# Rate of heat transfer / specific heat

1. Dec 29, 2009

### charlotte7070

I am digging back thorugh thermodynamics textbooks, and am seeing insulation values etc. but none on how to solve for time required to heat a material.

Here is the specific problem. I am trying to fgure out how long its takes to cool (or alternately heat) a concrete slab.

The slab is 8 in. with initial temp 60 degrees F. Temperature on top and bottom of slab is ambient air temp, 20 degrees.

k, concrete = 1.7 W / m - deg K
c, concrete = 750 Joules

the area of slab can be assumed to be very large (an acre? infinity?)
and the air can be assumed to not change as a result of the heat loss of the concrete.

So I have rate of heat transfer: dependent on the surface area,
H = -k A (dT / dX)

and specific heat of concrete, dependent on mass:
dQ = m c dT = Joules

t (seconds) to cool one inch thickness concrete by 22.2 deg C or K
= dQ / H = 637 seconds (this seems a little fast to me since concrete is fairly insulating material).

Does this look correct? (Assuming I have the heat capacity and conductivity of concrete correct). This would be derived into an integral to get the temp at a certain time and depth, but as a consultant would be laughed out of the room, and will simplify to some finite steps in time and thickness.
No need to get to technical! I won't get it! Thank you for your input.

2. Dec 29, 2009

### Mapes

Hi charlotte7070, welcome to PF. A good rule of thumb for characteristic heat diffusion times is $t\approx L^2/D$, where L is a characteristic length (I'd take 4 in, or 0.1 m, for this problem) and $D=k/c\rho$ is the thermal diffusivity, which looks like it's around 10-6 m2 s-1 for concrete. So I'd estimate it would take at least 104 s, or several hours, to heat most of the slab to close to 60°F.

3. Dec 29, 2009

### charlotte7070

thank YOU!!!!