Effective thermal conductivity of submicron powders: A numerical
Weijing Dai1,a, Yixiang Gan1,b*, Dorian Hanaor1,c
1The School of Civil Engineering, The University of Sydney, Sydney, Australia
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
Keywords: effective thermal conductivity, gas heat transfer, finite element analysis, compacted
Abstract: Effective thermal conductivity is an important property of granular materials in
engineering applications and industrial processes, including the blending and mixing of powders,
sintering of ceramics and refractory metals, and electrochemical interactions in fuel cells and Li-ion
batteries. The thermo-mechanical properties of granular materials with macroscopic particle sizes
(above 1 mm) have been investigated experimentally and theoretically, but knowledge remains
limited for materials consisting of micro/nano-sized grains. In this work we study the effective
thermal conductivity of micro/nano powders under varying conditions of mechanical stress and gas
pressure via the discrete thermal resistance method. In this proposed method, a unit cell of contact
structure is regarded as one thermal resistor. Thermal transport between two contacting particles
and through the gas phase (including conduction in the gas phase and heat transfer of solid-gas
interfaces) are the main mechanisms. Due to the small size of particles, the gas phase is limited to a
small volume and a simplified gas heat transfer model is applied considering the Knudsen number.
During loading, changes in the gas volume and the contact area between particles are simulated by
the finite element method. The thermal resistance of one contact unit is calculated through the
combination of the heat transfer mechanisms. A simplified relationship between effective thermal
conductivity and loading pressure can be obtained by integrating the contact units of the compacted
In energy systems, granular media are commonly used to store, convert, capture and produce
energy, including lithium-ion batteries, solid oxide fuel cells, and thermal storage systems. Heat
transport in these energy systems is a key issue necessitating extensive research to 1) establish a
fundamental understanding of the thermo-mechanical properties and 2) provide useful knowledge
for process optimisation [1-3]. Due to heterogeneous material properties, complex packing
structures and various inter-particle interactions, different theoretical methods have been developed
to study the heat transport mechanisms and the influencing external factors [4, 5].
Most research in this field has focused on the effective thermal conductivity (ETC) of powder
beds, employing three types of models to predict this parameter. Type I models are based on the
materials’ constitution and porosity and can be used to calculated the ETC by considering powder
beds as dispersion media . Type-II models regard powder beds as thermal circuits in order to
obtain the effective thermal resistance through an analogy with electrical circuits [4, 6-9]. Type-III
models discretise powder beds into many unit cells and integrate the cells’ properties to derive the
ETC [5, 10, 11]. The former two types of models can provide relatively simple equations to predict
the ETC while the type-III models usually involve considerable numerical calculation. However, it
is more convenient to study the effects of external factors on the ETC and to approximate the real
situation in practical application by applying type-III models. To realize a useful type-III model, a
proper description of the unit cell combing different heat transfer mechanism and external factors is
required. An analytical solution for a contact unit involving two spheres was proposed by Batchelor
and O’Brien in order to describe the relationship between ETC and contact area . Baharami et
al. employ the thermal contact resistance model to include the effects of rough contacts in the
Applied Mechanics and Materials Online: 2016-07-25
ISSN: 1662-7482, Vol. 846, pp 500-505
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spherical powder beds . The particle size of the powder beds considered in these studies has
typically been above 1 mm, thus the heat transfer through the interparticle gap is regarded as the
same as through a continuum gas phase. Furthermore, Gusarov et al. considered the effect of the
Knudsen number on the ETC of powder beds and the results shows that for millimetre- and
micrometre-sized particles the increase of gas pressure increases the ETC with saturation occurring
above a certain gas pressure . However, these studies have not addressed the powder beds
containing submicron or nano particles which are widely used in sintering process, thermal
insulation, and solid oxide fuel cells. In this study, the finite element method (FEM) is utilised to
calculate heat transport processes in the submicron units with different particle sizes. By altering the
gas phase pressure and mechanical deformation, the relationship between the ETC and loading
condition can be obtained.
