A numerical and experimental study to evaluate performance of vascularized cooling plates

International Journal of Heat and Fluid Flow 32 (2011) 1186–1198

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International Journal of Heat and Fluid Flow

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A numerical and experimental study to evaluate performance of vascularizedcooling plates

Kee-Hyeon Cho a,c, Won-Pyo Chang b, Moo-Hwan Kim a,⇑a Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk 790-784, Republic of Koreab KAERI, Daeduk-daero 1045, Dukjin-dong, Yuseong-gu, Daejeon 305-353, Republic of Koreac Energy and Resources Research Department, Research Institute of Industrial Science and Technology (RIST), Namgu, Pohang, Kyungbuk 790-600, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 March 2011Received in revised form 29 June 2011Accepted 19 September 2011Available online 14 October 2011

Keywords:ConstructalVascularCooling platesCooling channelSelf-healingSelf-cooling

0142-727X/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2011.09.006

⇑ Corresponding author. Tel.: +82 54 279 2165.E-mail address: [email protected] (M.-H. Kim)

The present paper reports on a numerical simulation and experimental validation of fluid flow and con-jugate heat transfer characteristics of new vascular channels, whose cross-sections are semi-circular. Thenumerical analysis covers the Reynolds number range of 30�2000, with a cooling channel volume frac-tion of 0.04, pressure drop range of 30�105 Pa. Six flow configurations were considered: first, second, andthird constructal structures with optimized hydraulic diameters and non-optimized hydraulic diameterfor each system size 10 � 10, 20 � 20, and 50 � 50, respectively. The numerical results of the proposedvascular channels show that the channel configurations of the optimized constructs show much lowerflow resistance and temperature distribution than those of the non-optimized constructs. It is also shownthat the power component in the power-law relationship between mass flow rate and pressure dropdecreases as the system size and mass flow rates increase. The numerical results are validated by exper-imental data, and with the two exhibiting excellent agreement in all cases. The validation study againstthe experimental data shows that the presented numerical model is a reliable tool for predicting the per-formance of cooling plates under practical operating conditions and for the design of self healing orcooling system.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

As electronic components have recently been designed withhigh power density, the development of compact and highlyefficient cooling geometries is of increasing interest for electronicsystems. Therefore, there is a demand for microelectronic packagesto facilitate an effective cooling device with an efficient coolingmethod to maintain an acceptable temperature for operating elec-tronic components. Microchannels are widely adopted in relevantindustries to design more compact geometries for heat transferapplications, because conventional cooling devices, using finnedheat sinks with air, are no longer adequate in responding to therecent heat load demand.

In general, the use of microchannels may improve thermalperformance due to their compactness, although such use may in-cur the disadvantages of increased pumping power and uneventemperature distribution. In this regard, it is desirable to increasethe temperature uniformity of the wall, while decreasing the pres-sure drop, to overcome the previously mentioned problems. How-ever, it is still a significant challenge achieving both improvedtemperature uniformity and lower pumping power across such

ll rights reserved.

.

compact devices for cooling plates of polymer electrolyte mem-brane fuel cell (PEMFC) or micro-electromechanical systems(MEMS) simultaneously.

Recent optimization technology used in multi-scale heat sinksor cooling plates has attracted increasing interest. Several typesof effective flow architectures have been proposed, and theirdesign has a significant effect on the operating performance ofengineering equipments. An important segment of this new litera-ture is based on a general principle—namely, the constructal law(Bejan, 2000; Bejan and Lorente, 2004, 2008), which Bejan ex-plained in 1997.

An increasing number of research studies have examined thedevelopment of flow architecture using the constructal law (e.g.,Chen et al., 2004; Brod, 2003; Hernandez et al., 2003; Kraus,2003; Bejan and Lorente, 2004, 2008). One active direction in engi-neering is the vascularization of smart materials, which may offernew or improved volumetric functions, such as self-cooling, self-healing, and variable transport properties (Wang et al., 2009). Abody with embedded tree-shaped flows that bathe the entirevolume is a vascularized body. Recently, Kim et al. (2006), andCho et al. (2010b) demonstrated how the method can be used todevelop vascular structures for self-healing materials, similar towidely encountered natural systems, and can be inspired fromthese systems for design. Saber et al. (2010) discussed the

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Nomenclature

Ac total internal wetted area of all the channels, m2, Eq. (6)Cp specific heat of coolant, J kg�1 K�1, Eqs. (5) and (6)d elemental length scale, mDh hydraulic diameter, m, Eq. (3)D1 semi-circular diameter of thin channels, m, Fig. 1D2 semi-circular diameter of thick channels, m, Fig. 1D1h hydraulic diameter of thin channels, mD2h hydraulic diameter of thick channels, mH, L outer dimensions of vascularized unit, mm, Figs. 2 and 5_m mass flow rate, kg/s

No number of thin channels, Fig. 6P local pressure, PaP pumping power, W, Eq. (12) and Figs. 9 and 10_Q sensible heat transfer rate, W, Eqs. (5), (6), and (10)q00 heat flux imposed on the cooling plate, W/m2, Fig. 2R thermal resistance, K W�1, Eq. (10)r flow non-uniformity ratio, Eq. (9)Re Reynolds number, Eq. (4)Sv svelteness number, Eq. (1) and Table 1T1, T2, T3 thermocouple locations, Fig. 5U total uncertainty, Eq. (8)Vin local mean velocity at inlet, m s�1, Eq. (4)

Vc total channel flow volume, m3, Eq. (1) and Table 1X, Y, W inner dimensions of vascularized unit, mm, Fig. 1

Greek symbolsa power component, Eq. (11)DP pressure difference, Pau channel porosityl fluid dynamic viscosity, kg/s mq fluid density, kg m�3

Subscriptsc channelf fluidi channel rankin inletm meanmax maximumONB onset of boilingout outlets surfacesat saturationw wall

K.-H. Cho et al. / International Journal of Heat and Fluid Flow 32 (2011) 1186–1198 1187

geometrical design of various channel networks based on a multi-scale approach, proposing the rapid design of channel multi-scalenetworks with minimum flow maldistribution.

