Thermal conductivity of some porous metals

IN HEAT AND MASS ~ Vol. 4, pp. 417 - 423, 1977

THERMAL

Perga~sn Press Printed in Great Britain

CONDUCTIVITY

OF

SOME

POROUS

METALS

J. Januszewski, M.I. Khokhar and A.S. MuJumdar Department of Chemical Engineering McGill University, Montreal, Canada

(C~m,unicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT Thermal conductivity measurements at room temperature are reported for commercially available sintered metal powders and felt metals. Results are compared with various analytical, semi-empirical and empirical models. Thermal conductivity of sintered metals was found to correlate with their tensile strength; finer particle size generally resulted in lowered conductivity values at constant voidage. No single model yielded satisfactory predictions for the various porous metal samples studied.

Introduction Porous metals find widespread applications in such diverse fields as nuclear power generation, food processing, foundry practice, filtration, thermal insulation, transpiration cooling, heat pipes, etc.

On a more fundamental level,

porous metals produced in a variety of metals and alloys with different internal structures provide good physical models to test predictive correlations for thermal properties of granular materials in general.

Vachon et al.

(1) and, very recently, Cheng and Vachon (2) have provided excellent reviews of the state-of-the-art of estimation of thermal conductivity of porous materials. Most of the predictive models tested in this work are discussed in detail in Reference (!).

A review of the methods for predicting the thermal conducti-

vity of polymer composite systems by Progelhof et al. (3) is also very relevant to this study.

Mendoza (4) has tested and extended a few of the models to

estimate thermal conductivity of model foodstuffs - a problem of great interest in thermal processing of foodstuffs.

417

418

J. Januszewski,

M.I. Khokhar and A.S. Muj~-ndar

Vol. 4, No. 6

The work reported in this paper is an extension of the work presented earlier by Biceroglu, Mujumdar,

van Heiningen and Douglas

(5).

Thermal con-

ductivity at 18°C ± 3°C was determined using the transient method developed by loffe and loffe

(6).

Heat sinks of different

thermal capacities were used,

depending on the size and conductivity of the sample, level of the measurement

to within 7%.

to maintain the accuracy

This was necessary because the sample

size was, in some cases, limited by the availability of test specimens from the manufacturers.

Biceroglu et al. have listed the pertinent

references,

giving details of the design and accuracy of the apparatus used. Although a large number of correlations were tested, this paper will confine its attention only to the following more successful models. Koh and Fortini

Ke/K f = (i - e) / (I + ne 2) n = ii for sintered metals, n = i0 for felt metals

Alexander

Ke/K f = (Ks/Kf)(l - C)~ = 0.53 for sintered metal powders

Bruggeman

K e

= K [(l-e)(l-Ke)] [l+K(l-e)] s Kf

where K = 1.5 (Kf-Ks)/(2/K f + ~ s ) ( 2 ~ Nielsen

s + /Kf)

Ke/K f = (i + ABe) / (I - Bc~)

where

B

A = 0.50;

(Ks/K f) - i (Ks/Kf) + A e

m

'

(l-em)e ~ ~ i +-------i--m

= 0.52 for felt metal TABLE

1

Samples Used in Present Study Material

Porosity Range

Sintered metal powders (316 stainless steel, bronze, Ni-Cr (80~ ~ 20)

0.19 - 0.71 (various particle sizes)

Felt metal (347 s.s., 430 s.s., Ni, Hastelloy X)

0.40 - 0.90

Table I lists the various samples used in present study.

Vol. 4, No. 6

T H ~ R M A L C D N D U C T I V I T Y O F POROUS MEPALS

419

Sintered 316 stainless steel samples in a wide range of porosity (0.19 0.71) and partlcle-slzes were provided by Panoramic Corp., Janesville, Wisconsin, U.S.A.

Four particle size ranges were available (-20 +50, -50 +80,

-80 +100 and -100 +200). 0.72 to 8.92 W/mK.

