International Journal of Heat and Fluid Flow 38 (2012) 126–132

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International Journal of Heat and Fluid Flow

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Experimental study of air natural convection on metallic foam-sintered plate

Zhiguo Qu a,⇑, Tiansong Wang a, Wenquan Tao a, Tianjian Lu b

a Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinab Key Laboratory of Strength and Vibration of MOE, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e i n f o

Article history:Received 23 October 2011Received in revised form 21 August 2012Accepted 23 August 2012Available online 27 September 2012

Keywords:Metallic foamNatural convectionInclinationHeat-transfer rateEmpirical correlation

0142-727X/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.ijheatfluidflow.2012.08.00

⇑ Corresponding author. Tel./fax: +86 29 82668036E-mail address: zgqu@mail.xjtu.edu.cn (Z. Qu).

a b s t r a c t

The natural convection on metallic foam-sintered plate at different inclination angles was experimentallystudied. Seven copper foam samples with different pore densities (10–40 pore per inch), porosities (0.90–0.95), and aspect ratios (the ratio of foam thickness to sample length, 0.1–0.5) were measured at inclina-tion angles of 0� (vertical orientation), 15�, 30�, 45�, 60�, 75�, 90� (horizontal orientation). The heatconduction and natural convection inside the foam both contributed to the total heat transfer. Although, theform and viscous drag, which are influenced by permeability and viscous friction in the thermal boundarylayer respectively, tend to suppress the natural convection, the heat transfer was finally enhanced by thefoam sintered surface due to large surface area extension. Optimum inclination range 60–75� correspond-ing to maximum average Nu number was found in the heat flux range of 600–1800 W/m2. The sinteredfoam surface with lower porosity and pore density was recommended for heat transfer enhancement.Particularly, the sample with porosity 0.9, pore density of 10 PPI, aspect ratio of 0.5 offered the highestaverage Nu number among the studied samples. An empirical correlation for modified Nusselt numberat isoflux boundary condition considering the foam morphology parameter and inclination angle wasproposed within deviation ±15% between the correlation and the experimental data.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction have been applied in cooling electronic components, jet engines,

Natural convection in saturated porous media has receivedincreasing attention over the last several decades due to its widerange of applications, such as for cooling electronic components.Natural convection in enclosure has been studied extensively.Nield and Bejan (2006) constructed a detailed overview of theexternal natural convection in an unbounded porous medium re-gion and the internal natural convection in a simple geometricenclosure saturated with porous media. Aldabbagh et al. (2008)experimentally investigated natural convection from an isothermalvertical plate embedded in a cubic basket with 0.8 cm widespheres saturated with water. They found that the Nusselt numberincreased linearly for modified Rayleigh number between 100 and500. Varol et al. (2008) and Oztop et al. (2009, 2011) numericallyinvestigated natural convection in various shapes of inclinedtwo-dimensional trapezoidal, triangular, and partially open enclo-sures filled with a fluid-saturated porous medium using the finite-difference method.

An effective way for natural convection enhancement is to ex-tend the surface area at a given volume due to the inherent disad-vantage of the low heat-transfer coefficient of natural convection.Degischer and Kriszt (2002) reported that metallic foams withattractive mechanical, electrical, acoustic, and thermal properties

ll rights reserved.5

.

adsorption chillers, sound absorber, and compact heat exchangers.Metallic foam has high porosity, high solid thermal conductivity,high surface area, and easy sintering to a tube or a wall, and is anideal heat-transfer surface for natural convection. The geometrystructure model is of great importance for the prediction the effec-tive thermal conductivity. Bhattacharya et al. (2002) representedthe structure by a model consisting of a two-dimensional hexagonalcells with a lump considered as a circular blob of metal at the inter-section. Due to the large thermal conductivity difference betweensolid matrix and saturated fluid for metallic foam, the local thermalnon-equilibrium should be considered. Hence, Zhao et al. (2005)conducted numerical studies for buoyancy-induced flows in high-porosity FeCrAlY foams in a cylindrical enclosure heated frombeneath based on the local thermal non-equilibrium. They also dis-cussed the effects of microstructural parameters (such as porosityand pore density) on heat and mass transfer in metallic foams. Kat-hare et al. (2008) and Davidson et al. (2009) experimentally studiednatural convection in a bottom-heated cylindrical enclosure filledwith water-saturated copper foam and reticulated vitreous carbonfoam of different pore densities (10 and 20 PPI) over a range ofDarcy and Rayleigh numbers, respectively. They suggested thatthe permeability has great influence on natural convection to leadthat lower pore density samples can provide superior performanceand the heat conduction and buoyant advection are the dominantmodes for heat transfer inside the foam. The using of metallic foamfor natural convection enhancement was further verified. Jamin and

