Heat through a semi-infinity solid

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The discussion revolves around a thermal analysis of a metallic bar at 100ºC placed in soil at 20ºC, focusing on heat conduction. The user seeks to determine the distance at which the temperature gradient in the soil reaches 20ºC and the location of this point, as well as the time it takes for the bar to cool back to 20ºC. They reference a heat conduction formula but are uncertain about the distance involved. The analysis neglects external factors like convection and radiation, and all material properties are known. The user also provides a link to a relevant paper for further reference.
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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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