International Journal of Heat and Mass Transfer 58 (2013) 135–151

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Thermofluidics and energetics of a manifold microchannel heat sinkfor electronics with recovered hot water as working fluid

Chander Shekhar Sharma a, Manish K. Tiwari a, Bruno Michel b, Dimos Poulikakos a,⇑a Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerlandb Advanced Thermal Packaging, IBM Research Zurich, 8803 Rueschlikon, Switzerland

a r t i c l e i n f o

Article history:Received 2 October 2012Received in revised form 31 October 2012Accepted 5 November 2012

Keywords:Microchannel heat sinkElectronics coolingEnergyExergyEfficiencyTurbulent flow

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11

⇑ Corresponding author. Tel.: +41 44 632 27 38; faxE-mail address: dimos.poulikakos@ethz.ch (D. Pou

a b s t r a c t

A detailed thermo-hydrodynamic analysis of a hot water cooled manifold microchannel heat sink forelectronic chip cooling is presented. The hot water cooling enables efficient recovery of heat dissipatedby the even hotter chip by using hot water recovered from a secondary application. Contrary to usualexpectation of laminar flow in electronic cooling, high flow rate and high fluid temperatures result in tur-bulent flow conditions in the inlet and outlet manifolds of the heat sink with predominantly laminar flowconditions in microchannels. To simulate these complex flow conditions, a three dimensional (3D) con-jugate heat transfer model with turbulent flow is developed. Microchannel heat transfer structure ismodeled as porous medium with permeability parameters extracted from a 3D model for a single micro-channel. The energetic performance of the heat sink is analyzed in terms of 2nd law efficiency andsources of exergy destruction are identified by detailed local entropy generation analysis at low and highReynolds number conditions of 2400 and 11200 respectively. This analysis shows that entropy generationdue to heat transfer dominates the net entropy generation in the heat sink for both conditions. Althoughentropy generation due to viscous dissipation increases significantly with increased Reynolds number, itstill contributes less than a third to the total entropy generated at high Reynolds numbers. Use of hotwater reduces the heat transfer component of entropy generation significantly, thus leading to higher2nd law efficiency.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Electronic chips have witnessed an exponential increase in cir-cuit density in the past few decades. This increase has largely keptpace with Moore’s law [1]. However, since about the year 2000,this has resulted in an exponential increase in heat dissipationfrom the chips due to lack of voltage scaling, thus bringing thermalpackaging of electronics into sharp focus. The ever-increasing heatdissipation from chips has put much more pressure for the replace-ment of air cooling with liquid cooling. Single phase liquid coolingfor microprocessors has been long recognized as an effective meth-od to replace conventional air-cooling to handle the increasingheat densities of current and future microprocessors. The liquidbest suited thermally for single-phase cooling is water due to itshigh specific heat and thermal conductivity, as well as high avail-ability and environmental friendliness.

Apart from effective cooling of high heat dissipating electronicchips, increasing energy consumption by large computing systemssuch as data centers has also become an issue of concern. Directelectricity consumption by data centers had already reached 1%

ll rights reserved..012

: +41 44 632 11 76.likakos).

of the total world electricity consumption by 2005 [2] due to in-creased demands for IT (Information Technology) related servicessuch as internet and telephony. This figure grew further, thoughat a significantly lower rate, to between 1.1% and 1.5% by 2010[3].The introduction of the Green500 list for supercomputers[4,5] emphasized that performance can no longer be the solemotivation for development of microprocessors and that perfor-mance per unit energy consumption is the more appropriate met-ric for better computing. This becomes especially important in caseof large air-cooled data centers where energy spent for coolingcomprises almost half of the total energy consumed by such sys-tems. This portion of energy use can be significantly reduced byswitching to liquid cooling. This is because the much lower ther-mal resistance inherent in liquid use enables cooling above the freecooling limit thereby eliminating the need for coolant chillers. Thefree cooling limit represents the minimum temperature at whichthe coolant can effectively transport heat from a chip to ambientconditions without the need for an additional chiller.

Additionally, and perhaps more importantly, if hot water in thetemperature range of 50–70 �C is used to cool electronic chips,direct utilization of the collected thermal energy for secondaryapplications, like district heating or specific industrial applications,becomes feasible [6–9]. In such a system, the hot water, after

Nomenclature

A area (m2)cp specific heat at constant pressure (J/kg K)closs coefficient of non-linear momentum loss termd length scale (lm)D diameter (lm)DHT porous medium sizeEx flow exergy (W)f CFD solution on a grid, flowrate (l/min)GCI grid convergence indexh specific enthalpy (J/kg), grid spacing (lm)H height, thickness (lm)k turbulent kinetic energy (m2/s2)Kperm permeability (m2/s2)L length (lm)_m mass flow rate (kg/s)

n number of microchannelsN level of granularityp observed order of convergenceP pressure (Pa)Prt turbulent Prandtl number_q heat flux vector (W/m2)_Q heat dissipation (W)R thermal resistance (�C cm2/W)Re Reynolds numbers specific entropy (J/kg K)_Sgen rate of entropy generation (W/m3 K)T temperature (�C,K)T Reynolds averaged temperature (K)T0 fluctuating temperature (K)U velocity vector (m/s)Us superficial velocity vector in porous medium (m/s)V volume (m3)W width (lm)_W power (W)

Greek letterse turbulent dissipationg efficiencyd kronecker delta

u viscous dissipation (s�2)c porosityk thermal conductivity (W/m K)l dynamic viscosity (Pa s)lt eddy viscosity (Pa s)x turbulent frequency (s�1)q density (kg/m3)s stress tensor (Pa)

Subscriptsavg averagec coldplatech channelD dissipationD direct dissipationD0 turbulent dissipationeff effectiveelect electricalf fluidhs microchannel heat transfer structureim inlet manifoldin inletintf solid–liquid interfacem manifoldsmax maximum temperature at chipmeas measuredmin minimumom outlet manifoldout outletpump pumpingpred predictedQ heat transferQ mean temperature gradientQ0 fluctuating temperature gradients solidth thermaltot totalTIM thermal interface material

136 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

performing the task of chip cooling, is used for the secondary appli-cation where it gives off a part of its energy. The cooling loop thenrecovers the water back, still hot but at a lower temperature, to thehotter chip to serve as the coolant.

