As the power lost through the stainless steel barrel in my previous post seemed very low I decided to calculate it as a sanity check. I ignored the heat lost from the barrel by convection and radiation. They may be significant now but when I insulate it they shouldn't be.
The outside diameter of the tube is 6.35mm and the bore is 3.6mm, so that gives a cross sectional area of 2.15x10-5m2. Its length is 0.05m. The temperature difference over that length is 240°C-50°C = 190°C. The conductivity of stainless steel is 17 W/mK. So the heat flow is 17 x 190 x 2.15x10-5 / 0.05 = 1.39W. That means it isn't very much compared to the total power required, so that matches my observation.
The amount of heat flowing into the heat sink is therefore 1.39W and it raises its temperature by 30°C, so the heat sink would have to be 22 °C/W. It was just a scrap one I had laying about so I don't have a spec but it is only 70 x 25 x 20mm so that seems in the right ball park.