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Influence of texture shape and arrangement on thermo-hydraulic performance of the textured microchannels
International Journal of Thermal Sciences 147 (2020) 106146
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Influence of texture shape and arrangement on thermo-hydraulic performance of the textured microchannels Himani Sharma a, Anvesh Gaddam b, Amit Agrawal a, *, Suhas S. Joshi a, Stefan S. Dimov b a b
Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India Department of Mechanical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
A R T I C L E I N F O
A B S T R A C T
Keywords: Effective slip length Microchannel Nusselt number Pressure-driven flow Temperature jump length Superhydrophobic surfaces Textured surfaces
Textured superhydrophobic surfaces (TSS) are purported to reduce flow friction in microchannels due to velocity slip at liquid-gas interface. At the same time, the liquid-gas interface inhibits heat transfer in textured micro channels due to the low thermal conductivity of entrapped gas phase. Despite significant understanding on fluid flow and thermal transport on the TSS, the interplay of texture shape and arrangement on thermo-hydraulic performance has not been investigated in detail hitherto. To this end, we have numerically investigated the pressure-driven flow through textured microchannels with an aim to enhance the thermo-hydraulic performance. The effective slip length and temperature jump length were estimated as a function of flow and geometry pa rameters for three types of micropillar shapes viz., square, triangular and herringbone, decorated in micro channels in regular and staggered manner. Scaling relations for the effective slip length and temperature jump length have been shown to be valid for triangular and herringbone shaped micropillars at different flow and geometry related parameters. Herringbone shaped micropillars exhibit more flow friction and allow a significant heat transfer in microchannels within the parameter range investigated, followed by triangular and square shaped micropillars. Although the arrangement of textures in microchannels was found to affect the flow friction substantially, its effect on heat transfer was found to be marginal. Subsequently, the overall thermo-hydraulic performance was observed to be superior in regularly arranged herringbone shaped micropillars, at moderate to high constriction ratios (a ratio of texture pitch to half channel height) and high Peclet numbers over the other texture shapes. The results presented in this work would serve as a useful guide to attain maximum thermohydraulic performance in textured microchannels.
1. Introduction Textured superhydrophobic surfaces (TSS) have garnered much attention in the last decade due to their anti-wetting characteristics. Theoretical [1,2] and experimental [3–5] works have shown that a significant amount of reduction in hydrodynamic drag is possible in both internal and external flow conditions by employing the TSS. Particu larly, in microscale laminar flows, such as flow through microchannels, a substantial decrease in pumping power can be achieved by surface texturing. When a liquid flows over the TSS, a liquid-gas interface forms between the asperities as a result of the balance between surface tension and pressure forces. In such heterogeneous Cassie-Baxter state, liquid flowing over the TSS supposedly experiences low shear stress due to the velocity slip at the liquid-gas interface. If the TSS is supplied with heat flux, the liquid-gas interface inhibits heat transfer to the flowing liquid.
Therefore, in microchannel-based heat sinks, thermal management is essential besides reducing the power to drive liquids. Consequently, improving the thermo-hydraulic performance of the textured micro channels in laminar flow conditions is vital. The analytical works to understand the effect of the TSS on flow friction in microchannels in early stages, were restricted to onedimensional rib geometries [6–8]. The effective slip length was employed as a common parameter to define the performance of TSS of different texture geometries and scales. Equations were formulated to predict the effective slip length as a function of gas fraction, a ratio of the area covered by the liquid-gas interface to the total projected area, under the creeping flow conditions. These expressions were applicable for ribs arranged parallel and normal to the flow direction in microchannels. Subsequently, scaling relations were derived for the effective slip length for other generic texture types, such as square shaped micropillars and microholes [9,10]. Numerical works further extended the investigations
* Corresponding author. E-mail address: [email protected] (A. Agrawal). https://doi.org/10.1016/j.ijthermalsci.2019.106146 Received 22 April 2019; Received in revised form 8 September 2019; Accepted 15 October 2019 Available online 24 October 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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International Journal of Thermal Sciences 147 (2020) 106146
Nomenclature Roman A, B C, D AC ALG Br Ca DH H k L M Nu Pe Po Q qt Re T TC TS uS U Us u
v w X Y Z
constants of linear regression for hydrodynamic slip length constants of linear regression for thermal slip length area of the composite interface, m2 area of the liquid-gas interface, m2 Brinkman number Capillary number hydraulic diameter of the microchannel, m half of the microchannel height, m thermal conductivity of water, W/m.K texture pitch, m mass flow rate, kg.s 1 Nusselt number Peclet number Poiseuille number average heat flux provided to the domain, W/m2 heat flux to the top surface of micropillar, W/m2 Reynolds number temperature, K average temperature on composite interface, K average temperature on top surface of micropillar, K slip velocity on the composite interface, m.s 1 average velocity, m.s 1 normalized slip velocity on the composite interface streamwise velocity, m.s 1
spanwise velocity, m.s 1 interfacial normal velocity, m.s streamwise coordinate transverse coordinate interfacial normal coordinate
1
Abbreviations RS regular square RT regular triangular RH regular herringbone SS staggered square ST staggered triangular SH staggered herringbone TSS textured superhydrophobic surfaces Greek Symbols φ gas fraction ε constriction ratio λ/L effective slip length λT/L temperature jump length ρ density of the liquid, kg.m 3 μ viscosity of the liquid, N.s.m 2 α thermal diffusivity of the liquid, m2.s 1 η goodness index θ non-dimensional temperature for constant heat flux condition, k(T-Tm)/qtDH ΔP pressure drop, N/m2
of flow friction on the TSS to high Reynolds number (Re) flows, and also to different geometry-dependent parameters (e.g. texture pitch/mi crochannel height) [11,12]. For instance, the effective slip length was found to be Re-dependent in the case of micropillars, microholes and transverse ribs (ribs normal to the flow direction) in microchannels [13]. In general, the textures that provide continuous liquid-gas interfaces, such as longitudinal ribs (ribs along the flow direction) and micropillars produce larger effective slip lengths than the TSS involving microholes and transverse ribs. It was observed that repetitive occurrence of acceleration-deceleration cycles in the liquid flow causes under performance of the latter texture types than the former [14]. At the same time, the textures with continuous liquid-gas interfaces (e.g. longitudi nal ribs and micropillars) have also been shown to negate the detri mental effects of surfactant [15] and particle [16] contamination on slippage as compared to textures with discontinuous liquid-gas in terfaces (e.g. transverse ribs and microholes). However, owing to the very reason of continuity of the liquid-gas interfaces, wetting transition occurs quickly in the former texture types than the latter ones [17]. Consequently, it appears there exists a conflict between these two design criteria: maximum slippage and stability against wetting transition. Despite these inferences, Lam et al. [18] showed that longitudinal ribs perform better than other texture types for a given pressure difference at the liquid-gas interface, under non-inertial flow settings. On the other hand, temperature jump length is analogous to the effective slip length from thermal transport viewpoint. It is a common metric to determine the thermal transport on the TSS. A larger value of the temperature jump length in diabatic slip flows corresponds to a greater resistance to the convective heat transfer. At the same time, a larger value of the effective slip length indicates a lesser resistance to the fluid flow. Ng and Wang [19] formulated expressions for the tempera ture jump length for a variety of textures, such as square/circular shaped micropillars, square/circular shaped microholes, and ribs for constant temperature boundary condition aimed at diffusion-dominated flows. Enright et al. [20] derived an expression for Nusselt number for micropillars in a parallel plate microchannel supplied with constant heat
flux, as a function of effective slip length and temperature jump length under purely diffusive conditions. They also demonstrated that longi tudinal ribs exhibit better thermo-hydraulic performance followed by micropillars in the Stokes flow limit. Using an effective medium approach, Moreria and Bandaru [21] obtained an analytical solution of Nusselt number for flow over transverse ribs in a parallel plate micro channel, by incorporating effective thermal conductivity of the TSS. Similar to adiabatic slip flows, analytical works have also incorporated the effect of interface curvature on the convective heat transfer over the TSS in diabatic flows [22,23]. Overall, a microchannel with the TSS decrease the heat transfer as compared to a plain microchannel due to the decrease in effective solid-liquid contact area. Numerical works predominantly analysed the effect of inertia [24, 25], texture type and their arrangement [26–28] on thermal transport in microchannels for improving thermo-hydraulic performance. For example, thermo-hydraulic performance of a microchannel with trans verse ribs can be increased for offsetting their positions on top and bottom walls [27]. Cheng et al. [28] investigated the influence of generic texture types such as micropillars, microholes, transverse and longitudinal ribs, on thermo-hydraulic performance under constant temperature boundary condition. They found that thermo-hydraulic performance of the square shaped micropillars surpasses the other texture types, at high gas fractions and high Reynolds numbers. While the experimental works in this domain are limited, a recent study has shown that the thermo-hydraulic performance can be enhanced, when the liquid-gas interface is nearly flat rather instead of protruding in/out of the gas cavity [29]. In summary, a considerable amount of research has been undertaken so far to understand the influence of TSS on both athermal and thermal transport in microchannels. The critical parameters that influence the flow and thermal resistance under the non-wetting conditions include texture type, arrangement, liquid-gas interface curvature, texture pitch/ microchannel height and Reynolds/Peclet number. The interplay be tween these parameters has shown to affect the thermo-hydraulic per formance of textured microchannels. Overall, two generic texture types 2
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standout in terms of thermo-hydraulic performance: longitudinal ribs and micropillars. Since, non-linear convective acceleration terms are absent in the governing equation for the fluid momentum for longitu dinal ribs (neglecting thermocapillary stresses), the effect of inertia on the flow and thermal resistance is insignificant for this texture type [13, 28]. Previous attempt to manipulate the flow field by tailoring the arrangement of longitudinal ribs in a microchannel has not yielded any positive effects [30]. However, inertial effects play an important role in athermal and thermal transport in the case of micropillars. At the same time, tailoring the shape/arrangement of longitudinal ribs is difficult due to their one-dimensional nature. On the other hand, bi-dimensional textures (e.g. micropillars and microholes) allow manipulation on flow field inside microchannels based on their shape/arrangement. In fact, several efforts have been made to understand the influence of micro pillars arrangement (e.g. regular, staggered and random) on thermal transport [26], flow friction [26,31] and interfacial stability [31] in microchannels. It is apparent from the current understanding that ve locity and temperature fields in a microchannel are influenced by bi-dimensional textures. Therefore, micropillars could have better advantage than longitudinal ribs to enhance the thermo-hydraulic per formance of microchannels. In the light of above discussion, it is apparent that the design and optimization of surface textures is necessary to enhance the thermohydraulic performance of microchannels. It also appears from the literature that the effect of non-generic texture shapes on flow and thermal transport has not been investigated thus far. Therefore, the main focus of this paper is on understanding the effect of micropillar shape and its arrangement in microchannels, on flow friction and thermal transport. The numerical simulations were performed for a pressuredriven flow through microchannels containing the TSS that were pro vided with a constant heat flux. Micropillars of different shapes viz.,
square, triangular and herringbone, arranged in either regular or stag gered manner, were considered in this study. The inspiration to use particular non-generic shapes such as triangular and herringbone came from their usage in enhancing single-phase heat transfer in micro channel heat sinks [32]. As discussed earlier, the sparsely spaced micropillars give larger effective slip lengths in microchannels. There fore, only micropillars with high gas fractions (>70%) have been investigated in this work. The effective slip length and temperature jump length were estimated for these TSS as a function of gas fraction, microchannel constriction and Reynolds number/Peclet number. This has helped comprehend the significance of flow and geometry param eters on fluid flow and thermal transport. Lastly, the goodness index was evaluated to compare the thermo-hydraulic performance of the various textured shapes considered here. 2. Methodology A steady, laminar, incompressible and pressure-driven flow through a parallel plate microchannel, with top and bottom walls decorated with micropillars, has been considered in this work. The flow is assumed to exist in the Cassie-Baxter state with the liquid-gas interface appearing to be flat between successive micropillars. Three types of micropillar shapes viz., square, triangular and herringbone have been investigated (see Fig. 1 a-c). The micropillars were arranged either in regular or staggered fashion, on the top and bottom walls of microchannel as illustrated in Fig. 1 d-e. The regions highlighted in Fig. 1 d-e are considered for the numerical simulations, due to the symmetry and periodicity of the micropillar arrangements in the microchannels, along Y/Z- and X-directions, respectively. A typical computational module for the regular arrangement of square shaped micropillars inside a micro channel is schematically depicted with appropriate nomenclature and
Fig. 1. (a–c) Micropillar shapes considered in this study are shown. a. Square b. Triangular and c. Herringbone. (d–e) An illustration of the herringbone shaped micropillars in a microchannel considered in this study is shown in d. regular and e. staggered arrangements. The regions highlighted in dashed lines represent computational area in the present numerical simulations. f. The computation module for a square shaped micropillar in a regular arrangement is shown. 3
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International Journal of Thermal Sciences 147 (2020) 106146
boundary conditions in Fig. 1f.
