ms

S.s Rd.

Keywords:ExperimentMetal foamPorous media

re lx, antsegim

employed the Darcy-extended Brinkman model for momentum, and the non-thermal-equilibrium

insideid ow

plex and random. Exact solutions of the complete transport equa-tions are virtually impossible [1,2]. Researchers have solvedsimplied forms of the governing equations, and relied on numer-ical simulations. Calmidi and Mahajan [3] numerically studiedforced convection of air ow in aluminum foam. The solid and uidtemperatures decayed gradually as the distance from the heatedwall increased. Hwang et al. [4] indicated that the local Nusselt

al equilibrium atperatures are allby uid che solid a

uid. Vafai and Kim [9] solved the local-thermal-equilgovernning equations for a porous medium between twoparallel plates.

Haji-Sheikh and Vafai [10] provided analysis of heat transfer inporous media imbedded inside ducts of different shapes, by solvingthe governing equations assuming local thermal equilibrium, andapplying a constant-wall-temperature boundary condition. Minko-wycz and Haji-Sheikh [11] solved the local-thermal-equilibriumequations for the case of parallel plates and circular porous pas-sages including the effect of axial conduction. Corresponding author. Tel.: +1 313 993 3285; fax: +1 313 993 1187.

International Journal of Heat and Mass Transfer 67 (2013) 877884

Contents lists availab

H

.eE-mail address: nihad.dukhan@udmercy.edu (N. Dukhan).uid due to the internal structure of these materials, which en-hances convection between the solid and the uid, and gives riseto an added mechanism of heat transfer called dispersion.

The internal architecture of open-pore foams in general is com-

two parallel plates. They assumed local thermthe heated base, i.e., the solid, uid and wall temequal. They identied three regimes dominatedtion, solid conduction or convection between t0017-9310/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.055onduc-nd theibriumheatedapplications such as heat exchangers and chemical reactors.Open-pore metal and graphite foams are excellent candidates forsuch designs. Among the important characteristics of these foams,from a heat transfer point of view, are the relatively high conduc-tivities of the solid phase and the large surface area per unit vol-ume. This is in addition to the vigorous mixing of the owing

cal results for a channel lled with a uid-saturated porous med-ium, also using local thermal equilibrium. Kim and Jang [7]investigated the validity of the local thermal equilibrium assump-tion itself.

Lee and Vafai [8] presented an analytical model for the solid andthe uid temperatures for Darcy ow in porous media betweenFluid temperatureBrinkmanConvection

1. Introduction

Convection heat transfer from theia (when the pores are open to utwo-energy-equation model for the temperatures of the solid and the uid phases inside the foam. Thesolution steps, which are not new, are summarized. A comparison shows good agreement between theexperimental and the analytical air temperatures, given the complexity of the foams morphology andthe rounding nature of the volume-averaging technique. However, the analysis seems to under-predictthe uid temperature over most of the cross section. The experimental technique can be used for valida-tion of other analytical solutions, numerical models and heat-exchange engineering designs based onmetal foam and similar porous media.

2013 Elsevier Ltd. All rights reserved.

surface of porous med-) has a wide range of

number increased with increasing Reynolds number for aluminumfoam. Angirasa [5] numerically studied convection heat transferdue to water ow in metal foam heat dissipaters. He invoked localthermal equilibrium. Poulikakos and Renken [6] provided numeri-Available online 21 September 2013tal pressure drop points that were obtained using the same experimental set-up. The experimental airtemperatures are compared to their analytical counterparts. The volume-averaged analytical formulationFluid temperature measurements insideBrinkmanDarcy ow convection analysi

Nihad Dukhan , Muntadher A. Al-Rammahi, AhmedDepartment of Mechanical Engineering, University of Detroit Mercy, 4001 W. McNichol

a r t i c l e i n f o

Article history:Received 22 April 2013Received in revised form 19 August 2013Accepted 21 August 2013

a b s t r a c t

Actual air temperatures wewall by a constant heat unique for such measuremespeeds were in the Darcy r

