Thermal Transport Phenomena in Porvair Metal … Report Thermal Transport Phenomena in Porvair Metal...

Final Report Thermal Transport Phenomena in Porvair

Metal Foams and Sintered Beds

C. Y. Zhao, T. Kim, T.J. Lu and H. P. Hodson

Micromechanics Centre & Whittle Lab Department of Engineering

University of Cambridge

August 2001

1

Abstract

This report presents experimental and numerical results on pressure drop and heat

transfer for Porvair metal foams and sintered bronze particle beds. The metal foams

considered in this study are made of two different materials - FeCrAlY and copper.

The microstructures of the foams have been characterized with SEM and image

analysis, and experiment measurements on heat transfer and pressure drop have been

performed for eight FeCrAlY samples and six copper samples with different pore

sizes (ppi) and different relative densities. The experimental results show that the heat

transfer of FeCrAlY samples is more sensitive to relative densities than cell size,

whereas for copper samples, the heat transfer is more sensitive to the cell size than the

relative density. This difference in transport phenomena is attributed to the thermal

resistance on the solid side and is heavily influenced by the solid conductivity. A

numerical model is developed to consider the transport of heat based on the measured

microstructural parameters for Porvair metal foams; the effects of foam cellular

microstructure on overall heat transfer are predicted. Generally, the heat transfer will

increase with increasing relative density, and decrease with increasing pore size (ppi).

Finally, forced convection through a sintered bronze bed is studied numerically. The

sintered bronze bed has a much smaller porosity compared with the metal foams, and

water was used as the coolant instead of air for metal foams. The results indicate that

the heat transfer of the sintered bed is about 30% higher than that of a packed bed.

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Contents

Nomenclature (4)

1 Introduction (6)

1.1 Background (6)

1.2 Objectives of the study (9)

2 Porvair metal foams (10)

2.1 Introduction (10)

2.2 Metal foam processing (13)

2.2.1 Metal foam fabrication and capabilities (13)

2.2.2 Capability of manufacturing complex assemblies (14)

2.3 Microstructure of the Porvair metal foams (15)

2.3.1 Specification of the microscopic parameters (15)

2.3.2 Measurement of Microstructures (19)

Procedures of the measurements

Results of the measured microstructures

2.3.3 Verification of the relationship among the microscopic parameters (25)

2.4 Experimental study for heat transfer in Porvair metal foams (26)

2.4.1 Experimental facility and procedures (26)

2.4.2 Measurement uncertainties (28)

2.4.3 Experimental results and analysis (28)

Pressure drop

Heat transfer

2.4.4 Product uncertainties (42)

2.5 Numerical modelling on forced convection in Porvair metal foams (44)

2.5.1 Mathematical formulations (44)

2.5.2 Boundary conditions (46)

2.5.3 Modelling on Porvair metal foams (47)

2.5.4 Numerical procedure (49)

2.5.5 Code validation (49)

2.5.6 Numerical results for Porvair metal foams (51)

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2.5.7 Effect of boundary conditions (58)

2.5.8 Effects of microstructural parameters: Optimisation (60)

Solid conductivity (ks)

Relative density (ρr)

Porosity (ε)

Pore size (dp)

3 Sintered particle bed (65)

3.1 Introduction (65)

3.2 Mathematical formulations (66)

3.3 Modelling on sintered bed (68)

3.4 Results and discussions (70)

4 Conclusions (73)

Appendix A SEM pictures of foam samples (79)

Appendix B Experimental data of Porvair metal foams (82)

4

Nomenclature

α~ specific surface area per unit volume, m-1

Cf heat capacity of fluid

CI inertial coefficient

df diameter of the fibre of the metal foam

dp diameter of the pore size for metal foam; particle diameter for sintered bed

Dh hydraulic diameter of the channel, )/(2 WHWH +

f friction resistance

FI inertial variable

hsf interfacial heat transfer coefficient

h overall heat transfer coefficient

H channel height

kd thermal dispersion conductivity

kf thermal conductivity of fluid

kfe effective thermal conductivity of fluid

ks thermal conductivity of solid

kse effective thermal conductivity of solid

K permeability

L length of the heat sink

Nuf,b local Nusselt number

Nu overall Nusselt number

P pressure

Pr Prandtl number

qw heat flux over the bottom surface

r inner-to-outer diameter ratio

R simplification quantity

T temperature

Tf fluid temperature

Tf,b bulk mean fluid temperature

Tin inlet fluid temperature

Ts solid temperature

Tw bottom wall temperature

5

u velocity

W width of heat sink

X dimensionless streamwise coordinate, y/H

Y dimensionless vertical coordinate, y/H

Greek symbols

ε porosity

optε optimised porosity

ρf density of fluid

µf viscosity of fluid

f volume-averaged value over the fluid region

s volume-averaged value over the solid region

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1 Introduction

1.1 Background

Recent advances in low-costing processes have enabled porous metals to be

manufactured at high volume. In many engineering applications, it is attractive to

design a primary mechanical structure to perform other functions such as heat

dissipation. For such multifunctional structures, a robust methodology allowing the

mechanical performance (stiffness, strength, weight) and thermal properties (heat

dissipation, pumping power) to be optimized simultaneously is desirable. On the other

hand, it is recognized that there is a diversity of thermal management issues. For

example, the increasing demand for execution speed in modern computers has led to

high heat fluxes (> 100 W/cm2) at the chip level. The high level of heat fluxes

provides a challenge of removing heat from the junctions of power electronics.

Therefore the use of porous metals as efficient compact heat exchangers for heat

removal has become one of the most promising cooling techniques. This is attributed

to the high surface-area-to-volume ratio intrinsic to these materials, resulting in

enhanced heat transport and miniaturization of thermal systems.

Forced convective heat transport in porous materials represents a rapidly growing

branch of thermal science. Several thermal engineering applications can benefit from

a better understanding of convection through porous materials exemplified by

geothermal systems, thermal insulations, microelectronic cooling system, filtering

devices and products manufactured in the chemical industry. The initial investigation

of fluid flow through a porous medium can be traced back to the nineteenth century.

Darcy [1] was the first to perform recorded experiments and to produce formulations

pertaining to a porous medium. He discovered that the area-averaged fluid velocity

through a column of porous material is proportional to the pressure gradient and

inversely proportional to the viscosity (µ) of the fluid seeping through the porous

material, represented by the Darcy flow law as follows,

−=

dx

dPKu

µ (1.1)

where K is a material constant called permeability.

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The Darcy flow model has subsequently been employed in numerous engineering

applications related to fluid flow and heat transfer in porous materials. However,

although the Darcy model is popular in porous medium convective heat transfer

investigations, it neglects several key physical effects of importance in channel flows.

For example, the Darcy flow model does not satisfy the no-slip condition on a solid

boundary by neglecting friction due to macroscopic shear, and the inertial forces

which are significant for relatively fast flows are disregarded.

To thoroughly understand the fluid flow and heat transfer characteristics in porous

medium is a challenging task. In this respect, the complex microscopic transport

phenomena at the pore level is important as these determine such macroscopic

phenomena as heat transfer augmentation and pressure loss increase. However, the

complexity of the cellular morphology typically found in commercial porous metals

usually precludes a detailed microscopic investigation of the transport phenomena at

the pore level. Therefore, the general transport equations are commonly integrated

over a representative elementary volume, which accommodates the fluid and the solid

states within a porous structure. Though the loss of information with respect to the

microscopic transport phenomena is inevitable with this approach, the integrated

quantities, coupled with a set of proper constitutive equations representing the effects

of microscopic interactions on the integrated quantities, do provide a rigorous and

effective basis for analysing the transport phenomena in porous materials.

There are two approaches available in applying the volume-averaging technique for

heat transfer investigations: one is averaging over a representative elementary volume

containing both the fluid and solid phases, and the other is averaging separately over

each of the phases, thus resulting in a separate energy equation for each individual

phase. These two models are referred to as the one-equation model and the two-

equation model, respectively. The one-equation model is valid when the thermal

communication is sufficiently effective so that the local temperature difference is

negligibly small between the fluid and the solid phases. In some applications,

however, the temperature differences between phases cannot be neglected. In these

situations the effects of the interfacial surface and interstitial heat transfer coefficients,

which are related to the internal heat exchange between the solid and fluid phases are

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major factors causing heat transfer augmentation in porous materials. In such cases,

the two-equation model needs to be utilized.

Although extensive studies have been conducted on heat transfer in a porous medium

channel [2-18], most of the research mainly focused on the packed bed as the porous

medium, often assuming a linear relationship between its effective conductivity and

porosity. In recent years, with advances in the processing technology, metal foams

and sintered metal particle beds can be mass-produced with high quality. The

microstructure of a metal foam produced via the metal sintering route by Porvair is

shown in Fig. 1.1, whereas Fig. 1.2 depicts a typical sintered particle bed. Because

metal foams and sintered beds are relatively new class of materials, investigations on

thermal transport in these materials are scarce compared to packed beds, and hence

the knowledge is not sufficient for engineering applications. For metal foams, due to

the complex cellular microstructures (Fig. 1.1), how to build a mathematical model

which can properly account for the heat transport remains a problem to be addressed.

For a sintered bed, how good is its heat transfer performance compared to that of a

packed bed if water is used as the coolant fluid? In addition, reliable test data is

urgently needed to provide the base for industrial design as well as validation of the

numerical models.

Fig. 1.1 Microstructure of a typical Porvair metal foam

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Fig. 1.2 Schematic of a sintered particle bed

1.2 Objectives of the study

The aim of this research is to experimentally and numerically investigate the transport

phenomena in Porvair metal foams and sintered bronze beds, and hence to provide

guidance on materials processing and compact heat exchanger designs. The focus of

this report will be placed on Porvair metal foams, for which both experimental and

numerical investigations have been performed. As for the sintered bed, only

numerical predictions have been carried out, with the aim to build a good heat transfer

model to assess the feasibility of using such porous material as efficient heat

dissipation medium.

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2 Porvair Metal Foams

2.1 Introduction

With recent developments in the processing technology, a range of novel materials

have been manufactured for advanced, compact, and lightweight thermal systems.

Metal foams with open cells, as one of the most promising emerging materials, have

received much attention in recent years. The microstructure of a typical Porvair foam,

shown in Fig. 2.1, consists of ligaments forming a network of inter-connected

dodecahedral-like cells. The cells are randomly oriented, and mostly homogeneous in

size and shape. Pore size may be varied from approximately 0.1 mm to 7 mm. The

relative density can be varied from 3% to 15%. Alloys and single-element materials

are available. Common materials include copper, aluminum, stainless steel, and high

temperature iron-based alloys (FeCrAlY). The distinctive feature of the Porvair metal

foams processed by the metal sintering technique is that the struts (ligaments) are

hollow (see Fig. 2. 1b) compared to the solid struts of ERG foams manufactured via

the expensive investment casting route.

(a) (b)

Figure 2.1 A typical Porvair metal foam: (a) cellular morphology; (b) cross-section of an individual strut

In an attempt to enhance convective thermal transport, the metal foam materials can

be used as an advanced compact heat exchanger. The motivation is attributed to the

high surface area to volume ratio as well as enhanced flow mixing due to the

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tortuosity of the metal foam. Therefore, it is believed that the overall performance of a

thermal system can be substantially enhanced by using metal foams. Furthermore,

metallic foams have attractive stiffness/strength properties and can be processed in

large quantity at low cost.

From the heat transfer point of view, metal foams can be considered as one type of

porous medium, so that the study on metal foams can be classified as thermal

transport in porous media. The transport phenomena in porous media have been of

continuing interest for the past five decades [2]. This interest stems from the

complicated and interesting phenomena associated with transport processes in porous

media. However, most studies on porous media have been restricted to the packed

beds and granular materials with porosities in the range 0.4 – 0.6 [2-18]. Relatively

few investigations of the transport phenomena have been conducted for very high

porosity media (porosity ε > 0.9) such as metal foams.

