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Old Sep 24th 2017, 07:10 AM   #1
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Diffusion equation

Hello!

I'm a physics student and i have a problem i can't solve. I was wondering if there is anyone here who would be willing to help me solve this?

House: a room (see attachment) has perfectly isolated walls, except the two windows where a convective heat exchange takes place (with the same transfer coefficient).
Outside temperature in front of a sun-faced wall-sized panoramic window is T1, while at the back it is T2. Calculate the stationary temperature field inside the room. You can also play by adding an additional energy flux through the front
window due to a sunlight at an angle φ.


Thank you so much for your help!
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File Type: pdf room.pdf (85.0 KB, 1 views)
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Old Sep 24th 2017, 08:35 AM   #2
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This has nothing at all to do with the "diffusion equation" If the temperature at one side (take it to be x= 0) is T1 and the temperature at the other side (take it to be x= d, the thickness of the glass) is T2 then the stationary temperature field inside the window is linear: T(x)= T1+ x(T2- T1)/d.

When x=0, that is T(0)= T1+ 0(T2- T1)d= T1. When x= d that is T(d)= T1+ d(T2- T1)/d= T1+ T2- T1= T2.

(Well, it does have a little to do with the "diffusion equation". The diffusion equation is $\displaystyle \frac{\partial^2 T}{\partial x^2}= \kappa\frac{\partial t}$. If the temperature field is "stationary" then its derivative with respect to t is 0 so the second derivative with respect to x must be 0 from which we conclude that the temperature is a linear function of x.)
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Old Sep 25th 2017, 06:56 AM   #3
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Thank you for your answer.

But isn't this the diffusion equation dT/dt=D (d^2 T)/dx^2 and if we are calculating stationary temperature the equation is actually Lapace ΔT=0 .
I think it is not the same in the x and y direction because the windows aren't the same size and the sun is only shining in the room on one side. So T=X(x)Y(y), and we get
1/X * (d^2 X)/dx^2=-1/Y * (d^2 Y)/dy^2 =C
But i am not sure what the boundary conditions should be?
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