Nomenclature

a thermal diffusivity (m/s2)ae effective thermal diffusivity (m/s2)Cp heat capacity (J/kg K)Dap Darcy number based on pore diameterdf fiber diameter (m)dp pore diameter (m)g gravitational acceleration (m/s2)hav average heat-transfer coefficient (W/m2 K)hy local heat-transfer coefficient (W/m2 K)ke effective thermal conductivity (W/m K)kf fluid thermal conductivity (W/m K)K permeability (m2)L length and width of metal foam sample/heater (m)Nuav average Nusselt numberNum modified Nusselt numberq heat flux (W/m2)Ra Rayleigh numberRam Darcy-modified Rayleigh numberTy wall temperature along the centerline in y direction

from the leading edge (K)Tw average temperature of the sintered wall (K)T1 ambient temperature (K)DT temperature difference (K)

Y position of the thermocouple (m)

Greek symbolsb coefficient of thermal expansion (K�1)d thickness of the metal foam sample (m)e porosityx pore density [(pore per inch (PPI)]/ power input to the film heater (W)h inclination angle with respect to the vertical (�)q density (kg/m3)mf kinematic viscosity of fluid (m/s2)

Subscriptsav averagec criticale effectivef fluidm modifiedp pores solidy distance from the leading edge in y direction1 ambient

Dataacqusition

DCpower

Foam sample

Film heaterKorean pine wood

Thermocouples

+-

g

Plexiglas room

Insulation

y

oz

Right-anglegeometry support

Supportbracket

Fig. 1. Schematic of the experimental setup (60� with the vertical orientation).

Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132 127

Mohamad (2008) experimentally studied natural convection heattransfer from a vertical cylinder pressed with carbon foam, locatedat the center of a rectangular enclosure by considering the effect ofradiation. They found that the Nusselt number was 2.5 times great-er than that of a bare copper pipe.

Aside from the natural convection in an enclosure, the naturalconvection on a wall sintered with metallic porous media has alsobeen studied. Bhattacharya and Mahajan (2006) presented experi-mental studies on buoyancy-induced convection in sintered alumi-num metal foams with different porosities (0.89–0.96) and poredensities (5–40 PPI) using air as the fluid medium. Their experi-ments were conducted both in vertical and horizontal orientations.The heat-transfer coefficients in these heat sinks were five to sixtimes higher than that on a smooth surface. Hetsroni et al.(2008) performed experimental measurements for natural convec-tion in aluminum metal foam strips with internal heat generationin the ambient using infrared camera to measure the thermal fieldof the strips. They found that the heat-transfer enhancement wasup to 18–20 times for the metal foam of 20 PPI with respect tothe flat plate.

The previous works mainly focused on natural convection in anenclosure or that on a vertical and horizontal metallic porous sin-tered surface. To the best knowledge of the present authors, theredo not appear any models or empirical correlation to predict thenatural convection on a plate sintered with foam layer at differentinclination angles in the published literatures. Such is just theobjective of present research. In the present study, the natural con-vection heat transfer of copper foam-sintered plates at variousinclination angles and morphology parameters was experimentallystudied. The effects of the inclination angle, foam morphologyparameter, Rayleigh number, and foam thickness were discussed.A correlation was finally proposed based on experimental data.

