WITHDRAWN: Numerical study of the interactions and merge of multiple bubbles during convective boiling in micro channels

International Communications in Heat and Mass Transfer 81 (2017) 116–123

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Numerical study of the interactions and merge of multiple bubbles during convective boiling in micro channels Qingming Liu ⁎, Wujun Wang, Björn Palm Department of Energy Technology, Royal Institute of Technology (KTH), 10044 Stockholm, Sweden

a r t i c l e

i n f o

Available online xxxx Keywords: CFD Flow boiling Bubbles Micro-channels

a b s t r a c t Multi bubbles interaction and merger in a micro-channel flow boiling has been numerically studied. Effects of mass flux (56, 112, 200, and 335 kg/m2 ∗ s), wall heat flux (5, 10, and 15 kW/m2) and saturated temperature (300.15 and 303.15 K) are investigated. The coupled level set and volume of fluid (CLSVOF) method and nonequilibrium phase model are implemented to capture the two-phase interface, and the lateral merger process. It is found that the whole transition process can be divided to three sub-stages: sliding, merger, and post-merger. The evaporation rate is much higher in the first two stages due to the boundary layer effects in. Both the mass flux and heat flux affect bubble growth. Specifically, the bubble growth rate increase with the increase of heat flux, or the decrease of mass flux. © 2016 Published by Elsevier Ltd.

1. Introduction As the miniaturization of electronic devices, cooling has become an urging challenge for the information, communication, and technology industry. The latest super computers can dissipate heat as high as a few hundred watts per centimeter, which makes traditional air cooling not capable. Liquid cooling has therefore become a promising solution [1]. There are two thermal management methods available nowadays: single phase and two phase flow micro-channel cooling. The single phase micro-channel cooling method has been studied extensively and already been used in certain applications. Its heat transfer mechanism has been found similar to that of macro-scale channels. On the contrary, the mechanism of two phase flow boiling in micro channels remains as an unsolved problem. Some researchers have argued that nucleate boiling is the dominant heat transfer mechanism in microchannels due to the heat transfer coefficient's dependency on heat flux [2]. Others, on the contrary, have argued that the evaporation of the thin film between the bubbles and the wall plays a more important role [3]. As new experimental phenomena emerge, many methods have been proposed for the estimation of pressure drop and heat transfer coefficient. Among these are three major groups: empirical correlations [4–6], superposition model [7], and mechanistic modeling based on flow regimes [8,9]. It has been found that all these methods, especially the last one, depend on a deep understanding of flow patterns. Experimental studies

⁎ Corresponding author. E-mail address: [email protected] (Q. Liu).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.12.011 0735-1933/© 2016 Published by Elsevier Ltd.

have shown that there are six major flow patterns in microchannel flow boiling. In addition to the four common flow patterns in macro channels -bubbly flow, slug flow, annular flow and mist flow, there are two new patterns in micro channels [10–13]: confined bubbly flow and elongated bubbly flow. These new flow patterns may play an important role to play in the heat transfer process. Advancements in multiphase algorithms and computing capacity have facilitated numerical investigations in micro-channel flows. Mukherjee and Kandlikar [14] simulated the growth of a confined bubble in a rectangular micro channel. Kunkelmann and Stephan [15] numerically studied the transient heat transfer during nucleate boiling and effect of contact line speed [16]. Li and Dhir [17] investigated a single bubble during flow boiling by means of the level set method. Gong and Cheng [18] examined periodic bubble nucleation, growth and departure from heated surface by using the lattice Boltzmann method. Agostini et al. [19] studied the velocity of an elongated bubble in an adiabatic micro-channel and proposed a predictive model based on these experiments. Furthermore, the collision process of elongated bubbles in micro channels without heat transfer was studied [20]. More recently, Consolini and Thome [21] proposed an one dimension model to predict the heat transfer coefficient of confined (or elongated) bubbles. Sun and Xu [22] developed a new model based on VOF method for FLUENT. Magnini et al. [23] developed a height function algorithm and investigated the effects of the leading elongated bubble. Liu et al. [24,25] examined the dynamics and heat transfer of the transition from nucleate to confined bubbles in a micro channel. They also studied the effect of contact angle and have found its importance on the bubble's shape. Most of the micro channel researches only focus on one individual flow regime. Interactions or transitions between flow regimes, however, have not been found in literatures that much. The aim of the present

