Numerical study of fluid flow and heat transfer phenomenon within microchannels comprising different superhydrophobic structures

International Journal of Thermal Sciences 124 (2018) 536–546

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Numerical study of fluid flow and heat transfer phenomenon within microchannels comprising different superhydrophobic structures

T

Masoud Kharati-Koopaee∗, Mohammad Reza Akhtari Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Superhydrophobic Microchannel Poiseuille number Nusselt number Thermal performance

This research aims at numerical study of fluid flow and heat transfer through microchannels having superhydrophobic surfaces consisting of aligned and staggered micropost patterns in the fully developed laminar flow regime. In this work, at the condition of constant surface heat flux, Poiseuille number, Nusselt number and also overall microchannel performance are examined at relative module width of Wm = 0.01, 0.1 and 1, cavity fraction range of Fc = 0.1 to 0.9 and Reynolds numbers of R e=10 and 100. In order to validate the current results, comparisons are made with theoretical and experimental approach and good agreements are observed. Numerical findings show that the staggered pattern is capable of producing higher frictional resistance and better thermal transport than the aligned structure. It is shown that an increase in the cavity fraction leads to a decrease in the Poiseuille and Nusselt numbers for the two micropost structures and this decrease becomes pronounced with increasing the relative module width. Results indicate that for the two micropost patterns, the role of increase in the relative module width is to decrease the Poiseuille and Nusselt numbers. It is found that the staggered arrangement could lead to higher overall performance than the corresponding aligned structure and enhancement in the performance becomes remarkable at high values of relative module width. Numerical findings indicate that for each micropost structure, an increase in the Reynolds number causes the microchannel overall performance to increase and the highest overall performance is attained at high relative module width and cavity fraction values.

1. Introduction Nowadays, the subject of heat removal from the electronic devices in the micro-scale is an important issue in the design process of such devices. In this context, employment of a microchannel which meets the desired conditions in terms of heat removal capacity and also frictional resistance has always been a challenging problem. Although some efforts are made to fulfill the desired heat removal ability of the microchannles such as employing of tree-shape [1], zigzag [2] or wavy [3,4] microchannels, however, the subject of pressure gradient associated to these microchannels has always been an open problem. Reviewing literature indicates that many researches are devoted for evaluation of microchannels performance in hydrodynamic point of view [5–8]. In the field of micro-scale devices, some researchers analyzed the flow through the microchannles which were used as the heat sink. Hung et al. [9] performed an optimization procedure to find the optimum microchannel heat sink among different geometric structures such as the single layered, double-layered or tapered microchannels. In their research, they concluded that the tapered microchannel had the

∗

best performance among the considered geometries. Wei et al. [10] assessed the pressure drop as well as heat transfer characteristic of the transversal elliptical microchannels. They found that the periodic transversal elliptical microchannel could be viewed as an alternative to the conventional microchannels since they had a potential to reduce pressure drop and also enhance the heat transfer rate. Xie et al. [11] conducted a research to examine the thermal performance of a transversal wavy microchannel. In their study, they found that for a same Reynolds number, this type of microchannel could have a higher thermal performance compared to the traditional straight rectangular microchannel especially at high wave amplitude. Chai et al. [12] studied the effect of dimensions and positions of rectangular ribs on the fluid flow and rate of heat transfer for the transverse microchambers. In this research, they indicated the optimum dimensions and position parameters of the ribs. Xie et al. [13] investigated the role of bifurcation on the thermal performance of the heat sink microchannels. Their results showed that the thermal performance of the microchannel with multistage bifurcation flow was better in comparison with straight microchannels. They also expressed that the utilization of multistage

Corresponding author. E-mail address: [email protected] (M. Kharati-Koopaee).

https://doi.org/10.1016/j.ijthermalsci.2017.11.004 Received 14 March 2017; Received in revised form 1 November 2017; Accepted 2 November 2017 Available online 15 November 2017 1290-0729/ © 2017 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

microchannels. As shown by Cheng et al. [27], in the laminar flow regime, the microchannels employing micropost-structured SHS in aligned form are a good alternative to the traditional microchannels for heat removal purposes in the mico-scale devices since they lead to a rather low frictional resistance and high heat transfer rate. In the research conducted by Cheng et al. [27], it was also shown that for the flow through microchannels, the capability of the channel in increasing the Nusselt number increased when the flow experienced flow acceleration and deceleration at each microchannel spanwise direction. So, one could infer that microchannels having the micropost-structured SHS in the staggered form (which, contrary to the aligned structure, causes the flow to experience flow acceleration and deceleration at each spanwise location) could be viewed as a promising alternative in order to have high Nusselt number with a rather low frictional resistance. In this context, the question may arise about the extent to which a microchannel which uses the micropost-structured SHS in the staggered could enhance the microchannel performance, which is the motivation for the present study. In this study, the frictional resistance, heat transfer phenomenon and also overall performance of different microchannels employing SHS in the form of aligned and staggered micropost are evaluated and compared in the laminar flow regime. In this research, for better understanding of role of each micropost pattern on the microchannel performance, results are obtained at different relative module widths, cavity fractions and Reynolds numbers.