2. Experimental program

2.1. Apparatus and test procedures

The schematic diagram of the experimental setup is shown inFig. 1. The setup, placed in a Plexiglas room, consisted of test sam-

ples, power supply system, data acquisition system, and right-anglegeometry support. The test sample was copper foam, withthickness ranging from 10 to 50 mm, sintered on a 2 mm thick cop-per substrate. Three typical samples with three pore densities wereshown in Fig. 2. The length and width of the sample were both100 mm. Seven samples with two distinct porosities (0.90 and0.95) and three different pore densities (10, 20, and 40 PPI) usingair as the fluid medium were examined. The specifications of thefoam samples are given in Table 1. The samples were commercialproducts and were fixed in milled groove (100 mm � 100 mm �10 mm). The length (L) and width (W) of the samples were both100 mm. The porosity and pore size have been tested and verifiedby the authors before the experimental study. The milled groovewas at the center of a pine wood block (200 mm � 200 mm �50 mm). The gap between the sample and the groove bottomsurface was packed with aluminum silicate fiber-insulated mate-rial with thermal conductivity of 0.034 W/m K. The film heater(50 V/50 W) connected to an adjustable DC power supply with asurface area identical to that of the sample substrate was pastedto the sample substrate.

(a) 10 PPI (b) 20 PPI (c) 40PPI

50 mm

L=

100

mm

Fig. 2. Three typical samples of copper foam with different pore densities.

Table 1specifications of copper foam samples used in the present work.

Sample# x(PPI)

e d/L dp

(mm)ke (Bhattacharya et al.,2002) (W m�1 K�1)

K (Calmidi,1998) (m2)

1 10 0.95 0.1 2.54 6.99 9.65E�82 10 0.95 0.2 2.54 6.99 9.65E�83 10 0.95 0.4 2.54 6.99 9.65E�84 10 0.95 0.5 2.54 6.99 9.65E�85 20 0.95 0.5 1.27 6.99 2.41E�86 10 0.90 0.5 2.54 13.96 7.44E�87 40 0.90 0.5 0.64 13.96 4.65E�8

Thermocouples

100

2030

100

30 30 20y

xo

Centerline

1010

30

Fig. 3. Schematic of the location of the thermocouples (unit: mm).

128 Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132

The average substrate temperature was calculated by nine T-type thermocouples with an interval of 30 mm fixed in threegrooves (1 mm in width and 1 mm depth) made in the substrate.To investigate the local heat-transfer coefficient distribution, twoadditional thermocouples were fixed along the centerline in they-direction. The locations of the thermocouples are shown inFig. 3. The ambient temperature was monitored by another T-typethermocouple. Another T-type thermocouple was located on theback side of pine wood block to estimate the heat loss. All the tem-peratures were monitored and obtained using a Keithley dataacquisition system. To apply the different inclination angles, thetest samples, including the film heater and the Korean pine wood,were fixed on a right-angle geometry support with inclinations of30�, 45�, and 60� so that the heat-transfer performance in the hor-izontal and vertical orientations, as well as that in the three afore-mentioned inclination angles, could be tested. The heat flux wasadjusted from 300 to 3000 W/m2 to obtain different Rayleigh num-bers and substrate temperatures. All measurements were per-formed under a steady-state condition, which was achieved afterat least 3 h, depending on the power input and the sample used.The steady-state conditions were verified if the fluctuations ofthe average substrate temperature, as well as the temperature dif-ference between the substrate and ambient, were within ±0.1 Kwithin 30 min.

2.2. Data processing and uncertainty analysis

The porous morphology parameter could be introduced into theNusselt number for a natural convection in porous media. The two

definitions of the Nusselt number are the average (Nuav) andmodified (Num) Nusselt numbers. Correspondingly, the modifiedRayleigh number (Ram) was also defined in addition to theconventional Rayleigh number (Ra).

The average Nusselt number Nuav and Rayleigh number Ra aredefined by the following equations,

Nuav ¼hav

kfL ð1Þ

and

Ra ¼ gbqL4

af mf kf: ð2Þ

It was found the temperature between the back side of pine woodblock and ambient was almost identical, Hence the heat loss canbe considerably ignored. The average and the local heat-transfercoefficients along the centerline in the y-direction are defined byEqs. (3) and (4), respectively.

hav ¼u

L2ðTw � T1Þð3Þ

Table 2Transition equations used in the present work.

