Transient 1D radial conduction

[jb26]

Morning all,

This is probably going to seem absurdly simple, however I need to do a very quick conduction calculation as part of a much larger problem I am working on. I was wondering if anyone would be able to give me some pointers as time is of the essence.

The scenario can be simplified as follows: I have a slab of homogeneous material 10mm thick that can be considered semi-inifinite with a series of discretised elements on top of it. The attached (terrible and uninformative) image shows two such elements; let's pretend they are 1mm apart. To further simplify we can consider the elements to not have area and simply act as points. All I need to know is how to calculate the time taken to raise the surface temperature at point 2 by a given amount (lets say 1degree K) when the surface temp of point 1 is raised above the ambient temperature of the bulk material by some amount (lets say 5 degree K) for a known transient heating "on" time (lets say 2s).

You can see I've been quite general, but this is because I need to try some different heating scenarios.

Thanks in advance for any help given.

J

EDIT: I know I put radial in the title, but I simplified the geometry when writing the post an forgot to change it, sorry!

 

[Chestermiller]

Hi JB26. Welcome to Physics Forums.

Some questions:

Is this a one shot deal, or is there an ongoing cycle?
Is it truly a linear array, or are the sources laid out on an area grid?
Are the elements truly points, or do they have area? This affects the solution.
What is the thermal diffusivity of the slab material?

Chet

 

[jb26]

Cheers chet,

I'm trying to keep it simple tbh as I am just after an estimate. In reality the elements aren't heat sources (there is a more complicated bioheat mechanism generating heat in the material from an external stimulus). The real array elements do have area, but again I am just wanting to estimate the time for heat to transfer from one point to another across the surface of a material for a transient case.

Thermal diffusivity is of the order 1.92E-7

Cheers

 

[Chestermiller]

Cheers chet,

I'm trying to keep it simple tbh as I am just after an estimate. In reality the elements aren't heat sources (there is a more complicated bioheat mechanism generating heat in the material from an external stimulus). The real array elements do have area, but again I am just wanting to estimate the time for heat to transfer from one point to another across the surface of a material for a transient case.

Thermal diffusivity is of the order 1.92E-7

Cheers

units of thermal diffusivity are m^2/s?

 

[Chestermiller]

See the following thread: https://www.physicsforums.com/showthread.php?t=664993

Also, check out Conduction of Heat in Solids by Carslaw and Jaeger

 

[jb26]

yes the thermal diffusivity I stated was in standard SI form of m^2/s

Funnily enough I have a copy of that book sat right next to me. I took it out for some reason a long time ago and only just realized I have it! That said, any help is still very gratefully received as that faster I can get a rough number the better.

I'll have a read of the thread and then try to find a similar problem in the book.

Cheers