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Applied Thermal Engineering 33-34 (2012) 157e166

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

Hydraulic-thermal performance of vascularized cooling plates with semi-circularcross-section

Kee-Hyeon Cho a,*, Chi-Woong Choi b

a Energy and Resources Research Department, Research Institute of Industrial Science and Technology (RIST), Namgu, Pohang, Kyungbuk 790-600, Republic of KoreabDepartment of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk 790-784, Republic of Korea

a r t i c l e i n f o

Article history:Received 26 April 2011Accepted 22 September 2011Available online 1 October 2011

Keywords:ConstructalVascularDendriticCooling platesSelf-healingSelf-cooling

* Corresponding author. Tel.: þ82 54 279 5267; faxE-mail address: khcho@postech.ac.kr (K.-H. Cho).

1359-4311/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.applthermaleng.2011.09.029

a b s t r a c t

Analytical and numerical studies are conducted to investigate the hydraulic and thermal performance ofnew vascular channels with semi-circular cross-sections. An analytical model is built to minimize theflow resistance in vascularized cooling plates and then vascular designs are created by using theproposed model. The analytical results show that the dimensionless global flow resistance for theconfigurations with semi-circular cross-sections, both the optimized and non-optimized constructs, issignificantly higher than it is for the configurations with circular cross-sections. A numerical model forthree-dimensional fluid flow and heat transfer characteristics of vascularized cooling plates is alsopresented. Then, to validate the analytical model, the flow resistances predicted by the analytical modelare compared with numerical data subject to a fixed volume and a fixed pumping power, and favorableagreements between the numerical and analytical results are obtained as system size increases. It is alsoshown that the thermal resistances for the first and second optimized constructs are closely competitiveacross all working conditions, whereas the best architecture in the non-optimized configurations is thethird construct among the cooling plates.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Over the last few decades, thermal management has becomea crucial issues in the design of electronic or automotive compo-nents because temperature affects their performance and reli-ability. Especially with the decrease in size of thermal systems, it isvital to dissipate their heat. Even the traditional convective coolingof these systems may lose its performance in some cases [1] andthus should be managed.

Effective flow architectures of several types are proposedbecause the design of the flow architecture has a significant effecton the operating performance of engineering equipment [2]. A newresearch direction for global optimization is based on the con-structal law [3e6]. Among the engineered flow architecturesderived from the constructal law are the tree-shaped (dendritic)designs. Tree-shaped flow configurations offer maximum accessbetween one point (inlet, or outlet) and an infinite number ofpoints (area, volume) [7]. Numerous studies [8e19] have beenconducted to examine the development of flow architecture usingthe constructal law. Wechsatol et al. [20] developed general rules to

: þ82 54 279 6309.

All rights reserved.

construct asymmetric trees. They also showed that asymmetricbifurcation provides lower flow resistance than symmetric bifur-cation. We have characterized the thermo-hydraulic performanceof vascular designs with circular cross-sections [2]. More recently,Kim et al. [21] showed that the main features of a steam generatorcan be determined based on themethod of constructal design. Theyalso showed that the total heat transfer rate increases pro-portionately to the length of the entire heat exchanger.

Polymer electrolyte membrane fuel cells (PEMFCs) are the typicalexamples where cooling is important: local hot spots due to animproperly designed cooling system can accelerate the mechanicaldamage of polymer electrolyte membranes, impairing the reliabilityof PEMFCs [22]. In reality, a temperature gradient of a magnitude ofw5 �C has been found across the cathode. This temperature gradienthas a significant influence on the water and heat transport ina PEMFC [23]. Thus, dealing with hot spots is one of the biggestchallenges for cooling plates in a PEMFC.

Chen et al. [24] studied the cooling performance of severalserpentine and parallel channel designs through a numerical simu-lation of fluid flow and heat transfer within the cooling plates. Theyconcluded that the cooling effect of serpentine-type cooling modescould be better than that of parallel-type cooling modes in terms oftemperature uniformity and maximum temperature. Similarly, Choi

Nomenclature

A area, m2

C Constant factor, m2 s�1, Eqs. (3) and (4)d elemental length scale, mDh hydraulic diameter, m, Eq. (6) and Table 1Di channel diameter of the ith channel, mD1h hydraulic diameter of thin channels, mD2h hydraulic diameter of thick channels, mLi length of the ith channel, m_m mass flow rate, kg s�1

_mi mass flow rate of the ith channel, kg s�1

P local pressure, PaP pumping power, W, Eq. (28)p perimeter, m, Eq. (5)Pin inlet pressure, PaPout outlet pressure, Paq total heat imposed on the cooling plate, W, Eq. (27)R thermal resistance, K W�1, Eq. (27)Re Reynolds number, Eq. (26)Sv svelteness number, Eq. (2)Tm mass-weighted average temperature, K, Eq. (25)Vin local mean velocity at inlet, m s�1, Eq. (26)V element volume, m3, Eq. (1)

