Thermal Time Constant and Specific Heat - John Dunn, Consultant, Ambertec P.E., P.C.

We have all kinds of information about finding maximum the temperatures that some particular things will get to, things like power supply heat sinks for example, but it may also be useful to look of how long will it take the temperature to get to that calculated extreme value.

With the specific heat of that particular something, we can make an estimate of the time constant of that something undergoing a thermal excursion:

Note: In this example, we look up the specific heat of aluminum at    http://hyperphysics.phy-astr.gsu.edu/hbase/tables/sphtt.html    where we find the value as 0.900 joules / °C / gram.

We examine in this example, a real life situation I once came across:  

 

Comments

[Frank Walker]

Nice one John!

[George Storm]

Not just "a bit of fun", but a nice illustration that "big" in natural systems often means "slow".
But I still felt the need to work out what the thermal resistance might mean in physical terms. So if we take the block to be about 20x20x10cm high, the thermal resistance given corresponds to the interface being...
. rather thin (about 1-mil or 30-um) if its a good insulator such as expanded polystyrene, or
. a few mm upwards if it's almost anything solid.

But, while we are on educational diversions...
One of the "tools" I find useful in nearly every area of work is knowledge of the relevant scaling laws. That is, what happens if you take a geometrically similar structure with all dimensions scaled linearly.
Of course you need to know where scaling works well and where it is at best a guide; speed of MOSFETs at small geometries is an example of where simple (purely dimensionally based) scaling laws break down.
But in many (maybe most) other situations basic scaling laws work very well. Heat transfer works is one of these, and for most cases right down to nanometer dimensions:- the time constants of thermal systems vary as the square of the linear dimensions.

Naturally we can confuse ourselves when the problem is presented (as originally here) in a mixed format. I.e. the thermal resistance as a "given", whereas thermal mass is calculated from geometry), so we need to 'factor in' that thermal resistance scales inversely with linear dimensions.

[Jean-Marc Nogier]

The aluminum mass has a temperature gradient between the base plate (bottom) and the ambiant (top).
I am wondering what temperature is reached at say 5*Tau, and where is this temperature located ?

[John Dunn]

Hi, Jean-Marc.

I actually haven't figured out how to deal with the real life issue of there being a thermal gradient in the aluminum block which is why I assumed that there was zero assumed thermal resistance in the aluminum block itself. For estimation purposes, this was good enough, but clearly incomplete.

Maybe someone can help us out here?