This is the transient temperature rise of outer surface above ambient temperature.

The transient response of the cable environment is calculated by an exponential integral formula which is the direct transient equivalent of the steady-state procedure adopted in IEC 60287, for groups of separate cables. The transient temperature rise, $Δθ_{e_t}$, above ambient of the outer surface of the hottest cables of a group of cables, and similarly loaded, is given by this formula.

If the duration of the complete transient period $τ$ results in the use of long-term formulae, then this is also true for all calculations done at partial time steps, even though it may be less than ⅓ of the time constant of the cable and the use of short-term formulae could be justified. However, this would lead to a discontinuity in the temperature rise curve that cannot be justified.

Note: The parameters $D_o$ and $L_cm$ are in meters.

$\Delta \theta_{\mathrm{e_{\mathrm{t}}}}$

K

$\frac{W_{\mathrm{I}} \rho_{\mathrm{4}}}{4 \pi} \left(- \operatorname{expi}{\left (- \frac{D_{\mathrm{o}}^{2}}{16 \delta _{\mathrm{soil}} \tau} \right )} + \operatorname{expi}{\left (- \frac{L_{\mathrm{cm}}^{2}}{\delta _{\mathrm{soil}} \tau} \right )}\right)$ | long-term transients |

$- \frac{W_{\mathrm{I}} \rho_{\mathrm{4}}}{4 \pi} \operatorname{expi}{\left (- \frac{D_{\mathrm{o}}^{2}}{16 \delta _{\mathrm{soil}} \tau} \right )}$ | short-term transients |

$\delta _{\mathrm{soil}}$

Soil thermal diffusivity [m²/s]

$L_{\mathrm{cm}}$

Depth of laying [m]

$\pi$

$\rho_{\mathrm{4}}$

Thermal resistivity of soil [K.m/W]

$\tau$

Transient period [s]

$W_{\mathrm{I}}$

Ohmic losses per phase [W/m]