Measured thermal conductivity values varied from

Disregarding effects of particle size, Figure 1 shows that

the measured k values compare favorably with the empirical models of Koh and Fortln (7) and Alexander (8) and the analytical model of Bruggeman in agreement with conclusions of Reference (5).

0.6 i

0.5 KOH & FORTINI

51NTERED

\X,

a4

316 SS

FELT ,AL o

Ill

~e

0.3

Q2 " ~

,NIELSEN

^'

0.1

-'~.-~(~-,.

0

/

I

I

I

I

I

I

I-

0.1

0.2

0.3

0.4

0.5

0.6

0,7

IE,

FORTIN, (FM.

I'0" '~, 0.8 09

VOID FRACTION FIG. i

Present data compared with theoretical and semi-empirical models.

Figure 2 shows the variation of K e with porosity for the 316 s.s. samples with particle size ranges as parameters. correspond to smaller particles.)

(Note that larger mesh sizes

Except for some scatter in the data around

0.4 < e < 0.5, the effect of particle size is seen to be negligible.

The

scatter may be attributable to variations in process conditions in manufacture which may lead to variation in inter-partlcle bonding.

The latter results in

variation in the tensile strength of the material, which correlates well with the measured Ke of the sample.

420

J. Januszewski,

M.I. Khokhar and A.S. Muj~ndar

Vol. 4, No. 6

316 SS $11NTERED POWDER

Partide Mesh S~ze

10

-20 +50

t

O

9

-50 +80

0

40

0

-I00 +200

+I00

8

>

r-i

I- 0

4D

u []

3

.I~-.

2

I

I

I

I

0.2

0.4

a6

0.8

POROSITY, |

FIG. 2 Effect of particle size on thermal conductivity

Table 2 gives the physical characteristics

and measured conductivity

porous alloy samples provided by Union Carbide Co., N.Y. (9) for conductivity of Ni - Cr (80 : 20) allo~,resulted with Koh and Fortini, Alexander, models

Bruggeman models;

(e.g. Nielsen, Lichteneker,

tunately,

Halpin-Tsai,

of

Using published data in poor comparison

comparison with other

etc.) was even poorer.

Unfor-

the solid alloy could not be obtained to measure its conductivity

use in the models.

However,

for

the measured K

values for the porous alloys e listed in Table 2 agree with independent measurements (i0). Although the sample size is too small to generalize, to the tensile strength of the sample.

it seems that K

e

is strongly correlated

Higher tensile strength implies better

inter-particle bonding, which also results in lowered thermal contact resistance

Vol. 4, No. 6

~

CONDUCTIVITY OF POROUS METALS

421

On physical grounds this effect would be more significant for smaller particle sizes

(i.e. more inter-particle

higher thermal conductivity

contacts per unit length) and materials of

(i.e. when the contact resistance is a limiting

factor). TABLE

2

Measured K e and Physical Property Data for Porous Ni - Cr Alloys

Composition

Voidage,

e

Particle Size

Tensile Strength

~m

N/m 2

K

e W/mK

Ni:Cr

(80

: 20)

0.66

i00

2.6 x 106

0.763

Ni:Cr

(80

: 20)

0.67

i00

12.4 x 106

0.962

Ni:Cr

(80

: 20)

0.67

i00

5.5 x 106

0.725

Ni:Cr : AI (67 : 22 : ii)

0.66

i00

11.7 x 106

1.08

Ni:Cr

(80

: 20)

0.62

30

4.8 x 106

0.813

Ni:Cr

(80

: 20)

0.70

30

3.4 x 106

0.647

Ni:Cr (80

: 20)

0.47

30

27.5 x 106

1.59

Table 3 presents physical data and measured K

for various felt metal e samples provided by Brunswick Corp., Technetics Div., Milford, Conn. These data, and indeed all data on felt metals or foam metals which are liable to change in physical structure as a result of application of pressure when held in the apparatus,

must be taken with caution.

For felt metals,

the porosity

was measured after the tests and it was observed to be generally lower than that prior to the tests.

