Steady State 2-D Heat Equation with Mixed Boundary Conditions

In summary, the conversation is about trying to solve the Laplacian Equation with mixed boundary conditions on a 1m x 1m rectangular square. The relevant equations are provided and the attempt at a solution involved separation of variables but the problem could not be solved. The person asking for help is also requested to provide some of their work for further assistance.
  • #1
NickD2
1
0

Homework Statement



I am trying to solve the Laplacian Equation with mixed boundary conditions on a rectangular square that is 1m x 1m.



Homework Equations



[tex]\nabla[/tex]2T=0

.....T=500K
....________
....|@@@@|
T=500K...|@@@@|...T=500K
....|@@@@|
....|______.|
....Convection
....dT
....-- = h(T(x,0)-300K)
....dy
The square is 1m x 1m
h = 10


The Attempt at a Solution



I started by doing separation of variables and ended up with something that I could not solve...

P.S. Please excuse the .'s and @'s as they are just spacers to keep the geometery of the problem in tact.

Thanks!
 
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  • #2
If you show some of your work, we might be able to figure out exactly what's giving you trouble.
 

What is the Steady State 2-D Heat Equation with Mixed Boundary Conditions?

The Steady State 2-D Heat Equation with Mixed Boundary Conditions is a mathematical model used to describe the steady state temperature distribution in a 2-dimensional system with different types of boundary conditions.

What are the key assumptions of the Steady State 2-D Heat Equation with Mixed Boundary Conditions?

The key assumptions of this equation include: the system is in a steady state, the temperature gradient is constant, the thermal conductivity is constant, and there is no internal heat generation.

How is the Steady State 2-D Heat Equation with Mixed Boundary Conditions solved?

The equation can be solved using various numerical methods such as finite difference method, finite element method, and boundary element method. Analytical solutions are also available for specific boundary conditions.

What are the applications of the Steady State 2-D Heat Equation with Mixed Boundary Conditions?

This equation is commonly used in fields such as engineering, physics, and environmental science to analyze heat transfer in various systems, such as buildings, electronic devices, and geological formations.

What are some limitations of the Steady State 2-D Heat Equation with Mixed Boundary Conditions?

While this equation is a useful tool for analyzing heat transfer, it does have some limitations. It assumes steady state conditions, which may not always be accurate in real-world scenarios. It also does not account for factors such as convection and radiation, which can significantly affect temperature distribution. Additionally, the accuracy of the solution depends on the accuracy of the boundary conditions and physical properties used in the equation.

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