However, the various studies on the constructal law or on amulti-scale approach solely focus on its theoretic aspects, whilestudies on numerical or experimental works to validate previousresearch results are limited. To the best of the authors’ knowledge,only Da Silva and Bejan (2006) among Bejan and his coworkersexperimentally investigated the hydraulic and thermal behaviorof a balanced counterflow heat exchanger. Moreover, very fewstudies related to the flow and heat transfer in the cooling platesof PEMFC have been reported in the literature. Several authors(Garrity et al., 2007; Choi et al., 2008; Fan et al., 2008) investigateda coupled cooling process involved in fluid flow and heat transfer

m

W

x

yzx

yz

m

m

Y

X

D1

D2

D1

D2

m

W

x

yzx

yz

m

m

Y

X

D1

D2

D1

D2

(a) Fig. 1. Schematic diagram of the three-dimensional geometry of cooling plates with trethird construct.

between a solid plate and coolant flow for the optimization ofthe cooling design of a fuel cell stack. However, they mainly pre-sented the design with a parallel-serpentine channel in the coolingplates of PEMFC. Furthermore, despite various studies conductedon the hydraulic and thermal performance of the cooling channels,none have arrived at a global optimized solution.

In light of this research background, the current work focuseson both numerical analysis and experimental investigation to eval-uate the hydraulic and thermal performance of cooling plates withsemi-circular cross-sections as an effective solution to fabricatingtest sections. The aim of this study is to present the experimentalvalidation of a proposed numerical model and to evaluate theperformance of cooling plates, based on constructal-based vasculardesigns. Three designs are considered on a square flat volume

W

m

m

Y

X

W

m

Y

X

D1

D2

W

m

m

Y

X

W

m

Y

X

D1

D2

(b) (c)e-shaped channel (10 � 10 elements): (a) first construct; (b) second construct; (c)

1188 K.-H. Cho et al. / International Journal of Heat and Fluid Flow 32 (2011) 1186–1198

consisting of 10 � 10 volume elements: a first-level construct(Fig. 1a), a second-level construct (Fig. 1b), and a third-level con-struct (Fig. 1c). The structures with 20 � 20 and 50 � 50 volumeelements in addition to the 10 � 10 elements are also studied.

2. Numerical model and simulation

2.1. Geometry

The constructal-based geometry considered was a vascularizedcooling body consisting of a square slab measuring X � Y and

Table 1Geometric dimensions for the constructal configurations.

System size Complexity Vc (m3) d (m) D

10 � 10 1st 4 � 10�6 10�2 12nd 4 � 10�6 10�2 13rd 4 � 10�6 10�2 1

20 � 20 1st 2 � 10�6 5 � 10�3 92nd 2 � 10�6 5 � 10�3 93rd 2 � 10�6 5 � 10�3 9

50 � 50 1st 8 � 10�7 2 � 10�3 32nd 8 � 10�7 2 � 10�3 33rd 8 � 10�7 2 � 10�3 3

m

X=100

Y=

100

10

10

m

Active area100 100

xy xym

X=100

Y=

100

10

10

Active area100 100

xy xy

(a)

inlePin

0n

T =∂∂

0n

T =∂∂

.

.

.

.

q

x zx z

z

z

yx yx

inlePin

0n

T =∂∂

0n

T =∂∂

.

.

.

.

q

x zx z

z

z

yx yx

(cFig. 2. Geometries and dimensions corresponding to the methods used in the present stuboundary conditions of cooling plates (2nd construct, 20 � 20 elements). All dimension

having thickness W, where W is the dimension of the solid bodyin the direction perpendicular to the plane X � Y, as shown inFig. 1. The size of the square domain was measured in terms ofN � N, where N is the number of small square elements countedalong one side (Cho et al., 2010b). All structures had the embeddedvasculatures of semi-circular cross-section channels filled withcoolants.

New vascular designs for the volumetric bathing of the smartstructures with volumetric functionalities (self-healing, cooling)were configured by using the methodology explored in the previ-ous research work (Cho et al., 2010b): one channel size versus

h (D1h = D2h) (m) D2h (m) D1h (m) Sv

.868 � 10�3 2.856 � 10�3 1.582 � 10�3 6.3

.803 � 10�3 2.418 � 10�3 1.276 � 10�3 6.3

.885 � 10�3 2.251 � 10�3 1.301 � 10�3 6.3

.530 � 10�4 1.833 � 10�3 8.100 � 10�4 7.9

.330 � 10�4 1.566 � 10�3 6.560 � 10�4 7.9

.550 � 10�4 1.417 � 10�3 6.490 � 10�4 7.9

.860 � 10�4 1.029 � 10�3 3.350 � 10�4 10.8

.830 � 10�4 9.000 � 10�4 2.780 � 10�4 10.8

.860 � 10�4 8.010 � 10�4 2.720 � 10�4 10.8

H=120

L=

120

Cell area (120 120)

Active area100 100

m

m

H=120

L=

120

Cell area (120 120)

Active area100 100

m

m

(b)

t

outlet

0n

T =∂∂

q Pout=0

0n

T =∂∂

t

outlet

0n

T =∂∂

q Pout=0

0n

T =∂∂

) dy: (a) numerical method; (b) experimental method; (c) computational domain ands are in mm.

Table 2Thermo-physical properties of DI water and AISI 304 (in SI units) at atmospheric pressure (273.15 < T < 373.15 K used in the simulations) (Cho et al., 2010a).

DI water AISI 304

q 998.2 8030.0k �0.829 + 0.0079 T � 1.04 � 10�5 T2 11.702649 + 0.012955 TCp 5348 � 7.42 T + 1.17 � 10�2 T2 114.227517 + 1.877902 T � 0.003234 T2 + 3.0 � 10�6 T3 � 8 � 10�10 T4

l 0.0194 � 1.065 � 10�4 T + 1.489 � 10�7 T2 –

Fig. 3. Experimental apparatus.


two channel sizes, increasing complexity (1st, 2nd, and 3rd con-structs), and increasing size (up to 50 � 50 elemental volumes).The detailed geometrical dimensions for each configuration aresummarized in Table 1. The svelteness (Sv) of the flow structure,which is a global property, is defined as the ratio between theexternal and internal length scales:

Sv ¼ external length scaleinternal length scale

¼ ðXYÞ1=2

V1=3c

ð1Þ

Sv plays an important role in the evolution the best or near-bestarchitecture in a fixed space (near the ‘‘equilibrium flow configura-tion’’ performance level (Bejan, 2000)). Optimized multiple scalesD1h and D2h were distributed non-uniformly through the availableflow volume, as shown in Fig. 1. The corresponding non-optimizedconfigurations with only one channel size were also defined on asquare domain composed of N � N elements. One channel size, Dh,existed with the channel volume, Vc, fixed on the same basis asthe results of the optimized configurations. The channel porosityu is fixed at 0.04 for all the configurations.