bifurcated plates could reduce the overall thermal resistance and the best number of stages of bifurcations was two. Chai et al. [14–16] carried out a parametric study to evaluate the heat transfer rate, pressure drop and overall performance of fluid flow through microchannels. They focused on the aligned and offset fan-shaped ribs mounting on the microchannel walls. Xie et al. [17] examined the thermal performance of the longitudinal and transversal wavy microchannels. Their numerical findings revealed that the longitudinal wavy microchannels were inferior while the transversal wavy microchannels were superior to the conventional rectangular microchannels in terms of overall thermal performance. In the context of fluid flow through the heat sink microchannels, the channels employing superhydrophobic surfaces (SHSs) as the channel walls could be viewed as a good alternative to overcome the high pressure loss associated with them. The SHSs are combination of hydrophobicity utilizing a thin hydrophobic layer as well as micro-sized protrusions emerging from the surface; or in the other form, microholes carved in the surface. One of the features associated to these surfaces is that they could lead to a contact angle of more than 150° for a water droplet resting on them [18]. Since the protrusions or holes are made in a low size, the penetrating of fluid in between the protrusions or holes is not possible due to surface tension effect and so, the air traps in the cavities. Accordingly, the wetted contact area between the fluid and solid surface decreases, resulting in the mitigation of frictional resistance for the fluid flowing through the channel [19]. Although some researches are performed to assess the effects of different parameters on the fluid flow and heat transfer phenomenon within the microchannels employing SHSs, however, they were rare and were also limited to few cases. In this field, Maynes et al. [20] carried out an analytical approach to analyze the thermal transport in microchannels with transverse rib and cavity structured superhydrophobic walls. In their work, it is shown that an increase in the cavity fraction causes the average Nusselt number to decrease and an increase in the relative rib-cavity module length led to a decrease in the Nusselt number. Maynes and Crockett [21] studied the apparent temperature jump and also thermal transport in microchannels consisting of rib and cavity aligned in the streamwise direction. Their results revealed that the relative size of cavity in comparison with the cavity fraction affected the overall thermal behavior. It was also shown that the relative size of the rib and cavity module width compared to the relative module width influenced the thermal behavior. Enright et al. [22] presented some expressions for the thermal and hydrodynamic slip lengths for the pillar and also ridge structures. In their research, they also described the conditions under which the heat transfer rate might be enhanced. Ng and Wang [23] performed numerical approach for determining of temperature jump coefficient for different SHSs. The SHSs they focused on were parallel grooves, circular and square posts and holes. Enright and Hodes [24] examined the thermal transport behavior of a specific microchannel consisting of pillar-structured superhydrophobic surface and demonstrated that the apparent slip length could increase against the adverse microchannel temperature gradient. Cowley et al. [25] investigated the heat transfer rate in a microchannel having micro-ribs and cavities which were aligned and perpendicular to the flow direction and ware also made of a highly conductive material. In this work, they declared the significance of axial heat conduction. Cowley et al. [26] conducted a numerical approach to investigate the effects of inertia on the flow through microchannels comprised of square array of square pillars which were aligned with the flow direction. They explored the effect of Peclet number, solid fractions and relative channel spacing size on the Nusselt number, friction factorReynolds number product and temperature jump length and hydrodynamic slip length. Cheng et al. [27] assessed frictional resistance and thermal performance of flow though microchannels employing square holes, posts and transverse and longitudinal grooves. They also evaluated the combined frictional and thermal performance of microchannels by calculation of goodness factors of the considered

1.1. Problem description In this work, the microposts shape is assumed to be square and flow through an infinite width rectangular microchannel is concerned. Fig. 1 depicts the aligned and staggered micropost configurations. In this figure, H denoted the microchannel height. In order to perform numerical calculation, owing to symmetry, only half of microchannel height is considered. Shown in Fig. 2, are the top and right side views for half of the channel height for the aligned and staggered micropost structures. In Fig. 2, dark regions represent the considered computational domain and WC and W represent the cavity width and combined post and cavity width, respectively. The microchannel hydraulic diameter is defined as Dh = 4A/ Pw where A and Pw represent flow area and total liquid perimeter, respectively. Flow area and total liquid perimeter are A = Hb and Pw = 2(H + b) where b stands for the microchannel width. Assuming infinite width microchannel (i.e. b → ∞), the microchannel hydraulic diameter would be Dh = 2H . In this work, relative module width and cavity fraction are defined as Wm = W / Dh and Fc = Af / At , respectively, where Af is the cavity area and At denotes the total area of the microchannel wall. Reynolds number is also defined as R e = um Dh / ν where um and ν are mean flow velocity through the channel and kinematic viscosity, respectively. To study the heat transfer phenomenon within the microchannel, the solid portion of the solid-cavity composite surface is assumed to be subjected to a uniform heat flux. In order to assess the effect of different micropost patterns on the frictional resistance and heat transfer phenomenon of flow through the

Fig. 1. Different microchannel arrangements, a) aligned and b) staggered arrangement.

537

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 2. Different views for the a) aligned and b) staggered micropost arrangements. Dark regions represent considered computational domain.

microchannel, Poiseuille number (Po) (i.e., friction factor-Reynolds number product) and Nusselt number were examined. The significance of friction factor is a well known issue since it dictates the required pumping power for the flow through the microchannel and is calculated via the following:

Wm = 0.01, 0.1 and 1, cavity fractions of Fc = 0.1, 0.3, 0.5, 0.7 and 0.9 and Reynolds numbers of R e=10 and 100.

f = 8τw /(ρum2 )

Assuming an incompressible fluid flow through the microchannel, the governing equations are the incompressible form of the continuity, momentum and energy equations. So, adopting a constant properties fluid flow, the governing equations are as the followings:

1.2. Numerical procedure

(1)

Where τw represent average shear stress on the channel wall and ρ denotes the fluid density. The local friction factor for each x station is also defined as:

fx = 8τw, x /(ρum2 )

∂ui =0 ∂x i

(2)

