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CTRC Research Publications Cooling Technologies Research Center

3-1-2008

Permeability and #ermal Transport in

Compressed Open-Celled Foams

Ravi Annapragada

asravi@purdue.edu

S V. Garimella

Purdue University, sureshg@purdue.edu

Jayathi Y. Murthy

School of Mechanical Engineering, Purdue University, jmurthy@purdue.edu

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<is document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for

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Annapragada, Ravi; Garimella, S V.; and Murthy, Jayathi Y., "Permeability and <ermal Transport in Compressed Open-Celled Foams"

(2008). CTRC Research Publications. Paper 95.

h=p://dx.doi.org/10.1080/10407790802154173

1

Permeability and Thermal Transport in Compressed Open-Celled Foams

*

S. Ravi Annapragada, Jayathi Y. Murthy, Suresh V. Garimella

†

,

School of Mechanical Engineering, 585 Purdue Mall, Purdue University,

West Lafayette, IN-47907 USA

Abstract

A computational methodology is proposed to describe the fluid transport in compressed open-

celled metallic foams. Various unit-cell foam geometries are numerically deformed under uniaxial

loads using a finite element method. An algorithm is developed and implemented to deform the fluid

domain mesh inside the unit-cell foam based on the deformed solid unit-cell geometry. Direct

simulations of the fluid transport in these deformed meshes are then performed over a range of

Reynolds numbers used in practical applications. The model is validated against available

experimental results and correlations. A corrected model is proposed for the permeability of

compressed foams as a function of strain for flows transverse to the direction of compression. The

thermal conductivity of fluid-saturated foams is also computed. Compression of foams increases the

conductivity transverse to the direction of compression and decreases the conductivity parallel to it.

*

Submitted for possible publication in Numerical Heat Transfer, March 2008

†

Corresponding author, Email: sureshg@purdue.edu, Tel: 765-494-5646, Fax: 765-494-0539

2

NOMENCLATURE

a edge length of the unit cell, m

A area, m

2

C

P

specific heat, J g

−1

K

−1

D diameter of the pore, m

Da Darcy number

E Young’s modulus, Nm

-2

f friction factor

J diffusion flux vector, m

2

s

-1

K permeability, m

2

k thermal conductivity, Wm

-1

K

-1

L length of the periodic module, m

Nu Nusselt number

q” heat flux, Wm

-2

P pressure, Nm

-2

Pr Prandtl number

Pe Peclet number

R radius of the pore, m

Re Reynolds number

s center-to-center distance, m

T temperature, K

t time, s

u,v,w velocities along x,y,z directions, ms

-1

V volume, m

3

x,y,z Cartesian coordinates

Greek

α

thermal diffusivity, m

2

s

-1

δ displacement, m

ε

strain

λ Lame’s constant

µ

dynamic viscosity, kg m

-1

s

-1

ρ density, kg m

-3

Φ porosity

Superscripts

- average or mean

Subscripts

0 uncompressed

B bulk

bc body center

comp compression

D Darcian

Eff effective value for the foam

f fluid, foam

in inlet

int intersection

K permeability

s solid

sa surface area

sc spherical cap

solid bulk value of solid

top top surface

unrestrained lateral sides free to move

3

1 INTRODUCTION

Open-celled foams possess a number of interesting physical properties which has led to a wide range

of research studies during the last two decades. These foams have been used as sound absorbents, as

scaffolds in tissue engineering, in hydrogen storage, catalysis and other applications [1,2]. More recently,

metal foams have been considered for electronics cooling applications [3]. In many of these applications,

low-porosity foams are created by compressing high-porosity open-celled foams. The mechanical,

thermal and fluid-dynamical properties of such compressed foams are of great industrial and research

interest.

