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Slug flow heat transfer in square microchannels
International Journal of Heat and Mass Transfer 62 (2013) 752–760
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Slug flow heat transfer in square microchannels V. Talimi a, Y.S. Muzychka a,⇑, S. Kocabiyik b a b
Microfluidics and Multiphase Flow Research Lab, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada A1B 3X5 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7
a r t i c l e
i n f o
Article history: Received 29 August 2012 Received in revised form 12 March 2013 Accepted 12 March 2013 Available online 16 April 2013 Keywords: Heat transfer Square microchannel Two-phase flow Slug flow Numerical simulation
a b s t r a c t While heat transfer in Taylor (slug) flows in circular microtubes has been examined exclusively by researchers, noncircular microchannels have not been widely considered in the literature. This is a large gap in research since noncircular microchannels are common structures in microcooling processes. Square and rectangular microchannels are the most important examples. In the present study the heat transfer process in slug flows in square microchannels has been investigated numerically under constant wall temperature boundary condition. The local heat flux for the moving slugs has been converted to total microchannel heat flux using the integration methods suggested recently by the authors. This leads to microchannel wall average heat flux which is the parameter of interest in heat sink problems. Potential effects of liquid film around bubbles on heat transfer process have been discussed by comparing the numerical results with experimental data. Finally, effects of Reynolds number, contact angle, and slug length on heat transfer in slug flow inside square microchannels have been studied and graphs for prediction of slug flow heat transfer in square microchannels have been provided for slug flows with no liquid film or very thin liquid film around the bubbles. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Non-boiling two phase slug flow has attracted researchers in recent years due to its capacity to enhance heat and mass transfer in small scales. This type of flow is typically a train of liquid slugs and bubbles and has also been named as plug flow, segmented flow, and Taylor flow [1]. There are still many large gaps in research based on several published reviews on slug flow hydrodynamics and heat transfer [2–5]. According to Talimi et al. [2] heat transfer in non-circular microchannels needs to be studied in detail. Muzychka et al. [6] showed that internal circulations inside moving slugs are the primary mechanism of heat transfer enhancement. Fig. 1 shows these circulations inside liquid slugs for two different microchannel wall materials: hydrophilic and hydrophobic walls. These circulations bring fresh liquid from center of the slug to the wall, where heat transfer process occurs. This provides a renewal mechanism to the thermal boundary layer and increases heat (or radial mass) transfer. Numerical simulation of slug flows in microchannels can be performed using different approaches. In the first approach, a fixed frame of reference is used to generate computational domain. This requires at least two inlets for the two phases, a certain initial ⇑ Corresponding author. E-mail addresses: [email protected] (V. Talimi), [email protected] (Y.S. Muzychka), [email protected] (S. Kocabiyik). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.035
length in order for slugs to be developed, and a longer length to examine pressure drop and heat transfer process [7–10]. In general, this approach is very time consuming, since a very fine mesh is required to capture all the important details inside the flow e.g. thin liquid film around the bubbles [11]. Some researchers used a moving frame of reference to make the computational domain smaller and accelerate numerical simulations [12–14]. There is a need for a careful setting of inlet and outlet. This approach is still computationally expensive since a two phase flow is considered, and interactions between the two phases must be captured. The fastest approach is using a single phase moving frame of reference computational domain with fixed leading and trailing interfaces [15–17]. Fig. 2 schematically shows the computational domains used in the aforementioned three approaches. In most two phase slug flows, there is a very thin liquid layer around bubbles [1]. Taylor [1] showed the importance of the film thickness on the internal circulations. As Fig. 3 shows streamline shape in the moving liquid slugs depend on Capillary number, Ca, and a dimensionless number, m, which is a relative velocity between the two phases, as follows:
m¼
UB UL UB
ð1Þ
Based on Fig. 3 at high Ca numbers a complete bypass occurs through the liquid film around the bubbles, while in slug flows with low Ca numbers internal circulations exist inside the liquid slugs. As reported in [18–21] the film thickness is a function of
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
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Nomenclature Bo Ca Cp D Dh f g h H k L Lw Ls LH s m _ m Nu p Pe Pr Q q qw rw R Re s_ t T u uw U
Bond number, gD2h ðql qg Þ=r Capillary number, lU/r specific heat capacity, J/kg K diameter, m hydraulic diameter of channel, m friction factor gravitational acceleration, m/s2 thermal convection coefficient, W/m2 K distance between plates, m thermal conductivity, W/m K microchannel length, m dimensionless microchannel length, L/DhPe length of liquid slug, m dimensionless length of slug dimensionless relative velocity mass flow rate, kg/s Nusselt number, hDh/k pressure, Pa Peclet number, UDh/a Prandtl number, m/a total heat flux, W mean heat flux, W/m2 dimensionless wall heat flux dimensionless position in diagonal direction radius, m Reynolds number, UDh/m energy (heat) generation, W/m3 time, s temperature, K horizontal velocity component, m/s dimensionless velocity in x-direction average liquid velocity, m/s
x y yw Y z Z
position in x-direction, m position in y-direction, m dimensionless position in y-direction microchannel width (y-direction), m position in z-direction, m microchannel height (z-direction), m
Greek symbols thermal diffusivity, m2/s liquid phase fraction d film thickness, m l dynamic viscosity, N s/m2 m kinematic viscosity, m2/s q fluid density, kg/m3 sw wall shear stress, Pa h contact angle r surface tension, N/m
a aL
Superscripts w dimensionless ðÞ mean value Subscripts B bubble i inlet lm logarithmic mean L liquid m mean s slug w wall
Ca number, so the film thickness is an important parameter and governs the flow pattern i.e. internal circulations. Therefore, the film thickness may have significant effects on heat transfer process. There are several correlations for film thickness as a function of Ca [1,22–24]. Gupta et al. [11] reported that the correlation suggested by Bretherton [23] is the most appropriate correlation:
ð2Þ
Recently, Luo and Wang [25] investigated the velocity field inside liquid slugs in rectangular microchannels. Sobieszuk et al. [26] measured bubble lengths in Taylor flows in square microchannels, and showed that the correlations suggested by Qian and Lawal [27] can predict bubble length with a good accuracy. As the existing reviews [2–5] reported, there are not many published works on heat (or mass) transfer in two phase slug flows in square microchannels. Mass transfer in slug flow in square microchannels
Fig. 1. Internal liquid slug circulation, (a) hydrophobic surface, (b) hydrophilic surface.
Fig. 2. Typical computational domains frequently used for slug flow numerical simulations, (a) fixed frame of reference two phase, (b) moving frame of reference two phase, (c) moving frame of reference single phase.
d ¼ 1:34 Ca2=3 R
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of interest. According to Shah and London [33] and Muzychka et al. [6], the goal in cooling processes such as heat sinks is wall average heat flux enhancement. There are different methods to nondimensionalize wall heat flux as reported by Muzychka et al. [6]: Wall to Bulk Mean, Tw Tm. Wall to Inlet, Tw Ti. Mean Wall to Inlet, T w T i . The Nusselt number, Nu, based on the first approach is:
NuDh ¼
qDh kDT lm
ð3Þ
where DTlm is the logarithmic mean temperature difference and defined as follows:
DT lm ¼
ðT w T i Þ ðT w T o Þ T i ln TTwwT o
ð4Þ
If one considers the wall to inlet temperature difference which is more appropriate for heat sinks with one fluid stream [6], the dimensionless wall heat transfer, qw, is as follows:
qH ¼
qDh kðT w T i Þ
ð5Þ
These two dimensionless heat transfers above are related in the single phase Poiseuille flow as follows [6]: Fig. 3. Streamlines in liquid slugs in front of a Taylor bubble, Taylor [1].
