# Thermal resistance of trough

Thermal resistance of trough to account for increase in the air temperature inside the trough to the ambient air temperature.

Symbol
$T_{tr}$
Unit
K
Formulae
 $\frac{1}{3 p_{tr}}$ IEC 60287-2-1 $\frac{0.2}{0.705 h_{t} + w_{t}}$ IEE Wiring Regulations, BS 7671 $\frac{\pi \rho_{4}}{2 \left(\frac{\pi w_{t}}{2 t_{t}} + \ln{\left(\frac{t_{t}^{2}}{\left(2 h_{t} + 2 t_{t}\right)^{2}} \right)} + 2\right)}$ Slaninka I with all equal resistivites $\frac{1}{\frac{\frac{\pi w_{t}}{t_{t}} - 4 \ln{\left(2 \right)} + 4}{\pi \rho_{t}} + \frac{2 \ln{\left(2 \right)}}{\pi \rho_{4}} - \frac{2 \ln{\left(\frac{t_{t}^{2}}{\left(2 h_{t} + 2 t_{t}\right)^{2}} \right)}}{\pi \rho_{4}}}$ Slaninka II with different resistivites $\frac{N_{sum}}{W_{sum}} + t_{t} \left(\rho_{4} - 0.9\right) + \frac{0.333}{p_{tr}}$ Anders I based on simplified finite element study $\frac{\rho_{4}}{H_{tc} + \phi_{b} + 0.3907}$ Anders II extending Slaninka II
Related
$h_{t}$
$p_{tr}$
$\rho_{4}$
$\rho_{t}$
$t_{t}$
$W_{sum}$
$w_{t}$
Used in
$\Delta \theta_{air}$
Choices
IdMethodInfo
0IEC 60287-2-1An empirical formula is used which gives the temperature rise of the air in the trough above the air ambient as Δθ being equal to the total power dissipated in the trough per meter length (W/m) divided by 3 times that part p of the trough perimeter which is effective for heat dissipation (m). Any portion of the perimeter, which is exposed to sunlight, is therefore not included in the value of p.
1IEE Wiring Regulations, BS 7671The IEE Wiring Regulations, BS 7671, is a United Kingdom standard that sets out the requirements for low-voltage electrical installations. Rating factors are given for standard troughs and are applied to the tabulated rating for cables in free air. The origin of the derating factors is not known but considered likely to derive from equations given in an anonymous document probably prepared in about 1950. It is stated that a ground temperature of 25$^{\circ}$C is used and the method is only applicable to troughs within buidings where conditions are generally drier, and thermal resistivites higher, than for outdoor troughs.
2Slaninka I with all equal resistivitesAs with the IEC 60287 method, Slaninka I leads to an additional temperature rise, which is added to the ambient air temperature. Isothermal conditions for both the ground surface and the inner surface of the enclosure are assumed and certain assumptions were made that are only valid for troughs of roughly square cross-section.
3Slaninka II with different resistivitesSlaninka II is an extended solution to a situation with non-isothermal conditions and troughs that are not roughly square. This work also attempted to develop a method for calculating the heat transfer between the cable surface and the inner surface of the trough. The method divided the trough along a horizontal axis that ran through the centreline of the cables. Heat transfer to the upper part of the enclosure was taken to be by convection and radiation, while that to the lower part was taken to be by radiation only. Heat transfer by conduction for cables laid on the base of the trough was not considered.
4Anders I based on simplified finite element studyThis is the proposed analytical solution by Anders et al based on finite element study. In the analysis, the thermal resistances of the materials of the trough and the surrounding soil were equal to each other and changed simultaneously. Changing the thermal resistance results in a significant change of temperatures of the cable. On the basis of the analysis, the influence of the soil thermal resistivity is taken into account by modifying the formula from IEC 60287-2-1.
5Anders II extending Slaninka IIThis is the proposed analytical solution by Anders et al based on previous work and is an extension of the Slaninka’s method. The calculated results were compared with with those obtained from test work performed at ERA Technology outside in 1968 and reported in Electricity Council Research Centre (ECRC) Report R219 and gives good agreement. The method is limited to only one system though.