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Conductive and Radiative Heat Transfer in Ceramic and Metal Foams at Fire Temperatures

Contribution to the Special Issue “Materials in Fire” Guest Editor K. Ghazi Wakili

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Abstract

In addition to the multiple actual or possible applications of metal and ceramic foams in various technological fields, their thermal properties make them a good candidate for utilization as fire barriers. Several studies have shown experimentally their exceptional fire retardance due to their low apparent thermal conductivity. However, while the thermal properties of this porous material have been widely studied at ambient temperature and are, at present, well-known, their thermal behaviour at fire temperatures remains relatively unexplored. Indeed, at such temperatures, the major difficulties are not only due to the fact that thermal measurements are rendered fussy since heavy equipments are required but also stem from the fact that a significant part of the heat transfer occurs by thermal radiation which is much more difficult to evaluate than conductive heat transfer. Therefore, the present chapter is written with a view to report progress on the knowledge of heat transfer in open cell foams and to enlighten the reader on the mechanisms of heat transfer at high temperatures. A first part is devoted to the review of the prior published works on the experimental or theoretical characterisations of radiative and conductive heat transfers from ambient to high temperatures. By taking inspiration from the concepts and models presented in these previous works, we propose, in a second part, a model of prediction of the conductive and radiative contributions to heat transfer at fire temperatures. This analytical model is based on numerical simulations applied to real foams and takes into account the structure of the foam and the optical and thermal properties of the constituents. In a third part, we propose an innovative experimental technique of characterization of heat transfer in foams at high temperatures which permit to evaluate independently the radiative and conductive contributions from a unique and simple measurement. The experimental results obtained on several metal and ceramic foams are compared to the results predicted by our numerical model. The good adequacy between experimental and theoretical results show the consistency of both approaches.

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Abbreviations

a :

Length of cell struts (m)

A :

Characteristic size of particles interacting with radiation (m)

d :

Thickness of cell struts (m)

D :

Characteristic size of cell lumps (m)

D cell :

Cell diameter (m)

E c , E h :

Emissivities of the hot and cold boundaries

f s :

Fraction of solid phase in struts (-)

f v  = 1 − ε:

Solid fraction (-)

k :

Thermal conductivity (W/m/K)

k c :

Effective thermal conductivity (W/m/K)

k equ :

Equivalent thermal conductivity (W/m/K)

k rad :

Radiative conductivity (W/m/K)

K R :

Rosseland extinction coefficient (m−1)

I λ :

Spectral radiative intensity (W/m2/Sr)

I 0 λ (T):

Spectral radiative intensity of the black body at temperature T (W/m2/Sr)

L :

Thickness of the foam slice (m)

L opt :

Optical thickness of the foam slice (-)

M :

Foam density (kg/m3)

P λ (μ′ → μ):

Scattering phase function (-)

\( \dot{Q} \) :

Heat flux density (W/m2)

q r , q c , q t :

Radiative, conductive and total heat flux densities (W/m2)

\( \vec{r} \) :

Position vector

sp :

Specularity parameter (-)

T :

Temperature (K)

T hot , T cold :

Temperatures of the hot and cold boundaries (K)

V 1, V 2 :

Volume of struts and lumps (m3)

x = πA/λ :

Size parameter (-)

z :

1D coordinate (m)

β, σ, κ :

Extinction, scattering and absorption coefficients (m−1)

β*:

Weighted extinction coefficient (m−1)

\( \vec{\Updelta } \) :

Direction vector

ε:

Porosity (-)

μ:

Directing cosine of the radiant intensity (-)

θ:

Scattering angle (rad)

ρ:

Reflectivity (-)

λ:

Wavelength (m)

σ = 5.67 × 10−8 :

Stefan–Boltzmann constant (W/m2/K−4)

ω:

Scattering albedo (-)

fluid, solid:

Relative to the fluid/solid phases

DOM, ROSS:

Calculated by the DOM or the Rosseland Approximation

⊥, //:

Relative to perpendicular or parallel polarization

dif, spec:

Relative to diffuse or specular reflection

rad:

Radiative

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Authors and Affiliations

  1. Société « Etude Conseils Calcul en Mécanique des Structures » (EC2MS), 66, boulevard Niels Bohr, 69603, Villeurbanne Cedex, France

    Rémi Coquard

  2. Commissariat à l’Energie Atomique (CEA)/Le Ripault, BP 16, 37 260, Monts, France

    Denis Rochais

  3. Université de Lyon, CNRS, INSA-Lyon, CETHIL, UMR5008, 69621, Villeurbanne, France

    Dominique Baillis

  4. Université Lyon 1, 69622, Villeurbanne, France

    Dominique Baillis

Authors
  1. Rémi Coquard
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  2. Denis Rochais
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  3. Dominique Baillis
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Correspondence to Rémi Coquard.

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Coquard, R., Rochais, D. & Baillis, D. Conductive and Radiative Heat Transfer in Ceramic and Metal Foams at Fire Temperatures. Fire Technol 48, 699–732 (2012). https://doi.org/10.1007/s10694-010-0167-8

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