ε¼
2.1. Mathematical formulations
L H
(5)
ρUDH μ
(6)
DH U
(7)
The following continuity, momentum and energy equations, Eq. (1) – (3), are applicable to the flow conditions inside the microchannel.
Re ¼
(1)
Pe ¼
(2)
where, ALG is area of the liquid-gas interface inside the domain, AC is area of the composite interface inside the domain, L is texture pitch, H is half of the microchannel height, ρ is density of the liquid, μ is viscosity of the liquid, α is thermal diffusivity of the liquid and U is average axial velocity inside the microchannel. The typical dimension of texture pitch considered is 50 μm. Consequently, the hydraulic diameter (DH ¼ 4H) of the microchannels range from 40 μm to 400 μm, corresponding to the parameter space of constriction ratio (ε: 0.25–5). In should be noted that the simulations were conducted with the assumption that liquid flow exists under the Cassie state and the airwater interface is considered to be flat. These assumptions are valid under the circumstances where capillary number (Ca, a ratio of viscous forces to the surface tension forces) for system is much less than unity. Based on the characteristic flow velocities through the microchannels of different constriction ratios for a Reynolds number of 1000, viscosity of the water (~0.001 Pa s) and surface tension of the air water interface (0.072 N/m), the Ca values are calculated. In all the cases, it was esti mated that Ca << 1. Therefore, it is reasonable to assume that surface tension plays a fairly dominant role tin maintaining the liquid-gas interface to be flat for the parameter range investigated in our simula tions. However, it is expected that modest deformation would take place at high constrictions ratios in the high-Re regime in practical situations. Furthermore, the zero-shear stress condition at the liquid-gas interface overpredicts the effective slip length as shown in previous works [11, 12]. However, the numerical simulations showed that the percentage change in the effective slip length between the viscosity contrast of 0.02 (e.g. air-water interface) and 0 is about 5% for micropillars arranged at a gas fraction of 0.75 [33]. Since this difference is not so significant and at the same time, this study focuses on comparing different texture shapes and arrangements on the frictional and thermal characteristics, it is reasonable to assume that the liquid-gas interface is shear-free. At the same time, we have also estimated the value of Brinkman number (Br), which characterizes the viscous dissipation in flow through micro channels. It is the ratio of heat generated due to the viscous dissipation and heat exchanged at the wall. The Brinkman number in our simula tions is estimated to be much less than unity (Br ≪1) for flow through microchannels with low constriction ratios (ε < 3). However, the value Brinkman number was found to be only slightly less than unity (Br < 1) for flow through microchannels with high constriction ratios (ε > 3). Therefore, it is reasonable to neglect viscous dissipation affects due to the fact that Br < 1 and also the simulations are performed with an aim to compare different texture shapes and their arrangement.
r:u ¼ 0
ρðu:rÞu ¼
rP þ μr2 u
ρcp u:rT ¼ kr2 T
(3)
where, the velocity field u ¼ [u, v, w] and temperature field T corre spond to the spatial field S ¼ [X, Y, Z]. The working liquid considered in this study is water with constant thermo-physical properties (thermal conductivity, k ¼ 0.6 W/m.K and viscosity, μ ¼ 0.001 Pa s). The geome tries of the computational module were discretized in GAMBIT and the above-mentioned coupled equations were solved numerically in a commercial computational framework, Ansys Fluent 16.0. The liquidgas interface was approximated as a shear-free boundary and adia batic (∂u/∂Z ¼ ∂v/∂Z ¼ w ¼ ∂T/∂Z ¼ 0), while the no-slip boundary condition (u ¼ v ¼ w ¼ θ ¼ 0) was imposed on the top surface of the highly conducting micropillar. Periodic boundary conditions (uin ¼ uout, vin ¼ vout, win ¼ wout, θin ¼ θout) were implemented at the inlet and outlet of the computational module. This was due to the periodically repetitive velocity and temperature fields inside the microchannel for fully developed hydrodynamic and thermal flow conditions. Symmetry boundary conditions (top surface: ∂u/∂Z ¼ ∂v/∂Z ¼ w ¼ ∂T/∂Z ¼ 0; side surfaces: ∂u/∂Y ¼ ∂v/∂Y ¼ v ¼ ∂T/∂Y ¼ 0) were employed on top and sides of the computational module, due to the mirror symmetry along Yand Z-directions, inside the microchannel. The abrupt changes in the velocity and temperature are expected on the TSS due to the change of boundary condition from no-slip to shearfree. As a consequence, the region near the composite interface (solid/ liquid/gas interface at Z ¼ 0) was locally refined before initializing the simulation. At the inlet, the mass flow rate corresponds to the desired Reynolds number. Initially, the computational domain was solved for the velocity field. Once the continuity and velocity residuals reached 10 4, a velocity gradient-based adaption method was used to obtain the grid independent solution. This method refines the grid in the vicinity of the highest velocity gradients in the computational domain based on a user-defined threshold. The refinement process continued until the continuity and velocity residuals reach 10 9. Once the velocity field was obtained, the computational domain was solved for the temperature field, by delivering heat flux (qt) to the top surface of the micropillar. Initially, the temperature field was obtained until the energy residuals reached 10 4. Thereafter, a temperature gradient-based adaption method was implemented to refine the grid for obtaining a gridindependent solution, until the energy residuals reach 10 8. Although the gradient-based mesh adaption for temperature field was employed over the adapted mesh for hydrodynamic solution, it has not yielded any significant change in number of cells except for a very few cases. The gradient-based grid adaption method used here, considerably reduces the computational time and resources, as discussed elsewhere [11,12, 16,25]. The final number of cells was ranged from 100,000 to 1,000,000 in our simulations after the adaption. Also, the user-defined threshold for gradient-based adaption of velocity and temperature fields was chosen between 10 and 30% in our simulations. It should also be noted that the total heat flux (q ¼ φqt) provided to the computational domain was kept constant in all the simulations. The relevant dimensionless geometric and flow quantities such as gas fraction (φ), constriction ratio (ε), Reynolds number (Re) and Peclet number (Pe) used in this study are defined as follows: φ¼
ALG AC
α
2.2. Calculation of flow parameters The pressure gradient (ΔP/L) across the computational module ob tained from the numerical simulations has been converted into Pois euille number (Po) using Eq. (8). Subsequently, the effective slip length (λ/L) was estimated from the Poiseuille number using Eq. (9) [16]. Po ¼
D3H ρΔP m_ μL
� λ 1 32 ¼ � L ε Po
(8) � 1 3
(9)
Here, m is the mass flow rate at the inlet of the microchannel. Similarly, the temperature jump length (λT/L) was evaluated using Eq.
(4) 4
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International Journal of Thermal Sciences 147 (2020) 106146
(10). This equation in turn is employed to calculate the Nusselt number (Nu) for the convective heat transfer inside the computational module using Eq. (11) [25]. λT k ¼ ðTS L qt Nu ¼
(10)
TC Þ
� λT ε 17 þ L 4 140
3 1 US þ US 2 70 210
�
1
(11)
Here, k is the thermal conductivity of the liquid, TS is the tempera ture on top surface of the micropillar, TC is the temperature of the composite interface (solid-liquid-gas interface at Z ¼ 0), Us ¼ (uS/U) is the normalized slip velocity on the composite interface and uS is slip velocity on the composite interface. It is apparent that the Nu reduces to 140/17 in the case of flow through a plain channel. It should be noted that the inexact form of Eq. (11) is analogous to the one presented in some previous works [20]. The goodness factor of plain and textured microchannels can be expressed by (NuP/PoP.Pr1/3) and (Nu/Po.Pr1/3), respectively [27,28]. Here, NuP (¼8.235) and PoP (¼ 96) correspond to Nusselt number and Poisueille number for a plain parallel plate microchannel, whereas Nu and Po correspond to Nusselt number and Poiseuille number for a textured microchannel, with Pr being the Prandtl number of the fluid. Consequently, the thermo-hydraulic performance of microchannels with the TSS with respect to a plain microchannel has been evaluated in terms of goodness index (η) using Eq. 12
η¼
Nu=Po NuP =PoP
Fig. 2. Scaling of the effective slip length and temperature jump length for square shaped micropillars in regular arrangement inside a microchannel with ε ¼ 0.01 at Re ¼ 0.1.
3. Results and discussion In the subsequent sections, we will discuss the influence of liquid inertia, microchannel constriction and gas fraction, on the flow friction and convective heat transfer, inside microchannels containing micro pillars of different shapes, arranged in either regular or staggered manner. The six types of TSS investigated are: regular square (RS), regular triangular (RT), regular herringbone (RH), staggered square (SS), staggered triangular (ST) and staggered herringbone (SH). The range of flow and geometrical parameters considered for this study are: 0.75 � φ � 0.98, 0.25 � ε � 5, 1 � Re � 1000 and 7 � Pe � 7000.
