Abstract
Diverse expressions for the thermal conductivity of cellular materials are reviewed. Most expressions address only the conductive contribution to heat transfer; some expressions also consider the radiative contribution. Convection is considered to be negligible for cell diameters less than 4 mm. The predicted results are compared with measured conductivities for materials ranging from fine-pore foams to coarse packaging materials. The dependencies of the predicted conductivities on the material parameters which are most open to intervention are presented graphically for the various models.
Similar content being viewed by others
Abbreviations
- a :
-
Absorption coefficient
- C itv(J mol−1 K−1):
-
Specinc heat
- E :
-
Emissivity
- E L :
-
Emissivity of hypothetical thin parallel layer
- E o :
-
Boundary surfaces emissivity
- f :
-
Fraction of solid normal to heat flow
- f s :
-
Fraction of total solid in struts of cell
- K(m−1):
-
Mean extinction coefficient
- k(Wm−1 K−1):
-
Effective thermal conductivity of foam
- k cd(Wm−1 K−1):
-
Conductive contribution
- k cr(Wm−1 K−1):
-
Convertive contribution
- k g(Wm−1K−1):
-
Thermal conductivity of cell gas
- k r(Wm−1 K−1):
-
Radiative contribution
- k s(Wm−1 K−1):
-
Thermal conductivity of solid
- L(m):
-
Thickness of sample
- L g(m):
-
Diameter of cell
- L s(m):
-
Cell-wall thickness
- n :
-
Number of cell layers
- r :
-
Reflection coefficient
- t :
-
Transmission coefficient
- T(K):
-
Absolute temperature
- T m(K):
-
Mean temperature
- T N :
-
Fraction of energy passing through cell wall
- T 1(K):
-
Temperature of hot plate
- T 2(K):
-
Temperature of cold plate
- V g :
-
Volume fraction of gas
- V w :
-
Volume fraction of total solid in the windows
- w :
-
Refractive index
- δ(m):
-
Effective molecular diameter
- η(Pa s):
-
Gas viscosity
- θ:
-
Structural angle with respect to rise direction
- σ(Wm−2 K−4):
-
Stefan constant
References
House of Commons Energy Committee, Sixth Report, Energy Policy Implications of the Greenhouse Effect (Her Majesty's Stationery Office, London 1989) Vol. 1, para. 102.
R. C. Progelhof, J. L. Throne and R. R. Ruetsch, Polym. Engng. Sci. 16 (1976) 615.
D. K. Hale, J. Mater. Sci. 11 (1976) 2105.
R. Taylor, in “International encyclopedia of composites”, edited by S. M. Lee Vol. 5, (VCH, New York, 1991) pp. 530–548.
J. T. Mottram and R. Taylor, ibid.“ Vol. 5, (VCH, New York, 1991) pp. 476–496.
R. A. Crane, R. I. Vachon and M. S. Khader, Proceedings of the Seventh Symposium on Thermophysical Properties, Galthersberg, MD, USA (American Society for Mechanical Engineers, NY, USA, 1977) pp, 109–123.
J. A. Valenzuela and L. R. Glicksman, in Thermal Insulation, Materials and Systems for Energy Conservation in the 80's, ASTM STP 789, edited by F. A. Govan, D. M. Greason, J. D. McAllister (American Society for Testing and Materials, Philadelphia, USA) pp. 688–702.