Even though metal foams can be broadly classified as porous media, they have very

distinctive features such as high porosities (ε > 0.9) and a unique open-celled

structure. Consequently, most of previous studies on packed beds and granular porous

media are not applicable to metal foams. Only during the last fifteen years, transport

phenomena in metal foams have started to receive attention [19-29]. Under the

assumption of local thermal equilibrium, Hunt and Tien [19] studied the effects of

thermal dispersion on forced convection in metal foams with water as the fluid phase,

and concluded that conduction of the metal foam may not be significant due to its thin

cell ligaments, and dispersion may dominate the heat transport for metal foams with

high porosity. Sathe et al. [20] studied combustion in metal foams as applied to

porous radiant burners. Younis and Viscanta [21] have measured the volumetric heat

transfer of ceramic foam materials, and developed a Nusselt number correlation fit to

the experimental data. The volumetric heat transfer rates measured were higher than

those for packed beds. Lee et al. [22] investigated the application of metal foams as

high-performance air-cooled heat sinks in electronics packaging. In their experimental

study, they demonstrated that aluminum foams could dissipate heat fluxes up to 100

W/cm2. Using the fin approach, Lu et al. [23] have developed an analytical model to

predict the metal foam-assisted heat transfer, where foam is modelled as inter-

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connected cylinders. Bastarows et al. [24] studied the single-sided heating of a foam-

filled channel for electronics cooling applications. The experimental method utilized

both conductive thermal epoxy bonding and brazing of the metal foam to a heated

plate. The test results revealed that brazed foam materials are much more effective at

heat removal than epoxy-bonded samples, and that the heat exchange performance is

three times more efficient compared to a conventional fin-pin array. Recently,

Calmidi and Mahajan [25,26] proposed an effective thermal conductivity model for

high porosity metal foams, and conducted an experimental and numerical

investigation on the forced conduction in ERG aluminum foams with air as the fluid

phase. In their numerical study, they employed two-equation heat transfer model and

found that the thermal dispersion effect was extremely small if the fluid phase was air,

which was quite different from the conclusion derived by Hunt and Tien [19] who

used water as the coolant. Kim et al. [27] experimentally studied laminar heat

transport in aluminum foams, and their results showed that the foam material offers a

better heat transfer performance compared to that of a louvered array, but at a greater

pressure drop. Paek et al. [28] studied the effective conductivity and permeability of

aluminum foams, and indicated that the effective conductivity (ke) increases as the

porosity decreases; however, no noticeable changes of ke were detected by varying the

cell size of the metal foam at a fixed porosity (ε). The permeability K is substantially

affected by both porosity and cell size. Boomsma and Poulikakos [29] put forward an

effective thermal conductivity model based on a three-dimensional structure of the

foam geometry. The results showed that despite the high porosity of the foam, the

heat conductivity of the solid phase controls the overall effective thermal conductivity

to a large extent, a fact that must be dealt with in the foam manufacturing process if

specific ranges of the foam effective conductivity are desired.

Though the above investigations have been conducted for the heat transport in metal

foams, the study is still incomplete. Firstly, because these studies mainly focused on

ceramic or aluminum foams with solid cell struts, for forced convection in metal

foams made of different materials with different cellular morphologies, there is little

information in the open literature and there is a lack of reliable experimental data.

Secondly, various models on the effective thermal conductivity of metal foams have

been proposed [25, 28, 29], but their effects on predicting the overall heat transfer

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performance of the foam in forced convection remain to be quantitatively studied. In

other words, question like if the effective thermal conductivity can dominate the

forced convection, whether heat is taken away directly by coolant or firstly conducted

by solid and then transferred to the coolant flow, need to be answered. Finally, no

investigation has been conducted for Porvair metal foams with hollow struts

(ligaments). Compared to foams with solid struts, how the hollowness affects the

overall heat transfer performance needs to be addressed.

In this section, the microstructures of Porvair metal foams will be characterized.

Experimental measurements on forced convection will subsequently be carried out for

eight FeCrAlY and six copper Porvair foams. A numerical model will then be

developed incorporating the measured microstructural parameters. The effect of

boundary conditions as well as the effects of microstructural foam parameters on

overall heat transfer performance will be studied. In other words, the parameters

which characterize thermal transport in metal foams need to be identified, and how

these parameters affect overall heat transfer will be examined.

2.2 Metal foam processing

2.2.1 Metal foam fabrication and capabilities

Metal foams have been manufactured for many years using a variety of novel

techniques. Metallic sintering, metal deposition through evaporation,

electrodeposition or chemical vapor decomposition (CVD), and investment casting

(among numerous other methods) have created open cell foams. In foam creation

through metal sintering, metallic particles are suspended in slurry and coated over a

polymeric foam substrate. The foam skeleton vaporizes during heat treatment and the

metallic particles sinter together to create the product. This method is thought to be

the most cost-effective method, and the most amenable to mass production. The CVD

method utilizes chemical decomposition of a reactive gas species in a vacuum

chamber onto a heated substrate (polymer or carbon/graphite, depending upon the

temperature of the deposition process). Production rates are limited in this method due

to the rate at which material may be deposited on the substrate. However, highly

refractory metals and ceramics may be created with this method (including rubidium

and silicon carbide), with high quality. Molten metal infiltration is utilized to make

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aluminum and copper foam materials [30]. In this method, the foam precursor is

coated with a mold casting and packed into casting sand. The casting assembly is

heated to decompose the precursor and harden the casting matrix. Molted metal is

then pressure infiltrated into the casting, filling the voids of the original matrix. After

solidification, the material is broken free from the mold. The method has the

advantage of being capable of producing a product of highly useful materials and

alloys (such as aluminum), and generate a product with solid struts. However, the

process is complex and expensive, requiring several processing steps and highly

specialized equipment. Of the methods suitable to produce metal foam materials, only

the metal sintering method offers promise as a method that is capable of economically

producing millions of components annually. The process is similar to the production

of ceramic foam materials that are used in molten metal filtration, except in the heat

treatment process. Heat treatment needs, however, are identical to those required in

power metal industry for sintering pressed and injection molded materials.

Production-designed equipment may be used effectively with automated lines,

eliminating the need for handling in the process.

2.2.2 Capability of manufacturing complex assemblies

To effectively use metal foam materials in heat exchange devices it is necessary to

combine the material with tubes and sheets for flow separation and heat transfer.

Development efforts have taken place at Porvair Fuel Cell Technology to

successfully combine a variety of metal foam materials with solid structures. Several

proprietary articles have been constructed combining tubes and sheets to construct

advanced, multifunctional heat exchange devices for a variety of customers. An

important consideration in the formation of the advanced heat exchangers is the

quality of the bond joint between foam and tube. Figure 2.2 is a photograph of a

developmental component consisting of tubes imbedded in a metal foam matrix.

Metallurgical bonding between the tube wall and the foam matrix was achieved in this

example by direct sintering. Figure 2.3 is an SEM micrograph of the joint region.

Assemblies have also been manufactured through a proprietary co-sintering

technique. Figure 2.4 shows an example of a foam-filled tube manufactured with this

method. Complex assemblies combining metal foam with metal packaging are in the

design stage to create an advanced two-phase heat exchange component for use in

fuel cell fuel processing systems at Porvair Fuel Cell Technology.

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Figure 2.2 Metal foam compact heat exchanger for high temperature service

(foam material is PFCT's FeCrAlY)

Fig. 2.3 SEM micrograph of a foam strut Fig. 2.4 Example assemblies manufactured

Sintered to a solid tube. in a proprietary co-sintering technique

2.3 Microstructure of the Porvair metal foams

2.3.1 Specification of the microscopic parameters

A typical Porvair metal foam structure shown in Fig. 2.1 is characterized by several

key geometrical and physical parameters, namely, porosity (ε, void volume fraction),

pore size (dp), fibre diameter (df), inner-outer diameter ratio (r), and relative density

(ρr). It should be noted that these parameters are not all independent of each other.

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Fig. 2.5 Close-up of a single open cell (from ONR workshop, Cambridge, UK)

Fig. 2.6 Model of the tetrakaidecahedron (from ONR workshop, Cambridge, UK)

dp

17

With a closer look at a single cell of a typical open-celled metal foam (Fig. 2.5), it can

be found that the cell has the approximate shape of a tetrakaidecahedron with roughly

12-14 pentagonal or hexagonal faces (Fig. 2.6). The fibres (ligaments) form the edges

of the tetrakaidecahedron, and there is a lumping of material at joints where the

ligaments meet. The cross-section of the ligaments is circular only for ε ∼ 0.85 or less.

As the porosity increases from this value, the cross-section of the ligament changes

from circular to triangular due to the different rate of metal sintering at high porosity

levels [31]. Since the structure is considerably complex, it may be approximated by an

cross-cylinder representation [23,31] as shown in Fig. 2.7.

Fig. 2.7 Open-cell representation of metal foam structure [23]

First, consider the idealised representation of an open-celled metal foam as shown in

Fig. 2.7. The ratio of the fibre diameter 'fd to the pore size '

pd can be derived as [23],

( )πε

3

12

'

' −=

p

f

d

d (2.1)

Now consider the more reasonable metal foam representation of Fig. 2.6. Calmidi

[31] obtained the relationship between 'pd and the pore size dp, as

pp dd 59.0' = (2.2)

'pd

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In other words, the pore size dp associated with the model of Fig. 2.6 can be converted

into an equivalent cell size 'pd for the cell model of Fig. 2.7 by using Eq. (2.2). In the

derivation of Eq. (2.1), it has been assumed that the fibres are circular. However, as

previously discussed, in practice the fibres of a metal foam are not circular at

porosities higher than 0.85. It is therefore necessary to introduce a shape-factor S to

account for this discrepancy, with [31]:

( )( )04.0/11 ε−−−= eS (2.3)

Thus, from Eqs. (2.1) to (2.3), an appropriate equation for metal foam structure is

written as [27],

( )( )

−−

=−− 04.0/11

1

3

118.1

επε

ed

d

p

f (2.4)

where dp and df in Eq. (2.4) now refer to the measured pore size and fibre diameter. It

should be pointed out that Eq. (2.4) is sufficient for ERG foams whose fibres

(ligaments) are solid and for which the simple relation between porosity ( ε) and

relative density (ρr) exits, ερ −= 1r . Thus, for ERG foams, there are only two

independent microscopic parameters, i.e., pore size (dp) or fibre diameter (df), and

porosity (ε) or relative density (ρr).

For Porvair metal foams, however, because the ligaments are hollow, another

parameter - the inner-to-outer diameter ratio (r) - is needed. The relationship between

the porosity and relative density then becomes

( )( )211 rr −−= ερ (2.5)

Consequently, the following cross-relationships exist between a ERG foam and a

Porvair foam:

( )2,, 1 rERGrPorvairr −= ρρ at same porosity (ε) (2.6)

2

2

1 r

rERGporvair −

−=

εε at same relative density (ρr) (2.7)

Both Eqs. (2.6) and (2.7) will be used in later calculations.

From Eqs (2.4) to (2.7), it can be seen that there are three independent parameters

characterizing Porvair metal foams, namely, porosity (ε) or relative density (ρr), pore

size (dp) or fibre diameter (df), and inner-to-outer diameter ratio of the fibre (r).

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2.3.2 Measurement of the microstructures

Accurate values of cell parameters are crucial to the numerical calculations to be

presented later. In this section, the relative density (ρr), pore size (dp), fibre diameter

(df), and inner-to-outer diameter ratio of the fibre (r) will be carefully measured. From

the previous section, it is known that only three independent parameters are needed,

so the pore size (dp) and fibre diameter (df) are dependent on each other. The aim of

measuring both of them is to use the test data to check the applicability of Eq. (2.4)

for Porvair metal foams. In this study, eight FeCrAlY samples and six copper

samples provided by Porvair are measured. The samples have been specified with

industrial terminologies: pore size in terms of ppi ( pores per inch), and relative

density (defined as the ratio of foam density to the density of the solid of which the

foam is made). Table 2.1 lists all samples and their industrial specifications.

