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.

Heat Flow

The defining equation of heat flow is governed by Fourier's law of heat conduction:

        $q_x=-kA\frac{\partial T}{\partial x}$    

Conduction, Convection & Radiation

For cable modelling, we typically assume constant thermal conductivity and the generalised heat flow equation (PDE) is given by:

        $\rho C\frac{\partial T}{\partial t}-\nabla ·\left(k\nabla T\right)=\dot{q}$    - parabolic form (transient)

        $-\nabla ·\left(k\nabla T\right)=\dot{q}$    -   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.

Power (heat generated)

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:

• conductor loss: Power Loss
• sheath loss: [work in progress]
• dielectric loss: Dielectric loss in cables

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. 

Solving the problem

To solve a cable installation problem, the following steps are carried out:

1. the installation is modelled as a series of heat flow PDE
2. boundary conditions are determined
3. the system of equations and boundary conditions are solved to give heat flow and temperatures

Boundary Conditions

Depending on the complexity of the installation, various boundary conditions may be relevant.  For cable installation, we typically come across the following boundary conditions.

Finite Element Analysis (FEA)

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:

1. defining the geometry of the cable installation
2. setting up the heat flow PDE(s) for the components of the installation
3. applying boundary (and initial) conditions
4. meshing the geometry
5. solving the installation system
6. plotting and visualizing the results

Symbols

A        - area, m²
c        - specific heat of material, J/kg.°C
k        - thermal conductivity, W/m.°C
T        - temperature, °C
T_a        - surface temperature of enclosure

q         - heat flow, W (or Joules/second)
$\dot{q}$        - heat generated per unit volume, W/m³

ρ        - density, kg/m³
t         - time, s   

Boundary conditions:
g       - heat flux, W/m² 
h       - weighting coefficient
r        - temperature, °C
q        - heat transfer coefficient