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One end of a small insulated rod is attached to a support which is maintained at a constant temperature of 473 K,and once the rod has reached a uniform temperature in its entire length the insulation is removed. Now the surrounding air is at 298 K with convective heat transfer coefficient h.

I am required to find the time elasped when the temperatures at some length of rod reaches a given temperature.The radius,thermal conductivity and thermal diffusivity are all given

This is a unsteady state heat conduction and convection problem.I got the following pde $$k \frac{\partial^2 T }{\partial z^2} - \frac k\alpha \frac{\partial T}{\partial t}=\frac{2h}{R} (T-T_0)$$

where $k$ is thermal conductivity $\alpha$ is thermal diffusivity and $T_0$ is surrounding ambient temperature.I am facing trouble finding out the two boundary conditions.The initial condition is that at $t=0$ the rod is at a uniform temperature of 473K.The first boundary condition is at $z=0$ the temperature is 473K.However i am unable to find out the second boundary condition.The tip of the fin is not insulated and is neither maintained at a constant temperature.

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  • $\begingroup$ The "second boundary condition" is the prescribed value of $T_0(x,t)$. In your problem it is a constant for all $x$ and $t$, but in general it could be an arbitrary function of both $x$ and $t$. $\endgroup$ – alephzero Feb 18 at 21:38
  • $\begingroup$ You have a temperature BC at one end of the rod. How does the other end of the rod behave? It has heat flux. Express that mathematically. This becomes your second required BC in $z$. $\endgroup$ – Jeffrey J Weimer Feb 18 at 23:23
  • $\begingroup$ @user1318499 I couldn't solve the pde with one initial and one boundary condition.can you suggest a way to solve it using finite difference method to find time elasped when temperature at 80cm becomes 323 K and the rod is 160 cm long. $\endgroup$ – user471651 Feb 19 at 3:36
  • $\begingroup$ Using explicit finite difference we need to extend the interval to 160+1.6 cm and 160+3.2 if we choose 16 mm as our differential length. The initial condition for the extended interval should be 298K and the boundary condition for all time it is at 298 K.Am I right? @user1318499 $\endgroup$ – user471651 Feb 19 at 3:39
  • $\begingroup$ @alephzero sir the temperature of the fin is not maintained at 298K and it will take time to reach that temperature.Is my approach in the comments above correct?However i am getting a time limit exceeded error when trying to implement it in c++. $\endgroup$ – user471651 Feb 19 at 4:04
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The BC at the far end is the flux condition.

$$ -k \left. \frac{dT}{dx}\right|_{x=L} = h(T - T_o) $$

See Chapter 17 in the book by Welty etal for examples of extended surfaces and their solutions at steady state. Finally, just for fun, here is an example temperature profile of an extended beam at steady state with the heat transfer BC. Your final answer will end at a comparable profile when $m^2 = 10$.

temperature profile in extend beam

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