Steady State Heat Transfer Understanding

[Thomas H.]

I'm doing some analysis for a project and am looking for some guidance related to steady state heat transfer.

Steady state heat transfer, as the name implies, is independent of time. Using Equation 16..20 as an example; the temperature distribution for slab via steady-state, one-dimensional conduction is only dependent on heat flux, thermal conductivity, and slab thickness. But logically, shouldn't the temperature profile for the slab become uniform (dT/dx = 0) as time approaches infinity? The energy should equalize and both sides of the slab should eventually be the same temperature. How long is this steady state heat transfer equation valid for? I feel like I'm missing something very obvious, the math does not match my intuition.

For some more information, I'm looking to design a composite wall layup for my project. Given a set heat flux and ambient temperature, I am choosing my wall properties such that the inside of my wall stays at a reasonable temperature. I'd like to know how long the inside of my wall can stay at this reasonable temperature.

 

[IRstuff]

That's Zeno's Paradox; if you need to get from Point A to Point B, you have to get to the midpoint first. To get to the midpoint, then, you have to get to the 3/4 point, then the 7/8 point, etc. You can never get to Point B, mathematically. However, the real world is full of inaccuracies and noise. If the thermal time constant is, say, 5 minutes, then by 5 times the thermal time constant, at 25 minutes, the temperature distribution is likely to be sufficiently close to the steady state that it's indistinguishable, due to noise and measurement error. And the true error is insignificantly tiny, relative to everything else.

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

faq731-376 forum1529 Entire Forum list

http://www.eng-tips.com/forumlist.cfm

 

[BrianE22]

One side of the slab is held at T1, the other at T2. If T1 and T2 differ then there will always be conduction from one side to the other hence dT/dx is never zero. I think you are assuming T1 and T2 eventually become the same temperature. For the example as shown something is preventing that from happening (may be convection is different on each side of plate).

 

[IRstuff]

sorry, I read wrong. The OP's assertion only applies if there is no thermal pathway for the heat arriving at T2 to go elsewhere, i.e., held at T2 by sinking the heat into somewhere else.

For the case where the slab was uniformly at T2 and one end is then dragged to T1, then thermal gradient establishes itself over time, wherein Zeno's Paradox applies. But for all practical purposes, that slope will be indistinguishable from the infinite time scenario within about 5 time constants, assuming that one isn't looking for nano-kelvin levels of distinguishability, in which case, more time would be needed. Aside from waiting for the thermal gradient to get there in the first place, it'll take you days, if not weeks to measure the difference, since measuring a nanokelvin is OK if you're a Nobel-prize winning physicist in a cryogenics lab, but under terrestrial temperatures, there are bunches of noise that make even measuring microkelvin differences challenging.

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

faq731-376 forum1529 Entire Forum list

http://www.eng-tips.com/forumlist.cfm

 

[IRstuff]

Someone just announced getting down to 38 picokelvins, but only briefly; nevertheless, as mentioned earlier, even microkelvins

https://www.popularmechanics.com/science/a37852132/coldest-temperature-recorded-in-lab/

https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.127.100401

This thermometer gets you down to 0.1 millikelvin resolution, and is not something you toss into the toolbox after use. In any case, more practical thermometers get down to the range of 1 to 10 millikelvin resolution, so if a thermal transient gets within that amount from the ideal, you're basically done.

https://www.isotechna.com/TTI22-High-Accuracy-Thermometer-p/tti22.htm

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

https://www.youtube.com/watch?v=BKorP55Aqvg

faq731-376 forum1529 Entire Forum list

http://www.eng-tips.com/forumlist.cfm