1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Heat equation applied to a rod

  1. Feb 11, 2009 #1
    1. The problem statement, all variables and given/known data

    http://img444.imageshack.us/img444/7641/20240456gw8.png [Broken]

    2. Relevant equations
    http://img14.imageshack.us/img14/5879/63445047rj2.png [Broken]

    Note that the rightside of the rod is insulated.

    3. The attempt at a solution
    I get this model:

    [tex] \frac{ \partial{u} }{ \partial{t} } = \kappa \frac{ \partial{ ^2 u} }{ \partial{x^2} } +s [/tex]

    [tex]\frac{ \partial{u}} { \partial{x} } = 0[/tex]

    In steady state this gives: [tex]u(x) = \frac{- s}{ \kappa} \frac{1}{2}x^2 + \frac{s}{ \kappa } L x + u_0 [/tex]

    But if I calcute than the asked u' at x=0:

    I get:

    [tex]\frac{du}{dx} = \frac{s}{ \kappa} L [/tex]

    Is this correct?
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 12, 2009 #2
    What I don't understand is what do they mean by "total heat supply"? I presume they mean s (=source). But I get a different answer out of my equation.
  4. Feb 12, 2009 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your answer looks fine. Note, though, that the equation

    \frac{ \partial{u}} { \partial{x} } = 0

    means nothing on its own; we need to specify a location:

    \left(\frac{ \partial{u}} { \partial{x} }\right)_{x=L} = 0

    For the heat supply question: we need to distinguish the total heat S from the heat per length [itex]s=S/L[/itex] that goes into the differential equation. By applying Fourier's conduction law, your answer indicates a total heat flow of [itex]sL=S[/itex], which is correct. The units will always confirm whether S or s is being used appropriately.
  5. Feb 12, 2009 #4
    You're right but I couldn't get this in latex. Note that the notation you are using isn't the right one either there should be a large bar at the right hand side something like this: [tex] |_{x=L} [/tex]

    Of ocurse, how could I overlooked that!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook