IEC 60364-5-54 gives the following formula for the calculation of the factor k:

$k=\sqrt{\frac{Q_c\left(\beta +20\right)}{{\rho }_{20}}\ln \left(\frac{\beta +{\theta }_f}{\beta +{\theta }_i}\right)}$

where Q_c = volumetric heat capacity of conductor at 20°C, J.K^-1.mm^-3
ß = reciprocal of temperature coefficient of resistivity at 0 °C 
ρ₂₀ =  electrical resistivity of conductor material at 20 °C, Ω.mm
ϴ_i = initial temperature of conductor, °C
ϴ_f = final temperature of conductor, °C

Note: strictly speaking ß is specfied at 0 °C.  Within myCableEngineering calcuations, ß is determined by taking the reciprocal of the 20 °C temperature coefficient.  This introduces a small, but negligible error. 

Typical Application

For manual calculations, the following will give realistic results:

Substituting the above values and rearranging the IEC equation slightly, gives:

$k=226\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{234.5+{\theta }_i}\right)}$- copper conductors

$k=148\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{228+{\theta }_i}\right)}$- aluminium conductors

$k=78\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{202+{\theta }_i}\right)}$- steel