Vc total channel flow volume, m3, Table 1x, y, z Cartesian coordinatesX, Y, W inner dimensions of vascularized unit, mm, Figs. 1

and 2

Greek symbolsDP pressure difference, Par fluid density, kg m�3

f porosity, Eq. (1)m fluid dynamic viscosity, kg s�1 mn fluid kinematic viscosity, m2 s�1

j non-dimensional global flow resistance, Eq. (15)

Subscriptscomp compensation of pressure drop along the inlet and

outlet channelsi channel rankin inletmax maximumout outlet

Superscript/ Vector quantity

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166158

et al. [25], using computational fluid dynamics (CFD) simulations,examined the performance of cooling plates for the PEMFC bycomparing the numerical results for serpentine channels with thosefor parallel channels. They also showed that modified serpentine-type and modified parallel-type models are more cooling thanthose that are unmodified. Yu et al. [26] studied several multi-passserpentine flow-field designs in order to achieve better heatmanagement by using cooling plates. They demonstrated thatmulti-pass serpentine flow-field designs had better cooling performancethan did conventional serpentine flow-field, in terms of both themaximum temperature and temperature uniformity. Recently, Kur-nia et al. [27] studied the thermal performanceof parallel, serpentine,wavy, coiled and novel hybrid channels. Likewise, despite severalstudies of the hydraulic and thermal performance of cooling plates,none of these models report the optimized cooling channel archi-tectures (or multi-scale based) based on the constructal law.

As noted above, while there have been some theoretical studieson the constructal law or on a multi-scale approach, the studiesmeant to validate previous research results or to optimize thegeometric configurations of the cooling plates with semi-circularcross-sections for more practical applications have been limited.This research therefore uses analytical and numerical investiga-tions to evaluate the hydraulic and thermal performance of coolingplates with semi-circular cross-sections which provide a solution tothe fabrication of test sections. In general, the circular section does

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

not represent the channel cross-sections being produced in engi-neering devices very well, while the stacked plates for etchedpassages is represented by a semi-circle. The impact of thesecharacteristics on the relative heat transfer and pressure dropperformance of such channels has not been quantified [28]. In thisstudy, three types of design are considered on a square flat volumeconsisting of 10 � 10, 20 � 20, and 50 � 50 volume elements:a first-level construct (Fig. 1a), a second-level construct, anda third-level construct, respectively.

2. Analytical model and method

2.1. Geometry

The considered geometry in the study is a vascularized coolingbody consisting of a square slab measuring X � Y and havingthickness W, where W is the dimension of the solid body in thedirection perpendicular to the plane X � Y, as shown in Fig. 1. Thesize of the square domain is measured in terms of N� N, where N isthe number of small square elements counted along one side (seeFig. 1b). New vascular designs for the volumetric bathing of thesmart structures with volumetric functionalities (self-healing,cooling) are optimized by using the methodology which wasexplored in the previous research work [7]: one channel size versustwo channel sizes, increasing complexity (1st, 2nd, and 3rd

h (D1h ¼ 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

Fig. 1. One typical physical model of cooling plates with tree-shaped channel of the first construct with 10 � 10 elements: (a) three-dimensional geometry with two hydraulicdiameters (D1h, D2h); (b) two-dimensional analytical model.

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166 159

constructs) and increasing size (up to 50 � 50 elemental volumes).The detailed geometrical dimensions for each configuration aresummarized in Table 1. All structures have the embedded vascu-latures of semi-circular cross-section channels filled with coolants.Optimized multiple scales D1 and D2 are distributed non-uniformlythrough the available flow volume, as shown in Fig. 1. The corre-sponding non-optimized configurations with only one channel sizeare also defined on a square domain composed of N � N elements.One channel size, D, exists with the channel volume Vc fixed on thesame basis as the results of the optimized configurations. Thechannel porosity f is fixed at 0.04 for all the configurations.

When different flow structures are compared, X and Y are fixedbut W (i.e., elemental length d) varies with increasing the systemsizednamely, N2 ¼ 10 � 10, 20 � 20, or 50 � 50. To investigate theeffects of geometric complexity on the behavior of the fluid flowand heat transfer characteristics of such vascularized networks, due

Fig. 2. Computational domain employed for the present study (the first construct with 10 �numerical model. All dimensions are in mm.

to computational limitations, the present study is limited to only 3configurations (1st, 2nd, and 3rd constructs) with differentcomplexity.

