On the basis of transformation thermodynamics, a thermal conductivity expression for cylindrical thermal cloak with arbitrary cross section is derived. Based on the expression, we design a thermal cloak with a non conformal section profile. Results of full wave simulation show that the cloak possesses heat protection function, since it is capable of guiding heat fluxes to travel around its inner domain, leaving the inner region untouched; meanwhile, the heat fluxes return to their original pathways, resulting in a perfect thermal invisible effect. Arbitrary shaped cylindrical thermal cloak with conformal or non-conformal cross section can be designed based on the derived thermal conductivity expression, and all of them have perfect thermal protection and invisibility functions. It is indicated that the derived conductivity expression is applicable to designing cylindrical thermal cloaks with an arbitrary cross section. The generality of this method has been confirmed, and it may have potential applications in the designing of thermal protection devices for computer chips and satellite, etc.