What is the derivation of Fourier's Law of Conduction?

[Bassalisk]

Hello,

Can anybody explain to me how does this:

dQ=-X*dS*grad(T)*dt

t being time and T being TD temperature.

transforms into this:

Q=-X*S*(T2-T1)*delta(t)/d

delta(t) being time interval

d being thickness length of material
X being heat conductivity

Thanks

 

[timthereaper]

Maybe I don't quite get the form you have it in...S is what exactly?

 

[Bassalisk]

Maybe I don't quite get the form you have it in...S is what exactly?

S=Surface or Area

 

[Studiot]

Is S area ?

 

[Bassalisk]

Is S area ?

Yes.

 

[Studiot]

The one dimensional form of the law is

$$\frac{{dQ}}{{dt}} = - kS\frac{{dT}}{{dx}}$$

Where dQ/dt is the rate of heat flow which is directly proportional to the cross sectional area and the temperature gradient.
T is temperature
You only need use grad (T) for 3 D where grad(T) is the direction of max gradient.
t is time
S is area cross section
k is the constant of proportionality (thermal conductivity) -I don't like X because x is the axis variable.

 

[Studiot]

You can integrate this if you assume the temperature fall to be a linear function of x through the material to obtain your second formula.

That is dT = ax and integrate between limits x =0 and x = thickness to eliminate constant of integration.

 

[timthereaper]

Well, the top equation you have is in differential form and it's a PDE. If you want to go through the math for the 1-D case, you'll find it equivalent to the second equation you have:

http://en.wikiversity.org/wiki/Solution_to_the_1-D_Heat_Equation

 

[Bassalisk]

Thanks all, I got it now.