The volume fraction occupied by all channels is held constant:

f ¼ total channel volumetotal volume

¼ Vc

V(1)

where the element volume is V ¼ XYW. The void fraction f is theaverage porosity of the whole structure assuming the channelvolume is distributed uniformly. Another geometric property of theflow structure was the svelteness (Sv), which is the ratio of theexternal length scale ((XY)1/2) divided by the internal length scale:

Sv ¼ external length scaleinternal length scale

¼ ðXYÞ1=2V1=3c

(2)

10 elements): (a) analytical domain; (b) numerical domain; (c) boundary conditions for

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166160

Sv is a global property of the flow architecture, playing animportant role in its evolution toward the best or near-best archi-tecture in a fixed space (near the “equilibrium flow configuration”performance level [3]).

2.2. Analytical model

Fig. 1b illustrates a square domain composed of 10 � 10elements for an analytical approach. The centers of the elementsare indicated by the black circles. The only degree of freedom in thedesign of Fig. 1b is the channel diameter ratio D1h/D2h. Theassumptions on which the analytical analysis was based, are fullydeveloped laminar flow, negligible pressure losses at junctions orbends, and fluid with constant properties.

An estimation of the pressure drop along a channel with lengthLi, diameter Dh,i, and mass flow rate _mi for an active area, as illus-trated in Figs. 1b and 2a, can be given as:

DPi ¼ C_miLiD4h;i

(3)

where C is a constant factor, for example,C ¼ 128p5n=ððpþ 2Þ4ðp2 � 8ÞÞ, if the shape of the duct cross-section is semi-circular and i is the channel rank, DPi representsthe corresponding pressure drop inside a duct that can bea channel, a distributor, or a collector portion [3,5].

In the present study, control volume for the analytical analysis isillustrated in Fig. 2a, which is different from the model (Fig. 2a)used in the numerical analysis. That is to say, since the analyticalmodel has no channel volume corresponded to the cell area, asshown in Fig. 2a and b, the inlet and outlet pressure drops acrossthe channel volumes are compensated by equation

DPcomp ¼ C_mLinD42h

þ C_mLoutD42h

(4)

whereDPcomp is the pressure drop along a channel with inlet lengthLin (0.01m), outlet length Lout (0.01m), inlet diameterD2h, andmassflow rate _m.

2.3. Hydraulic diameter optimization for low flow resistance

To increase the freedom to morph the flow architecture, we addmore length scales that can be varied, in this case twodiameter sizesinstead of one [6]. The ratio of diameters D1/D2 is the additionaldegree of freedom. The optimized multiple scales D1 and D2 will bedistributed non-uniformly through the available flow volume.

Here we show how to build the first constructs with twodiameter sizes on a square domainwith 10 � 10 elements. It is wellknown that the thinner channels (D1) should be placed in thecanopy and the thicker (D2) in the stem and main branches.

The hydraulic diameter of any duct is given by

Dh ¼ 4Ap

(5)

where A is the cross-sectional area and p is the perimeter. Thehydraulic diameter for the semi-circular cross-sectional area can berewritten as

Dh ¼ pDpþ 2

(6)

where D is the diameter of the semi-circular cross-sectional area.The first construct (Fig. 1b) is optimized by using the method

presented in our previous work [7]. The calculation of DP= _m

consists of elements involved in Eq. (3) for all channels, taking intoaccount the mass continuity at every junction, and adding all theDPi ’s along one flow path, from the inlet to the outlet of the flowstructure. For Fig. 1b, we calculate the overall pressure drop acrossthe entire flow structure by writing

DP ¼ DP1 þ DP2 þ DP3 þ DP4 þ DP5 þ DP6 þ DP7 þ DP8 þ DP9þ DP9 þ DP10 þ DP11 þ DP30

¼ Cd

12

_m1

D42h

þ _m2

D42h

þ _m3

D42h

þ _m4

D42h

þ _m5

D42h

þ _m6

D42h

þ _m7

D42h

þ _m8

D42h

þ _m9

D42h

þ _m10

D42h

þ 9 _m11

D41h

þ 12

_m30

D42h

!

(7)

where:

_m1 ¼ _m2 þ _m20 ¼ _m11 þ _m29 ¼ _m30 ¼ _m_m2 ¼ _m3 þ _m19_m3 ¼ _m4 þ _m18_m4 ¼ _m5 þ _m17_m5 ¼ _m6 þ _m16_m6 ¼ _m7 þ _m15_m7 ¼ _m8 þ _m14_m8 ¼ _m9 þ _m13_m9 ¼ _m10 þ _m12_m10 ¼ _m11

(8)