When different flow structures are compared, X and Y are fixed,but W (i.e., elemental length d) varies with increasing system size—namely, N2 = 10 � 10, 20 � 20, or 50 � 50. To investigate the effectsof geometric complexity on the behavior of the fluid flow and heattransfer characteristics of these vascularized networks, due tocomputational and experimental limitations, the present study is

limited to only three configurations (1st, 2nd, and 3rd constructs)with different complexity.

In addition, the numerical model was modified so that thenumerical results could be compared with those obtained throughexperimental observations: outer part including the inlet and out-let are added to the basis of Fig. 1 to solve the fabrication problemof test sections, as shown in Fig. 2. The control volume for thenumerical analysis is illustrated in Fig. 2a, which is different fromthe model (Fig. 2b) used in the experimental analysis. The flowpath (straight channel for simplicity, see Fig. 2a) of numerical mod-el along the outer volume of active area different from the path ofexperimental model (curved channel, see Fig. 2b). However, inreality the pressure drop difference between the two models isnegligible, as the resulting relative error falls within 3%.

2.2. Numerical model

The heat transfer and fluid flow performance of the constructalchannel architectures were simulated numerically using a modelfor three-dimensional conjugated heat transfer for each configura-tion. To simplify the numerical simulation, only the cell area(1.44 � 10�2 m2) including the active area (10�2 m2) of the coolingplate for PEMFC was included in the computational domain. Theframe (x, y, z) is aligned with the (X, Y, W) directions, as shownin Fig. 1. Cooling is provided by an embedded three-dimensionalsemi-circular channel network. The boundary conditions areapplied as shown in Fig. 2c. The bottom surfaces are subjected to

Top plate Bottom plate

(a)


constant heat fluxes, q00 = 5000, 1500, and 500 W/m2 to maintainthe single-phase through the available flow volume for 10 � 10,20 � 20 and 50 � 50 elements, respectively. The other surfacesare adiabatic, as shown in Fig. 2c.

The numerical work covered the overall pressure drop rangeDP = 30�105 Pa, which corresponded to the mass flow rate range1.7 � 10�5 to 6.9 � 10�3 kg/s. The inlet temperature Tin is fixed at18 �C. The material properties of DI water and AISI 304 steel usedin this study were determined according to the correlations listedin Table 2. To focus on the effect of the optimized and non-optimizedchannel configurations on the cooling plate performance, the sameassumptions and governing equations as presented in previouswork (Cho et al., 2010a) were used.

(b)

Bonded line Top plate

2.3. Numerical method

Computations were performed using a finite-volume package(Fluent Inc., 2006) with the pressure-based solver, the node-basedgradient evaluation, the SIMPLE algorithm for pressure–velocitycoupling, and the second order upwind scheme for momentumand energy equations. In grid generation, hexagonal grids areadopted. The independence of the solution with respect to the gridsize was checked by examining the values of the mass flow rate,maximum temperature, temperature differences between the inletand outlet, and convective heat transfer coefficient between chan-nel walls and fluid for each geometrical configuration. Convergenceis achieved when the residuals for the mass and momentum equa-tion are smaller than 10�4, and the residual of the energy equationis less than 10�11.

Bottom plate

(c)

Fig. 4. Test sections: (a) photograph of the test section (before bonding, D1h = D2h,3rd constructs with 10 � 10 elements); (b) photograph of the test section (afterbonding, 50 � 50 elements); (c) SEM image of the bonded test sections (cross-sectional view, left: 10 � 10 elements, right: 50 � 50 elements).

3. Experimental apparatus and procedure

3.1. Flow loop

Fig. 3 shows the experimental setup and the correspondingapparatus constructed to supply deionized (DI) water to the vascu-larized cooling plate test section at the desired operating condi-tions. The experimental apparatus can be split into three mainsystems: the pressurizing system, the cooling plate test sectionsystem, and the data acquisition system. A pressure vessel con-nected to high-pressure helium gas (He) was used to push the DIwater through 1/4-in. tube. Two T-type (Copper/Constantan) ther-mocouples with an accuracy of ±0.5 �C in the range of �40 to125 �C and a diameter of 6.35 mm were installed at the cold sideinlet and hot side outlet of the test section to measure the inletand exit temperatures of the DI water, respectively.

The differences in pressure of DI water flowing through test sec-tion were measured using three differential pressure transducerswith different working ranges (7, 35, and 350 kPa, respectively).A Setra pressure transducer (Model 230) with an accuracy of±0.25% over the full range of each model, respectively, was cali-brated against reference standards of a known and well-character-ized measurement uncertainty in the pressure measurements(AMETEK PK II). Meanwhile, the mass flow rate was determinedby weighing the mass increment over a longer given period of time(30 s) using a high precision electronic balance (ED6202S-CW)with an accuracy of 0.01 g, connected to a PC by RS232C to recordthe real-time data.

The heating power was supplied by a power supply unit, con-sisting of an AC power supply (the Silicon Controlled Rectifiers[SCR] controller TPR-2S), accepting a control signal (e.g.,4–20 mAdc) from a signal conditioning device (power controller,MT4W). The thermal load was provided by a silicone rubber heatercomposed of Cu–Ni thermic wires with a diameter of 0.22 mm(12 X). All temperatures and differential pressures were collected

using the HP high speed data acquisition system (Agilent34970A), displayed on a PC monitor, and stored in the PC’s memoryfor further analysis.

3.2. Test section

The test section shown in Fig. 4 was manufactured from AISI304 plates, silicon rubber heaters, and 3 K-type thermocouples.The cooling plate test sections for channel configurations with10 � 10 and 20 � 20 elements were fabricated on the bottom platewith a 5- and 3-mm thick AISI 304 plate, respectively, by directmechanical milling using a 30-lm diameter end mill on a precisioncomputer numerical control (CNC) milling machine (see Fig. 4a).Subsequently, it was assembled by diffusion bonding with a topplate 5- and 2-mm thick, respectively. Concurrently, the test sec-tions for the largest system size in the present study, 50 � 50 ele-ments with a 120 � 120 mm2 cross-section 1.5 mm thick, wereetched chemically and assembled with a 0.5-mm thick AISI plateusing diffusion bonding. The inlet and outlet section of the testassembly were made of AISI 316 1/4-in. tube and 1/4-in. fittings,which are easy to assemble and disassemble (see Fig. 4b). Fig. 4cshows a cross-sectional view (SEM image) of a portion of thebonded test section structure. The positions of the thermocouples,marked from T1, T2, and T3, along the diagonal in the direction ofthe fluid flow, and silicon rubber heater are illustrated in Fig. 5. Asshown, thermocouples are embedded in the top wall of the plate,