A significant body of work has analyzed liquid cooled micro-channel heat sinks. Tuckerman and Pease [10] were the first to re-port that a heat sink with microchannels is ideal for liquid coolingof electronic chips because the heat transfer coefficient scales in-versely with the characteristic channel dimension for a fully devel-oped laminar flow. A maximum power dissipation density of790 W/cm2 with a thermal resistance of 0.1 �C cm2/W but at theexpense of a high pressure drop of 2 bar was reported. Most ofthe later studies have concentrated on traditional microchannelheat sinks in which liquid enters axially at one end of long micro-channels. These studies have utilized single microchannel modelsto successfully capture the thermodynamic performance of suchheat sinks. Lee et al. [11] studied heat transfer in rectangularmicrochannels and concluded that conventional numerical analy-sis can be used to model thermal performance of microchannels.Modeling of entire heat sinks with limited number of channelshas also been reported [12,13].

Traditional liquid cooled microchannel heat sinks have been fol-lowed by the development of manifold microchannel (MMC) heat

sinks. These heat sinks differ from the traditional heat sinks asthe coolant fluid is delivered and returned through alternating,uniformly spaced slot nozzles across the chip plane. This hierarchi-cal design greatly reduces the travel length of the coolant fluidthrough the microchannel heat transfer structure and hence, sig-nificantly reduces the pressure drop for flow of coolant throughthe heat sink, thus making MMC heat sinks a very viable electroniccooling solution. Copeland et al. [14] and Ryu et al. [15] undertooknumerical studies of MMC heat sinks with multiple inlet and exitmanifolds supplying liquid to microchannels from vertical direc-tion. Based on the geometric and flow symmetry, they used a singlemicrochannel model for analysis of the heat sink.

Modeling of flow and heat transfer in the area of electronic cool-ing is generally made simpler by the fact the flow is predominantlylaminar due to small physical dimensions of the heat sinks. Thus aconjugate heat transfer analysis of a heat sink involves solvinglaminar incompressible Navier–Stokes equations. Most of the stud-ies reported so far have analyzed such scenarios. With rising chipheat fluxes, however, the flow rate through the heat sink is contin-uously being pushed and there is now emphasis on developingheat sinks using scalable manufacturing approaches [16]. This,along with the use of hot water for electronic cooling for efficientchip heat recovery, pushes up the flow Reynolds number of coolant

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 137

fluid in heat sink. As a result, fluid flow can reach transitional andturbulent flow regimes. It is possible to encounter Reynolds num-bers beyond the laminar range of 1000 and thus laminar flowequations are no longer sufficient to model the flow. There are afew studies that report turbulent flow analysis in the area of elec-tronic cooling. Dhindsa and Pericleous [17,18] have compared var-ious turbulent models with regards to prediction of flow andtemperature profiles in electronic cooling with air as the coolantfluid. They compared various turbulence models for the flow re-gimes encountered in electronic cooling. It was reported that x-equation based turbulence models like k–x turbulence model orthe Shear Stress transport (SST) turbulence models perform betterin such low Reynolds number flows as compared to the e-equationbased k–e model that is designed typically for high Reynolds num-ber flows. Kasten et al. [9] performed preliminary analysis of aMMC heat sink using turbulence modeling to model the fluid flowin the heat sink.

Apart from consideration of turbulence in fluid flow, numericalmodeling of MMC heat sinks also involves other challenges. Due tothe use of a manifold to supply coolant fluid across the microchannelheat transfer structure, such heat sinks can suffer from non-uniformdistribution of coolant fluid [9,19]. This, in turn, can lead to non-uniformity in temperature distribution in the chip. As a result, thesingle channel assumption, as used in some previous studies[14,15], no longer suffices and modeling of the full MMC heat sinkbecomes imperative. The need for such modeling has been high-lighted by Sharma et al. [8] where it was concluded that it is not pos-sible to closely predict the maximum temperature at the chip forMMC heat sink by representing heat transfer in the manifold micro-channel heat sink through a single microchannel model. Thisbecomes difficult in a single microchannel model due to non-uniform distribution of coolant and heat dissipation from chip.Hence, a complete numerical analysis of the entire heat sink isrequired.

Modern day MMC heat sinks can contain many tens of micro-channels. Thus, unlike a few previous studies [12,13], it is not

Fig. 1. Schematic of the manifold microchannel heat sink (arrows indicate direction ofcomprises the computational domain using hydrodynamic and thermal symmetry.

possible to model each channel separately while considering anumerical analysis of the entire manifold plus microchannelsystem. In such a scenario, microchannels can be modeled asporous medium. Kim and Kim [20] showed analytically that it ispossible to numerically analyze the thermal and hydrodynamicperformance of microchannel heat sinks by modeling the micro-channel heat transfer structure as fluid-saturated porous medium,thus making the consideration of entire heat sink feasible. Severalstudies have analyzed manifold microchannel heat sinks followingthis approach. However, in these studies, either the flow was fullylaminar [19] or the analysis was limited to hydrodynamics withoutconsideration of the thermal aspects [9].

In this paper, we present a detailed computational fluid dynam-ics (CFD) study of conjugate heat transfer inside a realistic MMCheat sink cooled by hot water at 30–60 �C. This heat sink is beingused for a liquid cooled data center supercomputer that employshot water to recover energy dissipated from the chip for buildingheating [16]. We adopt a hierarchical modeling approach by firstanalyzing the microchannel heat transfer structure at single micro-channel level and using the information from that analysis to mod-el the entire heat sink through the porous media approach. Wemodel the turbulent flow in the manifold structure (encountereddue to the large size of manifolds, high flow rates through the heatsink and high fluid temperatures). We also account for the possibil-ity of non-uniformity in the heat dissipation in the chip, in order toenable a closer prediction of the maximum temperature at chip.The results from the CFD analysis are then utilized to characterizethe 2nd law efficiency of heat recovery from chip. Finally, we ana-lyze the local entropy generation in the heat sink to identify themajor sources of exergy destruction.

2. Manifold microchannel heat sink

A schematic of the manifold microchannel heat sink analyzed inthis study is shown in Fig. 1. The heat sink material was copper andit was made by diffusion bonding the manifold and microchannel

coolant flow): (a) Isometric view and (b) top view. The shown half of the heat sink

Table 1Microchannel dimensions.