Material pore size (ppi) relative density

Sample #1 10 ppi 5% Sample #2 10 ppi 10% Sample #3 30 ppi 5% Sample #4 FeCrAlY 30 ppi 10% Sample #5 60 ppi 5% Sample #6 60 ppi 10% Sample #7 30 ppi 7.5%

Sample #9 10 ppi 5% Sample #10 10 ppi 10% Sample #11 Copper 30 ppi 5% Sample #12 30 ppi 10% Sample #13 60 ppi 5% Sample #14 60 ppi 10%

Table 2.1. Industrial specifications of Porvair foam samples

Measurement procedures

The measurement of cell and foam parameters was conducted by using the SEM

(scanning electronic microscope) and an image analysis software KS 400 v3.0, from

Karl Zeiss Vision GmbH. In order to get good images with the SEM, the Electrical

20

Discharge Machine (EDM) was used to cut the samples, which can provide a good

quality cutting section with smooth surfaces. Twenty or more images of each sample

were taken with the SEM. Measurements of the pore size (dp), and the diameter of the

fibres (ligaments) as well as the inner holes in the ligaments were then made by using

the image analysis software. After a geometric calibration, the software can give the

area and the perimeter of the selected open cell, as illustrated in Fig. 2.8.

Fig. 2.8 The image analysis software used in this study

From the measured area and perimeter, the equivalent pore size can be calculated.

Similarly, the average diameters of the fibres and the inner holes can be obtained.

Comparing different hand-drawn perimeters for the same cell, the uncertainty is

estimated to be less than 10%. A summary of the SEM images for each sample is

given in Appendix A. As for the relative density, it is obtained straightforwardly by

weighing the foam sample and measuring its overall dimensions.

Results of the measured microstructures

a) Cell size (dp)

The nominal cell (pore) size of a metal foam can be calculated directly from the

product specifications based on ppi (Table 2.1). In this project, metal foams having

three different industrial specifications are studied: 10 ppi, 30 ppi and 60 ppi. The

corresponding nominal cell sizes are 2.54 mm, 0.847 mm and 0.423 mm, respectively.

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For each sample, ten open cells were measured, thus the averaged cell size with its

standard deviation can be obtained. The measured results for seven FeCrAlY samples

and six copper samples are shown in Table 2.2 and Table 2.3, respectively. Sample

eight for the FeCrAlY foam has identical industrial specifications as those of sample

seven but manufactured from a different batch job, and will be examined later for

uncertainties in product quality due to processing.

Sample #1 Sample #2 Sample #3 Sample #4 Sample #5 Sample #6 Sample #7

Nominal 2.54 2.54 0.847 0.847 0.423 0.423 0.847 Average 3.131 3.109 1.999 2.089 0.975 0.959 1.998

Std deviation 0.304 0.201 0.108 0.118 0.0229 0.0245 0.145

Table 2.2 Cell size (mm) of FeCrAlY samples

Sample #9 Sample #10 Sample #11 Sample #12 Sample #13 Sample #14 Nominal 2.54 2.54 0.847 0.847 0.423 0.423 Average 2.645 2.697 1.284 1.431 0.554 0.657

Std deviation 0.181 0.240 0.046 0.081 0.0269 0.0457

Table 2.3 Cell size (mm) of copper samples.

It is noted from Tables 2.2 and 2.3 that there is significant difference between the

nominal and measured cell sizes, and the cell size of the FeCrAlY sample is quite

different from that of a copper sample with the same ppi. The difference can be

attributed to the manufacturing process and properties of different materials.

b) Diameters of the fibre and its inner hole

The other parameter to be measured from the SEM images is the diameter ratio for the

hollow struts. This is more difficult due to the small dimensions of the fibres. For

FeCrAlY samples, reasonably good measurements can be made because of the images

taken for various struts have relatively good quality. However, for copper samples,

SEM images show that nearly all the struts are solid, with no presence of inner holes,

as shown in Fig. 2.9. This may be attributed to a better sintering process of copper

after the evaporation of the polymeric skeleton. As a result, only the averaged fibre

diameters were measured for copper samples. The measured results are summarized

in Table 2.4 and Table 2.5 for FeCrAlY and copper samples, respectively.

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Figure 2.9 Solid struts of a copper sample

Sample #1 Sample #2 Sample #3 Sample #4 Sample #5 Sample #6 Sample #7 hole diameter 161 154 110 107 43.5 41.6 116 strut diameter

287 351 215 267 124 154 241

ratio 56% 44% 51% 40% 35% 27% 46%

Table 2.4 Strut diameters (µm) and hole ratio of FeCrAlY samples.

Sample #9 Sample #10 Sample #11 Sample #12 Sample #13 Sample #14 strut diameter 263 270 122 127 88.8 93.2

Table 2.5 Strut diameters (µm) of copper samples.

c) Relative density

The foam samples used in the heat transfer experiments are of the sandwich type,

consisting of a foam core brazed onto two thin copper plates. Depending on the

quality of processing and the material being used, the weight of the brazing material

could take up as much as 25% of the weight of the foam itself: two relative densities

were therefore measured for each sample. The first measurement represents the

23

relative density of the whole foam core with the presence of the brazing material. The

second measurement was made by cutting a small cubic sample from the central

portion of the foam core in order to eliminate the effect of brazing. The measured

results are given in Table 2.6.

Relative density

Materials Sample # pore size Nominal Measurement Measurement (core only) Sample #1 10 ppi 5% 5.7% 4.6% Sample #2 10 ppi 10% 14.3% 12.5%

FeCrAlY Sample #3 30 ppi 5% 6.1% 4.1% Sample #4 30 ppi 10% 10.2% 9.3% Sample #5 60 ppi 5% 9.0% 5.5% Sample #6 60 ppi 10% 13.7% 9.2% Sample #7 30 ppi 7.5% 8.1% 5.4% Sample #9 10 ppi 5% 6.7% 7.44% Sample #10 10 ppi 10% 8.2% 11.5% Sample #11 30 ppi 5% 4.4% 6.0%

Copper Sample #12 30 ppi 10% 10.0% 11.9% Sample #13 60 ppi 5% 5.7% 7.3% Sample #14 60 ppi 10% 9.5% 8.5%

Table 2.6 Measured relative density for all samples

Note that substantial discrepancy exists between the two relative densities for a given

sample. Interestingly, the first measurement is consistently larger than the second

measurement for FeCrAlY samples, whereas the reverse holds for copper samples.

This indicates that whilst there is a high concentration of brazing material near the

foam-skin interface for FeCrAlY samples, the residues of the brazing material have

migrated away from the interface to the central portion of the foam core in copper

24

samples. For subsequent calculations, only the first measurement will be used to

represent the relative density of each sample.

The above study on microstructures has led to a better physical picture of the Porvair

metal foams. Table 2.7 summarizes all the results on microstructural parameters for

both FeCrAly and copper samples.

Properties FeCrAlY S-1 S-2 S-3 S-4 S-5 S-6 S-7

Pore size (ppi) 10 10 30 30 60 60 30 Nominal relative density (%) 5 10 5 10 5 10 7.5 measured relative density (%) 5.7 14.3 6.1 10.2 9.0 13.7 8.1 measured relative density (%)-core only 4.6 12.5 4.1 9.3 5.5 9.2 5.4 Nominal cell size (mm) 2.54 2.54 0.847 0.847 0.423 0.423 0.847 Measured cell size (mm) 3.131 3.109 1.999 2.089 0.975 0.959 1.998 Hole diameter (µm) 161 154 110 107 43.5 41.6 110 Fibre diameter (µm)

287 351 215 267 124 154 241 Ratio (%)

56 44 51 40 35 27 46

Properties Copper S-9 S-10 S-11 S-12 S-13 S-14

Pore size (ppi) 10 10 30 30 60 60 Nominal relative density (%) 5 10 5 10 5 10 measured relative density (%) 6.7 8.2 4.4 10.0 5.7 9.5 measured relative density (%)-core only 7.44 11.5 6.0 11.9 7.3 8.5 Nominal cell size (mm) 2.54 2.54 0.847 0.847 0.423 0.423 Measured cell size (mm) 2.645 2.697 1.284 1.431 0.554 0.657 Hole diameter (µm)

n/a n/a n/a n/a n/a n/a strut diameter(µm)

263 270 122 127 88.8 93.2 ratio (%)

n/a n/a n/a n/a n/a n/a

Table 2.7 Summary of measured microstructures of all foam samples

25

2.3.3 Verification of the relationship among the microscopic parameters

From the measured relative densities (ρr), diameter ratio (r) and fibre diameters (df) of

the seven FeCrAlY and six copper samples, the pore size (dp) can be calculated by

using Eqs. (2.4) and (2.5). The comparison between the calculation and test data is

shown in Fig. 2.10.

0 0.1 0.2 0.3 0.40

1

2

3

4

measured valueEq. (3.4)

Fig. 2.10 Comparison of measured pore size and that calculated from Eq. (2.4)

From Fig. 2.10, it can be seen that there is certaindeviation between the measured and

calculated pore sizes, but the deviation is considered reasonable given the complexity

of the foam microstructures and the uncertainty of the measurements.

df

dp

26

2.4 Experimental study for heat transfer in Porvair metal foams

The pressure drop and heat transfer of the seven FeCrAlY foams and six copper

samples listed in Table 2.7 are measured in this section. The foam shown in Fig. 2.11

was sandwiched between two 1 mm thick copper plates using nickel based brazing,

and was subsequently trimmed to fit into the test section of a heat sink channel of size

0.127 m (W)× 0.127 m (L) × 0.012 m (H).

Fig. 2.11 Sandwiched foam sample ready for heat transfer experiment

2.4.1 Experimental procedures and data acquisition

The experimental apparatus consists of four sections: coolant supplier, test section,

flow channel, and data acquisition system. Air is used as a coolant and is forced

through the channel inlet by a suction type air blower. One wire screen and one

honeycomb are inserted in the channel before its cross-section starts to contract. The

coolant then flows through a 9:1 contraction section and a flow developing channel

region to ensure that the flow is hydraulically fully-developed when it reaches the test

section. As shown in Fig. 2.12, the foam sample is encapsulated by two acrylic

sidewalls. A flow rate regulator is located between the exit of the test section and the

suction device.

27

Figure 2.12 Photograph of test rig for pressure drop and heat transfer experiment

For pressure drop measurements, 4 static pressure taps of inner diameter 0.68 mm

were placed on each sample along the flow direction on the upper cover plate with a

uniform spacing of 25 mm. A Scanivalve is calibrated with a digital micromanometer

before performing measurements.

For heat transfer experiments, an asymmetrical isoflux (constant wall heat flux)

boundary condition was imposed on the lower copper skin by a heating element

(silicone-rubber etched foil from Watlow™ Inc.). The amount of heat released from

the heating element was adjusted by changing the voltage of V6HMTF Zenith™ AC

variac, which was monitored by a Fluke 73 series II digital multimeter. The variac

controls the heating element with a voltage range of 0 – 240 V, corresponding to a

heat input of 0 – 150 W. For this investigation, input voltages of 120 and 168 V were

used, generating 75 W of heat (or 4000 W/m2 heat flux intensity) and 105 W (or 8000

W/m2). The external surface of the heating element was covered by a 45mm thick

Tancast 8™ thermal insulation material to minimize heat loss from direct contact.

Because the electrical resistance foil was etched in a zigzag pattern on the silicone

rubber pads, there is a possible gap (unspecified by the manufacturer) between two

columns of the foil. Pure copper heat spreader plate (0.9 mm) was inserted between

the heating element and the metal foam skin to ensure uniformity of heat flux entering

the heat sink. Four thin foil (0.05 mm thickness) T-type copper-constantan

thermocouples (from Rhopoint Inc.) were installed on the lower copper plate along

the longitudinal direction (i.e. the flow direction). There were two additional T-type

thermocouples, positioned separately at the inlet and the outlet of the test section to

measure the coolant temperature at each location. A temperature scanner with reading

28

resolution of ± 0.1 K was used to measure the thermocouple output, with the

capability to record temperature from all thermocouples simultaneously.

All measurements were performed under steady state conditions, and it usually takes

10-40 minutes to reach steady state after each change of the Reynolds number. A Pitot

tube was positioned before the test section to measure stagnation pressure and static

pressure at the inlet of the test section. Because the blockage ratio, i.e., the ratio of

channel height (12 mm) to the tube outer diameter (0.51 mm), is 23.5, wall

interference to the Pitot tube is expected to be negligible.