has been investigated by Raimondi and Prat [28] and Yue et al. [29], for liquid–liquid and gas–liquid two phase flows, respectively. Yue et al. [29] showed that pressure drop in two phase slug flows with very short liquid slugs deviates from correlations suggested by Kreutzer et al. [30]. Majumder et al. [31] performed experiments on heat transfer in two phase slug flows in square minichannels. They reported a heat transfer enhancement up to 1.2–1.6 times when compared to fully developed laminar single phase flows. In their experiments, they applied heat only from one side of the 3.3 (mm) 3.3 (mm) square channel. They also concluded that the heat transfer enhancement is a function of geometrical parameters on the slug flow such as bubble length or slug length. Betz and Attinger [32] performed experiments on heat transfer in slug flow in square microchannels under constant wall heat flux boundary condition. They provided the present authors with their data including flow inlet and exit temperatures and other flow conditions. These data have been used in the present study in order to perform a comparison between present study results and their experimental results. In the present study, a constant wall temperature has been considered as a thermal boundary condition. This required some adjustments in Betz and Attinger [32] data using a goal seek process to find the constant wall temperature which gives the same Nu number. Based on the wall material and its high heat conduction coefficient, k, there is a very small gradient in wall temperature which enabled us to convert data from isoflux boundary condition to isothermal thermal boundary condition. More information on data adoption has been reported in Appendix A. In the present study, a moving liquid slug with the same length and shape as Betz and Attinger [32] reported data has been simulated numerically, using the moving frame of reference simulation approach. 2. Theory In internal heat convection under constant wall temperature the heat flux through the wall to the liquid stream is the parameter
qH ¼
1 4LH
½1 expð4NuDh LH Þ
ð6Þ
where Lw is dimensionless channel length and defined as follows:
LH ¼
L=Dh Re Pr
ð7Þ
As discussed by Muzychka et al. [6] the first approach (wall to bulk mean temperature difference) is appropriate for heat exchanger models and the second approach (wall to inlet temperature difference) is more appropriate for heat sink applications where there is only one fluid stream as coolant. Therefore, the wall to inlet temperature difference has been used to calculate dimensionless heat transfer under constant wall temperature boundary condition in the present study. Muzychka et al. [34] suggested a heat transfer predictive model based on an asymptotic analysis and the available data in the literature. They suggested to use the slug length instead of the tube length as the length scale when nondimensionalizing heat transfer data. Their model is as follows:
2
1:614 q ¼ 4 H1=3 Ls
!3=2
H
þ
1 4LH s
!3=2 32=3 5
ð8Þ
where
qH ¼
Q=ðaL pDLÞD kðT w T i Þ
ð9Þ
and
LH s ¼
paL Ls k _ p 4mC
ð10Þ
The correlation above is applicable when a system includes more than five liquid slugs. Two approaches for integrating heat transfer results of moving frame of reference simulation were compared by Talimi et al. [15]. Based on what they argued, it is more straightforward if one uses difference between inlet and outlet temperatures to measure total heat removal rate. In a moving frame of reference simulation, this
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
means temperature difference between initial and final temperatures. This approach has been used in the present study. In the present study, the liquid slug outlet temperature will be calculated using the moving frame of reference numerical simulation. Once the inlet, wall, and outlet temperatures are known, the mean wall Nu number can be calculated using Eq. (5). The mean heat flux can be calculated using enthalpy change in moving liquid slugs (during simulation time) to be used in this equation. 3. Numerical modeling The simulations have been carried out using a single phase moving frame of reference, i.e. only a liquid slug moving inside a square microchannel has been considered as computational domain. The computational fluid dynamic software package ANSYS Fluent has been used in the present study. Only a brief description of points of direct relevance to the computations will be provided here, further details of the implementation can be found in the literature and Fluent manuals [35,36]. The governing equations for an incompressible and Newtonian fluid with constant properties are: The continuity equation:
~¼0 OU
ð11Þ
The momentum equation:
q
~ DU ~ ¼ q~ g Op þ lO2 U Dt
ð12Þ
The energy equation:
qC p
DT ¼ kO2 T þ s_ Dt
ð13Þ
The equations above have been expanded in three dimensions, (x, y, and z), in the present study. The surface tension force in the momentum equation (Eq. (12)) has not been included since the interface between two phases are fixed. 3.1. Model setup A square microchannel with a hydraulic diameter of 0.0005 (m) has been considered in the present study. The microchannel geometry is the same as what Betz and Attinger [32] used in their experiment in order to make a direct comparison. Fig. 4 shows the
755
computational domain with fixed leading and trailing interfaces. The interface shape has been build by intersecting a sphere with the rectangular cube of the microchannel. The center position and radius (curvature) of the sphere have been adjusted for different mean contact angles, from 60° to 120°. Two symmetry planes in vertical and horizontal directions (X–Z and X–Y, respectively) help to make the computational domain smaller. The top and side walls are moving walls and their velocities have been calculated based on the data provided by Betz and Attinger [32]. The fluid is water with fixed physical and transport properties. The fluid properties have been examined at the bulk temperature (average of inlet and exit temperatures) of Betz and Attinger experiments [32]. The simulations have been performed as steady state simulation in order to gain a fully developed hydrodynamics solution showing internal circulations. Zero shear stress and no slip boundary conditions have been applied to the interfaces and walls, respectively. The energy equation (Eq. (13)) was solved using unsteady state solver and thermal boundary conditions have been applied. Constant wall temperature and no heat transfer have been applied to walls and interfaces, respectively. When a two phase slug flow in a thin wall microchannel is desired, the constant wall temperature boundary condition is hard to achieve. Mehdizadeh et al. [37] performed a conjugated numerical simulation considering wall material and reported axial temperature gradient in the wall. However, in most applications, microchannels are manufactured in thick and conductive plates (see Fig. 8 in Betz and Attinger [32] for example) where these gradients could be damped significantly. Therefore, a constant wall temperature has been applied to the walls in the present study. The numerical simulations have been performed for the data points in Betz and Attinger [32] experiments which have Reynolds numbers less than 1000 in order to have laminar flow. These Re numbers are lower than the critical Reynolds value of turbulent flow. The critical value has been reported to 2300 for continues single phase Poiseuille flows [38] or 1000 for moving single droplets [39]. The gravitational effects have also been neglected since the Bond number, Bo, is lower than the criterion reported by Bretherton [23] in this study. In the present stydy, the PRESTO! interpolation scheme was used to compute the face pressure because of its higher accuracy [36] especially near the interfaces. The momentum and energy equations were discretized using a second-order upwind scheme. The pressure and velocity were coupled using the SIMPLE algorithm. The temporal discretization method used was first-order implicit method. The Green–Gauss node based averaging scheme was used to evaluate the gradients and derivatives as recommended in [36] for its higher accuracy. A similar experience has been also reported by Gupta et al. [11]. The models were tested to determine the appropriate convergence residual criteria which give good results. The residual criteria for convergence were set to 1 104 for continuity and all of the directions of momentum equations, and 1 107 for energy equation which are ten times smaller than their default values. 3.2. Grid independency and validation study
Fig. 4. The computational domain used in the present study.
Three dimensional structured mesh was used in the present study. As reported by Talimi et al. [15] a very high mesh resolution is required in the corners in order to achieve grid independence wall shear stress results. This could lead to very time consuming simulations in three dimensional cases. Three mesh resolutions have been studied first to check if grid independence heat transfer results could be achieved using lower mesh resolutions. The width and height of the computational domain have been divided into 20, 40, and 80 mesh in the three cases. Table 1 presents total mesh number and numerical results for these three cases.
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Table 1 Different mesh sizes examined for the present study. Mesh size
Mesh number
Iterations
sw (Pa)
Change in sw %
Change in qw %
1 20 1 40 1 80
32,000
291
2.43
–
–
256,000
760
2.76
11.9%
1.13%
2,048,000
1918
3.07
10.1%
0.27%
As can be seen in Table 1, required iterations increases by increase in the mesh resolution which means higher simulation time. The average wall shear stress has also been reported in Table 1. Based on these values, there is around 12% change in the average wall shear stress when one uses the second case instead of the first case. This will be around 10% using the third case instead of the second case. While the change has not been approached to an insignificant order, there are more than 2 million elements in the simulations. This is mainly due to the singularities close to the corners as reported by Talimi et al. [15]. While sw still increases using finer mesh sizes, the heat transfer results show grid independency. This is due to the fact that the flow in the corners do not contribute significantly in the heat transfer process. However, wall shear stress and pressure drop is not in the scope of the present work. Therefore, based on Table 1, the second mesh size has been selected as the appropriate mesh size since it gives grid independence heat transfer results, qw, (using Eq. (5)) with reasonable amount of computational efforts. Since qw has been calculated for different simulation times (slug positions through the channel length), they have not been reported in Table 1. The change in qw in Table 1 is average of dimensionless heat transfer differences for all the simulation times. It is expected that as the length of a liquid slug increases, the velocity profile across the cross section of the slug approaches to a single phase Poiseuille velocity profile. The single phase Poiseuille velocity profile can be achieved analytically using separation of variables. The single phase dimensionless velocity profile as a function of y and z is as follows: [33]
uH ðy; zÞ ¼
1 X
n1 1 ð1Þ 2 cos 3 n n¼1;3...
npy 2Y
pz ! cosh n2Y 1 pZ cosh n2Y
ð14Þ
Figs. 5 and 6 show analytical and numerical velocity profile for a long slug (Ls = 10Dh) along vertical and diagonal directions. The analytical profiles in these figures have been achieved using 50 terms in the series of Eq. (14). As can be seen there is a good agreement and the mean and maximum differences are 0.3% and 0.7%, respectively.
Fig. 5. Dimensionless velocity along dimensionless vertical axis (yw = y/Y), numerical and theoretical.
