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Finite Difference method to solve diffusion equation

  1. Aug 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Plot the transient conduction of a material with k = 210 w/m K, Cp = 350 J/kg K, ρ = 6530 kg/m3

    Where the material is a cylinder, with constant cross sectional area and is well insulated. The boundary conditions for the cylinder:
    T(0,t) = 330K
    T(l,t) = 299K
    [itex]\frac{∂T}{∂t}(x,17)=0[/itex]

    2. Relevant equations
    Diffusion equation:
    [itex]\frac{∂T}{∂t}=\alpha\frac{∂^{2}T}{∂x^{2}}[/itex]

    rearranged diffusion equation in finite difference form
    [itex]T(x,t+Δt)=\frac{\alphaΔt}{Δx^{2}}[T(x+Δx, t)-2T(x,t)+T(x-Δx,t)][/itex]


    3. The attempt at a solution
    Hi all

    I've never used the finite difference equation before to solve a PDE and I'm unsure how to use it. I know how to find values such as α (thermal diffusivity), but I'm unsure on how to sub in my initial and boundary conditions. And which values would I use for T(x+Δx, t) or T(x-Δx,t)? Since I'm trying to find change in temperature over time at a fixed distance x, I would assume that Δx = 0? (which I know is incorrect).

    I've tried looking online for PDE finite analysis techniques, but they're all either ODE examples or mesh analysis (something we haven't covered).
     
  2. jcsd
  3. Aug 23, 2014 #2

    AlephZero

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    Try Googling the Crank-Nicholson method for the 1D heat conduction equation (or the diffusion equation). This is a standard example in courses on finite difference, numerical method, and PDEs.

    I don't understand your boundary condition ##\frac { \partial T} {\partial t}(x,17) = 0##. Why t= 17? If the time derivative is constant everywhere at one time, isn't it constant everywhere at all times - i.e. the solution is steady state heat conduction along a uniform rod with the end temperatures given, so there is nothing to solve for numerically!
     
  4. Aug 23, 2014 #3
    Sorry I made a mistake with that, I was meant to say that the system reaches steady state at approximately t = 17.

    We haven't yet covered the Crank-Nicholson method, so I have a feeling we may have to stick with the explicit method.
     
  5. Aug 24, 2014 #4

    AlephZero

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    If you want an explicit method you could use http://en.wikipedia.org/wiki/FTCS_scheme

    With the constant temperatures at T(0,t) and T(l,t) the solution will converge to steady state as t increases for any starting temperature distribution T(x,0). It seems a bit strange that the question doesn't specify the starting temperature, so you can have something to check your results against. Just saying that it approximately converges by t = 17 doesn't tell you much.
     
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