Top plate Bottom plate

Tap jig

m

m

Pressure tap

(a)

10010

0120

OD= 6.35

1515

ID= 4.57

Heater

Cooling plate

10010

0120

OD= 6.35

1515

ID= 4.57

Heater

Cooling plate

(b)

2 3 5

Top plate

Heater

100

3Tap jig

10

STS tube OD: 6.35 20 10 20

10

Bottom plate

Flow in Flow out

paterusserPpaterusserP

1202 3 5

100

3

10

20 20

10rat

1202 3 5

100

3

10

20 20

120

(c)

Y=

100

mm

X=100 mm

H=120 mm

Active area100 mm x 100 mm

Cell area 120 mm x 120 mm

L=120

Inlet

Outlet

T1

T2

T3

50 mm

50 mm

Flow direction

Inlet

Outlet

10 mm

10 m

m

10 m

m

10 mm

10 mm

10 m

m

Y=

100

mm

X=100 mm

H=120 mm

Active area100 mm x 100 mm

Cell area 120 mm x 120 mm

L=120

Inlet

Outlet

T1

T2

T3

50 mm

50 mm

Flow direction

Inlet

Outlet

10 mm

10 m

m

10 m

m

10 mm

10 mm

10 m

m

(d)

Fig. 5. Details of assembled test section: (a) isometric view of assembled test section; (b) bottom view of test section containing a silicone rubber heater; (c) cross-sectionalview showing the dimensions of the inlet and outlet fittings; (d) thermocouple locations on cooling plates. All dimensions are in mm.


iav

er

m/m

=

0 10 20 30 40 50

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

D1 = D

2

D1

D2

Channel number

m

m

i=1, 2 3, ………………………………………. No

Flow direction

Fig. 6. Comparison of the flow uniformity between the optimized and non-optimized results at DP ffi 1 kPa for the first constructs with 50 � 50 elements.


while the silicon rubber heater is attached to the bottom plate sur-face (see Fig. 5c).

3.3. Test procedure

The fluid was degassed prior to each experimental run. Degas-sing was achieved by boiling the DI water in the reservoir vigor-ously using an imbedded 1 kW cartridge heater for one and ahalf hours. Such work is not expected to affect the generation ofbubbles, which result in an greater pressure drop. In each experi-ment, the power supply to the heaters was set to the desired valueafter the flow rate and the inlet fluid temperature were stabilized.Steady state was achieved within approximately 20–30 min for thetest section with optimized channels in each test run, and within2 h for the test sections with the non-optimized channels, whenall temperature readings were within ±0.1 �C. All power, tempera-ture, pressure, and mass flow measurements were averaged over a10-min period. The flow rate was then increased for the next test,in increments of approximately 10–30 SCCM (standard cubic cen-timeter per minute), and the experimental procedure wasrepeated.

During experiments, the flow loop’s components were first ad-justed to yield the desired inlet temperature Tin and mass flow rate_m. The water inlet temperature Tin was set to 18 �C. The experi-

mental parameters covered the following ranges: flow rates of10–400 ml/min and heat fluxes of 500–5000 W/m2 (with the threeheated flux levels, q00 = 5000 W/m2, q00 = 1500 W/m2, andq00 = 500 W/m2 for channel configurations with 10 � 10, 20 � 20,and 50 � 50 elements, respectively). To yield acceptable values ofthe heat flux that can sustain liquid single-phase flow, we used acorrelation derived for predicting the onset of nucleate boiling(ONB). This widely used correlation was proposed by Bergles andRohsenow (1964):

q00ONB ¼ 5:30S1:156 1:8ðTw � TsatÞONB

� �2:41=S0:0234

ð2Þ

where q00ONB is the heat flux at the onset of nucleate boiling (W/m2), Sis the system pressure (kPa), Tw is the wall temperature (K), and Tsat

is the saturation temperature (K). Based on Eq. (2), no boiling willoccur in our experiments, because an applied heat flux(q00 = 500 W/m2) is sufficiently small compared to that of the esti-mated ONB heat flux q00ONB (7.2 � 105 W/m2 at DP = 1000 Pa for thenon-optimized constructs with 50 � 50 elements). In addition, thepower input across the test section was controlled such that the

surface temperature was below the saturation temperature of theDI water, especially more careful experiments were needed to ob-serve a single bubble or a few bubbles close to the outlet at low flowrates.

3.4. Data reduction

The hydraulic diameter for the semi-circular cross-sectionalarea can be rewritten as

Dh ¼pD

pþ 2ð3Þ

where D is the diameter of the semi-circular cross-sectional area.The Reynolds number is determined using the average inlet

velocity calculated from the measured mass flow rate. The highestReynolds numbers for all structures did not exceed 2000, whichensured that the flow regime was laminar. The Reynolds numberis defined as:

Re ¼ qf V inDh

lfð4Þ

where Vin is the average frontal velocity at the inlet and Dh is thehydraulic diameter.

The sensible heat transfer rate of DI water can be determinedfrom the inlet and outlet temperatures across the test sections,specifically:

_Q ¼ _mCpðTout � T inÞ ð5Þ

where _m is the mass flow rate, Cp is the specific heat of coolant atthe mean DI water temperature Tave, and Tin and Tout are the inletand outlet temperature of coolant, respectively. The mean temper-ature is defined as the arithmetic average of the DI water tempera-ture, Tm = (Tout + Tin)/2. The above sensible heat transfer rate can beused for calculating an average heat transfer coefficient:

have ¼_Q

Ac � DT lm¼

_mCPðTout � T inÞAc � DTm

ð6Þ

where Ac is the wetted surface area of the cooling channels. In Eq.(6), the mean temperature difference between the fluid and the wallis defined as:

DTm ¼ Ts �ðT in þ ToutÞ

2¼ ðT1þ T2þ T3Þ

3� ðT in þ ToutÞ

2ð7Þ

where T1, T2, T3 are the top wall temperatures measured by threethermocouples; Tin and Tout are the bulk temperature of fluid atthe inlet and outlet of the cooling plates, respectively. As it is diffi-cult to measure the exact wall surface temperature in the currentexperiments, Ts was estimated by taking the average reading ofthe three thermocouples embedded in the top plate of the test sec-tions, as illustrated in Fig. 5.