2.4.2 Measurement uncertainties

An uncertainty analysis is performed following the method of Kline and McClintock

[32]. The maximum heat loss through the insulation materials is estimated to be less

than 2% (2.8W) of the 150W heat input. The heat loss through the side-walls is

assumed to be negligible due to the small conduction area. Here, only random errors

are considered for heat input, temperature and pressure measurements; analysis of

systematic errors has not been conducted due to the shortage of relevant information

and its complexity. The thermal conductivity kf of aire varies slightly in the operating

temperature range of 300.0 to 350.0 K. An arithmetic mean value is used for kf, with

uncertainty estimated to be within 6.6%. From these, the uncertainty in the measured

heat transfer coefficient and Nusslet number is estimated to be less than 7.0% and

9.6%, whilst the uncertainety in the pressure drop and fiction factor measurements is

estimated to be less than 5.0% and 7.8%, respectively.

2.4.3 Experimental results and analysis

The pressure drop and heat transfer results will be presented in this section.

Pressure drop

The measured results of pressure drop are plotted in Fig. 2.13 and Fig. 2.14 for

FeCrAlY and copper samples, respectively, as a function of the mean flow velocity

Um of air.

29

2 4 6 8 10 12 14 160

20,000

40,000

60,000

80,000

100,000

120,000

140,000

S - 7

S - 2

S - 1

S - 6

S - 5

S - 4

S - 3

FeCrAlY samples

Fig. 2.13 Pressure drop results for all FeCrAlY samples

0 2 4 6 8 10 120

20,000

40,000

60,000

80,000

100,000

120,000

140,000

S - 11

S - 10

S - 9

S - 12

S - 13

S - 14

Copper samples

Fig. 2.14 Pressure drop results for all copper samples

)m

Pa(

dL

dP

Um (m/s)

m

Pa

dL

dP

Um (m/s)

30

The pressure drop increases with relative density, while decreases with pore size (dp).

It can also be noted that the pressure drop of a copper sample is in general much

higher than that of a FeCrAlY sample with identical nominal relative density and ppi.

This may be attributed to the much irregular and smaller cell size of the copper

sample compared to the FeCrAlY sample.

10,0001

10

100

S - 1S - 2S - 3S - 4S - 5S - 6S - 7

500

3000 5000 20000

FeCrAlY

Fig. 2.15 Friction factors of all FeCrAlY samples

1,000 10,000

10

100

S - 11

S - 10

S - 9

S - 12

S - 13

S - 14200

500

50

20

2000 5000 20000

Copper

Fig. 2.16 Friction factors of all copper samples

f

ReDh

f

ReDh

31

From the pressure drop measurements, the friction factor for each sample, defined as

fP

L

D

Uh

m

=∆

ρ 2 2/, can be calculated. The results are shown in Fig. 2.15 and Fig. 2.16

for FeCrAlY and copper samples, respectively. It can be seen that the friction factor

approach to a constant for each sample, as expected in turbulence flows at relatively

high Reynolds numbers.

Heat Transfer

Skin temperature TW(x) measured from thermocouples placed on the foam skin plates

was recorded. The temperature distribution for one case (Sample #4, FeCrAlY with

30 ppi and 10% relative density) at an input heat flux q of 8000 W/m2 is plotted in

Fig. 2.17 for selected Reynolds numbers. The typical linear variation of wall surface

temperature along the flow direction is observed for fully developed convection with

isoflux boundary condition.

x [m]

TW

(x)

[K]

0 0.02 0.04 0.06 0.08 0.1 0.12300

310

320

330

340

350

360

370

380

ReDh = 3360ReDh = 5450ReDh = 7590ReDh = 10500ReDh = 12500ReDh = 14270

Fig. 2.17 Substrate surface temperature distribution along the flow direction for FeCrAlY S-4 (q = 8000 W/m2).

32

The thermal performance of Porvair metal foams as a heat sink medium can be

assessed by calculating the overall Nusselt number defined as follows:

∫=L

dxxNuL

Nu0

)(1

(2.8)

where f

h

inw k

D

TxT

qxNu

−=

)()( (2.9)

Here, q is the applied heat flux, TW is the local substrate temperature, Tin is the coolant

inlet temperature, kf is the thermal conductivity of the coolant, and Dh is the hydraulic

diameter of the heat sink channel.

The overall Nusselt numbers for seven FeCrAlY samples are summarized in Fig.

2.18.

10,000

100

S - 1S - 2S - 3S - 4S - 5S - 6S - 7

3,000 2,00005,000

200

300

80

400

500

FeCrAlY

Fig. 2.18 Overall Nusselt numbers of all FeCrAlY samples

Sample #1 (10 ppi, 5% nominal relative density) has the lowest value of the overall

Nusselt number, while sample #2 (10 ppi, 10% nominal relative density) achieves the

highest value. At a fixed nominal cell size for FeCrAlY samples, the effect of relative

density on the overall Nusselt numbers is shown in Figs. 2.19 to 2.21 for 10ppi, 30ppi

and 60 ppi, respectively. Similarly, at a given nominal relative density, the effect of

cell size on the overall heat transfer is presented in Figs. 2.22 and 2.23 for 5% and 10

% nominal relative densities, respectively.

Nu

33

10,000

100

S-2, 10% relative density(14.3% measured)


200

300

3,000 5,000 2,0000

10 ppi

400

500

FeCrAlY

Fig. 2.19 Effect of relative density on overall Nusselt number for a fixed cell size (10 ppi)

10,000

100

S-4, 10%(10.2% measured)


200

300

3,000 5,000 2,0000

S-7, 7.5%(8.1% measured)

30 ppi

FeCrAlY

Fig. 2.20 Effect of relative density on overall Nusselt numbers for a fixed cell size (30 ppi)

Nu

Nu

ReDh

ReDh ReDh

34

10,000

100



200

300

3,000 5,000 2,0000

60 ppi

FeCrAlY


10,000

100

3,000 2,00005,000

200

300

70

Nominal relative density = 5%

60 ppi(9.0% measured)


10 ppi (5.7% measured)

FeCrAlY

Fig. 2.22 Effect of cell size on overall Nusselt numbers for a fixed nominal relative density (5%)

Nu

ReDh

Nu

ReDh

35

10,000100

3,000 2,00005,000

200

300





400

500

FeCrAlY

Fig. 2.23 Effect of cell size on overall Nusselt numbers

for a fixed nominal relative density (10%)

From the above results, it can be seen that the overall heat transfer in general

increases with increasing relative density and decreasing cell size. but the Fig. 2.23

doesn’t show the same situation. However, Fig. 2.23 suggests that increasing the

relative density is more important than reducing the cell size for heat transfer

enhancement. In other words, variations in the cell size influence the pressure drop

more than heat transfer, whilst heat transfer is more sensitive to the variations in the

relative density.

The overall Nusselt numbers for six copper samples are presented in Fig. 2.24. As for

the FeCrAlY samples, the effects of relative density on the overall heat transfer of

copper samples are shown in Figs. 2.25 to 2.27, and effects of cell size are given in

Fig. 2.28 and Fig. 2.29.

Nu

ReDh

36

1,000 10,000

100

S-9 S-10 S-11S-12 S-13 S-14

200

2,000 6,000 2,00004,000

300

400

500

600

50

Copper

Fig. 2.24 Overall Nusselt numbers of all copper samples

1,000 10,000100

200

2,000 6,000 2,00004,000

300

400

500

600

10 ppi

Copper




Nu

ReDh

Nu

ReDh

37

1,000 10,000

100

200

2,000 6,000 2,00004,000

300

400

500

600

50

30 ppi

Copper

S-12, 10% relative density(10% measured) S-11, 5% relative density

(4.4% measured)


1,000

100

200

2,000 6,0004,000

300

400

500

508,000

60 ppi

Copper




Nu

ReDh

Nu

ReDh

38

1,000 10,000

100

200

2,000 6,000 2,00004,000

300

400

500

600

50

Nominal Relative density = 5%

Copper





1,000 10,000

100

200

2,000 6,000 2,00004,000

300

400

500

600

50


Copper


30 ppi(10% measured)



Nu

ReDh

Nu

ReDh

39

From these results, it can be seen that the overall heat transfer of a copper sample is

not as sensitive to the relative density as that for a FeCrAlY samples, whereas the cell

size effect on heat transfer of the copper sample is a bit more significant than that for

the FeCrAlY sample. Again, smaller cell sizes (at a fixed relative density) lead to

higher overall heat transfer. Some explanations will be given below.

For FeCrAlY samples, the solid thermal conductivity (ks) is around 20 W/mK, so the

thermal resistance at the solid side is large. Consequently, increasing the relative

density of these foams can cause significant reduction of the thermal resistance in the

solid part, resulting in a strong effect on the overall heat transfer. However, for copper

samples, the thermal conductivity (ks) is approximately 300 W/mK, implying that the

thermal resistance in the solid part is relatively small and the resistance in fluid part is

relatively large. Therefore, increasing the relative density of copper sample would not

lead to a dramatic effect on heat transfer, whereas reducing the cell size could to a

certain extent enhance overall heat transfer duo to the reduction of thermal resistance

in the fluid part. Fig. 2.30 compares the heat transfer between FeCrAlY and copper

samples having the same measured relative density (10%) and ppi (30). It is seen that

the slope of the Nu versus ReDh curve for the copper sample is larger than that for

the corresponding FeCrAlY sample, confirming the above explanation.

1,000 10,000

100

S-4, FeCrAlY(10.2% measured)

200

300

2,000 5,000 2,0000

30 ppi, 10% relative density

S-12, Copper(10% measured)

400

500

Fig. 2.30 Comparison of overall Nusselt number for FeCrAlY and copper samples

Nu

ReDh

40

At a given Reynolds number, the Nusselt numbers of copper samples are

approximately 2 to 4 times those of FeCrAlY samples. For both types of foam,

increasing the relative density and decreasing the cell size will lead to heat transfer

enhancement. However, it should be pointed out that the increase in pressure drop

will become even larger as a consequence of the heat transfer enhancement. To give

an overall assessment, it is often necessary to introduce a nondimensional efficiency

index, j Nu f≡% , which represents the ratio between heat transfer and flow

resistance. Fig. 2.31 and Fig. 2.32 plot j~

as a function of the Reynolds number for

FeCrAlY and copper samples, respectively. Both figures indicate that the j~ number

decreases while increasing relative density and decreasing cell size, confirming that

pressure drop increases at a faster pace than heat transfer does. The comparison

between FeCrAlY and copper samples are given in Fig. 2.33. Note that sample 1

(FeCrAlY) and sample 9 (copper) achieve the highest efficiency while sample 6

(FeCrAlY) exhibits the lowest efficiency. On the whole, there is no major difference

between FeCrAlY and copper samples with similar relative density and pore size.

0 4,000 8,000 12,000 16,0000

5

10

15

20

25

30

S - 1S - 2S - 3S - 4S - 5S - 6

Fig. 2.31 The efficiency numbers for FeCrAlY samples

j~

ReDh

41

0 4,000 8,000 12,000 16,0000

5

10

15

20

25

S - 9S - 10S - 11S - 12S - 13S - 14

Fig. 2.32 The efficiency numbers for copper samples

0 4,000 8,000 12,000 16,0000

5

10

15

20

25

30

S - 1S - 2S - 3S - 4S - 5S - 6S - 9S - 10S - 11S - 12S - 13S - 14

Fig. 2.33 The comparison of the efficiency numbers

between FeCrAlY and copper samples

j~

ReDh

j~

ReDh

42

2.4.4 Product uncertainties

Up until now the experiments on forced convection have been conducted for seven

FeCrAlY samples and six copper samples. It is noted that the measured

microstructures are quite different from the industrial specifications. It would also be

interesting to compare the performance of two FeCrAlY samples (S-7 and S-8) with

identical industrial specifications, i.e., 7.5% relative density and 30 ppi, but

manufactured from two different batches. With the effect of brazing, the measured

relative densities are 8.1% and 8.9% for sample 7 and sample 8, respectively. Because

of the difference in relative densities, their pore sizes will also be different (although

this is yet to be measured). As a result, the measured pressure drop and heat transfer

are somewhat different for the two samples, as shown in Fig. 2.34 and Fig. 2.35.

Therefore, how to improve the product quality and reduce the variability from

different batches is a problem to be addressed.

2 3 4 5 6 7 8 9 100

10,000

20,000

30,000

40,000

50,000

60,000

S - 7

S - 8

FeCrAlY samples

Fig. 2.34 Comparison of pressure drop results between samples 7 and 8.