In equation (2), τw, x is the area weighted average of shear stress along the y direction on the channel wall at the concerned x station. In the present work, Nusselt number, consistent with the definition presented in Ref. [22], is defined as:

Nu =

q Dh k (Tw − Tm)

ρ

qx Dh k (Tw, x − Tm, x )

∂x i

=μ

∂p ∂ ⎛ ∂uj ⎞ − ∂x j ∂x i ⎝ ∂x i ⎠ ⎜

⎟

∂ (ui T ) k ∂ ⎛ ∂T ⎞ = ∂x i ρCp ∂x i ⎝ ∂x i ⎠ ⎜

(3)

In equation (3), q and k stand for the average surface heat flux on the channel wall (i.e., the composite solid-cavity surface) and fluid thermal conductivity, respectively. In this equation, Tw is denoted as the average temperature of the solid portion of the composite solid-cavity surface and Tm represents fluid mean temperature within the microchannel. In this study, for each x station, denoting the area weighted average of temperature on the solid surface and heat flux on the composite surface along the y direction as Tw, x and qx , respectively, the local value of Nusselt number for each x station is calculated via the following expression:

Nu x =

∂ (ui uj )

(5)

(6)

⎟

(7)

In these equations, ui represents the velocity component, T is fluid temperature, p is pressure and μ , k and Cp denote the fluid viscosity, thermal conductivity and specific heat, respectively. The working fluid is water and cavities are filled with air. Shown in Fig. 3, are different boundary conditions types used for the two micropost configurations, with red areas representing the solid surface. As presented in Fig. 3, the left and right sides of the computational domain and also the flow inlet and outlet boundaries are chosen as the periodic boundaries for both micropost patterns. In this work, the liquid-cavity interface is assumed to be adiabatic and also flat with zero shear stress assumption on it [28]. The assumption of flat cavity surface is an idealization which simplifies the numerical calculation. In order to realize the order of error generated using this assumption, one could refer to the research conducted by Maynes et al. [29], who showed that for the fully developed laminar flow through microchannel having ridge and cavity parallel to the flow direction, the results for the flat interface model differed by less than 4% compared to the model assumes a curved interface. Since half of the channel height is considered for the numerical calculations, the symmetry boundary condition is also used for top of the computational domain. In this research, the grid generator software, GAMBIT, is used for generation of the grid and commercial flow solver software, FLUENT 6.3.26, is used for numerical calculations. In the considered numerical model, the second order upwind scheme is used for discretization of momentum and energy equations and SIMPLEC algorithm is implemented for the pressure-velocity coupling. The residuals are also considered as the convergence criterion and iteration is stopped as they reach to the value of less than about 10−8.

(4)

Where Tm, x is the fluid mean temperature at the considered x station. This research focuses on the condition under which the fluid does not penetrate into the cavities. Generally, the fluids flowing through the patterned microchannels experience lower frictional resistance than those flowing through the conventional microchannels employing solid walls. This means that the fluid pressure within the patterned microchannels could be low enough to avoid penetration of the fluid into the cavities. Furthermore, as also mentioned earlier, since the spacing between the micropost is in the order of micro meter, the surface tension effect could also help to avoid penetration of fluid into the cavities. In the research undertaken, hydraulic diameter is chosen to be Dh = 1 mm and a constant surface heat flux of q = 70000 w / m2 is applied at the solid portion of the microchannel wall for the all considered microchannels. For better understanding of fluid flow through the microchannel, results are obtained at relative module widths of 538

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 3. Computational domain with different boundary conditions used for numerical calculations. a) aligned and b) staggered arrangement.

1.3. Grid study and validation

Table 1 Poiseuille and Nusselt numbers for the coarse and refined grids for aligned micropost configuration.

In order to obtain grid independent results, at Reynolds number of R e=100 (i.e., the flow condition for which the highest flow variables gradient occurs) and for each microchannel geometry, a comprehensive grid study is carried out. To do this, the coarse grids correspond to each microchannels are generated firstly. Then, the refined girds for each coarse grid are generated by doubling the node number at each direction. Exploring the generated grids reveals that, for the cells adjacent to the channel wall, the distance of cell center to the channel wall for the coarse and refined grids are 10−7 and 5×10−8 m, respectively. To illustrate more regarding the topology of the generated grids, Fig. 4 presents the coarse grids for the aligned and staggered arrangements when relative module width and cavity fraction are Wm = 1 and Fc = 0.9, respectively, with red region representing opposite area to the solid portion of composite surface at the bottom of the computational domain. In this figure, denoting the first, second and third indexes as the number of computational cells in the x, y and z directions, the number of computational cells for the aligned and staggered configurations are 60 × 60 × 15 and 113 × 96 × 15, respectively. In order to perform grid independency study, Poiseuille Number and also Nusselt number obtained from the coarse grids are compared with those of the refined grids. The grid study reveals that for each microchannel geometry, the coarse grids have adequate resolution to be used at Reynolds number of R e=100. Since the flow at this Reynolds number leads to the highest velocity gradient through the channel, the appropriateness of the coarse grids in this Reynolds number ensure the adequacy of the coarse grids in analyzing the fluid flow at Reynolds number of R e=10. For more explanation, Tables 1 and 2 present Poiseuille number as well as Nusselt number for flow at Reynolds number of R e=100 at the lower and upper limits of the considered relative module width and cavity fraction values for the two micropost structures. These tables exhibit that the maximum differences between the coarse and refined grids results for the Poiseuille and Nusselt numbers are about 1%, which implies the adequacy of the generated coarse grids for the numerical calculations. In order to do validation, the present work results are compared to