3.5. Experimental uncertainties

Detailed uncertainties of the various calculated parameterswere estimated according to Holman (1984). In estimating theuncertainty in measured and calculated quantities, both the biasand precision errors need to be considered. The uncertainty U is gi-ven as follows:

U ¼ e2bias þ e2

precision

h i1=2ð8Þ

where ebias and eprecision are bias and precision uncertainties, respec-tively. The total uncertainties for the calculated parameters for theReynolds number, pumping power, and average heat transfer coef-ficient were ±2.0�7.4%, ±3.7�10.1%, and ±2.1�14.5%, respectively.

(a) D1h D2h )b( D1h = D2h

(c) D1h D2h (d) D1h = D2h

(e) D1h D2h (f) D1h = D2h

Fig. 7. Dimensional distribution of temperature in the designs for the first constructs: (a) 10 � 10 elements, D1h – D2h, Sv = 6.3, DP ffi 300 Pa; (b) 10 � 10 elements, D1h = D2h,Sv = 6.3, DP ffi 300 Pa; (c) 20 � 20 elements, D1h – D2h, Sv = 7.9, DP ffi 1 kPa; (d) 20 � 20 elements, D1h = D2h, Sv = 7.9, DP ffi 1 kPa; (e) 50 � 50 elements, D1h – D2h, Sv = 10.8,DP ffi 10 kPa; (f) 50 � 50 elements, D1h = D2h, Sv = 10.8, DP ffi 10 kPa.


The calculated uncertainties are shown as error bars on the flowcurves plotted in the following section (Figs. 8–10).

4. Results and discussion

The results of the fluid flow and heat transfer characteristicsbased on the numerical and experimental data are presented anddiscussed in this section for a wide range of Reynolds numbers.Using the experimental procedure described in Section 3.3, thenumerical results of the fluid flow and heat transfer analysis ob-tained in optimized and non-optimized constructal test sectionsare compared with those from the experiments.

4.1. Numerical analysis

To capture the effect of maldistribution on the thermal perfor-mance, flow maldistribution is evaluated by using the flow non-uniformity ratio in the mass flow rate distribution along the thinchannels. The non-uniformity ratio r is defined as:

r ¼_mi

_maveð9Þ

where _mi is the mass flow rate, only along the thin channels (D1h),and _mave is the average value of mass flow rates along the thinchannels (D1h), with a given number of channels (No) on the square

Table 3Comparison of maximum temperature difference (Armax) between optimized and non-optimized constructs.

System size Complexity DP (Pa) Optimized Tmax (K) DTmax (K) (a) DP (Pa) Non-optimized Tmax (K) DTmax (K) (b) b�ab

�� (%)

10 � 10 1st 94.8 307.8 13.9 94.4 327.9 31.5 55.9277.9 301.0 6.7 276.9 314.5 15.4 56.5

2nd 92.5 310.2 16.2 93.7 326.6 31.7 48.9270.9 301.6 8.2 273.6 311.2 15.4 46.8

3rd 91.8 314.0 18.1 93.4 322.1 28.4 36.3266.7 305.8 8.9 272.8 309.3 13.9 36.0

20 � 20 1st 291.0 299.5 7.8 292.7 325.2 31.8 75.5947.0 295.1 3.2 960.3 309.1 13.5 76.3

2nd 285.0 300.7 8.3 291.6 323.7 30.9 73.1916.2 295.2 3.4 951.5 306.9 12.7 73.2

3 rd 285.3 302.6 10.3 290.8 320.6 28.3 63.6917.4 296.8 4.3 948.3 304.2 11.8 63.6

50 � 50 1st 2939.3 294.3 3.2 2971.3 326.1 33.6 90.59591.9 292.4 1.2 9835.1 308.8 13.6 91.2

2nd 2874.6 294.6 2.9 2966.4 323.5 31.5 90.89217.7 292.4 1.1 9791.2 305.6 12.3 91.1

3 rd 2877.1 295.5 3.7 2961.3 321.2 29.3 87.49248.4 292.9 1.5 9760.0 303.8 11.4 86.8


domains of Fig. 6. The variable i indicates the channel rank, wherechannel number (No) is 50 for the first construct with 50 � 50 ele-ments. In most instances, it is very important to spread the flowas uniformly as possible, because the difference among the massflow rates creates undesirable hot spots directly linked to the ther-mal resistance. This phenomenon will be discussed in the nextsection.

Fig. 6 shows a comparison of the flow uniformity between theoptimized and non-optimized results at DP ffi 1 kPa for the firstconstructs with 50 � 50 elements. In comparing the maldistribu-tion of optimized and non-optimized constructs, the design ofthe optimized constructs is superior to that of the non-optimizedconstructs. The flow maldistribution becomes undesirable for boththe optimized and non-optimized configurations, due to the minorlosses, as the mass flow rate increases, thus, resulting in a morenon-uniform flow and formation of hot spots. These findings arein agreement with previously published works (Cho et al., 2010a,2010b) considering vascular designs with circular cross-sections.

Fig. 7 presents the temperature distribution of the active areaonly on the top surface of the cooling plates. Fig. 7a–f shows thetemperature distribution at the top plane (active area only) ofthe cooling plate with 10 � 10, 20 � 20, and 50 � 50 elements,respectively. The maximum temperature for optimized channels(D1h – D2h) (Fig. 7a, c, and d) was much lower than for non-opti-mized channels (D1h = D2h) (Fig. 7b, d, and f) under the same leg-end. Most notably, clear and intensive hot spots near the uppercentral area are evident for the non-optimized configurations asthe system size continues to increase.

In addition, the thermal resistance is the most effective physicalquantity to evaluate the heat dissipation performance of a coolingplate, and is defined as:

R ¼ Ts;max � Ts;min

_Q¼ DTmax

_Qð10Þ

where Ts,max is the maximum temperature at the cooling plate andTs,min the coolant inlet temperature. To reduce the thermal resis-tance R for a given heat transfer rate, the temperature at hotspotsmust be reduced. The maximum temperature differences DTmax ofthe optimized channel configurations were much lower than thoseof the non-optimized channel configurations. For example, the max-imum temperature difference of the 1st optimized constructs with

10 � 10 elements was approximately 56.5% lower than that of thenon-optimized construct in the vicinity of DP = 277 Pa, whereasthe maximum temperature difference of the 2nd optimized con-structs with 50 � 50 elements was approximately 90.8% lower thanthat of the non-optimized construct in the vicinity of DP = 2900 Pa.These results are summarized in Table 3.