)m

Pa(

dL

dP

U (m/s)

43

3,000 6,000 9,000 12,000 15,0000

100

200

300

400

S-7, 7.5%(8.1% measured)

30 ppi

FeCrAlYS - 8, 7.5%(8.9% measured)

Fig. 2.35 Comparison of overall Nusselt numbers between samples 7 and 8 The experimental data of pressure drop, Nusselt number and efficiency index for all

14 Porvair metal foam samples are listed in Appendix B.

Nu

ReDh ReDh

44

2.5 Numerical modelling on forced convection in Porvair metal foams

In this section the modelling and numerical simulation will be conducted for Porvair

metal foams. With the measured relative density, pore size and strut hollowness as

input, the predicted heat transfer performance will be compared with test data to

check the validity of the numerical model.

2.5.1 Mathematical formulations

The problem under investigation is forced convection of incompressible fluid flow

through open-celled metal foams. As previously discussed, the complexity of the

cellular structure usually precludes a detailed microscopic investigation of the

transport phenomena at the pore level in porous media. Therefore, the general

transport equations are commonly integrated over a representative elementary

volume, which accommodates the fluid and the solid phases within a porous structure.

There are two approaches available in applying the volume-averaging technique for

heat transfer investigations: one is averaging over a representative elementary volume

containing both the fluid and the solid phases, and the other is averaging separately

over each of the phases, thus resulting in a separate energy equation for each

individual phase. These two models are referred to as the one-equation model and the

two-equation model, respectively. The one-equation model is valid when the local

temperature difference between the fluid and solid phases is negligibly small. Because

the temperature difference between the solid and fluid phases cannot be neglected in

metal foams with air as the coolant fluid, the two-equation model will be used below.

The geometry of the compact heat sink with metal foams is depicted in Fig. 2.11 and

Fig. 2.12. For simplicity, the width of the channel is assumed to be sufficiently long

such that the problem can be considered as two dimensional, and the other

assumptions upon which the numerical model is based are summarized as follows:

(1) The medium is homogeneous and isotropic.

(2) Forced convection dominates in the metal foams, i.e., natural convection

effects are negligible.

(3) Variation of the thermophysical properties with temperature is ignored.

(4) Due to the relatively low operating temperature (< 100C) considered in the

present study, radiation heat transfer is neglected.

45

(5) Fluid flow and heat transfer reach steady state in the channel.

Under these assumptions, the governing equations for the velocity and temperature

fields in the metal foam material are established by applying the volume-averaging

technique. The extended Darcy equation proposed by Vafai and Tien [3,4] is used in

place of the Darcy equation in order to account for the boundary and inertial effects.

The governing equations and boundary conditions are

Continuity equation

0=•∇ V (2.10)

Momentum equation

( ) [ ]JFK

P Ifff

f

f VVVVVV •−−∇+−∇=∇• ρµ

ε

µ

ε

ρ 2 (2.11)

The last term in Eq. (2.11) was first introduced by Forchheimer to account for the

inertial effects (non-Darcy flow). Similarly, the second term accounts for the

boundary effects on the velocity distribution, and was first introduced by Brinkman.

Solid phase energy equation

( ){ } ( )fssfsdse TTahTkk −−∇•+•∇= ~0 (2.12)

Fluid phase energy equation

( ){ } ( )fssffdfefffTTahTkkTC −+∇•+•∇=∇• ~Vρ (2.13)

where means a volume-averaged value and fefffsfse kCahTk and ,,,~,,, µρ are

effective thermal conductivity of the solid, temperature, interfacial heat transfer

coefficient, wetted area per volume, density, viscosity, heat capacity and effective

thermal conductivity of fluid, respectively; kd is the thermal dispersion conductivity,

K is the permeability of the porous medium, ε is the porosity, V is the velocity vector,

and PPJ VV /= is the unit vector aligned along the pore velocity vector, Vp; FI is the

inertial variable with unit m-1, which depends on the microstructure of the porous

medium. If the Reynolds number is small such that laminar flow prevails,

KCF II /= , where CI is the inertial coefficient.

An order of magnitude analysis on the momentum equation shows that the momentum

boundary layer thickness is of the order of ( ) 2/1/εK and that the convective term

( )VV ∇• responsible for boundary layer growth is significant only over a length of

46

the order of ( )ν/cKu [3,9]. Therefore, a fully developed momentum boundary layer

is in force beyond a very short developing length. In the present study, the fully

developed velocity field is assumed, and hence the momentum equation (2.11) and

energy equations (2.12) and (2.13) can be simplified as

220 uFuK

uP Ifff

f ρµ

ε

µ−−∇+−∇= (2.14)

( )fssfs

ses

se TTahy

Tk

yx

Tk

x−−

∂∂

∂∂

+

∂∂

∂∂

= ~0 (2.15)

( ) ( ) ( )fssff

dfef

dfef

f TTahy

Tkk

yx

Tkk

xx

TuC −+

∂

∂+

∂∂

+

∂

∂+

∂∂

=∂

∂ ~ρ (2.16)

For brevity, the bracket has been dropped in these equations.

2.5.2 Boundary conditions

When a heat flux is directly applied to the outer surface of the metal foam, the applied

heat is transferred to the solid and fluid phases by conduction and convection. As

discussed in references [33, 34], the wall heat flux boundary condition may be viewed

in two different ways. The first is to assume that each representative elementary

volume at the wall surface receives a prescribed heat flux that is equal to the wall heat

flux qw. As a result, the heat will be divided between the two phases on the basis of

the physical values of their effective conductivities and their corresponding

temperature gradients. The second approach is to assume that each of the individual

phases at the wall surface will receive an equal amount of heat flux qw [33].

In the present study, a 1mm copper substrate with high thermal conductivity is

attached to the metal foam as shown in Fig. 2.11, and the heat flux is applied to the

external wall of the substrate instead of being applied directly to the outer surface of

the metal foam. In this case, the temperature at the interface between the metal foam

and the copper substrate can be considered to be uniform regardless of whether it is in

contact with the solid or fluid phase due to the high thermal conductivity of the

copper cover plate. Consequently, the boundary condition at the heating side of the

channel can be written as

WHysHyf TTT ≅≅==

(2.17)

47

where Tw implies the temperature at the interface. This temperature is not known a

priori and must be obtained as part of the solution.

Finally, the boundary conditions are specified as follows,

00 ==∂∂

+∂

∂==

y

sse

y

ffew y

Tk

y

Tkqq and fs TT = at y = 0;

0=q at y = H (2.18)

inf TT = and 0=∂∂

x

Ts at x = 0

0=∂∂

x

Ts and 0=∂

∂

x

T f at x = L

where H and L are the height and length of the channel, respectively.

2.5.3 Modelling on Porvair metal foams

Before proceeding further, the permeability K and inertial variable FI appearing in the

momentum equation (2.14) must be known in order to calculate the velocity field. The

permeability K and inertial variable FI of a metal foam have been investigated by

several researchers [19,28,31,35], although only Calmidi [31] gave specific

formulations of K and FI based on experimental data. In this study, the formulations

proposed by Calmidi [31] will be used, with

( )11.1

224.0

2100073.0

−

−

−=

p

f

p d

d

d

Kε (2.19)

( ) Kd

dF

p

fI /100212.0

63.1

132.0

−

−

−= ε (2.20)

Similarly, in energy equations (2.15) and (2.16), the effective conductivity kse, kfe and

the dispersion conductivity kd need to be determined in order to close the equations.

An analytical model for the effective conductivity of open-celled metal foams based

on the three-dimensional cellular morphology has recently been proposed by

Boomsma and Poulikakos [29], yielding

( )DCBAeff RRRR

k+++

=2

2 (2.21)

where

48

( )( ) ( )( ) fs

A keekeeR

−−−+−+=

12412

422 πλπλ

λ (2.22a)

( )( ) ( )( ) fs

Bkeeekee

eR

22

2

2422

2

λλλλ

−−−+−−

= (2.22b)

( )( ) ( )( ) fs

Ckeeke

eR

2212222212

2222

2

−−−+−−

=πλπλ

(2.22c)

( ) fs

D keke

eR

22 4

2

−+= (2.22d)

with ( )( )

( )ee

e

−−−−

=243

228/522 3

πε

λ , e = 0.339 (2.23)

Thus, the effective solid conductivity, kse, is obtained by setting kf = 0 in (2.21) and

(2.22). Similarly, the fluid conductivity, kfe, is obtained by setting ks = 0 in both

equations. It should be noted that the above effective conductivity model is developed

for metal foams having solid struts (e.g., ERG foams). For Porvair foams, the cell

ligaments are typically hollow, and hence the porosity and effective solid conductivity

need to be changed accordingly. If the Porvair foams has the same relative density as

the ERG material, then its porosity can be calculated according to:

2

2

1 r

rporvair −

−=

εε (2.24)

where r is the inner and out diameter ratio of the hollow struts in Porvair foams.

Similarly, the effective solid conductivity of a Porvair foam should be modified as

( )2, 1 rkk seporvairse −= (2.25)

As for the dispersion conductivity, kd, a widely adopted expression is [26],

( ) em

ekDd ku

uCk PrRe= (2.26)

where ν

Kumk =Re , the Reynolds number based on permeability,

e

fe k

Cµ=Pr is the

Prandtl number based on effective conductivity, um is the average flow velocity

entering the foam sample, and CD ( 0.1≈ ) is the thermal dispersion coefficient.

Prior to numerical calculations, the surface area density, a~ , and interstitial heat

transfer coefficient, hsf, need to be known for the porous medium. The solid-fluid

49

interfacial surface area for arrays of parallel cylinders intersecting in three mutually

perpendicular directions (Fig. 2.7) is

2

3~

p

f

d

da

π= (2.27)

For metal foams, this expression is modified by taking the 3D microstructure into

account (i. e., open cells shaped like dodecahedra, and noncircular fiber cross-

sections). By doing so, the pore size dp and solid strut diameter, df, are multiplied by a

factor 0.59 and ( )( )04.0/11 ε−−− e , respectively [31]. As for the interstitial heat transfer

coefficient, hsf, Wakao et al. [10] proposed one of the most comprehensive models for

packed beds. For foamed materials, however, no such general model exists. Here,

based on a correlation developed by Zukauskas [36] for staggered cylinders in

crossflow, the following correlation [26] is employed in the present study,

ffsf dkh /PrRe52.0 37.05.0= (2.28)

where, ( )ενfud=Re .

2.5.4 Numerical procedure

Because a fully developed velocity condition is assumed at the cross-section, the

momentum equation (2.14) becomes a second-order nonlinear ODE, and its solution

can be numerically calculated. The iteration will be terminated when the condition

that the integral of the non-dimensional velocity over the height of the channel is

unity within a specified error of 10-5.

The energy equations were solved by using the ADI finite difference scheme, with

127 46× uniform grid spacings used in the x and y directions, respectively. The

convergence criterion was that the change in the solid-phase and fluid-phase

temperatures was less than 10-5 between successive iterations.

2.5.5 Code validation

In order to validate the numerical code, the heat transfer performance of a ERG metal

foam is calculated and compared with the experimental data obtained by Calmidi and

Mahajan [22]. In their experiment, the boundary condition was same as in the present

study, and the permeability and effective solid conductivity kse were given. The given

values of K and kse are directly input the numerical programme. The predicted

50

velocity distribution and Nusselt number for the ERG foam tested in [22] are shown

in Fig. 2.36 and Fig. 2.37, respectively. The agreement between prediction and

experimental data [26] is good (Fig. 2.37).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

Rek = 140

20

80

Fig. 2.36 Fully developed velocity distribution along the vertical direction of a ERG foam channel

0 20 40 60 80 100 120 1400

5

10

15

20

Experimental data [26]Numerical result

Fig. 2.37 Comparison between numerical calculation and experimental data [26]

for a ERG foam channel

Nu

Rek

u

Y

51

2.5.6 Numerical results for Porvair metal foams

For Porvair metal foam sample 1, Fig. 2.38 shows the solid and fluid temperature

distributions at the cross-section of X = 0.5. It can be seen that the temperature

difference between the solid and fluid cannot be neglected, especially for smaller

Reynolds numbers and small values of Y (i.e., near the thermal boundary zone).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

40

60

80

100

120

140

160

Sample 1

X = 0.5

Fig. 2.38 Solid and fluid temperature distributions

along the vertical direction at X = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

30

40

50

60

70

80

90

T

T f,b

s,b

T s,b

Tf,b

Re = 2000 Dh

Re = 10000Dh

Sample 1

Fig. 2.39 Averaged solid and fluid temperature distributions along the streamwise direction

T

Y

ReDh = 2000

10000

Ts Tf

T

X

52

The streamwise distributions of solid and fluid temperatures averaged over the cross-

section of the channel, ,s bT and ,f bT , are presented in Fig. 2.39 for sample 1. As

expected from a constant heat flux boundary condition, the solid and fluid

temperature lines are parallel to each other. The variation of local Nusselt number,

f

h

bfw

wbf k

D

TT

qNu

,, −

= , along the streamwise direction is shown in Fig. 2.40 for three

Reynolds numbers.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160

80

100

120

140

160

180

200

220

10000

Re = 2000

6000

Dh

Sample 1

Fig. 2.40 Variation of local Nusselt number along streamwise direction

From Fig. 2.40, it is seen that the effect of thermal entrance is significant if X is small

(< 0.2), and the thermal entrance length becomes longer with increasing Reynolds

number.

