A twodimensional numerical simulation based on the finite element method was used to solve the steady state conductive heat transfer problem. The radiative and convective heat transfers are negligible. The objective of this study is to estimate the effective conductivity of the PSU/BaTiO3 composite material considering the geometrical configuration shown in Fig. 2. The chosen configuration is represented by an elementary cell modeled by a BaTiO3 particle of circular shape with a radius r between 0.3µm and 0.8µm, centered in the matrix (PSU) of square shape of L = 2µm dimension. These are polymer matrices loaded with different particles at different volume concentrations.
The temperature domain in the composite material was defined by numerically solving the Laplace equation (Eq. 1) using a finite element formulation, imposing the following boundary conditions:

The two faces perpendicular to the direction of heat flow are assumed to be isothermal with temperatures T1 = 298 K and T2 = 323 K.

The faces parallel to the direction of the heat flow are adiabatic;

The transfers by radiation and convection are negligible;

The thermal contact resistance between matrices and loads is negligible;

The dispersion of the spherical particles in the matrices is homogeneous;
Heat transfer in a material is due to the phenomenon of conduction described by the equation:
$$0=\rho cp\times \frac{\partial T}{\partial t}\nabla (\text{k}\nabla \text{T})$$
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The heat flow through the hot face to the cold face is calculated from the integration of the Laplace equation (Eq. 1) applied to the boundaries of the elementary cell of the composite.
$$Q={\int }_{A}k\frac{\partial T}{\partial z}dxdy$$
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Where:
Q is the heat flux; k is the thermal conductivity of the composite;
\(\frac{\partial T}{\partial z}\) is the temperature variation along the z direction.
x and y represent the exchange surface.
For the calculation of the effective thermal conductivity, we introduce the value of the flux calculated from the COMSOL software, in Eq. (3). Thus, if we know the flux, the effective thermal conductivity is calculated from Eq. (3):
$$k= \frac{Q}{A}\frac{{L}_{z}}{({T}_{1}{T}_{2})}$$
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Where the exchange area A = x × y (m²); z is calculated,
ΔT = T1 T2 (°K) is the temperature variation
The effective thermal conductivity of the composite is given by the following formula: [23]
$$k= \frac{(Q\times Z)}{(x\times y\times ({T}_{1}{T}_{2}\left)\right)}$$
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