Numerical study on heat transfer and pressure drop in laminar-flow multistage mini-channel heat sink

Numerical study on heat transfer and pressure drop in laminar-flow multistage mini-channel heat sink

International Journal of Heat and Mass Transfer 108 (2017) 1197–1206 Contents lists available at ScienceDirect International Journal of Heat and Mas...

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International Journal of Heat and Mass Transfer 108 (2017) 1197–1206

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Numerical study on heat transfer and pressure drop in laminar-flow multistage mini-channel heat sink Yeongseok Kim a,b, Myungjoon Kim a, Chisung Ahn c, Hyeong U. Kim c, Sang-Woo Kang b, Taesung Kim a,c,⇑ a

School of Mechanical Engineering, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi 440-746, Republic of Korea Vacuum Center, Korea Research Institute of Standards and Science, 267 Gajeong-ro, Yuseong-gu, Daejeon 305-340, Republic of Korea c SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi 440-746, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 10 February 2016 Received in revised form 7 December 2016 Accepted 10 December 2016

Keywords: Multistage Mini-channel Heat transfer Pressure drop Numerical study

a b s t r a c t Mini-channel has been more studied recently than micro-channel to optimize the heat emission and pressure drop by regulating the channel size and length. In this work, a multistage mini-channel heat sink using water coolant was designed to obtain a larger cooling rate in a small area with a lower pressure drop. To confirm the performance of the structure, we conducted numerical simulations under laminar and single-phase conditions. The diameter and length of the channel were 2 and 530 mm, respectively. From the simulation, the local convection coefficient, coolant temperature, channel-wall temperature, effectiveness, and pressure drop were analyzed in relation to the mass flux, heat-source temperature, and number of stage stacks. To obtain valid simulation results on the heat transfer, we used wellmatched conventional correlation. The result of the pressure drop was compared with the experimental result to confirm the validity of the hydrodynamic model. The simulation result shows that the maximum cooling rate was 40 W/cm2 at a pressure drop of 1383 Pa in a quintuple-stage model. However, the triplestage structure had the best effectiveness of 0.83 under the same simulation conditions. The pressure drop of the multistage structure was higher than that of the single-stage structure. However, the increase of the total pressure drop was small as against the increase of the cooling rate. Ó 2016 Published by Elsevier Ltd.

1. Introduction With the advancement in nanotechnology, integrated circuits (ICs) have steadily become smaller to reduce electrical energy consumption and to achieve high calculation performance. In particular, the state-of-the-art IC has gradually become smaller toward 10 nm or less to achieve high density and high performance in large-scale integration (LSI). The downscaling of ICs suffers from various concerns in terms of physical limitation related on the leakage current, leading to heterogeneous three-dimensional (3D) integration from homogeneous two-dimensional (2D) integration [1]. Even though the heterogeneous 3D integration consumes lesser power than the 2D integration, LSIs still require effective cooling applications to discharge the generated heat from the surface of different chips such as semiconductor and photonic device chips. Liquid cooling process using micro-/mini-channel is one of the efficient candidates for small-scale cooling applications owing to their high heat-transfer performance. As described in [2], Kandlikar ⇑ Corresponding author at: School of Mechanical Engineering, Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi 440-746, Republic of Korea. E-mail address: [email protected] (T. Kim). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.025 0017-9310/Ó 2016 Published by Elsevier Ltd.

et al. classified micro- and mini-channels in terms of the smallest channel dimension and suggested applications depending on the ranges of channel diameters. The dimension range of a microchannel is within 10–200 lm, and that of a mini-channel is 200– 3000 lm. The micro-/mini-channel was suggested by Tuckerman and Pease [3] in 1981 as a novel heat sink for high-powerconsumption IC devices. They experimentally obtained heat dissipation of 790 W/cm2 at a maximum temperature difference of 71 °C where no phase transition occurred between the substrate and input water temperature. Wang and Peng [4] also conducted similar experiments on heat transfer and flow behavior according to the Reynolds number (Re). According to their work, a fully developed turbulent regime started at Re of from 1000 to 1500. For further study, Peng and Peterson [5] conducted additional experiments on heat transfer and flow in relation to the hydraulic diameter of a micro-channel in which the investigation ranged between 133 and 367 lm. In particular, the effects on the heat transfer and pressure drop according to the shape and size of the micro-/mini-channel were investigated by Morini [6]. According to his work, various conditions such as coolant [7,8] and geometry [3–5,9,10] were used to establish a conventional theory for micro-/ mini-channels. And study on various channel shapes was con-