No. Variable name Expression Reference

1 Effective thermal conductivity ke ¼ 0:35½ekf þ ð1� eÞks� þ 0:65½e=kfþð1�eÞ=ks �

Bhattacharya et al. (2002)

2 Permeability K ¼ 0:00073ð1� eÞ�0:224ðdf =dpÞ�1:11d2P

Calmidi (1998)

3 Pore diameter dp ¼ 0:0254x

Lu et al. (2006)

4 Fiber diameter df ¼ 1:18dP

ffiffiffiffiffiffiffiffiffið1�eÞ

3p

q1

1�e�ðð1�eÞ=0:04Þ

� � Bhattacharya et al. (2002)

5 Effective thermal diffusivity ae ¼ keqeCpe

¼ ke½eqfþð1�eÞqs �½eCpfþð1�eÞCps �

Bhattacharya and Mahajan (2006)

Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132 129

hy ¼/

L2ðTy � T1Þ: ð4Þ

The average temperature between substrate temperature and ambi-ent temperature is ranging from 20.73 �C to 98.93 �C. In the modi-fied Nusselt and Rayleigh numbers, the fluid thermal conductivitykf and its thermal diffusivity af are replaced by the effective thermalconductivity ke and effective thermal diffusivity ae. Thus, the formu-lation of Num and Ram are expressed as

Num ¼hav

keL ð5Þ

and

Ram ¼gbqKL2

aemf ke; ð6Þ

where K, ke, and ae were obtained from the empirical correlationfrom available literature, and the equations are summarized in Ta-ble 2. The thermal properties in Eqs. (1)–(6) were evaluated withthe characteristic temperature, (Tw + T1)/2. In the present study,Nuav was applied for the parametric study on the heat-transfer per-formance, whereas Num was applied for empirical heat transfer cor-relation development in which the morphology parameter could beintroduced for application purposes.

The maximum uncertainty of hav from Eq. (3) was caused byuncertainties in the measurements of power input /, temperaturedifference DT, and heated area A. The maximum uncertainty ofNuav from Eq. (1) was caused by uncertainties in the averageheat-transfer coefficient hav and the edge length L of the sinteredplate, neglecting the uncertainty of kf. By neglecting the effect ofheat loss, the maximum uncertainty of hav and Nuav was deter-mined by the root sum square method (Moffat, 1988) in Eqs. (7)and (8), respectively,

dhav

hav¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid//

� �2

þ dAA

� �2

þ dDTDT

� �2s

ð7Þ

and

dNuav

Nuav¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidhav

hav

� �2

þ dLL

� �2s

: ð8Þ

The uncertainty of power input / provided by the film-heatproducer was ±4%. The uncertainty of the present T-type thermo-couples was 0.3 �C, and the maximum temperature differenceobserved in the present study was 20.73 �C. The uncertainties inthe estimation of length of the sintered plate and heated area were±0.5% and ±1% based on the sample manufacturing precision.Hence, the total uncertainties of hav and Nuav were 4.37% and4.4%, respectively.

3. Results and discussion

3.1. Heat-transfer performance

3.1.1. Overall heat transferThe effects of the inclination angle on the average Nusselt num-

ber Nuav along the heated base with different parameters, includ-ing heat flux, porosity, pore density, and aspect ratio, wereinvestigated. The heat conduction and buoyant advection insidethe foam both contributed to the total heat transfer. The heat con-duction is determined by the effective thermal conductivity. Thenatural convection inside the foam matrix was influenced by formdrag and viscous drag. The form drag is induced by winding flowpath and pore structure formed by the obstructing ligaments. Var-ious morphology parameters including porosity and pore densityassociated with different permeability and pore structures can leadto various form drag. The viscous drag was induced by the fluid vis-cous friction inside the thermal boundary layer. The variation inmorphology parameters and inclination angles can affect the abovetwo heat transfer modes to display different variation trend.