A two-hemisphere contact unit cell is used to evaluate the effective thermal conductivity of
individual element as shown in Figure 1. The hemispheres, with constant material properties, form a
contact along the axial direction concentrically. The gap between the hemispheres is filled with air,
thus completing a cylindrical unit. In this simulation unit, a constant temperature difference
between the top surface and bottom surface is applied, and uniform temperatures of the upper and
lower hemispheres are established at the initial state as and , respectively. An axial
compression is applied from the top surface and the bottom surface is fixed. In this study, the
commercial finite element package ABAQUS is utilised to perform the FEM calculation. The
temperature difference ∆ is maintained to be 1K and −=∆ is created at initial state. The
properties of the air between the gap of the two hemispheres are assumed to be independent of the
temperature distribution of the air phase because ∆T is relatively small. Thus, in the numerical
model, a distance-dependent the gap thermal conductance (*GAP CONDUCTANCE in ABAQUS)
is used to approximate the thermal conductance via the gas gap.
(a) (b) (c)
Figure 1: Unit cell model: (a) schematic drawing of two spherical particles nearly in contact;
(b) Two particles in contact; (c) The FEM model with temperature profile with an arbitrary unit.
Here, ∆T, ∆x, hg and hs denote the temperature difference, particle deformation, heat flux through
the gas phase and through the solid contact, respectively.
Gas heat transfer at submicron-scale depends on the the ratio of the mean free path, , of gas
molecules to the gap distance, as given by the Knudsen number, . For the powder beds
comprised of submicron particles, the gaps between particles become comparable to the mean free
path of the gas molecules, leading to a transition of the heat transfer modes of gas phase. In other
words, the equation ℎ=/ to calculate the heat conductance of gas is invalid in this range,
where is the thermal conductivity of gas and is the gap distance. An approximation based on
∆T hg ∆T
Applied Mechanics and Materials Vol. 846 501
the Knudsen number is used to predict the gas phase heat conductance. The mean free path is
calculated based on the dynamic viscosity
and the dynamic viscosity is calculated by
where µ0 is dynamic viscosity at =296.15K. At a certain temperature , the mean free path
depends on gas pressure and gas constitution, where is the molar mass of the gas. When the
Knudsen number is very large ( ≫100), the heat conductance is approaching the free
molecular limit 
Here is the gas constant and is the adiabatic exponent. For 0.01<< 100, the heat
conductance at different Knudsen number is approximated by
It can be concluded that the gas heat conductance at ≫100 depends on not only the
temperature but also the gas pressure while the heat conductance is not affected by the gas pressure
for small values of Knudsen number.
Figure 2: Air heat conductance as a function of gap distance at different gas pressure, at = 298.
The solid line indicate the conductance calculated based on the continuum equation, ℎ=/.
502 Advances of Computational Mechanics in Australia
Further to the heat transfer of gas phase, the heat transfer through the contact region of two
hemispheres in one unit cell needs to be defined. Thermal contact resistance is commonly used to
describe the heat transfer across the solid contact between two relatively large hemispheres. In such
situation, the contact region consists of a finite number of nanoscale contact spots. However, this
contact model is not applicable for situation where the contact region itself falls into nanoscale. Due
to the consistent material properties of the two contact hemispheres, the nano contact region is
considered to have the same thermal properties as the bulk material. Thus the heat conductance of
the contact region is defined as the heat conductance of 1nm thick material in this simulation.
In ABAQUS, the gap conductance is varied with Knudsen number to simulate the gas heat
transport at different gas pressure. Deformation of the unit cell can be achieved by applying
displacement as the boundary condition. Table 1 summarizes the materials and model parameters
used in the ABAQUS simulations. The hemispheres are assumed to be SiO2 powders and the gas
phase is air. The negative sign of axial deformation ratio means that the two hemispheres are
separated and a finite gap is left between two hemispheres.
Table 1: Materials and model parameters used in ABAQUS simulation
Powder material: SiO2
Thermal conductivity ks=1.3 nW/nm·K, specific heat c=7*1011 nJ/kg·K, density
ρ=2.65*10-24 kg/nm3, Young’s modulus=7*10-8 N/nm2, Poisson’s ratio=0.17.