Note also that

DP20 þ DP21 ¼ DP2 þ DP19DP19 þ DP22 ¼ DP3 þ DP18DP18 þ DP23 ¼ DP4 þ DP17DP17 þ DP24 ¼ DP5 þ DP16DP16 þ DP25 ¼ DP6 þ DP15DP15 þ DP26 ¼ DP7 þ DP14DP14 þ DP27 ¼ DP8 þ DP13DP13 þ DP28 ¼ DP9 þ DP12DP12 þ DP29 ¼ DP10 þ DP11

(9)

Combining Eqs. (7)e(9) we find that themass flow rates throughthe element centers are

_m2 ¼ _m29 ¼ 112201138722

_m

_m3 ¼ _m28 ¼ 4760069361

_m

_m4 ¼ _m27 ¼ 83941138722

_m

_m5 ¼ _m26 ¼ 3796169361

_m

_m6 ¼ _m25 ¼ 12_m

_m7 ¼ _m24 ¼ 3140069361

_m

_m8 ¼ _m23 ¼ 54781138722

_m

_m9 ¼ _m22 ¼ 2176169361

_m

_m10 ¼ _m11 ¼ _m20 ¼ _m21 ¼ 26521138722

_m

(10)

The flow volume constraint is

Vc ¼ p8

�19d� D2

2 þ 90d� D21

�(11)

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166 161

Combining Eqs. (6) and (10)e(11) we find that the flow resis-tance can be expressed non-dimensionally as

DPCd _m

¼ 238689138722

1D41

!þ 11

2

1D42

!!�pþ 2p

�4

(12)

By minimizing the expression (12) with respect to D1 and D2

subject to constraint (11) we obtain the optimal ratio of diametersand the minimized flow resistance

�D1

D2

�opt

¼ 0:554 (13)

DPCd _m

�Vc

d

�2

¼ 38:950 (14)

2.4. Analytical solution: pressure drop vs. mass flow rate

Eq. (14) can be rewritten in terms of the dimensionless globalflow resistance [7] form as below:

j ¼ DPC _m

f2d3 (15)

where d is the elemental length and f is the porosity of the coolingplate [2,7]. Here the corresponding mass flow rate for eachconfiguration can be determined from Eq. (15) for a given pressuredrop, DP.

3. Numerical model and method

3.1. Numerical model

The heat transfer and fluid flow performance of the con-structal channel architectures are simulated numerically usinga model for three-dimensional conjugated heat transfer for eachconfiguration. To simplify the numerical simulation, only the cellarea (1.44 � 10�2 m2) including the active area (10�2 m2) of thecooling plate for PEMFC is included in the computational domain.The frame (x, y, z) is aligned with the (X, Y, W) directions, asshown in Fig. 1. Cooling is provided by an embedded three-dimensional semi-circular channel network. Coolant is pumpedinto the cooling plate with a specified pressure at the inlet. Thispressure has to be high enough to overcome the pressure losseswhen the cooling plate is under operation. The porosity f is fixedat 0.04 for all the configurations. The boundary conditions areapplied as shown in Fig. 2c. The bottom surfaces are subjected toconstant heat fluxes, q00 ¼ 5000, 1500, and 500 W/m2 in order tomaintain the single phase through the available flow volume for10 � 10, 20 � 20 and 50 � 50 elements, respectively. The othersurfaces are adiabatic.

The numerical work covers the overall pressure drop rangeDP ¼ 30�105 Pa, which corresponded to the mass flow rate range1.7 � 10�5�6.9 � 10�3 kg/s. The inlet temperature Tin is fixed at291 K. The material properties of DI water [29] and AISI 304 steel[30] used in this study are determined according to the correlationslisted in an earlier work [2].

To focus on the effect of the optimized and non-optimizedchannel configurations on the cooling plate performance, thesame assumptions as presented in previous work [2] were made.Based on the assumptions, the governing equations for mass,

momentum, and energy were solved numerically in the solid andfluid domains of the cooling plate, as follows:

(1) Mass conservation:

vuvx

þ vv

vyþ vw

vz¼ 0 (16)

(2) Momentum equation:

rf

�vðuuÞvx

þ vðuvÞvy

þ vðuwÞvz

�¼ �vp

vxþ�v

vx

�mfvuvx

�

þ v

vy

�mfvuvy

�þ v

vz

�mfvuvz

��(17)

rf

�vðuvÞvx

þvðvvÞvy

þvðvwÞvz

�¼�vp

vyþ�v

vx

�mf

vv

vx

�þ v

vy

�mf

vv

vy

�

þ v

vz

�mfvv

vz

��(18)

rf

�vðuwÞvx

þ vðvwÞvy

þ vðwwÞvz

�¼ �vp

vzþ�v

vx

�mfvwvx

�

þ v

vy

�mfvwvy

�þ v

vz

�mfvwvz

��(19)

(3) Energy equation:- Energy in the fluid domain:

rf

v��

cp�f uT�

vxþv��

cp�f vT�

vyþv��

cp�f wT

�vz

!