Fig. 4 demonstrates the effects of the inclination angle on theoverall heat-transfer rate on two porosities (0.9 and 0.95) andthree pore densities (10, 20, and 40 PPI) with fixed aspect ratio of0.5 at given heat flux of 1000 W/m2. There exists optimal inclina-tion angle range of 60–75� associated with maximum Nuav. Theaverage Nusselt numbers Nuav for all the samples gradually peakedto a value corresponding to the turning of the inclination angle andthen decreased when the angle was above the optimal inclinationangle. The reason was explained in the following. The form dragcaused by permeability was unchanged because the foam mor-phology parameter was fixed for each sample. However, whenthe inclination varied, the components of gravity and buoyancyalong and perpendicular to the foam surface were changed corre-spondingly to result in variation of viscous force inside the foam.It can then be speculated that the optimal inclination range wasassociated with lowest viscous force. Finally, the enhancement ra-tio for the sample with optimized morphology parameter (e = 0.9,x = 10 PPI) compared with that of the smooth plate was in therange of 2.25–2.4 at the corresponding heat flux.

The influences of porosity and pore density on Nuav at fixedinclination angle can be also displayed in Fig. 4. The averageNusselt number Nuav for the sample (e = 0.95) is lower than thatfor the sample (e = 0.9) at fixed pore density and aspect ratio (d/L = 0.5, x = 10 PPI). The increasing of porosity lead to an increasein permeability, resulting in reduced form drag, as plays a positiverole in heat-transfer enhancement. However, the effective thermalconductivity is reduced as indicated in Table 2, as tends to weakenthe heat transfer. The effect for porosity on heat transfer perfor-mance reveals that heat conduction inside the foam matrixdominated the heat transfer process. As the porosity is increased,the decrease in the effective thermal conductivity exceeds theincrease in permeability.

On the other hand , the average Nusselt number Nuav also de-creased with increasing of pore density the two fixed porosities

-10 0 10 20 30 40 50 60 70 80 90 100

495051

96

99

102

105

108

111

114

117

120

123

Nu av

θ (°)

ε =0.90,10PPIε =0.90,40PPISmooth Plateε =0.95,10PPIε =0.95,20PPI

Fig. 4. Effects of inclination angle on the overall heat transfer rate with differentgeometry-related properties (d/L = 0.5, q = 1000 W/m2).

5.0x107

1.4x108

2.3x108

3.2x108

4.1x108

5.0x108

40

50

60

70

80

90

100

110

120

130

Nu av

Ra

δ /L=0.1,ε =0.95,10PPIδ /L=0.2,ε =0.95,10PPIδ /L=0.4,ε =0.95,10PPIδ /L=0.5,ε =0.95,10PPIδ /L=0.5,ε =0.95,20PPIδ /L=0.5,ε =0.90,10PPIδ /L=0.5,ε =0.90,40PPI Smooth Plate

Fig. 6. Experimental heat transfer data in terms of Nusselt and Rayleigh numbers invertical orientation, including the smooth plate.

130 Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132

(e = 0.9, e = 0.95) . As the porosity is increased, the effective thermalconductivity was unchanged because the effective thermal con-ductivity was determined by porosity when the ks, kf are knownas indicated in Table 2, however, the permeability was increased,resulting in increased form drag, which can suppress the naturalconvection inside the foam. Hence, the heat transfer performancebecame inferior at higher pore density values.

Fig. 5 shows the average Nusselt number versus inclination an-gle at different heat fluxes for a specific sample (e = 0.95, d/L = 0.5,x = 10 PPI). The average Nusselt number Nuav increased withincreasing heat flux at fixed inclination angle. Similar trend inFig. 4 was found, an optimal inclination angle existed at each givenheat flux. The optimal inclination angle shifted from 60� to 75�with an increase in the heat flux from 600 to 1800 W/m2, andthe effect of the inclination angle on the average Nusselt numberNuav was more evident at a higher heat flux. Al-Bahi et al. (2006)have studied the natural convection in a tilted rectangular enclo-sure with a single discrete heater on the side wall with numericalmethods. They found that the optimized inclination angle range for60–75� existed at Ra = 106 since the flow streamline transition froma two cells pattern to a large cell pattern occurs. The detailednumerical study for natural convection on present foam sinteredplate can be further studied. However, the Ra number in Fig. 5for the foam surface is in the order of 108. Hence it can also bespeculated that the present optimized inclination angle was corre-sponding to the similar flow transition inside the foam matrix.

