A cylindrical composite structure is formed from epoxy resin $($yellow$)$ and Aramid fabric

$($black$)$ as shown below. If the thermal conductivity, thickness and number of epoxy resin and

fabric layers are $k_e,t_e,N_e$ and $k_f,t_f,N_f,$ respectively,

a$)$ Derive a symbolic expression for the effective thermal conductivity $(k_{\text{e}\text{f}\text{f},r})$ of the Aramid fiber

reinforced composite structure with respect to the heat transfer in the r$-$direction.

reinforced composite structure with respect to the heat transfer in the z$-$direction.

c$)$ If the $\frac{r_{in}}{r_{\text{o}\text{u}\text{t}}}=0\mathrm{.}8,$ the $t\frac{h_e}{t}{h}_f=2,\frac{L}{r_{\text{o}\text{u}\text{t}}}=2$ and $\frac{k_e}{k_f}=5,$ calculate the ratio of heat transfer rate in

the r$-$direction to z$-$direction for $4$ Aramid fabric layers. $($Assume $1$D steady state heat transfer

with constant thermal conductivity and surface temperatures$)$

Effective thermal conductivity is used to model the composite structures as a homogenous material.

Top and bottom surfaces of the composite are coved by the epoxy layers.