Symbol Description Value (lm)

Hch Height of channel 1473Wch Width of channel 159Wfin Width of fin (i.e. microchannel side wall) 181Lch Length of channel/fin 17000Wnozzle,in Width of inlet nozzle 2000Wnozzle,out Width of outlet nozzle 2000Hc Thickness of coldplate above TIM 1200HTIM TIM thickness 12LTIM Heated length of TIM 12550

138 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

layers together (Wolverine, Inc). This heat sink was manufacturedusing a micromachining process rather than micro-fabrication andthus is a completely scalable design. The heat sink consisted of oneinlet manifold and two outlet manifolds. Fig. 1 depicts half of theheat sink with the flow path through the heat sink indicated by ar-rows. Coolant entered through the inlet pipe into the inlet mani-fold as a jet. The flow impinged on the microchannels throughthe slot nozzles in the base of the inlet manifold and dividedequally into two streams. Each stream then travelled through themicrochannels absorbing heat coming from the chip underneaththe TIM. A commercial TIM with thermal conductivity of 2.7 W/m K was used. Finally, the two streams of coolant entered the out-let manifold, merged and exited the heat sink via the outlet pipe.

3. Single microchannel model

As discussed earlier, to simulate the thermofluidics of the fullheat sink, it was necessary that the microchannel heat transferstructure was modeled as a fluid saturated porous medium. Thisapproach requires specification of numerical coefficients for themomentum loss term in the porous medium model (see Section 4for further details). In order to extract the values of these coeffi-cients, flow through a single microchannel of the heat transferstructure was investigated in detail. The microchannels were de-signed to be rectangular. However, the channel walls can becomecurved due to the nature of the micromachining-based manufac-turing process. The curved shape of the machined microchannelis shown in Fig. 2(b). In order to analyze and quantify any effectsof this curved shape on the hydrodynamic and thermal character-istics of the microchannel heat transfer structure, a single curvedmicrochannel was also investigated numerically together withthe rectangular microchannel, both kinds having the same averagewidth. The models set-up is similar to what was used in Ref. [8]and is briefly described below.

The computational domains, major dimensions and boundaryconditions for both the channels are shown in Fig. 2 and listed inTable 1. The TIM thickness, HTIM, was small compared to the restof the channel and hence is not depicted in Fig. 2. At the inlet,the coolant mass flux and inlet temperature were imposed. Atthe outlet nozzle, an ‘opening’ type of boundary condition (ac-

Fig. 2. Computational domain and boundary conditions

counts for flow recirculation across the boundary) was imposed.This takes care of both inflow and outflow of the fluid at the outletboundary, because it lies in recirculating flow zone [21]. Thesymmetric boundary conditions as shown in Fig. 2 were exploitedto minimize computational cost. Uniform heat flux was imposed atthe lower surface of the TIM and an adiabatic condition wasimposed at all external boundaries of the solid domain. A block-structured, hexagonal, non-uniform Cartesian mesh, with a finermesh near the walls to adequately resolve the boundary layer,was used to discretize the computational domain. Grid indepen-dence was checked using the Grid Convergence Index (GCI)method. Grids consisting of 7.27 and 7.45 million cells were usedfor the curved channel and the rectangular channel simulationsrespectively. The GCI method is described in detail in Section 4.3.

Inside the channels, the flow is laminar and governed by the fol-lowing 3D conservation equations for mass (Eq. (1)), momentum(Eqs. (2) and (3)) and energy (Eq. (4) for fluid and Eq. (5) for solid)[22].

$ � ðqUÞ ¼ 0; ð1Þ$ � ðqU� UÞ ¼ �$P þ $ � s; ð2Þ

s ¼ l $Uþ ð$UÞT � 23

dð$ � UÞ� �

; ð3Þ

$ � ðqUhÞ ¼ $ � ðkfrTÞ þ s : $U; ð4Þ$ � ðks$TÞ ¼ 0: ð5Þ

The model takes into account the temperature dependence of inten-sive water properties such as density (q), dynamic viscosity (l) and

for (a) rectangular channel and (b) curved channel.

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 139

specific heat (cp) [23]. Continuity of temperature and heat flux wereimposed at the fluid–solid interface, as shown in Eqs. (6) and (7),where n is the direction normal to the interface.

� ksðð$TÞs;intf � nÞ ¼ �kf ðð$TÞf ;intf � nÞ; ð6ÞTs ¼ Tf : ð7Þ

The conjugate heat transfer problem was solved using the com-mercial solver Ansys CFX� version 12.1. The solver uses the finitevolume approach to discretize the governing equations into mostlysecond-order accurate, coupled linear algebraic equations. The dis-cretized governing equations were solved in a coupled mannerusing the algebraic multi-grid method. The steady-state equationswere solved using a pseudo-transient term to evolve the steady-state solution [22]. Convergence of the equations being solvedwas tracked by monitoring the normalized equation residuals aswell as global imbalances for conserved quantities of mass,momentum and energy [21,22]. We imposed 1 � 10�6 as the nor-malized residual target and 0.1% as the conservation target forour simulations.

4. Full heat sink model

4.1. Computational domain and boundary conditions

The schematic shown in Fig. 1 forms the computational domainfor the analysis of the entire heat sink. The boundary conditions forthe heat sink model are shown in Fig. 3. The analysis involves con-jugate heat transfer, turbulent flow and flow through a porousmedium. The microchannel heat transfer structure was modeledas a porous medium [20] as mentioned earlier. The fluid, solidand porous domains are also outlined in Fig. 3. At the inlet,mass-flow and inlet temperature and at outlet, atmospheric pres-sure (arbitrary choice of constant pressure boundary condition)was imposed. Fully developed turbulent flow was assumed at theinlet. Symmetry boundary condition for the flow was imposed asshown. At the outer walls of heat sink, heat loss to environment,as obtained from measurements, was imposed. Heat flux dissi-pated from the chip was imposed over a part of the lower face ofheat sink (chip area, as shown in Fig. 4(c)). The major dimensionsof the heat sink are shown in Fig. 4 and their numerical values usedin the model are listed in Table 2.

4.2. Governing equations

The flow in the two manifolds is beyond the laminar range,because the Reynolds number based on the hydraulic diameter ofthe inlet pipe, Rein, is in the ranges of 2000–11,000 for theoperating range of flow rates (0.3–1.0 l/min) and inlet watertemperature (30–60 �C). In addition, due to the large variation in

Fig. 3. Boundary conditions for full heat sink model.

the characteristic length scale in the heat sink, the flow regimechanges from turbulent in the inlet manifold to predominantlylaminar inside the microchannels and then back to turbulent inthe outlet manifold (see Fig. 3). This becomes clear from the widevariation in flow Reynolds number as the fluid flows through theheat sink. For example, for the operating condition correspondingto the highest flow rate and fluid inlet temperature, the Reynoldsnumber changes from 11200 at the inlet of the heat sink to lessthan 600 inside the microchannels and then back to 11200 at theoutlet of the heat sink. This makes numerical analysis of the heatsink especially challenging. The effect of turbulence is modeledusing the k–x model. The x-equation based models are knownto perform better than the e-equation based models for low Rey-nolds number flows [18]. The k–x model belongs to the class of2-equation models, which are based on the eddy-viscosity assump-tion. This assumption is not strictly valid in case of flows withstrong streamline curvatures. Streamlines can have significant cur-vature in a flow through manifold microchannel heat sinks andthus a case can be made for use of Reynolds stress models. Thesemodels do not depend upon the eddy-viscosity assumption andsolve six additional equations for the Reynolds stresses [22,24].However, a higher number of equations increase the computa-tional effort and achieving convergence with these models be-comes difficult for complex flows. The same was observed duringthe course of the present study. Hence, the basic k–x model wasused to model the flow in this work. The conservation equationsfor the full heat sink model are as described below [22].