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ducted by Khan and Kim [11], recently. These results were numerically verified by Dang and Teng [12]. They compared the hydrodynamics and heat-transfer behavior between micro- and minichannels using numerical methods, which agreed well with the experimental data. Their work demonstrated that a microchannel heat exchanger could be obtained with 1.2–1.53 times higher effectiveness (defined as the ratio of actual and maximum heat-transfer rates) than the mini-channel. However, despite their high heat-transfer performance, microchannels are inappropriate because of their huge pressure drop. Therefore, mini-channel has been more studied recently than the micro-channel. Wang et al. [13] demonstrated that the influence of friction characteristics can be neglected when the hydraulic diameter is larger than 1.0 mm, and their measurements agreed well with the conventional correlations. On the other hand, Reynaud et al. [14] measured the friction and heat-transfer coefficients in minichannels with diameters of 0.3–1.12 mm. The experimental results on heat transfer agreed well with the classical correlations in a minichannel. Because the heat-transfer behavior and hydrodynamics in a mini-channel agreed with the conventional correlations, numerical studies were conducted to design a small-size heat sink with simultaneous relatively high heat-transfer performance and low pressure drop. Xie et al. [15,16] conducted numerical simulation on laminar and turbulent mini-channel heat sink whose bottom size was 20 mm  20 mm to analyze the effect of channel dimensions, channel-wall thickness, bottom thickness, and inlet velocity on the pressure drop, thermal resistance, and maximum allowable heat. On the other hand, the design of the micro-/mini-channels in these works was similar to the 2D model suggested by Tuckerman and Pease. The geometric conditions of this model, such as the channel size and length, could control the heat transfer and pressure drop. To obtain high heat-transfer performance in a small cooling area, the model required a micro-scale channel size that generates a huge pressure drop because a longer channel should occupy a larger area. As a result, alternative designs were suggested by various research groups. Kandlikar and Upadhye [17] suggested split flow arrangement to extend the multiple inlets and outlets as originally recommended by Tuckerman and Pease [3]. This design could reduce the flow length of the fluid stream in the micro-channel by half, and the fluid flow rate through the micro-channels was also halved. This structure could increase the heat transfer coefficient near the multiple channel entrance regions due to the thermally developing flow. Colgan et al. [18] also studied the offset strip-fin arrangement in micro-channels. They obtained the high heat flux dissipation by applying the staggered fin arrangements, although this structure had the high pressure drop caused by high associated friction factors. In the present work, we designed a long mini-channel that is stacked multiple times, as an alternative heat transfer concept, to maintain high heat-transfer performance without a large increase in the pressure drop. Subsequently, we performed 3D numerical simulation to determine its heat transfer and pressure drop characteristics to analyze the interaction between stages. In the following discussion, the outline of a multistage mini-channel heat sink is introduced, followed by its modeling to perform numerical simulation. Then, simulation results are presented relative to the heatsource temperature, coolant mass flux, and number of stacked stages. Finally, a brief conclusion is presented.

2. Modeling 2.1. Schematics The concept of a multistage mini-channel heat sink is a 3D model that appears like a folded stacked 2D model (single stage

