ANALYSIS AND CHARACTERIZATION OF METALFOAM-FILLED DOUBLE-PIPE HEAT EXCHANGERS

Xi Chen, Fatemeh Tavakkoli, and Kambiz VafaiDepartment of Mechanical Engineering, University of California,Riverside, California, USA

The effect of using metal foams in double-pipe heat exchangers is investigated in this work.The advantages and drawbacks of using metal foams in these types of heat exchanger arecharacterized and quantified. The analysis starts with an investigation of forced convectionin metal foam-filled heat exchangers using the Brinkman-Forchheimer-extended Darcymodel and the Local Thermal Equilibrium (LTE) energy model. An excellent agreementis displayed between the present results and established analytical results. The presentedwork enables one to establish the optimum conditions for the use of metal foam-filleddouble-pipe heat exchangers.

1. INTRODUCTION

Metal foam is a porous medium that has a solid metal matrix with emptyor fluid-filled pores. Because metal foams have both functional and structuralproperties, they are utilized in a wide range of sectors and industries, includingtransportation, defense, aerospace, architectural designs, and energy industries.Metal foams can be classified as closed- or open-cell type. Open-cell metal foamshave the following characteristics: (1) large surface area, which improves energyabsorption and heat transfer [1]; (2) extended foam ligaments normal to the flowdirection, which results in fluid mixing, boundary layer disruption, and an increasein fluid turbulence [2]; and (3) lightweight, with high strength and rigidity. Therefore,metal foams have excellent potential to enhance heat transfer. For example, Lin et al.[3] demonstrated that the solid foam radiator is significantly more efficient thanthe fin radiator in enhancing heat transfer performance. Boomsma et al. [4] showedthat the compressed aluminum foam heat exchanger has nearly half the thermalresistance and significantly higher heat transfer efficiency as compared with theregular heat exchanger. Mahjoob and Vafai [5] indicated that although there is anincrease in pressure drop, the use of metal foam results in substantial heat transferenhancement that can compensate increased pressure loss.

There has been increased interest in establishing metal foam properties, suchas effective thermal conductivity, permeability, and inertial coefficient. Calmidi

Received 8 August 2014; accepted 25 January 2015.Address correspondence to Kambiz Vafai, Department of Mechanical Engineering, University of

California, Riverside, California 92521, USA. E-mail: Vafai@engr.ucr.edu

Numerical Heat Transfer, Part A, 68: 1031–1049, 2015Copyright # Taylor & Francis Group, LLCISSN: 1040-7782 print=1521-0634 onlineDOI: 10.1080/10407782.2015.1031607

1031

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

and Mahajan [6] developed a two-dimensional model to obtain the effective ther-mal conductivity of metal foams. A one-dimensional analytical heat conductionmodel was developed based on the three-dimensional geometry of metal foam cellsby Boomsma and Poulikakos [7]; this model demonstrated good agreement withthe experimental data. Bhattacharya et al. [8] researched the same topic based onthe analysis of Calmidi and Mahajan [6] and found that effective thermal conduc-tivity is more dependent on porosity than pore density. Their model was moreapplicable to high-porosity metal foams. Permeability and inertial coefficient,which are two key parameters of open-cell metal foams, have been studied byvarious researchers. These two parameters are highly structure dependent, andmuch of existing research on packed beds and granular porous media (with aporosity of 0.3–0.6) has been utilized to investigate their dependence on pore sizeand porosity. However, packed bed attributes may not be directly applicable tometal foams. As such, Calmidi [9] proposed a model based on his study of Porvairfoams to derive a specific formulation that relates the permeability and inertialcoefficient in terms of the porosity and pore size of metal foams. Zhao et al. [10]also investigated the determination of permeability and inertial coefficient forhighly porous metal foams.