(12)
2.3. Validation of numerical procedure Ybert et al. [9] derived scaling relations for the effective slip length for a shear-driven flow in the limit of Stokes flow, for a surface sparsely covered with square shaped micropillars (φ> 70%) in a regular manner. Eq. (13) gives the scaling law for the effective slip length for this sce nario, where A, B are constants. At the same time, Enright et al. [20] provided scaling relation for the temperature jump length for the similar conditions for regularly arranged micropillars supplied with constant heat flux. Eq. (14) gives the scaling law for the temperature jump length for this case, where, C, D are constants. λ ¼ Að1 L λT ¼ Cð1 L
φÞ
φÞ
1=2
1=2
B
D
3.1. Scaling laws We begin our analysis by examining the validity of scaling laws for the effective slip length and temperature jump length. The scaling laws, as explained earlier, were originally derived for creeping shear-driven flows past TSS (see Eq. (13)). Previous works have successfully extended their validity to inertial pressure-driven flows through textured microchannels [13]. In our earlier works, we have investigated the scaling laws for even particle-laden liquid-gas interfaces, finite aspect ratio microchannels, and liquid-infused textured surfaces [16,33, 34]. Here, we examine scaling laws for micropillars of different shapes that are arranged in a regular manner (RS, RT and RH) in microchannels having different constrictions, in inertial pressure-driven flows. Fig. 3 shows plots of the effective slip length as a function of gas fraction, for two different microchannel constrictions (ε ¼ 1, 5) and Reynolds numbers (Re ¼ 1, 1000). It appears that a linear variation of effective slip length is observed with the parameters for all the micropillar shapes investigated here. This indicates that the scaling laws are valid for different texture shapes. Furthermore, the effective slip length was found to be less in herringbone shaped micropillars, followed by triangular and square
(13) (14)
In order to validate the numerical procedure adopted in this work, simulations were performed using microchannels with square shaped micropillars that are arranged in a regular manner on top and bottom walls with high gas fractions (φ ¼ 75%, 90%, 95% and 98%). The effective slip length and temperature jump length were evaluated in microchannels with a large separation between the top and bottom walls (ε ¼ 0.01) at Re ¼ 0.1, to mimic the creeping shear-driven flow condi tions. A plot of the numerically obtained effective slip length and tem perature jump length at different gas fractions is shown in Fig. 2, which depicts a linear relationship with the function of gas fraction. The con stants of this linear relationship of the effective slip length and tem perature jump length were obtained by regression analysis, and are given in Table-1 and Table-2, respectively. These constants (A, B, C and D), obtained through numerical simulations, were compared with the results in literature under different flow and geometrical conditions as shown in Table-1 and Table-2. Based on the comparison, it appears that the numerical procedure followed in this work is valid for the different scenarios presented below.
Table 1 A comparison of values of coefficients in Eq. (13) obtained in the present work with the literature.
5
Reference
A
B
Conditions
Present work Ybert et al. [9] Davis and Lauga [10]
0.335 0.325 0.332
0.425 0.44 0.42
Re ¼ 0.1, ε ¼ 0.01 – –
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International Journal of Thermal Sciences 147 (2020) 106146
numbers are shown in Fig. 4. The plot of temperature jump length re sembles that of the effective slip length. It also exhibits a linear rela tionship, thus extending the validity of the scaling laws to different texture shapes. Further, it is evident that the temperature jump length is lower for herringbone shaped micropillars, followed by triangular and square shaped micropillars. This implies that the herringbone shaped micropillars enhance the convective heat transfer in microchannels. Furthermore, an increase in Peclet number enhances the convective heat transfer as evident by the decrease in the temperature jump length; see Fig. 4 b, d. At the same time, the effect of Peclet number on the tem perature jump length is increased with an increase in microchannel constriction. As explained earlier, the constants (C and D) of Eq. (14) correspond to the cases shown in Fig. 4, were evaluated through regression analysis and are presented in Table-4. The correlation coef ficient in the regression analysis is more than 99.5% for all the cases presented here.
Table 2 A comparison of values of coefficients in Eq. (14) obtained in the present work with the literature. Reference
C
D
Conditions
Present work Enright et al. [20] Cowley et al. [25]
0.465 0.471 0.469
0.599 0.603 0.593
Re ¼ 0.1, ε ¼ 0.01 Re < 1, ε ¼ 0.0125 Re ¼ 0.14, ε ¼ 0.04
shaped micropillars. Therefore, it is evident that the shape of the micropillars has a substantial influence on the flow friction in the microchannels. In addition, the liquid inertia considerably decreases the effective slip length for all the micropillar shapes. The constants (A and B) of Eq. (13) for the square, triangular and herringbone shaped micropillars for the scenarios simulated here, were evaluated through regression analysis, as listed in Table-3. For similar conditions as above, numerical simulations were also performed to obtain scaling relations for the temperature jump length, when a constant heat flux is delivered to the micropillars. The plots of the temperature jump length as a function of gas fraction for micro pillars of different geometries, microchannel constrictions and Peclet
3.2. Influence of liquid inertia It is apparent from the previous section that the values of the
Fig. 3. (a–d) Scaling of the effective slip length for square, triangular and herringbone shaped micropillars at different constriction ratios and Reynolds numbers. 6
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International Journal of Thermal Sciences 147 (2020) 106146
Fig. 4. (a–d) Scaling of the temperature jump length for square, triangular and herringbone shaped micropillars at different constriction ratios and Rey nolds numbers. Table-3 Numerically evaluated values of coefficients in Eq. (13) for three micropillar shapes at different microchannel constrictions and Reynolds numbers. Microchannel constriction
ε¼1 ε¼5
Texture shape Square Triangular Herringbone Square Triangular Herringbone
Re ¼ 1
Table-4 Numerically evaluated values of coefficients in Eq. (14) for three micropillar shapes at different microchannel constrictions and Reynolds numbers.
Re ¼ 1000
A
B
A
B
0.348 0.338 0.292 0.342 0.339 0.316
0.484 0.487 0.403 0.467 0.494 0.488
0.233 0.225 0.214 0.184 0.174 0.167
0.282 0.293 0.304 0.219 0.231 0.244
Microchannel constriction
ε¼1 ε¼5
effective slip length and temperature jump length reduce significantly with an increase in Reynolds number for all the micropillar shapes. Consequently, we investigated the effect of inertia on flow friction and convective heat transfer in the textured microchannels. Fig. 5 shows a variation in the effective slip length with Reynolds numbers, for all the textures shapes, arranged either in regular or in staggered manner for a
Texture shape Square Triangular Herringbone Square Triangular Herringbone
Re ¼ 1
Re ¼ 1000
C
D
C
D
0.473 0.451 0.431 0.463 0.438 0.425
0.595 0.519 0.575 0.527 0.447 0.557
0.185 0.190 0.174 0.098 0.077 0.072
0.130 0.228 0.263 0.154 0.101 0.117
gas fraction of 0.75, and at different microchannel constrictions. The effective slip length is observed to decrease gradually with an increase in Reynolds number for all the micropillar shapes in regular arrangement. However, the influence of inertia on flow friction is more pronounced at large microchannel constrictions; see Fig. 5a, c. This is due to the fact that flow field will be greatly influenced when top and bottom walls are 7
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International Journal of Thermal Sciences 147 (2020) 106146
Fig. 5. (a–d) Effective slip length as a function of Reynolds number for square, triangular and herringbone shaped micropillars at different constriction ratios and in regular and staggered configurations.
close to each other [11,12,33]. On the other hand, the staggered arrangement enacts minor influence on the effective slip length for herringbone shaped micropillars, whereas the square and triangular micropillars, in the staggered manner, exhibit more flow friction at high Reynolds numbers as shown in Fig. 5b. At the same time, the influence of inertia on the effective slip length for micropillars in staggered arrangement is shown with an increase in microchannel constriction (see Fig. 5d). Interestingly, the inertial effects appear to nullify the dif ference in the performance of micropillars in staggered arrangement, at large microchannel constrictions. Fig. 6 shows plots of the temperature jump length with respect to Reynolds number for micropillars in regular and staggered arrange ments, at different microchannel constrictions. The behaviour of tem perature jump length is analogous to the effective slip length. It decreases gradually with an increase in Reynolds number, for all the micropillars shapes in regular arrangement. It indicates an increase in the convective heat transfer with the inertia as shown in Fig. 6 a, c. The apparently large difference between the values of temperature jump length for herringbone and square shaped micropillars, at low and high
microchannel constrictions, signify higher heat transfer characteristics on the former than the latter. At the same time, the microchannel constriction influences more on the triangular and herringbone shaped micropillars as compared to square shaped micropillars, in the regular arrangement. It is also apparent from Fig. 6c and d that the micropillar shape exhibits marginal effect on the convective heat transfer in microchannel, in the staggered arrangement. The texture arrangement has a far more profound influence on the heat transfer characteristics of only square shaped micropillars, than on the flow friction. On the other hand, the triangular and herringbone shaped micropillars appear to be relatively unaffected, as evident from Fig. 6. The surface temperature profiles on the TSS in Fig. 7 also verify that a drastic variation in the temperature distribution for square shaped micropillars occurs between regular and staggered arrangements. It may be noted that the minimum and maximum temperatures on the com posite surface were normalized between 0 and 1. It is also observed that the micropillar shape has minimal influence on the temperature distri bution as discussed earlier. Furthermore, the triangular micropillars fared worse in enhancing the convective heat transfer in microchannels 8
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Fig. 6. (a–d) Temperature jump length as a function of Peclet numbers for square, triangular and herringbone shaped micropillars at different constriction ratios and in regular and staggered configurations.
in the low inertia conditions (Re < 10). Overall, the herringbone shaped micropillars outperformed other shapes over the parameter range investigated in this work.
distance between the upper and lower walls in the microchannels. This indicates that, the interaction of flow field between upper and lower walls of the textured microchannels intensifies with a decrease in dis tance between them and magnifies with the aid of fluid inertia. Conse quently, the temperature jump length is also obeserved to be a function of microchannel constriction only at high Peclet numbers, whereas, it remains relatively unchanged both in regular and staggered arrange ments as shown in Fig. 9. This also implies that, the texture arrangement has only a marginal influence on flow and heat transfer characteristics at low inertia conditions. On the other hand, the texture arrangement greatly affects the flow friction in inertial flows, while the heat transfer characteristics remain unchanged for the same settings.