49505152

90

95

100

105

110

115

120

125

q =600W/m2

q =1000W/m2

q =1200W/m2

q =1800W/m2

Smooth Plate,q =1000W/m2

-10 0 10 20 30 40 50 60 70 80 90 100

θ (°)

Nu av

Fig. 5. Nusselt number versus inclination angle at different heat fluxes for a specificsample (e = 0.95, d/L = 0.5, x = 10 PPI).

Fig. 6 shows the relation between average Nusselt and Rayleighnumbers in vertical orientation for the studied seven samples inthe vertical orientation. The smooth plate was also included forcomparison. It was found that the foam surface can be operatedin higher heat flux range. Although the form drag and viscous dragtend to suppress the natural convection and weaken the heattransfer, the heat transfer is finally enhanced by any one of thestudied foam sintered surface due to significantly extended heattransfer surface area. The sample (e = 0.9, d/L = 0.5, x = 10 PPI) of-fered the most superior heat transfer performance among the sam-ples and its average Nusselt number was around 2.16 times higherthan that for the smooth plate in the range at the heat flux of1.85 � 108.

The effects of aspect ratio (d/L) or foam thickness on the averageNusselt number at various inclination angles at fixed parameter(e = 0.95, x = 10 PPI, q = 1000 W/m2) are shown in Fig. 7. As the as-pect ratio was increased, the heat conduction and form drag can beconsidered as unchanged because of fixed porosity and pore den-sity. The heat transfer surface area was enlarged with associatedwith increased viscous drag. The heat-transfer performance in-creased with the aspect ratio d/L, varied from 0 (flat copper plate)to 0.5 because of the obvious surface extension, which exceededthe negative effect of increased viscous drag. The increasing trendbecame milder at higher aspect ratio (d/L > 0.4) because increasedviscous drag to suppress natural convection became apparent. Theviscous drag is influenced by inclination angle and the viscous drag

-0.05 0.10 0.25 0.40 0.5535

50

65

80

95

110

125

Nu av

δ/L

Angle (°) 0° 15° 30 °

45° 60° 75° 90°

Fig. 7. Effects of aspect ratios on Nuav at different inclination angles (e = 0.95,x = 10 PPI, q = 1000 W/m2).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

+15%

-15%

Nu m

(Cor

rela

tion)

Num(Experimental)

Fig. 9. Prediction values of the dimensionless correlation versus experimental dataat isoflux condition.

Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132 131

effect is comparatively significant with larger d/L. Hence, the effectof inclination angle on Nuav becomes substantial at large d/L.

3.1.2. Local heat transferTo understand the local convective heat transfer coefficient dis-

tribution along the heated substrate plate centerline along the y-direction, the above local heat-transfer coefficient distributions inthe horizontal and vertical orientations as a function of the dimen-sionless height are shown in Fig. 8a and b, respectively, for twosamples with pore density of 10 and 40 PPI at fixed porosity (0.9)and aspect ratio (0.5). The local heat-transfer coefficient for thefoam with pore density (10 PPI) was higher than that of foam withpore density (40 PPI) at fixed porosity. This trend was similar to theaverage Nusselt number shown in Fig. 6. Under the horizontal ori-entation, the local heat convection coefficients hy exhibited sym-metrical distributions, and a minimum value was obtained at thecenter location. This is due to the fact that the convection in thehorizontal orientation is a Rayleigh–Bénard convection character-ized by a fluid that moves up through the center of the Bénard celland then spreads out into the edges of the cell, resulting in thesymmetrical hy distribution. However, the local convective heattransfer coefficient hy in the vertical orientation decreased gradu-ally and encountered an slight increase near the top edge. The min-imum values were reached at dimensionless location (y/L = 0.5).

