Solution to heat equation with Neumann boundary conditions and lateral heat loss

Question

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:

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):

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:

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:

Any help will much appreciated.