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Heated square rod

  1. Feb 12, 2015 #1
    1. The problem statement, all variables and given/known data
    A long squared rod with sides ##l##, thermal conductivity ##\lambda ##, specific heat ##c_p## and density ##\rho ## is cooled with water with constant temperature. Inside our squared rod we have a long square heater with sides ##l/2## and is completely centered inside the rod. Heat sources per unit volume in this is equal to ##q [W/m^3]##. (When calculating the temperature, you can use only the biggest three terms of the sum)
    What is the maximum possible temperature of the surrounding water , before the heater starts to melt down? Melting temperature is ##T_c##.

    2. Relevant equations


    3. The attempt at a solution
    I am really not sure this is correct, but at least it is something:
    So we have two squares, one with sides ##l## and the other one centered in the first one with sides ##l/2##. I have put the origin of my coordinate system in the left bottom corner of the bigger square. This may not be a really good idea but there is a good reason why I chose this option.

    And the reason is, that if the origin is there, than the function that solves ##\nabla ^2T(x,y)=0## is $$T(x,y)=\sum_{n,m}A_{n,m}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)$$ Of course the equation I am solving is not ##\nabla ^2T=0## instead I have to consider the heater! Therefore ##\nabla ^2 T(x,y)=-\frac{q}{\rho c_p D}##.

    Meaning I should write ##\frac{q}{\rho c_p D}## in terms of this sum ##T(x,y)=\sum_{n,m}C_{n,m}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)##, so let's do that: $$\frac{q}{\rho c_p D}\int_{l/4}^{3l/4}\int _{l/4}^{3l/4}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)dxdy=C_{n,m}(\frac l 2)^2$$ Which brings me to $$C=\frac{8q}{\rho c_p D\pi ^2}\frac{1}{mn}$$ but only for (both) odd ##m## and ##n##.
    So finally $$\frac{q}{\rho c_p D}=\sum_{m,n}\frac{8q}{\rho c_p D\pi ^2}\frac{1}{mn}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)$$

    Now I can start solving ##\nabla ^2 T(x,y)=-\frac{q}{\rho c_p D}## $$\sum _{m,n}A_{m,n}\frac{\pi ^2}{l^2}(m^2+n^2)sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)=-\sum_{m,n}\frac{8q}{\rho c_p D\pi ^2}\frac{1}{mn}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)$$ Which brings me to
    $$A_{m,n}=\frac{8qL^2}{\rho c_pD\pi^4}\frac{1}{mn(m^2+n^2)}$$ Therefore the solution should be $$ T(x,y)=\sum_{\text{odd} m,n}\frac{8qL^2}{\rho c_pD\pi^4}\frac{1}{mn(m^2+n^2)}sin(\frac{n\pi}{l}x)sin(\frac{n\pi}{l}y)$$

    Now to calculate the temperature I think I should do this, but I am really not so sure about this part: $$P=\lambda S\frac{\Delta T}{L/4}$$ where ##P=\int qdV## so $$\int qdV=\lambda S\frac{T_c-T_x}{L/4}$$ where I used notation ##T_x## for the temperature that I hope the problem is asking after...

    Hmm. What do you think?
     
  2. jcsd
  3. Feb 12, 2015 #2

    BvU

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    This is not a contribution, just some questions:
    I imagine the rod is solid and the inner rod, same material properties, is the heat source. I would expect from symmetry that it's enough to consider one eigth, e.g. (with the origin at the axis of the rod) from x-axis to the line y=x. x-axis points to the middle of a side, y=x to an edge.
    Is that the right idea ?

    Heat is generated from x=0 to x=l/4 and the boundary conditions are: T gradient ##\perp## the lines mentioned and T = Twater at x =l/2 .
    Sound reasonable ?
     
  4. Feb 13, 2015 #3
    I can't see any problems with it. By the way, that is an interesting idea.
    Sounds very reasonable! Hopefully I will have some time left to check if the results match with the original post.
     
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