Results 1 to 7 of 7

Math Help - Heat Equation

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Heat Equation

    A rod of length L, cross section A, whose lateral surface is insulated, is made of a material of thermal constants c, p, k. Heat is produced electrically at a rate of \beta per unit volume. The ends are kept at temperature T and initially the rod is at temperature zero. Formulate the initial-boundary value problem for the temperature in the rod.

    My book is of no help on how to do this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    See if your school's library has Carslaw and Jaeger's Conduction of Heat in Solids. That should help quite a bit.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by dwsmith View Post
    A rod of length L, cross section A, whose lateral surface is insulated, is made of a material of thermal constants c, p, k. Heat is produced electrically at a rate of \beta per unit volume. The ends are kept at temperature T and initially the rod is at temperature zero. Formulate the initial-boundary value problem for the temperature in the rod.

    My book is of no help on how to do this.
    Here is what I got:

    \displaystyle \text{D.E.} \ \ u_{t}=ku_{xx}, \ \ \ 0\leq x\leq L, \ \ \ t>0

    \displaystyle \text{B.C.} \ \ u(0,t)=T \ \ \text{and} \ \ u(L,t)=T \ \ \ t>0

    \displaystyle \text{I.C.} \ \ u(x,0)=0, \ \ \ 0\leq x\leq L
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Ackbeet View Post
    See if your school's library has Carslaw and Jaeger's Conduction of Heat in Solids. That should help quite a bit.
    I will have to look for that later. Is it long or short and sweet?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    I would definitely agree with your initial and boundary conditions. As for the DE itself, I'm not sure that the thermal constant k of the rod is necessarily the same as the heat transfer coefficient, k, in the Heat Equation. In addition, I'm not seeing where the heat generation comes into your equations. It strikes me that your DE might not be homogeneous: don't there have to be source terms in there somewhere? I suppose the question to be asked is this: is the heat being generated all along the rod (as seems likely)? Or is it just being generated at the ends?

    Here's the Carslaw and Jeager book on Amazon. It's about 520 pages.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by Ackbeet View Post
    I would definitely agree with your initial and boundary conditions. As for the DE itself, I'm not sure that the thermal constant k of the rod is necessarily the same as the heat transfer coefficient, k, in the Heat Equation. In addition, I'm not seeing where the heat generation comes into your equations. It strikes me that your DE might not be homogeneous: don't there have to be source terms in there somewhere? I suppose the question to be asked is this: is the heat being generated all along the rod (as seems likely)? Or is it just being generated at the ends?
    I wish could answer that question but I don't know. My book tends to throw the user to the wolves.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    5
    Awards
    2
    The Wiki on the Heat Equation and Thermal Diffusivity show you how to construct the constant you need multiplying the second-order spatial derivative.

    Based on Model Problem XX.6 on this webpage, I would hazard a guess at something like

    u_{t}=\alpha\,u_{xx}+\dfrac{\beta}{AL}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. The Heat Equation
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: April 19th 2010, 01:42 PM
  2. 2D heat equation
    Posted in the Advanced Applied Math Forum
    Replies: 4
    Last Post: April 17th 2010, 11:41 AM
  3. Heat Equation Help
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: March 16th 2010, 03:29 PM
  4. Heat problem with a heat equation
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: November 25th 2009, 10:40 AM
  5. heat equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: June 1st 2009, 07:57 AM

Search Tags


/mathhelpforum @mathhelpforum