The calculated results for the overall Nusselt number ( Nu ) are shown in Figs. 2.41 to

2.51 for all FeCrAlY and copper foam samples and compared with the corresponding

experimental data. In these figures, the solid lines represent predictions based on the

solid conductivity ks = 16 W/mK for FeCrAlY samples and ks = 372 W/mK for

copper samples. Given the complexity of heat transfer in metal foams, the predictions

appear to be in reasonable agreement with measured values, although the predictions

somewhat underestimate the Nusselt numbers for FeCrAlY samples and overestimate

the overall heat transfer rate for copper samples. The reason may be attributed to the

bfNu ,

x

53

effects of nickel based brazing on the solid conductivity ks. Brazing tend to increase

the solid conductivity for FeCrAlY samples and decrease the solid thermal

conductivity for copper samples. In the aforementioned figures, the dashed lines

represent the predicted results by simply changing the solid conductivity ks for both

FeCrAlY and copper. It was found that the solid conductivity ks ranging from 16 ~ 26

W/mK for different FeCrAlY samples and 200 ~ 310 W/mK for different copper

samples led to improved predictions in comparison with test data.

3,000 5,000 7,000 9,000 11,000 13,000 15,0000

100

200

300

400

500



10 ppi

FeCrAlY

k = 20 W/mKs

k = 26s

Fig. 2.41 Nusselt numbers for sample 1 and sample 2: predictions versus experiments

3,000 6,000 9,000 12,000 15,0000

100

200

300


30 ppi

FeCrAlY

k = 26 W/mKs

Fig. 2.42 Nusselt numbers for sample 3: predictions versus experiments

Nu

ReDh

Nu

ReDh

54

3,000 5,000 7,000 9,000 11,000 13,0000

100

200

300

S-4, 10%(10.2% measured)

30 ppi

FeCrAlY

k = 20 W/mKs

Fig. 2.43 Nusselt number for sample 4: prediction versus experiment

3,000 5,000 7,000 9,000 11,000 13,0000

50

100

150

200

250

300



60 ppi

FeCrAlY

k = 18 W/mKs

Fig. 2.44 Nusselt numbers for sample 5 and sample 6: predictions versus experiments

Nu

ReDh ReDh

Nu

ReDh

55

3,000 6,000 9,000 12,000 15,0000

100

200

300

S-7, 7.5%(8.1% measured)

30 ppi

FeCrAlY

k = 23 W/mKs


2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,0000

200

400

600

800

10 ppi

Copper


k = 310 W/mKs

Fig. 2.46 Nusselt numbers for sample 9: prediction versus experiment

Nu

ReDh

Nu

ReDh ReDh

56

2,000 4,000 6,000 8,000 10,000 12,000 14,0000

200

400

600

800

10 ppi

Copper


k = 272 W/mKs

Fig. 2.47 Nusselt numbers for sample 10: prediction versus experiment

1,000 3,000 5,000 7,000 9,000 11,0000

200

400

600

800

30 ppi

Copper


k = 300 W/mKs


Nu

ReDh

Nu

ReDh

57

1,000 3,000 5,000 7,000 9,0000

200

400

600

800

30 ppi

Copper

S-12, 10% relative density(10% measured)

k = 200 W/mKs


1,000 2,000 3,000 4,000 5,000 6,0000

100

200

300

400

500

600

700

60 ppi

Copper S-13, 5% relative density(5.7% measured)

k = 320 w/mKs


Nu

ReDh

Nu

ReDh

58

1,000 2,000 3,000 4,000 5,000 6,0000

100

200

300

400

500

600

700

60 ppi

CopperS-14, 10% relative density(9.5% measured)

k = 230 w/mKs


2.5.7 Effect of boundary conditions

The numerical calculations thus far have been conducted for the constant heat flux

boundary condition. In this section, the effect of changing the boundary condition to

the constant wall temperature condition upon heat transfer will be examined briefly.

For constant wall temperature, the boundary will become more direct and easy to

implement in the numerical programme. The predicted local and overall Nusselt

numbers for sample 1 are shown in Fig. 2.52 and Fig. 2.53, respectively, together with

the results based on the constant heat flux condition. Note that the boundary effect is

small, as expected. In general, the heat removal capability of the foam with constant

heat flux is 5 ~ 6% larger than that associated with the constant wall temperature

boundary condition.

Nu

ReDh

59

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 180

100

120

140

160

180

200

Re = 6000Dh

Sample 1

constant heat flux

constant wall temperature

Fig. 2.52 Effect of boundary conditions on local heat transfer in sample 1

2,000 4,000 6,000 8,000 10,000 12,00040

60

80

100

120

140

160

180

constant heat flux

constant wall temperature

Sample 1

Fig. 2.53 Effect of boundary conditions on overall heat transfer in sample 1

bfNu ,

x

Nu

ReDh

60

2.5.8 Effects of microstructural parameters: Optimisation

Solid thermal conductivity (ks)

From Eqs. (2.21) to (2.25), it is known that the solid conductivity (ks) will directly

affect the effective solid conductivity (kse) of a metal foam, and subsequently affect its

heat transfer behavior. Fig. 2.54 presents the variations of the effective solid

conductivity with the solid material conductivity for a fixed foam porosity 0.95ε = .

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

ε = 0.95ε = 0.95

Fig. 2.54 Linear relationship between ks and kse

The effect of solid conductivity ks on the overall Nusselt number is shown in Fig. 2.55

for a fixed microstructure (sample 4) and three values of the Reynolds number. The

results all reveal that the overall Nusselt number increases sharply when ks is small,

and then gradually approaches a plateau with increasing ks. However, it should be

noted that, for small Reynolds numbers ( Re 1000= ), heat transfer saturation occurs at

small thermal conductivity levels (ks ≈ 50 W/mK), while for larger Reynolds numbers

Re = 4000, the overall heat transfer reaches the saturation stage when ks ≈ 200

w/mK. This implies that the main thermal resistance for small Reynolds numbers lies

on the fluid side when the solid thermal conductivity ks exceeds a critical value (≈ 50).

Here, we define this value as max,sk , beyond which the overall heat transfer is

independent of the solid thermal conductivity. For higher Reynolds number, max,sk

kse

ks

61

becomes larger because the thermal resistance in the fluid side decreases.

Consequently, in practical applications, there is no need to use foam materials with

high thermal conductivities, if the Reynolds number is small. The variation of

max,sk with the Reynolds number for sample 4 is presented in Fig. 2.56.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1,000

Re = 10000Dh

1000

4000

Fig. 2.55 Effect of solid conductivity on overall Nusselt number for sample 4

1,000 4,000 7,000 10,000 13,000 16,0000

100

200

300

400

500

600

700

800

Sample 4 microstructure

Fig. 2.56 Variation of max,sk with Reynolds number for sample 4

Nu

ks

ReDh

max,sk

62

Pore size (dp)

For fixed values of porosity ( 0.95ε = ) and Reynolds number ( Re 10000Dh = ), the

effect of varying pore size (dp) on the overall Nusselt number is presented in Fig. 2.57

for one FeCrAlY sample, but with the assumption that the inner-to-outer diameter

ratio r = 0. The figure shows that the heat transfer increases with decreasing pore size.

0 1 2 3 4 5 6 7 8 9 10100

120

140

160

Re = 10000

ε = 0.95ε = 0.95

FeCrAlY (r = 0)

Dh

Fig. 2.57 Effect of pore size (dp) on overall heat transfer

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

Re = 10000

ε = 0.95ε = 0.95

FeCrAlY (r = 0)

Dh

Fig. 2.58 Effect of pore size (dp) on heat transfer efficiency index

Nu

dp (mm)

j~

dp (mm)

63

However, because the pressure drop is reduced more significantly by increasing the

pore size (dp), the efficiency index ( fNuj /~ = ) still increases with increasing pore

size (dp), as shown in Fig. 2.58.

Porosity (ε)

For simplicity, assume the struts are solid (i.e., the inner-to-outer diameter ratio

0r = ). The relationship between porosity (ε) and relative density (ρr) can then be

simplified as ρr = 1-ε. For a fixed pore size of 4pd = mm and Re 10000Dh = , the

effect of porosity on the overall Nusselt number is shown in Fig. 2.59. The figure

shows that overall heat transfer decreases as the porosity is increased. In other words,

heat removal will be enhanced by increasing the relative density (ρr).

0.88 0.9 0.92 0.94 0.96 0.9840

80

120

160

200

Re = 10000

d = 4 mmp

FeCrAlY (r = 0)

Dh

Fig. 2.59 Effect of porosity (ε) on overall heat transfer

The effects of porosity on the efficiency index ( j~

) are shown in Fig. 2.60 for three

Reynolds numbers, with 0r = and 4pd = mm. For a given Reynolds number, there

exists an optimised porosity, εopt, which maximizes the efficiency index (Fig. 2.60).

This optimised porosity decreases with increasing Reynolds number. The variation of

εopt with the Reynolds number is presented in Fig. 2. 61.

Nu

ε

64

0.75 0.8 0.85 0.9 0.95 10

2

4

6

8

10

12

14

Re = 10000

2000

6000

Dh

d = 4 mmp

FeCrAlY (r = 0)

Fig. 2.60 Effect of the porosity (ε) on efficiency number ( j~

)

0 2,000 4,000 6,000 8,000 10,000 12,0000.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

d = 4 mmp

FeCrAlY (r = 0)

Fig. 2.61 Variations of the εopt with Reynolds numbers

j~

ε

ReDh

optε

65

3 Sintered Particle Bed

3.1 Introduction

Packed particle beds have been widely used in the chemical industry and for energy

storage purposes. Among the common applications are catalytic reactors, absorption

and adsorption operations, as well as packed bed heat storage units. Due to their

intrinsic high surface area ratio, the heat transfer rate will be increased by several

orders of magnitude. Therefore, they are now also being used as fixed-bed nuclear

propulsion systems, spacecraft thermal management systems and heat sinks for high

performance cooling in microelectronics. The forced convection in packed bed has

been extensively investigated [2-18] due to its many important engineering

applications.

Sintered beds with bronze spheres or other materials can now be produced in a large

scale with high quality. A sintered bed can provide much better contact between the

beads than a packed bed, and hence its effective thermal conductivity (kse) could be

dramatically improved. As a result, heat transfer could be significantly enhanced by

using sintering techniques. However, there is little study on forced convection in

sintered beds in comparison with the extensive investigations on packed beds. Hwang

and his co-workers [37,38] conducted experimental and numerical investigations on

the transport of heat in sintered bronze beads. However, in their numerical

calculations, the measured effective conductivity keff was used as the input: no model

was put forward to calculate the effective thermal conductivity for sintered particle

beds. In addition, in their experimental and numerical studies, air was used as the

coolant fluid. Since the porosity (ε) of a sintered bed is relatively small (ε ≈ 0.3 -0.4),

its surface area density should be large and hence the thermal resistance in the solid

part is relatively small compared to that on the fluid side. Consequently, if water is

used as the coolant, the thermal resistance in the fluid part will be reduced

dramatically, resulting in significant heat transfer enhancement. As very few studies

have been conducted for forced water convection in sintered beds, a thorough

investigation combining both modelling and testing is warranted.