As mentioned above, because metal foams with open cells can be regardedas a porous medium, fluid flow through these foams can be described by Darcy’slaw. However, Darcy’s law is limited to laminar flow and only when theReynolds number based on pore size is in the range 1–10. An increase in velocityresults in deviation from Darcy’s law [11]. To solve this problem, Vafai and Tien[12] proposed a generalized model to incorporate boundary and inertial effects.For modeling heat transfer in porous media, the energy equation model pro-posed by Vafai and Tien [12] is utilized. It should be noted that some additionalassumptions are required when this energy equation is used: (1) small tempera-ture differences between the fluid and solid phases on a local basis (i.e., bothphases are in local thermal equilibrium (LTE)); and (2) natural convection

NOMENCLATURE

A surface area, m2

cf specific heat, J=(kg K)D pipe diameter, mDa Darcy numberdp pore size, mdf fiber diameter of metal foam, mF inertial coefficienth convective heat transfer coefficient,

W=(m2 K1)K permeability, m2

k thermal conductivity, W=(m K)m mass flow, kg=sNu Nusselt numberP pressure, PaQ power, Wq heat flux, W=m2

R pipe radius, mT temperature, KU overall heat transfer coefficient,

W=(m2 K)u velocity, m=se porosity of the porous mediumm viscosity, kg=(m s)q density, kg=m3

Subscriptse effective valuef fluid phases solid phasem mean valuew wall

1032 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

and radiation in the porous medium can be ignored. The LTE model has beenutilized by various investigators in the analysis of forced convection through aporous medium [13–15].

Heat transfer performance of open-cell metal foams has been investigated byseveral researchers. These works have mainly focused on the fundamental rec-tangular channel geometry, but forced convection in metal foam-filled pipes hasalso been investigated [16]. However, few studies have investigated double-pipeheat exchangers filled with metal foams, which are the most widely used metal foamheat exchangers in various industrial applications. This could be due to the com-plexity of the coupling process between the fluids, which results in a temperatureand heat flux distribution which is not constant at the interface of the two pipes.To simplify the model and reduce computational time, prior works have considereda constant heat flux or a constant temperature at the interface. However, this doesnot reflect the actual conditions pertaining to the operation of double-pipe heatexchangers. Therefore, in this work, the Brinkman-Forchheimer-extended Darcyand LTE equations are utilized to simulate metal foam-filled heat exchangers whileincorporating the coupling effect between the inner and outer pipes and theintervening fluids. The influence of various parameters, such as different Darcynumbers, on the fluid and thermal performance of metal foam-filled pipes isaddressed. The heat transfer performance of practical double-pipe heat exchangersfilled with metal foams is also analyzed and compared to that of plain tube heatexchangers in this study, to quantify several aspects of optimum operating con-ditions for these types of device.

2. PHYSICAL PROBLEM

The problem under consideration is based on forced convective flow through apipe filled with metal foams, as shown in Figure 1a. The liquid flows through themetal foam-filled pipe (diameter 2 R, length L). The pipe wall is impermeable andsubject to a variable heat flux qw(z), which is determined based on the couplingconditions.

The inner pipe is inside an outer pipe constituting a simple double-pipe heatexchanger, as shown in Figure 1b. The inner and outer pipes, of diameter R1 andR2, respectively, are both filled with metal foam. The cold and hot liquids flow inopposite directions in a counterflow arrangement through the inner and theannular sections. The heat is transferred from the hot water in the outer annularpipe to the cold water in the inner pipe (or the other way around) through theinner pipe’s wall subject to the coupling conditions. The outer pipe is assumedto be insulated.

3. MATHEMATICAL FORMULATION AND BOUNDARY CONDITIONS

3.1. Mathematical Formulation

As mentioned earlier, the Brinkman-Forchheimer-extended Darcy flow modeland the LTE-based energy equation are utilized in our analysis. It is assumed thatthe properties of the metal foam and the fluid are homogeneous and isotropic.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1033

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

The radiation and natural convection are considered to be negligible. The governingequations can be written as [12, 15, 17]:

mf

eq2u

qrþ 1

r

qu

qr

!

