Partial differential equation boundary

[ficku1]

Homework Statement

I have to calculate the stationary field inside a room.

Homework Equations

The Attempt at a Solution

I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Ee^k_nx+Fe^-k_nx)cos(k_ny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary conditions
-λdT/dx(x=0,y)=h((T(0,y)-T1) and
-λdT/dx(x=a,y)=h((T2-T(a,y)).

h is convective heat coefficient.
I am wondering if this is possible to solve without the use of Matlab or some programme?
I can't figure out how to get constants E and F with these two BCs.

I tried and i got

-λk(E-F)cos(ky)=h((E+F)cos(ky)-T1) but i don't know how to continue. Do you mulltiply it by cos(k_my) and then integrate it? But I can't come to any reasonable result, i can't get the E or F out of it.

If anybody has any advice i would be very happy to hear it.
Thank you.

 

[Chestermiller]

What is the actual problem that you are solving?

 

[ficku1]

I am solving this problem:
House: a room (see figure) 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.

 

[Chestermiller]

I guess you are supposed to assume that this is a pure heat conduction problem, with no natural convection currents in the air. Suppose both windows were the same size. Would you be able to solve the problem then? What would you get?

 

[Ray Vickson]

Homework Statement

I have to calculate the stationary field inside a room.

Homework Equations

The Attempt at a Solution

I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Ee^k_nx+Fe^-k_nx)cos(k_ny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary conditions
-λdT/dx(x=0,y)=h((T(0,y)-T1) and
-λdT/dx(x=a,y)=h((T2-T(a,y)).

h is convective heat coefficient.
I am wondering if this is possible to solve without the use of Matlab or some programme?
I can't figure out how to get constants E and F with these two BCs.

I tried and i got

-λk(E-F)cos(ky)=h((E+F)cos(ky)-T1) but i don't know how to continue. Do you mulltiply it by cos(k_my) and then integrate it? But I can't come to any reasonable result, i can't get the E or F out of it.

If anybody has any advice i would be very happy to hear it.
Thank you.

In your boundary conditions, what is $\lambda$? Why do you have no "$n$" on your E and F; that is, should the solution be
$$T(x,y) = \sum_{n} \left( E_n e^{k_n x} + F_n e^{-k_n x} \right) \cos (k_n y) ? $$

Also, you should explain briefly how the $k_n = n \pi/a$ arise, and show briefly how you obtained your formula for $T(x,y)$. For all we know you might have made a mistake, but we cannot tell if you don't supply more details.

 

[ficku1]

Yes, the equation should be like that.
λ is thermal conductivity.
I have attached a document where you can see my calculations.

 

[Ray Vickson]

Yes, the equation should be like that.
λ is thermal conductivity.
I have attached a document where you can see my calculations.

Why are two of your boundary conditions given as
$$-\lambda \frac{\partial}{\partial x} T(x=a,y) = h [T(a,y)-T_1]\;\; \text{and} \; -\lambda \frac{\partial}{\partial x} T(x=0,y) = h[T_2 - T(0,y)]$$
instead of just $T(a,y) = T_1$ and $T(0,y) = T_2$?

 

[Nidum]

Is this the same problem ? :

https://www.physicsforums.com/threads/stationary-temperature-field-inside-a-house.926542/