Wm = 0.01 Fc = 0.1

Coarse grid Refined grid

Wm = 1 Fc = 0.9

Fc = 0.1

Fc = 0.9

Po

Nu

Po

Nu

Po

Nu

Po

Nu

95.6 95.7

8.18 8.18

89.1 89.3

7.80 7.80

93.3 93.9

7.99 8.02

13.8 14.0

1.54 1.55

Table 2 Poiseuille and Nusselt numbers for the coarse and refined grids for staggered micropost configuration. Wm = 0.01 Fc = 0.1

Coarse grid Refined grid

Wm = 1 Fc = 0.9

Fc = 0.1

Fc = 0.9

Po

Nu

Po

Nu

Po

Nu

Po

Nu

95.6 95.7

8.18 8.18

89.2 89.3

7.80 7.80

93.4 93.9

8.05 8.08

18.3 18.6

2.60 2.64

those of the theoretical and experimental approach. In Enright et al. [22], defining the Reynolds number based on the slip velocity and combined post and cavity width (i.e., R ec = ρuslip W / μ , where uslip stands for planar average of slip velocity on the channel wall), some expressions are derived theoretically for calculation of Poiseuille and also Nusselt number for flow through microchannels subjected to constant surface heat flux having micropost in aligned pattern at low values of R ec (i.e., when R ec → 0 ). So, to validate the current numerical model, for some selected cases at which a low value of R ec is attained, Poiseuille and also Nusselt numbers obtained from the current research are compared with those of Enright et al. [22]. Exploring numerical findings reveals that the microchannels operating at Reynolds number and relative module width of R e=10 and Wm = 0.1, respectively (which at most, at cavity fraction of Fc = 0.9, result in R ec = 0.4), are qualified to be used for validation of the numerical model. Fig. 5 presents the Poiseuille and Nusselt numbers associated to the present study and those of Enright et al. [22]. This figure reveals reasonable agreement between the present work results and those of theoretical approach at each cavity fraction, indicating correctness of the numerical model in prediction of Poiseuille and Nusselt numbers in the laminar flow regime. Fig. 6 compares slip length (=uslip/(∂U / ∂y ) y = 0 , where (∂U / ∂y ) y = 0 represent gradient of planar average of fluid velocity on the channel wall) which is non-dimensionalized by the pitch (i.e., the distance between the two adjacent microposts) for the staggered pattern obtained from the current research at different relative module widths and those of experimental work of Lee et al. [30] at various cavity fractions at

Fig. 4. Coarse grids used for numerical calculations. a) aligned and b) staggered configuration.

539

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 5. Comparison of Poiseuille and Nusselt numbers obtained from present work and those of theoretical approach [22].

the effect of increase in the shear stress on the channel and so, as presented in Fig. 7, Poiseuille number decreases with increasing the cavity fraction. Fig. 7 reveals that at high relative module width (i.e., Wm = 1) and at each Reynolds number, the difference between the staggered and aligned structure in terms of Poiseuille number is of little significance at low cavity fractions whereas at high cavity fraction, the staggered structure leads to higher Poiseuille number than the aligned configuration. To illustrate this phenomenon, for relative module widths of Wm = 1, cavity fraction of Fc = 0.9 and Reynolds number of R e=100, variation of local friction factor versus non-dimensional microchannel length is depicted in Fig. 9 for the two micropost patterns. Referring to this figure, one could conclude that the staggered pattern is capable of producing higher frictional resistance than the aligned structure. The cause for this could be explained base on the slip velocity on the channel wall. Exploring the work conducted by Koopaee et al. [32], one could see that the aligned pattern (which permits a continuous cavity surface) is capable of producing higher slip velocity than the staggered structure (For more illustration, one could also refer to Fig. 10 which provides the planar average slip velocity which is non-dimensionalized by the mean flow velocity through the channel versus cavity fraction for channel with relative module of Wm = 1 and Reynolds number of R e=100 for the two micropost structures). This means that for a fixed channel hydraulic diameter and Reynolds number, the aligned configuration leads to a more uniform planar average velocity distribution along the channel height than the staggered pattern. This phenomenon causes the staggered pattern to lead to a higher planar average velocity gradient on the channel wall and thus higher friction factor than the aligned arrangement. Fig. 7 also indicates that for each Reynolds number, the difference between the two micropost arrangements decreases as the cavity fraction decreases owing to mitigation of cavity role in reducing the friction factor. Fig. 7 shows that for each cavity fraction and Reynolds number, with decreasing the relative module width, Poiseuille number for each micropost structure increases and difference between the two micropost arrangements tends to decrease. As expected, for a fixed hydraulic diameter channel (i.e., fixed channel height), a decrease in the relative module width causes the combined post and cavity width to decrease, resulting in an increase in the number of posts for a specified microchannel length. This effect leads to the increase in the number of times of flow acceleration and deceleration for a fixed microchannel length and so increase in the Poiseuille number. To explain more, the local friction factor versus non-dimensional microchannel length for relative module widths of Wm = 0.1, cavity fraction of Fc = 0.9 and Reynolds number of R e=100 is shown in Fig. 11 for the two micropost patterns. Comparison of Fig. 9 with Fig. 11 shows that for a specified channel length, contrary to the case with relative module width of Wm = 1, the channel with relative module width of Wm = 0.1 confronts frequent high friction factor at the leading and trailing edges of each post due to frequent flow deceleration and acceleration. This effect results in the

Fig. 6. Comparison of non-dimensionalized slip length obtained from present work and those of experiment [30].