International Journal of

journal homepage: wwwetal foam and comparison to

Suleiman, Detroit, MI 48221, USA

ocally measured inside commercial aluminum foam cylinder heated at thend cooled by forced air ow. The specially-developed experimental tech-is described in detail, and is shown to produce reasonably good data. Aire. The permeability of the foam was directly determined from experimen-

le at ScienceDirect

eat and Mass Transfer

l sevier .com/locate / i jhmt

HeaLu et al. [12] analyzed the forced convection problem in a tubelled with a porous medium subjected to constant wall heat ux.The two-equation model, which relaxes the thermal equilibriumassumption, was solved. A closed-form solution for the solid andthe uid temperatures was presented. They used their results tostudy tubes lled with metal foams as heat exchangers. In a fol-low-up study, Zhao et al. [13] presented an analytical solution fora tube-in-tube heat exchanger, for which the inner tube and theannulus were lled with a porous medium (metal foam). Both ofthese studies employed the Brinkman-extended Darcy momentummodel. The researches in these two studies utilized the solutionsfor investigating the effect of various foam parameters in practical

Nomenclature

b1 constant, Eq. (16)Bi cylindrical porous media Biot number hrr2o=2ks

(dimensionless)c specic heat of uid (J kg1 K1)d1 constant, Eq. (16)d2 constant, Eq. (16)d3 constant, Eq. (16)Da Darcy number K=r2o (dimensionless)h convection heat transfer coefcient (Wm2 K1)k effective thermal conductivity (Wm1 K1)K permeability (m2)q00 heat ux (Wm2)T temperature (K)u volume-averaged pore velocity (m s1)U non-dimensional velocityr radial coordinate (m)R non-dimensional radial coordinatero radius of porous medium (m)z axial coordinate along ow direction (m)Z non-dimensional axial coordinate along ow direction

878 N. Dukhan et al. / International Journal ofheat-exchange designs employing metal foam.Analytical solutions in porous media continue to be sought after

due to their utility in practical engineering design, identifyingtrends of critical variables, parametric studies and for validatingnumerical models, see for example Xu et al. [14], Qu et al. [15]and Xu et al. [16]. Direct comparisons to experimental values ofkey variables, e.g., the uid and solid temperatures inside the foam,is lacking in [1416], and seem to be non-existent in the literatureconcerning heat transfer in metal foam. Experimental verications,when possible, add condence to analytical and numerical solu-tions. In a recent comprehensive review, Zhao [17] indicated thatthere is a lack of reliable experimental heat transfer data foropen-cell metal foam.

In the existing experimental studies concerning heat transfer inmetal foam, researchers typically measure substrate (wall) tem-perature, surface (skin) temperature and/or the temperatures atthe inlet and outlet of the foam. From such measurements, averagethermal transport parameters are determined, e.g., surface heattransfer coefcient and volumetric heat transfer coefcient. Often,these coefcients are used in obtaining Nusselt number whichserves as a basis for comparing analytical and numerical resultsto their experimental counterparts. Calmidi and Mahajan [3], forexample, measured the wall temperature of aluminum foam sam-ple bounded by substrates and heated. For comparing to theirnumerical results, they used the measured wall temperature to ob-tain the average surface heat transfer coefcient. Bhattacharyaet al. [18] used the same kind of temperature measurements toobtain the effective conductivity of metal foam, while Bhattach-arya and Mahajan [19] used similar measurements to assess theoverall thermal performance of a nned-metal-foam heat sink.Boomsma et al. [20] and Kurbas and Celik [21] conducted similarmeasurements and calculations to investigate the performance ofa metal-foam compact heat exchanger. Kim et al. [22] measuredthe wall, bulk inlet and outlet temperatures to obtain the space-averaged heat transfer coefcient and the Nusselt number for airow in metal foam. Similar measurements and calculations wereconducted by Zhao et al. [23]. Tzeng and Jeng [24] measured theskin, inlet and outlet temperature for air ow in metal foam chan-nel with 90-degree turned ow. The average Nusselt number was

