The transient heat transfer code  allows one to model multiple materials in a cylinder, each with temperature dependant properties. It assumes the barrel is axisymmetric, with no thermal gradient in the axial direction. The solution uses a finite difference method with the governing differential equation being:

The initial conditions are:

The boundary conditions used at the inner diameter are:

and at the outer diameter are:

Finally the interface conditions between adjacent radial layers require that:

and

Where T is the temperature, t is time, kr is thermal conductivity, ρ is density, c is specific heat, h is specific enthalpy, and the f and a superscripts stand for the surface subjected to internal and external, respectively. The property input to this model can be constant or temperature dependent. Linear interpolation is carried out for each property to determine the property value at a specific temperature in time.