This is the temperature rise at the surface of the cable produced by the total power $W_{tot}$ dissipated in cable k.

When the cables or ducts are embedded in concrete (backfill), the thermal resistivity of soil $\rho_4$ is replaced with the thermal resistivity of the backfill material $\rho_b$ or with the mean value of the thermal resistivities around cable p and k.

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. In this case, the parameters $d_{pk1}$ and $d_{pk2}$ are in meters.

Symbol
$\Delta \theta_{\mathrm{kp}}$
Unit
K
Formulae
 $\frac{W_{\mathrm{tot}} \rho_{\mathrm{4}}}{2 \pi} \ln{\left (\frac{d_{\mathrm{pk1}}}{d_{\mathrm{pk2}}} \right )}$ steady-state $\frac{W_{\mathrm{tot}} \rho_{\mathrm{4}}}{4 \pi} \left(\operatorname{expi}{\left (- \frac{d_{\mathrm{pk1}}^{2}}{4 \delta _{\mathrm{soil}} \tau} \right )} - \operatorname{expi}{\left (- \frac{d_{\mathrm{pk2}}^{2}}{4 \delta _{\mathrm{soil}} \tau} \right )}\right)$ long-term transients $- \frac{W_{\mathrm{tot}} \rho_{\mathrm{4}}}{4 \pi} \operatorname{expi}{\left (- \frac{d_{\mathrm{pk2}}^{2}}{4 \delta _{\mathrm{soil}} \tau} \right )}$ short-term transients
Related
$d_{\mathrm{pk1}}$
$d_{\mathrm{pk2}}$
$\delta _{\mathrm{soil}}$
$\Delta \theta_{\mathrm{p}}$
$\pi$
$\rho_{\mathrm{4}}$
$\tau$
$W_{\mathrm{tot}}$