¼ v

vx

�kfvTvx

�þ v

vy

�kfvTvy

�þ v

vz

�kfvTvz

�(20)

- Energy in the solid domain:

v

vx

�ksvTvx

�þ v

vy

�ksvTvy

�þ v

vz

�ksvTvz

�¼ 0 (21)

The constant heat flux imposed on the upper and lower X � Yplanes is:

q00 ¼ �ksvTvz

(22)

where ks is the thermal conductivity of the solid. The continuity ofthe temperature and heat flux at the interface between the solidand fluid require:

Tjs ¼ T jf

(23)

ksvTvn

����s ¼ kfvTvn

����f (24)

where kf is the fluid thermal conductivity and n is the vector normalto the surface.

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166162

3.2. Numerical method

Computations are performed using a finite-volume package [31]with the pressure-based solver, the node-based gradient evalua-tion, the SIMPLE algorithm for pressureevelocity coupling, and thesecond order upwind scheme for momentum and energy equa-tions. In grid generation, hexagonal grids are adopted. The inde-pendence of the solution with respect to the grid size is checked byexamining the values of themass flow rate, maximum temperature,temperature differences between the inlet and outlet, convectiveheat transfer coefficient between channel walls and fluid, and skinfriction coefficients for each geometrical configuration.

Grid-independence tests are carried out for the six configura-tions presented in the preceding sections. The number of cellsvaries from one case to another; for example, the smallest numberwas 3.0 � 106 for the optimized first construct with 10 � 10elements. Convergence is achieved when the residuals for the massand momentum equation are smaller than 10�4, and the residual ofthe energy equation is less than 10�11. In addition, the differencebetween the incoming and outgoing mass flow rates was examinedclosely with the sensible heat balance which is the differencebetween the incoming and outgoing enthalpies. For all of the casessimulated in this study, the relative difference is bounded within0.01%.

4. Results and discussion

In this section, vascular designs of the cooling plates with semi-circular cross-sections are first optimized analytically. Then, thevalidity of the analytical methodology for the flow resistances isexamined by comparing the numerical results with the analyticaldata. Finally, detailed descriptions of the thermal performance ofcooling plates are discussed based on the numerical results.

4.1. Analytical solution

Fig. 3 demonstrates the results of the comparison of thedimensionless global flow resistance j between circular [7] andsemi-circular cross-sections. The j value estimated for eachconfiguration has a single value. For example, the j value for theoptimized first constructs with 10� 10 elements (Fig. 1) with semi-circular channels, reaches the minimum value of 3.895 (Fig. 5).Optimal flow resistances of the first, second, and third construct of

000,4000,10010.5

1

10

60

1st constrcut

2nd construct

3rd construct 21 DD =21 DD =

construct

construct opt2

1DD

⎟⎟⎠

⎞⎜⎜⎝

⎛circular

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡semi-circular

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ 1st construct

2nd construct

3rd construct

1st constrcut

2nd

3rd

opt2

1DD

⎟⎟⎠

⎞⎜⎜⎝

⎛

N2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ semi-circular

1st construct

2nd construct

3rd construct

circular

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡[7]

Fig. 3. Comparison of the global flow resistances between the channel configurationswith circular and semi-circular cross-section: (a) D1 s D2; (b) D1 ¼ D2.

Fig. 4. Comparison of the pressure drop vs. mass flow rate determined numerically(closed and open symbols for the optimized constructs and non-optimized constructs,respectively) and the analytical solution (solid and dashed lines for the optimizedconstructs and non-optimized constructs, respectively): (a) 10 � 10 elements; (b)20 � 20 elements; (c) 50 � 50 elements.

a semi-circular cross-section with one (Dh) and two diameter sizes(D1h, D2h) for each configuration, are reported in Fig. 3.

For the optimized designs, the global flow resistance j decreasesslowlyas the system sizeN2 increases. Note, however, that the globalflow resistance j increases steeply as the system size N2 increases

Fig. 5. Dimensional distribution of temperature in the designs for each channel configuration: (a)e(b) 2nd construct with 10 � 10 elements, Sv ¼ 6.3, DP y 300 Pa; (c)e(d) 3rdconstruct with 10 � 10 elements, Sv ¼ 6.3, DP y 300 Pa; (e)e(f) 2nd construct with 20 � 20 elements, Sv ¼ 7.9, DP y 1 kPa; (g)e(h) 3rd construct with 20 � 20 elements, Sv ¼ 7.9,DP y 1 kPa; (i)e(j) 2nd construct with 50 � 50 elements, Sv ¼ 10.8, DP y 10 kPa; (k)e(l) 3rd construct with 50 � 50 elements, Sv ¼ 10.8, DP y 10 kPa.

