Abstract
The properties of thermal networks are examined to understand the effective thermal conductivity of heterogeneous two-phase composite materials and systems. At conditions of high contrast in thermal conductivity of the individual phases (k1 and k2), where k1 << k2 or k1 >> k2, the effective thermal conductivity of individual networks of the same composition was seen to be highly sensitive to the distribution of the phases and the presence of percolation paths across the network. However, when the contrast in thermal conductivities of the two phases was modest (k1/k2 ~ 10−2 to 102), the thermal networks were observed to exhibit an emergent response with a low variability in the effective thermal conductivity of mixtures of the same composition. A logarithmic mixing rule is presented to predict the network response in the low variability region. Excellent agreement between the model, mixing rule and experimental data is observed for a range two-phase porous and granular media. The modelling approach provides new insights into the design of multi-phase composites for thermal management applications and the interpretation or prediction of their heat transfer properties.
Highlights
Interest in the thermal characteristics of multi-phase and composite materials has continued to grow, as the need for such materials across a broad spectrum of areas has increased; sectors of interest include aerospace, packaging, electronics, construction and processing [1,2,3,4,5,6,7,8,9]
Thermal networks with randomly distributed elements were modelled with a range of k1 and k2 volume fractions (α1 = 0.3, 0.5 and 0.7) and twelve individual random networks were analysed at each volume fraction, to examine the variability of the network thermal conductivity as a consequence of the different random distributions of the k1 and k2 phases within each network
At conditions of high contrast in thermal conductivity of the individual phases, the thermal conductivity of each network is sensitive to the distribution of the phases across the network, as it prefers to flow through k1 or k2; see Figure 3b,f
Summary
Interest in the thermal characteristics of multi-phase and composite materials has continued to grow, as the need for such materials across a broad spectrum of areas has increased; sectors of interest include aerospace, packaging, electronics, construction and processing [1,2,3,4,5,6,7,8,9]. For heterogeneous composites and two-phase systems the situation is more complex, since the thermal properties are strongly dependant on a number of factors; these include the volume ratio of the constituents, their individual thermal properties and the distribution of constituents within the material This represents a challenge since, in many cases, a multi-phase system cannot be characterised by a single value of thermal conductivity and random mixtures of the same volume fraction can exhibit different conductivities, due to small differences in their spatial distribution. In order to provide a large range of contrasting k1 and k2 magnitudes for the thermal networks and explore mixtures containing phases of different thermal conductivity, the thermal contrast (k = k1/k2) between the two phases was varied for each network This spanned the conditions of: (i) k = 10−4, where heat is likely to flow through the k2 phase, (ii) k ~ 1, where heat is likely to flow. This was achieved by setting k2 to a constant value of 1 W·m−1·K−1 and varying k1 from 10−4 to 104 W·m−1·K−1
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