Stress-induced evolution of anisotropic thermal conductivity of dry granular materials

Abstract

Various factors, such as the volumetric fraction of constituents, mineralogy, and pore fluids, affect heat flow in granular materials. Although the stress applied on granular materials controls the formation of major pathways for heat flow, few studies have focused on a detailed investigation of its significance with regard to the thermal conductivity and anisotropy of the materials. This paper presents a numerical investigation of the stress-induced evolution of anisotropic thermal conductivity of dry granular materials with supplementary experimental results. Granular materials under a variety of stress conditions in element testing are analyzed by the three-dimensional discrete element method, and quantitative variations in their anisotropic effective thermal conductivity are calculated via the network model and conductivity tensor measurements. Results show that the directional development of contact area and fabric under anisotropic stress conditions leads to the evolution of anisotropy in thermal conductivity. The anisotropy induced in thermal conductivity by shear stress is higher than that induced by compressive stress because shear stress causes more significant changes in microstructural configurations and boundary conditions. The shear-stress-induced evolution of anisotropy between principal thermal conductivities depends on dilatancy as well as shearing mode, and the shear-driven discontinuity localizes the conductivity. Factors involved in the stress-induced evolution and their implications on the thermal conductivity characterization are discussed.

Appendices

Appendix A: Estimation of thermal conductivity via the thermal network model

Suppose that the granular assembly comprises a web of particles interconnected by thermal conductors. When the sum of heat flow (q _ij ) of all the connected thermal conductors from particles i and j is in equilibrium with a temperature difference (T _i  − T _j ) and thermal conductance (C _ij ) between the particles, energy conservation leads to the following steady-state equation:

$$ \sum\limits_{i \to j} {q_{ij} } = \sum\limits_{i \to j} {C_{ij} (T_{i} - T_{j} ) = 0} $$

(a1)

The initial temperature at each particle is determined by imposing constant temperature at the two boundary planes (e.g., top and bottom planes for $k_{zz}^{\text{eff}}$) and then performing successive iterations until the variation of temperature in each step becomes lower than a prescribed value. The thermal energy between particles is determined by the particle conductance ($C_{i}^{\text{p}}$) and contact conductance ($C_{ij}^{\text{c}}$), given by the following equation:

$$ q_{ij} = C_{ij}^{\text{eff}} \cdot \Updelta T = \left[ {\frac{1}{{C_{i}^{\text{p}} }} + \frac{1}{{C_{ij}^{\text{c}} }} + \frac{1}{{C_{j}^{\text{p}} }}} \right]^{ - 1} \left[ {T_{i} - T_{j} } \right] $$

(a2)

The neighboring particles can be considered to be either connected by a circular contact (i.e., overlapped) or separated (i.e., not overlapped), depending on the distance between particles (h _ij ), which is given by the following relation:

$$ h_{ij} = D - (r_{i} + r_{j} ) $$

(a3)

where D is the distance between the particle centers, and r _i and r _j are the radii of particles i and j, respectively. The particles are interconnected when h _ij  ≤ 0, and separated otherwise. It is assumed that heat transfer occurs between particles when h _ij is lower than ε^eff:

$$ \varepsilon^{\text{eff}} = \varepsilon \cdot \frac{{2r_{1} r_{2} }}{{r_{1} + r_{2} }} $$

(a4)

where ε is the cut-off range parameter to define the effective zone of heat transfer. Following the calculation of the temperature of particles in a random assembly by iterating Eq. (a1) according to conductance as described in [3, 46], the total heat flow of all the particles is determined by Eq. (a2). Finally, the effective thermal conductivity of a given granular assembly k ^eff is given by using Fourier’s law.

$$ k^{\text{eff}} = \frac{Q \cdot H}{A} \cdot \frac{1}{\Updelta T}\left[ \frac{W}{mK} \right] $$

(a5)

where Q is the total heat flow out of the boundary plane; H and A are the height and cross-sectional area of the assembly, respectively; and ΔT is the temperature difference between the two boundary planes.

Appendix B: Numerical formulation of the thermal conductivity tensor measurements

Assume a particle and a contact represent a heat reservoir and a one-dimensional thermal pipe, respectively. If we regard that heat flow occurs only in the thermal pipes, the average heat flux in a volume V of granular material is defined by a sum over all M pipes (see [17] for detailed derivation):

$$ {q_{i}} = \frac{1}{V}\sum\limits_{p = 1}^{M} {q_{i}^{(p)} A^{(p)} l^{(p)} }$$

(b1)

where A ^(p) and l ^(p) are the length and cross-sectional area of pipe p, respectively, and the heat flux in the pipe $q_{i}^{(p)}$ is given by the following relation:

$$ q_{i} = - \frac{{\Updelta Tn_{i} }}{\eta LA} $$

(b2)

where ΔT is the temperature difference between the two reservoirs on each end of the pipe; η _i is the outward unit normal vector; and L and η are the length and the thermal resistance per unit length of the pipe, respectively.

If the mean temperature gradient in the material, $\partial T/\partial x_{j}$, is applied at the microlevel within each pipe, then the temperature difference is given as shown below.

$$ \Updelta T = Ln_{j} \frac{\partial T}{{\partial x_{j} }} $$

(b3)

Substituting Eqs. (b2) and (b3) into (b1) gives the following equation:

$$ q_{i} = - \left[ {\frac{1}{V}\sum\limits_{p = 1}^{M} \frac{{l^{(p)}} n_{i}^{(p)} n_{j}^{(p)}} {\eta^{(p)}}} \right] \frac{\partial T}{{\partial x_{j} }} $$

(b4)

The heat flux vector and the temperature gradient are related by Fourier’s law for a continuum, as shown below:

$$ q_{i} = - k_{ij} \frac{\partial T}{{\partial x_{j} }} $$

(b5)

By comparing Eq. (b4) with Fourier’s law, Eq. (b5), the thermal conductivity tensor of the granular material is given by the following equation:

$$ k_{ij}^{\text{eff}} = \frac{1}{V}\sum\limits_{p = 1}^{M} \frac{{l^{(p)}} n_{i}^{(p)} n_{j}^{(p)}} {\eta^{(p)}}$$

(b6)