structure), as shown in Fig. 1a. So the channel length of each stages Lc,n, which belongs to nth stage mini-channel, decreases depending on n. However, this concept can maintain a long channel length Lc and decreases the target cooling area. The heat sink is heated by heat source, which is at the bottom and has uniform temperature. As shown in Fig. 1b, the target cooling area, which is same with the heat source area As, is the bottom surface area, and the height of the stage is designated as h. The mini-channel height and width are hc and wc, and the distance between channels is w. Similar to the 2D model, each mini-channel does not cross each other and has equal geometrical configuration such as the channel length and size. Water coolant with a constant temperature of 290 K is injected into the inlet located at the top stage, and the hot coolant, heated by internal convection from the channel surface, is ejected at the outlet from the bottom stage exhaust of an aluminum solid body heat sink. This configuration can reduce the temperature gradient at the bottom surface because the heat transfer from the channel wall to the coolant is reduced by the small temperature difference between the channel wall and coolant. To exclude the influence from the surrounding, such as convection and radiation, we assume that the surface of the heat sink is insulated except for As. Multistage mini-channel heat sink could be fabricated by assembling the caps and divider as shown in Fig. 1c. Caps are top and bottom units surrounding the divider to generate the insulated channels. Stacking the several dividers fabricates above the triplestage mini-channel. 2.2. Design To simulate the water coolant mini-channel heat sink, three types of physical processes were considered. First, the hydrodynamics in the water coolant needs to be defined to simulate the velocity profile and pressure drop in the channel. Then, the conduction in the heat-sink body must be considered. Maranzana et al. [19] demonstrated that the channel-wall temperature becomes largely non-uniform under small Re, which indicates a laminar-flow regime, because most of the heat flux is transferred to the coolant at the entrance of the micro-/mini-channel. Finally, convection between the coolant and channel wall is also a key factor in the modeling. In particular, the conventional correlation agrees well in the laminar-flow regime, as demonstrated in [5–12]. In this work, we designed a partial 3D structure to simulate the hydrodynamics and heat transfer behavior in the coolant and channel wall, as shown in Fig. 2b. This partial structure can be made available as a model that is expected to yield symmetric simulation results, especially in the middle of the heat sink. The geometrical conditions of the mini-channel multistage structure are listed in Table 1. To confirm the effect of the multistage on the heat transfer and pressure drop, we used a long channel length to minimize the local convection decrease and to obtain a sufficient hydrodynamically and thermally developed region which is favorable to perform the CFD on the internal flow, because the condition is robust for calculation. When the hydraulic diameter of the channel (Dh = hc  wc/2 (hc + wc)) is 2 mm, which is a large but typical diameter of a mini-channel. Generally, the channel length Lc is dozens of times of Dh to develop the flow fully in case of laminar internal flow. So, Lc was determined based on the minimum stage channel length Lc,5 (106 mm = 53 Dh) of the quintuple stage that had the shortest stage channel length to develop the flow. So, the Lc was defined as 530 mm (= 5 Lc,5). In addition, we assumed that the coolant flow is incompressible and laminar to simplify the calculation model. The mean of the injected coolant and the heat-source temperature (Tc,m = (Tc,i + Ts)/2) was used to define the thermophysical properties such as viscosity and density, which are strongly dependent on the temperature. Heat-source temperature Ts of under 370 K was used to

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Lc,1 = Lc

Multistage Mini-channel Heat sinks (n = 1 ~ 5)

Coolant Inlet

Heat

Convection

Cold

Lc,2

Hot emission Lc,3

Coolant Outlet Conduction

Lc,4

Lc,5

Heat source Lc,n = 1/n · Lc

a As

Inlet outlet

h

z

x

hc

y

wc

Lc,n z

w

x y

b

c

Fig. 1. Simplified concept of the multistage mini-channel heat sinks is panel a. And multistage mini-channel heat sink stacked two times. b: Schematic diagram. c: Assembly diagram.

a

A*

z y

c

D

*

A E*

E

I*

I

L*

L

H*

H

b

z x

D

y B

A E

z

I

J

L

K

H x

F

D

G C

Fig. 2. Symmetric 3D model for a mini-channel heat sink stacked two times. Panels a–c show the front, isometric, and right-side views, respectively. White arrows indicate the coolant directions in the panel b. The labels in panels a and c are used in Table 2 to define the boundaries. The gray and white portions indicate solid and fluid parts, respectively.

