Question about heat conduction equation.

In summary, the conversation is about solving a 1-D heat conduction equation with one end kept at zero temperature and the other insulated at x=L. The equation is \frac{\partial^2 u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2} and the condition for insulation is \frac{\partial U}{\partial x}(L, u)= 0. The speakers discuss the equation and provide pointers on how to solve it.
  • #1
DrKareem
101
1
I know how to solve a 1-D heat conduction equation when both ends are kept at 0 temperature(using separation of variables and Fourier series.). In the notes, the prof asked us to solve for one end kept at zero and the other insulated (at x=L) and referred to a non-existent chapter in our textbook for the solution. Can someone give me a few pointers on how to proceed? I don't know what to equate U(L,t) to, and that kinda gets u stuck...
 
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  • #2
Well, firstly, post the equation you're trying to solve!
 
  • #3
I'm sorry, I (falsely) assumed everyone would recognize the equation i had in mind. The equation is

[tex]\frac{\partial^2 u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2}[/tex]

With U(0,t)=0. U(L,t)= ??
 
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  • #4
Firstly, that equation has no function in it! Secondly, are you sure that's the correct expression? I thought the heat equation looked like [tex]\frac{\partial U}{\partial t}=a^2\frac{\partial ^2U}{\partial x^2}[/tex] in one dimension?

Anyway, I think the condition that says that at x=L the rod is insulated, means that the flux at x=L is zero; i.e.[tex]\left. \frac{\partial U}{\partial x}\right|_{x=L}=0[/tex]
 
  • #5
yes, i made those horrible typos sorry again (been up very early having nothing to do but solve). I will try solving it, i think it should be easy. Thanks for the input.
 
  • #6
Cristo is right. Saying that one end, x= 0, is held at 0 means, of course, U(0,t)= 0. Saying that the other end, x=L, means that
[tex]\frac{\partial U}{\partial x}(L, u)= 0[/tex]
 
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  • #7
Thanks HallsofIvy. I already solved it, and it was easy as expected (i just didn't know the mathematical interpretation of the word 'insulated' in that context).
 

1. What is the heat conduction equation and why is it important?

The heat conduction equation is a mathematical representation of how heat is transferred through a material. It is important because it allows us to predict and understand how heat will move through different materials, which is crucial for many industrial and scientific applications.

2. What factors affect heat conduction?

The three main factors that affect heat conduction are the thermal conductivity of the material, the temperature difference across the material, and the cross-sectional area of the material.

3. What are the units of the heat conduction equation?

The units of the heat conduction equation vary depending on the system of units being used. In the SI system, the units are watts per meter per Kelvin (W/mK). In the English system, the units are BTU per hour per foot per degree Fahrenheit (BTU/hr-ft-°F).

4. Can the heat conduction equation be solved analytically?

Yes, the heat conduction equation can be solved analytically for simple geometries and boundary conditions. However, for more complex systems, numerical methods must be used.

5. How does the heat conduction equation differ from the heat transfer equation?

The heat conduction equation specifically describes the transfer of heat through a material, while the heat transfer equation is a more general equation that can also account for other forms of heat transfer such as convection and radiation.

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