The following are given: Thermal Conductivity of Barrel Material of 37.67 W/(m·K), Barrel Length of 290mm, Barrel Inner Diameter of 16mm, Barrel Wall Thickness of 9.5mm, Barrel Outer Diameter of 0.035m.
The question as posed cannot be answered definitively, other than to say that you have already stated the temperature. Your problem needs to be clearly defined if you want advice.