4.2.1. Momentum and energy conservation in fluid domainThe Reynolds averaged continuity equation for the manifold is

the same as that in laminar flow i.e. Eq. (1). In tensorial notation,the Reynolds averaged momentum equation for the ith velocitycomponent with the eddy viscosity assumption is

@

@xjðqUiUjÞ ¼ �

@

@xipþ 2

3qk

� �þ @

@xjðlþ ltÞ

@Ui

@xjþ @Uj

@xi

� �� �; ð8Þ

where the turbulent viscosity is given by

lt ¼ qkx: ð9Þ

The two additional equations for the k–omega model are

@

@xjðqUjkÞ ¼

@

@xjlþlt

rk

� �@k@xj

� �þlt

@Ui

@xjþ@Uj

@xi

� �@Ui

@xj�b0qkx;

ð10Þ@

@xjðqUjxÞ ¼

@

@xjlþ lt

rx

� �@x@xj

� �þa

xklt

@Ui

@xjþ@Uj

@xi

� �@Ui

@xj�bqx2:

ð11Þ

The model constants have standard values: b0 = 0.09, a = 5/9,b = 0.075, rk = 2 and rx = 2.

Reynolds averaged energy conservation equation is

@

@xjqUj hþ1

2UkUk

� �� �¼ @

@xjkf@T@xjþ lt

Prt

@h@xj

� �

þ @

@xjUi ðlþltÞ

@Ui

@xjþ @Uj

@xi

� ��2

3dijk

� �� �;

ð12Þ

where the turbulent Prandtl number Prt is set to 0.9.For modeling near wall flow and heat transfer, a modified for-

mulation of the standard wall functions was used. This formulationautomatically switches from the wall function approach to thelow-Re formulation as the mesh is refined near the wall. Addition-ally, it utilizes scalable wall functions which allow y+ insensitivemesh refinement [22].

Fig. 4. Major dimensions of heat sink (a) top view, (b) view in symmetry plane, (c) bottom view and (d) end view. The color codes are same as Fig. 3. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

Table 2Heat sink dimensions.

Symbol Description Value (lm)

L1 Length of heat sink 35500L2 Length of outlet manifold 25250W1 Width of heat sink 35500W2 Width of manifold structure 23000Wom Width of outlet manifold 5000Wim Width of inlet manifold 5000D Diameter of inlet and outlet pipes 4000Hms Height of manifold structure 7000Hm Height of manifolds 4500Lhs Length of microchannel heat transfer structure 21250Lchip Length of chip 16886Wchip Width of chip 12550n Number of microchannels 62

140 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

4.2.2. Conservation equations in the porous domainSince the microchannel heat transfer structure was modeled as

a porous medium with spatially uniform porosity, the continuityequation inside the porous medium reduces to same equation asEq. (1). The following equation was solved for conservation ofmomentum inside porous medium

r � qc2 Us � Us

� ��r � lþ lt

cðrUs þ ðrUsÞTÞ

� �¼ SM �rp; ð13Þ

where Us is the superficial velocity and is equal to c U with U beingthe true velocity in the porous medium. Turbulence viscosity forflow inside porous medium, lt, is obtained by solving the same tur-bulence equations as in fluid domain (see Eqs. (10) and (11)). c isthe volume porosity and is defined as

c ¼ nWch

nWch þ ðnþ 1ÞWfin: ð14Þ

SM represents the directional momentum loss. Due to the highvelocities encountered, this term takes into account non-linear ef-fects through Forchheimer’s extension of the basic Darcy law [25]and is formulated as below

SM;x ¼ �l

Kperm;xUs;x �

clossffiffiffiffiffiffiffiffiffiffiffiffiffiffiKperm;x

p qjUsjUs;x; ð15Þ

SM;y ¼ �l

Kperm;yUs;y �

clossffiffiffiffiffiffiffiffiffiffiffiffiffiffiKperm;y

p qjUsjUs;y; ð16Þ

SM;z ¼ �l

Kperm;zUs;z �

clossffiffiffiffiffiffiffiffiffiffiffiffiffiKperm;z

p qjUsjUs;z: ð17Þ

Kperm,i represents the permeability along coordinate i. As shownin Figs. 1–3, the fluid enters the microchannel heat transferstructure predominantly in the negative z-direction and flowsalong the y-direction through the structure. In order to inhibit flowin the x- direction, the permeability in x direction was assumed tobe 100 times smaller than in other two directions. The permeabil-ity in the y direction was assumed to be equal to that in the zdirection. Kim and Kim [20] have suggested an expression forpermeability applicable to microchannels. However, since flowthrough microchannels in the heat sinks changes direction at theslot nozzles, the permeability values in the flow direction are ob-tained by matching the pressure drop through the porous mediumwith that in a single rectangular microchannel. This is explainedfurther in Section 5. The coefficient for the non-linear term in theabove equations is obtained as below [19,26]

closs ¼ 0:55 1� 5:5dch

DHT

� �; ð18Þ

Table 3Discretization error and grid independence test using DP as solution functional (for flow rate of 1 l/min and Tf,in of 60 �C).

Mesh Number of cells (million) Mean grid spacing (lm) r f = DP (Pa) p GCIfine GCIcoarseGCIcoarse

rp21 GCIfine

Fluid Solid

1 79.98 79.98 121.78 – 13223.82 36.62 39.16 121.78 1.31 12960.1 4.01 0.0127 0.0384 1.02033 17.57 51.31 121.78 1.31 12181.3

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 141

where the characteristic length scales for pore size (dch) and overallporous medium size (DHT) are determined following the references[27,19]

dch ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHchWch

p; ð19Þ

DHT ¼ffiffiffiffiffiffiffiffiffiffiHchL

p: ð20Þ

A 1-equation model for heat transfer was used for conservationof energy inside the porous medium.