Reynolds number of R e = 11.85. As this figure shows, the present work results and those of experiment are in a good concordance. In the present work, a straight channel is concerned whereas in the experimental work of Lee et al. [30], a circular channel is used and as also presented in Ref. [31], the more deviation between the numerical approach and experimental work results around the cavity fraction of Fc = 0.5 could be attributed to the geometry effect. 1.4. Numerical results and discussion 1.4.1. Effect of micropost patterns on the Poiseuille number Fig. 7 represents the effect of cavity fraction on Poiseuille number for the aligned and staggered micropost patterns at the considered relative module widths and Reynolds numbers. This figure shows that as the cavity fraction decreases to the lower limit, the value of Poiseuille number approaches to the value of Po = 96, which corresponds to that of the smooth channel. With increasing the cavity fraction, the free shear area increases and so the fluid particles travel a longer passage as they leave a post toward the next one. This effect, causes the flow deceleration and acceleration at the leading and trailing edges of the solid surface to become significant, resulting in a higher local shear stress at the leading and trailing edges of the solid surface and consequently on the solid surface. For more explanation, one could refer to Fig. 8 which presents the distribution of area weighted average of shear stress along the y direction versus non-dimensionalized channel length for the two micropost arrangements for different cavity fractions when relative module width and Reynolds number are Wm = 1 and R e=100, respectively. Although an increase in the cavity fraction leads to an increase in the local shear stress on the solid portion of the microchannel wall, however, the effect of increase in the cavity surface is superior to 540

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 7. Effect of cavity fraction on Poiseuille number for different micropost pattern at different Reynolds numbers and relative module widths.

micropost patterns in terms of the friction factor and so Poiseuille number increases as the relative module width increases. This issue could be confirmed by comparison of Fig. 9 with Fig. 11, indicating a negligible discrepancy for the local friction factor between the two micropost patterns for the channel at relative module width of Wm = 0.1 than the channel having relative module width of Wm = 1. Referring to Fig. 7, one could see that for each cavity fraction, at low relative module widths (i.e., Wm = 0.01 and 0.1), an increase in the Reynolds number leads to a low change in Poiseuille number for each micropost pattern due to low cavity width and thus low contribution of cavity in reducing the frictional resistance. Results show that at high relative module width, the effect of increase in Reynolds number on the Poiseuille number becomes significant for the two patterns owing to high cavity width and thus high influence of cavity in frictional resistance reduction. As presented in Fig. 7, at high relative module width, the impact of increase in the Reynolds number on the Poiseuille number is higher in the staggered pattern than the aligned arrangement. To illustrate this, the streamline around the two adjacent microposts on the channel

increase in Poiseuille number as the relative module width decreases for a certain microchannel length. Comparison of Fig. 9 with Fig. 11 also indicates that with decreasing the relative module width, the local friction factor increases for the two micropost arrangements. Referring to the research carried out by Koopaee et al. [32], one could conclude that a decrease in the relative module width causes the slip velocity to decrease and so, for a fixed channel hydraulic diameter and Reynolds number, the gradient of planar average velocity on the channel wall to increase. Consequently, one could conclude that the local friction factor increases with decreasing the relative module width. Fig. 7 indicates that for each cavity fraction and Reynolds number, the difference between the two micropost configurations increases with increasing the relative module width. This could be explained by recognizing that, for a fixed cavity fraction, a microchannel with a high relative module width yields higher cavity width in comparison with that having a low relative module width. This means that the contribution of cavity in reducing the friction factor is higher in microchannels with high relative module width than those having low relative module width. Therefore, the difference between the two

Fig. 8. Local shear stress versus non-dimensionalized channel length for different cavity fraction for the a) aligned and b) staggered arrangements when relative module width and Reynolds number are Wm = 1 and R e=100, respectively.

541

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 12. Streamlines at relative module of Wm = 0.1, cavity fractions of Fc = 0.3 to 0.9 and Reynolds number of R e=100 for aligned configuration. White regions represent the solid area.

Fig. 9. Local friction factor versus non-dimensionalized channel length for relative module width of Wm = 1, cavity fraction of Fc = 0.9 and Reynolds number of R e=100 for the two micropost arrangements.

provides a continuous cavity surface along the streamwise direction. Fig. 13 represents that for the staggered structure, contrary to the aligned arrangement, no continuous cavity surface in the direction of flow is observed for cavity fraction of less than Fc = 0.7. Thus, the staggered pattern leads to the frequent flow deceleration and acceleration appearing at the leading and trailing edges of the solid portion of the channel wall at each spanwise (y-direction) location (a condition which could not be observed in the aligned pattern). Accordingly, the effect of increase in the Reynolds number on the Poiseuille number in the staggered arrangement is more significant than the aligned one. Fig. 7 shows that at cavity fraction of Fc = 0.9, the impression of increase in the Reynolds number on the Poiseuille number for the staggered pattern tends to decrease. Exploring Fig. 13 indicates that, the causes for this is the appearance of continuous cavity surface at cavity fraction of Fc = 0.9 which leads the effects of flow deceleration and acceleration phenomena to decrease, resulting in the lower impact of Reynolds number on the Poiseuille number. Fig. 7 also reveals that at high relative module width, an increase in the Reynolds number causes the Poiseuille number of the two micropost configurations to increase and difference between the two micropost structures increases with increasing the Reynolds number. Physically, these are due to increase in the inertial effect as the Reynolds number increases and thus, higher impact of flow deceleration and acceleration phenomena on the Poiseuille number.

