.36 for constant wall heat flux), C is a model constant, and ReLS is the liquid slug Reynolds number ReLS defined based on the pipe diameter L and the velocity length-scale of the inclusions VGB: ReLS ¼ LVGB =mL ;

Fig. 14 Nusselt number distribution along the axis

where VGB ¼ LGB =s

ð11Þ

The liquid slug Reynolds number refers to that of the wall-adjacent liquid layer drifted by the gas inclusions responsible for surface renewal. The velocity length-scale of the inclusions VGB is thus defined as the ratio of the inclusion length scale LGB to the mean time between

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bubble or slug passage s. The best fit to the computational data is obtained for a model constant C = 0.022, as shown in Fig. 15, which, very surprisingly, is of the same order of magnitude as the original Dittus–Beolter correlation for turbulent heat transfer in single-phase pipe flows (C = 0.023), without the Chen’s (1966) two-phase flow enhancement factor. This model can be used to determine the Nusselt number in similar situations involving well defined gaseous/vapour inclusions such as bubbles, and slugs evolving in microfluidic devices, where L O (mm), and Pr > 1 liquids. As shown in Fig. 15, the proposed model fits remarkably well the simulated Nusselt number data (using Eq. 7) for a range of flow topology configurations. This is a new valuable result for practical applications, although it is intriguing to see how it behaves for churn flows.

6 Pressure drop Another key parameter in microfluidic transport is the pressure drop. It may indeed happen that this quantity is such that fluid pumping becomes ‘‘expensive’’ for a particular design process that promotes heat transfer. Figure 16 compares the pressure drop distribution in the pipes. Again, the two-phase flow data are compared to equivalent water-bubbly and water-slug results, i.e. using the correct inflow velocity profiles as discussed previously. While the bubbly flow seems to feature a lower pressure drop than the single-phase water flow, the slug flow requires, however, 14–15% larger pressure drop than the single phase. The key contributions here are the mass flow rate, which is obviously lower in the two-phase flows, and the inclusion-induced stresses as discussed previously. These two features in play, which tend to increase with the void fraction and interfacial velocity, compete indeed in a very subtle way: in the bubbly flow regime the mass flow rate is lower, but

Fig. 15 Nusselt number correlation as a function of Reynolds number

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the increase in the bubble-induced shear stress is not sufficient to considerably affect the pressure drop, whereas in the slug flow (and the slug-bubbly flow) the mass flow rate is even lower but the shear stress is way larger in proportion, which explains the increase in pressure drop. In the light of these results, it seems that the best operating conditions can be reached for a flow regime in which the void fraction lies in the interval 0.2–0.36, for the same superficial velocity, which would in turn provide the same pressure drop as in single phase, while enhancing heat transfer by a factor of 3.5–3.8.

7 Conclusions Detailed computational microfluidics flow simulations have been performed to study the effect of varying flow regime on the heat transfer in small tubes. Interface tracking methods were employed for the purpose, providing local detailed information about heat transfer and fluid flow. Overall the heat removal rate in two-phase flow is higher than in single phase. Subtle differences were revealed

Fig. 16 Pressure drop distribution for slug and bubbly flows


between slug flow, which dissipates more heat in the bulk, and bubbly flow, which has a higher wall heat flux due a pronounced blockage effect. It is in particular shown that the pressure induced by the early train dislocation into slugs or bubbles acts as a precursor to heat removal enhancement along the pipe. But more important is the defect-flow induced by the presence of bubbles and slugs, which has been revealed to substantially increase the wall shear and in turn heat transfer. The average Nusselt number distribution shows that the bubbly and slug patterns transport three to four times more heat from the tube wall to the bulk flow than pure water flow. From an engineering standpoint, the present results may be helpful in designing intelligent flow control systems. The simple correlation proposed is meant to be used as a guideline for design purposes. It is finally useful to insist on the splendid role of such direct computational microfluidics, which will pave the way to transcend the era of empiricism to knowledge-based design for new control technologies for microfluidic systems. Acknowledgments This work has been granted by the European Space Agency ESA, through the MAP research program MAP AO1999-045. The authors are thankful to the Coordinators, Olivier Minster (ESA) and Prof. Lounes Tadrist (University of Marseille) for the valuable comments. A part of the work has been conducted at the LAV (ETH Zurich), headed by Prof. K. Boulouchos, to whom the authors are grateful.

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