Radiative heat transport in porous materials
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1162-J03-03
Radiative heat transport in porous materials Valery Shklover1, Leonid Braginsky1, 2, Matthew Mishrikey2 and Christian Hafner2 1 Laboratory of Crystallography, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland 2 Laboratory of Electromagnetic Fields and Microwave Electronics, ETH Zürich, 8092 Zürich, Switzerland ABSTRACT The effect of porosity on the radiation component of the thermal conductivity of the thermal barrier coatings is studied. Heat transfer in the disordered porous structures as well as the porous photonic bandgap structures is investigated. The pores, which size is comparable with the characteristic radiation wavelength λmax=2897.8/T µm, were found to be most efficient obstacles for the heat radiation. I. INTRODUCTION Modern industry requires the development of thermal barrier coatings with service temperatures of about 1000–1500 oC and higher. Nano- and micro-grained metal oxide materials are commonly used (e.g., α-Al2O3). However, being good thermal insulators at intermediate temperatures (T1000 oC) due to radiative effects. On the contrary, porous materials (e.g., plasma sprayed yttrium stabilized zirconia coatings) do not demonstrate any increase of the thermal conductivity at high temperatures. The aim of this report is to understand the reason of such difference and suggest possible conceptions of the coating engineering. We analyze the radiative component of the thermal conductivity and consider the problem of heat photon propagation in non-homogeneous media. In particular, photon propagation in nano-grained and porous materials is investigated. Dependence of the radiation component on the pore size and total porosity is investigated. It is shown that materials having pores of size comparable with wavelength of the thermal photons are most efficient for insulating radiative heat. II. RADIATIVE HEAT TRANSPORT AT HIGH TEMPERATURES Consider the heat radiation incident on the coated surface. Suppose the black-body radiation, whose frequency distribution obeys the Planck's law. The radiation part of the thermal conductivity can be estimated as 1 3
κ r = CV SL
(1)
Here CV is the heat capacity per unit volume, S = c / ε is the velocity of light in the media, ε is the dielectric constant, and l is the mean free path (MFP) of the photons. Taking into account the dependence of these values on the photon frequency ω, we rewrite (1) as follows:
κr =
c 3 ε
∫
∞
0
CV (ω )l (ω )dω ,
where CV (ω ) =
(2)
h ∂ ω3 . 2 3 hω / k B T π c ∂T e − 1
Here h and kB are the Planck and Boltzmann constants, respectively. Behavior of the integrand 2) at small frequencies is very important for the integral value. Indeed, if we assume that the mean free path (MFP) is determined by the structure disorder, then the MFP should follow the Rayleigh law l (ω ) ∝ ω −4 , CV (ω ) ∝ ω 2 , and the integrand (2) has a singularity at ω → 0 . The simplest way to avoid this difficulty has been suggested in [1]. The authors introduce the cutoff frequency ω A , assuming l (ω ) = 0 , if ω ≤ ω A ,
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