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166 163

for the non-optimized designs. Fig. 3 also shows that larger archi-tectures flow more easily if they are configured according to thedeveloped method. The dimensionless global flow resistance j forthe configurations with semi-circular cross-sections, both theoptimized andnon-optimized constructs is significantlyhigher thanthat of the configurations with circular cross-sections. For example,the dimensionless global flow resistance of the semi-circularchannel design is larger by 79.4% based on the circular channeldesign for the optimized first construct with 10 � 10 elements.

4.2. Comparison between analytical and numerical solution

The results from the numerical work are compared with theanalytical results in this section. The overall pressure drop acrossthe cooling plates is defined as DP¼ Pin�Pout, where Pin and Pout arethe inlet and outlet pressures, respectively.

All temperature-dependent properties are based on the mass-weighted average temperatures, Tm, which is calculated based onthe mass-weighted average, as follows:

Tm ¼Xn

j¼1Tjrj��� v/

j$A/

j

���Pn

j¼1rj

��� v/j$A/

j

��� (25)

where Tj, rj, and v/

j are respectively the fluid temperature, densityand velocity vector in the jth cell along the cooling channel; whileA/

j is the flow area of the jth cell. All temperature-dependentproperties as well as all parameters used in analytical results arecalculated based on Eq. (25).

The Reynolds number is determined using the average inletvelocity calculated from the measured mass flow rate. The highestReynolds numbers for all structures did not exceed 2000, whichensured that the flow regime must be laminar. The Reynoldsnumber is defined as:

Re ¼ rf VinDh

mf(26)

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

Each graph in Fig. 4 illustrates how the imposed pressuredifference (DP) varies with the mass flow rate for the three systemsizes (10 � 10, 20 � 20, and 50 � 50). The results of the comparisonbetween the analytical and numerical results for three constructaltypes (1st, 2nd, and 3rd constructs) with the optimized and non-optimized diameters are also shown. Here closed and opensymbols represent the numerical results for the optimizedconstructs (first construct in black, second construct in red, and

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166164

third construct in green) and non-optimized constructs (same coloras optimized constructs), respectively. Solid and dashed lines alsopresent the analytical solution for the optimized constructs andnon-optimized constructs (same color as numerical results),respectively.

As Fig. 4 indicates, the pressure drop (DP) increases with themass flow rate ( _m) while the pressure drop variation from thenumerical results is more steep than from the analytical results,with the gradient increasing with the mass flow rate. This non-linear relationship of the pressure drop and mass flow rate can beattributed to the junctions and the developing of the fluid flow ateach junction or bend. In other words, with increasing the massflow rate, there are strong vortices which affect the local pressureloss, such as pressure drops caused by inlets, outlets, enlargements,and contractions. The formation of these vortices results in pres-sure variations and it becomes more relevant, especially for theconfigurations with 10 � 10 elements. Such a tendency can be alsorelated to the svelteness (Sv) for each configuration (see Table 1). Itwas shown in Refs. [5,32] that when Sv exceeds the order of 10, thepressure losses are dominated by Poiseuille fluid friction along thestraight channels, while the losses due to bends and junctions arenegligible. That is, decreasing Sv will increase the importance oflocal losses as is the case for 10� 10 elements. In contrast, at low Rein which there is no vortex, friction losses caused by wall frictiondue to the viscosity of DI water in motion are dominant. Also, itshould be noted that such a trend for the design of the optimizedconstructs is more significant due to the lower flow resistance thanit of the non-optimized constructs. Note, however, that the differ-ence between the analytical and numerical results becomes smalleras the system size (N2) increases in both optimized and non-optimized designs. In particular, the difference decreases sharplyfor the system size, 50 � 50, because of the large Sv (10.8) andresulting small vortices. For example, the relative differencesbetween the pressure drops of the optimized and non-optimizedstructures are approximately 17.9% at a mass flow rate of4.9� 10�4 kg/s and about 56.2% at amass flow rate of 2.0� 10�3 kg/s based on the non-optimized construct for the first construct with10 � 10 elements, respectively, as illustrated in Fig. 4a. Meanwhile,the relative differences between the pressure drops of the opti-mized and non-optimized structures are only approximately 4.6%at a mass flow rate of 3.0 � 10�5 kg/s for the first construct with50 � 50 elements, as shown in Fig. 4c. Consequently, the results of

Table 2Comparison of pressure drop, mass flow rate, and the maximum temperature of the firs