3.3. Influence of microchannel constriction In the previous sections, it was noticed that microchannel constric tion affects both the effective slip length and temperature jump length in textured microchannels. Here we examine the role of varying the dis tance between upper and lower walls of the textured microchannels on the flow friction and heat transfer characteristics. Fig. 8 shows the effective slip length plotted as a function of microchannel constriction, for micropillars in regular and staggered arrangements, at different Reynolds numbers. It is observed that the effective slip length for all micropillar shapes both in regular and staggered arrangements is a function of microchannel constriction, only at high Reynolds number. Furthermore, the inertia comes into effect and plays a dominant role with an increase in microchannel constriction, i.e., by decreasing the
3.4. Thermo-hydraulic performance of the TSS So far it is noted that both flow friction and convective heat transfer reduce with the aid of the TSS in microchannels. Therefore, a metric to compare the effectiveness of the TSS-based microchannels with respect 9
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Fig. 7. Surface temperature profiles on square, triangular and herringbone shaped micropillars in regular and staggered arrangements with a gas fraction of 0.75 for ε ¼ 5 and Pe ¼ 7000.
to the plain microchannels is necessary. The goodness index (η) as explained by Eq. (12) in section 2, gauges the combined hydraulic and thermal performance of the TSS-based microchannels. A value of goodness index greater than unity signifies beneficial effect of intro ducing the TSS in the microchannels. Fig 10a shows a plot of goodness index with the function of gas fraction for the micropillars arranged in staggered manner at different microchannel constrictions for highly inertial flows. The goodness index steadily increases with the gas frac tion at low microchannel constrictions for all the micropillar shapes. At the same time, it is also noticed that the texture shape has only minor influence on the goodness index at low constriction ratios. However, the thermo-hydraulic performance is found to increase rapidly with the gas fraction and reaches a plateau after a gas fraction of 0.95 for all the micropillar shapes at high microchannel constrictions. Interestingly, the performance of the non-generic texture shapes (herringbone and trian gular) is much higher than the generic texture shape (square). Furthermore, the herringbone shaped micropillars in staggered arrangement exhibit a goodness index value greater than 3, which is far higher than what has been achieved in the literature so far. Despite the high flow friction, 80–100% improvement in the goodness index is observed due to herringbone shaped micropillars over the square shaped
micropillars. It has been previously shown that texture orientation could generate secondary flows transverse to the flow direction in micro channels, thereby improve fluid mixing [35] and particle migration [36]. In a similar vein, herringbone and triangular shaped micropillars could produce transverse flow due to their no-slip/shear-free boundaries positioned oblique to the flow direction. In order to appreciate this effect on temperature field, the velocity field normal to the flow direction is visualized for herringbone and square shaped micropillars. Fig. 10b shows evolution of transverse velocity vectors (v, w) along the flow di rection for herringbone and square shaped micropillars. Apparently, herringbone shaped micropillars are exhibiting complex flow patterns in lateral direction, thus aiding in significant heat transfer in micro channels. This signifies the importance of texture shape optimization in enhancing the thermo-hydraulic performance. As explained in previous sections, the herringbone shaped micro pillars significantly outclass square shaped micropillars in increasing the convective heat transfer and flow friction. However, the substantial ef fects are realized only at high constriction ratios. Fig. 10b shows a plot of goodness index with the microchannel constriction for square and herringbone shaped micropillars in regular and staggered arrangements, for a gas fraction of 0.95 at Pe ¼ 7000. In all the cases, the benefit of 10
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Fig. 8. (a–b) Effective slip length as a function of constriction ratio for square, triangular and herringbone shaped micropillars at different Reynolds numbers and in regular and staggered configurations.
Fig. 9. (a–b) Temperature jump length as a function of constriction ratio for square, triangular and herringbone shaped micropillars at different Peclet numbers and in regular and staggered configurations.
arrangements. In line with the previous observations, micropillar arrangement has not shown any significant effect on goodness index at low constriction ratios. However, regularly arranged herringbone sha ped micropillars have shown better performance than those in staggered arrangement for Re > 200. After understanding the behaviour of different micropillar shapes, we now compare their performance with longitudinal ribs. As explained in section 1, longitudinal ribs have shown to exhibit lesser flow friction than the other generic texture types, thereby demonstrating better thermo-hydraulic performance [20]. Consequently, numerical simula tions performed for microchannels containing longitudinal ribs with the settings as described in section 2. Fig. 11b shows the plot of goodness index as a function of Reynolds number for longitudinal ribs and
tailoring shape/arrangement is only realizing at moderate-high constriction ratios (ε > 1). This also indicates that, the transverse flow patterns are intense only for ε > 1, so as to produce meaningful benefits. At the same time, the regularly arranged herringbone shaped micro pillars enhance the thermo-hydraulic performance of the microchannels as compared to those in staggered arrangement. That means, the flow friction is becoming overwhelmingly high in staggered arrangement at high flow rates. However, the similar behaviour is not noticed in the case of square shaped micropillars. This implies that the texture arrangement plays a crucial role in determining the thermo-hydraulic performance of the microchannels. To understand the influence of inertia on thermohydraulic performance, goodness index is plotted against Reynolds number (Fig. 11a) for herringbone shaped micropillars in different 11
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Fig. 10. a. Goodness index scaling for square, triangular and herringbone shaped micropillars at different constriction ratio in staggered configuration. b. Transverse velocity field (v, w) showing velocity vectors along the flow direction for herringbone and square shaped micropillars. c. Goodness factor as a function of constriction ratio for square and herringbone shaped micropillars in regular and staggered configurations corresponding to φ ¼ 0.95.
different micropillar shapes for φ ¼ 0.75 and ε ¼ 5. It is noted that, longitudinal ribs have performed better than all micropillar shapes in the low Re-regime (Re < 10). However, the goodness index of herring bone shaped and triangular micropillars has dramatically increased with the fluid inertia. Particularly, the fluid inertia has come into effect at Re > 10 (herringbone shaped) and Re > 100 (triangular) for these micropillar shapes in order to better their thermo-hydraulic perfor mance over longitudinal ribs. At the same time, square shaped micro pillars and longitudinal ribs have exhibited similar performance over the entire Re-regime. As noted earlier, the oblique no-slip/shear-free boundaries present in the non-generic texture shapes instigate the
cross flow patterns in the flow field with the aid of inertia. These ob servations emphasize the importance of tailoring texture shapes in improving the thermo-hydraulic performance of microchannels. 4. Conclusions In this work, the pressure-driven flow through microchannels con taining the TSS supplied with constant heat flux was numerically simulated. The micropillar textures with three different shapes such as square, triangular and herringbone that are arranged either in regular or staggered manner, on top and bottom walls of the microchannels were 12
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International Journal of Thermal Sciences 147 (2020) 106146
Fig. 11. a. Goodness index as a function of Reynolds number for herringbone shaped micropillars in regular and staggered configurations at different constriction ratios for φ ¼ 0.75. b. Comparison of goodness index for different micropillar shapes with longitudinal ribs arranged with φ ¼ 0.75 at ε ¼ 5.
investigated. The main conclusions drawn from this work are as follows:
References
1. Scaling relations for the effective slip length and temperature jump length, originally derived for the square shaped micropillars in the Stokes flow limit, were found to correlate well with other micropillar shapes such as triangular and herringbone. Thus, the validity of the scaling laws was extended to intricate geometries and inertial pressure-driven flows. 2. A discernible difference in the values of the effective slip length and temperature jump length was noticed among the micropillar shapes in the regular arrangement. Whereas, the difference was found to be negligible in the staggered arrangement with an increase in Reynolds number and microchannel constriction. This implies that the effec tive slip length and temperature jump length are rather weak func tions of micropillar shape in the staggered arrangement. 3. An increase in the Reynolds number and microchannel constriction leads to an increase in flow friction and an increase in the convective heat transfer, for all the micropillar shapes in both regular and staggered arrangements. However, the micropillar arrangement in side the microchannel has substantial influence on the flow friction, whereas its effect on the convective heat transfer was minor. 4. Overall, the herringbone shaped micropillars exhibits more flow friction and less thermal resistance. It is followed by triangular and square shaped micropillars respectively, in both regular and stag gered arrangements. 5. The influence of transverse flow has led to improvement in thermohydraulic performance of the non-generic texture shapes over the generic texture shapes. However, the positive benefits of this trans verse flow are realized only at moderate-high constriction ratios and high Reynolds/Peclet numbers.
[1] J.R. Philip, Flows satisfying mixed no-slip and no-shear conditions, Z. Angew. Math. Phys. 23 (3) (1972) 353–372. [2] S.C. Hendy, M. Jasperse, J. Burnell, Effect of patterned slip on micro-and nanofluidic flows, Phys. Rev. E 72 (1) (2005), 016303. [3] J. Ou, B. Perot, J.P. Rothstein, Laminar drag reduction in microchannels using ultrahydrophobic surfaces, Phys. Fluids 16 (12) (2004) 4635–4643. [4] R. Truesdell, A. Mammoli, P. Vorobieff, F. van Swol, C.J. Brinker, Drag reduction on a patterned superhydrophobic surface, Phys. Rev. Lett. 97 (4) (2006), 044504. [5] P. Muralidhar, N. Ferrer, R. Daniello, J.P. Rothstein, Influence of slip on the flow past superhydrophobic circular cylinders, J. Fluid Mech. 680 (2011) 459–476. [6] E. Lauga, H.A. Stone, Effective slip in pressure-driven Stokes flow, J. Fluid Mech. 489 (2003) 55–77. [7] C.J. Teo, B.C. Khoo, Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves, Microfluid. Nanofluidics 7 (3) (2009) 353. [8] C.O. Ng, C.Y. Wang, Stokes shear flow over a grating: implications for superhydrophobic slip, Phys. Fluids 21 (1) (2009), 087105. [9] C. Ybert, C. Barentin, C. Cottin-Bizonne, P. Joseph, L. Bocquet, Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries, Phys. Fluids 19 (12) (2007) 123601. [10] A.M. Davis, E. Lauga, Hydrodynamic friction of fakir-like superhydrophobic surfaces, J. Fluid Mech. 661 (2010) 402–411. [11] J. Davies, D. Maynes, B.W. Webb, B. Woolford, Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs, Phys. Fluids 18 (8) (2006), 087110. [12] D. Maynes, K. Jeffs, B. Woolford, B.W. Webb, Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction, Phys. Fluids 19 (9) (2007), 093603. [13] Y.P. Cheng, C.J. Teo, B.C. Khoo, Microchannel flows with superhydrophobic surfaces: effects of Reynolds number and pattern width to channel height ratio, Phys. Fluids 21 (12) (2009) 122004. [14] A. Gaddam, M. Garg, A. Agrawal, S.S. Joshi, Modeling of liquid–gas meniscus for textured surfaces: effects of curvature and local slip length, J. Micromech. Microeng. 25 (12) (2015) 125002. [15] F.J. Peaudecerf, J.R. Landel, R.E. Goldstein, P. Luzzatto-Fegiz, Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces, Proc. Natl. Acad. Sci. 114 (28) (2017) 7254–7259. [16] A. Gaddam, A. Agrawal, S.S. Joshi, M.C. Thompson, Slippage on a particle-laden liquid-gas interface in textured microchannels, Phys. Fluids 30 (3) (2018), 032101. [17] V. Bahadur, S.V. Garimella, Preventing the Cassie Wenzel transition using surfaces with noncommunicating roughness elements, Langmuir 25 (8) (2009) 4815–4820. [18] L.S. Lam, M. Hodes, R. Enright, Analysis of galinstan-based microgap cooling enhancement using structured surfaces, J. Heat Transf. 137 (9) (2015), 091003. [19] C.O. Ng, C.Y. Wang, Temperature jump coefficient for superhydrophobic surfaces, J. Heat Transf. 136 (6) (2014), 064501. [20] R. Enright, M. Hodes, T. Salamon, Y. Muzychka, Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels, J. Heat Transf. 136 (1) (2014), 012402.
