Cable performance is fundamentally a thermal problem. Current flowing in the conductor creates heat, and that heat must be transferred through the cable construction and into the surrounding installation environment.
Thermal modelling checks that steady-state or transient cable temperatures remain within the limits of the conductor, insulation, sheath, surrounding materials and any personnel-access requirements. It provides the engineering basis for current rating, installation assessment and more detailed studies where standard tabulated ratings are not enough.
Heat flow in cables
The defining equation for heat flow by conduction is Fourier’s law:
For cable modelling, the thermal conductivity is often treated as constant over the temperature range of interest. The general heat-flow equation can then be written in transient form as:
For steady-state analysis, the time-dependent term is removed:
These equations are three-dimensional. Many cable problems can be simplified to a cross-section, using the x and y directions only, where longitudinal temperature variation is not significant.
Conduction, convection and radiation
Conduction governs heat transfer through solid cable layers and surrounding materials such as ducts, backfill or soil. Practical installations may also require convection at exposed surfaces and radiation where surface temperature and enclosure geometry make it significant.
The appropriate heat-transfer mechanisms depend on the installation: buried cables are usually dominated by conduction through soil or backfill, while cables in air may be affected by natural or forced convection and by solar radiation. For more on external heating, see Solar Radiation Effects on Cable Current Rating.
Heat generated in the cable
The main heat source in a cable is the conductor loss, calculated from I2R. Where required, sheath, armour and dielectric losses can be estimated using IEC 60287 methods.
The heat generation term, , is expressed in W/m3. A calculated loss in W/m therefore needs to be divided by the volume over which the heat is dissipated before it is used as a volumetric source term in a thermal model.
Related loss calculations are covered in IEC 60287 Cable Current Capacity and Cable Sheath and Armour Losses.
Solving the thermal model
A cable thermal analysis normally follows three steps:
- Model the installation as a set of heat-flow equations for the cable, surrounding materials and boundaries.
- Apply the relevant boundary conditions, including fixed temperature, heat flux, convection or mixed boundaries.
- Solve the equations to obtain heat flow and temperature throughout the installation.
Boundary conditions
| Boundary type | Condition | Typical setup |
|---|---|---|
| Constant temperature | Dirichlet | h = 1, r = desired temperature |
| Constant heat flux | Neumann | q = 0, g = desired heat flow |
| Convection or mixed | Neumann | q = convection coefficient, g = environment temperature multiplied by the convection coefficient |
Finite element analysis
Analytical solutions for real cable installations are usually difficult, and often impractical. Finite element analysis (FEA) solves the problem by dividing the installation geometry into a mesh of small elements, solving the heat-flow equations for those elements and assembling the result for the complete installation.
- Define the geometry of the cable installation.
- Set up the heat-flow equations for each material region.
- Apply boundary and initial conditions.
- Mesh the geometry.
- Solve the system and review temperatures, heat flow and margins.
Symbols
| A | Area, m2 |
| c or C | Specific heat of material, J/kg.°C |
| k | Thermal conductivity, W/m.°C |
| T | Temperature, °C |
| Ta | Surface temperature of enclosure |
| q | Heat flow, W |
| Heat generated per unit volume, W/m3 | |
| ρ | Density, kg/m3 |
| t | Time, s |
| g | Heat flux, W/m2 |
| h | Weighting coefficient |
| r | Temperature, °C |
| q | Heat transfer coefficient in boundary-condition notation |
Thermal analysis is closely related to current rating and fault-temperature checks. For short-circuit heating, see Cable Thermal Withstand Under Fault Conditions.
For the effect of sustained operating temperature on insulation ageing, see Estimating Cable Life.
For conductor heat generation and cable thermal modelling, see also Cable Power Loss.