"mf

Ku" qf

Fe

K1=2u2 " dp

dz¼ 0 ð1Þ

qf cf udT

dz¼ ke

q2T

qr2þ 1

2

qT

qr

!

ð2Þ

3.2. Boundary Conditions

3.2.1. Boundary conditions for metal foam-filled pipes. For a metalfoam pipe subjected to a constant heat flux, the gradients of the velocity andtemperature in the r direction vanish at the centerline. The appropriate boundaryconditions are as follows:

du

dr¼ 0;

qT

qr¼ 0 at r ¼ 0 ð3Þ

Figure 1. Schematic of the double-pipe heat exchanger filled with metal foams: (a) a single pipe filled witha metal foam, (b) cross sectional view of the double-pipe heat exchanger.

1034 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

u ¼ 0; qw ¼ keqT

qrat r ¼ R ð4Þ

3.2.2. Boundary conditions for double-pipe heat exchangers filled withmetal foams.

1. Inlet boundary conditions

a. Inner pipe: T¼Tf1,in; and the velocity (uf1,in) is calculated based on thespecified inner Reynolds number, ReD1.

b. Outer pipe: T¼Tf2,in; and the velocity (uf2,in) is calculated based on thespecified outer Reynolds number, ReD2.

2. Boundary conditions at the walls

a. Inner pipe: The wall thickness of the inner pipe is considered to be negligibleand the coupling conditions are applied across the inner wall.

b. Outer pipe: The outer pipe is considered to be adiabatic.

3.3. Parameter Specifications

Several parameters, such as permeability (K), effective thermal conduc-tivity (ke), and inertial coefficient (F), need to be calculated to initiate thenumerical simulation. The permeability for open-cell metal foams can beobtained through a formulation proposed by Calmidi [9] based on his experimentaldata:

K

d2p

¼ 0:00073 1" eð Þ"0:224 df =dp

! ""1:11 ð5Þ

where dp is the pore size and df is the fiber diameter of the metal foam. These quan-tities are in turn obtained from:

dp ¼0:0254

PPIð6Þ

and:

df

dp¼ 1:18

ffiffiffiffiffiffiffiffiffiffiffi1" e

3p

r1

1" e"ðð1"eÞ=0:04Þ

$ %ð7Þ

where PPI signifies pores per inch. The effective thermal conductivity of a fluid-filledporous medium can be estimated by [1]:

ke ¼ ekf þ 1" eð Þks ð8Þ

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1035

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

To determine the inertial coefficient, the following correlation, which is based onexperimental data for metal foams, was proposed by Zhao et al. [10]:

F ¼ C 1" eð Þn=dp

& '& K1=2 ð9Þ

where C¼ 29.613 and n¼ 1.5226 for the alumina alloy foams while C¼ 7.861 andn¼ 0.5134 for the copper foams.

The Reynolds number for the inner and annular pipes is represented as:

ReD ¼Dh & um & qf

mf

ð10Þ

where Dh is the hydraulic diameter equaling 2R1 for the inner section and 2(R2"R1)for the annular section.

The following are two accepted definitions of Reynolds number for a porousmedium. The first is the permeability-based Reynolds number given by Eq. (11)[12, 18, 19]:

ReK ¼qu

ffiffiffiffiKp

mð11Þ

The second is the pore size-based Reynolds number given by Eq. (12) [5]:

Redp ¼qudp

mð1" eÞ ð12Þ

Darcy number, Da is defined as:

Da ¼ K

H2ð13Þ

All the constant parameters needed in simulations are presented in Table 1.

Table 1. Pertinent parameters for metal foam-filled heat exchangers

Parameter Value Units

cf1 4.179' 103 J=(kg(K)kf1 0.609 W=m=Kqf1 997 kg=m3

mf1 0.92' 10"3 Pa(scf2 4.179' 10"3 J=(kg(K)kf2 0.609 W=m=Kqf2 997 kg=m3

mf2 0.92' 10"3 Pa(sR1 0.01 mR2 0.02 mTf1 in 273.15 KTf2 in 373.15 K

1036 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

4. VALIDATION

The dimensionless velocity, u( is defined as:

u( ¼ u

u1ð14Þ

where u1 is the velocity outside the momentum boundary layer. The dimensionlesstemperature h is expressed as:

h ¼ Tw " T

Tw " Tm¼ Tw " T

qw=hð15Þ

where Tw, Tm, qw, and h are the wall temperature, mean temperature of the fluid, heatflux at the wall, and mean convective heat transfer coefficient, respectively.