Acknowledgments The research reported in this paper is within the framework of the UKIERI-DST programme “Surface functionalisation for food, packaging, and healthcare applications”.
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[21] D. Moreira, P.R. Bandaru, Thermal transport in laminar flow over superhydrophobic surfaces, utilizing an effective medium approach, Phys. Fluids 27 (5) (2015), 052001. [22] L.S. Lam, M. Hodes, G. Karamanis, T. Kirk, S. MacLachlan, Effect of meniscus curvature on apparent thermal slip, J. Heat Transf. 138 (12) (2016) 122004. [23] T.L. Kirk, M. Hodes, D.T. Papageorgiou, Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature, J. Fluid Mech. 811 (2017) 315–349. [24] D. Maynes, B.W. Webb, J. Crockett, V. Solovjov, Analysis of laminar slip-flow thermal transport in microchannels with transverse rib and cavity structured superhydrophobic walls at constant heat flux, J. Heat Transf. 135 (2) (2013), 021701. [25] A. Cowley, D. Maynes, J. Crockett, Inertial effects on thermal transport in superhydrophobic microchannels, Int. J. Heat Mass Transf. 101 (2016) 121–132. [26] M. Kharati-Koopaee, M.R. Akhtari, Numerical study of fluid flow and heat transfer phenomenon within microchannels comprising different superhydrophobic structures, Int. J. Therm. Sci. 124 (2018) 536–546. [27] W. Ren, Y. Chen, X. Mu, B.C. Khoo, F. Zhang, Y. Xu, Heat transfer enhancement and drag reduction in transverse groove-bounded microchannels with offset, Int. J. Therm. Sci. 130 (2018) 240–255. [28] Y. Cheng, J. Xu, Y. Sui, Numerical study on drag reduction and heat transfer enhancement in microchannels with superhydrophobic surfaces for electronic cooling, Appl. Therm. Eng. 88 (2015) 71–81.
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Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts
Influence of texture shape and arrangement on thermo-hydraulic performance of the textured microchannels Himani Sharma a, Anvesh Gaddam b, Amit Agrawal a, *, Suhas S. Joshi a, Stefan S. Dimov b a b
Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India Department of Mechanical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
A R T I C L E I N F O
A B S T R A C T
Keywords: Effective slip length Microchannel Nusselt number Pressure-driven flow Temperature jump length Superhydrophobic surfaces Textured surfaces
Textured superhydrophobic surfaces (TSS) are purported to reduce flow friction in microchannels due to velocity slip at liquid-gas interface. At the same time, the liquid-gas interface inhibits heat transfer in textured micro channels due to the low thermal conductivity of entrapped gas phase. Despite significant understanding on fluid flow and thermal transport on the TSS, the interplay of texture shape and arrangement on thermo-hydraulic performance has not been investigated in detail hitherto. To this end, we have numerically investigated the pressure-driven flow through textured microchannels with an aim to enhance the thermo-hydraulic performance. The effective slip length and temperature jump length were estimated as a function of flow and geometry pa rameters for three types of micropillar shapes viz., square, triangular and herringbone, decorated in micro channels in regular and staggered manner. Scaling relations for the effective slip length and temperature jump length have been shown to be valid for triangular and herringbone shaped micropillars at different flow and geometry related parameters. Herringbone shaped micropillars exhibit more flow friction and allow a significant heat transfer in microchannels within the parameter range investigated, followed by triangular and square shaped micropillars. Although the arrangement of textures in microchannels was found to affect the flow friction substantially, its effect on heat transfer was found to be marginal. Subsequently, the overall thermo-hydraulic performance was observed to be superior in regularly arranged herringbone shaped micropillars, at moderate to high constriction ratios (a ratio of texture pitch to half channel height) and high Peclet numbers over the other texture shapes. The results presented in this work would serve as a useful guide to attain maximum thermohydraulic performance in textured microchannels.
1. Introduction Textured superhydrophobic surfaces (TSS) have garnered much attention in the last decade due to their anti-wetting characteristics. Theoretical [1,2] and experimental [3–5] works have shown that a significant amount of reduction in hydrodynamic drag is possible in both internal and external flow conditions by employing the TSS. Particu larly, in microscale laminar flows, such as flow through microchannels, a substantial decrease in pumping power can be achieved by surface texturing. When a liquid flows over the TSS, a liquid-gas interface forms between the asperities as a result of the balance between surface tension and pressure forces. In such heterogeneous Cassie-Baxter state, liquid flowing over the TSS supposedly experiences low shear stress due to the velocity slip at the liquid-gas interface. If the TSS is supplied with heat flux, the liquid-gas interface inhibits heat transfer to the flowing liquid.
Therefore, in microchannel-based heat sinks, thermal management is essential besides reducing the power to drive liquids. Consequently, improving the thermo-hydraulic performance of the textured micro channels in laminar flow conditions is vital. The analytical works to understand the effect of the TSS on flow friction in microchannels in early stages, were restricted to onedimensional rib geometries [6–8]. The effective slip length was employed as a common parameter to define the performance of TSS of different texture geometries and scales. Equations were formulated to predict the effective slip length as a function of gas fraction, a ratio of the area covered by the liquid-gas interface to the total projected area, under the creeping flow conditions. These expressions were applicable for ribs arranged parallel and normal to the flow direction in microchannels. Subsequently, scaling relations were derived for the effective slip length for other generic texture types, such as square shaped micropillars and microholes [9,10]. Numerical works further extended the investigations
* Corresponding author. E-mail address: [email protected] (A. Agrawal). https://doi.org/10.1016/j.ijthermalsci.2019.106146 Received 22 April 2019; Received in revised form 8 September 2019; Accepted 15 October 2019 Available online 24 October 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.
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Nomenclature Roman A, B C, D AC ALG Br Ca DH H k L M Nu Pe Po Q qt Re T TC TS uS U Us u
v w X Y Z
constants of linear regression for hydrodynamic slip length constants of linear regression for thermal slip length area of the composite interface, m2 area of the liquid-gas interface, m2 Brinkman number Capillary number hydraulic diameter of the microchannel, m half of the microchannel height, m thermal conductivity of water, W/m.K texture pitch, m mass flow rate, kg.s 1 Nusselt number Peclet number Poiseuille number average heat flux provided to the domain, W/m2 heat flux to the top surface of micropillar, W/m2 Reynolds number temperature, K average temperature on composite interface, K average temperature on top surface of micropillar, K slip velocity on the composite interface, m.s 1 average velocity, m.s 1 normalized slip velocity on the composite interface streamwise velocity, m.s 1
spanwise velocity, m.s 1 interfacial normal velocity, m.s streamwise coordinate transverse coordinate interfacial normal coordinate
1
Abbreviations RS regular square RT regular triangular RH regular herringbone SS staggered square ST staggered triangular SH staggered herringbone TSS textured superhydrophobic surfaces Greek Symbols φ gas fraction ε constriction ratio λ/L effective slip length λT/L temperature jump length ρ density of the liquid, kg.m 3 μ viscosity of the liquid, N.s.m 2 α thermal diffusivity of the liquid, m2.s 1 η goodness index θ non-dimensional temperature for constant heat flux condition, k(T-Tm)/qtDH ΔP pressure drop, N/m2
of flow friction on the TSS to high Reynolds number (Re) flows, and also to different geometry-dependent parameters (e.g. texture pitch/mi crochannel height) [11,12]. For instance, the effective slip length was found to be Re-dependent in the case of micropillars, microholes and transverse ribs (ribs normal to the flow direction) in microchannels [13]. In general, the textures that provide continuous liquid-gas interfaces, such as longitudinal ribs (ribs along the flow direction) and micropillars produce larger effective slip lengths than the TSS involving microholes and transverse ribs. It was observed that repetitive occurrence of acceleration-deceleration cycles in the liquid flow causes under performance of the latter texture types than the former [14]. At the same time, the textures with continuous liquid-gas interfaces (e.g. longitudi nal ribs and micropillars) have also been shown to negate the detri mental effects of surfactant [15] and particle [16] contamination on slippage as compared to textures with discontinuous liquid-gas in terfaces (e.g. transverse ribs and microholes). However, owing to the very reason of continuity of the liquid-gas interfaces, wetting transition occurs quickly in the former texture types than the latter ones [17]. Consequently, it appears there exists a conflict between these two design criteria: maximum slippage and stability against wetting transition. Despite these inferences, Lam et al. [18] showed that longitudinal ribs perform better than other texture types for a given pressure difference at the liquid-gas interface, under non-inertial flow settings. On the other hand, temperature jump length is analogous to the effective slip length from thermal transport viewpoint. It is a common metric to determine the thermal transport on the TSS. A larger value of the temperature jump length in diabatic slip flows corresponds to a greater resistance to the convective heat transfer. At the same time, a larger value of the effective slip length indicates a lesser resistance to the fluid flow. Ng and Wang [19] formulated expressions for the tempera ture jump length for a variety of textures, such as square/circular shaped micropillars, square/circular shaped microholes, and ribs for constant temperature boundary condition aimed at diffusion-dominated flows. Enright et al. [20] derived an expression for Nusselt number for micropillars in a parallel plate microchannel supplied with constant heat
flux, as a function of effective slip length and temperature jump length under purely diffusive conditions. They also demonstrated that longi tudinal ribs exhibit better thermo-hydraulic performance followed by micropillars in the Stokes flow limit. Using an effective medium approach, Moreria and Bandaru [21] obtained an analytical solution of Nusselt number for flow over transverse ribs in a parallel plate micro channel, by incorporating effective thermal conductivity of the TSS. Similar to adiabatic slip flows, analytical works have also incorporated the effect of interface curvature on the convective heat transfer over the TSS in diabatic flows [22,23]. Overall, a microchannel with the TSS decrease the heat transfer as compared to a plain microchannel due to the decrease in effective solid-liquid contact area. Numerical works predominantly analysed the effect of inertia [24, 25], texture type and their arrangement [26–28] on thermal transport in microchannels for improving thermo-hydraulic performance. For example, thermo-hydraulic performance of a microchannel with trans verse ribs can be increased for offsetting their positions on top and bottom walls [27]. Cheng et al. [28] investigated the influence of generic texture types such as micropillars, microholes, transverse and longitudinal ribs, on thermo-hydraulic performance under constant temperature boundary condition. They found that thermo-hydraulic performance of the square shaped micropillars surpasses the other texture types, at high gas fractions and high Reynolds numbers. While the experimental works in this domain are limited, a recent study has shown that the thermo-hydraulic performance can be enhanced, when the liquid-gas interface is nearly flat rather instead of protruding in/out of the gas cavity [29]. In summary, a considerable amount of research has been undertaken so far to understand the influence of TSS on both athermal and thermal transport in microchannels. The critical parameters that influence the flow and thermal resistance under the non-wetting conditions include texture type, arrangement, liquid-gas interface curvature, texture pitch/ microchannel height and Reynolds/Peclet number. The interplay be tween these parameters has shown to affect the thermo-hydraulic per formance of textured microchannels. Overall, two generic texture types 2