J. C. Maxwell, “A treatise on electricity and magnetism”, Vol. 1 (Clarendon Press, Oxford 1892) p. 440.
W. Woodside and J. H. Messmer, J. Appl. Phys. 32 (1961) 1688.
A. V. Liukov, A. G. Shashkor, L. L. Vasiliev and Yu. E. Fraisman Int. J. Heat Mass Transfer 11 (1968) 117.
R. L. Hamilton and O. K. Crosser, Ind. Engng. Chem. Fundam. 1 (1962) 187.
D. Tabor, “Gases, liquids and solids”, 2nd Edn (Cambridge University Press, Cambridge, 1979) p. 57.
M. E. Stephenson and M. Mark, Amer. Soc. Heat., Refrig. Air Conditioning Engineers J., 3 February (1961) 75.
G. T-N. Tsao, Ind. Engng. Chem. 53 (1961) 395.
A. Sugawaru and Y. Yoshizawa, J. Appl. Phys. 33 (1962) 3135.
A. W. Pratt, in “Thermal conductivity“, edited by R. P. Tye, Vol. 1 (Academic Press, London 1969) p. 319.
C. H. Lees, Phil. Mag. 49 (1900) 221.
Z. Hashin and S. Shtrikman, J. Appl. Phys. 33 (1962) 3125.
D. J. Doherty, R. Hurd and G. R. Lester, Chem. Ind. July (1962) 1340.
S. Baxter and T. T. Jones, Plastics Polym. 40 (1972) 69.
F. K. Brockhagen and W. Schmidt, in “Polyurethane Foams”, edited by T. T. Healy (Iliffe, London, 1964) pp. 93–144.
E. Kerner, Proc. Phys. Soc. B 369 (1956) 802.
A. D. Brailsford and K. G. Major, Brit. J. Appl. Phys. 15 (1964) 313.
T. Zhang, J. R. G. Evans and K. K. Dutta, J. Euro. Ceram. Soc. 5 (1989) 303.
R. Hamilton and O. Crosser, Ind. Engng. Chem. Fundam. 1 (1962) 187.
M. H. Kuok, H. K. Sy and K. L. Tan, Reg. J. Energy, Heat Mass Transfer 7 (1985) 17.
S. Oka and K. Yamone, Jpn. J. Appl. Phys. 6 (1967) 469.
R. M. Barrer, in “Diffusion in polymers” Edited by J. Crank and G. S. Park, (Academic Press, London 1968) pp. 165–216.
Von D. A. G. Bruggeman, Ann. Phys. 5 (1935) 636.
R. C. Progelhof and J. L. Throne, J. Cell. Plast. 11 (1975) 152.
A. Eucken, Forsch. Gebiete Ingenieuru B3 Forschurgshaft No. 353 (1932) 16.
H. W. Russell, J. Amer. Ceram Soc. 18 (1939) 1.
B. Budiansky, J. Compos. Mater. 4 (1970) 286.
T. B. Jefferson, O. W. Witzell and W. L. Sibbitt, Ind. Engng. Chem. 50 (1958) 1589
L. Topper, Ind. Engng. Chem. 47 (1955) 1377.
S. C. Cheng and R. I. Vachon, Int. J. Heat Mass Transfer 13 (1970) 537.
D. Bedeaux and R. Kapral, J. Chem. Phys. 79 (1983) 1783.
L. E. Nielsen, J. Appl. Polym. Sci. 17 (1973) 3819.
Idem., Ind. Engng. Chem. Fundam. 13 (1974) 17.
Idem., Appl. Polym. Symp. 12 (1966) 249.
J. M. Peterson and J. J. Hermans, J. Compos. Mater. 3 (1969) 338.
M. M. Levy, J. Cell. Plastics 2 (1966) 37.
D. Bhattacharjee, J. A. King and K. N. White-Head, ibid. 27 (1991) 240.
R. J. Harding, ibid. 1 (1965) 224.
Idem. ibid. 1 (1965) 385.