The numerical results for the rectangular duct were compared to theanalytical results of Vafai and Kim [15]. Figure 2 shows the comparisons amongthe present dimensionless velocity, temperature distribution, and Nusselt number

Nu ¼ h&Dh

ke

( )and those given in Ref. [15]. The numerical results were found to be

in very good agreement with the analytical solutions of Vafai and Kim [15].A further comparison was done for a circular pipe. Our results were comparedto those of Hooman and Gurgenci [14] and Hooman and Ranjbar-Kani [20].Figure 3 shows the comparisons between the present dimensionless velocity andtemperature distributions and those given in Refs. [14, 20]. The maximumdeviation was found to be less than 2%.

There is a scarcity of numerical and experimental data for forced convectionin double-pipe heat exchangers filled with metal foams. Figure 4 shows thecomparison between our results and prior works for velocity distribution ininner pipes. As can be seen, we demonstrated excellent agreement.

5. RESULTS AND DISCUSSION

5.1. Case 1: Metal Foam-Filled Pipes

5.1.1. Pressure drop. Figure 5a shows the variation in pressure drop per unitlength in metal foam pipes at different Reynolds and Darcy numbers. As expected,the metal foam pipe showed a substantially larger pressure drop than a plain pipeunder the same operating conditions. In addition, the pressure drop increasedsubstantially with a decrease in Darcy number due to the greater resistance to theflow at lower Darcy numbers.

Figure 5b shows the relationship between the friction factor f ¼ Dhpi2HLqf u2

m=2

( )and

permeability-based Reynolds number (ReK). As can be seen from this figure, thefriction factor (f) is nearly invariable after ReK> 20. This is because the inertial force(Forchheimer term which is proportional to the square of velocity) becomes thedominant part of the pressure drop when ReK is larger than 20.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1037

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

Figure 2. Comparison of fully developed flow and temperature fields in a rectangular duct: (a) velocityfield, (b) temperature field, (c) Nusselt number.

1038 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

5.1.2. Heat transfer performance. To examine and assess heat transferperformance, the overall Nusselt number, defined as Nuf ¼ hD

kf, is used. Figure 6a

shows the effects of Reynolds number (ReD) and the Darcy number (Da) on theoverall heat transfer coefficient (Nuf) for a metal foam-filled pipe. As can be seen,the Nusselt number increases gradually with an increase in the fluid velocity whilea decrease in Darcy number (permeability) leads to an increase in Nusselt number.Comparisons of the heat transfer performance of a plain pipe and a metalfoam-filled pipe are also presented in Figure 6a. As expected, the metal foam-filledpipe has a substantially higher Nusselt number (up to 50 times) as compared witha plain pipe.

Figure 3. Comparison of fully developed flow and temperature fields in a circular pipe: (a) velocity field,(b) temperature field.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1039

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

To investigate the effect of different materials, metal foams, such as titanium(k¼ 7 W=m-K), stainless steel (k¼ 16 W=m-K), nickel (k¼ 91 W=m-K), alumina(k¼ 202 W=m-K), and copper (k¼ 399 W=m-K), were simulated at Redp¼ 5 and20. The results are presented in Figure 6b, which illustrates the almost linear increasein heat transfer with increase in the thermal conductivity of the solid skeleton.

5.2. Case 2: Double-Pipe Heat Exchanger Filled with Metal Foam

5.2.1. Pressure drop. Figure 7a shows the effect of Reynolds number on thepressure drop per unit length for metal foam pipes with Da¼ 10"4 and Da¼ 10"2.The relationship between the friction factor (f) and permeability-based Reynoldsnumber (ReK) is shown in Figure 7b.