System size Complexity Optimized

DP (Pa) _m (kg/s) (a) Tmax

10 � 10 1st 28.8 3.999 � 10�4 323913.5 3.555 � 10�3 296

2nd 28.3 3.463 � 10�4 329872.2 3.094 � 10�3 297

3rd 28.3 3.023 � 10�4 333857.5 2.813 � 10�3 300

20 � 20 1st 98.3 1.963 � 10�4 3092764.7 2.406 � 10�3 293

2nd 97.1 1.863 � 10�4 3122261.5 2.217 � 10�3 293

3rd 97.0 1.543 � 10�4 3152646.4 1.753 � 10�3 294

50 � 50 1st 990.4 1.459 � 10�4 2999591.9 9.859 � 10�4 292

2nd 979.5 1.633 � 10�4 2999217.5 1.041 � 10�3 292

3rd 979.3 1.301 � 10�4 3019248.4 8.093 � 10�4 292

analytical and numerical approaches agree relatively well, if thesystem size, N2, is big enough to neglect the minor losses. Theresults also come out that the flow conductance variation with theincrease in the mass flow rate has a similar trend within themonitored range.

In comparing the mass flow rate of optimized and non-optimized constructs, it results in that the design of the first opti-mized constructs is superior to that of the first non-optimizedconstructs for 10 � 10 elements, as shown in Fig. 4. For example,the flow resistance of the optimized design is about 126.5% smallerin the vicinity of DP y 28.8 Pa, as listed in Table 2. The largest flowrate occurs in the first construct configuration among the threeoptimized constructs, as shown in Fig. 4a. However, the flow accessperformance of the optimized second construct is better than thatof the first and third constructs when N is larger than 20, as shownin Fig. 4b and c. The maximum relative percentage differencebetween the mass flow rates of the optimized and non-optimizedstructures is also about 249.2% in the vicinity of DP y 2800 Pa for20 � 20 elements. This trend manifests more clearly as the systemsize increases: The optimized second construct design for 50 � 50elements provides a greater flow access (i.e., smaller by a factor of1/10 in the vicinity of DP y 9700 Pa based on the non-optimizedconstruct). These results are summarized in Table 2, which pres-ents the detailed performance of the architecture examined in thepresent study on square domainsN�N that increased in size all theway up to 50 � 50.

It is also important to realize that the third constructal designsfor the non-optimized channel configurations provide smaller flowresistance among all system sizes (see Fig. 4aec). Fig. 4 shows thatalthough with the additional pressure drop at bifurcations theoverall pressure drop of the third constructs are smaller than that ofthe first and second constructal structures: the third constructsdistribute the flow better than the first and second constructsacross all working conditions. These results are in agreement withpreviously reported studies [2,7], considering vascularized channelnetworks with circular cross-section.

4.3. Thermal resistance

Each graph in Fig. 5 shows the dimensional temperaturedistribution of the active area only on the top surface of the coolingplates. Fig. 5aed, eeh, and iel show the temperature distribution at

t, second, and third construct with two diameters and one diameter.

Non-optimized ����b� ab

���� (%)(K) DP (Pa) _m (kg/s) (b) Tmax (K)

.9 28.8 1.766 � 10�4 362.3 126.5

.9 906.6 1.576 � 10�3 306.8 125.6

.4 28.7 1.724 � 10�4 363.2 100.8

.1 892.7 1.576 � 10�3 303.7 96.3

.9 28.6 1.950 � 10�4 354.9 55.0

.3 873.5 1.869 � 10�3 302.7 50.5

.9 98.3 5.426 � 10�5 358.6 261.8

.5 2812.0 5.826 � 10�4 303.9 312.9

.3 98.0 5.506 � 10�5 357.1 238.3

.4 2756.8 6.347 � 10�4 300.8 249.2

.5 97.9 6.010 � 10�5 351.6 156.8

.4 2752.6 6.711 � 10�4 298.0 161.2

.3 993.3 1.701 � 10�5 362.0 757.6

.4 9835.1 8.811 � 10�5 308.8 1019.0

.8 992.2 1.789 � 10�5 358.3 813.0

.4 9791.2 9.710 � 10�5 305.6 972.5

.1 991.1 1.906 � 10�5 354.3 582.3

.9 9760.0 1.046 � 10�4 303.8 673.6

Fig. 6. Numerical results for global thermal resistance vs. pumping power: (a) 10 � 10elements; (b) 20 � 20 elements; (c) 50 � 50 elements.

