1 dimensional heat flow boundary conditions


by hfitzgerald
Tags: boundary conditions, heat flow equation, laplace
hfitzgerald
hfitzgerald is offline
#1
May23-11, 05:50 AM
P: 2
1. The problem statement, all variables and given/known data
n is given by:
2Θ/∂x2=1/α2 ∂Θ/∂t
, where Θ(x, t) is the
temperature as a function of time and position, and α2
is a constant characteristic for the
material through which the heat is flowing.
We have a plate of infinite area and thickness d that has a uniform temperature of 100◦C.
Suddenly from t = 0 onwards we put both sides at 0◦C (perhaps by putting the plate between
two slabs of ice).
Write down the four boundary conditions for this plate.


2. Relevant equations

I can't think of any relevant equations to this

3. The attempt at a solution
so far I have got
Θ(0, t)=0
Θ(d, t)=0 where d is the thickness of the bar.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution
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HallsofIvy
HallsofIvy is offline
#2
May23-11, 06:10 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,904
Well, so far, all you have done is write down the problem!

thetaxx= (1/a^2)thetat
theta(0, t)= theta(d, t)= 0, theta(x, 0)= 100.

Now, attempt a solution. What methods have you learned for solving such problems? Most common are "separation of variables" and "Fourier series", both of which will work here, but no one can make any suggestions until we know which methods you know and where you are stuck with this problem.
hunt_mat
hunt_mat is offline
#3
May23-11, 07:18 AM
HW Helper
P: 1,585
I would try as separation of variables method, so write:
[tex]
\theta (t,x)=T(t)X(x)
[/tex]


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