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Table 1 Geometrical conditions. Number of stage stacks n Target cooling area As Channel diameter hc, wc, Dh Channel length Lc Channel distance w Stage channel length Lc,n Stage height h Material//thermal conductivity

[cm2] [mm] [mm] [mm] [mm] [mm]

1 2 10.6 5.3 2 530 2 530 265 4 Aluminum//160 W/m K

prevent boiling of the coolant. The wall roughness was neglected to focus on the effects of the multistage. We conducted simulations on heat emission and pressure drop in terms of Ts, coolant mass flux M, and number of stage stacks n. And the simulation was conducted under steady state condition to evaluate the performance of the heat transfer. These independent variables are listed in Table 2. 2.3. Governing equations and parameters To simulate the phenomenon introduced in Section 2.2, computational fluid dynamics (CFD) was implemented using the commercial code COMSOL Multiphysics, which uses finite element method and can apply various physical processes to the model. The boundary conditions listed in Table 3 for a periodic mini-channel model were adapted. The heat equation was adopted to describe the conduction in the heat-sink body as given by

qC p @[email protected] ¼ r  ðkrTÞ þ fe

ð1Þ

In steady-state condition, the first term @T/@t is ignored, and the conduction heat flux term q ¼ krT can be obtained by line integration of specific-heat emission fe at a direction normal to the mini-channel wall, which describes cooling rate q due to the convection from the interface between the channel wall and coolant. Therefore, heat emission Qe is described by surface integration of q with respect to the surface area of channel wall Acw. k, q and Cp were thermal conductivity, density and specific heat of heat-sink body, respectively. Generally, q can be calculated by

q ¼ hx ðT w  T c Þ ¼ Q e =As

3 3.533333

4 2.65

5 2.12

176.6667

132.5

106

Table 3 Boundary conditions for numerical simulation. Face

Condition

Solid part

Conduction

A⁄-B⁄-C⁄-D⁄ A-B-C-D-H-G-F-E I-J-K-L E⁄-F⁄-G⁄-H⁄-H-G-F-E I-J-K-L-L⁄-K⁄-J⁄-I⁄ E⁄-F⁄-G⁄⁄-H⁄-L⁄-K⁄-J⁄-I⁄ A⁄-A-E-E⁄-I⁄-I-L-L⁄-H⁄-H-D-D⁄ A⁄-B⁄-C⁄-C-B-A D⁄-C⁄-C-D

Symmetry Symmetry Symmetry Convection Convection Convection Insulation Insulation Constant temperature

Fluid part

Convection

E-F-G-H-L-K-J-I E⁄-E-I-I⁄ L⁄-L-H-H⁄ E⁄-F⁄-G⁄-H⁄-H-G-F-E I-J-K-L-L⁄-K⁄-J⁄-I⁄ E⁄-F⁄-G⁄-H⁄-L⁄-K⁄-J⁄-I⁄

Symmetry Constant temperature Outflow Convection Convection Convection

Fluid part

Laminar flow

E-F-G-H-L-K-J-I E⁄-E-I-I⁄ L⁄-L-H-H⁄ E⁄-F⁄-G⁄-H⁄-H-G-F-E I-J-K-L-L⁄-K⁄-J⁄-I⁄ E⁄-F⁄-G⁄-H⁄-L⁄-K⁄-J⁄-I⁄

Symmetry Inlet Outlet Wall Wall Wall

2 Nux þ 1 5:364½1 þ ðGz=55Þ10=9 

3=10

¼ 41 þ

Gz=28:8 1=2

½1 þ ðPr=0:0207Þ2=3 

½1 þ ðGz=55Þ10=9 

3=5

!5=3 33=10 5

ð3Þ

ð2Þ

where hx is defined the convective coefficient along the channellength direction and we assumed that hx is normal direction value of the channel wall-surface. Tw, and Tc denote wall and coolant temperatures at the interface between wall and coolant. q has various values along the position at the channel-wall, because hx and temperature different between Tw and Tc are also decreased as along with coolant flow in the mini-channel. Also, q is determined at the target cooling area As, which is the bottom surface of the mini-channel heat sink. Muzychka and Yovanovich [20] developed models to describe the pressure and heat-transfer behavior in the combined region of non-circular ducts and channels. In particular, they compared the models with the correlation, which is proposed by Churchill and Ozoe [21,22], to verify the heat-transfer behavior as follows:

This model described the local Nusselt number Nux, which defines the thermal behavior depending on the Graetz number Gz, and this model is valid for all Prandtl numbers Pr over an entire range of channel length Lc. The local convective coefficient hx and Gz are obtained by

Nux ¼ hx x=kf ;

Gz ¼

p

x 4 Lc RePr

ð4Þ

where x, kf and Re denote the displacement from the inlet of each stage in the mini-channel, thermal conductivity of the coolant and Reynolds number, respectively. Therefore, the heat emission generated by convection from the channel wall is expressed in this model as follows:

I

Qe ¼

!

