External thermal resistance of tunnel

External thermal resistance of the surrounding soil of a tunnel.

In order to use the deep tunnel thermal inertial correction also for rectangular shaped tunnels, the outer diameter is calculated approximately using formula from Japanese Cable Standard JCS 0501. Then the parameter $u$ is calculated using this outer diameter and the formula for circular shape below is used. The thermal resistance of the wall of a rectangular tunnel or channel is calculated acc. Heinhold.

$\frac{\rho_4}{2\pi} \ln\left(u+\sqrt{u^2-1}\right)+T_{tw}$circular tunnel and rectangular tunnel (JCS)
$\frac{\rho_4}{2\pi} \ln\left(\frac{3.388L_{cm}}{\sqrt{A_t}}\right)+T_{tw}$rectangular tunnel (IEC)
$\frac{\rho_4}{\frac{4}{\pi} \left(1+\ln\left(\frac{\frac{h_t}{d_t+d_{im}}+1}{\sqrt{2}}+\sqrt{\left(\frac{\frac{h_t}{d_t+d_{im}}+1}{\sqrt{2}}\right)^2-0.5}\right)\right)+\frac{w_t}{d_t+d_{im}}}$rectangular channel (Heinhold)