Greeke porosity (dimensionless)c ratio of effective viscosity to actual uid viscosity = l/lek ratio of effective thermal conductivities = kf/ksh dimensionless temperatureq density of uid (kg m3)r surface area per unit volume of porous medium (m1)x dimensionless parameter =

c=Da

pw dimensionless parameter =

Bik 1=kp

Subscriptsf uidm mean values solidw wall1 free stream

t and Mass Transfer 67 (2013) 877884calculated based on these temperatures. Hetsroni et al. [25] mea-sured the wall temperature of a volumetrically-heated metal foamsample using an infrared camera, and used this temperature to cal-culate the average heat transfer coefcient. Hwang et al. [4] usedwall and exit temperature measurements for air ow through me-tal foam, followed by iteration to back calculate the interstitial heattransfer coefcient inside the foam. Noh et al. [26] measured thewall temperature for an annulus lled with aluminum foam in or-der to calculate the local surface heat transfer coefcient. Kim et al.[27] measured the inlet, outlet and wall temperatures for an alumi-num-foam n in a plate-n heat exchanger. They used these tem-peratures to assess the thermal performance of the n.

In the current study, direct measurements of the uid temper-atures inside commercial aluminum foam using a specially-devel-oped technique are presented. Such measurements have anintrinsic value. To the knowledge of the present authors, the mea-suring technique and the experimental data are novel, and havenot been performed or published previously. The experimentaltechnique for measuring the uid temperature can be used for val-idation of analytical and numerical models of heat transfer inmeso-scale porous media and for assessing the performance ofheat-exchange engineering designs based on such media.

The results of the current study are compared to their analyticalcounterparts. For completeness, the analytical results aredemonstrated by concisely presenting and explaining the stepsand solution of the volume-averaged thermal- non-equilibriumgoverning equations for forced convection heat transfer in a

cylindrical porous media due to Brinkman-extended Darcy ow. Itmust be noted here that the solution and its steps are not new, buthave been reported in [1215]

2. The volume-average formulation

Consider a cylindrical isotropic porous medium of radius robounded by an impermeable wall as shown schematically inFig. 1. The system is heated at the wall by a constant heat uxq00. There is fully-developed one-dimensional ow of a Newtonianuid in the z-direction with a volume-averaged pore velocity com-ponent u. Upon solving the Brinkman-Darcy momentum equation

N. Dukhan et al. / International Journal of Heainside and outside the porous boundary layer [9], the velocity canbe obtained [10,11,28] as:

U 1 IoxRIox 1

where R = r/ro Da K=r2o (the Darcy number), K is the permeability,c = l/le, l is the uid viscosity, le is the effective viscosity, U = u/u1, u1 is the velocity outside the boundary layer, x

c=Da

pand

Io is the modied Bessel function of order zero.For thermally fully-developed conditions, the volume-averaged

two-energy-equation model is

ksr

@

@rr@Ts@r

hrTs Tf 0 2

kfr

@

@rr@Tf@r

hrTs Tf qcu @Tf

@z3

where ks and kf are the effective thermal conductivities, Ts and Tf arethe volume-averaged temperatures of the solid and the uid insidethe open-pore porous medium, respectively, h is the interstitial heattransfer coefcient between the solid and the uid, r is the surfacearea per unit volume of the porous medium and q and c are the den-sity and the specic heat of the uid, respectively. The Pclet num-ber is assumed to be high such that the longitudinal conduction forboth the solid and the uid are negligible. For inclusion of theseterms, see Minkowycz and Haji-Sheikh [11]. Equations (2) and (3)are well known [3,8,2934]. The following boundary conditionsapply:

at r 0 @Ts@r

@Tf@r

0 4

at y roks @Ts@r

kf @Tf@r

q00 and Ts Tf Tw 5

where Tw is the wall temperature (a function of the ow direction),which is not known a priori [8]. The equality of the temperatures atthe heated wall as described in Eq. (5) has been used in the litera-ture [1214]. Eqs (2)(5) are made non-dimensional:

u

q

z

r

Ts

Tf

roq

Fig. 1. Schematic of the porous-media heated cylinder.1R

@

@RR@hs@R

Bihs hf 0 6

kR

@

@RR@hf@R

Bihs hf 2U 7

at R 0 @hs@R

@hf@R

0 8

at R 1 @hs@R

k @hf@R

0 and hs hf 0 9

where R = r/ro, Z = z/ro, hs Ts Tw=q00ro=ks; hf Tf Tw=q00ro=ks,k kf =ks and Bi hrr2o=ks is the Biot number. The derivative ohf/oZ,which is constant for thermally fully-developed conditions, is ob-tained by integrating the sum of Eqs. (2) and (3) over the cross-sec-tional area and applying the boundary conditions. This integrationresults in @hf =@Z 2q00=qcumro

3. Solution

The two energy Eqs. (6) and (7) are added, and U is substitutedfor from Eq. (1) to yield:

1R

@

@RR

@

@Rhs khf 2 1 IoxR

Iox

10

which is a non-homogenous ordinary differential equation (ODE)with the dependent variable hs khf . The solution is

hs khf 12 R2 1

2x2

IoxRIox 1

11

Eqs. (6) and (7) are de-coupled by (a) solving Eq. (11) for hs in termsof hf and (b) substituting for hs, ohf/oZ and U in Eq. (7). The followingequation is obtained after rearranging:

d2hfdR2

1RdhfdR

Bik 1k

hf 1k Bi2x2

12

2 Bi

2R2

2 x2 Bix2

IoxRIox

12

which is a single non-homogenous ODE for the uid temperature.The general solution of Eq. (12) is obtained by rearranging the

homogenous part to take the general form of Bessel equation[35,36]. The particular solution to Eq. (12) is sought by assumingan appropriate form (according to the forcing function, i.e., theright-hand side of Eq. (12)). The complete solution of Eq. (12) is

hf b1I0wR d1I0xR d2R2 d3 13where w Bik 1=kp and the constants are given byb1 d1 Iowd2d3Iow , d1

2Bix2x2 Ioxkx2Bik1, d2 12k1 and d3 2kBik12

2Bi1=22=x2Bik1 . In obtaining the homogenous solution, a complexroot i

Bik 1=k

pis encountered. When the root is found to be

imaginary, as the case here, Bessel function Jo is replaced by themodied Bessel function Io in the solution [35,36], as was done in[14,15] in which a cylindrical system related to current problemwas solved, and was not done in [12,13] when solving an identicalproblem to the one in hand. More details regarding this issue areavailable in [37].

The solid temperature can readily be obtained by substitutingfor hf from Eq. (13) into Eq. (11), which gives

hs 12 kd2

R2 2x2Iox kd1

IoxR kb1I0wR

t and Mass Transfer 67 (2013) 877884 879 12 2x2

kd3 14

4. Experiment

An experimental heat transfer model was designed, fabricatedand tested, in order to allow for direct measurement of the uidtemperature, and subsequently for comparing the measured valuesto the predictions of the volume-averaged analytical solution. Themodel was essentially a cylindrical aluminum tube lled andbrazed to an aluminum foam core. Brazing of the two similar mate-rials minimized the thermal contact resistance. The tube had alength of 15.24 cm (6.0 in) in the ow direction, an inside diameterof 25.56 cm (10.1 in) and a tube thickness of 6.4 mm (0.25 in). Thefoam was obtained commercially (ERG Materials and Aerospace[38]), and was made from aluminum alloy 6101-T6. It was markedby an approximate industrial designation as 20-ppi (pores per lin-ear inch). The porosity of the foam was 91% (calculated from mea-surements of its volume and weight), while other geometricparameters were based on manufacturers data.