Fig. 6. Dimensionless velocity along .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 2 þ Z 2 , numerical and theoretical. r H ¼ y2 þ z2
diagonal
direction
4. Results and discussion The numerical simulation results has been compared to experimental data in the first section, and then effects of different parameters (including Reynolds number, contact angle, and slug length) on slug flow heat transfer in square microchannels have been discussed. 4.1. Comparison to experimental data The flow pattern and streamlines at the vertical symmetry plane of the simulated moving slug at ReL = 512 are shown in Fig. 7. As can be seen, the streamlines are approximately parallel in a large portion of front half of the slug. This leads the circulation core toward the trailing interface and closer to the microchannel wall. According to Talimi et al. [40] this is expected at Re numbers at the order of 500 and upper. The streamlines shown in Fig. 7 are different from what has been schematically shown by Taylor in Fig. 3. The reason is bypass flow through the thin liquid film around gas bubbles which has not been simulated in the present study, therefore the liquid is completely circulating inside moving slugs in Fig. 7. The difference between hydrodynamics and its effects on heat transfer process will be discussed in more details in later sections of the present study. Due to the flow pattern shown in Fig. 7, as the liquid slug travels inside the microchannel, fresh liquid from the center is brought to the wall and keeps the heat transfer rate at a high level. Fig. 8 shows temperature distribution on the vertical symmetry of the moving slug at different simulation times. As expected, the heated liquid in the thermal boundary layer is moving toward the slug center and the thermal boundary layer remains thin.
Fig. 7. Streamlines inside a moving liquid slug at vertical symmetry, ReL = 512.
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300 (K)
304.6(K)
309.2(K)
313.8(K)
318.4(K)
757
323 (K)
(a)
(b)
(c) Fig. 9. Nusselt number as a function of liquid phase Reynolds number.
(d)
Fig. 8. Contours of temperature in a moving liquid slug at vertical symmetry for ReL = 512 at different simulation times (positions): (a) t = 0.0034 s, (x = 2.5 mm), (b) t = 0.0102 s, (x = 7.5 mm), (c) t = 0.0171 s, (x = 12.5 mm), (d) t = 0.0239 s, (x = 17.5 mm).
The circulation core can be observed in Fig. 8(a) close to the upper corner at trailing interface. Since this region is so close to the wall, it will be heated fast due to thermal diffusion process. It can be understood from the boundary between hot and cold regions in Fig. 8(b), that the temperature contours exactly obeys the streamlines shown in Fig. 7. After this time, diffusion of thermal energy from the center of slug toward the wall, helps the heat convection through the streamlines to squeeze the cold region inside the moving slug (Fig. 8(c)). After traveling a distance, the slug is going to be thermally saturated which means that the average temperature of the liquid inside the slug approaches the wall constant temperature. At this time, the heat transfer rate drops due to the small temperature difference between wall and flow temperatures. This is shown in Fig. 8(d). The simulation time which is the time a slug has traveled inside the microchannel and is about to exit can be determined using the slug velocity and the microchannel length from Betz and Attinger [32]. Heat transfer rates can then be nondimensionalized using Eq. (5) (wall to inlet temperature difference) and slugs temperature rise i.e. the difference between initial and final slug temperatures in the transient numerical simulations. The Nu numbers for different Re numbers from the present study and Betz and Attinger [32] are shown in Fig. 9. The Re numbers less than 1000 have been considered in the comparison in order to ensure that the flow is not turbulent. As can be seen in Fig. 9, while there is a same trend, the Nu numbers from the present numerical study are much higher than what experiments show. Different Re numbers lead to different microchannel dimensionless lengths, Lw. The Nu numbers from the present study and experiments by Betz and Attinger [32] are presented in Fig. 10. This figure also includes ‘‘Leveque’’ limit of Nu [33] (see Eq. (16) in Appendix A) in channels with a constant wall temperature versus microchannel dimensionless length, Lw. As can be seen, present study data points show significant enhancement in heat transfer compared to Betz and Attinger [32] data points. The average difference is around 220% which means that the Nu numbers achieved from the present numerical study is around three times more than the Nu numbers from the experiments.
Fig. 10. Nusselt number as a function of microchannel dimensionless length.
This high difference is important and should be investigated in more detail. There might be one possibility for increase in heat transfer when moving slugs are considered individually in a moving frame of reference numerical simulations: liquid film effects. As discussed earlier (Fig. 3), a portion of the liquid inside a moving slug bypassed through the thin liquid film around the coming bubble and finally enters the next liquid slug. This bypassing liquid receives heat from the wall quickly (due to its small mass) and become almost thermally saturated when is about to enter the next liquid slug. This hot liquid then decreases the heat removal capacity of the next slug once it enters that slug. The other negative effect of liquid film is decrease in the portion of liquid phase circulating inside moving slugs. These effects might be changed by change in the thickness of liquid film around the bubbles. This suggests that the effect of film thickness on heat transfer enhancement is an impediment. In other words, while the heat transfer in the film region has been reported around 30 to 40% of the total heat transfer [37,41], this portion is out of the total heat transfer of a slug flow with the liquid film around bubbles, and should not be compared with present case i.e. the total heat transfer of a slug flow with no (or very thin) film around bubbles. In the present numerical study, moving slugs have been simulated individually in a moving frame of reference without considering any inlets or outlets to or from the adjacent films. This leads to complete circulation inside the simulated moving slugs. As a result, all the liquid is contributing in circulations which are the responsible mechanism for heat transfer enhancement. This may show that more enhancement in heat (or radial mass) transfer can be achieved using slug flows with no liquid film around the bubbles. This type of flows may be generated under certain conditions or using hydrophobic (or super hydrophobic) materials as microchannel wall. Further studies (numerical and experimental)
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are required to prove or disprove this discussion and provide more details on the heat transfer and pressure drop in this type of two phase slug flows. 4.2. Heat transfer prediction As discussed in the previous section, heat removal process might be enhanced using separate slugs or a slug flow with a very thin liquid film around the bubbles. In this section, effects of different parameters on flow pattern and heat transfer in moving slugs inside square microchannels have been studied. These parameters are Reynolds number, contact angle, and slug length. Fig. 11 shows streamlines inside moving slugs (on horizontal symmetry plane, see Fig. 4) with Ls/Dh = 3 and h = 120° for different Reynolds numbers of 100, 200, and 500. As can be seen in Fig. 11(a), in low Re slug flows the streamlines are parallel and circulation core is located approximately in the middle of the slug (horizontally). By an increase in Re the flow pattern changes and the circulation core moves toward the rear interface close to the microchannel walls. The flow pattern at Re = 500 in Fig. 11(c) is quite similar to the flow patterns in Fig. 7 for the same Re but a different contact angle. Effect of Reynolds number on dimensionless heat transfer, qw, for the two moving slugs of Fig. 11(a) and (c) is shown in Fig. 12. In both slug flows, qw begins and ends to same asymptotes and the difference is in the region in between. At the beginning of the process, the heat transfer is due to thermal diffusion into the thin thermal boundary layer (see Fig. 8(a)) and the circulations have not started their contribution in the cooling process yet. After a while, the moving slug with higher Re starts to receive more thermal energy and shows higher qw. This seems to be due to the stronger circulations shown in Fig. 11(c) compared with Fig. 11(a). Both slugs follow the fully developed asymptote after xw 0.05. Fig. 13 shows effects of the other parameter of interest, contact angle, on the dimensionless heat transfer in slug flows in square microchannels. As can be seen, the change in qw is not significant when contact angle, h, varies. The comparison in this figure has been conducted for three slug flows in square microchannels with same slug lengths (Ls/Dh = 3) and Reynolds number (Re = 500). The different contact angles of 60, 90, and 120 have been considered, and as can be seen the difference is visible in the developing region compared to fully developed region. The qw for slugs with h = 90 is 4% higher than the slugs with h = 120 in the developing region where xw < 0.05. The difference is around 0.2% in the fully developed region where xw > 0.05. The qw difference in developing and fully developed regions between slugs with h = 60 and h = 90 are 9% and 0.5%, respectively. This shows that the dimensionless heat transfer, qw, for slug flows in square microchannels for different
Fig. 12. Dimensionless heat transfer, qw, for moving liquid slugs with same lengths and contact angles (Ls/Dh = 3 and h = 120) at different Reynolds numbers (Re = 100 and 500).
Fig. 13. Dimensionless heat transfer, qw, for moving liquid slugs with same lengths and Reynolds numbers (Ls/Dh = 3 and Re = 500) at different contact angles (h = 60, 90, and 120).
contact angles could be approximated using a right angle slug, which is quite simpler when developing computational domain for numerical studies. As such, the dimensionless heat transfer, qw, has been reported for slug flows with right angles at different slug lengths and Reynolds numbers in the present study. It has been accepted that there is a difference between advanced and receding contact angles of a moving slug in microchannels, but based on the lack of models for predicting this difference (except for the circular microchannels [42]), studying the effects of different advanced and receding contact angles on heat transfer has been left for future studies. Dimensionless heat transfer, qw, for slug flows with different slug lengths for two Reynolds numbers of 100 and 500 have been shown in Figs. 14 and 15, respectively. These two figures could be used for heat transfer prediction in slug flows in square microchannels, when film thickness is very small or there is no film around the gas phase, which is the case in hydrophobic microchannels. The single phase slug flow qw has also been presented in these figures, to show the potential enhancements in heat transfer using slug flow in microchannels. The single phase dimensionless heat transfer can be determined using the following equation suggested by Muzychka et al. [34] and Eq. (6) for square microchannels:
2
NuDh
Fig. 11. Streamlines inside moving liquid slugs with Ls/Dh = 3, h = 120 (horizontal symmetry plane) at different Reynolds numbers: (a) Re = 100, (b) Re = 200, and (c) Re = 500.