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standout in terms of thermo-hydraulic performance: longitudinal ribs and micropillars. Since, non-linear convective acceleration terms are absent in the governing equation for the fluid momentum for longitu dinal ribs (neglecting thermocapillary stresses), the effect of inertia on the flow and thermal resistance is insignificant for this texture type [13, 28]. Previous attempt to manipulate the flow field by tailoring the arrangement of longitudinal ribs in a microchannel has not yielded any positive effects [30]. However, inertial effects play an important role in athermal and thermal transport in the case of micropillars. At the same time, tailoring the shape/arrangement of longitudinal ribs is difficult due to their one-dimensional nature. On the other hand, bi-dimensional textures (e.g. micropillars and microholes) allow manipulation on flow field inside microchannels based on their shape/arrangement. In fact, several efforts have been made to understand the influence of micro pillars arrangement (e.g. regular, staggered and random) on thermal transport [26], flow friction [26,31] and interfacial stability [31] in microchannels. It is apparent from the current understanding that ve locity and temperature fields in a microchannel are influenced by bi-dimensional textures. Therefore, micropillars could have better advantage than longitudinal ribs to enhance the thermo-hydraulic per formance of microchannels. In the light of above discussion, it is apparent that the design and optimization of surface textures is necessary to enhance the thermohydraulic performance of microchannels. It also appears from the literature that the effect of non-generic texture shapes on flow and thermal transport has not been investigated thus far. Therefore, the main focus of this paper is on understanding the effect of micropillar shape and its arrangement in microchannels, on flow friction and thermal transport. The numerical simulations were performed for a pressuredriven flow through microchannels containing the TSS that were pro vided with a constant heat flux. Micropillars of different shapes viz.,
square, triangular and herringbone, arranged in either regular or stag gered manner, were considered in this study. The inspiration to use particular non-generic shapes such as triangular and herringbone came from their usage in enhancing single-phase heat transfer in micro channel heat sinks [32]. As discussed earlier, the sparsely spaced micropillars give larger effective slip lengths in microchannels. There fore, only micropillars with high gas fractions (>70%) have been investigated in this work. The effective slip length and temperature jump length were estimated for these TSS as a function of gas fraction, microchannel constriction and Reynolds number/Peclet number. This has helped comprehend the significance of flow and geometry param eters on fluid flow and thermal transport. Lastly, the goodness index was evaluated to compare the thermo-hydraulic performance of the various textured shapes considered here. 2. Methodology A steady, laminar, incompressible and pressure-driven flow through a parallel plate microchannel, with top and bottom walls decorated with micropillars, has been considered in this work. The flow is assumed to exist in the Cassie-Baxter state with the liquid-gas interface appearing to be flat between successive micropillars. Three types of micropillar shapes viz., square, triangular and herringbone have been investigated (see Fig. 1 a-c). The micropillars were arranged either in regular or staggered fashion, on the top and bottom walls of microchannel as illustrated in Fig. 1 d-e. The regions highlighted in Fig. 1 d-e are considered for the numerical simulations, due to the symmetry and periodicity of the micropillar arrangements in the microchannels, along Y/Z- and X-directions, respectively. A typical computational module for the regular arrangement of square shaped micropillars inside a micro channel is schematically depicted with appropriate nomenclature and
Fig. 1. (a–c) Micropillar shapes considered in this study are shown. a. Square b. Triangular and c. Herringbone. (d–e) An illustration of the herringbone shaped micropillars in a microchannel considered in this study is shown in d. regular and e. staggered arrangements. The regions highlighted in dashed lines represent computational area in the present numerical simulations. f. The computation module for a square shaped micropillar in a regular arrangement is shown. 3
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boundary conditions in Fig. 1f.
ε¼
2.1. Mathematical formulations
L H
(5)
ρUDH μ
(6)
DH U
(7)
The following continuity, momentum and energy equations, Eq. (1) – (3), are applicable to the flow conditions inside the microchannel.
Re ¼
(1)
Pe ¼
(2)
where, ALG is area of the liquid-gas interface inside the domain, AC is area of the composite interface inside the domain, L is texture pitch, H is half of the microchannel height, ρ is density of the liquid, μ is viscosity of the liquid, α is thermal diffusivity of the liquid and U is average axial velocity inside the microchannel. The typical dimension of texture pitch considered is 50 μm. Consequently, the hydraulic diameter (DH ¼ 4H) of the microchannels range from 40 μm to 400 μm, corresponding to the parameter space of constriction ratio (ε: 0.25–5). In should be noted that the simulations were conducted with the assumption that liquid flow exists under the Cassie state and the airwater interface is considered to be flat. These assumptions are valid under the circumstances where capillary number (Ca, a ratio of viscous forces to the surface tension forces) for system is much less than unity. Based on the characteristic flow velocities through the microchannels of different constriction ratios for a Reynolds number of 1000, viscosity of the water (~0.001 Pa s) and surface tension of the air water interface (0.072 N/m), the Ca values are calculated. In all the cases, it was esti mated that Ca << 1. Therefore, it is reasonable to assume that surface tension plays a fairly dominant role tin maintaining the liquid-gas interface to be flat for the parameter range investigated in our simula tions. However, it is expected that modest deformation would take place at high constrictions ratios in the high-Re regime in practical situations. Furthermore, the zero-shear stress condition at the liquid-gas interface overpredicts the effective slip length as shown in previous works [11, 12]. However, the numerical simulations showed that the percentage change in the effective slip length between the viscosity contrast of 0.02 (e.g. air-water interface) and 0 is about 5% for micropillars arranged at a gas fraction of 0.75 [33]. Since this difference is not so significant and at the same time, this study focuses on comparing different texture shapes and arrangements on the frictional and thermal characteristics, it is reasonable to assume that the liquid-gas interface is shear-free. At the same time, we have also estimated the value of Brinkman number (Br), which characterizes the viscous dissipation in flow through micro channels. It is the ratio of heat generated due to the viscous dissipation and heat exchanged at the wall. The Brinkman number in our simula tions is estimated to be much less than unity (Br ≪1) for flow through microchannels with low constriction ratios (ε < 3). However, the value Brinkman number was found to be only slightly less than unity (Br < 1) for flow through microchannels with high constriction ratios (ε > 3). Therefore, it is reasonable to neglect viscous dissipation affects due to the fact that Br < 1 and also the simulations are performed with an aim to compare different texture shapes and their arrangement.
r:u ¼ 0
ρðu:rÞu ¼
rP þ μr2 u
ρcp u:rT ¼ kr2 T
(3)
where, the velocity field u ¼ [u, v, w] and temperature field T corre spond to the spatial field S ¼ [X, Y, Z]. The working liquid considered in this study is water with constant thermo-physical properties (thermal conductivity, k ¼ 0.6 W/m.K and viscosity, μ ¼ 0.001 Pa s). The geome tries of the computational module were discretized in GAMBIT and the above-mentioned coupled equations were solved numerically in a commercial computational framework, Ansys Fluent 16.0. The liquidgas interface was approximated as a shear-free boundary and adia batic (∂u/∂Z ¼ ∂v/∂Z ¼ w ¼ ∂T/∂Z ¼ 0), while the no-slip boundary condition (u ¼ v ¼ w ¼ θ ¼ 0) was imposed on the top surface of the highly conducting micropillar. Periodic boundary conditions (uin ¼ uout, vin ¼ vout, win ¼ wout, θin ¼ θout) were implemented at the inlet and outlet of the computational module. This was due to the periodically repetitive velocity and temperature fields inside the microchannel for fully developed hydrodynamic and thermal flow conditions. Symmetry boundary conditions (top surface: ∂u/∂Z ¼ ∂v/∂Z ¼ w ¼ ∂T/∂Z ¼ 0; side surfaces: ∂u/∂Y ¼ ∂v/∂Y ¼ v ¼ ∂T/∂Y ¼ 0) were employed on top and sides of the computational module, due to the mirror symmetry along Yand Z-directions, inside the microchannel. The abrupt changes in the velocity and temperature are expected on the TSS due to the change of boundary condition from no-slip to shearfree. As a consequence, the region near the composite interface (solid/ liquid/gas interface at Z ¼ 0) was locally refined before initializing the simulation. At the inlet, the mass flow rate corresponds to the desired Reynolds number. Initially, the computational domain was solved for the velocity field. Once the continuity and velocity residuals reached 10 4, a velocity gradient-based adaption method was used to obtain the grid independent solution. This method refines the grid in the vicinity of the highest velocity gradients in the computational domain based on a user-defined threshold. The refinement process continued until the continuity and velocity residuals reach 10 9. Once the velocity field was obtained, the computational domain was solved for the temperature field, by delivering heat flux (qt) to the top surface of the micropillar. Initially, the temperature field was obtained until the energy residuals reached 10 4. Thereafter, a temperature gradient-based adaption method was implemented to refine the grid for obtaining a gridindependent solution, until the energy residuals reach 10 8. Although the gradient-based mesh adaption for temperature field was employed over the adapted mesh for hydrodynamic solution, it has not yielded any significant change in number of cells except for a very few cases. The gradient-based grid adaption method used here, considerably reduces the computational time and resources, as discussed elsewhere [11,12, 16,25]. The final number of cells was ranged from 100,000 to 1,000,000 in our simulations after the adaption. Also, the user-defined threshold for gradient-based adaption of velocity and temperature fields was chosen between 10 and 30% in our simulations. It should also be noted that the total heat flux (q ¼ φqt) provided to the computational domain was kept constant in all the simulations. The relevant dimensionless geometric and flow quantities such as gas fraction (φ), constriction ratio (ε), Reynolds number (Re) and Peclet number (Pe) used in this study are defined as follows: φ¼
ALG AC
α
2.2. Calculation of flow parameters The pressure gradient (ΔP/L) across the computational module ob tained from the numerical simulations has been converted into Pois euille number (Po) using Eq. (8). Subsequently, the effective slip length (λ/L) was estimated from the Poiseuille number using Eq. (9) [16]. Po ¼
D3H ρΔP m_ μL
� λ 1 32 ¼ � L ε Po
(8) � 1 3
(9)
Here, m is the mass flow rate at the inlet of the microchannel. Similarly, the temperature jump length (λT/L) was evaluated using Eq.