Gent and Rusch made the first attempt to study the effect of compression of an open-cell

polyurethane foam on the resulting fluid transport [4]. The foam was represented by an array of circular

tubes. Based on this assumption, the effective cell diameter associated with the compressed foam was

related to the strain (

ε

) by the relationship,

1/2

0

(1 )

d d

ε

= + . A simple model based on scaling was defined

to arrive at the permeability of compressed polyurethane foams as a function of

ε

. They also

experimentally observed that Darcy flow (with viscosity dominating) was valid until Re

D

≈

1, beyond

which inertial forces were found to dominate. An extension to the model was proposed by Hilyard and

Collier [5], who used packed-bed theory to relate permeability to compression and porosity in

polyurethane foams. Mills and Lyn [6] used the Hilyard and Collier model to account for the effect of air

pressure during the deformation under impact of a polyurethane foam.

Recently, Dawson et al. [7] performed controlled experiments in which foams were compressed to

80% of their original linear dimension. The flow direction was always the same as the direction of

compression. For polyurethane foams, the foam was observed to exhibit elastic behavior for small

amounts of compression. Beyond a certain strain (

ε

= 7.5%), the foams buckled along bands and the

densification was observed to occur along these regions. The experiments also showed that the

permeability is independent of the cell size. A model was proposed to predict the effect of applied strain

on permeability, with the experiments providing an empirical model constant in the densified region.

Schulenburg et al. [8] computed pore-scale velocities in an impacted foam using the lattice-

Boltzmann method and compared the results to measurements made via Magnetic Resonance Imaging

(MRI). Three-dimensional MRI images of the velocity fields of water flow inside the foam were

obtained. The regions with velocities below a particular threshold level were assumed to constitute the

foam walls. These images provided the geometry used in the lattice-Boltzmann fluid simulations. The

modeling technique is specific to the pore geometry obtained from the MRI images and cannot be

generalized. Also, no effective parameters such as permeability were computed from the simulations.

4

The compressive behavior of metal foams is different from that of polyurethane foams. Kwon et al.

[9] showed that the mode of failure in metals is through plastic collapse at the joints, finally leading to

complete collapse of the cell. The flow through these metal foams has received little attention until

recently. Experimental results [10, 11, 12] for permeability of compressed metal foams available through

about the year 2005 were summarized in Dukhan et al. [10]. Boomsma et al. [13] measured the thermal

performance of metal foams compressed to strains of -0.5, -0.75, -0.83 and -0.88. Metal foams were

recommended for use in compact heat exchangers since the thermal resistance was shown to be lower,

and the efficiency greater, than that of existing heat exchangers. Klein et al. [14] constructed a heat

exchanger using compressed foams and studied the thermal performance relative to the extent and

direction of compression for Reynolds number of 100-1000. The thermal performance was observed to

be independent of the direction of compression. No effective parameters such as permeability were

reported.

The research to date has concentrated on the performance of foams compressed well beyond the

plastic limit and into the crushed regime. In the present work, we investigate the performance of

compressed foams at compressions within the plastic-collapse limit (≤ 10% compression). The unit cell

modeling approach of Krishnan et al. [15] forms the starting point for this work. We develop a new

coupled unit-cell model to understand the effect of compression on the flow and thermal characteristics of

these foams. The model is validated against experiments and correlations for polyurethane foams and is

extended to model the fluid and thermal transport in metal foams.

2 GEOMETRIC MODEL

In this work, we use the same methodology for geometry creation as discussed in [15]. The shape of

the pore is assumed to be spherical and spheres of equal volume are arranged according to the following

three lattice structures: (i) BCC, body-centered cubic, (ii) FCC, face-centered cubic, and (iii) A15 lattice,

which is similar to the Weaire- Phelan (WP) structure [16, 17]. The periodic foam unit-cell geometry is

obtained by subtracting the spheres at the various lattice points from the unit cell cube as shown in Figure

1a. The cross-section of the foam ligaments is a set of convex triangles (Plateau borders), all of which

meet at symmetric tetrahedral vertices [16]. It is noted that there is a non-uniform distribution of metal

mass along the length of the ligament, with more mass accumulating at the vertices (nodes) and resulting

in a thinning at the center of the ligament as experimentally observed in foam samples by many authors

(e.g., [18]). Figure 1b shows sample open-cell structures formed for three different lattice arrangements.

The distinguishing features of this approach are that: (i) the geometry creation is simple; (ii) it captures

many of the important features of real foams; and (iii) meshing of the geometry is easier compared to the