31=2 !2 1:128 2 ¼ 4 pffiffiffiffiffiffi þ 4:935 5 LH
ð15Þ
As Figs. 14 and 15 show, slug flows with shorter slugs provide higher heat transfer rates, and an increase in slug length leads to a decrease in total heat transfer rate. The fluctuations in the curves are due to fluid circulations inside the moving slugs, which are not
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759
discussed that the hydrodynamics and heat transfer of slug flows might be simulated numerically using right angle computational domains (slugs), which are much simpler to be developed. Finally, dimensionless heat transfer has been reported for different slug lengths and Reynolds numbers and the results could be used in prediction of heat transfer rates, when there is a very thin liquid film around the gas phase or there is no film which is the case in hydrophobic microchannels. Acknowledgments
Fig. 14. Dimensionless heat transfer, qw, for moving liquid slugs for different slug lengths (Ls/Dh = 1, 2, 3, and 5) at same Reynolds numbers (Re = 100).
The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Auto21 NCE. The authors also acknowledge Amy R. Betz and Daniel Attinger for providing their experimental data [32]. Appendix A. Experimental data adoption
Fig. 15. Dimensionless heat transfer, qw, for moving liquid slugs for different slug lengths (Ls/Dh = 1, 2, 3, and 5) at same Reynolds numbers (Re = 500).
As mentioned in ‘‘introduction’’, experiments by Betz and Attinger [32] have been conducted under constant wall heat flux boundary condition. However, a constant wall temperature boundary condition was of interest in the present study, therefore, there was a need to adopt isoflux experimental data to isothermal condition. As mentioned, the microchannel wall material in Betz and Attinger [32] is highly conductive hence a small temperature gradient in axial direction is expected. Based on the aforementioned assumptions, a goal seek method has been applied to find wall temperatures (constant) which give same Nu numbers under same flow condition. Based on Leveque solution, the Nu number under constant wall temperature is [33]: 1
NuT ¼ 0:641 the case in the single phase slug flow. Furthermore, a slug flow approaches to the thermally saturated situation sooner than a single phase slug flow, which shows more heat removal from microchannel walls.
ðfReÞ3 L
ð16Þ
1 H3
where:
LH ¼
L Dh Pe
ð17Þ
and the local Nu number under constant wall heat flux is: 5. Conclusions
1
NuH ¼ 0:517 Heat transfer process under constant wall temperature boundary condition in two phase slug flows inside square microchannels has been considered in the present study. Individual moving slugs have been simulated numerically and results have been compared to experimental data provided by Betz and Attinger [32]. The numerical Nu numbers showed higher values but similar general trends compared with experimental Nu numbers. This difference requires more in-depth investigation. The film thickness might be responsible to decrease total heat transfer in a two phase slug flow inside microchannels. This is due to negative effects of liquid film around bubbles on thermal boundary layer zrenewal and amount of liquid in circulations inside moving slugs. This may show that heat transfer could be further increased in microchannel heat sinks using two phase slug flows under dry-out flow condition i.e. two phases are in direct contact with microchannel wall. This type of slug flow could be generated using hydrophobic or superhydrophobic materials in microchannels. The effects of Reynolds number, contact angle, and slug length on slug flow heat transfer have also been studied in the present work. It has been shown that Reynolds number, Re, has a significant effect on slug flow heat transfer due to a change in flow pattern and circulation shape in moving slugs, and with an increase in Reynolds number the heat transfer increases. Contact angle shows some small and negligible effects on slug flow heat transfer, and it has been
ðfReÞ3
ð18Þ
1
xH 3
where:
xH ¼
x Dh Pe
ð19Þ
If one uses the definition of Nu number, then:
NuH ¼
qw Dh kDT
ð20Þ
Combining Eqs. (18) and (20) the local wall to bulk temperature difference is: 1
DT ¼ 1:93
q w D h xH 3
ð21Þ
1
kðfReÞ3
which could be expanded using Eq. (18) as follows:
DT ¼ 1:93
qw Dh 1
kðfReÞ3
13 x Dh Pe
ð22Þ
Integrating the expression above over 0 ? L one can determine the mean wall to bulk temperature difference. In general, in the boundary layer region (thermal entrance length in internal flows) the bulk temperature does not vary significantly from inlet temperature, therefore (Tw Tm) approaches to (Tw Ti) as x ? 0.
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Based on the small values of Lw in the present study (see Fig. 10), it could be assumed that the boundary layer theory is applicable. Integrating DT using:
DT mean ¼
1 L
Z
L
DTdx
ð23Þ
0
gives: 1
NuH ¼ 0:689
ðfReÞ3 L
1 H3
ð24Þ
which is only 7% different from NuT in Eq. (16). References [1] G.I. Taylor, Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech. 10 (1961) 161–165. [2] V. Talimi, Y.S. Muzychka, S. Kocabiyik, A review on numerical studies of slug flow hydrodynamics and heat transfer in microtubes and microchannels, Int. J. Multiphase Flow 39 (2012) 88–108. [3] R. Gupta, D.F. Fletcher, B.S. Haynes, Taylor flow in microchannels: a review of experimental and computational work, J. Comput. Multiphase Flow 2 (2010) 1–31. [4] P. Angeli, A. Gavriilidis, Hydrodynamics of Taylor flow in small channels: a review, Proc. IMechE Part C J. Mech. Eng. Sci. 222 (2008) 737–750. [5] N. Shao, W. Salman, A. Gavriilidis, P. Angeli, CFD simulations of the effect of inlet conditions on Taylor flow formation, Int. J. Heat Fluid Flow 29 (2008) 1603–1611. [6] Y.S. Muzychka, E.J. Walsh, P. Walsh, Heat transfer enhancement using laminar gas–liquid segmented plug flows, J. Heat Transfer 133 (2011). [7] L. Wu, M. Tsutahara, L.S. Kim, M.Y. Ha, Three-dimensional lattice Boltzmann simulations of droplet formation in a cross-junction microchannel, Int. J. Multiphase Flow 34 (2008) 852–864. [8] M.N. Kashid, F. Platte, D.W. Agar, S. Turek, Computational modelling of slug flow in a capillary microreactor, J. Comput. Appl. Math. 203 (2007) 487–497. [9] V. Kumar, S. Vashisth, Y. Hoarau, K.D.P. Nigam, Slug flow in curved microreactors: hydrodynamic study, Chem. Eng. Sci. 62 (2007) 7494–7504. [10] D. Lakehal, G. Larrignon, C. Narayanan, Computational heat transfer and twophase flow topology in miniature tubes, Microfluid. Nanofluid. 4 (2008) 261– 271. [11] R. Gupta, D.F. Fletcher, B.S. Haynes, On the CFD modelling of Taylor flow in microchannels, Chem. Eng. Sci. 64 (2009) 2941–2950. [12] D.M. Fries, Ph.R. von Rohr, Liquid mixing in gasliquid two-phase flow by meandering microchannels, Chem. Eng. Sci. 64 (2009) 1326–1335. [13] T. Taha, Z.F. Cui, CFD modelling of slug flow in vertical tubes, Chem. Eng. Sci. 61 (2006) 676–687. [14] M. Worner, B. Ghidersa, A. Onea, A model for the residence time distribution of bubble-train flow in a square mini-channel based on direct numerical simulation results, Int. J. Heat Fluid Flow 28 (2007) 83–94. [15] V. Talimi, Y.S. Muzychka, S. Kocabiyik, A numerical study on shear stress and heat transfer of segmented flow between parallel plates, in: Proceedings of the 9th International Conference on Nanochannels, Microchannels, and Minichannels, ICNMM9, June 2011, Edmonton, Canada, 2011. [16] P. Young K. Mohseni, The effect of droplet length on Nusselt numbers in digitized heat transfer, in: Proceedings of I-THERM, 2008, 11th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems, 2008, pp. 352–359.