(4) 4
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(10). This equation in turn is employed to calculate the Nusselt number (Nu) for the convective heat transfer inside the computational module using Eq. (11) [25]. λT k ¼ ðTS L qt Nu ¼
(10)
TC Þ
� λT ε 17 þ L 4 140
3 1 US þ US 2 70 210
�
1
(11)
Here, k is the thermal conductivity of the liquid, TS is the tempera ture on top surface of the micropillar, TC is the temperature of the composite interface (solid-liquid-gas interface at Z ¼ 0), Us ¼ (uS/U) is the normalized slip velocity on the composite interface and uS is slip velocity on the composite interface. It is apparent that the Nu reduces to 140/17 in the case of flow through a plain channel. It should be noted that the inexact form of Eq. (11) is analogous to the one presented in some previous works [20]. The goodness factor of plain and textured microchannels can be expressed by (NuP/PoP.Pr1/3) and (Nu/Po.Pr1/3), respectively [27,28]. Here, NuP (¼8.235) and PoP (¼ 96) correspond to Nusselt number and Poisueille number for a plain parallel plate microchannel, whereas Nu and Po correspond to Nusselt number and Poiseuille number for a textured microchannel, with Pr being the Prandtl number of the fluid. Consequently, the thermo-hydraulic performance of microchannels with the TSS with respect to a plain microchannel has been evaluated in terms of goodness index (η) using Eq. 12
η¼
Nu=Po NuP =PoP
Fig. 2. Scaling of the effective slip length and temperature jump length for square shaped micropillars in regular arrangement inside a microchannel with ε ¼ 0.01 at Re ¼ 0.1.
3. Results and discussion In the subsequent sections, we will discuss the influence of liquid inertia, microchannel constriction and gas fraction, on the flow friction and convective heat transfer, inside microchannels containing micro pillars of different shapes, arranged in either regular or staggered manner. The six types of TSS investigated are: regular square (RS), regular triangular (RT), regular herringbone (RH), staggered square (SS), staggered triangular (ST) and staggered herringbone (SH). The range of flow and geometrical parameters considered for this study are: 0.75 � φ � 0.98, 0.25 � ε � 5, 1 � Re � 1000 and 7 � Pe � 7000.
(12)
2.3. Validation of numerical procedure Ybert et al. [9] derived scaling relations for the effective slip length for a shear-driven flow in the limit of Stokes flow, for a surface sparsely covered with square shaped micropillars (φ> 70%) in a regular manner. Eq. (13) gives the scaling law for the effective slip length for this sce nario, where A, B are constants. At the same time, Enright et al. [20] provided scaling relation for the temperature jump length for the similar conditions for regularly arranged micropillars supplied with constant heat flux. Eq. (14) gives the scaling law for the temperature jump length for this case, where, C, D are constants. λ ¼ Að1 L λT ¼ Cð1 L
φÞ
φÞ
1=2
1=2
B
D
3.1. Scaling laws We begin our analysis by examining the validity of scaling laws for the effective slip length and temperature jump length. The scaling laws, as explained earlier, were originally derived for creeping shear-driven flows past TSS (see Eq. (13)). Previous works have successfully extended their validity to inertial pressure-driven flows through textured microchannels [13]. In our earlier works, we have investigated the scaling laws for even particle-laden liquid-gas interfaces, finite aspect ratio microchannels, and liquid-infused textured surfaces [16,33, 34]. Here, we examine scaling laws for micropillars of different shapes that are arranged in a regular manner (RS, RT and RH) in microchannels having different constrictions, in inertial pressure-driven flows. Fig. 3 shows plots of the effective slip length as a function of gas fraction, for two different microchannel constrictions (ε ¼ 1, 5) and Reynolds numbers (Re ¼ 1, 1000). It appears that a linear variation of effective slip length is observed with the parameters for all the micropillar shapes investigated here. This indicates that the scaling laws are valid for different texture shapes. Furthermore, the effective slip length was found to be less in herringbone shaped micropillars, followed by triangular and square
(13) (14)
In order to validate the numerical procedure adopted in this work, simulations were performed using microchannels with square shaped micropillars that are arranged in a regular manner on top and bottom walls with high gas fractions (φ ¼ 75%, 90%, 95% and 98%). The effective slip length and temperature jump length were evaluated in microchannels with a large separation between the top and bottom walls (ε ¼ 0.01) at Re ¼ 0.1, to mimic the creeping shear-driven flow condi tions. A plot of the numerically obtained effective slip length and tem perature jump length at different gas fractions is shown in Fig. 2, which depicts a linear relationship with the function of gas fraction. The con stants of this linear relationship of the effective slip length and tem perature jump length were obtained by regression analysis, and are given in Table-1 and Table-2, respectively. These constants (A, B, C and D), obtained through numerical simulations, were compared with the results in literature under different flow and geometrical conditions as shown in Table-1 and Table-2. Based on the comparison, it appears that the numerical procedure followed in this work is valid for the different scenarios presented below.
Table 1 A comparison of values of coefficients in Eq. (13) obtained in the present work with the literature.
5
Reference
A
B
Conditions
Present work Ybert et al. [9] Davis and Lauga [10]
0.335 0.325 0.332
0.425 0.44 0.42
Re ¼ 0.1, ε ¼ 0.01 – –
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numbers are shown in Fig. 4. The plot of temperature jump length re sembles that of the effective slip length. It also exhibits a linear rela tionship, thus extending the validity of the scaling laws to different texture shapes. Further, it is evident that the temperature jump length is lower for herringbone shaped micropillars, followed by triangular and square shaped micropillars. This implies that the herringbone shaped micropillars enhance the convective heat transfer in microchannels. Furthermore, an increase in Peclet number enhances the convective heat transfer as evident by the decrease in the temperature jump length; see Fig. 4 b, d. At the same time, the effect of Peclet number on the tem perature jump length is increased with an increase in microchannel constriction. As explained earlier, the constants (C and D) of Eq. (14) correspond to the cases shown in Fig. 4, were evaluated through regression analysis and are presented in Table-4. The correlation coef ficient in the regression analysis is more than 99.5% for all the cases presented here.
Table 2 A comparison of values of coefficients in Eq. (14) obtained in the present work with the literature. Reference
C
D
Conditions
Present work Enright et al. [20] Cowley et al. [25]
0.465 0.471 0.469
0.599 0.603 0.593
Re ¼ 0.1, ε ¼ 0.01 Re < 1, ε ¼ 0.0125 Re ¼ 0.14, ε ¼ 0.04
shaped micropillars. Therefore, it is evident that the shape of the micropillars has a substantial influence on the flow friction in the microchannels. In addition, the liquid inertia considerably decreases the effective slip length for all the micropillar shapes. The constants (A and B) of Eq. (13) for the square, triangular and herringbone shaped micropillars for the scenarios simulated here, were evaluated through regression analysis, as listed in Table-3. For similar conditions as above, numerical simulations were also performed to obtain scaling relations for the temperature jump length, when a constant heat flux is delivered to the micropillars. The plots of the temperature jump length as a function of gas fraction for micro pillars of different geometries, microchannel constrictions and Peclet
3.2. Influence of liquid inertia It is apparent from the previous section that the values of the
Fig. 3. (a–d) Scaling of the effective slip length for square, triangular and herringbone shaped micropillars at different constriction ratios and Reynolds numbers. 6
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Fig. 4. (a–d) Scaling of the temperature jump length for square, triangular and herringbone shaped micropillars at different constriction ratios and Rey nolds numbers. Table-3 Numerically evaluated values of coefficients in Eq. (13) for three micropillar shapes at different microchannel constrictions and Reynolds numbers. Microchannel constriction
ε¼1 ε¼5
Texture shape Square Triangular Herringbone Square Triangular Herringbone
Re ¼ 1
Table-4 Numerically evaluated values of coefficients in Eq. (14) for three micropillar shapes at different microchannel constrictions and Reynolds numbers.
Re ¼ 1000
A
B
A
B
0.348 0.338 0.292 0.342 0.339 0.316
0.484 0.487 0.403 0.467 0.494 0.488
0.233 0.225 0.214 0.184 0.174 0.167
0.282 0.293 0.304 0.219 0.231 0.244
Microchannel constriction
ε¼1 ε¼5
effective slip length and temperature jump length reduce significantly with an increase in Reynolds number for all the micropillar shapes. Consequently, we investigated the effect of inertia on flow friction and convective heat transfer in the textured microchannels. Fig. 5 shows a variation in the effective slip length with Reynolds numbers, for all the textures shapes, arranged either in regular or in staggered manner for a
Texture shape Square Triangular Herringbone Square Triangular Herringbone
Re ¼ 1
Re ¼ 1000
C
D
C
D
0.473 0.451 0.431 0.463 0.438 0.425
0.595 0.519 0.575 0.527 0.447 0.557
0.185 0.190 0.174 0.098 0.077 0.072
0.130 0.228 0.263 0.154 0.101 0.117
gas fraction of 0.75, and at different microchannel constrictions. The effective slip length is observed to decrease gradually with an increase in Reynolds number for all the micropillar shapes in regular arrangement. However, the influence of inertia on flow friction is more pronounced at large microchannel constrictions; see Fig. 5a, c. This is due to the fact that flow field will be greatly influenced when top and bottom walls are 7
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Fig. 5. (a–d) Effective slip length as a function of Reynolds number for square, triangular and herringbone shaped micropillars at different constriction ratios and in regular and staggered configurations.