Fig. 10. Non-dimensional planar average slip velocity versus cavity fraction for relative module width of Wm = 1 and Reynolds number of R e=100 for the two micropost arrangements.

surface at relative module of Wm = 0.1, cavity fractions of Fc = 0.3 to 0.9 and Reynolds number of R e=100 are presented in Figs. 12 and 13 for both micropost patterns. Fig. 12 reveals that the aligned pattern

Fig. 11. Local friction factor versus non-dimensionalized channel length for relative module widths of Wm = 0.1, cavity fraction of Fc = 0.9 and Reynolds number of R e=100 for a) aligned and b) staggered pattern.

542

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

1.4.2. Effect of micropost patterns on the Nusselt number Fig. 14 depicts variation of Nusselt number versus cavity fraction for the two micropost configurations for different relative module widths and Reynolds numbers. With decreasing the cavity fraction, this figure represents the value of 8.235 for Nusselt number which is associated to the smooth channel. This figure indicates that, since the cavity area is treated as an adiabatic surface, Nusselt number decreases as the cavity fraction increases. Fig. 14 reveals that at each Reynolds number, the staggered pattern leads to higher Nusselt number than the aligned one and this phenomenon becomes noticeable with increasing the relative module width. To illustrate this phenomenon, the contour of velocity magnitude on the channel surface at relative module width of Wm = 1, cavity fractions of Fc = 0.7 and Reynolds number of R e=100 for the two micropost arrangements are presented in Fig. 15. This figure shows that the staggered pattern allows a longer passage than the aligned structure for some fluid particles when they leave a post toward the next one. Referring to Fig. 15, one could see that this effect causes these fluid particles in the staggered structure to experience a higher velocity than the aligned pattern when they reach a micropost, resulting in a thinner thermal boundary layer and so higher channel Nusselt number. For more explanation, the local values of Nusselt numbers for the staggered and aligned configurations for relative module width of Wm = 1, cavity fraction of Fc = 0.5 and Reynolds number of R e=100 are compared in Fig. 16. As presented in this figure, for the two micropost structures, the local Nusselt number in the leading and trailing edge of solid surface increases which is due to flow deceleration and acceleration. Fig. 16 affirms the capability of the staggered arrangement in producing higher Nusselt number than the aligned structure. Fig. 14 shows that at each cavity fraction, Reynolds number and for each micropost pattern, Nusselt number increases with decreasing the relative module width. As discussed before, for a fixed hydraulic diameter channel (i.e., fixed channel height), a decrease in the relative

Fig. 13. Streamlines at relative module of Wm = 0.1, cavity fractions of Fc = 0.3 to 0.9 and Reynolds number of R e=100 for staggered configuration. White regions represent the solid area.

Fig. 14. Effect of cavity fraction on Nusselt number for different micropost pattern at different relative module widths and Reynolds numbers.

543

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 15. Contours of velocity magnitude (m/s) on the channel surface for relative module width of Wm = 1, cavity fraction of Fc = 0.7 and Reynolds number of R e=100 for a) aligned and b) staggered structure. White region represents the solid surface.

flow deceleration and acceleration at each spanwise location, a condition which could not be observed in the aligned arrangement due to existence of continuous free shear area along the streamwise direction. This phenomenon causes the effect of increase in the Reynolds number on the Nusselt number for the staggered arrangement to become more significant than the aligned one. Fig. 14 shows that at cavity fraction of Fc = 0.9, the effect of increase in the Reynolds number on the Nusselt number for the staggered pattern tends to decrease due to appearance of continuous cavity surface along the streamwise direction (see Fig. 13). Fig. 14 represents that for high relative module width and for two micropost patterns, an increase in the Reynolds number leads to an increase in the Nusselt number. As the Reynolds number increases, one could infer that a thinner thermal boundary layer is formed where the flow decelerates (leading edge of the solid surface) or accelerates (trailing edge of the solid surface), resulting in the increase in the Nusselt number. Results show that as the Reynolds number increases, the difference between the two micropost patterns increases due to higher role of boundary layer thinning.

Fig. 16. Local Nusselt number for relative module width of Wm = 1, cavity fraction of Fc = 0.5 and Reynolds number of R e=100 for the two micropost arrangements.

module width leads to an increase in the number of posts for a fixed microchannel length. This phenomenon causes the number of times of flow acceleration and deceleration to increase for a specified microchannel length, resulting in the increase in the channel Nusselt number. As presented in Fig. 14, at each cavity fraction and Reynolds number, the difference between the two micropost arrangements increases with increasing the relative module width. Similar to the reasoning presented for Poiseuille number, for each cavity fraction, this is due to higher cavity width and so higher contribution of cavity in influencing the Nusselt number. Fig. 14 indicated that at each cavity fraction and also at low relative module widths, an increase in the Reynolds number leads to a low change in Nusselt number for each micropost pattern due to low cavity width and so little impact of cavity. Numerical findings indicate that with increasing the relative module width, due to high cavity width and thus high role of cavity in the heat transport phenomenon, the impression of increase in the Reynolds number on the Nusselt number becomes significant. Referring to Fig. 14, one could see that for high relative module width, the effect of increase in the Reynolds number on the Nusselt number is higher in the staggered pattern than the aligned configuration. To justify this phenomenon, one could refer to Figs. 12 and 13 which depicts the streamlines on the channel surface for the two patterns. As mentioned earlier, the staggered pattern leads to the frequent

1.4.3. Effect of micropost patterns on the overall microchannel performance In the previous sections, the effect of different relevant parameters on the Poiseuille number as well as Nusselt for the flow through microchannels employing microposts in aligned and staggered structures as the channel walls were discussed. One of the most significant issues in the field of fluid flow through the heat sink microchannels is to identify the condition at which the channel may operate with high overall performance (i.e., the condition of high heat transfer coefficient along with the low pumping power requirement). Consistent with the definition presented in [33], the goodness factor, which compares somehow the channel performance in terms of heat removal capacity and frictional resistance, is defined as the following:

φ=

Nu fRe Pr 1/3

(8)

Where Pr denotes the fluid prandtl number. The channel thermal performance index (η), is defined as the ratio of the channel goodness factor to that of the laminar fully developed flow through a two parallel-plate channel. So, devoting the values of 8.235 and 96 for the Nusselt number and Poiseuille number, respectively, for the laminar fully developed flow through a two parallel-plate channel, the microchannel thermal performance index is defined as the following: 544

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

Fig. 17. Effect of cavity fraction on the thermal performance index for different micropost patterns at different relative module widths and Reynolds numbers.

