# Anisotropic Heat Transfer in Plane of Carbon Fabrics Reinforced Geopolymer Composite
## 2 Model Layout and Properties for Simulation  
### The Model for the Fiber Reinforced Geopolymer Composite  
Figure 1 displays the model for the fiber reinforced geopolymer composite. The carbon infinite element domain is presented in the boundary with the geopolymer composite. The mesh size of the outer domain and inner arrangement is shown with a maximum element size of 0.1 cm and with a minimum element size of 0.018 cm. The maximum element growth rate was 1.5 with a curvature factor of 0.6.  
The diffusion of the outer wall is zero in stationary condition as below  
\[-n\cdot\
abla u=0 \tag{9}\]  
The heat transfer within the solid of the composite is defined as below  
\[\rho C_{p}\cdot\
abla T+\
abla\cdot q=Q+Q_{ted} \tag{10}\]  
\[q=-K\
abla T\]  
Here, \(\rho\) is the density of the composite, \(C_{p}\) is the heat capacity at constant pressure, \(K\) is the thermal conductivity of composite, with \(q\) as the heat source.  
Figure 2 shows the cross section of the carbon fiber reinforced geopolymer composite. The infinite domains that present within the boundary wall of the geometry truncate to model a boundary stage of fibers. Based on the heat transfer module, infinite element domains could suppress boundary effects.  
Figure 1: Model geometry of carbon fibers reinforced in geopolymer matrix with infinite element domain on the left side (highlight fiber, matrix hidden) and right side (matrix highlight, fiber hidden).  
Figure 2: Outer domain wall of the fiber reinforced in geopolymer composite (**a**) and (**b**) mesh size of the composite.  
The material properties are summarized in Table 1 for the carbon fiber reinforced geopolymer composite.  
Anisotropic thermal conductivity is represented as second order tensor. The tensor is defined as  
\[K=\begin{cases}K_{xx}&K_{xy}&K_{xz}\\ K_{yx}&K_{yy}&K_{yz}\\ K_{zx}&K_{zy}&K_{zz}\end{cases}=\begin{bmatrix}60&0&0\\ 0&4&0\\ 0&0&4\end{bmatrix} \tag{11}\]  
In general, thermal conductivity of a composite is highest along the plane direction compared to thickness. The matrix represented for \(K\) is provided x-component depending upon the shape of the fibers. The curvilinear coordinate provides such vector system for the interface. Three predefined methods could be able to function in the curvilinear coordinate system. In this system, the diffusion method has been used in a scalar potential. The streamline flow of heat represents along the direction of the carbon fiber. The base vector system for a single fiber is shown in Figure 3. The parameters used for the fiber axis are chosen with the specification as below a number of four loops considered along the fiber axis that determines a number of seven fibers per axis. The length of the loop is considered as 0.2 mm with an amplitude of 0.05. The curvilinear coordinates allow the inlet and outlet boundaries that allow the heat transfer in the direction of the composite (Figure 4). All other boundaries are thermally insulated walls. A Gaussian field of heat distribution in the center of the geometry is applied with a convective cooling boundary condition on both sides.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Property** & **Geopolymer** & **Carbon Fiber (Core)** & **Carbon (Infinite** \\  & **Ref [6]** & **Ref [6]** & **Domain**) \\ \hline Thermal conductivity & 0.1141 W/(m K) & \{60, 4, 4\} W/(m K) & 60 W/(m K) \\ Density & 2000 kg/m\({}^{3}\) & 1500 kg/m\({}^{3}\) & 1500 kg/m\({}^{3}\) \\ Heat capacity at constant pressure & 1650 J/(kg K) & 1000 J/(kg K) & 1000 J/(kg K) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermal conductivity properties of carbon fiber reinforced geopolymer composite.  
Figure 3: A three-dimensional curvilinear coordinate system for a single fiber.