close to each other [11,12,33]. On the other hand, the staggered arrangement enacts minor influence on the effective slip length for herringbone shaped micropillars, whereas the square and triangular micropillars, in the staggered manner, exhibit more flow friction at high Reynolds numbers as shown in Fig. 5b. At the same time, the influence of inertia on the effective slip length for micropillars in staggered arrangement is shown with an increase in microchannel constriction (see Fig. 5d). Interestingly, the inertial effects appear to nullify the dif ference in the performance of micropillars in staggered arrangement, at large microchannel constrictions. Fig. 6 shows plots of the temperature jump length with respect to Reynolds number for micropillars in regular and staggered arrange ments, at different microchannel constrictions. The behaviour of tem perature jump length is analogous to the effective slip length. It decreases gradually with an increase in Reynolds number, for all the micropillars shapes in regular arrangement. It indicates an increase in the convective heat transfer with the inertia as shown in Fig. 6 a, c. The apparently large difference between the values of temperature jump length for herringbone and square shaped micropillars, at low and high
microchannel constrictions, signify higher heat transfer characteristics on the former than the latter. At the same time, the microchannel constriction influences more on the triangular and herringbone shaped micropillars as compared to square shaped micropillars, in the regular arrangement. It is also apparent from Fig. 6c and d that the micropillar shape exhibits marginal effect on the convective heat transfer in microchannel, in the staggered arrangement. The texture arrangement has a far more profound influence on the heat transfer characteristics of only square shaped micropillars, than on the flow friction. On the other hand, the triangular and herringbone shaped micropillars appear to be relatively unaffected, as evident from Fig. 6. The surface temperature profiles on the TSS in Fig. 7 also verify that a drastic variation in the temperature distribution for square shaped micropillars occurs between regular and staggered arrangements. It may be noted that the minimum and maximum temperatures on the com posite surface were normalized between 0 and 1. It is also observed that the micropillar shape has minimal influence on the temperature distri bution as discussed earlier. Furthermore, the triangular micropillars fared worse in enhancing the convective heat transfer in microchannels 8
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Fig. 6. (a–d) Temperature jump length as a function of Peclet numbers for square, triangular and herringbone shaped micropillars at different constriction ratios and in regular and staggered configurations.
in the low inertia conditions (Re < 10). Overall, the herringbone shaped micropillars outperformed other shapes over the parameter range investigated in this work.
distance between the upper and lower walls in the microchannels. This indicates that, the interaction of flow field between upper and lower walls of the textured microchannels intensifies with a decrease in dis tance between them and magnifies with the aid of fluid inertia. Conse quently, the temperature jump length is also obeserved to be a function of microchannel constriction only at high Peclet numbers, whereas, it remains relatively unchanged both in regular and staggered arrange ments as shown in Fig. 9. This also implies that, the texture arrangement has only a marginal influence on flow and heat transfer characteristics at low inertia conditions. On the other hand, the texture arrangement greatly affects the flow friction in inertial flows, while the heat transfer characteristics remain unchanged for the same settings.
3.3. Influence of microchannel constriction In the previous sections, it was noticed that microchannel constric tion affects both the effective slip length and temperature jump length in textured microchannels. Here we examine the role of varying the dis tance between upper and lower walls of the textured microchannels on the flow friction and heat transfer characteristics. Fig. 8 shows the effective slip length plotted as a function of microchannel constriction, for micropillars in regular and staggered arrangements, at different Reynolds numbers. It is observed that the effective slip length for all micropillar shapes both in regular and staggered arrangements is a function of microchannel constriction, only at high Reynolds number. Furthermore, the inertia comes into effect and plays a dominant role with an increase in microchannel constriction, i.e., by decreasing the
3.4. Thermo-hydraulic performance of the TSS So far it is noted that both flow friction and convective heat transfer reduce with the aid of the TSS in microchannels. Therefore, a metric to compare the effectiveness of the TSS-based microchannels with respect 9
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Fig. 7. Surface temperature profiles on square, triangular and herringbone shaped micropillars in regular and staggered arrangements with a gas fraction of 0.75 for ε ¼ 5 and Pe ¼ 7000.
to the plain microchannels is necessary. The goodness index (η) as explained by Eq. (12) in section 2, gauges the combined hydraulic and thermal performance of the TSS-based microchannels. A value of goodness index greater than unity signifies beneficial effect of intro ducing the TSS in the microchannels. Fig 10a shows a plot of goodness index with the function of gas fraction for the micropillars arranged in staggered manner at different microchannel constrictions for highly inertial flows. The goodness index steadily increases with the gas frac tion at low microchannel constrictions for all the micropillar shapes. At the same time, it is also noticed that the texture shape has only minor influence on the goodness index at low constriction ratios. However, the thermo-hydraulic performance is found to increase rapidly with the gas fraction and reaches a plateau after a gas fraction of 0.95 for all the micropillar shapes at high microchannel constrictions. Interestingly, the performance of the non-generic texture shapes (herringbone and trian gular) is much higher than the generic texture shape (square). Furthermore, the herringbone shaped micropillars in staggered arrangement exhibit a goodness index value greater than 3, which is far higher than what has been achieved in the literature so far. Despite the high flow friction, 80–100% improvement in the goodness index is observed due to herringbone shaped micropillars over the square shaped
micropillars. It has been previously shown that texture orientation could generate secondary flows transverse to the flow direction in micro channels, thereby improve fluid mixing [35] and particle migration [36]. In a similar vein, herringbone and triangular shaped micropillars could produce transverse flow due to their no-slip/shear-free boundaries positioned oblique to the flow direction. In order to appreciate this effect on temperature field, the velocity field normal to the flow direction is visualized for herringbone and square shaped micropillars. Fig. 10b shows evolution of transverse velocity vectors (v, w) along the flow di rection for herringbone and square shaped micropillars. Apparently, herringbone shaped micropillars are exhibiting complex flow patterns in lateral direction, thus aiding in significant heat transfer in micro channels. This signifies the importance of texture shape optimization in enhancing the thermo-hydraulic performance. As explained in previous sections, the herringbone shaped micro pillars significantly outclass square shaped micropillars in increasing the convective heat transfer and flow friction. However, the substantial ef fects are realized only at high constriction ratios. Fig. 10b shows a plot of goodness index with the microchannel constriction for square and herringbone shaped micropillars in regular and staggered arrangements, for a gas fraction of 0.95 at Pe ¼ 7000. In all the cases, the benefit of 10
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Fig. 8. (a–b) Effective slip length as a function of constriction ratio for square, triangular and herringbone shaped micropillars at different Reynolds numbers and in regular and staggered configurations.
Fig. 9. (a–b) Temperature jump length as a function of constriction ratio for square, triangular and herringbone shaped micropillars at different Peclet numbers and in regular and staggered configurations.
arrangements. In line with the previous observations, micropillar arrangement has not shown any significant effect on goodness index at low constriction ratios. However, regularly arranged herringbone sha ped micropillars have shown better performance than those in staggered arrangement for Re > 200. After understanding the behaviour of different micropillar shapes, we now compare their performance with longitudinal ribs. As explained in section 1, longitudinal ribs have shown to exhibit lesser flow friction than the other generic texture types, thereby demonstrating better thermo-hydraulic performance [20]. Consequently, numerical simula tions performed for microchannels containing longitudinal ribs with the settings as described in section 2. Fig. 11b shows the plot of goodness index as a function of Reynolds number for longitudinal ribs and
tailoring shape/arrangement is only realizing at moderate-high constriction ratios (ε > 1). This also indicates that, the transverse flow patterns are intense only for ε > 1, so as to produce meaningful benefits. At the same time, the regularly arranged herringbone shaped micro pillars enhance the thermo-hydraulic performance of the microchannels as compared to those in staggered arrangement. That means, the flow friction is becoming overwhelmingly high in staggered arrangement at high flow rates. However, the similar behaviour is not noticed in the case of square shaped micropillars. This implies that the texture arrangement plays a crucial role in determining the thermo-hydraulic performance of the microchannels. To understand the influence of inertia on thermohydraulic performance, goodness index is plotted against Reynolds number (Fig. 11a) for herringbone shaped micropillars in different 11
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Fig. 10. a. Goodness index scaling for square, triangular and herringbone shaped micropillars at different constriction ratio in staggered configuration. b. Transverse velocity field (v, w) showing velocity vectors along the flow direction for herringbone and square shaped micropillars. c. Goodness factor as a function of constriction ratio for square and herringbone shaped micropillars in regular and staggered configurations corresponding to φ ¼ 0.95.
different micropillar shapes for φ ¼ 0.75 and ε ¼ 5. It is noted that, longitudinal ribs have performed better than all micropillar shapes in the low Re-regime (Re < 10). However, the goodness index of herring bone shaped and triangular micropillars has dramatically increased with the fluid inertia. Particularly, the fluid inertia has come into effect at Re > 10 (herringbone shaped) and Re > 100 (triangular) for these micropillar shapes in order to better their thermo-hydraulic perfor mance over longitudinal ribs. At the same time, square shaped micro pillars and longitudinal ribs have exhibited similar performance over the entire Re-regime. As noted earlier, the oblique no-slip/shear-free boundaries present in the non-generic texture shapes instigate the
cross flow patterns in the flow field with the aid of inertia. These ob servations emphasize the importance of tailoring texture shapes in improving the thermo-hydraulic performance of microchannels. 4. Conclusions In this work, the pressure-driven flow through microchannels con taining the TSS supplied with constant heat flux was numerically simulated. The micropillar textures with three different shapes such as square, triangular and herringbone that are arranged either in regular or staggered manner, on top and bottom walls of the microchannels were 12
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Fig. 11. a. Goodness index as a function of Reynolds number for herringbone shaped micropillars in regular and staggered configurations at different constriction ratios for φ ¼ 0.75. b. Comparison of goodness index for different micropillar shapes with longitudinal ribs arranged with φ ¼ 0.75 at ε ¼ 5.
investigated. The main conclusions drawn from this work are as follows:
References
1. Scaling relations for the effective slip length and temperature jump length, originally derived for the square shaped micropillars in the Stokes flow limit, were found to correlate well with other micropillar shapes such as triangular and herringbone. Thus, the validity of the scaling laws was extended to intricate geometries and inertial pressure-driven flows. 2. A discernible difference in the values of the effective slip length and temperature jump length was noticed among the micropillar shapes in the regular arrangement. Whereas, the difference was found to be negligible in the staggered arrangement with an increase in Reynolds number and microchannel constriction. This implies that the effec tive slip length and temperature jump length are rather weak func tions of micropillar shape in the staggered arrangement. 3. An increase in the Reynolds number and microchannel constriction leads to an increase in flow friction and an increase in the convective heat transfer, for all the micropillar shapes in both regular and staggered arrangements. However, the micropillar arrangement in side the microchannel has substantial influence on the flow friction, whereas its effect on the convective heat transfer was minor. 4. Overall, the herringbone shaped micropillars exhibits more flow friction and less thermal resistance. It is followed by triangular and square shaped micropillars respectively, in both regular and stag gered arrangements. 5. The influence of transverse flow has led to improvement in thermohydraulic performance of the non-generic texture shapes over the generic texture shapes. However, the positive benefits of this trans verse flow are realized only at moderate-high constriction ratios and high Reynolds/Peclet numbers.
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Acknowledgments The research reported in this paper is within the framework of the UKIERI-DST programme “Surface functionalisation for food, packaging, and healthcare applications”.
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