4.2. Experimental verification

The validity of the numerical model for the flow resistances inthe vascularized cooling plates was examined first by comparingthe numerical results with the experimental results. Subsequently,the dependence of the top surface temperature on pumping powerwas compared for three locations and the heat transfer coefficientperformance was evaluated. Both comparisons showed that thepressure drop and pumping power in the test section gave the larg-est impact on the performance of the cooling plates.

The pressure drop as a function of mass flow rate is presented inFig. 8 for all architectures studied. Each graph in Fig. 8 presents theflow resistances evaluated from 10 � 10 to 50 � 50 elements inthe system size N2. The mass flow rate ( _m) changed according tothe pressure difference imposed to the test sections. There is anapproximately power-law relationship between the mass flow rate( _m) and overall pressure drop (DP) in every flow channel:

DP � _ma ð11Þ

where the power component, a, varies with _m. From Fig. 8 it is obvi-ous that increasing the system size will decrease the power compo-nent for both the optimized and non-optimized constructs. Athigher mass flow rates the non-linearity on a logarithmic scale be-comes larger for 10 � 10 elements, i.e., the pressure drop differencebetween two optimized and non-optimized cases for the givenmass flow rate is reduced significantly. However, it became smallerunder similar circumstances as the system size increased, as illus-trated in Fig. 8b and c. For example, the power component (a) ran-ged from 1.542�1.561 to 1.290�1.360 for the optimized constructsas the system size increases, whereas it ranged from 1.644�1.647 to1.239�1.255 for the non-optimized constructs. The reason is that athigher mass flow rates (or high Re) there are strong vortices whichaffect the local pressure loss, such as pressure drops caused by in-lets, outlets, enlargements, and contractions. Such a tendency may

(a)

(b)

(c)

Fig. 8. Comparison of the pressure drop versus mass flow rate determinednumerically (closed and open symbols for the optimized constructs and non-optimized constructs, respectively) and the experimental solution (solid anddashed lines for the optimized constructs and non-optimized constructs, respec-tively): (a) 10 � 10 elements; (b) 20 � 20 elements; (c) 50 � 50 elements. Theclosed and open symbols represent the experimental data for the optimized andnon-optimized constructs, respectively, and the results from the numerical analysesfor the optimized and non-optimized constructs are plotted in the solid and dashedlines, respectively.


be also related to the Svelteness (Sv) for each configuration (seeTable 1). It was shown in reference work (Lorente and Bejan,2005; Bejan and Lorente, 2008) that when Sv (Cho et al., 2010b) ex-ceeds the order of 10, the pressure losses are dominated by Poiseu-ille fluid friction along the straight channels, while the losses due to

bends and junctions are negligible. That is, decreasing Sv will in-crease the significance of local losses as is the case for 10 � 10elements. In contrast, at low Re in which there is no vortex, fric-tion losses caused by wall friction due to the viscosity of DI waterin motion are dominant. Also, it should be noted that such atrend for the design of the optimized constructs is more signifi-cant due to the lower flow resistance than that of the non-optimized constructs, as seen in Fig. 8a. Additionally, Fig. 8acompares the numerical prediction of pressure drop versus massflow rate with the experimental data for the constructal configu-rations with 10 � 10 elements. For the optimized construct de-signs, the experimental results were observed to be slightlyhigher than those from the numerical analysis for both optimizedand non-optimized architecture, while the experimental resultsfor the non-optimized configurations fell within the range ofnumerical analyses. However, the tendency is similar among allthe optimized and non-optimized configurations; the flow con-ductance of the first construct turned was superior to that ofthe second and third constructal structures. On the other hand,the experimentally as well as numerically observed flow resis-tance of the third construct was the lowest among the non-opti-mized channel configurations, as illustrated in Fig. 8a. For the10 � 10 elements, the numerical results agreed relatively wellwith the experimental data over the range of mass flow rate ex-plored in this study. However, discrepancies were also found be-tween the comparisons for the optimized constructs, most likelydue to geometric uncertainties, which certainly increased theflow resistances as a consequence.

A comparison in Fig. 8b gave a decrease of the pressure dropsfor the optimized third construct with 20 � 20 elements of 76.0%,compared with those of the non-optimized third construct at amass flow rate of 0.001 kg/s. The difference between the optimizedand non-optimized pressure drops across the test sections in theexperimentally measured data is overestimated by about 16.1%according to the numerical analysis results calculated at a massflow rate of 0.0013 kg/s. As shown in Fig. 8b, the pressure dropspredicted by numerical analyses also agree well with the experi-mental data. Moreover, the performance variation with the in-crease of the mass flow rates demonstrated a similar trendwithin the experimental error.

Similarly, Fig. 8c compares the pressure drop to mass flowrate between the numerical prediction and experimental datafor the constructal configurations with 50 � 50 elements. Forexample, a significant decrease (about 1/17) in the pressure dropacross the test sections for the optimized second construct with50 � 50 elements was observed, compared with that of the non-optimized third construct at a mass flow rate of 0.0001 kg/s. Asseen in Fig. 8a–c, a good agreement between the numerical re-sults and experimental data is present for the flow resistance.

Fig. 9 compares the measured wall temperatures, T1, T2, and T3,at the three surface locations in the test section (see Fig. 5) withthose predicted by the numerical analysis with increasing pump-ing power P. The pumping power for these comparisons was esti-mated based on the measured mass flow rate and pressure drops.The pumping power required by the HLW element is given as:

P ¼_mDPqf

ð12Þ

where _m is the total mass flow rate through the inlet. Fig. 9a–fillustrates the temperature distribution at the top plane (activearea only) of the cooling plate for the first constructs with10 � 10, 20 � 20, and 50 � 50 elements, respectively. The errorbars are the standard deviations of the measured surfacetemperatures.

