1 dimensional heat flow boundary conditions

  1. 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
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,370
    Staff Emeritus
    Science Advisor

    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.
     
  4. hunt_mat

    hunt_mat 1,583
    Homework Helper

    I would try as separation of variables method, so write:
    [tex]
    \theta (t,x)=T(t)X(x)
    [/tex]
     
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