As expected, pressure drop increases with an increase in Reynolds number anda decrease in Darcy number. The viscous term (Darcy term) is the dominant factorwhen ReK< 20, and the inertia force (Forchheimer term) becomes dominant forpressure drop beyond ReK> 20.

5.2.2. Heat transfer performance. The logarithmic mean temperaturedifference, which is the appropriate average temperature difference for the heatexchanger design, is employed. For the counterflow arrangement, we present:

DT1 ¼ Tf 2 in " Tf 1 out ð16Þ

DT2 ¼ Tf 2 out " Tf 1 in ð17Þ

Figure 4. Comparison of velocity fields in a metal foam-filled double-pipe heat exchanger.

1040 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

DTm ¼DT1 " DT2

In DT1DT2

ð18Þ

and

Q ¼ U & A & DTm ð19Þ

where A is the total surface area of the inner pipe.

Figure 5. Variation in pressure drop with Darcy number in a metal foam-filled pipe: (a) pressure drop,(b) friction factor.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1041

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

Heat transfer, Q can also be presented as:

Q ¼ _mm1cp1 Tf 1;out " Tf 1;in

! "¼ _mm2cp2 Tf 2;in " Tf 2;out

! "ð20Þ

where _mm1 and _mm2 are the mass flow rate of cold and hot fluids (Kg=s); cp1 and cp2 arethe specific heat of cold and hot fluids (J=Kg-K); and Tf1,in, Tf1,out, Tf2,in, and Tf2,out

are the average temperature of cold and hot fluids flowing in and out of the pipes.

Figure 6. Effect of Reynolds number, Darcy number, and thermal conductivity variation in Nusseltnumber for a metal foam-filled pipe: (a) effect of ReD and Da variation, (b) effect of ks variation.

1042 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

Figure 8a shows the heat transfer performance for a plain double pipe anda double pipe filled with metal foam. Again, as can be seen, total heat transferincreases when Darcy number decreases and Reynolds number increases.

Figure 8b shows the ratio of the metal foam heat exchanger heat transfercoefficient for a double pipe to that of the plain double-pipe heat exchanger. It canbe seen that metal foams have a substantial effect on heat transfer enhancementat lower Reynolds numbers. For example, the overall heat transfer coefficient is

Figure 7. Variation in pressure drop and friction factor variation with Darcy number in a metalfoam-filled double-pipe heat exchanger: (a) pressure drop, (b) friction factor.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1043

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

increased 18 times when Reynolds number is close to 1,000. However, the influence ofmetal foams diminishes as Reynolds number increases.

5.2.3. Comprehensive performance evaluation. Heat transfer performanceand energy consumption are two important factors that should be considered when

Figure 8. Comparison of heat transfer performance in a metal foam-filled double-pipe heat exchangerwith a plain double-pipe heat exchanger: (a) total heat transfer coefficient, (b) ratio of total heat transfercoefficient.

1044 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

Tab

le2.

Co

mp

reh

ensi

vep

erfo

rman

cech

arac

teri

stic

so

fm

etal

foam

hea

tex

chan

gers

com

par

edto

pla

intu

be

hea

tex

chan

gers

Cas

eD

aC

old

wat

erve

loci

ty(m=s

)C

old

wat

erfl

ow

rate

(m3=s

)

Fo

am-fi

lled

hea

tex

chan

ger

Pla

intu

be

hea

tex

chan

ger

DP L!" m

etal

foam

DP L!" pl

ain

tube

Q L!" m

etal

foam

Q L!" pl

ain

tube

I(%

)D

P=L

(Pa=

m)

Pp=L

(W=m

)Q=L

(W=m

)D

P=L

(Pa=

m)

Pp=L

(W=m

)Q=L

(W=m

)

110"

40.

13.

14'

10"

528

,918

0.90

819

,986

15.9

40.

0005

1,78

31,

814

11.2

174

82

10"

40.