!

q d A ¼ Mcp ðT c;o  T c;i Þ

ð5Þ

Ach

Table 2 List of independent variables. Heat source temperature Ts Coolant mass flux M Number of stage stack n

[K] [g/s]

300 0.1 1

310 0.3 2

330 0.6 3

350 1 4

370 5

Y. Kim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1197–1206

where Ach is the wetted surface area of the channel. Arrow marks of the q and A indicated vector in normal direction of the channel wall surface. In addition, the coolant of mass flux M, is heated by Qe from inlet temperature Tc,i to outlet temperature Tc,o. cp is the specific heat of the coolant. To compare the Qe performance of the multistage mini-channel by normalizing the scale, we define effectiveness e as follows:

e ¼ Q e;n =Q e;1

ð6Þ

where subscript n represents the number of stage stacks and Qe,1 and Qe,n are the Qe representation of the single and n-time stacked multistage mini-channels. Above parameters were calculated by commercial program (COMSOL Multiphysics). Also, pressure drop Dp was obtained by solving the Incompressible Navier-Stokes equation with the No Slip condition. 2.4. Grid test As presented at Eq. (4) in Section 2.3, we used conventional correlation equation verified by Muzychka and Yovanovich [20] to obtain a reasonable calculation result without experiment, because the multistage model was unique design, which couldn’t be compared with the previous experiment. However, the calculation result could be contaminated by grid effect because we used the discrete calculation scheme to calculate the physical values of each node. Therefore, reducing the grid effect could lead to the simulation of accurate physical phenomenon. To reduce the grid effect due to the propagation of residuals in the grid system, we conducted numerical simulations on the heat emission of the multistage mini-channel heat sink according to the grid size, as shown in Fig. 3. The simulation results were obtained according to the grid size under M = 1 g/s, Ts = 370 K, and n = 1–5, which was the largest Reynolds number condition requiring the most appropriate mesh in this study. Each grid had a hexahedron shape, and the size was approximately determined. Fig. 3 shows that the deviations in each multistage model become smaller after a grid size of 80,000 because the grid quality improves as the grid size increase. In this model, we used a grid size of 120,000, which has small difference (under 1.5%) with 170 k simulation, to obtain the reasonable solutions by eliminating the grid effect.

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3. Results and discussion 3.1. Convection mechanism hx was calculated using Eqs. (3) and (4) in accordance with the channel geometry. Fig. 4a and b, which show the hx transition in the single-stage mini-channel according to the dimensionless position x⁄ = x/LcRePr, show that mass flux M is more effective than Ts in controlling hx. Meanwhile, the curves of the hx transition are constant, and the difference in hx between the maximum and minimum values was within 10% in the arrangement of Ts shown in Fig. 4b. The curves gradually decreases as M increases, as shown in Fig. 4a, which means that the heat transfer mainly occurs within the thermal entrance length. Similar tendencies of M and Ts were also adapted, but intrinsic characteristics were found in the multistage mini channel, as shown in Fig. 4c. In contrast to the single-stage heat sink, the multistage heat sink had a corner, as shown in the faces between J-KK⁄-J⁄ and F-G-G⁄-F⁄ in Fig. 2c. Because the thermal and hydrodynamic boundary layers were developed anew, a sharp peek in hx was generated at the corner, and convection effectively occurred. Similar to roadmap suggested by Kandlikar [23], multistage minichannel also focused on increasing of the heat transfer coefficient to accommodate the high heat flux by expanding the entrance region. The coolant exchanges momentum rapidly with the channel wall passing through the corner, and then, the coolant flow is stabilized after corner with the rebuilding the thermal and hydrodynamic boundary layers. Therefore, each stage of multistage has the entrance region, which has large local convection coefficient similar with channel inlet, next to the corner. Hence, the coolant temperature of the multistage mini-channel increased with the varying trends in the single-stage, as shown in Fig. 5. Whereas the coolant temperature of the single-stage minichannel continuously increased with the shape of the logarithmic curve, that of the multistage mini-channel steeply increased with the shape of the step function curve. The major reason for the unique curve is the coolant mixing effect at the corner, as shown in Fig. 6. At the corner, the coolant was mixed by the rapid change in the flow direction. Thus, the coolant temperature sampled at the center line of the mini-channel steeply increased. Further, different temperature regions were generated similar to the step function because the upper stage channel wall lost large heat to the coolant without any supplement from the heat source due to the heat loss in the lower stage. Convection also occurred more effectively at the corner than in the straight channel due to the large convection coefficient and surface area at the corner as shown in Fig. 6 by red arrow indicating the heat flux. The heat flux heads toward channel wall because of huge convection at the corner.