Q. Liu et al. / International Communications in Heat and Mass Transfer 81 (2017) 116–123

Nomenclature Latin letters A area C coefficient specific heat cp Ca capillary number D diameter Eo Eotvos number F force G mass flux H enthalpy k curvature n normal vector L length P pressure Pr Prandtl number q heat flux thermal resistance Rint Rg gas constant Re Reynolds number T temperature U velocity vector Z vertical distance We Weber number Greek letters α volume fraction β growth constant δ thickness θ contact angle λ thermal conductivity μ viscosity ρ density σ surface tension Φ level set function Subscripts b bubble c condensation d diffusion e evaporation f fluid g gas l liquid gr grid int interface v vapor w wall sat saturation

paper is to study one of these processes: the transition from isolated bubbly flow to confined bubbly flow, thus enhance the understanding of heat transfer mechanism of micro channel flow boiling. 2. Numerical model 2.1. Interface reconstruction Interface reconstruction is one of the major challenges in two phase flow simulation. The interface is either “captured” or “tracked” by different numerical means. Tracking method is usually more time consuming and harder to implementation. Among capturing methods, volume of

117

fluid (VOF) and level set (LS) method are two of the most widely used numerical tools. VOF [26] is a one-fluid algorithm deriving from continuum equations which enables it to have a mass-conservation nature. However, this causes relatively poor interface reconstruction due to a less accurate estimation of interface curvatures. The level set method has a better estimation of interface curvature because it is a smooth function, but a poorer mass conservation, specifically when the interface experiences severe stretching or tearing. The complementary features of these two methods lead to a new method, developed by Sussman and Puckett [27] and called coupled level set and VOF method (CLSVOF). With this new tool, both level set function and volume fraction equations are solved, the interface is linearly reconstructed every time step from the volume fraction and surface tension is calculated by the level set function. This new algorithm significantly enhances mass conservation and curvature estimation, albeit with the price of complication. The governing equations of CLSVOF method are summarized as follows: ∂ρ þ ∇∙ðρuÞ ¼ 0 ∂t

ð1Þ


    ∂u ρ þ u∙∇u ¼ −∇P þ ∇ μ ∇∙u þ ∇∙uT þ ρg þ F σ ∂t

ð2Þ

ρC p


 ∂T þ u∙∇T ¼ Φ þ ∇∙ðλ∇TÞ ∂t

ð3Þ

∂α þ u∇α ¼ 0 ∂t

ð4Þ

∂ϕ þ u∇ϕ ¼ 0 ∂t

ð5Þ

They are the continuity Eq. (1), the momentum Eq. (2), the energy Eq. (3), the volume fraction (4), and the level-set Eq. (5). Physical properties, such as density, viscosity, and thermal conductivity are the volume averaged value of all the phases in the cell defined by Eq. (6) n

1

1

0

Φ ¼ ∑ Φi α i ∀α ðx; t Þ ¼ ð0; 1Þ

if x ∈ primary phase if x ∈ interface Γ if x ∈ secondary phase

ð6Þ

where α is the volume fraction of the primary phase (gas in the present paper) in each computational cell. The level-set function ϕ is a signed distance to the interface. Accordingly, the interface level function is defined as below. ϕðx; t Þ ¼

þ if x ∈ primary phase 0 if x ∈ interface Γ − if x ∈ secondary phase

ð7Þ

Solve the level set equation to get the curvature and normal to interface n¼

∇ϕ ∇ϕ k ¼ ∇∙ j∇ϕj j∇ϕj

ð8Þ

Then get the surface tension force by the following equation F σ ¼ −σkδðϕÞ∇ϕ

ð9Þ

where   1− cos 3πϕ=2Lgr δðϕÞ ¼ 3Lgr 0

if jϕjb1:5 Lgr otherwise

where Lgr is the minimum grid spacing.

ð10Þ

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2.2. Evaporation model The heat flux transferred at the interface is calculated from Eq. (11). qe0 ¼

T int −T sat hlg Rint

ð11Þ

where Tint is the interface temperature and Rint is the interfacial resistance defined by Eq. (12) Rint

2−C e ¼ 2C e

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2πRgas T sat =2 2 ρv hlg

ð12Þ

where Ce is the evaporation (or accommodation) coefficient. The mass source can be calculated as
 q ρ_0 ¼ N e0 j∇αj hlg

ð13Þ

where N is a normalization factor to be determined as N¼

α∮ Ω j∇αjdΩ ∮ Ω αj∇αjdΩ

ð14Þ

Since the interface is not a sharp surface the spatial distribution across the interface needs to be distributed appropriately. Hardt and Wondra [28] has developed a smeared phase change model. The first step is to get a smeared mass source ρ_1 by Eq. (15) ρ_1 −ρ_0 ¼ Cd ∇2 ρ_0 Δt