Molar mass M=0.028966 kg/mol, dynamic viscosity µ
Axial deformation ratio -1% 0% 1% 2% 5%
Particle diameter (nm) 100 200 500 1000
Gas pressure (Pa) 1 100 10000 100000 1000000 1000000
Result and discussion
In these models, the reaction heat fluxes are calculated according to the temperature boundary
conditions. At the equilibrium state, the effective thermal conductivity of individual unit cells can
be calculated based on the Fourier’s law
where is the reaction heat flux, denotes the effective thermal conductivity of unit cell, is the
axial -length of the unit cell and ∆ is the temperature difference.
Figure 3 presents the change of ETC of the unit cells versus the change of the gas pressure for
different particle sizes. When the two hemispheres are not in contact (-1% deformation), the heat
transfer inside the unit cell takes place entirely through the gas phase. It is clear that increasing the
gas pressure continuously increases the ETC of the unit cells which is consistent with the trend in
the heat conductance of air versus pressure in Figure 2. For conditions where the heat transfer occur
solely through the gas phase, larger particle sizes result in a larger ETC of the unit cells. Note that
the gap size is relative to the size of the unit cell, and in Figure 3(a) a displacement of -1% is
considered. When the two hemispheres approach to each other to form a point contact, the effect of
gas pressure diminishes at low gas pressure region. The gas pressure begins to affect ETC of unit
cells at value above 10 kPa and this effect becomes more significant as the gas pressure increases.
However, for conditions of particle contact the relationship between the particles size and ETC is
opposite to the situation of separated hemispheres shown in Figure 3(a). Compared with the ETC of
non-contact group, the ETC of this point contact group is much larger due to the presence of solid-
solid contact conductance. The contact point provides a thermal shortcut for the heat transport
because the heat conductance of the contact region is several orders (103~109) larger than the gas
Applied Mechanics and Materials Vol. 846 503
Figure 3: The effective thermal conductivity as a function of air pressure in different sized powders,
at T=298K: (a) non contact, -1% deformation, (b) point contact, 0% deformation,
(c) 2% deformation, and (d) 5% deformation
When the deformation increases as shown in Figure 3 (b-d), with an expanding contact region,
the ETC of the unit cells rises and follows a similar pattern as a function of the gas pressure. The
magnitude of the increase in the ETC due to the change in pressure in the three contact groups (0%,
2%, 5%) actually corresponds to the increment of ETC in the non-contact group (-1%). Only when
the gas heat transfer is comparable to the heat conduction through the contact region, the gas
pressure starts to significantly alter the ETC. This phenomenon indicates these two heat transport
mechanisms, i.e., heat conduction by solid-solid contact and heat transfer through gas phase, are
parallel, which is a general consideration in many other methods used to predict the ETC of powder
beds. Figure 4 shows the effect of deformation on the ETC of unit cells. As the deformation
increases, the ETC of unit cells of different converges. The reason is that the increasing contact area
dominates the heat transport in the unit cell, and the gas gap becomes less significant.
Figure 4: The effective thermal conductivity as a function of axial deformation in different sized
powders, at P=100 kPa and T=298 K
504 Advances of Computational Mechanics in Australia
The effective thermal conductivity of compacted submicron powders was studied for various
loading conditions, including gas pressure, temperature and deformation. A microscopic unit cell
was established to provide the basis for evaluating overall conductivity of the packed beds and was
analysed using finite element method. The Knudsen number has been incorporated in the gas
conduction model to include the influence of gap distance, gas pressure and temperature. The
numerical result shows a clear dependency of the effective thermal conductivity on parameters of
particle size, gas pressure and deformation. Increasing gas pressure tends to elevate the overall
conductivity at high pressure. When interparticles contacts are form, smaller sizes lead to larger
overall conductivity of unit cells. With increasing compaction, the overall conductivity increases
and the solid-solid contact conduction dominates the heat transfer. This study provides a theoretical
basis to consider the heat transfer in compacted beds with submicron-sized particles under a wide
range of loading conditions.
Financial support for this research from the Australian Research Council through grants
DE130101639 is greatly appreciated.
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