30

31

32

33

34

35

36

37

38

39

40

41

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0y /L

h y (W

/m2 K

)

q = 1200W/m2,10PPI

q = 1500W/m2,10PPI

q = 1800W/m2,10PPI

q = 1200W/m2,40PPI

q = 1500W/m2,40PPI

q = 1800W/m2,40PPI

(a) Horizontal orientation

29

30

31

32

33

34

35

36

37

38

39

40

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

q = 1200W/m2,10PPI

q = 1500W/m2,10PPI

q = 1800W/m2,10PPI

q = 1200W/m2,40PPI

q = 1500W/m2,40PPI

q = 1800W/m2,40PPI

y /L

h y (W

/m2 K

)

(b) Vertical orientation

Fig. 8. Distribution of the local heat convection coefficient along the dimensionlessheight from the leading edge (e = 0.9, d/L = 0.5). (a) Horizontal orientation and (b)vertical orientation.

This trend was different from that for the smooth vertical plate.The local heat transfer coefficient for the vertical plate graduallydecreased and approached to a constant value with increasingthe distance from the leading edge because the boundary layergrew from laminar to turbulent flow and the corresponding localheat transfer coefficient was independent of the distance fromthe leading edge. The local heat transfer coefficient for the verticalfoam sintered plate was markedly different from that of the plainplate because the boundary layer of the foam sintered surfacewas deteriorated and recreated at the top part (0.8 < y/L < 0.9),leading to local heat transfer augmentation.

3.2. Dimensionless heat transfer correlation

Based on the present experimental data, the heat transfer corre-lation with the modified Nusselt number Num at the isoflux bound-ary condition is plotted and expressed as a function of fourparameters (Eq. (9)),

Num ¼ f ðRam; d=L;Dadp ; hÞ: ð9Þ

The empirical dimensionless correlation is proposed as (Eq. (10)),

Num ¼ CRaam

dL

� �b

Daep 1þ cos

ph180

� �� �n

; ð10Þ

where Dap is the Darcy number based on the pore diameter and isexpressed as

Dap ¼ K=d2p: ð11Þ

The coefficients C, a, b, e, and n in Eq. (10) are fitted by regressiontechnique with the present 98 sets of experimental data shown inFigs. 4–7 by using software of MATLAB. The final correlation expres-sion and its application scope are expressed in Eqs. (12) and (13),respectively,

Num ¼ 5844:5Ra0:0459m ðdLÞ0:1957Da2:2487

dp1þ cos

ph180

� �� ��0:0503

ð12Þ

and

300 W=m26 q 6 3000 W=m2;0:1 6 d=L 6 0:5;0

�6 h

6 90�: ð13Þ

The modified Nusselt numbers Num based on the experimentaldata in Eq. (5) and predicted by the proposed dimensionless corre-lation of Eq. (12), are compared in Fig. 9, and the deviation is with-in ±15%.

132 Z. Qu et al. / International Journal of Heat and Fluid Flow 38 (2012) 126–132

4. Conclusion

As the porosity is increased, the effective heat conductivity andform drag was both reduced. The heat transfer was weakened withincreasing of porosity because the heat conduction prevailed in theprocess. Increased pore density only resulted in additional formdrag to suppress natural convection inside foam. Hence, foam sur-face with lower porosity and pore density owns superior heattransfer performance. The variation in inclination angle resultedvarious viscous drag, and the existed optimal inclination range60–75� associated with lowest viscous drag. The increased aspectratio resulted in improved heat transfer performance because thepositive effect for heat transfer enhancement of the extension forheat transfer surface area exceeded the negative effect of inducedincreased viscous drag. Generally, the heat-transfer performancewas enhanced for each of the studied foam samples compared withsmooth plate. The sample (e = 0.9, d/L = 0.5, x = 10 PPI) offered themost superior heat transfer performance. The local heat-transfercoefficient distribution along the centerline of the horizontal foamsintered plate was symmetric. However, the local heat transfercoefficient gradually first decreased and then encountered a localimprovement at the top part of the foam along the centerline inthe vertical opposite direction of gravitational acceleration. Anempirical dimensionless correlation was proposed with a deviationerror of ±15% to provide a helpful guide for natural convectiveheat-transfer surface design.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China (No. 51176149), the National Key Projects of Fun-damental R/D of China (973 Project: 2011CB610306) and theFundamental Research Funds for the Central Universities.

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