51.

57'

10"

454

4,58

685

.500

43,2

6819

3.84

0.03

046,

518

2,80

96.

6356

23

10"

20.

13.

14'

10"

52,

214

0.06

919

,629

15.9

40.

0005

1,78

313

911

.00

732

410"

20.

51.

57'

10"

451

,645

8.10

842

,074

193.

840.

0304

6,51

826

66.

4654

5

1045

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

designing heat exchangers. For the evaluation of the heat transfer performance, thethermal load per unit length (Q=L) and the pumping power (Pp=L¼ (DP=L) &Vf),where Vf is the flow rate of the fluid, are utilized. The performance factor (I), givenby Eq. (21), is introduced to assess the comprehensive performance:

I ¼Q=L" Pp=L! "

metal foam" Q=L" Pp=L! "

pla in tube

Q=L" Pp=L! "

pla in tube

ð21Þ

Table 2 shows the comprehensive performance characteristics of double-pipe heatexchangers with and without metal foams. As can be seen, inserting metal foams sig-nificantly increases the pressure drop. However, there is a marked improvement inthe management of thermal load (as much as 11 times) while the performance factorcan increase by more than 700%.

5.2.4. Effects of radius and dispersion on performance. Figure 9 showsthe effect of radius ratio on heat transfer performance for a metal foam-filleddouble-pipe heat exchanger at different Reynolds numbers. As can be seen, totalheat transfer increases when radius ratio increases. The effect of thermal dispersion[19, 21] on heat transfer performance for a plain double pipe and a double pipe filledwith metal foam is shown in Figure 10. As expected, total heat transfer increaseswhen Reynolds number increases. The ratio of the metal foam heat exchanger heattransfer coefficient for a double pipe to that of a plain double-pipe heat exchanger isshown in Figure 10b. There is an almost linear increase in heat transfer with anincrease in Reynolds number.

Figure 9. Effect of the radius ratio on the heat transfer performance for a metal foam-filled double-pipeheat exchanger at different Reynolds numbers.

1046 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

6. CONCLUSIONS

Forced convection in metal foam heat exchangers has been analyzed using theBrinkman-Forchheimer-extended Darcy model and local thermal equilibrium energymodel for a porous medium. Numerical results for the velocity and temperatureprofiles in metal foam pipes are first obtained for constant heat flux boundary

Figure 10. Comparison of heat transfer performance in a metal foam-filled double-pipe heat exchangerwith a plain double-pipe heat exchanger incorporating thermal dispersion: (a) total heat transfercoefficient, (b) ratio of total heat transfer coefficient.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1047

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

conditions. The comparison between the numerical and analytical solutionsdemonstrates the accuracy of our results. It can be seen that a lower Darcy numberresults in enhancement of heat transfer performance.

The effect of using metal foams in double-pipe heat exchangers, as well as theirconsiderable heat transfer enhancement over plain-tube exchangers, was investigated.Attributes related to heat transfer enhancement at the expense of additional pressuredrop were categorized. In this respect, a performance factor was introduced andquantified to further help the evaluation and assessment of the compromise betweenheat transfer enhancement and pressure drop in designing a metal foam heat exchanger.

REFERENCES

1. B. Alazmi and K. Vafai, Analysis of Fluid Flow and Heat Transfer Interfacial ConditionsBetween a Porous Medium and a Fluid Layer, Int. J. Heat Mass Transfer, vol. 44,pp. 1735–1749, 2001.

2. E. C. Ruiz, Modelling of Heat Transfer in Open Cell Metal Foams, University of PuertoRico, Mayaguez Campus, Puerto Rico, 2004.

3. W. Lin, J. Yuan, and B. Sunden, Review on Graphite Foam as Thermal Material for HeatExchangers, World Renewable Energy Congr. 2011, WREC 2011, 2011.

4. K. Boomsma, D. Poulikakos, and F. Zwick, Metal Foams as Compact High PerformanceHeat Exchangers, Mech. Mater., vol. 35, pp. 1161–1176, 2003.