@

@xjðqUjhÞ ¼

@

@xikeff ;i

@T@xiþ c

lt

Prt

@h@xi

� �: ð21Þ

The effective thermal conductivities in the three energy directionsare calculated based on whether solid and fluid thermal resistancesto heat conduction are in parallel or series. Accordingly

keff ;x ¼kf ks

cks þ ð1� cÞkf; ð22Þ

keff ;y ¼ keff ;z ¼ ckf þ ð1� cÞks: ð23Þ

4.2.3. Momentum and energy conservation at domain interfacesContinuity of temperature and heat flux is imposed at both

fluid–solid and porous-solid interfaces similar to Eqs. (6) and (7)with an equivalent conductivity used for porous medium.Similarly, at the fluid-porous interface, continuity of mass flux,momentum, heat flux as well as temperature is imposed.

The discretized governing equations for the full heat sink modelwere also solved using Ansys CFX� version 12.1. For the full heatsink model, we imposed 1 � 10�6 as the normalized residual targetand 0.1% as the conservation target for our simulations.

4.3. Mesh and grid independence

Flow modeling in general and turbulence modeling in particularis sensitive to the mesh used for discretization of the computa-tional domain. A block-structured, hexagonal, Cartesian meshwas used for all the domains in the full heat sink model. The meshwas non-uniform with finer mesh near the walls to adequately re-solve the boundary layer. Grid independence was checked usingGCI method which is based on Richardson extrapolation [28] andhas been recommended as a uniform method for reporting of gridconvergence [29]. GCI is defined as

GCIfine ¼Fs

rp21 � 1

f2 � f1

f1

��������; ð24Þ

where Fs is a factor of safety and is set to 1.25. The solutions on thethree grids are denoted by f1, f2 and f3 and the mean grid spacing byh1, h2 and h3, respectively, with h1 being the finest and h3 being thecoarsest grid. The parameter r21 represents the grid refinementfactor and is defined as the ratio of grid spacing h2 to h1. Theresulting order of convergence, p, is obtained by the solution of atranscendental equation [29]. The p value so computed is valid onlyif the grids are in asymptotic range i.e. the following condition issatisfied [30]

GCIcoarse

rp21GCIfine

¼ 1: ð25Þ

As mentioned earlier, the flow field to be modeled was turbulent.The mesh density close to the wall was kept sufficiently dense sothat y+ for wall adjacent mesh cells was below 10 on all walls. Anumber of meshes were prepared with average grid size in fluidand porous domains ranging from 120.36 lm to 29.89 lm. A setof three grids was selected that satisfied the asymptotic range testof Eq. (25) as well as exhibited monotonic convergence. The threegrids and results are described in Table 3, wherein DP is used asthe solution functional (i.e. f) [30].

The ratio GCIcoarserp

21GCIfinewas close to one indicating that three grids

were in asymptotic range. The GCIfine and the GCIcoarse were smallfor both the solution functionals. The relatively large value of p,as compared to the formal order of convergence of the CFD solver,can be explained by the fact that the ratio of mean grid sizes in dif-

ferent directions hy

hx; hz

hx

� was not constant for the three grids [31].

Following the modified procedure for calculation of p [31], a set offive grids was used and the resulting value of p came out to be1.154, which agrees better with the less than second order conver-gence typical of CFD codes. Also, the relative change in solutionfunctional values was smaller between mesh 2 and mesh 1 as com-pared to the corresponding change between mesh 3 and mesh 2.Therefore, mesh 2 was used for all further simulations.

5. Results and discussions

5.1. Extracting the permeability from single microchannel model

As already discussed in Section 2, single microchannel modelsfor curved and rectangular channels have been set up and usedto compare the hydrodynamic and thermal characteristics. Thesimulations were performed for flow rates ranging from 0.3 to1.0 l/min and fluid inlet temperature of 60 �C. The water tempera-ture at the outlet slot-nozzle and the maximum temperature of thecold plate are compared in Fig. 5(a). Fig. 5(b) compares pressuredrop from the inlet to the outlet slot-nozzles. As evident fromFig. 5, the thermal performance for both channel types is very sim-ilar. In terms of the pressure drop, the curved channels, in fact, per-form slightly better than the rectangular channels by guiding theflow and, thereby, reducing the pressure drop due to fluid impinge-ment on the channel walls acting as fins. Hence, the curvature inchannels does not affect the performance of the microchannel heattransfer structure.

The coefficients in the momentum loss term of Eqs. (15)–(17)require specification of the permeability in three directions Kperm,i.We extract these permeability values by tuning the basic perme-ability values from Kim and Kim [20] so that the pressure dropthrough the microchannel cooling structure matches the pressuredrop through a single microchannel. This is required to accountfor the effect of flow turning by 90 degrees inside the porous med-ium, after entry through the inlet nozzle and before exit throughthe outlet slot nozzle. To determine the permeability, we used areduced heat sink model consisting of only the inlet and outlet

Fig. 5. Comparison of rectangular and curved channel (a) Fluid outlet temperature(Tf,out) and maximum cold plate temperature (Tmax) and (b) Overall pressure drop(DPoverall).

142 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

nozzles and the porous medium structure as shown in Fig. 6. At theinlet, massflow and at the outlet, atmospheric pressure were im-posed. The symmetry boundary condition for flow was adoptedas shown in Fig. 6. The rest of the bounding surfaces were modeledas no-slip walls. In this reduced model, the flow was assumed to beisothermal and only the continuity and momentum equationswere solved.

Fig. 6. Boundary conditions for reduced heat sink model.

Since the channel width varies in the actual heat sink, an aver-age channel width was used in the rectangular microchannel mod-el. By tuning the pressure drop through the microchannel structureagainst the microchannel model, a quadratic fit was obtained forthe permeabilities in y and z directions, as a function of the flowrate, f in l/min, through heat sink as below

Kperm;y ¼ Kperm;z ¼ ð6:5065� 10�9Þf 2 þ ð8:1561� 10�9Þfþ 7:715� 10�10: ð26Þ

A constant low permeability was used in x direction and is given by

Kperm;x ¼1

100cd2

ch

12

!: ð27Þ

Pressure drop through the reduced heat sink model, by usingpermeability values from Eqs. (26) and (27), is compared againstpressure drop from rectangular microchannel model in Fig. 7. Asevident, the tuned permeability helps to closely approximate thepressure drop through the microchannel.