[17] H. Oprins, J. Danneels, B. Van Ham, B. Vandevelde, M. Baelmans, Convection heat transfer in electrostatic actuated liquid droplets for electronics cooling, Microelectron. J. 39 (2008) 966–974. [18] M.J. Martinez, K.S. Udell, Boundary integral analysis of the creeping flow of long bubbles in capillaries, J. Appl. Mech. 56 (1989) 211–217. [19] Y. Chen, R. Kulenovic, R. Mertz, Numerical study on the formation of Taylor bubbles in capillary tubes, Int. J. Therm. Sci. 48 (2009) 234–242. [20] E.I. Shen, K.S. Udell, A finite element study of low Reynolds number two phase flow in cylindrical tubes, J. Appl. Mech. 52 (1985) 253–256. [21] M.D. Giavedoni, F.A. Saita, The axisymmetric and plane cases of a gas phase steadily displacing a newtonian liquid – a simultaneous solution of the governing equations, Phys. Fluids 9 (8) (1997) 2420–2428. [22] F. Fairbrother, A.E. Stubbs, The bubble-tube method of measurement, J. Chem. Soc. 1 (1935) 527–529. [23] F.P. Bretherton, The motion of long bubbles in tubes, J. Fluid Mech. 10 (1961) 166–188. [24] P. Aussillous, D. Quere, Quick deposition of a fluid on the wall of a tube, Phys. Fluids 12 (10) (2000) 2367–2371. [25] R. Luo, L. Wang, Liquid velocity distribution in slug flow in a microchannel, Microfluid. Nanofluid. 12 (2012) 581–595. [26] P. Sobieszuk, P. Cyganski, R. Pohorecki, Bubble lengths in the gas–liquid Taylor flow in microchannels, Chem. Eng. Res. Des. 88 (2010) 263–269. [27] D. Qian, A. Lawal, Numerical study on gas and liquid slugs for Taylor flow in a T-junction microchannel, Chem. Eng. Sci. 61 (2006) 7609–7625. [28] N.D.M. Raimondi, L. Prat, Numerical study of the coupling between reaction and mass transfer for liquid–liquid slug flow in square microchannels, AIChE J. 57 (7) (2011) 1719–1732. [29] J. Yue, L. Luo, Y. Gonthier, G. Chen, Q. Yuan, An experimental study of air–water Taylor flow and mass transfer inside square microchannels, Chem. Eng. Sci. 64 (2009) 3697–3708. [30] M.T. Kreutzer, F. Kapteijn, J.A. Moulijn, Inertial and interfacial effects on pressure drop of Taylor flow in capillaries, AIChE J. 9 (2005) 2428–2440. [31] A. Majumder, B. Mehta, S. Khandekar, Local Nusselt number enhancement during gas–liquid Taylor bubble flow in a square mini-channel: an experimental study, Int. J. Therm. Sci. 66 (2013) 8–18. [32] A.R. Betz, D. Attinger, Can segmented flow enhance he at transfer in microchannel heat sinks?, Int J. Heat Mass Transfer 53 (2010) 3683–3691. [33] R.K. Shah, A.L. London, Laminar Flow Forced Convection in Ducts, Academic Press, 1978. [34] Y.S. Muzychka, E. Walsh, P. Walsh, Simple models for laminar thermally developing slug flow in noncircular ducts and channels, J. Heat Transfer (2010) 132. [35] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, second ed., Pearson, 2007. [36] Fluent Inc. 2006, Fluent 6.3 User Guide, FLUENT INC. [37] A. Mehdizadeh, S.A. Sherif, W.E. Lear, Numerical simulation of thermofluid characteristics of two-phase slug flow in microchannels, Int. J. Heat Mass Transfer 54 (2011) 34573465. [38] Incropera, DeWitt, Bergman, Lavine, Fundamentals of Heat and Mass Transfer, sixth ed., John Wiley and Sons, 2007. [39] H. Ren, R.B. Fair, M.G. Pollack, E.J. Schayghnessy, Dynamics of electrowetting droplet transport, Sens. Actuators B 87 (2002) 201–206. [40] V. Talimi, Y.S. Muzychka, S. Kocabiyik, On the validity of two-dimensional heat transfer simulation of moving droplets between parallel plates, in: Proceedings of the 9th International Conference on Nanochannels, Microchannels, and Minichannels, ICNMM9, June 2011, Edmonton, Canada, 2011. [41] R. Gupta, D.F. Fletcher, B.S. Haynes, CFD modeling of flow and heat transfer in the Taylor flow regime, Chem. Eng. Sci. 65 (2010) 2094–2107. [42] D. Yu, Ch. Choi, M.H. Kim, Pressure drop and dynamic contact angle of tripleline motion in a hydrophobic microchannel, Exp. Therm. Fluid Sci. 39 (2012) 60–70.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Slug flow heat transfer in square microchannels V. Talimi a, Y.S. Muzychka a,⇑, S. Kocabiyik b a b
Microfluidics and Multiphase Flow Research Lab, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada A1B 3X5 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7
a r t i c l e
i n f o
Article history: Received 29 August 2012 Received in revised form 12 March 2013 Accepted 12 March 2013 Available online 16 April 2013 Keywords: Heat transfer Square microchannel Two-phase flow Slug flow Numerical simulation
a b s t r a c t While heat transfer in Taylor (slug) flows in circular microtubes has been examined exclusively by researchers, noncircular microchannels have not been widely considered in the literature. This is a large gap in research since noncircular microchannels are common structures in microcooling processes. Square and rectangular microchannels are the most important examples. In the present study the heat transfer process in slug flows in square microchannels has been investigated numerically under constant wall temperature boundary condition. The local heat flux for the moving slugs has been converted to total microchannel heat flux using the integration methods suggested recently by the authors. This leads to microchannel wall average heat flux which is the parameter of interest in heat sink problems. Potential effects of liquid film around bubbles on heat transfer process have been discussed by comparing the numerical results with experimental data. Finally, effects of Reynolds number, contact angle, and slug length on heat transfer in slug flow inside square microchannels have been studied and graphs for prediction of slug flow heat transfer in square microchannels have been provided for slug flows with no liquid film or very thin liquid film around the bubbles. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Non-boiling two phase slug flow has attracted researchers in recent years due to its capacity to enhance heat and mass transfer in small scales. This type of flow is typically a train of liquid slugs and bubbles and has also been named as plug flow, segmented flow, and Taylor flow [1]. There are still many large gaps in research based on several published reviews on slug flow hydrodynamics and heat transfer [2–5]. According to Talimi et al. [2] heat transfer in non-circular microchannels needs to be studied in detail. Muzychka et al. [6] showed that internal circulations inside moving slugs are the primary mechanism of heat transfer enhancement. Fig. 1 shows these circulations inside liquid slugs for two different microchannel wall materials: hydrophilic and hydrophobic walls. These circulations bring fresh liquid from center of the slug to the wall, where heat transfer process occurs. This provides a renewal mechanism to the thermal boundary layer and increases heat (or radial mass) transfer. Numerical simulation of slug flows in microchannels can be performed using different approaches. In the first approach, a fixed frame of reference is used to generate computational domain. This requires at least two inlets for the two phases, a certain initial ⇑ Corresponding author. E-mail addresses: [email protected] (V. Talimi), [email protected] (Y.S. Muzychka), [email protected] (S. Kocabiyik). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.035
length in order for slugs to be developed, and a longer length to examine pressure drop and heat transfer process [7–10]. In general, this approach is very time consuming, since a very fine mesh is required to capture all the important details inside the flow e.g. thin liquid film around the bubbles [11]. Some researchers used a moving frame of reference to make the computational domain smaller and accelerate numerical simulations [12–14]. There is a need for a careful setting of inlet and outlet. This approach is still computationally expensive since a two phase flow is considered, and interactions between the two phases must be captured. The fastest approach is using a single phase moving frame of reference computational domain with fixed leading and trailing interfaces [15–17]. Fig. 2 schematically shows the computational domains used in the aforementioned three approaches. In most two phase slug flows, there is a very thin liquid layer around bubbles [1]. Taylor [1] showed the importance of the film thickness on the internal circulations. As Fig. 3 shows streamline shape in the moving liquid slugs depend on Capillary number, Ca, and a dimensionless number, m, which is a relative velocity between the two phases, as follows:
m¼
UB UL UB
ð1Þ
Based on Fig. 3 at high Ca numbers a complete bypass occurs through the liquid film around the bubbles, while in slug flows with low Ca numbers internal circulations exist inside the liquid slugs. As reported in [18–21] the film thickness is a function of
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
753
Nomenclature Bo Ca Cp D Dh f g h H k L Lw Ls LH s m _ m Nu p Pe Pr Q q qw rw R Re s_ t T u uw U
Bond number, gD2h ðql qg Þ=r Capillary number, lU/r specific heat capacity, J/kg K diameter, m hydraulic diameter of channel, m friction factor gravitational acceleration, m/s2 thermal convection coefficient, W/m2 K distance between plates, m thermal conductivity, W/m K microchannel length, m dimensionless microchannel length, L/DhPe length of liquid slug, m dimensionless length of slug dimensionless relative velocity mass flow rate, kg/s Nusselt number, hDh/k pressure, Pa Peclet number, UDh/a Prandtl number, m/a total heat flux, W mean heat flux, W/m2 dimensionless wall heat flux dimensionless position in diagonal direction radius, m Reynolds number, UDh/m energy (heat) generation, W/m3 time, s temperature, K horizontal velocity component, m/s dimensionless velocity in x-direction average liquid velocity, m/s
x y yw Y z Z
position in x-direction, m position in y-direction, m dimensionless position in y-direction microchannel width (y-direction), m position in z-direction, m microchannel height (z-direction), m
Greek symbols thermal diffusivity, m2/s liquid phase fraction d film thickness, m l dynamic viscosity, N s/m2 m kinematic viscosity, m2/s q fluid density, kg/m3 sw wall shear stress, Pa h contact angle r surface tension, N/m
a aL
Superscripts w dimensionless ðÞ mean value Subscripts B bubble i inlet lm logarithmic mean L liquid m mean s slug w wall
Ca number, so the film thickness is an important parameter and governs the flow pattern i.e. internal circulations. Therefore, the film thickness may have significant effects on heat transfer process. There are several correlations for film thickness as a function of Ca [1,22–24]. Gupta et al. [11] reported that the correlation suggested by Bretherton [23] is the most appropriate correlation:
ð2Þ
Recently, Luo and Wang [25] investigated the velocity field inside liquid slugs in rectangular microchannels. Sobieszuk et al. [26] measured bubble lengths in Taylor flows in square microchannels, and showed that the correlations suggested by Qian and Lawal [27] can predict bubble length with a good accuracy. As the existing reviews [2–5] reported, there are not many published works on heat (or mass) transfer in two phase slug flows in square microchannels. Mass transfer in slug flow in square microchannels
Fig. 1. Internal liquid slug circulation, (a) hydrophobic surface, (b) hydrophilic surface.