ð15Þ

where Cd is the diffusive coefficient, Δt is the time step. Choice of Cd value is a tradeoff between heat flux smearing and mass conservation. This is because a small Cd leads to mass conservative but a large Cd ensure heat flux smearing. ρ_1 needs to be normalized to ensure gas only generates in gas phase and liquid only evaporates in liquid. ρ_ ¼ Nl ð1−α Þρ_1 −N v αρ_1

ð16Þ

where Nl and Nv are normalization factors to be determined from Eq. (17) Nv ¼

∮ Ω ρ_1 dΩ ∮ Ω ρ_1 dΩ N ¼ ∮ Ω αρ_1 dΩ l ∮ Ω ð1−αρ_1 ÞdΩ

ð17Þ

The detail of the implementation of this model can be found in our previous paper [25]. 2.3. Boundary and initial conditions The micro-channel studied in the present paper has a diameter of 0.64 mm and a length of 5 mm. As Fig. 1 shows, the channel is vertically placed and the gravity vector points downwards. Cartesian coordinates are used with hexahedral grid. Two perpendicular and symmetric boundaries are defined at the center of the tube, therefore only a quarter of the tube is simulated. Three pseudo thermo-couples are placed on the wall corresponding to each bubble in order to measure the temperature and heat transfer coefficients. The size of each thermo-couple is 0.2 × 0.2 mm in order to at least cover the smallest bubble. A single phase steady laminar simulation is performed at the beginning and the result is used as the initial condition for the multiphase simulation. A constant heat flux (5, 10 or 15 kW/m2) is supplied at the wall. No-slip velocity boundary condition is applied for the wall. The system operating pressure is 7.0 bar. At the inlet, uniform velocity and temperature boundary conditions are chosen for the single phase simulation. After that, these results are

Fig. 1. The simulation domain is a 5 mm long cylindrical with a diameter of 0.64 mm. Bubbles' diameters are 0.1, 0.12 and 0.14 mm for cases with high saturated temperature (305.15 K) and 0.08, 0.1, and 0.12 mm for cases with low saturated temperature (300.15 K), respectively.

used as the inlet condition for the multiphase simulation. Some of the important parameters are listed in Table 1. The numbers indicates that the inertia force is negligible and it is a laminar flow regime. The value of contact angle θmax is based on an experimental result of R134a on an Aluminum wall [29]. For laminar flow, the thermal entry length is a function of Reynolds number and Prandlt number. The thermal entry lengths corresponding to the two given inlet velocities are 67D and 34D, respectively, which means all cases of this paper are in the thermal entry region. The wall temperature is slightly above the saturated temperature depending on the position. Three pairs of spherical “seed bubble” are placed close to the wall at the beginning of simulations. Their distances to the inlet are 1, 1.5 and 2 mm, respectively. As only bubbles bigger than certain size (critical radius) can survive [30], all the embryo bubbles are initialized above this limit. It is also observed from experiments that bubbles grow along the flow, which makes bubbles downstream have greater volumes.

Table 1 Simulation parameters. Saturated temperature (K) Initial radius (mm) Mass flux(kg/m2 ∗ s) Heat flux (kW/m2) Initial volume (10−12 m3) Reynolds Number

300.15 0.08/0.1/.12

305.15 0.1/0.12/0.14

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 56 56 112 200 200 200 335 5 10 5 5 10 15 10 6.7 6.7 6.7 6.7 6.7 6.7 6.7 198

198

396

712

712

712

1098

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119

Therefore the three pairs of bubble are initialized with radius of0.1, 0.12 and 0.14 mm respectively. The operating pressures are 7 and 7.7 bar (corresponding saturated temperatures are 300.15 and 305.15 K, respectively) and the temperature and pressure inside the bubbles are kept constant and equal to these saturated values for the duration of the simulation.

3. Verification and validation The validation has been done in the authors' previous paper [25]. In summary, a validation of the phase change model against the wellknown Sriven theory [31], a validation of the surface capturing method against lubrication law [32], and a validation of the coalescence model against an experiment [33]. The independency of mesh size has also been proved. Three cases with different mesh sizes have been studied D is choose for all cases. and a minimum mesh sizes of 128