5.2. Heat sink model simulations

5.2.1. Model validationExperimental measurements from Sharma et al. [8] for the

water outlet temperature (Tf,out) and the maximum temperatureat the base of TIM (Tmax) were used to validate the full heat sinkmodel. These measurements were available for heat sink operatingconditions of flow rates from 0.3 to 1 l/min in steps of 0.1 l/min andat 4 different water inlet temperatures, Tf,in, between 30 �C and60 �C. The heat dissipation from the chip, _Qchip, was kept constantat 100 W. Fig. 8 shows the validation of the model in terms of pre-dictions for the water temperature at the heat sink outlet,Tf,out. Asevident from Fig. 8, the model is able to capture well the heat ab-sorbed by the water as it flows through the microchannel heat sink.

It was reported previously that the microchannel only model ofthe heat sink is not able to predict accurately the chip level maxi-mum temperature, Tmax, with the predictions deviating fromexperimental measurements by as much as 7.2 �C, which amountsto an error of nearly 19% [8]. This deviation is caused by theunavoidable error introduced by two key assumptions in the singlemicrochannel model: that the coolant is distributed uniformlyamong all the channels and that the heat dissipation from the chipis also spatially uniform. As will be explained further in Sec-tion 5.2.2, the coolant fluid enters the inlet manifold as a jet, which

Fig. 7. Matching the pressure drop between microchannel and reduced geometrymodel by tuning the permeability.

Fig. 8. Model validation for fluid outlet temperature (Tf,out). Legend subscripts ‘meas’ and ‘pred’ indicate measured and simulated values for Tf,out respectively.

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 143

leads to non-uniform distribution of coolant among the channels.The full heat sink model captures this effect. In addition to this,non-uniformity in heat dissipation from the chip is modeled byassuming a checkerboard pattern of power granularity in the chip.Yazdani et al. [32] have shown that a minimum chip power gran-ularity needs to be modeled for accurate predictions of chip levelmaximum temperatures. We investigated this aspect for the pres-ent heat sink model. We characterized the granularity by the num-ber of alternating rows or columns in the checkerboard pattern (N)for the heat flux shown in Fig. 9a. Each row or column of the pat-tern consists of rectangular blocks with alternating heat fluxboundary conditions. Adiabatic boundary condition is assumedfor the blue colored blocks while heat flux boundary condition isassumed for red colored blocks, such that the total heat flux forall red blocks is equal to the total heat dissipated by the chip. Onlyodd integer values are chosen for N in order to satisfy the symmet-ric boundary condition shown in Fig. 3. It is observed that a patterncorresponding to N = 3 is sufficient, as model predictions for Tmax

for N > 3 do not change significantly. Fig. 9(a) depicts the imposi-tion of the heat flux boundary condition for N = 3. Fig. 9(b) com-pares model predictions for Tmax against experimentalmeasurements. The full heat sink model predicts Tmax accurate towithin 2.7 �C at all operating condition. This amounts to a maxi-mum of 6.3% deviation of predictions from experiments.

5.2.2. Hydrodynamic performance of the heat sinkFig. 10 shows the velocity field in two perpendicular planes for

the operating condition corresponding to the highest Reynoldsnumber (flow rate of 1 l/min and Tf,in of 60 �C). The geometry ofthe heat sink presents a backward facing step at the inlet and a for-ward facing step at the outlet from the microchannel heat transferstructure (see Fig. 10(b)). Fig. 10(a) clearly depicts that the flow en-ters the inlet manifold as a jet and impinges either directly on themicrochannel heat transfer structure through slot nozzle or on thebackward facing step forming the inlet slot nozzle. The jet-like flowcauses large flow separation and the backward facing step leads totwo independent recirculation zones in the upper and lower part ofthe inlet manifold. Fig. 10(b) shows that the flow enters the chan-nels as a slot jet, impinges on the channel floor and then reattachesto the channel upper wall after travelling a significant fraction ofthe channel length. Hence, the flow is developing over most ofthe channel axial length. In the outlet manifold, the flow comingout of the slot nozzle flows over the forward facing step that leadsto the formation of a large vortex, that fills almost the entire outlet

manifold, along with a small, counter-rotating vortex in the upperhalf of the manifold. Similar flow field patterns are observed forother operating conditions as well.

Most of the turbulence in the flow is generated in the inlet man-ifold and in the inlet slot nozzle. Fig. 11 shows the variation of theturbulent kinetic energy (TKE) in the plane of fluid symmetry andthe mid cross-sectional plane. In the inlet manifold, strong sheardevelops between the jet and the large recirculation zone in theupper part of the inlet manifold. Another source of large shear isthe fluid impingement on the microchannel heat transfer structure(visible in both the planes). Both these zones contribute most of theTKE generation with the latter contributing significantly more thanthe former. Fig. 11(b) also shows that as the fluid enters and flowsalong the channels, most of that TKE is dissipated within a short tra-vel length as the flow laminarizes due to small channel widths. Inthe outlet manifold, although the flow enters by forming a largeand a small vortex, the shear zone between the two is rather limitedand it does not lead to any significant TKE generation.

5.2.3. Thermal and energetic performance of the heat sinkThe thermal performance of the heat sink can be analyzed in

terms of the net thermal resistance to heat flow (Rth) from chipto coolant fluid. The thermal resistance of the heat sink is definedas

Rth ¼AchipðTmax � Tf ;inÞ

_Qchip

: ð28Þ

Fig. 12(a) shows how Rth changes with the flow rate at differentfluid inlet temperatures, Tf,in. As the flow rate increases, the heattransfer coefficient values also increase across the channel lengthbecause of the predominantly developing laminar flow regime overmost of the channel axial length [8]. Fig. 12(b) shows temperaturecontours in the mid cross-sectional plane and demonstrates thethermally developing nature of flow inside the channels. This ex-plains the decreasing thermal resistance with increasing flow rates.

It was discussed in Section 1 that hot water is used for heatrecovery from the chip so that it can be utilized for other purposes,such as building or district heating. After this utilization the stillwarm but at lower temperature water returns to the (hotter) chipand serves as the coolant. Based on this it becomes important thatthe heat recovery takes place in an exergetically efficient manner.The exergetic or the 2nd law efficiency of the heat sink can be cal-culated as below [33]

Fig. 9. (a) Checkerboard pattern of heat flux boundary condition (for N = 3) and (b) model validation for maximum temperature at the base of TIM (Tmax).