To isolate and measure the uid temperature inside the foamusing common thermocouples, each thermocouple was shieldedin a specially designed small perforated aluminum tube having

wall of the aluminum cylinder to house thermocouple beads formeasuring the wall temperature at various locations in the owdirection (not shown). On the outer surface of the aluminum cylin-der, thermofoil heaters (made by Minco Products) were attached

880 N. Dukhan et al. / International Journal of Heat and Mass Transfer 67 (2013) 877884an inner diameter of 3.15 mm and an outer diameter of 4.55 mm.The tube was perforated with circular holes 3.8 mm in diameter,Fig. 2(a). A thermocouple was inserted in each perforated tubeand xed in place using high-temperature epoxy, blocking as fewof the perforated tubes holes as possible. The bead of each thermo-couple was positioned such that it did not extend out of the smalltube, and remained shielded and protected by the wall of the smalltube, as seen in Fig. 2(a). As such, when the tube and its thermo-couple are inserted in the foam, the bead would not touch the solidligaments of the foam or the wall of the small tube.

Two sets of ten-holes were drilled through the wall of the alu-minum cylinder and the foam core reaching pre-determined radialdistances inside the foam. The rst set was at a distance of 3.81 cm,while the other set was at 6.35 cm form the foam entrance. The ra-dial depth of the holes, measured from the outer surface of thetube, were 2.03, 3.30, 4.57, 5.84, 7.11, 8.38 9.56, 10.92, 12.19 and13.46 cm (0.8, 1.3, 1.8, 2.3, 2.8, 3.3, 3.8, 4.3, 4.8 and 5.3 in). Theholes were arranged around the cross section with an angle of36o between each two adjacent holes (Fig. 3), and were organizedin order to minimize alteration of the internal structure of the foamand the interference with air ow through the foam. The diametersof these holes were slightly larger than the diameters of the smallperforated tubes to allow tight t. The small perforated tubes withFig. 2. Experimental aluminum foam heat transfer model construction: (a) Thermocoupleassemblies inserted into holes with the leads directed outward in their slot, (c) A slot lleheaters attached to outer surface of the heat transfer model.their thermocouples were inserted in these holes, as seen in Fig. 2(b). This allowed for air temperature measurements at differentknown radial distances inside the foam. The perforation madethe small tubes more compatible with the internal structure ofthe foam and allowed air to ow through while minimizing block-age to the ow.

Slots were machined on the outer surface of the aluminum cyl-inder to house the thermocouple wires, Fig. 2(b). After the thermo-couples in their perforated small tubes were inserted in the foam,the lead wires of the thermocouples were placed in the especially-machined slots and directed away. Thermal epoxy then lled theremaining volume of the slots, to end up with a smooth surfacematching the original outer surface of the aluminum cylinder,Fig. 2(c). This insured that there were no air pockets trapped inthe wall thickness. Other small shallow holes were drilled in the

Fig. 3. Location and arrangement of thermocouple holes around the cross section;numbers represent the depth of each hole in cm.in its small peroferated aluminum tube (thermocouple assembly), (b) Themocoupled with thermal epoxy; Two holes for pressure drop measurement and (d) Thermofoil

using thermal adhesive. The heaters covered the outside area,Fig. 2(d). Each heater had a resistance of 22.3X, and could provideup to 645W of power. The whole assembly was then covered witha 2.54-cm-thick insulation sheet.

Experiments were performed in an open-loop wind tunnel asshown in Fig. 4. A suction unit (SuperFlo 600 Bench), which couldproduce air ow rates up to 17 m3/min (600 ft3/min) was used toinduct room air into the tunnel and through the foam sample.The suction unit was powered by eight fans; they were electricallymodied so that two can be on at a time, which was suitable forproducing low ow rates through the tunnel. A variable ow con-troller provided further adjustment of the airow through the tun-nel and the test section.

Two holes were drilled at the top of the test section at a distanceof 5.08 cm (2 in), Fig. 2(c). The diameters of these holes were7.9 mm (0.31 in) and they housed pressure measurement tubing.