F. J. Norton, ibid. 3 (1967) 23.
C. J. Hilado and W. R. Proops, ibid. 5 (1969) 299.
G. W. Ball, R. Hurd and M. G. Walker, ibid. 6 (1970) 66.
R. R. Dixon, L. E. Edelman and D. K. McLain, ibid. 6, (1970) 44.
M. Bomberg, ibid. 26 (1990) 275.
T. T. Jones, Plastics Polym. 40 February (1972) 33.
R. Boetes and C. J. Hoogendoorn, Proc. Int. Cent., Heat Mass Trans. 24 (1987) 14.
L. Glickman, M. Schuetz and M. Sinofsky, Int. J. Heat Mass. Trans. 30 (1987) 187.
M. A. Schuetz and L. R. Glicksman, J. Cell. Plast. 20 (1984) 114.
L. R. Glicksman, Cellular Polym. 10 (1991) 276.
A. Cunningham, Proc. Int. Cent. Heat Mass Trans. 24 (1987) 32.
R. J. J. Williams, C. M. Aldao, Polym. Engng. Sci. 23, (1983) 293.
M. Garbuny, “Optical Physics”, (Academic Press, NY, 1965) p. 257.
A. L. Loeb, J. Amer. Ceram. Soc. 37 (1954) 96.
J. Francl and W. D. Kingery, ibid. 37 (1954) 99.
W. J. Batty, S. D. Probert and P. W. O'Callaghan, Appl. Energy 18 (1984) 117.
R. J. Harding, J. Cell. Plast. 2 (1966) 206.
R. Caps, A. Trunger, D. Buttner and J. Fricke, Int. J. Heat and Mass Transf. 27 (1984) 1865.
J. Fricke, R. Caps, D. Buttner, U. Heinemann and E. Hummer, J. Non-Cryst. Sol. 95–96 (1987) 1167.
R. Caps and J. Fricke, Solar Energy 36 (1986) 361.
T. W. Tong and C. L. Tien, Trans. ASME Ser. C. J. Heat Transfer 105 (1983) 70.
T. W. Tong, Q. S. Yang and C. L. Tien, ibid. 105 (1983) 76.
R. W. Skochdopole, Chem. Engng. Prog. 57 (10) (1961) 55.
H. Jeffreys, Proc. Roy. Soc. A. 118 (1928) 195.
T. T. Healy, “Polyurethane foams” (Iliffe, London, 1964) p. 121.
Anon, The properties of Dupont Vespel Parts, Technical Data Sheet (Dupont, Wilmington, DE, Undated).
J. Fricke, J. Non-Cryst. Solids 100 (1988) 169.
R. C. Weast (editor) Handbook of Chemistry and Physics”, 55th Edn (CRC Press, Cleveland, Ohio 1974) p. E2.
C. R. Wilke, J. Chem. Phys. 18 (1950) 517.
Loc. cit. [12], pp. 69–70.
J. Fricke in Proceedings of the First International Symposium on Aerogels at Wurzberg 1985, edited by J. Fricke (Springer Verlag, Heidelberg, 1986) pp. 94–103.
D. Buttner, R. Caps, U. Heinemann, E. Hummer, A. Kadur and J. Fricke, Solar Energy 40 (1988) 13.
D. Buttner, R. Caps and J. Fricke, High Temp. High Pressures 17 (1985) 375.
J. Brandrup and E. H. Immergut, “Polymer Handbook”, 2nd Edn (Wiley, New York, 1974) p. VIII-9.
C. A. Dostal (editor) “Engineered Materials Handbook”, Vol. 2 (American Society for Metals, Metals Park, Ohio, 1988) p. 260.
Loc. cit. [80], p. 133.
BS874 Part 2, Tests for Thermal Conductivity and Related Properties, Section 2.1, Guarded hot plate method, (British Standards Institute, London, 1986).
K. M. Haunton, J. K. Wright and J. R. G. Evans, Brit. Ceram. Trans. J. 89 (1990) 53.
P. Hammond and J. R. G. Evans, J. Mater. Sci. Lett. 10 (1991) 294.
J. Greener and J. R. G. Evans, J. Mater. Sci. in press.
W. Schramm, Bull. Amer. Ceram. Soc. 60 (1987) 1194.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Collishaw, P.G., Evans, J.R.G. An assessment of expressions for the apparent thermal conductivity of cellular materials. JOURNAL OF MATERIALS SCIENCE 29, 2261–2273 (1994). https://doi.org/10.1007/BF00363413
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF00363413