heat diffusion

[logistic_guy]

here is the question

Derive the solution of the two dimensional steady-state heat diffusion equation using the method of separation of variables with the boundary conditions in figure 1.16. Use this solution to find the temperature at the midpoint \(\displaystyle T(1,0.5)\) with the boundary temperatures in figure 1.17. Show that the temperature at the midpoint is \(\displaystyle 94.5^{\circ}C\) by considering the first five nonzero terms of the infinite series that must be evaluated.

Hint: Use the transformation \(\displaystyle \theta \equiv \frac{T - T_1}{T_2 - T_1}\) and solve \(\displaystyle \frac{\partial^2 \theta}{\partial x^2} + \frac{\partial^2 \theta}{\partial y^2} = 0.\)

figure 1.16


figure 1.17



my attemb

this is the solution i must get



i think they want me to use the first part

\(\displaystyle \theta(x,y) = \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1} + 1}{n}\)

which look like infinite series of calculus

 

[mario99]

Read this thread, it will help you a lot to derive the solution.

Laplace PDE that will Blow your Mind: partial^2-u/partial-x^2 + partial^2-u/partial-y^2 = 0,...

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 0 < x < 1, \ \ \ \ \ 0 < y < 2 u(x,0) = x, \ \ \ \ \ u(x,2) = x^2 u(0,y) = y, \ \ \ \ \ u(1,y) = y^2 This is a Laplace partial differential equation. I don't know how to solve this problem. I looked for similar...

www.freemathhelp.com

 

[logistic_guy]

Read this thread, it will help you a lot to derive the solution.

Laplace PDE that will Blow your Mind: partial^2-u/partial-x^2 + partial^2-u/partial-y^2 = 0,...

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 0 < x < 1, \ \ \ \ \ 0 < y < 2 u(x,0) = x, \ \ \ \ \ u(x,2) = x^2 u(0,y) = y, \ \ \ \ \ u(1,y) = y^2 This is a Laplace partial differential equation. I don't know how to solve this problem. I looked for similar...

www.freemathhelp.com

thank

it look different and it don't explain any steps

 

[mario99]

thank

it look different and it don't explain any steps

Did you check the links given by professor Khan?

 

[khansaheb]

Derive the solution of the two dimensional steady-state heat diffusion equation

Heat is "transported" by (i) conduction, (ii) convection and (3) radiation. What does "heat diffusion" include?

 

[logistic_guy]

Did you check the links given by professor Khan?

no

Heat is "transported" by (i) conduction, (ii) convection and (3) radiation. What does "heat diffusion" include?

i don't understad heat diffusion but i think the question is about the two dimensional steady state conduction

 

[mario99]

no

Open the link in post #3. Example 2 is exactly the same as the partial differential equation in your OP. Just follow the steps. If you got stuck anywhere, tell us.

 

[logistic_guy]

Open the link in post #3. Example 2 is exactly the same as the partial differential equation in your OP. Just follow the steps. If you got stuck anywhere, tell us.

i open link, example 2 is very different

 

[mario99]

i open link, example 2 is very different

It is the same, it just uses different variables. I will show you.

This is your partial differential equation:

[imath]\displaystyle \frac{\partial^2 \theta}{\partial x^2} + \frac{\partial^2 \theta}{\partial y^2} = 0[/imath]

[imath]\displaystyle \theta(0,y) = 0[/imath]
[imath]\displaystyle \theta(L,y) = 0[/imath]
[imath]\displaystyle \theta(x,0) = 0[/imath]
[imath]\displaystyle \theta(x,W) = 1[/imath]

This is the partial differential equation in Example 2:

[imath]\displaystyle \frac{\partial^2 u_3}{\partial x^2} + \frac{\partial^2 u_3}{\partial y^2} = 0[/imath]

[imath]\displaystyle u_3(0,y) = 0[/imath]
[imath]\displaystyle u_3(L,y) = 0[/imath]
[imath]\displaystyle u_3(x,0) = 0[/imath]
[imath]\displaystyle u_3(x,H) = f_2(x)[/imath]

Compare between them and you will notice only this difference:

[imath]\displaystyle u_3 = \theta[/imath]
[imath]\displaystyle H = W[/imath]
[imath]\displaystyle f_2(x) = 1[/imath]

Read the solution of Example 2, and anywhere you see [imath]\displaystyle u_3, H, \ \text{or} \ f_2(x),[/imath] just replace it with [imath]\displaystyle \theta, W, \ \text{or} \ 1,[/imath] respectively.

Note: I took the boundary conditions of your partial differential equation from figure 1.16.

 

[logistic_guy]

It is the same, it just uses different variables. I will show you.

This is your partial differential equation:

[imath]\displaystyle \frac{\partial^2 \theta}{\partial x^2} + \frac{\partial^2 \theta}{\partial y^2} = 0[/imath]

[imath]\displaystyle \theta(0,y) = 0[/imath]
[imath]\displaystyle \theta(L,y) = 0[/imath]
[imath]\displaystyle \theta(x,0) = 0[/imath]
[imath]\displaystyle \theta(x,W) = 1[/imath]

This is the partial differential equation in Example 2:

[imath]\displaystyle \frac{\partial^2 u_3}{\partial x^2} + \frac{\partial^2 u_3}{\partial y^2} = 0[/imath]

[imath]\displaystyle u_3(0,y) = 0[/imath]
[imath]\displaystyle u_3(L,y) = 0[/imath]
[imath]\displaystyle u_3(x,0) = 0[/imath]
[imath]\displaystyle u_3(x,H) = f_2(x)[/imath]

Compare between them and you will notice only this difference:

[imath]\displaystyle u_3 = \theta[/imath]
[imath]\displaystyle H = W[/imath]
[imath]\displaystyle f_2(x) = 1[/imath]

Read the solution of Example 2, and anywhere you see [imath]\displaystyle u_3, H, \ \text{or} \ f_2(x),[/imath] just replace it with [imath]\displaystyle \theta, W, \ \text{or} \ 1,[/imath] respectively.

Note: I took the boundary conditions of your partial differential equation from figure 1.16.

give me some time to read this