N. Dukhan et al. / International Journal of HeaThe pressure drop was measured using an Omega differential pres-sure transmitter with a range 0746 Pa (0 to 3 in H2O).

As mentioned above, the test section of the tunnel was formedby brazing the foam sample to the inside surface of an aluminumcylinder. A reducing nozzle connected the exit of the test sectionto a ow-measurement section. The tunnel section that housedthe speed measurement device was circular with a diameter of63.5 mm (2.5 in). As such, the measured air speed at the owmeterhad to be adjusted, using conservation of mass and the ratio of thediameters of the test section and the speed measurement section,in order to obtain the mean velocity in the test section (andthrough the foam). The low velocities in the foam of the currentexperiment were achieved due to the relatively large cross-sectional area of the experimental model. For velocity measure-ment, an Extech gas ow meter that could measure speeds up to35 m/s (29 ft/min) was used.

The test section was secured in place and sealed. The pressuretransducer and the ow meter were connected to a data acquisi-tion system (DAQ by Omega Engineering) which delivered thereadings to a computer. The ow rate was then varied to realizedifferent velocities in the test section. For each velocity, the stea-dy-state static pressure drop was measured using the pressure tapsand the differential pressure transducer. The measurements wererepeated three times and the average of the three runs was re-ported. Form these data, the permeability was determined, as willbe shown later.

The heat transfer experiment proceeded as follows. The freeleads of all thermocouples were connected to the data acquisitionsystem, where the temperatures could be displayed and recordedusing a computer. The volumetric ow rate of room air was ad-justed using controls on the suction unit, such that the desiredFig. 4. Photograph of the experimental set-up.speed was realized inside the foam. The surface heaters were pow-ered, and the power input was adjusted using a variac, so that thedesired power, and hence the heat ux to the foam, was achieved.The air temperatures inside the foam, as well as the wall temper-ature, were monitored until there were no noticeable variationsin their readings, indicating steady-state conditions, which tookabout 40 min. The steady-state air and wall temperatures wererecorded, as was the ambient air temperature.

5. Uncertainty analysis

The uncertainty in the velocity measurement had a contributionfrom a xed error, ef = 2% (provided by the manufacturer) and arandom estimated error, er = 10% in each reading. For the pressuretransducer ef was 5% and er was 7% (these were provided in a cal-ibration certicate). The total uncertainties in the pressure andvelocity were calculated by the root-sum-squares method accord-ing to Figliola and Beasley [39]:

d e2f e2r

q15

This resulted in a total uncertainty in the pressure drop dDp of 8.6%,and in the air velocity dum of 10.2%. The length and diameter ofeach sample were measured using a caliber. The uncertainties inthese readings were dL = dD = 1.0 mm, or 1.9%.

Further uncertainty analysis was performed in order to obtainthe uncertainty in the computed values of the permeability K fromcurve-tting of the experimental pressure drop and velocity data.According to Darcys law, the pressure drop per unit length ofthe porous medium is linear with the velocity, i.e.:

DpL AV 16

where A is a curve-t constant, and K = l/A. The percent uncertaintyin K is a result of uncertainties in its constituents l and A, and isgiven by [39]

dKK

dll

2 dA

A

2s17

where dl and dA are the uncertainties in l and A, respectively. Theuncertainty in l was estimated as 1 107 N s/m2, taken as theaccuracy of the reported values in property tables [40]. This valueis small enough to cause negligible impact on the overall uncer-tainty in K, therefore it was ignored. Therefore, the uncertainty inK is simply given by:

dKK dA

A 100% 18

The average uncertainty in A is the same as the uncertainty in DpLV ;according to Eq. (16). So it is given by

dAA

dDPDp

2 dL

L

2 dV

V

2s19

Using a reference velocity of 0.2 m/s and a reference pressure dropof 1.6 Pa, the uncertainty in A, which is the same as the uncertaintyin K, was obtained as 13.4%.