Once again, because of limited sample size, the

variation of K e with metal composition, be identified individually.

voidage as well as particle size cannot

It may be noted, however,

that higher tensile

strength, which couples effects of pore size, bonding and voidage,

is

associated with higher K . e As may be seen from Figure i, the felt metal data agree reasonably well, considering Nielsen

the experimental difficulties,

(10) models.

with the Koh and Fortlni

(7) and

Models assuming isotropy are clearly unsuitable for

estimation of K e of such anisotropic materials as the felt metals.

422

J. Januszewski,

M.I. l
TABLE

Vol. 4, No. 6

3

Felt Metal Physical Properties and Measured Thermal Conductivity Composition

e

Pore Size Range, pm

347 s.s.

0.68

i0 - 42

347 s.s.

0.62

7 - 27

347 s.s.

0.40

3 - 12

430 s.s.

0.89

250 - I000

Tensile Strength

K

2.07 x 107

1.16

4.14 x 107

1.22

ii.0

e

x 107

3.61

3.45 x 106

0.37

430 s.s.

0.58

41 - 150

3.52 x 107

2.73

Ni

0.87

45 - 182

3.1

x 106

1.27

Ni

0.83

32 - 128

5.24 x 106

1.26

Ni

0.61

12 - 48

2.34 x 107

6.24

Hastelloy X

0.81

-

i0.0

x I0 I0

0.48

Closure New data for a number of recently available commercial samples of sintered metal powders and felt metals are presented and compared with available predictive correlations.

Particle size is observed to have negligible influence

on thermal conductivity of porous metals.

Tensile strength is observed to

be a key physical property directly correlated to the thermal conductivity the specimen.

of

More tests using a large number of specially manufactured

specimens are required to isolate the influence of voidage,

particle size and

shape on both the tensile strength and K e of porous metals. Acknowledgements The authors gratefully acknowledge mentioned

the support of the various manufacturers

in the text for providing the samples used in this study. Nomenclature

K

effective thermal conductivity e

Kf

thermal conductivity of fluid

K

thermal conductivity of solid

s

porosity

(voidage) other symbols explained in text, where used

Vol. 4, No. 6

THERMAL CONDUCTIVITY OF POROUS METALS

423

References

(i)

Vachon, R.I., Prakouras, A.G., Crane, R. and Khader, M.S., "Thermal Conductivity of Heterogeneous Mixtures and Lunar Soils", Final Report, NASA 8-26579, Auburn University (1973).

(2)

Cheng, S.C. and Vachon, R.J., "A Technique for Predicting the Thermal Conductivity of Suspensions, Emulsions and Porous Materials", International Journal of Heat and Mass Transfer, Vol. 13, p. 537 (1970).

(3)

Progelhof, R.C., Throne, J.L. and Ruetrch, R.R., "Methods for Predicting the Thermal Conductivity of Composite Systems: A Review", Polymer Engineering and Science, Vol. 16, No. 9, p. 615 (1976).

(4)

Mendoza, E., "Thermal Conductivity of Model Food Systems", Ph.D. Thesis, University of Massachusetts (1975).

(5)

Biceroglu, 0., Mujumdar, A.S., van Heiningen, A.R.P. and Douglas, W.J.M., "Thermal Conductivity of Sintered Metal Powders at Room Temperature", Letters in Heat and Mass Transfer, Vol. 3, p. 183 (1976).

(6)

loffe, A.V. and loffe, A.F., "Measurement of Thermal Conductivity of Semiconductors in the Vicinity of Room Temperature", Soviet Physics, Technical Physics, Vol. 3, No. ii, p. 2163 (1958).

(7)

Koh, T.C.Y. and Fortini, A., "Thermal Conductivity and Electrical Resistivity of Porous Materials", NASA - CR 120854 (1972).

(8)

Alexander, E.G., "Structure-Property Relationship in Heat Pipe Wicking Materials", Ph.D. Thesis, N.C. State University (1972).

(9)

Hoyt, S.L., Ed., ASME Handbook, "Metals Properties", McGraw-Hill Co. Inc., New York - Toronto - London, p. 381 (1954).