Fig. 2. Typical computational domains frequently used for slug flow numerical simulations, (a) fixed frame of reference two phase, (b) moving frame of reference two phase, (c) moving frame of reference single phase.
d ¼ 1:34 Ca2=3 R
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of interest. According to Shah and London [33] and Muzychka et al. [6], the goal in cooling processes such as heat sinks is wall average heat flux enhancement. There are different methods to nondimensionalize wall heat flux as reported by Muzychka et al. [6]: Wall to Bulk Mean, Tw Tm. Wall to Inlet, Tw Ti. Mean Wall to Inlet, T w T i . The Nusselt number, Nu, based on the first approach is:
NuDh ¼
qDh kDT lm
ð3Þ
where DTlm is the logarithmic mean temperature difference and defined as follows:
DT lm ¼
ðT w T i Þ ðT w T o Þ T i ln TTwwT o
ð4Þ
If one considers the wall to inlet temperature difference which is more appropriate for heat sinks with one fluid stream [6], the dimensionless wall heat transfer, qw, is as follows:
qH ¼
qDh kðT w T i Þ
ð5Þ
These two dimensionless heat transfers above are related in the single phase Poiseuille flow as follows [6]: Fig. 3. Streamlines in liquid slugs in front of a Taylor bubble, Taylor [1].
has been investigated by Raimondi and Prat [28] and Yue et al. [29], for liquid–liquid and gas–liquid two phase flows, respectively. Yue et al. [29] showed that pressure drop in two phase slug flows with very short liquid slugs deviates from correlations suggested by Kreutzer et al. [30]. Majumder et al. [31] performed experiments on heat transfer in two phase slug flows in square minichannels. They reported a heat transfer enhancement up to 1.2–1.6 times when compared to fully developed laminar single phase flows. In their experiments, they applied heat only from one side of the 3.3 (mm) 3.3 (mm) square channel. They also concluded that the heat transfer enhancement is a function of geometrical parameters on the slug flow such as bubble length or slug length. Betz and Attinger [32] performed experiments on heat transfer in slug flow in square microchannels under constant wall heat flux boundary condition. They provided the present authors with their data including flow inlet and exit temperatures and other flow conditions. These data have been used in the present study in order to perform a comparison between present study results and their experimental results. In the present study, a constant wall temperature has been considered as a thermal boundary condition. This required some adjustments in Betz and Attinger [32] data using a goal seek process to find the constant wall temperature which gives the same Nu number. Based on the wall material and its high heat conduction coefficient, k, there is a very small gradient in wall temperature which enabled us to convert data from isoflux boundary condition to isothermal thermal boundary condition. More information on data adoption has been reported in Appendix A. In the present study, a moving liquid slug with the same length and shape as Betz and Attinger [32] reported data has been simulated numerically, using the moving frame of reference simulation approach. 2. Theory In internal heat convection under constant wall temperature the heat flux through the wall to the liquid stream is the parameter
qH ¼
1 4LH
½1 expð4NuDh LH Þ
ð6Þ
where Lw is dimensionless channel length and defined as follows:
LH ¼
L=Dh Re Pr
ð7Þ
As discussed by Muzychka et al. [6] the first approach (wall to bulk mean temperature difference) is appropriate for heat exchanger models and the second approach (wall to inlet temperature difference) is more appropriate for heat sink applications where there is only one fluid stream as coolant. Therefore, the wall to inlet temperature difference has been used to calculate dimensionless heat transfer under constant wall temperature boundary condition in the present study. Muzychka et al. [34] suggested a heat transfer predictive model based on an asymptotic analysis and the available data in the literature. They suggested to use the slug length instead of the tube length as the length scale when nondimensionalizing heat transfer data. Their model is as follows:
2
1:614 q ¼ 4 H1=3 Ls
!3=2
H
þ
1 4LH s
!3=2 32=3 5
ð8Þ
where
qH ¼
Q=ðaL pDLÞD kðT w T i Þ
ð9Þ
and
LH s ¼
paL Ls k _ p 4mC
ð10Þ
The correlation above is applicable when a system includes more than five liquid slugs. Two approaches for integrating heat transfer results of moving frame of reference simulation were compared by Talimi et al. [15]. Based on what they argued, it is more straightforward if one uses difference between inlet and outlet temperatures to measure total heat removal rate. In a moving frame of reference simulation, this
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
means temperature difference between initial and final temperatures. This approach has been used in the present study. In the present study, the liquid slug outlet temperature will be calculated using the moving frame of reference numerical simulation. Once the inlet, wall, and outlet temperatures are known, the mean wall Nu number can be calculated using Eq. (5). The mean heat flux can be calculated using enthalpy change in moving liquid slugs (during simulation time) to be used in this equation. 3. Numerical modeling The simulations have been carried out using a single phase moving frame of reference, i.e. only a liquid slug moving inside a square microchannel has been considered as computational domain. The computational fluid dynamic software package ANSYS Fluent has been used in the present study. Only a brief description of points of direct relevance to the computations will be provided here, further details of the implementation can be found in the literature and Fluent manuals [35,36]. The governing equations for an incompressible and Newtonian fluid with constant properties are: The continuity equation:
~¼0 OU
ð11Þ
The momentum equation:
q
~ DU ~ ¼ q~ g Op þ lO2 U Dt
ð12Þ
The energy equation:
qC p
DT ¼ kO2 T þ s_ Dt
ð13Þ
The equations above have been expanded in three dimensions, (x, y, and z), in the present study. The surface tension force in the momentum equation (Eq. (12)) has not been included since the interface between two phases are fixed. 3.1. Model setup A square microchannel with a hydraulic diameter of 0.0005 (m) has been considered in the present study. The microchannel geometry is the same as what Betz and Attinger [32] used in their experiment in order to make a direct comparison. Fig. 4 shows the
755
computational domain with fixed leading and trailing interfaces. The interface shape has been build by intersecting a sphere with the rectangular cube of the microchannel. The center position and radius (curvature) of the sphere have been adjusted for different mean contact angles, from 60° to 120°. Two symmetry planes in vertical and horizontal directions (X–Z and X–Y, respectively) help to make the computational domain smaller. The top and side walls are moving walls and their velocities have been calculated based on the data provided by Betz and Attinger [32]. The fluid is water with fixed physical and transport properties. The fluid properties have been examined at the bulk temperature (average of inlet and exit temperatures) of Betz and Attinger experiments [32]. The simulations have been performed as steady state simulation in order to gain a fully developed hydrodynamics solution showing internal circulations. Zero shear stress and no slip boundary conditions have been applied to the interfaces and walls, respectively. The energy equation (Eq. (13)) was solved using unsteady state solver and thermal boundary conditions have been applied. Constant wall temperature and no heat transfer have been applied to walls and interfaces, respectively. When a two phase slug flow in a thin wall microchannel is desired, the constant wall temperature boundary condition is hard to achieve. Mehdizadeh et al. [37] performed a conjugated numerical simulation considering wall material and reported axial temperature gradient in the wall. However, in most applications, microchannels are manufactured in thick and conductive plates (see Fig. 8 in Betz and Attinger [32] for example) where these gradients could be damped significantly. Therefore, a constant wall temperature has been applied to the walls in the present study. The numerical simulations have been performed for the data points in Betz and Attinger [32] experiments which have Reynolds numbers less than 1000 in order to have laminar flow. These Re numbers are lower than the critical Reynolds value of turbulent flow. The critical value has been reported to 2300 for continues single phase Poiseuille flows [38] or 1000 for moving single droplets [39]. The gravitational effects have also been neglected since the Bond number, Bo, is lower than the criterion reported by Bretherton [23] in this study. In the present stydy, the PRESTO! interpolation scheme was used to compute the face pressure because of its higher accuracy [36] especially near the interfaces. The momentum and energy equations were discretized using a second-order upwind scheme. The pressure and velocity were coupled using the SIMPLE algorithm. The temporal discretization method used was first-order implicit method. The Green–Gauss node based averaging scheme was used to evaluate the gradients and derivatives as recommended in [36] for its higher accuracy. A similar experience has been also reported by Gupta et al. [11]. The models were tested to determine the appropriate convergence residual criteria which give good results. The residual criteria for convergence were set to 1 104 for continuity and all of the directions of momentum equations, and 1 107 for energy equation which are ten times smaller than their default values. 3.2. Grid independency and validation study
Fig. 4. The computational domain used in the present study.
