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.
Place a thermocouple inside the barrel, or heat it slowly enough or soak it long enough to that you are sure the surface temperature is representative of the temperature you need within 10°C.