144 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

gII ¼Exout

Exin þ _Wpump þ _Welect

¼ Exout

Exin;tot; ð29Þ

where the flow exergy Exj at jth location, with j indicating either in-let or outlet, is defined as

Exj ¼ _m hj � h0 � T0ðsj � s0Þ þ12jUj2

� �: ð30Þ

Ambient reference conditions of T0 = 25 �C and P0 = 1 atm wereused to evaluate flow exergy in Eq. (30). In essence, the control vol-ume for exergy analysis includes both the heat sink and the elec-tronic components generating heat. The variation of the 2nd lawefficiency at various operating conditions is shown in Fig. 13. Forany given fluid inlet temperature Tf,in, the temperature differential,Tw,avg(x) � Tf,bulk(x) required for same amount of heat transfer from

solid to fluid is reduced as the flow rate increases. Thus, exergydestruction due to heat transfer over a finite temperature differen-tial is also reduced [33]. On the other hand, the exergy loss due tofluid pressure drop increases at higher flow rates. However, in thecase of liquid cooling, the contribution of the pressure drop term tothe exergy destruction (in the form of pumping power, _Wpump, seeEq. (29)) is less than 4% for all operating conditions. Hence, as theflow rate increases, the exergetic efficiency increases at any fluidinlet temperature. The exergetic efficiency also increases with in-crease in fluid inlet temperature at any given flow rate.

5.2.4. Local entropy generation in the heat sinkThe 2nd law efficiency quantifies the amount of exergy

destruction in a particular system. Exergy destruction is directly

Fig. 10. Velocity vectors for flow rate of 1 l/min and Tf,in of 60 �C in (a) plane of fluid symmetry and (b) mid cross-sectional plane (i.e. the plane perpendicular to plane of fluidsymmetry at half length of heat sink).

Fig. 11. TKE for flow rate of 1 l/min and Tf,in of 60 �C in (a) plane of fluid symmetry and (b) mid cross-sectional plane.

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 145

146 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

proportional to entropy generation in the system via the Gouy–Stodola theorem [33]. Hence, major sources of exergy destructioncan be identified by analyzing the amount of entropy generatedin various parts of the system. This analysis can be performed byutilizing the entropy transport equation [34] and it forms a part

Fig. 12. (a) Thermal resistance of the heat sink and (b) Temperature contours inmid cross-section plane for flow rate of 1 l/min and Tf,in of 60 �C.

Fig. 13. Variation of 2nd law efficiency w

of post-processing analysis of the CFD results. The net entropy gen-eration per unit volume, for incompressible flow with no bulk heatsources, is expressed as

_Sgen ¼ �_q

T2 � rT þ 2l/T

; ð31Þ

where / represents the viscous dissipation and q, the local heat dif-fusion as per Fourier’s law. The first and the second terms in theabove equation quantify the local entropy generation due to heat

transfer ð _Sgen;Q Þ and fluid friction ð _Sgen;DÞ respectively. For turbulentflows, Eq. (31) needs to be Reynolds averaged. Kock and Herwig[35,36] have presented the Reynolds averaging of the above equa-tion and modeling of the resulting unclosed terms for high Reynoldsnumber flow using k–e model. However, to the best of our knowl-edge, the corresponding extensions factoring the volume porosityand the anisotropic thermal conductivity have not been reportedin the literature. Therefore, the equations from reference [36] weremodified to account for the volume porosity and the anisotropicthermal conductivity inside the porous medium and the use of k–x model for turbulence. Reynolds averaged entropy generation

due to heat transfer, _Sgen;Q , can be calculated as below

_Sgen;Q ¼ _Sgen;Q þ _Sgen;Q 0 ; ð32Þ

_Sgen;Q represents entropy generation due to mean temperature gra-

dients. For a material volume with anisotropic conductivity, _Sgen;Q isgiven by

_Sgen;Q ¼1T2

keff ;i@T@xi

!224

35: ð33Þ

keff is equal to kf for manifolds and ks for the solid domain and forporous domain, it is obtained from Eqs. (22) and (23). _Sgen;Q 0 repre-sents entropy generation due to fluctuating temperature gradientsand is given by

_Sgen;Q 0 ¼c

T2keff ;i

@T 0

@xi

� �224

35; ð34Þ

where c was included to account for volume porosity in porous do-main and is equal to 1 for fluid domain, 0 for solid domain and isgiven by Eq. (14) for porous domain. Using the value of eddy viscos-ity from Eq. (9), _Sgen;Q 0 can be modeled as

ith change in operating conditions.

Fig. 14. _Sgen;D for (a) flow rate of 0.3 l/min and Tf,in of 40 �C and (b) flow rate of 1 l/min and Tf,in of 60 �C (both figures are plotted to same magnitude scale and use logarithmiccolor scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 147

Fig. 15. _Sgen;Q for (a) flow rate of 0.3 l/min and Tf,in of 40 �C and (b) flow rate of 1 l/min and Tf,in of 60 �C (both figures are plotted to same magnitude scale and use logarithmiccolor scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

148 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151 149

_Sgen;Q 0 ¼cqcpkxPrtT2

@T@xi

!2

: ð35Þ

The Reynolds averaged entropy generation due to fluid friction,_Sgen;D, is given by

_Sgen;D ¼ _Sgen;D þ _Sgen;D0 ; ð36Þ

where, _Sgen;D represents entropy generation due to direct dissipation(i.e. due to gradient in mean velocity) and is given by

_Sgen;D ¼clT

2@Ux

@x

!2

þ @Uy

@y

!2

þ @Uz

@z

!28<:

9=;þ @Ux

@yþ @Uy

@x

!224

þ @Ux

@zþ @Uz

@x

!2

þ @Uy

@zþ @Uz

@y

!235: ð37Þ

For calculating _Sgen;D inside the porous medium, true velocities areused. _Sgen;D0 represents the entropy generation due to turbulent dis-

Table 4Aggregate entropy generation at low and high Rein condition.

Aggregate entropy generation (W/K) Rein = 2420 Rein = 11200

_Sgen;D;tot1.4 � 10�6 3.1 � 10�6

_Sgen;D0 ;tot9.6 � 10�5 1.4 � 10�3

_Sgen;D;tot ¼ _Sgen;D;tot þ _Sgen;D0 ;tot9.7 � 10�5 1.4 � 10�3

_Sgen;Q ;tot1.5 � 10�3 6.3 � 10�4

_Sgen;Q 0 ;tot1.9 � 10�4 1.6 � 10�4

_Sgen;Q ;tot ðsolidÞ 3.2 � 10�3 2.6 � 10�3

_Sgen;Q ;tot ¼ _Sgen;Q ;tot þ _Sgen;Q 0 ;tot þ _Sgen;Q ;tot ðsolidÞ 4.9 � 10�3 3.4 � 10�3

_Sgen;tot ¼ _Sgen;D;tot þ _Sgen;Q ;tot 5.0 � 10�3 4.8 � 10�3

Fig. 16. jrTj for flow rate of 0

sipation (i.e. due to gradients in velocity fluctuations) and is givenby

_Sgen;D0 ¼clT

2@U0x@x

� �2

þ@U0y@y

!2

þ @U0z@z

� �28<:

9=; @U0x

@yþ@U0y@x

!224

þ @U0x@zþ @U0z

@x

� �2

þ@U0y@zþ @U0z@y

!235: ð38Þ

Again, using the definition of viscosity from Eq. (9), _Sgen;D0 , can mod-eled as

_Sgen;D0 ¼cqkx

T: ð39Þ

Volume porosity is included in Eqs. (37)–(39) to account for the factthat true velocities are used for computation of entropy generationin porous domain. Wall functions for entropy generation from refer-ence [36] were not included in this analysis. This is because, as dis-cussed further ahead, most of the entropy generation was found totake place in TIM and solid domain of the heat sink.