The uncertainty in the temperature measurement was based onthe accuracy of the thermocouple 0.2 C, provided by manufac-turer. The effective thermal conductivity of the solid aluminumligaments of the foam was obtained from an analytical one-dimen-sional model given by Calmidi and Mahajan [41]. These researchersindicated that this model was excellent in matching their

t and Mass Transfer 67 (2013) 877884 881measured values of the effective conductivity. Actually, similarfoam (20 pores per inch, 90.6% porous foam, made by the samemanufacturer) was tested in [41], and an effective thermal

Table 1Parameters for the metal-foam lled model.

Parameter Value

ro 0.128 mum 0.2 m/sq00 8494.5 W/m2

r 1313.8 m2/m3

K 1.16 1007 m2ks 6.62 W/m Kh 207.8 W/m2 KBi 682.9x 340.8w 303.1k 0.0044

882 N. Dukhan et al. / International Journal of Heaconductivity was obtained as 6.9 W/m K which is only 4.0% differ-ent from the value obtained for the foam used in the current study,as reported in Table 1, which is encouraging. The uncertainty in theeffective conductivity was conservatively assumed to be 10%. Theuncertainty in the heat ux was assumed to be 10%, while theuncertainty in the radius was 1.9%. These values along with the dif-ference between the uid temperature and the wall temperature atthe center of the channel (79.7 C), which is the maximum differ-ence that would result in the highest uncertainty, were substitutedin the following equation

dhfhf

dTfTw Tf

2 dTw

Tw Tf

2 dks

ks

2 dq00

q00

2 dro

ro

2s

20Eq. (20) yielded an uncertainty in the non-dimensional temperatureof the uid equal to 14.3%.

6. Results

6.1. Pressure drop and permeability

The Forchheimer-extended Darcy equation for the pressuredrop, after dividing by the Darcy velocity is

DpLV

lK qCV 21

where, as before, K is the permeability and C is a form drag coef-cient. The measured pressure drop is plotted against the Darcyvelocity, according to Eq. (21), in Fig. 5. The Darcy regime is identi-ed by the absence of the last term (the Forchheimer form dragterm) in Eq. (21). In the purely Darcy regime, Eq. (21) is a horizontalline with a y-intercept equal to l/K. When the form drag is impor-

tant, Eq. (21) applies and we get a line with a slope equal to qC. Thischange from Darcy to Forchheimer regime is clearly shown in Fig. 5,

Fig. 5. Experimental pressure drop versus Darcy velocity.and is occurring at a Darcy velocity of about 0.3 m/s. For velocitiesbelow 0.3 m/s, the ow regime is purely Darcian. So, for heat trans-fer measurements, a velocity of 0.2 m/s was chosen to insure thatthe ow was well within the purely Darcy regime.

The Darcy-regime permeability was determined from theDarcy-regime pressure-drop data points only, according to Eq.(21), as was done by Dukhan and Ali [42]. The air viscosity, l,was taken as 1.8 105 kg/m s.

6.2. Heat transfer results and comparison

For the sample of the current study, the parameters of Table 1were obtained by direct measurement or by computation, as fol-lows. The surface area density of the foam was calculated form acorrelation given by the manufacturer, as shown in [43]. The effec-tive thermal conductivity of the solid, ks, was obtained from amod-el given in Calmidi and Mahajan [41]. The interstitial heat transfercoefcient inside the foam, h, was computed based on a correlationgiven by Kuwahara et al. [44].

Fig. 6 shows the measured wall temperature along the owdirection for ve different locations: 1.27, 3.81, 6.35, 10.16 and12.7 cm. This temperature increases in the ow direction as ex-pected. There is a subtle but identiable change of slope occurringat 6.35 cm from the entrance- a sign of change from thermallydeveloping to fully-developed region. This trend in the wall tem-perature of heated metal foam is very similar to what has been pre-sented in Calmidi and Mahajan [3], with the change in slopeoccurring at an axial distance of about 8.3 cm from the entrance.This difference in the thermal entry length is most likely due tothe difference in air speeds: 0.2 m/s in the current study and0.61 m/s in Calmidi and Mahajan [3] and the number of poresper inch: 20 in the current study and 5 in [3]. In the current inves-tigation, the experimental case for the axial location of 6.35 cmwas taken to be in the thermally fully-developed region, whilethe case at 3.81 cm was taken as nearly thermally fully-developed,as will be shown next.

