Hypothetical Scenario (Steel Rod + Heat + Vacuum)

[LKMW]

One end of 1m steel rod of 3cm diameter is welded to the interior wall of a steel container (steel: 310s or 2520). The other end of the steel rod is not in contact with anything. The pressure in the container is below 0.01mpa (vacuum). A heat source of 1100 Degree celsius is applied externally to the attached point. After two 30 minute intervals (first at 30 minutes, then at 60 minutes), what is the temperature of the hanging end of the steel rod?

---OR (perhaps simplified)---

Steel Rod: 1m length, 3cm diameter

It is placed in a vacuumed environment (no air to conduct away the heat). If one end is heated externally (outside of the vacuum) at 1100 Degree celsius, what is the temperature of the other end of the rod after a) 30 minutes and b) 1 hour.

 

[Mapes]

Hi LKMW, welcome to PF. Perhaps the simplest way of tackling the problem is to estimate the thermal diffusion time from the equation $t\approx L^2/D$. This will give the approximate time for a temperature change to propagate (i.e., the approximate time until the temperature at the end of the bar is a large fraction of 1100°C). From looking online, it looks like the thermal diffusivity of steel is about 10^-5 m² s^-1, which implies a time of about 1700 minutes. Since we'd expect a continuous increase, approximately linear, over this time, our first estimate is a temperature increase on the order of 10°C at the end of the rod after 30-60 minutes .

A crude approximation, but at least it tells us whether the answer is closer to room temperature or 1100°C (I'd go with the first). A better analysis might use finite element analysis to look at the conductive heat transfer through the rod, and a still better analysis would incorporate radiative transfer from the rod to and from the surroundings.