Calculate Final Temp of Thin Steel Piece: Heat Transfer Problem

[JJ420]

First off I would like to say that this is not a homework question. I am trying to calculate the temperature of a very thin piece of steel.

Assumptions

The underside of the metal is perfectly insulated
To = 20C (initial temperature of material)
Q"=3000W/m^2 (heat flux)
K = 51.9W/m*K (thermal conductivity)
C = .472 KJ/Kg*K ( specific heat)
x = 1mm ( thickness of metal)
t = 0.1s

What I need to know is how to calculate the final temperature at the surface.

 

[Benindelft]

Am I right in thinking you are doing some laser heating here? if so you should look at
the paper written by Bechtel "heating of solid targets with laser pulses" . Also check out the citations there are a lot of papers out there that solve this kind of problem

 

[Mapes]

This is a transient conduction problem that can be made much easier by assuming you only need to know the temperature about a second after the heat flux is turned off, not before. The reason is that the characteristic diffusion time through 1 mm of steel is about a tenth of a second:

$$t\approx\frac{h^2}{D}=\frac{h^2 c \rho}{k}\approx 0.1\,\mathrm{sec}$$

where D is the thermal diffusivity and h is the thickness, and I've used your numbers. After about a second, the temperature will have become nearly uniform in the plate, and the temperature increase can be calculated by considering the amount of input energy: 300 J/m2.

$$\Delta T=\frac{E}{c\rho h}$$

I get about a tenth of a degree Celsius for the temperature increase. Does this help?