# Thermal conductivity problem

1. Dec 20, 2007

### thistle

Forgive me if I am posting in the wrong place.

I have a problem dealing with thermal conductivity.

I have an aluminum plate, approx. 12 in/ long, and 1.5 in. x 4 in on the end which is being applied with 200 W. What I'm interested in is how long it will take before the plate reaches a thermal steady state.

I've never done problems in thermodynamics before so I hope I'm using the right terms. I want to know the time it will take for the plate to reach its max temperature. How do I find this? I'm not sure what equations to use. I realize that environmental conditions factor in so it's a somewhat difficult problem but if anyone can lead me in the right direction I would appreciate the help very much.

2. Dec 20, 2007

### Hootenanny

Staff Emeritus
How much mathematics have you done? Solutions to such a problem require some elementary knowledge of differential equations.

3. Dec 20, 2007

### thistle

I have done differential equations. If I knew the equations to use to solve this problem I can handle the math. I just can't find anywhere that tells which ones to use.

4. Dec 20, 2007

### Hootenanny

Staff Emeritus
Okay, firstly we have Fourier's law which describes that heat flux ($\vec{\phi}$) through a material of thermal conductivity ($\kappa$);

$$\vec{\phi} = -\kappa\nabla T$$

However, as I'm sure you know there will be some heat flux through the lateral surfaces of the plate. Although not entirely accurate, Newton's law of cooling is the only law I have met which describes that rate of cooling;

$$q^\prime = h\cdot A\left(T - T_s\right)$$

Where h is the heat transfer coefficient, A is the area (which in your case will be a function of time while the plate is at a non-uniform temperature), T is the surface temperature and Ts is the ambient temperature.

Perhaps this problem would be better addressed by an Engineer, if one would like to chime in....

5. Dec 23, 2007

### IITian

It's not that simple. You have to say the fluid (air) flow conditions in the room. Moreover, the solution of Fourier's equation is not straightforward unless you have done fair bit of partial differential equations.

6. Dec 24, 2007

### GT1

Maybe the most easy way will be to run a test and measure the time it takes
But if you wish to solve it, you can start with energy balance on the plate:
E(in) - E(out)=E(stored)
E(conduction) - E(Convection + Radiation)= E(stored)
For conduction use the 3D Fourier's law
For convection use Newton's law of cooling.you need to estimate H according to your problem's conditions, if H is for natural convection, then it changes as the temperature of the plate changes so you might have solve the problem numerically anyway.
for radiation use the Stepan-Bolzmann law.
and E(stored )= (1/alpha)*d(T)/d(t).
and the solution will be T(X,Y,Z,t).
You can find those equations at any heat transfer book.

7. Dec 24, 2007

### vector3

Your bar has a total 6 sides. The top is receiving heat at a rate of 200 W. Steady state conditions will prevail when the remaining 5 sides are emitting a total of 200 W. In otherwords, the heat flowing in (200 W) equals the heat flowing out. A thermogradient will develop throughout the bar. This represents the basic physics of the problem (Heat Transfer and or Quantity of Heat). The sides exposed to the environment, ie air, will give off heat at the rate in which the air will receive the heat, which depends on the temperature of the surrounding air. The foregoing only addresses heat transfer, leaving radiation and the other heat transfer mechanism out of the problem. (Generally speaking, they are also dependent on the intial temperature or system state.)

Going up a notch mathematically, and using the heat equation, we need initial conditions for the problem to be mathmatically speaking, "well posed". The intital conditions are typically temperatures somewhere in the system.