η=

(Nu/fRe ) (8.235/96)

the patterned microchannels lead to the higher performance index than the conventional ones.

(9)

Fig. 17 represents the thermal performance index at different cavity fractions for the two micropost configurations for different relative module widths and Reynolds numbers. Fig. 17 shows that at low relative module width, the change in the cavity fraction, Reynolds number and also micropost pattern has a negligible effect of the thermal performance index and the channel thermal performance is the same as the two parallel-plate channel. This figure shows that as the relative module width increases, the effect of change in the microchannel parameters on the thermal performance index becomes manifest. As presented in Fig. 17, the staggered pattern could yield a higher thermal performance index than the aligned one. One could also see that for each micropost structure, the high thermal performance index is achieved at high cavity fractions and an increase in the Reynolds number causes the thermal performance index to increase. Fig. 17 represents that the highest thermal performance corresponds to the staggered structure at relative module width of Wm = 1, cavity fraction of Fc = 0.9 and Reynolds number of R e=100 and the thermal performance index in this case is about 1.65 times better than that of the simple two parallel-plate channel. If one focuses on the heat exchange capability in the patterned microchannels, it could be seen that the patterned microchannels always lead to a lower heat transfer rate than the conventional microchannels (which used solid surface as the channel wall) due to lower solid surface occurs in the patterned microchannels. In fact, employing the micropost structure will never lead to the heat transfer enhancement. However, the patterned microchannels result in a lower drag than the conventional ones. So, if one compares the fluid flow and the heat transfer phenomenon within the patterned microchannels with those within the conventional microchannels, it could be seen that in the patterned microchannels, the reduction in the pumping power is higher than the reduction observed in the heat transfer rate, and thus,

2. Conclusion In this research, the effect of change in the micropost arrangement from the aligned pattern to the staggered one on the Poiseuille, Nusselt number and also overall performance of the heat sink microchannels are investigated in the laminar flow regime at various relative module widths, cavity fractions and Reynolds numbers. In the considered range of parameters, following results could be drawn.

• At each relative module width and Reynolds number, Poiseuille and

•

•

• 545

Nusselt numbers for each micropost configuration decreases with increasing the cavity fraction. Results show that for each cavity fraction and Reynolds number, an increase in the relative module width results in a decrease in Poiseuille and Nusselt numbers of each micropost pattern. For each micropost configuration, the effect of change in the Reynolds number on the Poiseuille and Nusselt numbers increases with increasing the relative module width. Numerical findings show that an increase in the Reynolds number causes the Poiseuille and Nusselt numbers of the two micropost patterns to increase and this increase is more pronounced for the staggered structure. At low values of relative module widths and for each Reynolds number and cavity fraction, the difference between Poiseuille and Nusselt numbers for the aligned and staggered arrangement is of little significance. With increasing the relative module width, the difference between Poiseuille and Nusselt numbers of the two micropost arrangements increases and the staggered pattern leads to higher Poiseuille and Nusselt numbers. At low relative module width, the effect of change in the cavity fraction and Reynolds number on the overall microchannel performance is negligible for each micropost pattern. Results declare that

International Journal of Thermal Sciences 124 (2018) 536–546

M. Kharati-Koopaee, M.R. Akhtari

• •

the effect of change in the aforementioned parameters on the microchannel performance increases with increasing the relative module width and at each cavity fraction and Reynolds number, the staggered pattern leads to a higher performance than the aligned structure. Results reveal that for each micropost structure and at each cavity fraction, an increase in the Reynolds number enhances the microchannel performance and the channel enhancement in the staggered arrangement is more significant than the aligned structure. It is shown that for each micropost pattern and at each Reynolds number, the highest overall microchannel performance is achieved at high values of relative module width and cavity fraction.