66

The objective of this chapter is to build a numerical model, and study the effect of

effective solid conductivity on the overall heat transfer of a sintered bed. Predictions

on forced water convection will be obtained for sintered beds made with bronze beads

having four different diameters.

3.2 Mathematical formulations

The problem under investigation is forced convection of incompressible fluid flow

through the sintered porous channel, as shown in Fig. 3.1.

Fig. 3.1 Forced convection in sintered bronze bed

The width of the channel is assumed to be long enough that the problem will

essentially be two dimensional and a fully developed velocity field is assumed,

because the momentum boundary layer becomes fully developed over a very short

developing length in the porous medium. The governing equations for the velocity

and temperature fields in the sintered bed are established by applying the volume-

averaging technique. The extended Darcy equation proposed by Vafai and Tien [3,4],

is used in place of the Darcy equation in order to account for the boundary effect and

inertial effect. The two-equation heat transfer model [4,6] is employed in the present

analysis. The governing equations and boundary conditions are therefore identical to

those described in Chapter 2, but will be presented below for easy reference:

Momentum equation

220 uK

Cu

KuP Ifff

f

ρµ

ε

µ−−∇+−∇= (3.1)

Solid phase energy equation

( )fssfs

ses

se TTahy

Tk

yx

Tk

x−−

∂∂

∂∂

+

∂∂

∂∂

= ~0 (3.2)

Fluid phase energy equation

qw

H

67

( ) ( ) ( )fssff

dfef

dfef

f TTahy

Tkk

yx

Tkk

xx

TuC −+

∂

∂+

∂∂

+

∂

∂+

∂∂

=∂

∂ ~ρ (3.3)

Boundary conditions

00 ==∂∂

+∂

∂==

y

sse

y

ffew y

Tk

y

Tkqq and fs TT = at y = 0;

0=q at y = H (3.4)

inf TT = and 0=∂∂

x

Ts at x = 0

0=∂∂

x

Ts and 0=∂

∂

x

T f at x = L

It should be pointed out that although equations (3.1) to (3.4) are the same as those of

chapter 2 for metal foams, the permeability K, inertial coefficient CI, effective solid

and fluid conductivity (kse, kfe), dispersion conductivity kd, interfacial heat transfer

coefficient hsf, and surface area density a~ are different from those of metal foams.

These parameters need to be separately modelled in order to close the equations.

3.3 Modelling on sintered bed

The permeability and inertial coefficient for a packed spherical particle bed were

obtained from the experimental results [39,40] as functions of porosity ε and particle

diameter dp:

( )2

23

1150 ε

ε

−= pd

K 2/3150

75.1

ε=IC (3.5)

These semi-empirical relations have been extensively validated, and can give reliable

predictions on the permeability and inertial coefficient of a packed bed.

For a sintered particle bed, Hwang and Chao [37] obtained its permeability and

inertial coefficient, and the results obtained appear to be less than those predicted

from the packed bed correlations of (3.5). So far, the test data are not sufficient to

correlate a formulation similar to (3.5) for sintered beds. However, it is known that

the permeability and inertial coefficient have small effects on the heat transfer of a

packed or sintered bed, although they have large effects on pressure drop. In the

68

present study only heat transfer will be studied, and hence the permeability and

inertial coefficient correlations of Eq. (3.5) will be adopted for numerical calculations.

The effective thermal conductivity (keff) of a fluid saturated particle bed has

significant effect on its heat transfer, and hence must be studied carefully. So far,

most researchers employed the linear relations ( ) sse kk ε−= 1 and ffe kk ε= in their

investigations; however, the linear relationship for effective solid conductivity ( )sek

could lead to overestimations of keff, as demonstrated below.

An analytical model for the effective conductivity of a packed bed was proposed by

Zehner and Schluender [41], yielding

( )( )

−

−−

+−

−

−

−−

+−−=sfp

pp

sfpsfp

psf

sfpfe kkB

BB

kkBkkB

Bkk

kkBkk

1

1

2

11ln

/1

1

1

1211

2

* εε (3.6)

where ( )[ ] 9/10/125.1 εε−=pB . Because the effective conductivity ek of a sintered

particle bed is expected to be higher than that of a packed bed *ek , a linear relationship

*ee Ckk = will be assumed in the present investigation, with the coefficient C

determined from experiments. Based on the measured data on equivalent conductivity

for a sintered bed [37], it is found that the selection of C = 1.8 leads to the best

agreement with the experimental data. Finally, the effective conductivity of the solid

part and fluid part in a sintered particle bed can be obtained as follows,

feese kkk −= and ffe kk ε= . (3.7)

Next consider the thermal dispersion effects in a porous channel. Thermal dispersion

results from the existence of the solid matrix, which forces the flow to undergo a

tortuous path around the solid particles. Several researchers [5-8] have investigated

the thermal dispersion effects on the heat transfer of packed bed. It is found that the

thermal dispersion conductivity is in general proportional to a product of the local

velocity and mixing length, namely,

uldCCk pftd ρ= (3.8)

where tC is an empirical constant, and l is the wall function for thermal dispersion

introduced by Cheng and co-workers [5,14]. To account for the wall effect on the

69

reduction of lateral mixing of the fluid, the Van Driest type of wall function l can be

written as

( )[ ]( ) ( )[ ] HyHwdyH

Hywdyl

p

p

≤≤−=

≤≤−−=

2/ /-exp-1

2/0 /exp1 (3.9)

where w is an empirical constant. In the present investigation, the empirical constants

375.0=tC and 5.1=w as suggested in [14] are used.

Finally, the heat transfer coefficient between the solid and fluid phases, hsf, modelled

by the following correlations [42], will be used in the present study:

( )( ) 100Re RePr/0.0156

100Re RePr/0040.0

d04.1333.0

d35.1333.0

≥=

<=

vf

vfsf

dk

dkh (3.10)

where 4 /vd aε= % is the average void diameter, Re and Red are the Reynold numbers

based on average void diameter dv and partical diameter, dp, respectively. The surface

area density of a sintered bed with spherical particles is given by

( )pd

a2

1346.20~ εε−= (3.11)

70

3.4 Results and discussions

For water-cooled convection in a packed bronze particle bed [13], Fig. 3.2 shows the

predictions on local heat transfer coefficient by using different effective solid

conductivity kse models. It can be seen that whilst the packed bed model (3.6) gives

quite good agreement with the experimental data, the commonly used linear model

(1 )se sk kε= − results in significant overestimations. This overestimation by the linear

conductivity model was also observed by Jiang and Ren [13], but they attributed it to

the boundary condition used, and attempted to remedy the discrepancy between model

and experiment by using different boundary conditions while keeping the linear

effective solid conductivity model. This approach is nonetheless difficult to justify, as

it will lead to the inconsistency of the effective solid conductivity kse used in the

boundary condition and energy equations.

0 1 2 3 4 5 60

10,000

20,000

30,000

40,000

Experimental data [5]

G = 0.0542 kg/s

partical diameter = 0.428 mmporosity = 0.365

packed model (Eq. 4.6)

sintered model

k = (1-ε)ε) kse s

packed bronze bed

Fig. 3.2 Effect of effective solid conductivity on local heat transfer coefficient in a packed particle bed

For the water-cooled heat transfer in a sintered bronze bed, the effect of kse on the

overall Nusselt number is shown in Fig. 3.3. It can be seen that the linear model

dramatically overestimates the Nusselt number. The results also show that sintering

can increase the heat transfer of a packed bed by approximately 30%, and increase its

effective solid conductivity kse by about 80%.

xh

(W/m2K)

Hx /

71

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000100

200

300

400

500

600

700


sintered bronze bed

packed model ( Eq. 4.6)

sintered model

k = (1- ε)ε) ksse

Fig. 3.3 Effect of effective solid conductivity kse on overall Nusselt number

of sintered bed

0 2 4 6 8 10 120

0.1

0.2

0.3

Red = 75

250

500


Fig. 3.4 Variation of dimensionless wall temperature of a sintered porous channel along streamwise direction for different Reynolds numbers

Nu

ReDh

se

inw

kqH

TT

/

−

Hx /

72

0 2,000 4,000 6,000 8,000 10,000 12,0000

100

200

300

400

500

600

Sintered copperPorosity = 0.38

1.5 mm

pd = 0.5 mm

1.0 mm

2.0 mm

Fig. 3.5 Effect of particle diameter on overall Nusselt number of a sintered bronze particle bed

The predicted dimensionless wall temperature distributions are shown in Fig. 3.4 for a

sintered bronze bed. Fig. 3.5 gives the predictions on the effect of particle diameter

upon overall heat transfer of the sintered bed. It is seen that overall heat transfer

increases as the particles become smaller. The numerical predictions will be compared

with experiment data in a future study.

Nu

ReDh

73

4 Conclusions

In this report, forced convection in heated porous channels made of metal foam and

sintered bed have been investigated. The microstructures of FeCrAlY and copper

foams have been measured, and the results show that the microstructures are quite

different from the industrial specifications, and the microstructures of FeCrAlY

samples are different from those of copper samples with same industrial

specifications. The experimental study on forced convection was conducted for eight

FeCrAlY and six copper foam samples. The results show that the heat transfer of

FeCrAlY samples is more sensitive to the relative density than the cell size, whereas

for copper samples, the effect of relative density is less significant in comparison with

the cell size effect. This can be attributed to the different thermal resistances on the

solid side for FeCrAlY and copper samples.

A numerical model was developed to incorporate the measured microscopic

parameters, and the selection of suitable boundary conditions was discussed. The

effects of microscopic parameters on overall heat transfer performance were

numerically investigated and optimizations for designing metal foam heat exchangers

were carried out. The optimal foam relative density increases as the Reynolds number

is increased. The predictions imply that there is no need to use high thermal

conductivity materials if the Reynolds number is small.

Finally, numerical modelling on forced convection across sintered particle beds was

carried out, and the effect of effective solid conductivity on heat transfer was

examined. The model was used to study the forced water convection in sintered

bronze particle beds The results show that overall heat transfer in a sintered bed is

about 30 percent higher than that of a packed bed, and that reducing the particle size

leads to enhanced heat removal capability of the porous medium.

74

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.

78

Acknowledgments

This work is supported partially by Porvair PLC, partly by the U.K. Engineering and

Physical Sciences Research Council (EPSRC grant number EJA/U83), and partly by

the U.S. Office of Naval Research (ONR/ONRIFO grant number N000140110271).

Alberic Du Chene carried out most of the microstructural studies with SEM and

image analysis, and participated in heat transfer measurements.