Three dimensional structured mesh was used in the present study. As reported by Talimi et al. [15] a very high mesh resolution is required in the corners in order to achieve grid independence wall shear stress results. This could lead to very time consuming simulations in three dimensional cases. Three mesh resolutions have been studied first to check if grid independence heat transfer results could be achieved using lower mesh resolutions. The width and height of the computational domain have been divided into 20, 40, and 80 mesh in the three cases. Table 1 presents total mesh number and numerical results for these three cases.
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Table 1 Different mesh sizes examined for the present study. Mesh size
Mesh number
Iterations
sw (Pa)
Change in sw %
Change in qw %
1 20 1 40 1 80
32,000
291
2.43
–
–
256,000
760
2.76
11.9%
1.13%
2,048,000
1918
3.07
10.1%
0.27%
As can be seen in Table 1, required iterations increases by increase in the mesh resolution which means higher simulation time. The average wall shear stress has also been reported in Table 1. Based on these values, there is around 12% change in the average wall shear stress when one uses the second case instead of the first case. This will be around 10% using the third case instead of the second case. While the change has not been approached to an insignificant order, there are more than 2 million elements in the simulations. This is mainly due to the singularities close to the corners as reported by Talimi et al. [15]. While sw still increases using finer mesh sizes, the heat transfer results show grid independency. This is due to the fact that the flow in the corners do not contribute significantly in the heat transfer process. However, wall shear stress and pressure drop is not in the scope of the present work. Therefore, based on Table 1, the second mesh size has been selected as the appropriate mesh size since it gives grid independence heat transfer results, qw, (using Eq. (5)) with reasonable amount of computational efforts. Since qw has been calculated for different simulation times (slug positions through the channel length), they have not been reported in Table 1. The change in qw in Table 1 is average of dimensionless heat transfer differences for all the simulation times. It is expected that as the length of a liquid slug increases, the velocity profile across the cross section of the slug approaches to a single phase Poiseuille velocity profile. The single phase Poiseuille velocity profile can be achieved analytically using separation of variables. The single phase dimensionless velocity profile as a function of y and z is as follows: [33]
uH ðy; zÞ ¼
1 X
n1 1 ð1Þ 2 cos 3 n n¼1;3...
npy 2Y
pz ! cosh n2Y 1 pZ cosh n2Y
ð14Þ
Figs. 5 and 6 show analytical and numerical velocity profile for a long slug (Ls = 10Dh) along vertical and diagonal directions. The analytical profiles in these figures have been achieved using 50 terms in the series of Eq. (14). As can be seen there is a good agreement and the mean and maximum differences are 0.3% and 0.7%, respectively.
Fig. 5. Dimensionless velocity along dimensionless vertical axis (yw = y/Y), numerical and theoretical.
Fig. 6. Dimensionless velocity along .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 2 þ Z 2 , numerical and theoretical. r H ¼ y2 þ z2
diagonal
direction
4. Results and discussion The numerical simulation results has been compared to experimental data in the first section, and then effects of different parameters (including Reynolds number, contact angle, and slug length) on slug flow heat transfer in square microchannels have been discussed. 4.1. Comparison to experimental data The flow pattern and streamlines at the vertical symmetry plane of the simulated moving slug at ReL = 512 are shown in Fig. 7. As can be seen, the streamlines are approximately parallel in a large portion of front half of the slug. This leads the circulation core toward the trailing interface and closer to the microchannel wall. According to Talimi et al. [40] this is expected at Re numbers at the order of 500 and upper. The streamlines shown in Fig. 7 are different from what has been schematically shown by Taylor in Fig. 3. The reason is bypass flow through the thin liquid film around gas bubbles which has not been simulated in the present study, therefore the liquid is completely circulating inside moving slugs in Fig. 7. The difference between hydrodynamics and its effects on heat transfer process will be discussed in more details in later sections of the present study. Due to the flow pattern shown in Fig. 7, as the liquid slug travels inside the microchannel, fresh liquid from the center is brought to the wall and keeps the heat transfer rate at a high level. Fig. 8 shows temperature distribution on the vertical symmetry of the moving slug at different simulation times. As expected, the heated liquid in the thermal boundary layer is moving toward the slug center and the thermal boundary layer remains thin.
Fig. 7. Streamlines inside a moving liquid slug at vertical symmetry, ReL = 512.
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
300 (K)
304.6(K)
309.2(K)
313.8(K)
318.4(K)
757
323 (K)
(a)
(b)
(c) Fig. 9. Nusselt number as a function of liquid phase Reynolds number.
(d)
Fig. 8. Contours of temperature in a moving liquid slug at vertical symmetry for ReL = 512 at different simulation times (positions): (a) t = 0.0034 s, (x = 2.5 mm), (b) t = 0.0102 s, (x = 7.5 mm), (c) t = 0.0171 s, (x = 12.5 mm), (d) t = 0.0239 s, (x = 17.5 mm).
The circulation core can be observed in Fig. 8(a) close to the upper corner at trailing interface. Since this region is so close to the wall, it will be heated fast due to thermal diffusion process. It can be understood from the boundary between hot and cold regions in Fig. 8(b), that the temperature contours exactly obeys the streamlines shown in Fig. 7. After this time, diffusion of thermal energy from the center of slug toward the wall, helps the heat convection through the streamlines to squeeze the cold region inside the moving slug (Fig. 8(c)). After traveling a distance, the slug is going to be thermally saturated which means that the average temperature of the liquid inside the slug approaches the wall constant temperature. At this time, the heat transfer rate drops due to the small temperature difference between wall and flow temperatures. This is shown in Fig. 8(d). The simulation time which is the time a slug has traveled inside the microchannel and is about to exit can be determined using the slug velocity and the microchannel length from Betz and Attinger [32]. Heat transfer rates can then be nondimensionalized using Eq. (5) (wall to inlet temperature difference) and slugs temperature rise i.e. the difference between initial and final slug temperatures in the transient numerical simulations. The Nu numbers for different Re numbers from the present study and Betz and Attinger [32] are shown in Fig. 9. The Re numbers less than 1000 have been considered in the comparison in order to ensure that the flow is not turbulent. As can be seen in Fig. 9, while there is a same trend, the Nu numbers from the present numerical study are much higher than what experiments show. Different Re numbers lead to different microchannel dimensionless lengths, Lw. The Nu numbers from the present study and experiments by Betz and Attinger [32] are presented in Fig. 10. This figure also includes ‘‘Leveque’’ limit of Nu [33] (see Eq. (16) in Appendix A) in channels with a constant wall temperature versus microchannel dimensionless length, Lw. As can be seen, present study data points show significant enhancement in heat transfer compared to Betz and Attinger [32] data points. The average difference is around 220% which means that the Nu numbers achieved from the present numerical study is around three times more than the Nu numbers from the experiments.
Fig. 10. Nusselt number as a function of microchannel dimensionless length.