3.2. Heat emission

Fig. 3. Grid test of the multistage mini-channel heat sink using heat emission transition according to the grid size.

In this study, q was not more than 256 W/cm2, which is the simulation results obtained by Xie et al. [15], due to the long channel length designed to verify the effects of multistage on the heat transfer and pressure drop. Because of the use of a long channel, the portion in the thermal entrance region where heat transfer mainly occurred was reduced, but the effects of the multistage were clarified, as shown in Fig. 7a. In this work, the maximum value of q was 40 W/cm2 under Ts = 370 K and M = 1 g/s at the quintuple-stage mini channel, as shown in Fig. 7b. Fig. 7a and b showed that q was increased by n, M and Ts. Especially, q increased steeply between the double- and triple-stage. And the increase in the curve was similar to that of the logarithmic shape depending on n because of the decrease in Qe. And the small increase of the cooling rate curve, in Fig. 7b, means that the large

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M

Fig. 4. Local convection coefficient. a and b: Single mini-channel heat sink. c: Multistage mini-channel heat sink. For reference, the single-stage local convection coefficient is described relative to the mass flux and heat-source temperature in a and b, respectively. The trend of multistage local convection coefficient is compared relative to the number of stage stacks in c.

320

Coolant temperature, Tc [ K ]

315 310 305 Num. of Stage stack, n

300

(1 g/s, 330 K)

n=1 n=2 n=3 n=4 n=5

295 290 0.0

0.2

0.4

0.6

0.8

1.0

Position ratio, x/Lc Fig. 5. Coolant temperature transition sampled at the center of the channel. The solid line indicates the single-stage mini channel as the reference, and the dashed lines show the multistage mini channels.

number of stage stack (n > 4) isn’t effective to enlarge the cooling rate under Ts = 310 K condition. It means optimal number of stage stack is influenced by the temperature of cooling target. Fig. 8 clearly describes transition of Qe depending on n. As defined by Eq. (6), e reflects the Qe of the each multistage. Qe of the multistage shows a tendency to decrease as increase of n. This decrease tendency explains alteration of the slope of the curve in Fig. 7. The increase slope between the double- and triple-stage, in Fig. 8, induces the steep increase of q in Fig. 7, because q is defined by Eq. (2). Generally, q is increased by decrease of As as increase of n in case of the multistage mini-channel. And the slope of the curve of q decreases as increase of n, because Qe decreases. However, q is increased in case of double- and triple-stage, because Qe increases as n increases in contrast with general multistage structure. The coolant and wall temperatures could explain the increase of Qe between the double- and triple-stage. Single stage structure has the largest e (Fig. 8). However, q increased as along with the increase of n (Fig. 7). It means multistage structure could cool down the hot spot rapidly than singlestage structure. That is because As decreased as increase of n.

