Kirchhoff Transform: Thermal Diffusivity Dependency?

[chavavic]

When applying Kirchhoff's transformation to heat conduction PDE with temperature dependent thermophysical properties (k,ρ , Cp) , one obtains a transformed energy variable
u=∫Cp(τ) dτ and a term for a thermal diffusivity (α=k/ρ*Cp), thus reducing the nonlinerarity of the equation. When consulting some texts about the method, I find that there is discrepancy around the thermal diffusivity term. Some authors define the diffusivity as a function of the original temperature varable (τ); while others declare that the diffusivity is now cast in terms of the transformed variable (u) . Which declaration is correct? I gather that being an intrinsic material property the thermal diffusivity function should not be altered by the transformation, and thus should remain dependent on the original temperature (τ) variable. Is this so?

 

[gtoguy97]

Yes, the thermal diffusivity should remain dependent on the original temperature (τ) variable. The transformation is meant to reduce the nonlinearity of the equation, not to alter the thermophysical properties.