10-5 10-4 10-3 10-2290

300

310

320

Experimental

Sur

face

tem

pera

ture

(K

)

Pumping power [W]

T1 T2 T3

T1 T2 T3

10 × 10, 1st construct, D1

≠ D2

Predicted

10-5 10-4 10-3 10-2

290

300

310

320

330

340

350

Pumping power [W]

Sur

face

tem

pera

ture

(K

)

10 × 10, 1st construct, D1 = D

2

Experimental

T1 T2 T3

T1 T2 T3

Predicted

(b)(a)

10-5 10-4 10-3 10-2 10-1290

295

300

305

310

Surf

ace

tem

pera

ture

(K

)

Pumping power [W]

20 × 20, 2nd construct, D1

≠ D2

Experimental

T1 T2 T3

T1 T2 T3

Predicted

10-5 10-4 10-3 10-2

290

300

310

320

330

340

350

Surf

ace

tem

pera

ture

(K

)

Pumping power [W]

20 × 20, 2nd construct, D1 = D

2

Experimental

T1 T2 T3

T1 T2 T3

Predicted

(d)(c)

10-6 10-5 10-4 10-3 10-2290

300

310

320

330

340

Sur

face

tem

pera

ture

(K

)

Pumping power [W]

50 × 50, 3rd construct, D1

≠ D2

Experimental

T1 T2 T3

T1 T2 T3

Predicted

10-5 10-4 10-3 10-2 10-1290

300

310

320

330

340

350

Surf

ace

tem

pera

ture

(K

)

Pumping power [W]

50 × 50, 3rd construct, D1 = D

2

Experimental

T1 T2 T3

T1 T2 T3

Predicted

(f)(e)Fig. 9. Comparison between predicted and measured surface temperature of the cooling plates at thermocouple locations T1, T2, and T3: (a) 10 � 10 elements, 1st construct,D1h – D2h; (b) 10 � 10 elements, 1st construct, D1h = D2h; (c) 20 � 20 elements, 2nd construct, D1h – D2h; (d) 20 � 20 elements, 2nd construct, D1h = D2h; (e) 50 � 50 elements,3rd construct, D1h – D2h; (f) 50 � 50 elements, 3rd construct, D1h = D2h. The closed symbols represented the experimentally measured data for both the optimized constructs(Fig. 9a, c, e) and non-optimized constructs (Fig. 9b, d, f). The results obtained from the numerical analysis for the first (in black), second (in red), and third (in green)optimized construct are plotted in solid lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


As shown in Fig. 9, at each thermocouple located the flowstream, the surface temperature decreased with pumping poweras expected; the surface temperature rise was moderate for a high

pumping power, because the heat transfer coefficient between thechannel wall and fluid was sufficiently high. In short, the local heattransfer coefficient increase led to a decrease in the surface

10-5 10-4 10-3 10-2200

1,000

10,000

Ove

rall

hea

t tra

nsfe

r co

effi

cien

t [W

/m2 K

]

Pumping power [W]

Measured

Predicted

10 × 10, 1st construct

D1

≠ D2

D1 = D

2

D1

≠ D2

D1 = D

2

(a)

10-5 10-4 10-3 10-2 10-170

100

1,000

8,000

Ove

rall

hea

t tra

nsfe

r co

effi

cien

t [W

/m2 K

]

Pumping power [W]

Measured

Predicted

20 × 20, 2nd construct

D1

≠ D2

D1 = D

2

D1

≠ D2

D1 = D

2

(b)

10-6 10-5 10-4 10-3 10-2 10-120

100

1,000

4,000

Ove

rall

hea

t tra

nsfe

r co

effi

cien

t [W

/m2 K

]

Pumping power [W]

Measured

Predicted

50 × 50, 3rd construct

D1

≠ D2

D1 = D

2

D1

≠ D2

D1 = D

2

(c)

Fig. 10. Comparison of measured and predicted average heat transfer coefficientsas a function of pumping power: (a) 10 � 10 elements, 1st construct; (b) 20 � 20elements, 2nd construct; (c) 50 � 50 elements, 3rd construct. The open symbolsrepresent the experimental data, while the closed symbols represent the numericalpredictions. Also, the results from the numerical analysis for the optimized andnon-optimized constructs are plotted in the solid and dotted lines, respectively.


temperature, which manifested itself in a lower global thermalresistance. This implies that a major part of heat from the bottomsurface of the test section is surely transferred to coolant throughthe broad conjugate wall.

As illustrated in Fig. 9a, the highest surface temperature moni-tored at three locations (T1, T2, and T3) for the optimized first con-

struct with 10 � 10 elements occurs at the T1 location; it is about306.6 K at a given pumping power of 7.1 � 10�5 W, based on thenumerical results, and the relative difference between the pre-dicted and measured values was about 3.0%. Meanwhile, the tem-perature at the location of T2 for the non-optimized first constructwith 10 � 10 elements was the highest at 316.5 K, as illustrated inFig. 9b. This is caused by a hot spot occurring near the location ofT2, as shown in Fig. 9b.

Fig. 9c and d displays the measured surface temperature at thelocations described above for the optimized and non-optimizedsecond constructs with 20 � 20 elements, respectively. The highestsurface temperature for the optimized second constructs with20 � 20 elements occurred at the T1 location across all workingconditions, whereas the heat transfer ability for the non-optimizedsecond constructs was closely competitive with each other in theentire range.

Similarly, Fig. 9e and f illustrate the measured surface temper-ature for the optimized and non-optimized third constructs with50 � 50 elements, respectively. As shown in Fig. 9e and f, the sur-face temperature at T3 location is higher than those of other loca-tions for both the optimized and non-optimized across all workingconditions. This is the reason the cooling capacity of the coolant islowest at the right side of the lower parallel channel part of thesecond constructs and the third constructs provide poor heattransfer ability over the entire range, because of the area uncov-ered by cooling channels, as shown in Fig. 1c. However, the poorheat transfer effect by the uncovered area is likely to weaken ifthe system size (N2) increases above 50 � 50.

Fig. 10 compares the measured average heat transfer coeffi-cients with those predicted by the numerical analysis with increas-ing pumping power P, based on the results obtained in Fig. 9. Theerror bars are the standard deviations of the measured heat trans-fer coefficients. The calculation of error bars on havg consideredmeasurements uncertainties on temperatures Ts, Tout, and Tin,water mass flow rate _m and test section dimension, assuming thatall uncertainties are independent and uncorrelated. The relative er-ror in havg increases with increasing pumping power P. The overalluncertainties in havg were in the range of 2.1–14.5%.