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166 165

the top plane (active area only) of the cooling plate for the 2nd and3rd construct with 10 � 10, 20 � 20, and 50 � 50 elementalvolumes, respectively. The maximum temperature for optimizedchannels (D1h s D2h) (Fig. 5a, c, e, g, i, and k) is much lower than fornon-optimized channels (D1h ¼ D2h) (Fig. 5b, d, f, h, j, and l) underthe same legend. For example, the maximum temperature of theoptimized designs for the second construct with 10 � 10 elementsis smaller by about 6 K in the vicinity of 6Py 300 Pa, while themaximum temperature of the optimized designs for the secondconstruct with 50 � 50 elements is smaller by about 13 K in thevicinity of DP y 10 kPa. These results are depicted in Table 2. Thetemperature distribution also varies as the system size (N2)increases in both optimized and non-optimized designs. In otherwords, clear and intense hot spots near the upper central area arefound for the non-optimized configurations. This is especially trueif the flow architecture model consists of parallel channels becausethe mass flow rate in the parallel channel is lowest at the middle ofthe channels.

The thermal performance of cooling plates at constant heatflux is evaluated by introducing the thermal resistance, R, definedas:

R ¼ Tmax � Tminq

(27)

where q and (Tmax�Tmin) are the total heat imposed on the coolingplate and the overall temperature difference across the HLWvolume, respectively. The thermal resistance is numerically calcu-lated under the fixed pumping power, P. The fixed pumping power,P, is calculated by multiplying the measured flow rate and pressuredrop, defined as:

P ¼ _mDPrf

(28)

where _m is the total mass flow rate through the inlet. Theconstraint for the fixed pumping power means that the powerrequired to drive the coolant through the vascularized coolingnetworks is fixed. The present numerical work covers the pump-ing power range, 8.5 � 10�6e1.8 � 10�2, 5.3 � 10�6e4.6 � 10�2,and 2.5 � 10�6e5.7 � 10�2 W for 10 � 10, 20 � 20 and 50 � 50elements.

The effect of the pumping power P on thermal resistance R isshown in Fig. 6. On a graph with thermal resistance vs. fluid flowresistance or pumping power, there is one curve for each flowarchitecture subject to specified global constraints. In forcedconvection configurations, the thermal resistance decreases as thepumping power increases. The third constructs provide superiorheat transfer ability across all working conditions in non-optimized configurations, as illustrated in Fig. 6. This trendbecomes more evident as the system size (N2) increases. In addi-tion, those designs show better performance than the other non-optimized configurations, as the pumping power P increases.This is very interesting because the complexity exerts a significanteffect on the thermal performance in non-optimized configura-tions. In contrast, the worst performance is obtained for the non-optimized first construct among the all cooling plates studied.Meanwhile, R for the optimized constructal structures remainrelatively straight lines as the result of the low flow resistancecompared with the non-optimized structures overall workingconditions. The third constructs with optimized diameters providepoor heat transfer ability over the entire range because of the areauncovered by cooling channels. On the one hand, as the system size(N2) increases, the heat transfer effect related to the uncoveredarea decreases, as depicted in Fig. 6aec. On the other hand, thethermal resistances for the first and second optimized construct

are closely competitive and good with each other, across allworking conditions. For example, the first optimized constructalcooling plate gives the best performance when P is less thanapproximately 1.0 � 10�3 W for the 20 � 20 elements, whereas theperformance of the second is the best when P is greater thanaround 1.0 � 10�3 W.

K.-H. Cho, C.-W. Choi / Applied Thermal Engineering 33-34 (2012) 157e166166

5. Conclusions

An analytical and a numerical model have been developed toinvestigate the hydraulic and thermal performance of new vascularchannels, whose cross-sections are semi-circular. The validationswere also conducted by comparing the analytical results with thenumerical results using a three-dimensional CFD approach. Themajor findings are summarized as follows:

1. Analytical results showed that the dimensionless global flowresistances for the configurations with semi-circular cross-sections are significantly higher both the optimized and non-optimized constructs than those of the configurations withcircular cross-sections.

2. The flow conductances predicted by the analytical model werecompared with numerical data subject to a fixed volume anda fixed pumping power, and favorable agreements between thenumerical and analytical results were obtained as the systemsize increased.

3. The numerical results reveal that the optimized constructalcooling plates served lower thermal distribution, much betterthermal and flow resistance, and much lower pressure dropsthan the non-optimized cooling plates (conventional typeswith one diameter).

4. The thermal resistances for the first and second constructswere closely competitive and good with each other in opti-mized configurations, whereas the best architecture in thenon-optimized configurations was the third construct across allworking conditions.

5. In general, a further increase in the system size for all the casescaused the system performance to enhance: the larger vascu-latures were relatively more efficient than the smallervasculatures.

6. From a comparison of numerical and analytical approaches forlow flow resistance and thermal resistance, the numericalincluding the analytical methodology can be used practicallyfor vascular designs. Consequently, it is expected that theconceptual design of microchannel devices or cooling plates ofPEMFC using the constructal vascularized channel configura-tions will lead to an improved design of thermal systems fora superior performance.

Acknowledgement

The authors would like to gratefully thank Dr. A. Bejan for hisfruitful discussions and helpful suggestions.

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