4. Results and discussions 4.1. Dynamics of the growing and merging process As Fig. 2 indicates, bubbles slide up the tube's wall with help of the drag force. They absorb heat from the surrounding super-heated liquid and grow. At certain time (6.74 ms in this case) the bubbles become so big that they reach each other, which further leads to a merger. Their shapes change dramatically during this very short period (about 1 ms) due to the strong surface tension. At first the new merged bubble expands in radial direction (7.04 ms), and then it contracts (7.26 ms). Eventually the new merged bubble departure from the wall and become elongated by the shear force (8.17 ms). A comparison of simulation bubble shapes to experiments at times 6.74 ms is made by Fig. 3. It shows that the simulation has successfully captured the merging process. Velocity field around the bubbles is affected by both bubble growth and movements. Velocities around the interface are higher than those of fluid; and velocity boundary layer is disturbed by bubbles (Fig. 4 a). It demonstrates that the CLSVOF method has successfully eliminated “spurious velocities” at the interface, which is a common numerical

Fig. 3. Comparison of an experimental visualization [46] and the current simulation at time 6.74 ms. Constant wall heat flux q = 5 kW/m2, mass flux G = q = 112 kg/m2 ∗ s, and saturated temperature Tsat = 300.15 K.

error when strong surface tension is involved. The pressure field (Fig. 4 b) inside the bubbles is always higher than that in the liquid. 4.2. Heat transfer The heat transfer is significantly affected by the bubble growth and merger. Fig. 5 shows the temperature contour around the leading bubble pair. The temperature surrounding bubbles' interface is always kept close to saturated temperature mainly due to evaporation. As bubbles grow and move upwards, it creates a low temperature region behind them. In the front of the bubble, however, liquid temperature becomes higher because of a distortion of the thermal boundary layer. Moving bubbles are pushing up the super-heated liquid in

Fig. 2. Bubbles shape changes at different times (from left to right: 2.05, 4.09, 6.74, 7.04, 7.26 and 8.17 ms). constant wall heat flux q = 5 kW/m2, mass flux G = q = 5 kg/m2 ∗ s, and saturated temperature Tsat = 300.15 K.

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a)

b)

Fig. 4. Velocity and pressure field of the leading bubble pair at different times. Constant wall heat flux q = 5 kW/m2, mass flux G = 112 kg/m2 ∗ s, and saturated temperature Tsat = 300.15 K.

their front during both the sliding and the beginning of merge process. This effect then becomes less significant in the end of merge stage, because at that time the new bubbles have already been out of the thermal boundary layer. The evaporation rate contour is illustrated in Fig. 5. Most evaporation occurs at the front part of moving bubbles due to a higher temperature. When the bubbles slide along the inner wall, most of the bubble's

surface is surrounding by super-heated liquid of the thermal boundary layer. As the bubbles grow, this evaporating region becomes larger and. In order to validate with existing experimental results, a local bubble heat transfer coefficient is calculated as: hðzÞ ¼

qw T w;b −T f

ð18Þ

Fig. 5. Temperature(top) and evaporation rate logarithmic contour (down) of the upper bubble pair at different stages. Constant wall heat flux q = 5 kW/m2, mass flux G = q = 112 kg/m2 ∗ s, and saturated temperature Tsat = 300.15 K.

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qw is the constant wall heat flux, Tw,b is the temperature of the pseudo thermo-couples, and Tf is the local bulk fluid temperature corresponding to thermos-couples' locations. The coefficients are calculated and compared with analytical single phase heat transfer coefficient and experimental results [34] under the same boundary and operating conditions. From Fig. 6 it can be seen that all the three bubble pairs have almost the same heat transfer coefficients at the beginning of merging process (6.74 ms). This is because all bubbles are in the thermal boundary thermal layer and have a similar evaporation rate which has been confirmed by previous analysis. These coefficients, however, varies after the merger (8.17 ms). Specifically, the heat transfer coefficient of the middle pair increases as the evaporation enhances. On the contrary, this value for the leading pair, which actually has become a single bubble, decreases as the evaporate rate drops. It should be noted that the experimental result, due to its temporal inaccuracy, has no indication of this change. 4.3. Effects of heat flux The growth of the bubbles is affected by the wall heat flux (Fig. 7). Under the same mass flux boundary conditions, bubbles grow faster as the heat flux increases. In the case with low heat flux q = 5 kW/m2, the leading bubble pair is even not big enough to merger until the end of calculation (2 ms). On the contrary, in the case with high heat flux of q = 15 kW/m2 they merger as early as 1.5 ms, which means the higher heat flux leads to higher evaporation. Three different wall heat fluxes are examined here, namely: 5, 10, and 15 kW/m2. According to Fig. 8, bubbles with the highest heat flux condition have a much higher growth rate than the other two due to a non-linear relation between these two variables. 4.4. Effect of mass flux

Fig. 7. Bubbles grow at different wall heat fluxes. The time span between two snapshots is 0.5 ms and the first snapshot is taken at time 1.0 ms. Mass flux G = q = 335 kg/m2 ∗ s, and saturated temperature Tsat = 305.15 K.

