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A Heat Conduction Problem (Final exam on Monday!)

  1. Jun 5, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to go into this test with great aplomb.

    In each of Problems 1 through 8 find the steady-state solution of the heat conduction equation a2uxx=ut that satisfies the given boundary conditions.

    1. u(0,t)=10, u(50,t)=40


    3. ux(0,t)=0, u(L,t)=0

    2. Relevant equations

    Will be using separation of variables; so assume u(x,t)=X(x)T(t)

    3. The attempt at a solution

    After a long time---that is, as t approaches ∞---I anticipate that a steady state temperature distribution v(x) will be reached.

    Then u(x,t) will just be a2v''(x)=v'(t). Since v is not a function of t, v'(t) = 0 and v''(x)=0.

    v(x) must be a 1st degree polynomial: v(x) = Ax + B.

    Notice that the initial conditions to problem one imply that

    X(0)T(t)=10, X(50)T(t)=40.

    I don't want T(t) to be something trivial, so

    X(0)=10, X(50)=40

    ----> 10= 0 + B ---> B = 10

    40=A*50 + 10 ----> A = (40-30) / 50 = 1/5

    -----> v(x) = x/5 + 10

    That's the steady state heat distribution, which is all the problem asked for.

    Now problem 3. Notice the sub x.

    Hmmmm..... need to put the pieces of this puzzle together.

    ux(0,t)=0 is saying that the rod is insulated at x=0.

    So X'(0)=0.

    The other initial condition says X(50)=40.

    Where do I go from here?
  2. jcsd
  3. Jun 6, 2010 #2


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    Well, as you concluded above, you have v''(x) = 0. This gives:

    v(x) = Ax + B

    0 = X'(0) = v'(0) = A

    so v(x) = B. Now v(L) = 0 = B so

    v(x) = 0x + 0 = 0.

    Doesn't that agree with your intuition if you insulate one end and hold the other at 0?
  4. Jun 6, 2010 #3
    In other words, the temperature will be zero throughout the tube because only 0-degree air can come in?

    If a question like this comes up on the final tomorrow, I'll just write: "Steady-state solution is zero because 0-degree air comes in one end"
  5. Jun 6, 2010 #4


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    It could be a solid metal rod insulated laterally and on one end with the other end in ice water. :smile:

    Good luck on your exam. Just be ready to solve the problem completely if you are asked to do so. :cry:
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