5. S. Mahjoob and K. Vafai, A Synthesis of Fluid and Thermal Transport Models for MetalFoam Heat Exchangers, Int. J. Heat Mass Transfer, vol. 51, pp. 3701–3711, 2008.

6. V. Calmidi and R. Mahajan, The Effective Thermal Conductivity of High PorosityFibrous Metal Foams, J. Heat Transfer, vol. 121, pp. 466–471, 1999.

7. K. Boomsma and D. Poulikakos, On the Effective Thermal Conductivity of a Three-Dimensionally Structured Fluid-Saturated Metal Foam, Int. J. Heat Mass Transfer,vol. 44, pp. 827–836, 2001.

8. A. Bhattacharya, V. Calmidi, and R. Mahajan, Thermophysical Properties of HighPorosity Metal Foams, Int. J. Heat Mass Transfer, vol. 45, pp. 1017–1031, 2002.

9. V. V. Calmidi, Transport Phenomena in High Porosity Fibrous Metal Foams, Ph.D.thesis, University of Colorado, Boulder, 1998.

10. C. Zhao, T. Kim, T. Lu, and H. Hodson, Thermal Transport in High Porosity CellularMetal Foams, J. Thermophys. Heat Transfer, vol. 18, pp. 309–317, 2004.

11. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed., pp. 29–52, Springer-Verlag, New York, NY, 1991.

12. K. Vafai and C. Tien, Boundary and Inertia Effects on Flow and Heat Transfer in PorousMedia, Int. J. Heat Mass Transfer, vol. 24, pp. 195–203, 1981.

13. R. Nazar, N. Amin, D. Filip, and I. Pop, The Brinkman Model for the Mixed ConvectionBoundary Layer Flow Past a Horizontal Circular Cylinder in a Porous Medium, Int.J. Heat Mass Transfer, vol. 46, pp. 3167–3178, 2003.

14. K. Hooman and H. Gurgenci, A Theoretical Analysis of Forced Convection ina Porous-Saturated Circular Tube: Brinkman–Forchheimer Model, Transport PorousMed., vol. 69, pp. 289–300, 2007.

15. K. Vafai and S. J. Kim, Forced Convection in a Channel Filled with a Porous Medium:An Exact Solution, J. Heat Transfer, vol. 111, pp. 1103–1106, 1989.

16. D. A. Nield, A. V. Kuznetsov, and M. Xiong, Thermally Developing Forced Convectionin a Porous Medium: Parallel-Plate Channel or Circular Tube with Isothermal Walls,J. Porous Med., vol. 7, pp. 19–27, 2004.

1048 X. CHEN ET AL.

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015

17. K. Vafai and C. Tien, Boundary and Inertia Effects on Convective Mass Transfer inPorous Media, Int. J. Heat Mass Transfer, vol. 25, pp. 1183–1190, 1982.

18. D. A. Nield and A. Bejan, Convection in Porous Media, 3rd ed., pp. 8–16, Springer,New York, NY, 2006.

19. A. Amiri and K. Vafai, Analysis of Dispersion Effects and Non-Thermal Equilibrium,Non-Darcian, Variable Porosity Incompressible Flow Through Porous Media, Int.J. Heat Mass Transfer, vol. 37, pp. 939–954, 1994.

20. K. Hooman and A. Ranjbar-Kani, Forced Convection in a Fluid-Saturated Porous-MediumTube with Isoflux Wall, Int. Commun. Heat Mass Transfer, vol. 30, pp. 1015–1026, 2003.

21. B. Alazmi and K. Vafai, Analysis of Variants Within the Porous Media TransportModels, J. Heat Transfer, vol. 122, pp. 303–326, 2000.

ANALYSIS AND CHARACTERIZATION OF METAL FOAM 1049

Dow

nloa

ded

by [T

exas

A&

M U

nive

rsity

Lib

rarie

s] a

t 14:

41 1

7 Ju

ly 2

015