Heat through a semi-infinity solid

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Hi there!

I run into a situation where I can't go on. It's about a thermal analysis, I already made a simulation using Ansys but I also want an approach made "by hand".

In order to simplify the case I made this example:

We have a metallic bar inside the soil. The bar is at 100ºC and the soil at 20ºC. Check the attached picture.

Once we stick the bar on the ground I understand there is a gradient of temperature radially in the soil, from 100ºC (perfect contact, Point A) to 20ºC (Point B).

I know all the heat goes by conduction according to:
q = 2 π k (ti - to) / ln(ro / ri), but I don't know the distance.

My two questions are:

1) where does the gradient in the soil stops? what's the distance up to the 20ºC? Where is the point B located? (ok, these are 3 questions, but it's always the same!)

2) How long does it take to the point A to get back to 20ºC?

We neglect the superficial conditions (convection, radiation)
We know all the material properties

Thank you
regards,
 

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