Therefore, the slopes of curves in Fig. 7 will decrease after critical n since e decrease rapidly as increase of n as shown in Fig. 8. In this work, the critical n was three. Also, the slope of the triple-stage structure was the maximum as shown in Fig. 7. These result means stage stacking is valid to increase the q, and triple-stage is the most efficient design in this geometric conditions, Lc = 530 mm and Dh = 2 mm. Fig. 9a shows that Tw increases following the coolant flow direction with the approach to the heat source. Tw of the multistage increases with the shape of the step function curve, because each stage is placed at a specific distance from the heat source and heat was transferred to the upper stages by conduction. However, Tw of each stage decrease temporally because the coolant deprives the heat from the channel wall by convection. The decrease of Tw corresponds the overall heat emission at the straight channel and is inversely proportional to n generally. On the other hand, additional heat emission in the total area of corners of the multistage increases effectively depending on the number of corners with the increase of n as shown in Fig. 9b. This trade-off-relation, between decrease of overall heat emission at the straight channel and increase of additional heat emission at the edge, causes the distinct feature in this model, Qe,3 > Qe,2. Figs. 9a and 10 show that the temperature scales of the channel wall and coolant were different among the stages, and Tc varied depending on the position ratio of M. Therefore, the optimized n of the heat transfer was determined using the channel geometry such as Dh and Lc, and M determines the difference between Tw and Tc. In this study, the triple-stage mini-channel structure was optimized for heat transfer because it has the largest e among the multistage mini-channel in this channel geometry. In this work, we designed long channel model to obtain a reliable calculation results on the influence of multistage structure. Therefore, the model had smaller q than model suggested by Xie et al. [15]. However, it is expected that optimization of the multistage mini-channel, which means applications of the short Lc and many n to extend the entrance region, could enlarge q. 3.3. Pressure drop To verify the validity of the CFD model on Dp, we compared the result with the experimental result obtained by Kandlikar et al. [24]. Although the experimental result was two times larger than the CFD result due to the difference between the geometrical conditions of each model, the increase trend of Dp was similar to each

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325.96 330

325

320

315

310

305

300

y

295

z x

290 289.98 Fig. 6. Coolant temperature transition and heat flux in the double-stage mini-channel. Red arrows indicate heat flux in the heat transfer body and the dashed line indicates the data sampling line shown in Figs. 5 and 10. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Logarithmic increase in cooling rate according to the number of stage stacks. a: Depending on mass flux. b: Depending on heat source temperature.

other in terms of Re, as shown in the small box in Fig. 11. The simulation result of Dp had the half scale of the experiment result of Ref. [24]. This difference could be explained by below,

Dp ¼ 2f qf v 2 Lc =Dh

ð7Þ

where f, qf, and v indicate the Fanning friction factor, coolant density, and velocity, respectively. From Eq. (7), Dp has inverse proportion relation with Dh. And we used almost 2 times bigger channel diameter than Ref. [24]. Therefore, we regarded our CFD result were similar to the Ref. [24], because the CFD result corresponded to the theory. As shown in Fig. 11, Dp of the double-stage has 50% larger than the single-stage mini-channel above Re  600. However, multistage mini-channels have small increase, maximum 20%, depending on the n in the laminar flow regime. And the deference trend

on the pressure drop between single and multistage, are clarified depending on Re determined by M. The additional pressure drop is caused by rapid change of the coolant direction, generating the momentum exchange between the channel wall and the coolant in the corner of the multistage mini-channel. Dp of the mini-channel consists of the combination of the pressure drop in the straight parts and the corner. As shown in Fig. 12, pressure contour of the corner is complex and dense as against uniform arrangement of the straight channel. The additional pressure drop is accumulated at the every corner. And the momentum exchange is proportional to M. Therefore, additional pressure drop of the multistage mini-channel increases exponentially depending on the increase of Re. Also, the difference of Dp among the multistage mini-channels are getting wide depending on the increase of Re as shown in Fig. 11.

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Fig. 8. Steep decrease of effectiveness relative to the number of stage stacks.

Fig. 10. Coolant temperature transition along with the flow direction.

In terms of energy efficiency, the multistage mini-channel generated the smaller pressure drop than single-stage structure depending on the cooling rate as shown in Fig. 13. Although the cooling rate was increased drastically depending on the increase of the number of stage stack, the pressure drop wasn’t. As shown in Fig. 7, the cooling rate is increased drastically as along with the increase of the number of stage stack, because the cooling rate is in inverse proportion to the target cooling area which is reduced by stage stacking. However, the influence of the additional pressure drop on the total pressure drop was small as against the increase of the cooling rate. 4. Conclusion We have introduced a multistage mini-channel heat sink to increase the cooling rate without reducing the channel diameter in a small cooling target area. The existing mini-channel heat sink required reduction in the channel diameter to cool down a small area, which induced a large pressure drop in the channel. Multistage means that a 3D structure appears similar to a folded

Fig. 11. Pressure drop in terms of the Reynolds number. The simulation result was compared with the experimental result [24].

