Understanding of how cables (and busbar) perform is at heart a thermal problem.
Heat is generated in the cable due to current flowing. This generated heat then interacts with the environment and is dissipated. Thermal modelling of a cable installation establishes that the steady state temperatures are below the safe operating conditions for the materials and personnel.
Note: for a list of symbols, see the bottom of the post.
The defining equation of heat flow is governed by Fourier's law of heat conduction:
For cable modelling, we typically assume constant thermal conductivity and the generalised heat flow equation (PDE) is given by:
ρC∂T∂t-∇·k∇T=q˙ - parabolic form (transient)
-∇·k∇T=q˙ - elliptic form (steady state)
Note: the above equation is for three dimensions. Often in cable problems, we are only concerned with a cable section and will use partial derivatives in the 'x' and 'y' plane only.
The above equations govern heat flow by conduction. In practical situations, we often also have to consider Convection and Radiation.
The power dissipated in the cable (or conductors) is calculated I2R. Where required sheath and dielectric losses can be estimated using IEC 60287. For further information, see:
It should be noted that heat generated Q, is in W/m3. Any calculated I2R power needs to be converted to W/m3, by dividing the value obtained by the volume over which the power is dissipated.
To solve a cable installation problem, the following steps are carried out:
Depending on the complexity of the installation, various boundary conditions may be relevant. For cable installation, we typically come across the following boundary conditions.
To solve the cables analytically, would be extremely difficult if not impossible. We there for use finite element analysis (FEA). FEA involves breaking the geometry into a mesh of small solvable grids (such as tetrahedrons). By solving all the small grids we are able to solve the complete cable installation.
Typical steps in FEA consist of:
A - area, m2
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 (or Joules/second)
q˙ - 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