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

For cable modelling, we typically assume constant thermal conductivity, see Cable Thermal Analysis.

To consider the effect of convection (natural or forced in a gas or liquid), Newton's law of cooling is used:

$q=hA\left(T-T_a\right)$

for air (a common cable installation medium), the convection heat transfer coefficient h, flowing at velocity v, can be obtained from:

$h=7.371+6.43v^{0.75}$

To set a convection boundary conditions, we can use a Neumann condition with  q = desired convection coefficient and g = temperature of environment times the convection coefficient.

Enclosed Spaces

Convection is a complex topic, requiring a detailed understanding of heat flow, fluid flow and the behaviour of gases as they flow across surfaces.  For cable sizing, to achieve this level of technical detail is not practical nor achievable in many situations. One effective technique when solving equations for enclosed spaces is the use of an effective thermal conductivity, k_e.  Using an effective thermal conductivity enables the air to be treated from a conduction view, eliminating much of the complexity in implements full connectivity solutions.

For an enclosed spaced, the effective thermal conductivity is given by:

$k_e=k{N}_u$ The Grashof number is given by: $G_{r\delta }=\frac{g\beta \left(T_1-T_2\right){\delta }^3}{{\nu }^2}$ with: $\beta =\frac{1}{T_f}$ and $T_f=\frac{T_1+T_2}{2}$ Note: Tf is in Kelvin (not °C). Experimental results for free convection in an enclosure give: $N_{u_{\delta }}=C{\left(G_{r_{\delta }}{P}_r\right)}^n{\left(\frac{L}{\delta }\right)}^m$ Typically we can take g = 9.80665 and for air: Tf, °C ν, 10-5 m2.s Pr 0 1.343 0.720 20 1.568 0.708 80 2.056 0.697 120 2.591 0.689 Note: for differing temperatures, exact factors can easily be looked up on the internet

Note: δ is the hydro-dynamic boundary layer thickness.  

Values of C, n and m can be obtained from the following table using the Grashof-Prandtl number product:

Symbols

A                - area, m²
g                -  acceleration of gravity, m/s²
h                - convection heat transfer coefficient, W/m².°C
k                - thermal conductivity, W/m.°C
k_e               - effective thermal conductivity, W/m.°C 
T, T₁, T₂, T_a        - temperature, K
v                - velocity of air, m/s 
ν                 - kinematic viscosity, m²/s

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

β        -  coefficient of thermal expansion
δ        - enclosure dimension, m

G_r         - Grashof number
N_u        - Nusselt number
P_r        - Prandtl number
R_a        - Rayleigh number 

Radiation is the transfer of heat by electromagnetic radiation. 

Luckily in cable installations, we rarely encounter the need to consider radiation.  One exception is that of bare conductors in an enclosure, where the effect of radiation can be taken into account by:

$q=\epsilon \sigma A\left(T^4-{T_a}^4\right)$

with ε and A properties of the busbar, and T_a the temperature of the enclosure

Busbar & Enclosures

Radiation heat flow between two plates (busbar surface to enclosure for example), can be calculated from:

$q=\frac{\delta A\left({T_1}^4-{T_2}^2\right)}{{\displaystyle \frac{1}{{\epsilon }_1}+\frac{1}{{\epsilon }_2}-1}}$

q is the heat flow in W between plates of area A
ε₁, ε₂ are the emissivities of the two surfaces
T₁, T₂ are in K (not °C)

The effect of radiation is included in any solution as a suitable boundary condition.  An alternative approach would be to adjust the heat flow for the volume of the busbar and subtract this directly from the heat generated within the busbar.

Symbols

A         - area, m²
T₁       -  temperature of surface (hottest), K
T₂       -  temperature of enclosure

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

ε         - emissivity of a surface