This high difference is important and should be investigated in more detail. There might be one possibility for increase in heat transfer when moving slugs are considered individually in a moving frame of reference numerical simulations: liquid film effects. As discussed earlier (Fig. 3), a portion of the liquid inside a moving slug bypassed through the thin liquid film around the coming bubble and finally enters the next liquid slug. This bypassing liquid receives heat from the wall quickly (due to its small mass) and become almost thermally saturated when is about to enter the next liquid slug. This hot liquid then decreases the heat removal capacity of the next slug once it enters that slug. The other negative effect of liquid film is decrease in the portion of liquid phase circulating inside moving slugs. These effects might be changed by change in the thickness of liquid film around the bubbles. This suggests that the effect of film thickness on heat transfer enhancement is an impediment. In other words, while the heat transfer in the film region has been reported around 30 to 40% of the total heat transfer [37,41], this portion is out of the total heat transfer of a slug flow with the liquid film around bubbles, and should not be compared with present case i.e. the total heat transfer of a slug flow with no (or very thin) film around bubbles. In the present numerical study, moving slugs have been simulated individually in a moving frame of reference without considering any inlets or outlets to or from the adjacent films. This leads to complete circulation inside the simulated moving slugs. As a result, all the liquid is contributing in circulations which are the responsible mechanism for heat transfer enhancement. This may show that more enhancement in heat (or radial mass) transfer can be achieved using slug flows with no liquid film around the bubbles. This type of flows may be generated under certain conditions or using hydrophobic (or super hydrophobic) materials as microchannel wall. Further studies (numerical and experimental)
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are required to prove or disprove this discussion and provide more details on the heat transfer and pressure drop in this type of two phase slug flows. 4.2. Heat transfer prediction As discussed in the previous section, heat removal process might be enhanced using separate slugs or a slug flow with a very thin liquid film around the bubbles. In this section, effects of different parameters on flow pattern and heat transfer in moving slugs inside square microchannels have been studied. These parameters are Reynolds number, contact angle, and slug length. Fig. 11 shows streamlines inside moving slugs (on horizontal symmetry plane, see Fig. 4) with Ls/Dh = 3 and h = 120° for different Reynolds numbers of 100, 200, and 500. As can be seen in Fig. 11(a), in low Re slug flows the streamlines are parallel and circulation core is located approximately in the middle of the slug (horizontally). By an increase in Re the flow pattern changes and the circulation core moves toward the rear interface close to the microchannel walls. The flow pattern at Re = 500 in Fig. 11(c) is quite similar to the flow patterns in Fig. 7 for the same Re but a different contact angle. Effect of Reynolds number on dimensionless heat transfer, qw, for the two moving slugs of Fig. 11(a) and (c) is shown in Fig. 12. In both slug flows, qw begins and ends to same asymptotes and the difference is in the region in between. At the beginning of the process, the heat transfer is due to thermal diffusion into the thin thermal boundary layer (see Fig. 8(a)) and the circulations have not started their contribution in the cooling process yet. After a while, the moving slug with higher Re starts to receive more thermal energy and shows higher qw. This seems to be due to the stronger circulations shown in Fig. 11(c) compared with Fig. 11(a). Both slugs follow the fully developed asymptote after xw 0.05. Fig. 13 shows effects of the other parameter of interest, contact angle, on the dimensionless heat transfer in slug flows in square microchannels. As can be seen, the change in qw is not significant when contact angle, h, varies. The comparison in this figure has been conducted for three slug flows in square microchannels with same slug lengths (Ls/Dh = 3) and Reynolds number (Re = 500). The different contact angles of 60, 90, and 120 have been considered, and as can be seen the difference is visible in the developing region compared to fully developed region. The qw for slugs with h = 90 is 4% higher than the slugs with h = 120 in the developing region where xw < 0.05. The difference is around 0.2% in the fully developed region where xw > 0.05. The qw difference in developing and fully developed regions between slugs with h = 60 and h = 90 are 9% and 0.5%, respectively. This shows that the dimensionless heat transfer, qw, for slug flows in square microchannels for different
Fig. 12. Dimensionless heat transfer, qw, for moving liquid slugs with same lengths and contact angles (Ls/Dh = 3 and h = 120) at different Reynolds numbers (Re = 100 and 500).
Fig. 13. Dimensionless heat transfer, qw, for moving liquid slugs with same lengths and Reynolds numbers (Ls/Dh = 3 and Re = 500) at different contact angles (h = 60, 90, and 120).
contact angles could be approximated using a right angle slug, which is quite simpler when developing computational domain for numerical studies. As such, the dimensionless heat transfer, qw, has been reported for slug flows with right angles at different slug lengths and Reynolds numbers in the present study. It has been accepted that there is a difference between advanced and receding contact angles of a moving slug in microchannels, but based on the lack of models for predicting this difference (except for the circular microchannels [42]), studying the effects of different advanced and receding contact angles on heat transfer has been left for future studies. Dimensionless heat transfer, qw, for slug flows with different slug lengths for two Reynolds numbers of 100 and 500 have been shown in Figs. 14 and 15, respectively. These two figures could be used for heat transfer prediction in slug flows in square microchannels, when film thickness is very small or there is no film around the gas phase, which is the case in hydrophobic microchannels. The single phase slug flow qw has also been presented in these figures, to show the potential enhancements in heat transfer using slug flow in microchannels. The single phase dimensionless heat transfer can be determined using the following equation suggested by Muzychka et al. [34] and Eq. (6) for square microchannels:
2
NuDh
Fig. 11. Streamlines inside moving liquid slugs with Ls/Dh = 3, h = 120 (horizontal symmetry plane) at different Reynolds numbers: (a) Re = 100, (b) Re = 200, and (c) Re = 500.
31=2 !2 1:128 2 ¼ 4 pffiffiffiffiffiffi þ 4:935 5 LH
ð15Þ
As Figs. 14 and 15 show, slug flows with shorter slugs provide higher heat transfer rates, and an increase in slug length leads to a decrease in total heat transfer rate. The fluctuations in the curves are due to fluid circulations inside the moving slugs, which are not
V. Talimi et al. / International Journal of Heat and Mass Transfer 62 (2013) 752–760
759
discussed that the hydrodynamics and heat transfer of slug flows might be simulated numerically using right angle computational domains (slugs), which are much simpler to be developed. Finally, dimensionless heat transfer has been reported for different slug lengths and Reynolds numbers and the results could be used in prediction of heat transfer rates, when there is a very thin liquid film around the gas phase or there is no film which is the case in hydrophobic microchannels. Acknowledgments
Fig. 14. Dimensionless heat transfer, qw, for moving liquid slugs for different slug lengths (Ls/Dh = 1, 2, 3, and 5) at same Reynolds numbers (Re = 100).
The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Auto21 NCE. The authors also acknowledge Amy R. Betz and Daniel Attinger for providing their experimental data [32]. Appendix A. Experimental data adoption
Fig. 15. Dimensionless heat transfer, qw, for moving liquid slugs for different slug lengths (Ls/Dh = 1, 2, 3, and 5) at same Reynolds numbers (Re = 500).
As mentioned in ‘‘introduction’’, experiments by Betz and Attinger [32] have been conducted under constant wall heat flux boundary condition. However, a constant wall temperature boundary condition was of interest in the present study, therefore, there was a need to adopt isoflux experimental data to isothermal condition. As mentioned, the microchannel wall material in Betz and Attinger [32] is highly conductive hence a small temperature gradient in axial direction is expected. Based on the aforementioned assumptions, a goal seek method has been applied to find wall temperatures (constant) which give same Nu numbers under same flow condition. Based on Leveque solution, the Nu number under constant wall temperature is [33]: 1
NuT ¼ 0:641 the case in the single phase slug flow. Furthermore, a slug flow approaches to the thermally saturated situation sooner than a single phase slug flow, which shows more heat removal from microchannel walls.
ðfReÞ3 L
ð16Þ
1 H3
where:
LH ¼
L Dh Pe
ð17Þ
and the local Nu number under constant wall heat flux is: 5. Conclusions
1
NuH ¼ 0:517 Heat transfer process under constant wall temperature boundary condition in two phase slug flows inside square microchannels has been considered in the present study. Individual moving slugs have been simulated numerically and results have been compared to experimental data provided by Betz and Attinger [32]. The numerical Nu numbers showed higher values but similar general trends compared with experimental Nu numbers. This difference requires more in-depth investigation. The film thickness might be responsible to decrease total heat transfer in a two phase slug flow inside microchannels. This is due to negative effects of liquid film around bubbles on thermal boundary layer zrenewal and amount of liquid in circulations inside moving slugs. This may show that heat transfer could be further increased in microchannel heat sinks using two phase slug flows under dry-out flow condition i.e. two phases are in direct contact with microchannel wall. This type of slug flow could be generated using hydrophobic or superhydrophobic materials in microchannels. The effects of Reynolds number, contact angle, and slug length on slug flow heat transfer have also been studied in the present work. It has been shown that Reynolds number, Re, has a significant effect on slug flow heat transfer due to a change in flow pattern and circulation shape in moving slugs, and with an increase in Reynolds number the heat transfer increases. Contact angle shows some small and negligible effects on slug flow heat transfer, and it has been
ðfReÞ3
ð18Þ
1
xH 3
where:
xH ¼
x Dh Pe
ð19Þ
If one uses the definition of Nu number, then:
NuH ¼
qw Dh kDT
ð20Þ
Combining Eqs. (18) and (20) the local wall to bulk temperature difference is: 1
DT ¼ 1:93
q w D h xH 3
ð21Þ
1
kðfReÞ3
which could be expanded using Eq. (18) as follows:
DT ¼ 1:93
qw Dh 1
kðfReÞ3
13 x Dh Pe
ð22Þ
Integrating the expression above over 0 ? L one can determine the mean wall to bulk temperature difference. In general, in the boundary layer region (thermal entrance length in internal flows) the bulk temperature does not vary significantly from inlet temperature, therefore (Tw Tm) approaches to (Tw Ti) as x ? 0.
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Based on the small values of Lw in the present study (see Fig. 10), it could be assumed that the boundary layer theory is applicable. Integrating DT using:
DT mean ¼
1 L
Z
L
DTdx
ð23Þ
0
gives: 1
NuH ¼ 0:689
ðfReÞ3 L
1 H3
ð24Þ
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