Fig. 7 shows the non-dimensional uid temperature as a func-tion of the radial distance R. The solution given by Eq. (13) is plot-ted using a solid line, while the experimental data points are givenusing the black solid circles. Generally speaking, the two tempera-ture curves decrease gradually as the distance from the heated wall(R = 1) increases, as expected.

The two temperature curves are relatively close to each other,with the analytical solution predicting a slightly lower tempera-ture, in general. In addition to experimental errors, other errorsmay have been introduced into the analytical solution. The solu-tion required the computation of the heat transfer coefcient in-side the foam. This was obtained from a correlation based on ageometrical modeling of the foam structure, Kuwahara et al. [44].Similarly, the effective thermal conductivities ks and kf were ob-tained from a correlation also based on geometrical modeling ofthe foam [41]. One possible error may be the existence of a temper-ature slip condition at the wall for the uid phase. Sahraoui andKaviany [45] indicated that there are physical non-uniformitiesnear the solid wall that bounds a porous medium, which altersthe value of the effective conductivities of the solid and uidphases Moreover, the nature of the wall boundary condition whena constant heat ux is applied is still a topic of debate, i.e., how thetotal heat ux is split between the solid and the uid phases, seefor example Vafai and Yang [46].

The two points closest to the center of the foam cylinder areseen to produce lower non-dimensional uid temperature, or high-er physical temperatures, noting that hf Tf Tw=q00ro=ks. This is

t and Mass Transfer 67 (2013) 877884true for both of axial locations z = 3.81 and 6.35 cm. Inspection ofFig. 3, which shows the arrangement of the thermocouples aroundthe cross section, indicates that these two points are marked by the

n of

R

Hea

fFig. 6. Wall temperature as a functio

N. Dukhan et al. / International Journal ofnumbers 13.46 and 12.19 cm. The gure clearly shows that theblockage caused by inserting the two small perforated tubes withtheir thermocouples is signicant enough to cause the temperatureat these two locations to be higher than what the temperaturewould have been if the structure of the foam was not disturbed.

7. Conclusion

Direct measurement of the uid temperature inside a heatedcylinder lled with metal foam was conducted using a specially-designed technique. Scrutiny of published literature on heat trans-fer in metal foam reveals that such measurement has been lacking.The data points were obtained in the Darcy regime and in the ther-mally fully-developed region. In order to compare to analyticalpredictions, the thermal-non-equilibrium two-equation model forconvection heat transfer in a heated cylinder lled with a porousmedium and heated at the wall was revisited. The solution stepswere summarized and the analytical uid temperature was pre-sented. The solution employed the Brinkman-extended Darcy owmomentum equation. Relatively good agreement between the ana-lytical and experimental uid temperature was displayed. How-ever, the analytical solution of the volume-averaged energyequations seemed to under-predict the uid temperature overmost of the cross section. Nonetheless, the measurements give anopportunity to look into the issue of how volume-averaged valuescompare to physically-measured local values of common quanti-ties such as the temperature inside porous media. The experimen-tal technique can be of utility for heat transfer designs, and for

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[11] W.J. Minkowycz, A. Haji-Sheikh, Heat transfer in parallel plates and circularvalidating complex analytical and numerical modeling of the heattransfer phenomenon in open-cell meso-porous media.

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884 N. Dukhan et al. / International Journal of Heat and Mass Transfer 67 (2013) 877884

Fluid temperature measurements inside metal foam and comparison to BrinkmanDarcy flow convection analysis1 Introduction2 The volume-average formulation3 Solution4 Experiment5 Uncertainty analysis6 Results6.1 Pressure drop and permeability6.2 Heat transfer results and comparison

7 ConclusionReferences