Fourier's Equations: difficult problem

In summary, the conversation discusses a problem involving a series and two parallel insulators that are used to bring a system from a temperature of Tzero to Ttwo. However, the two temperatures have a large difference, leading to a varying kappa with temperature. The question is how to solve for Q/t of the whole system, taking into account the linear relationship between kappa and temperature change. The discussion also mentions a cylinder in series and two disks in parallel. The solution involves separating and integrating, but the issue is determining the general form and using it to solve the problem. It is unclear if the problem is a transient or steady state one. A picture may be helpful in understanding the problem.
  • #1
argpirate
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Homework Statement


So I have a problem where I have a series and two other parallel insulators bringing a system from a temperature of Tzero to Ttwo. However, the two aforementioned temperatures have a huge difference thus causing kappa to vary with temperature. I am trying to know how to solve for Q/t of the whole system accounting for the varying kappa with temperature (assume the kappa function is linear with temperature change (y=mx+b form)). So I have a cylinder that is in series (2*pi*r*L) and top and bottom that are parallel but not series (disks with thickness).

Homework Equations


Q/t=kAdT/dx

The Attempt at a Solution


I serperated and integrated. I set Q/t=Q/t for the series from where they change. The difficulty I have is coming up with the general form (T-t)/R and using that to do stuff.
 
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  • #2
Is this a transient problem or a steady state problem?
 
  • #3
I think we need a picture ...
 

1. What are Fourier's equations used for?

Fourier's equations are used to describe the transfer of heat or energy in a system. They are commonly used in physics, engineering, and other scientific fields to solve problems related to heat conduction, diffusion, and other processes.

2. Why are Fourier's equations considered a difficult problem?

Fourier's equations can be difficult to solve because they involve complex mathematical concepts such as partial differential equations and Fourier transforms. The equations also rely on a number of assumptions and simplifications, which can make them challenging to apply to real-world situations.

3. What is the history behind Fourier's equations?

Joseph Fourier, a French mathematician and physicist, first developed the equations in the early 19th century. He used them to study the flow of heat in solid objects, and his work laid the foundation for the field of mathematical physics.

4. What are some applications of Fourier's equations?

Fourier's equations have a wide range of applications, including predicting the temperature distribution in materials, analyzing heat transfer in electronic devices, and studying fluid dynamics. They are also commonly used in image and signal processing, as well as in solving problems related to sound and vibration.

5. Are there any limitations to Fourier's equations?

While Fourier's equations are a powerful tool for solving many problems, they do have some limitations. For example, they assume that the material being studied is homogeneous and isotropic, and that heat transfer occurs only through conduction. In reality, these assumptions may not always hold, leading to potential inaccuracies in the results.

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