# Heat problem with a heat equation

• Nov 24th 2009, 02:10 PM
arbolis
Heat problem with a heat equation
Consider a thermally isolated copper bar whose extremities are maintained to 0°C. The initial distribution of temperature is given by $T(x)=100 \sin \left ( \frac{\pi x}{L} \right )$ where $L$ is the length of the bar.
L=10cm.
Cross section of the bar : 1cm^2. (I call it $A$)
Thermal conductivity of the copper : 0.92 cal/(s cm °C). (I call it $K$)
Specific heat of the copper : 0.093 cal/(g °C). ( I call it $c$)
Density of the copper : 8.96 g/cm^3. (I call it $\rho$)
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1)Graph the initial distribution of temperature : Done.

2)What will be the final distribution of temperature after a very large time. Done. (The bar almost reaches 0°C everywhere).

3)Do a sketch about the distribution of the temperature after different amounts of time. Done. (Basically $T(x)$ becomes equal to $Y\sin \left ( \frac{\pi x}{L} \right )$ where $Y<100$.)

4)What is the gradient of the initial temperature in the extremities of the bar. Done : What I did was to derivate $T(x)$ and evaluating it in $x=0$. I reached $\frac{100 \pi}{L}$.

5)What is the initial heat flux through the extremities of the bar?
What I did was to write down $\frac{dq}{dt}=KA \frac{ \partial T}{ \partial x} \big |_{x=0}$.

6)What is the initial heat flux in the center of the bar? (I reached 0).
What will be this value for any posterior time. (Always 0.)
Analyze this answer. (I believe that although there is a heat flux in the right direction, it cancels out with the heat flux in the left direction, hence the total net flux in the center of the bar always remains 0.)

7)What is the initial value of the derivative of the temperature with respect to time in the center of the bar.
(What I did was : Oh oh. I think I fell over the diffusion heat equation, namely $\frac{ d Q}{dt}=K \nabla ^2 T$. I've no idea about how to solve this. It seems like an ODE of second order with given initial conditions. So I'm stuck here.

8)Give an estimation of the necessary time for the bar to cool off (at around $0.01T_0$).

9)The derivative of the temperature with respect to time in the center of the bar must : stay constant, increase or decrease?
(What I answer : It must decrease otherwise the bar would not cool off).
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I have a doubt about 5), shouldn't it be $\frac{dq}{dt}=cA \frac{ \partial T}{ \partial x} \big |_{x=0}$? I guess I should check out the units.
Also I guess that $\rho$ appears in part 7)...

Can you confirm all what I did, or any part of it?
I thank you for any help, as tiny as it may seems to you, it is greatly appreciated by me.
• Nov 25th 2009, 10:40 AM
shawsend
Hi Arbolis. I'm a little rusty but let me try to answer some of them. First write it precisely:

$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2},\quad 0\leq x \leq 10,\quad t\geq 0$

$u(0,t)=0,\quad u(10,t)=0$

$u(x,0)=100\sin\left(\frac{\pi x}{10}\right)$

where $k=\frac{K}{CD}$ (I think).

To solve this PDE, you generally use separation of variables which is not too bad to learn. Doing that I get:

$u(x,t)=100 e^{-\left(\frac{\pi}{10}\right)^2 kt} \sin(\frac{\pi x}{10})$

Now that I have the solution, it's easy to calculate $\frac{\partial u}{\partial x}$ and substitute the point $(5,0)$ right?

Ok, so you got (1) and (2) and as far as three, then just do a Plot3D[u(x,t),{x,0,10},{t,0,20}] in Mathematica looks to me. and (4) is just $\nabla u$. Isn't (5) just $\frac{\partial u}{\partial x}$ at the end points for t=0? (6) is the same dif at x=5 looks to me. (7) and (8) you can do right? As far as (9), just take partials of u with respect to t I think would answer that one.