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\rho \, c \, \frac{\partial T}{\partial t} = \frac{d}{dl} \, \bigg( \lambda \, \frac{dT}{dl} \bigg)  - \rho \, c \, v \, \frac{dT}{dl} + \frac{2 \lambdaU}{\lambda_e} \cdot \frac{r_fw}{r_w^2} \cdot U \cdot \left[ big( T_e(t, l, r_w) - T \right]big)

\rho_e \, c_e \, \frac{\partial T_e}{\partial t} = \nabla ( \lambda_e \nabla T_e)

T(t=0, l) = T_{e0}(l)

T_e(t=0, l, r) = T_{e0}(l)

T(t, l=0) = T_0(t)

T_e(t, l, r \rightarrow \infty) = T_{e0}(l)

2 \pi \, \lambda_e \, r_w \, \frac{\partial T_e}{\partial r} \, \bigg|_{r=r_w} = 2 \pi \, r_f \, U \cdot \left( T_e \, \bigg|_{r=r_w} - T \right)