Fig. 9. Wall temperature transition along with flow direction. a: The wall temperature was sampled at the center of the channel surface. b: Additional heat emission of total area of the corners.

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1350.4

y

1200

z x

1000 800 600 400 200 0

0 [Pa]

Fig. 12. Pressure contour of quintuple stage heat sink under M = 1 g/s and Ts = 370 K condition. The gray gradation and white line indicate the pressure. Red arrow means coolant flow direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In terms of the pressure drop, multistage had a higher value than the single-stage because of the additional pressure drop at the corner. This additional pressure drop gradually increased as the coolant mass flux increased, and the difference of pressure drop among the multistage was not obvious because the coolant flowed with a small mass flux in the channel in the laminar-flow regime. However, we could confirm the increase of the pressure drop was not critical when we used the multistage mini-channel structure, because increase of the pressure drop was small as against with the increase of the cooling rate. Acknowledgement

Fig. 13. Increase of pressure drop in terms of cooling rate.

single-stage structure, and this structure could maintain a long channel length by filing multiple times. In this work, we conducted numerical simulations to verify the performance of the multistage mini-channel. A 530-mm-long and 2-mm-diameter mini-channel was designed, and laminar and non-boiling single-phase coolant flux was used to determine the validity of the CFD model. The numerical simulation result confirmed that the cooling rate increased with the number of stage stacks. However, this improvement in the heat-transfer performance was gradually reduced according to the number of stage stacks due to unique temperature distributions (similar to the step function) in the channel wall and coolant according to the number of stage stacks. Therefore, the optimum structure for heat transfer was a triple-stage, which yielded the largest effectiveness among the multistage structures, because the largest temperature difference between the coolant and channel wall was induced in this model. However, the optimum number of stage stacks could be varied depending on the channel length and coolant mass flux because the coolant temperature gradient varied. Also, it is expected that the cooling rate of the multistage mini-channel could be enlarged by applying the shorter channel and more stage stacking than this model.

This work was supported by Low-dimensional Materials Genome Development by Korea research Institute of Standards and Science. (KRISS – 2016 - 16011070). And, This work (C0397484) was supported by Business for Cooperative R&D between Industry, Academy, and Research Institute funded Korea Small and Medium Business Administration in 2016. Reference [1] M. Koyanagi, T. Fukushima, K.-W. Lee, T. Tanaka, Applications of threedimensional LSI, MRS Bull. 40 (3) (2015) 242–247. [2] S.G. Kandlikar, W.J. Grande, Evolution of microchannel flow passages – thermohydraulic performance and fabrication technology, Heat Transf. Eng. 24 (1) (2003) 3–17. [3] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE Electr. Dev. Lett. 2 (5) (1981) 126–129. [4] B.X. Wang, X.F. Peng, Experimental investigation on liquid forced convection heat transfer through microchannels, Int. J. Heat Mass Transf. 37 (1) (1994) 73–82. [5] X.F. Peng, G.P. Peterson, Convective heat transfer and flow friction for water flow in microchannel structures, Int. J. Heat Mass Transf. 39 (12) (1996) 2599– 2608. [6] G.L. Morini, Single-phase convective heat transfer in microchannels: a review of experimental results, Int. J. Therm. Sci. 43 (7) (2004) 631–651. [7] J. Judy, D. Maynes, B.W. Webb, Characterization of frictional pressure drop for liquid flows through microchannels, Int. J. Heat Mass Transf. 45 (17) (2002) 3477–3489. [8] G.R. Warrier, V.K. Dhir, L.A. Momoda, Heat transfer and pressure drop in narrow rectangular channels, Exp. Therm. Fluid Sci. 26 (1) (2002) 53–64. [9] W.L. Qu, G.M. Mala, D.Q. Li, Pressure-driven water flows in trapezoidal silicon microchannels, Int. J. Heat Mass Transf. 43 (3) (2000) 353–364. [10] Z.X. Li, D.X. Du, Z.Y. Guo, Experimental study on flow characteristics of liquid in circular microtubes, Microscale Therm. Eng. 7 (3) (2003) 253–265. [11] A.A. Khan, K.-Y. Kim, Evaluation of various channel shapes of a microchannel heat sink, Int. J. Air-Cond. 24 (2016) 1650018.

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