Thermal resistance of riser/J-tubes in air/water

The heat transfer through risers and the J-tube air section is calculated considering:

  1. convection and radiation between the cable surface and the duct inner surface
  2. the conduction through the duct
  3. convection and radiation between the duct outer surface and ambient plus optionally solar radiation and wind.

Above the J-tube air section, the thermal network only needs to consider one of the separated power cores as the thermal influence of the hang off itself is considered negligible. The thermal network for a single power core is defined by a sub set of the thermal network designed for the J tube air section, with only $T_1$ and $T_2$, because due to the separation of the power cores, the amour layer is no longer present i.e. no need for $T_3$.

The radial thermal network below the sea level is comparable to the J-tube air section, with the only difference being that the convection and radiation terms within the air section are replaced by a solid water domain. The thermal resistance of the water between the cable and the J-tube is calculated using the standard thermal resistance equation for an annuls.

$\frac{\zeta_w}{2\pi} \ln\left(\frac{Di_D}{D_e}\right)$
0ERA empirical methodThe first method is an empirically derived method and was published by ERA in 1988. The cables had outer diameters 75 to 130 mm and the tubes 160 to 400 mm. For other values, the results must be used with caution. The method is intended for use with J tubes which are sealed at the top. The continuous rating is calculated by recognizing that under steady state conditions, the permissible heat flux across each radial component must be the same. Inherent within the balanced permissible heat flux statement is the assumption that minimal heat is produced within the armour and sheath which is a further limitation of this rating approach.
1Hartlein & BlackThe second method is an analytical method proposed by R.A. Hartlein and W.Z. Black in 1983. It is based around a thermal network model and is more general in its applicability than the ERA empirical method. It considers tubes which are both open and sealed at the top. The ladder network for the cable is akin to the network layout within IEC 60287-2, with the temperature difference between two radial positions being given as Δθ = qT where q is the heat flux passing through the region which has a thermal resistance, T.
2AndersThe third method was published by G.J. Anders in 1996 and is an extension of the pioneering work by Hartlein and Black which missed formulae for computation of heat transfer coefficients under certain conditions and required assumptions which were sometimes incompatible with typical cable-riser geometry. The paper by Anders updated the work of Hartlein and Black by redefining the mathematical model and supplementing information lacking in their work. Careful comparison of both models was made and reported in a previous paper by Anders, published 1995.
3ChippendaleThe fourth method is an analytical method proposed by R.D. Chippendale et al. in 2017 more in line with IEC 60287 approaches. Three component models are used, one for each of the three thermal sections: Central air section with the section below sea level before and the section with individual phases after. The results obtained demonstrate that a 2D approximation is acceptable for cases where the length of the tube air section is greater than 10 m. For lengths less than this, the use of the 2D methods becomes conservative and the 3D calculation is recommended, particularly for wind farm export cables with large conductor sizes.