As a previous study has shown [11], mass fluxes do not have a significant effect on the heat transfer of nucleate boiling dominant phenomenon in micro-channels. Nevertheless, it may not be true if the nucleate boiling influence is not so strong. As the current paper is studying flow boiling which has a mixture characteristic of both nucleate boiling and convection, the effect of mass flux is of our special interest. Four different mass fluxes are examined (56, 112, 200, and 335 kg/m2 ∗ s). All have a corresponding inlet Reynolds number smaller than 2100 which implies that they are laminar flows. The velocity streamlines and bubble growth rates under different mass fluxes are illustrated in Figs. 8 and 9, respectively. As the mass

flux decreases, the bubble growth rate increases. This is not a surprise because a lower mass flux leads to a higher liquid temperature field which further leads to a higher evaporation rate. Therefore bubbles with lower inlet mass flux usually will grow faster and merge earlier (Fig. 9). However, as the inlet mass flux increases, the bubble growth rate decreases. At certain point the lifting force which is partly caused by the inertial force become more significant than the growth rate in the merging process (Fig. 10). As is shown in Fig. 9, bubbles with higher mass fluxes (335 kg/m2 ∗ s) merger slightly earlier.

Tsat=305.15 K 2

5

2

htc (kw/m *s)

4 Leading bubbles 3

Dimensionless Volume(Vb/Vin)

Single Phase Experimental Simulation:6.74 ms Simulation:8.17 ms Trailing bubbles

2

Middle bubbles

1 0 0

1.8

2

3

4

5

Q=5 kw/m *s 2 Q=10 kw/m *s 2 Q=15 kw/m *s

2

Q= 5 kw/m *s 2 Q=10 kw/m *s

2.5

1.6 2 1.4

Coalescing

Coalescing

1.5

1.2 1 0

1

Tsat=300.15 K 3

2

0.5

1

Dimensionless Time (t/t

1.5 c1

)

1 0

0.5

1

Dimensionless Time (t/t

1.5 c2

)

Z (m) Fig. 6. The local bubble heat transfer coefficients compare with experimental results (Owhaib et al. 2010). Constant wall heat flux q = 5 kW/m2, mass flux G = q = 112 kg/m2 ∗ s, and saturated temperature Tsat = 300.15 K.

Fig. 8. Bubbles volume versus heat fluxes. Vb and Vin are the bubble volume and initial total bubble volume of six bubbles, respectively. tc1 and tc2 are the coalescing time for two case groups and has a value of 1.5 and 6.7 ms, respectively.

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transient process plays a critical role in the heat transfer of flow boiling in micro channels. The following conclusions can be drawn based on the study: The three stages (sliding, merger and post-merger stages) have been studied. Bubbles grow when they sliding along the inner wall of channels, then they merge with each other laterally when they are big enough to reach each other. After that they quickly form a new bubble-usually within one mini second-and eventually departure from the wall to the fluid. Bubbles have a higher heat transfer rate in the sliding and merging stages due to a greater contact area with super-heated liquid, which enhances the evaporation. Heat transfer enhancement is a consequence of both convection and evaporation. The convection happens around the vicinity of bubble and disturbs both velocity and thermal boundary layer. The evaporation, however, mostly occurs in the sliding and merging stages. Both the mass flux and heat flux affect the bubble growth and merger. A lower mass flux or a higher wall heat flux leads to a higher bubble growth rate and mostly an earlier merger under low Reynolds numbers. Acknowledgement The support of the Swedish Energy Agency through its research program Effsys Expand is gratefully acknowledged. References

Fig. 9. Bubbles grow with different inlet mass fluxes. Boundary conditions: constant wall heat flux q = 5 kW/m2, saturated temperature Tsat = 303.15 K.

5. Conclusions The present paper studies the transition of convective boiling from nucleate boiling to bubbly flow in a micro-channel with a diameter of 0.64 mm. Reconstruction of interface is done by the CLSVOF method and the interface heat flux jump is calculated by a non-equilibrium phase change model. The bubble coalescence process is studied. The

Fig. 10. Bubbles grow with different mass fluxes. Vb and Vin are the bubble volume and initial total bubble volume of three bubble pairs, respectively. tc1 and tc2 are the coalescing time for two case groups and has a value of 1.5 and 6.7 ms, respectively.

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