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I am conducting an experiment on heat conduction in a copper rod. To do so I am using a thin copper rod with several temperature probes, then I am going to compare the data with a solution to the heat equation accounting for lateral heat loss. The bar is $0.2 \textrm{m}$ long and the thermal diffusivity ($\alpha$) of copper is $1.12*10^{-4} m^2 s^{-1}$ and the coefficient of lateral heat loss ($\kappa$) is $\ 2.61*10^{-3} s^{-1}$. First I calculated the initial temperature distribution of the bar and got:

  data = 
  {{0, 26.75}, {0.01, 26.5}, {0.02, 28}, {0.03, 28.75}, {0.04, 30}, {0.05,  32}, 
   {0.06, 34.25}, {0.07, 37.5}, {0.08, 43.25}, {0.09, 59.5}, {0.1, 102}, 
   {0.11, 67.75}, {0.12, 40.5}, {0.13, 35.75}, {0.14, 34}, {0.15, 31.75}, 
   {0.16, 30}, {0.17, 27.5}, {0.18, 28}, {0.19, 26.5}, {0.2, 28}}

The graph of this data:

enter image description here

From here, I tried to use Mathematica to find a Fourier series with help from this. And I also calculated it by hand which is:

0.7607201329841713`/2 + 
Sum[(1/10 ((0.912752572216549` Sin[3.141592653589793` n])/
    n + (-0.002189478133985855` + 
     5.795634757369874` Cos[3.141592653589793` n] + 
     2.310248484743582` n Sin[3.141592653589793` n])/(
    6.293389749792516` + 
     1.` n^2) + (-0.888276236822269` Sin[3.141592653589793` n] + 
     0.888276236822269` Sin[6.283185307179586` n])/n + 
    1/(5.3549458751438115` + 
      1.` n^2) (5.51162427937814` Cos[3.141592653589793` n] - 
       0.003837157071497423` Cos[6.283185307179586` n] - 
       2.3817822905923016` n Sin[3.141592653589793` n] + 
       0.0016581813809965903` n Sin[6.283185307179586` n])))*
Cos[10 (n) π (x)] + (1/
   10 ((5.74431 + 
       0.913625 n^2 + (-5.74431 - 3.223 n^2) Cos[3.14159 n] + 
       5.79563 n Sin[3.14159 n])/(n (6.29339 + 1. n^2)) + 
    1/(n (5.35495 + 1. n^2)) ((4.75667 + 3.27006 n^2) Cos[
         3.14159 n] + (-4.75667 - 0.889934 n^2) Cos[6.28319 n] + 
       5.51162 n Sin[3.14159 n] - 0.00383716 n Sin[6.28319 n])))*
Sin[10 (n) π (x)], {n, 1, 100}]

The graph of this fit (100 partial sums):

enter image description here

Finally I used Mathematica to evaluate the PDE here is my code:

κ = 2.61*10^-3;
α = 1.12*10^-4;
NDSolve[{
  Derivative[0, 1][u][x, t] == α*Derivative[2, 0][u][x, t] - κ*(u[x, t] - 17), 
  Derivative[1, 0][u][0, t] == 0, 
  Derivative[1, 0][u][0.2, t] == 0, 
  u[x, 0] == (0.7607201329841713`/2 + 
     Sum[(1/10 ((0.912752572216549` Sin[3.141592653589793` n])/
        n + (-0.002189478133985855` + 
         5.795634757369874` Cos[3.141592653589793` n] + 
         2.310248484743582` n Sin[3.141592653589793` n])/(
        6.293389749792516` + 
         1.` n^2) + (-0.888276236822269` Sin[
           3.141592653589793` n] + 
         0.888276236822269` Sin[6.283185307179586` n])/n + 
        1/(5.3549458751438115` + 
          1.` n^2) (5.51162427937814` Cos[3.141592653589793` n] - 
           0.003837157071497423` Cos[6.283185307179586` n] - 
           2.3817822905923016` n Sin[3.141592653589793` n] + 
           0.0016581813809965903` n Sin[6.283185307179586` n])))*
    Cos[10 (n) π (x)] + (1/
       10 ((5.74431 + 
           0.913625 n^2 + (-5.74431 - 3.223 n^2) Cos[3.14159 n] + 
           5.79563 n Sin[3.14159 n])/(n (6.29339 + 1. n^2)) + 

        1/(n (5.35495 + 1. n^2)) ((4.75667 + 3.27006 n^2) Cos[
             3.14159 n] + (-4.75667 - 0.889934 n^2) Cos[
             6.28319 n] + 5.51162 n Sin[3.14159 n] - 
           0.00383716 n Sin[6.28319 n])))*
    Sin[10 (n) π (x)], {n, 1, 100}])}, 
  u, {t, 0, 10}, {x, 0, 0.2}]

However the plot of this solution using Mathematica, it gives me a very funny graph:

enter image description here

This is completely wrong as the bar is supposed to loose heat not gain heat. Any ideas as to what is going on? And the PDE comes with this error:

NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. >>

Here is the plot of the actual data, the solution to the PDE should look something like this: enter image description here

Any help will much appreciated.

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  • $\begingroup$ Your definition of u[x, 0] is definitely inconsistent with your other equations. Given your expression for u[x, 0], I get 1.67605 for Derivative[1, 0][u][0., 0] and 0.198103 for Derivative[1, 0][u][0.2, 0]. So the error message you are getting seems cogent, and your first task is to specify a consistent system of equations. $\endgroup$ – m_goldberg Jul 5 '15 at 19:35
  • $\begingroup$ Inconsistent boundary conditions certainly is undesirable but is unlikely to be causing u to grow with time, because this problem is localized to small t. Instead, - κ*(u[x, t] - 17) appears to be the problem, acting as a source of heat as long as u < 17. If this term is meant to represent lateral heat loss, perhaps it should be part of the boundary conditions in x. Even then, it still may be a source of heat until for u < 17. $\endgroup$ – bbgodfrey Jul 5 '15 at 22:27
  • $\begingroup$ As for the boundary conditions, they are zero because the ends of the bar are insulated. Therefore the spatial derivative (heat flux) at the 2 ends are zero. But how do I make a function that would be consistent with the initial temperature distribution? $\endgroup$ – daniel17027 Jul 6 '15 at 1:00

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