Soapstone Conductivity Question

[scott123]

Soapstone's Thermal conductivity: 6.4 W/mK

Specific heat capacity: 0.98 J/gK

Density: 2,980 kg/m³

Thickness of slab: 1.25" (3.18 cm)

Distance to center: .625" (1.59 cm)

Ambient temperature: 500 deg. F. (260 C.)

Core temp: 70 deg. F (21.11 C.)

I'm looking for the time it takes for the core to reach 500 deg (while maintaining a 500 deg. ambient temp).

For simplicity's sake, I don't care about radiant heat or convection, just conduction. I'm also aware that soapstone, being a natural material, varies in conductivity. 6.4 W/mK is a good enough ballpark for my present needs. Lastly, I'm also, for now, not taking thermal diffusivity into account- I'm only thinking linearly/one dimension, so area/volume shouldn't play a role. At least, I don't think it should. The center of a 1.25" x 10" x 10" slab should hit 500 deg. about the same time as a 1.25" x 30" x 40" slab (1.25 will always be the smallest dimension). I know McGee played around a bit with meat and the time it takes for energy to reach it's core, but I'm thinking stone is a little different (simpler, I would think) due to the exact dimensions and lack of water evaporation/convection.

The application for this is extremely practical. If I can get a half decent ballpark on the time it takes for the energy to transfer, I can save considerable money on electricity costs.

Cumulatively, I've probably spent about 24 hours on this over the last 6 months. I'm calling 'Uncle' on this one. I'm sure the answer is probably staring me in the face, but for now, it has me beat.

And, please, I'm looking for the math, not "I heat my soapstone for x minutes and it works great for me." I need to see the formula- not just to satisfy my 6 month long curiosity, but to be able to apply it to other materials (like firebrick).

[ray goud]

If you have the internet capability (DSL or equiv) download the .pdf at the following link:

http://web.mit.edu/lienhard/www/ahttv131.pdf

It is a textbook (ten megabytes/ 796 pages) just for heat transfer problems and analyses. The Lienhard's are on public radio, are profs., and their shows are usually interesting.

I don't have the time right now to dig out the relevant formula, probably later; in the meantime maybe you can find it there before I do. A while ago I used it to prove the fallacy of in-floor radiant hot water heating.

Ray

[scott123]

Thanks for the link. That's extremely comprehensive.

I spent some time looking through it and an appropriate formula doesn't seem to be jumping out at me.

It seems like everything is either area based or somehow involves Q, which I can only seem to resolve if I have a figure for the mass- which I don't. I'm trying to go from a single point on the exterior to the single point in the center of the core. A point with no mass, no area.

At least that's how I'm picturing it.

If you do find the time to dig out a relevant formula, I'd be incredibly appreciative.

[ChefCrash]

Hi Scott

Try using this:

H = kA (dT/dx) from here.

H = heat flow rate Cal/s

k = thermal conductivity

A = area of slab

dT = oven temperature minus center of slab temperature

dx = 1/2 thickness of slab

Then use that to calculate time to reach equilibrium.

[ray goud]

Thanks for the link. That's extremely comprehensive.

I spent some time looking through it and an appropriate formula doesn't seem to be jumping out at me.

It seems like everything is either area based or somehow involves Q, which I can only seem to resolve if I have a figure for the mass- which I don't. I'm trying to go from a single point on the exterior to the single point in the center of the core. A point with no mass, no area.

At least that's how I'm picturing it.

If you do find the time to dig out a relevant formula, I'd be incredibly appreciative.

Seems like an important factor is being ignored: the input of heat to the surface of the stone. If it was a blowtorch (or pair of) it would provide gobs more heat than a simple "infinitely restoring" steady state of oven environment.

[tomdarch]

This is a realm of physics and math that I've never directly dealt with, but... if I understand correctly, heat transfer is a function of temperature differential. (At least that's how we treat it designing buildings.) As the temperature differential approaches zero, the rate of heat transfer would approach zero. Just looking at your initial problem statement, I see an issue that might be causing perfectly good equations to throw weird results - if you're trying to find out how long it takes for the core temperature to come up to equal to the surrounding temperature, then any normal equation should result in infinity (or close to it), right? Lots of equations "misbehave" in this situation. Have you tried solving for an ambient temp of, say, 510 or 520 and a target core temp of, say, 490 or 480? In other words, try a target temp that is less than the ambient temp. I hope I'm being helpful and not a smarta@@!