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Thermal resistance between one conductor and sheath

All three-core cables require fillers to fill the space between insulated cores and the belt insulation or a sheath. The fillers of extruded cables usually have higher thermal resistivities than the insulation, unlike paper-insulated cables In a paper by G. Anders from 1998, a formula was developed to take into account the different thermal resistivities. This formula has been included in the new editions of the IEC standards.

For three-core cables with a touching metallic screen made of copper tapes each core the thermal resistance of the insulation is obtained in two steps:

  1. First, the cables are considered as belted cables for which $t_1/t$ = 0.5.
  2. Second, the resulting $T_1$ is multiplied by a factor $K$, called the screening factor.
In a paper by G. Anders from 1999, a more precise formula was developed. This formula has not been included in the IEC, but is used by Cableizer.

Type of screen/sheath of multi-core cables

For calculation of the cyclic and emergency current rating of cables acc. IEC 60853, the multi-core cable is replaced by an equivalent single-core construction dissipating the same total conductor losses. The space between the equivalent single-core conductor and the sheath is considered to be completely occuied by insulation (for oil-filled cables, this space is filled partly by the total volume of oil in the ducts and the remainder is oil-impregnated paper).

Symbol
$T_1$
Unit
K.m/W
Formulae
$\frac{\rho_i}{2\pi} \ln\left(1+\frac{2t_1}{d_c}\right)$Single-core cables
$\frac{\rho_i}{2\pi} G_1$Multi-core cables, general formula
$K_1 \frac{\rho_i}{2\pi} G_1$Three-core cables, screened, round conductors
$0.89T_1+K_1 \left(\frac{\sqrt{3} \rho_f}{\rho_i}-0.12d_c+2.25\right) e^{0.13+\frac{t_1}{d_c}-7K_1}$Three-core cables, screened, round conductors (Anders1999)
$\frac{\rho_i}{2\pi} G_1+0.031\left(\rho_f-\rho_i\right) e^{\frac{0.67t_1}{d_c}}$Three-core cables, unscreened, round conductors
$0.385\rho_i \frac{2t_i}{d_c+2t_i}$Three-core cables, oil-filled, round conductors, with metallized paper core screens & oil ducts between the cores
$0.35\rho_i \left(0.923-\frac{d_c}{d_c+2t_i}\right)$Three-core cables, oil-filled, round conductors, with metal tape core screens & oil ducts between the cores
$\frac{475}{{D_{sc}}^{1.74}} \left(\frac{0.5\left(\frac{D_{shb}+D_{sh}-2t_{sh}}{2}-2.16D_{sc}\right)}{D_{sc}}\right)^{0.62}+\frac{\rho_i}{2\pi} \ln\left(\frac{d_c-2t_{sc}}{d_c}\right)$Three-core cables, oil-filled, round conductors, without fillers and oil ducts, having a copper woven fabric tape binding the cores together and a corrugated aluminium sheath
$T_{ct}+T_{cs}+T_{ins}+T_{is}+T_{scb}+T_{scs}+T_{dsh}$Single-core cables, Cigre Guide 880 GP15
$\frac{1}{F_{lay,3c}} T_1$Three-core cables, Cigre Guide 880 GP44
$1.07T_1$Single-core cables, part-metallic sheathed, trefoil, up to 35 kV
$1.16T_1$Single-core cables, part-metallic sheathed, trefoil, from 35 to 150 kV
$\frac{\rho_i}{2\pi} \ln\left(\frac{D_{ins}}{d_c}\right)$4-loop method
$\frac{\theta_c-\theta_{encl}}{W_{conv,ce}+W_{rad,ce}}$PAC/GIL
Related
$d_c$
External diameter of conductor [mm]
$D_{ins}$
Diameter over insulation [mm]
$D_{sc}$
Diameter over screen [mm]
$D_{sh}$
Diameter over sheath [mm]
$D_{shb}$
Diameter below sheath [mm]
$F_{lay,3c}$
Effective length per unit lay length of twisted conductors
$G_1$
Geometric factor $G_1$
$K_1$
Screening factor
$\pi$
Archimedes' constant $\pi$
$\rho_f$
Thermal resistivity of filler [K.m/W]
$\rho_i$
Thermal resistivity of insulation [K.m/W]
$t_1$
Thickness of insulation to sheath [mm]
$T_{cs}$
Thermal resistance of conductor shield [K.m/W]
$T_{ct}$
Thermal resistance of conductor tape [K.m/W]
$T_{dsh}$
Thermal resistance of corrugation filling [K.m/W]
$t_i$
Thickness of insulation [mm]
$T_{ins}$
Thermal resistance of insulation [K.m/W]
$T_{is}$
Thermal resistance of insulation screen [K.m/W]
$t_{sc}$
Thickness of the screen [mm]
$T_{scb}$
Thermal resistance of screen bedding [K.m/W]
$T_{scs}$
Thermal resistance of screen serving [K.m/W]
$t_{sh}$
Thickness of the sheath [mm]
$\theta_c$
Temperature of conductor [°C]
$\theta_{encl}$
Temperature of enclosure [°C]
$W_{conv,ce}$
Convection heat transfer, conductor to enclosure [W/m]
$W_{rad,ce}$
Radiation heat transfer, conductor to enclosure [W/m]
Used in
$d_{ct}$
External diameter of conductor for transient calculations [mm]
$I_c$
Conductor current [A]
$Q_{B,i}$
Element B of two-part thermal circuit, insulation [J/(m.K)]
$Q_{B,s}$
Element B of two-part thermal circuit, screen/sheath [J/(m.K)]
$R_{c1}$
Thermal resistance, 1st loop [K.m/W]
$T_A$
Element A of equivalent thermal circuit [K.m/W]
$T_B$
Element B of equivalent thermal circuit [K.m/W]
$T_d$
Internal thermal resistance for dielectric losses [K.m/W]
$T_{eq}$
Equivalent thermal resistance [K.m/W]
$T_{int}$
Internal thermal resistance for current losses [K.m/W]
$T_r$
Total thermal resistance [K.m/W]
$T_{tot}$
Total thermal resistance, transient [K.m/W]
$\theta_{ar}$
Temperature of armour [°C]
$\theta_{c,z}$
Conductor temperature at z [°C]
$\theta_e$
External temperature of the object [°C]
$\theta_f$
Temperature of filler for multi-core cables type SS with sheath [°C]
$\theta_s$
Temperature of screen/sheath [°C]
$\theta_{sc}$
Temperature of screen [°C]
$\theta_{sh}$
Temperature of sheath [°C]
$\theta_{sp}$
Temperature of steel pipe [°C]