Figs. 14 and 15 compare the Reynolds averaged entropy gener-ation due to fluid friction, and heat transfer for two operating con-ditions that correspond to the low and the high Reynolds numberflow investigated: flow rate of 0.3 l/min and Tf,in of 40 �C (Rein =2420) and flow rate of 1 l/min and Tf,in of 60 �C (Rein = 11200). Forthe sake of comparison, plots for both the operating conditionsare shown for same magnitude range. Since the entropy generationlevels traverse over a large range of order of magnitudes, the entro-py generation plots use a logarithmic color scale for ease ofvisualization.

Fig. 14(a) and (b) show that as the operating condition changes

from low to high Rein, _Sgen;D increases. This is expected because the

.3 l/min and Tf,in of 40 �C.

150 C.S. Sharma et al. / International Journal of Heat and Mass Transfer 58 (2013) 135–151

flow becomes more turbulent with increase in Rein thus increasing

viscous dissipation in the flow. Fig. 15(a) and (b) show that _Sgen;Q

decreases as the operating condition is changed from low to highRein. This is because the high Rein condition corresponds to highheat transfer coefficient in microchannel heat transfer structureand thus low temperature gradients in the fluid. The temperaturegradient in the solid part also decreases with increase in Tf,in and

thus _Sgen;Q decreases in the entire heat sink.The change in the various components of the entropy genera-

tion with change in operating conditions can be analyzed by calcu-lating the aggregate entropy generation over the entire heat sink.This can be done by integrating the entropy generation compo-nents over the volume of the heat sink as shown below

_Sgen;D;tot ¼Z

fluid;porous

_Sgen;D dV ð40Þ

_Sgen;Q ;tot ¼Z

fluid;porous

_Sgen;Q dV þZ

solid

_Sgen;Q dV ð41Þ

Table 4 shows calculated values of aggregate entropy genera-tion components at low and high Rein. At both the operating condi-tions, most of the entropy generation due to viscous dissipation,_Sgen;D;tot is contributed by the turbulent dissipation component,_Sgen;D0 ;tot and most of the entropy generation due to heat transfer,_Sgen;Q ;tot is contributed by solid due to low thermal conductivity ofTIM.

As operating condition is changed from low to high Rein, _Sgen;D;tot

increases due to significant increase in turbulent dissipation at

high Rein. On the other hand, _Sgen;Q ;tot is reduced due to lower overalltemperature gradients at higher Tf,in. However, the decrease in_Sgen;Q ;tot more than compensates the increase in _Sgen;D;tot thus reduc-

ing _Sgen;tot ¼ _Sgen;D;tot þ _Sgen;Q ;tot . As a consequence, the 2nd law effi-ciency also increases with increase in Rein.

Most of _Sgen;D takes place in regions of high turbulence at bothlow and high Reinconditions. This becomes evident from compari-

son of Fig. 14(b) with Fig. 11. Most of _Sgen;Q occurs in regions of hightemperature gradients. This becomes evident by comparingFig. 15(a) with Fig. 16. Fig. 16 shows the net temperature gradientin manifold and microchannels for the low Rein condition.

In addition, _Sgen;Q ;tot dominates _Sgen;D;tot at both low and high Rein

conditions. This indicates that viscous dissipation plays a lesserrole in the overall entropy generation in the heat sink. At low Rein

condition, _Sgen;Q ;tot is about an order of magnitude higher than_Sgen;D;tot . At high Rein conditions, although _Sgen;D;tot increases signifi-cantly; it contributes less than a third of the total entropygeneration.

6. Conclusions

The thermofluidic and energetic performance of an MMC heatsink, operated with hot recovered water for the cooling of evenhotter electronic chips, was investigated. 3D conjugate heat trans-fer models for microchannels, were developed and combined witha full heat sink model.

In the full heat sink model, the fluid turbulence was modeledusing the k–x model, the microchannel system heat transfer wasmodeled as a fluid-saturated porous medium, and the heat dissipa-tion from the chip was imposed as a realistic checkerboard pattern.The model predicts Tmax to within 2.7 �C of literature experiments,which is significantly better than the state of the art. This showsthat the full MMC heat sink needs to be modeled in order to closelypredict the chip level maximum temperature.

The energetic performance of the heat sink was analyzed usingthe 2nd law of thermodynamics. Sources of exergy destruction in

the heat sink were identified by a detailed local entropy generationanalysis at low and high Rein operating conditions. The analysis

shows that the turbulent dissipation, _Sgen;D0 ;tot , contributes most of

the entropy generation arising from the fluid friction, _Sgen;D;tot . How-

ever, the total entropy generation due to heat transfer, _Sgen;Q ;tot , ismuch higher than the total entropy generation due to viscous dis-sipation, _Sgen;D;tot , at both low and high Rein conditions. _Sgen;D;tot in-creases significantly at high Rein but still contributes less thanone-third of the total entropy generated in the heat sink. Reduction

in _Sgen;Q ;tot with increase in Rein leads to reduction in overall entropy

generation _Sgen;tot . Our results conclusively show that the use of hotwater as coolant can lead to a high 2nd law efficiency, which isessential to successfully utilize the heat recovered from coolingof electronic chips and data centers for secondary usage such asbuilding heating. Our results also show that moderately increasedinvestment of pumping energy, while increasing viscous dissipa-tion can reduce the entropy generation due to heat transfer andthus minimize the overall entropy generation.

Acknowledgments

The work was supported in part through AQUASAR projectfunded by the Competence Center Energy and Mobility (CCEM).The support is gratefully acknowledged. The authors thank SeverinZimmermann (ETH Zurich) for providing experimental data on theheat sink performance and Ingmar Meijer (IBM Research, Zurich)for useful discussions and comments on this work.

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