Temperature rise by buried object k

This is the temperature rise of cable p due to the total power $W_{tot}$ dissipated in cable k.

When both cables p and k are embedded in the same concrete (backfill), the temperature rise by buried object k is adjusted with the difference between the thermal resistivity of the backfill material $\rho_b$ and the thermal resistivity of the soil $\rho_4$: $$\Delta\theta_{kp}=\Delta\theta_b+(\Delta\theta_{kp}-\Delta\theta_b)*\rho_b/\rho_4$$ $\Delta\theta_b$ is the temperature rise by buried object k on the top border of the backfill. With this method, the difference in temperature rise inside the backfill due to the difference in thermal resistivity is taken into consideration.

For transient calculations, this is the transient temperature rise at the surface of the cable produced by the total power $W_{tot}$ dissipated in cable k.

Symbol
$\Delta \theta_{kp}$
Unit
K
Formulae
$\frac{\rho_4 W_{tot}}{2\pi} \ln\left(\frac{d_{pk1}}{d_{pk2}}\right)$steady-state
$\frac{\rho_4 W_{tot}}{4\pi} \left(-\operatorname{expi}\left(\frac{-\left(\frac{d_{pk2}}{1000}\right)^2}{4\tau \delta_{soil}}\right)+\operatorname{expi}\left(\frac{-\left(\frac{d_{pk1}}{1000}\right)^2}{4\tau \delta_{soil}}\right)\right)$transient