An internal gas sampling probe is located within a galvanised steel chimney, extending into the hot combustion exhaust gases.

An internal gas sampling probe is located within a galvanised steel chimney, extending into the hot combustion exhaust gases. The probe is constructed from stainless steel (thermal conductivity, 14 W/(m K) and may be considered to be of circular cross section, with a diameter of 12 mm. 

The temperature of the chimney wall and the probe base may be assumed to be a constant, equal to the external ambient temperature of 15 °C. 

The exhaust gases are at a temperature of 420 K, and the surface heat transfer coefficient may be assumed to be equal to 32 W/(m2 K). The effect of radiative heat transfer may be assumed to be negligible. 

part 1 - can you show the temperature of the probe at a distance of 30 mm from the chimney wall. 

part 2 - can you show the rate at which energy is gained by convection heat transfer to the probe 

part 3 - can you show the minimum length for the probe to be considered thermally infinite.