Appl Therm Eng 62 (2) (2014) 791–802. [14] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 1: heat transfer, Int J Heat Mass Transf 97 (2016) 1069–1080. [15] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 2: pressure drop, Int J Heat Mass Transf 97 (2016) 1081–1090. [16] L. Chai, G.D. Xia, H.S. Wang, Parametric study on thermal and hydraulic characteristics of laminar flow in microchannel heat sink with fan-shaped ribs on sidewalls – Part 3: performance evaluation, Int J Heat Mass Transf 97 (2016) 1091–1101. [17] G. Xie, J. Liu, Y. Liu, B. Sunden, W. Zhang, Comparative study of thermal performance of longitudinal and transversal-wavy microchannel heat sinks for electronic cooling, ASME J Electron Packag 135 (2) (2013) 9 021008. [18] W. Barthlott, C. Neinhuis, Purity of the sacred lotus, or escape from contamination in biological surfaces, Planta 202 (1) (1997) 1–8. [19] J. Ou, J.P. Rothstein, Direct velocity measurements of the flow past drag-reducing ultrahydrophobic surfaces, Phys Fluids 17 (10) (2005) 10 103606. [20] 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, ASME J Heat Transf 135 (2) (2013) 10 021701. [21] D. Maynes, J. Crockett, Apparent temperature jump and thermal transport in channels with streamwise rib and cavity featured superhydrophobic walls at constant heat flux, ASME J Heat Transf 136 (1) (2014) 10 011701. [22] R. Enright, M. Hodes, T. Salamon, Y. Muzychka, Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels, ASME J Heat Transf 136 (1) (2014) 9 012402. [23] C.-O. Ng, C.Y. Wang, Temperature jump coefficient for superhydrophobic surfaces, ASME J Heat Transf 136 (6) (2014) 6 064501. [24] R. Enright, M. Hodes, Analysis and simulation of heat transfer in a superhydrophobic microchannel, Proceedings of the 14th International Heat Transfer Conference, August 8-13, Washington, DC, USA, 2010. [25] A. Cowley, D. Maynes, J. Crockett, Effective temperature jump length and influence of axial conduction for thermal transport in superhydrophobic channels, Int J Heat Mass Transf 79 (2014) 573–583. [26] A. Cowley, D. Maynes, J. Crockett, Inertial effects on thermal transport in superhydrophobic microchannels, Int J Heat Mass Transf 101 (2016) 121–132. [27] 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. [28] Y.P. Cheng, C.J. Teo, B.C. khoo, Microchannel flow with superhydrophobic surfaces: effects of Reynolds number and pattern width to channel height ratio, Phys Fluids 21 (12) (2009) 12 122004. [29] D. Maynes, K. Jeffs, B. Woolford, B.W. Webb, Laminar flow in a microchannel with hydrophobic surface patterned micro-ribs oriented parallel to the flow direction, Phys Fluids 19 (9) (2007) 12 093606. [30] C. Lee, C.-H. Choi, C.-J. Kim, Structured surfaces for giant liquid slip, Phys Rev Lett 101 (2008) 4 064501. [31] M.A. Samaha, H.V. Tafreshi, M. Gad-el-Hak, Modeling drag reduction and meniscus stability of superhydrophobic surfaces comprised of random roughness, Phys Fluids 23 (2011) 8 012001. [32] M.K. Koopaee, M. Jahanmiri, H.M. Zadeh, Numerical investigation of drag reduction in microchannels with superhydrophobic walls consist of aligned and staggered microposts, International Proceedings of Computer Science and Information Technology, vol. 33, 2012, pp. 104–109 (Singapore). [33] A.L. London, Compact heat exchangers, Mech Eng ASME 86 (1964) 31–34.

References [1] P. Xu, X.Q. Wang, A.S. Mujumdar, C. Yap, B.M. Yu, Thermal characteristics of threeshaped microchannel nets with/without loops, Int J Therm Sci 48 (11) (2009) 2139–2147. [2] Z.Y. Zheng, D.F. Fletcher, B.S. Haynes, Chaotic advection in steady laminar heat transfer simulations: periodic zigzag channels with square cross-sections, Int J Heat Mass Transf 57 (1) (2013) 274–284. [3] Y. Sui, C.J. Teo, P.S. Lee, Direct numerical simulation of fluid flow and heat transfer in periodic wavy channels with rectangular cross-sections, Int J Heat Mass Transf 55 (1–3) (2012) 73–88. [4] M. Kharati-Koopaee, M. Zare, Effect of aligned and offset roughness patterns on the fluid flow and heat transfer within microchannels consist of sinusoidal structured roughness, Int J Therm Sci 90 (2015) 9–23. [5] T.V. Nizkaya, E.S. Asmolov, J. Zhou, F. Schmid, O.I. Vinogradova, Flows and mixing in channels with misaligned superhydrophobic walls, Phys Rev E 91 (3) (2015) 7 033020. [6] Q.T. Nguyen, D.V. Papavassiliou, Turbulent plane Poiseuille-Couette flow as a model for fluid slip over superhydrophobic surfaces, Pysical Rev E 88 (6) (2013) 11 063015. [7] C.T. Fairhall, R. Garcia-Mayoral, Influence of surface texture on drag reduction of superhydrophobic surfaces, European drag reduction and flow control meeting, march 23-26, 2015 (Cambridge, UK). [8] B. Woolford, D. Maynes, B.W. Webb, Liquid flow through microchannels with grooved walls under wetting and superhydrophobic conditions, Microfluid Nanofluidics 7 (1) (2009) 121–135. [9] T.-C. Hung, T.-S. Sheu, W.-M. Yan, Optimal thermal design of microchannel heat sinks with different geometric configurations, Int Commun Heat Mass Transf 39 (10) (2012) 1572–1577. [10] B. Wei, M. Yang, Z. Wang, H. Xu, Y. Zhang, Flow and thermal performance of a water-cooled periodic transversal elliptical microchannel heat sink for chip cooling, J Nanosci Nanotechnol 15 (4) (2015) 3061–3066. [11] G. Xie, J. Liu, W. Zhang, B. Sunden, Analysis of flow and thermal performance of a water-cooled transversal wavy microchannel heat sink for chip cooling, ASME J Electron Packag 134 (4) (2012) 6 041010. [12] L. Chai, G. Xia, M. Zhou, J. Li, J. Qi, Optimum thermal design of interrupted microchannel heat sink with rectangular ribs in the transverse microchambers, Appl Therm Eng 51 (1–2) (2013) 880–889. [13] G. Xie, F. Zhang, B. Sundén, W. Zhang, Constructal design and thermal analysis of microchannel heat sinks with multistage bifurcations in single-phase liquid flow,

546