79

Appendix A SEM pictures of the foam samples

FeCrAlY samples #1-2 : 10 ppi, 5% (left) and 10% (right)



80

FeCrAlY sample #7 : 30 ppi, 7.5%

Copper samples #9-10 : 10 ppi, 5% (left) and 10% (right)


81


82

Appendix B Experimental data of the Porvair metal foams

Sample 1

DhRe f Nu fNuj =~

4905.06 7.18875 99.56 13.84942 5621.53 7.05263 110.12 15.61403 5993.54 6.98079 115.56 16.554 6613.56 7.20445 122.023 16.93717 7330.03 7.11745 133.37 18.73845 8570.08 7.16491 145.17 20.26125 9134.98 7.08157 150.23 21.21422 9603.44 7.08307 154.45 21.80552 10140.8 7.07803 166.23 23.48535 10802.2 7.44879 168.12 22.57011 13461.4 7.34777 195.56 26.61488 14274.3 7.42381 203.05 27.35118 15486.8 7.44736 212.79 28.57254 3926.8 7.76595 88.87 11.44355

4767.28 7.32992 96.89 13.21843 5304.63 7.25995 107.12 14.75492 5869.54 7.2772 113.23 15.55956 6324.22 7.52397 120.25 15.98225 7026.91 7.20246 128.67 17.86473 8101.61 7.16949 140.3 19.56903 9203.87 7.20398 150.28 20.86069 11504.8 7.16491 175.45 24.4874 12193.8 7.38319 184.82 25.03254 15211.2 7.47569 212.09 28.37063

Sample 2

DhRe f Nu fNuj =~

3706.35 17.458 132.887 7.61181 4271.26 17.2535 144.297 8.36335 5235.74 16.7985 154.35 9.18832 6117.55 17.5226 162.758 9.28846 6778.9 17.4031 172.721 9.92473

8198.06 17.7353 181.828 10.2523 9038.54 16.8768 193.041 11.4382 9369.21 17.7603 206.509 11.6276

10375 17.5875 217.848 12.3865 11229.3 17.9454 227.213 12.6614 11711.5 18.1443 238.929 13.1683 12400.4 18.1101 252.58 13.9469 13006.7 18.4008 268.957 14.6166

13544 18.4487 278.846 15.1147 13971.2 18.5505 286.883 15.465 14425.8 18.6186 292.242 15.6962

83

Sample 3

DhRe f Nu fNuj =~

3954.36 15.8226 92.9947 5.87733 4781.05 15.6805 106.993 6.82332 5401.08 14.998 117.169 7.81231 5786.87 14.9907 126.213 8.41942 6765.12 15.1869 139.136 9.16158 7026.91 15.9952 143.868 8.99445 7812.27 14.915 152.257 10.2083 8280.73 15.0013 158.384 10.558 8680.3 16.2924 159.315 9.77849

9038.54 15.7879 161.942 10.2573 9410.55 15.6121 172.904 11.075 10154.6 16.0084 180.888 11.2996 11022.6 15.8978 191.553 12.049 11780.4 16.0196 201.654 12.588 12979.1 15.6231 215.451 13.7905

14853 16.2005 238.189 14.7026 16079.2 16.0834 253.496 15.7613

Sample 4

DhRe f Nu fNuj =~

4202.37 21.3704 119.431 5.58862 5235.74 21.1366 133.12 6.29808 5786.87 21.5944 145.45 6.73554 6613.56 21.3731 156.976 7.34456 7109.58 22.6095 168.113 7.4355 7578.04 22.3172 176.01 7.88674 8129.17 22.5308 182.152 8.08458 8680.3 22.7571 188.692 8.29157

9093.65 23.0914 195.046 8.44669 9782.56 23.0829 204.307 8.85101 10526.6 22.6909 215.215 9.48464 11022.6 23.4462 224.852 9.59013 3720.13 22.1426 106.268 4.79926 5649.08 22.0679 138.307 6.26734 6820.24 23.5611 159.198 6.75682 7798.49 22.121 171.932 7.77234 8198.06 23.4757 182.024 7.75372 8749.19 23.3384 190.348 8.156 9231.43 23.4747 196.734 8.38068 9479.44 24.2011 202.315 8.35974 10016.8 23.5841 209.223 8.87136 10333.7 23.8414 213.687 8.96285 11160.4 24.5806 224.342 9.12679 11711.5 24.346 231.419 9.50542 12083.5 24.6721 235.264 9.53563 12538.2 24.0183 239.997 9.99226

84

Sample 5

DhRe f Nu fNuj =~

6613.56 40.1808 138.229 3.43956 8625.19 39.1782 170.619 4.35495 9231.43 39.8823 180.328 4.5215 9644.78 41.1242 188.264 4.57794 10333.7 40.4633 200.143 4.94628 5373.52 41.0703 114.33 2.78376 6406.89 40.9808 135.908 3.31638 6820.24 40.7629 143.631 3.52357 7784.71 40.4557 157.054 3.88212 8818.08 39.7719 174.146 4.37862 9438.11 41.1382 185.198 4.50185 3789.02 40.3517 85.5631 2.12043 4202.37 42.2906 93.2572 2.20515 4615.72 41.5652 100.092 2.40807 5166.85 40.9471 108.397 2.64724 5855.76 41.6983 124.658 2.98952 6200.22 42.3124 128.183 3.02944 6958.02 41.9967 143.747 3.42282 7274.92 43.0993 151.934 3.52521 7991.39 42.4674 162.762 3.83263 8363.4 43.5743 170.207 3.90613

9093.65 43.7361 181.092 4.14056 9920.34 43.8194 194.41 4.43662

Sample 6

DhRe f Nu fNuj =~

3306.78 126.685 111.091 0.876907 3582.35 130.981 117.572 0.897626 4271.26 143.723 133.361 0.927903 4684.61 143.295 143.498 1.00142 5166.85 142.659 150.174 1.05268 5649.08 141.449 161.396 1.14102 6062.43 151.871 171.539 1.1295 6613.56 151.632 186.023 1.22681 6889.13 155.064 191.979 1.23806 7440.26 151.37 199.631 1.31883 7991.39 144.76 208.721 1.44184 3582.35 131.639 119.407 0.907079 3857.91 143.598 127.629 0.888794 4133.48 147.773 129.924 0.879213 4409.04 147.685 136.959 0.927372 4684.61 156.137 143.253 0.917483 4960.17 153.623 148.299 0.965344 5235.74 154.521 154.848 1.00212 5786.87 153.559 167.305 1.08952 5993.54 150.976 171.925 1.13876

85

Sample 7

DhRe f Nu fNuj =~

3375.67 21.44 95.34 4.446828 4491.71 21.72 114.07 5.251842 4781.05 21.48 118.246 5.504935 5263.29 21.44 126.49 5.89972 5456.19 20.56 127.96 6.223735 5828.2 21.32 136.366 6.396154 6021.1 20.96 135.102 6.445706

6351.78 20.92 143.137 6.842113 7054.47 21.04 151.348 7.193346 7509.15 21.48 158.895 7.397346 8638.97 21.32 178.843 8.388508 8914.53 21.12 180.64 8.55303 9920.34 21.76 194.23 8.926011 10223.5 21.84 197.464 9.041392

10499 21.92 201.184 9.178102 11725.3 22.32 215.198 9.641487 12483.1 21.8 233.01 10.68853

Sample 8

DhRe f Nu fNuj =~

5924.65 26.2613 180.156 6.86012 6475.78 30.2866 201.608 6.65668 7798.49 28.3476 224.551 7.92134 9176.32 26.7997 246.613 9.20208 10388.8 27.1713 253.572 9.33235 12400.4 27.7587 304.67 10.9756 2893.43 44.2979 123.854 2.79593 3444.56 40.5491 134.19 3.30933 5097.95 31.1843 164.587 5.27789 6062.43 30.5055 186.822 6.12421 7440.26 26.3319 204.98 7.78447 8266.95 26.022 226.826 8.71669 8859.42 26.4526 238.013 8.9977

9507 27.4392 254.664 9.28102 10195.9 27.3826 263.828 9.63488 10843.5 26.9619 270.288 10.0248 11711.5 26.8298 284.451 10.602 12400.4 27.2566 296.808 10.8894

86

Sample 9

DhRe f Nu fNuj =~

2237.9 21.495 135.6 6.308443824 2781.984 20.6573 170.379 8.247883315

3898.2 16.2698 212.782 13.07834147 4422.02 19.7374 253.162 12.8265121 5262.57 19.9737 282.41 14.13909291 6090.94 20.1939 331.291 16.40549869 6675.4 20.7094 349.219 16.86282558 7206.8 22.4478 371.195 16.53591889

7698.96 21.2106 383.532 18.08209103 8649.13 22.4972 418.662 18.60951585 9745.5 23.262 461.821 19.8530221

10659.14 23.8816 478.437 20.03370796 11658.06 24.4405 504.308 20.63411141 12547.33 24.5972 524.185 21.31075895 13095.52 25.1464 538.037 21.39618395 13668.06 25.6405 550.275 21.46116495 14289.04 25.5724 563.351 22.02964915 14837.5 26.1063 569.428 21.81189981

15166.43 26.0218 575.296 22.10823233

Sample 10

DhRe f Nu fNuj =~

2650.58 17.0038 132.738 7.806373 3749.58 19.0628 188.846 9.90652 4141.84 20.9139 206.622 9.879649 4446.39 21.1785 237.755 11.22624 4943.41 23.8759 265.865 11.13529 5445.3 25.4975 285.892 11.21255

5910.65 26.4133 302.882 11.46703 6334.58 27.1141 321.723 11.86552 6505.12 27.4537 327.34 11.92335 6943.67 27.8504 351.769 12.63066 7796.4 31.6219 386.623 12.22643

8673.49 31.4459 421.23 13.39539 9648.05 32.0186 440.222 13.74895 10318.1 33.6777 456.104 13.54321 11146.4 34.0906 474.6 13.92173 12035.7 34.3622 496.453 14.44765 12547.3 36.1275 509.164 14.09353 13400.1 35.6643 526.577 14.76482

87

Sample 11

DhRe f Nu fNuj =~

2465.86 32.652 102.327 3.133866

3758.27 39.8319 158.356 3.975607

4420.46 46.6983 198.684 4.25463

5201.35 55.4059 242.124 4.370004

6047.21 60.8869 289.035 4.74708

7137.96 66.9585 332.565 4.966733

7841.38 71.8474 357.121 4.970549

8834.67 76.8855 385.539 5.014457

9543.09 77.5951 394.746 5.087254

10089.1 78.2565 409.66 5.234837

10820 78.847 425.659 5.398544

Sample 12

DhRe f Nu fNuj =~

1218.19 66.7853 60.8549 0.9112021888.19 76.6088 111.646 1.4573522558.19 83.9382 152.369 1.8152523142.92 87.4918 185.115 2.1157983825.11 90.2598 221.406 2.4529864391.57 94.4846 260.621 2.7583444994.57 94.2893 286.959 3.0433895559.81 95.2244 312.906 3.2859866192.05 96.468 354.724 3.6771166764.6 97.9562 375.861 3.837031

7410.23 96.4191 394.52 4.091727906.04 98.3219 405.643 4.1256638015.67 98.1632 419.132 4.2697471449.64 68.0189 79.8763 1.1743251705.46 74.8236 101.268 1.3534232338.92 80.355 140.017 1.742482801.83 84.9213 169.731 1.9986863515.69 89.0332 215.029 2.4151554166.2 91.8521 242.153 2.636336

4970.21 93.5729 300.24 3.2086215883.85 95.0973 342.3 3.5994716388.18 96.2807 360.863 3.748037120.31 96.5266 386.843 4.0076317698.95 97.4158 406.942 4.1773728174.04 97.4873 425.548 4.365163

88

Sample 13

DhRe f Nu fNuj =~

1722.52 136.839 142.335 1.0401642046.56 132.484 164.548 1.2420222279.23 134.117 174.092 1.2980612583.78 132.758 199.95 1.5061242941.92 124.288 213.039 1.7140753373.16 121.019 232.381 1.9202033787.35 119.763 253.661 2.1180254219.8 117.34 268.148 2.285222

4530.44 114.803 278.755 2.4281164897.11 114.571 302.77 2.6426415216.28 112.819 324.25 2.8740735777.86 110.396 344.956 3.1247155988.61 111.374 356.875 3.2042946388.18 111.078 394.397 3.5506316854.74 110.94 421.75 3.8016047292.07 108.075 428.991 3.969382

Sample 14

DhRe f Nu fNuj =~

1340.01 155.951 69.9748 0.448697 1572.68 157.083 89.7347 0.571257 1925.95 159.969 126.372 0.789978 2486.32 164.872 169.809 1.029944 2814.01 161.402 221.408 1.37178 3145.36 165.689 245.257 1.480225 3551.02 163.845 269.135 1.64262 3898.2 160.447 294.035 1.832599

4278.27 159.879 320.429 2.004197 4558.46 160.994 337.747 2.097886 4922.7 160.933 355.729 2.210417

5379.52 158.209 385.723 2.43806 6050.74 156.577 415.666 2.654707 1218.19 159.5 45.0412 0.28239 1572.68 155.908 81.772 0.524489 1722.52 159.024 93.5456 0.588248 1989.3 166.021 127.988 0.770915

2385.21 166.275 163.023 0.980442 2723.87 163.219 201.627 1.235316 3145.36 166.451 229.204 1.377006 3620.45 162.33 269.727 1.661597 3978.6 163.423 296.388 1.813625

4335.53 160.849 307.559 1.912098 4665.66 164.164 346.711 2.111979 5126.13 157.249 372.088 2.366234 5664.57 155.318 399.912 2.574795

Thermal Transport Phenomena in Porvair Metal … Report Thermal Transport Phenomena in Porvair Metal...

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