As depicted in Fig. 10a–c, the numerical heat transfer coeffi-cients are in the range of 60–6654 W/m2 K. With increasing pump-ing power the heat transfer coefficients increase for all channelconfigurations. The increasing rate of the average heat transfercoefficient with the increase in the pumping power is relativelyfast at the lower pumping power, but becomes slow with the in-crease in the pumping power, which is caused by the higher pres-sure losses due to the minor losses. Notably, the difference in heattransfer coefficient between optimized and non-optimized struc-tures was found to increase as the system size increases, whichis consistent with the results in Refs. (Cho et al., 2010a; Cho andKim, 2010). Alternatively, from the perspective of a fixed pumpingpower input, the data shows that more heat is transferred as thesystem size increases. For example, the average heat transfer coef-ficients of the 1st optimized constructs with 10 � 10 elements islarger by 89.8% based on the non-optimized construct at a massflow rate of 1.0 � 10�4 kg/s, while the maximum relative percent-age difference between the average heat transfer coefficients of theoptimized and non-optimized structures is about 376.6% based onthe non-optimized construct for 50 � 50 elements. Furthermore,for the same amount of applied pumping power the optimizedchannel configurations always have higher heat transfer coeffi-cients than those of the non-optimized channel configurations.For example, the average heat transfer coefficients of the 2ndoptimized constructs with 20 � 20 elements was larger by 173%based on the non-optimized construct in the vicinity of_m ¼ 1:0� 10�4 kg=s. Therefore, the optimized constructs are al-

ways most effective. On the other hand, a comparison of the results


of both numerical and experimental data shows good agreementboth in the optimized and non-optimized constructs. This confirmsthat classical Navier–Stokes and energy equations introduced hereare valid for the numerical modeling of the proposed vascularchannel networks. In addition, the experimental and numerical re-sults show the same trends but the numerical results slightly over-estimate the heat transfer coefficients, especially in Fig. 10a and b.The discrepancies of flow resistance as discussed above, especiallyfor the optimized constructs with 10 � 10 and 20 � 20 elements,may be the reason for this difference, across all working conditions.In summary, a close scrutiny of Figs. 8–10 revealed that the predic-tions from the numerical analysis were readily consistent with theexperimental data within the experimental uncertainty over arange of experimental conditions, and the flow and thermal perfor-mances increased significantly as the system size increased.

5. Conclusions

In the present study, the hydraulic and thermal behavior of con-structal vascularized cooling plates with semi-circular cross-sections were investigated experimentally as well as numerically,and the best suitable strategy for the thermal management of cool-ing plates was also suggested.

Based on our numerical results, the optimized constructal cool-ing plates served lower thermal distribution, much better thermaland flow uniformity, and much lower pressure drops than the non-optimized cooling plates (conventional types with one diameter). Itwas also found that a further increase in the system size, for allcases, causes the system performance to enhance: the larger vascu-latures are relatively more efficient than the smaller vasculatures.The best architecture in the non-optimized configurations was thethird construct over all working conditions.

In addition, the proposed numerical model was validated by theexperimental data, and exhibited an excellent agreement in allcases. It was demonstrated from the comparison with the experi-mental data that the numerical simulations were sufficiently accu-rate and the numerical approach considered may be usedpractically for vascular designs. Thus, the use of these optimizedsystems as cooling plates for PEMFC or microelectronic coolingapplications has great future potential.

References

Bejan, A., 1997. Advanced Engineering Thermodynamics, second ed. Wiley, NY.Bejan, A., 2000. Shape and Structure: From Engineering to Nature. Cambridge

University Press, UK.Bejan, A., Lorente, S., 2004. The constructal law and the thermodynamics of flow

systems with configuration. International Journal of Heat and Mass Transfer 47,3203–3214.

Bejan, A., Lorente, S., 2008. Design with Constructal Theory. Wiley, Hoboken, NJ.Bergles, A.E., Rohsenow, W.M., 1964. The determination of forced-convection

surface boiling heat transfer. ASME Journal Heat Transfer C 86, 365–372.Brod, H., 2003. Residence time optimized choice of tube diameters and slit heights

in distribution systems for non-Newtonian liquids. Journal of Non-NewtonianFluid Mechanics 111, 107–125.

Chen, F.C., Gao, Z., Loutfy, R.O., Hecht, M., 2004. Analysis of optimal heat transfer in aPEM fuel cell cooling plate. Fuel Cells 3, 181–188.

Cho, K.H., Lee, J., Bejan, A., Kim, M.H., 2010a. Fluid flow and heat transfer invascularized cooling plates. International Journal of Heat and Mass Transfer 53,3607–3614.

Cho, K.H., Lee, J., Kim, M.H., Bejan, A., 2010b. Vascular design of constructalstructures with low flow resistance and nonuniformtiy. International Journal ofThermal Sciences 49, 2309–2318.

Cho, K.H., Kim, M.H., 2010. Fluid flow characteristics of vascularized channelnetworks. Chemical Engineering Science 65, 6270–6281.

Choi, J., Kim, Y., Lee, Y., Lee, K., Kim, Y., 2008. Numerical analysis on the performanceof cooling plates in a PEFC. Journal of Mechanical Science and Technology 22,1417–1425.

Da Silva, A.K., Bejan, A., 2006. Dendritic counterflow heat exchanger experiments.International Journal of Thermal Sciences 45, 860–869.

Garrity, P., Klausner, J.F., Mei, R., 2007. A flow boiling microchannel evaporator platefor fuel cell thermal management. Heat Transfer Engineering 28, 877–884.

Fan, Y., Boichot, R., Goldin, T., Luo, L., 2008. Flow distribution property of theconstructal distributor and heat transfer intensification in a mini heatexchanger. AIChE Journal 54, 2796–2808.

Fluent Inc., 2006. FLUENT Version 6.3 User’s Guide. Fluent Inc., Lebanon, NH.Hernandez, G., Allen, J.K., Mistree, F., 2003. Platform design for customizable

products as a problem of access in a geometric space. Engineering Optimization35, 229–254.

Holman, J.P., 1984. Experimental Methods for Engineers, fourth ed. McGraw-Hill,NY.

Kraus, A.D., 2003. Constructal theory and the optimization of fin arrays. In: ASMEInternational Mechanical Engineering Congress and Exposition. Washington, DC.

Kim, S.W., Lorente, S., Bejan, A., 2006. Vascularized materials: tree-shaped flowarchitectures matched canopy to canopy. Journal of Applied Physics 100,063525.

Lorente, S., Bejan, A., 2005. Svelteness, freedom to morph, and constructalmultiscale flow structures. International Journal of Thermal Sciences 44,1123–1130.

Saber, M., Commenge, J.M., Falk, L., 2010. Microreactor numbering-up in multi-scalenetworks for industrial scale applications: impact of flow maldistribution onthe reactor performances. Chemical Engineering Science 65, 372–379.

Wang, K.M., Lorente, S., Bejan, A., 2009. Vascular materials cooled with grids andradial channels. International Journal of Heat and Mass Transfer 52, 1230–1239.

A numerical and experimental study to evaluate performance of vascularized cooling plates

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