Solving the Heat Equation: Exploring Solutions to Temperature Differentials

[MisterX]

I've been teaching myself some thermodynamics, and I've been thinking about solving the heat equation.

$\frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2}$

I haven't taken a course in PDEs.

I have noticed that if I assume an exponential solution, there are not non-decaying traveling wave solutions; the phase and group velocities are imaginary.

$i\omega = k^2K$
$\omega = -ik^2K$

non-traveling space oscillations decay in time, which isn't surprising since temperature tends to equalize:
$e^{i(kx - \omega t) } = e^{ikx }e^{-k^2K t}$

If we wanted to make omega real, we could have

$k = \pm a(1+i)$

$e^{\mp ax} e^{\pm iax }e^{-i2a^2K t}$

This indicates a phase speed of $2aK$ and a group speed of $4aK$ for the non-decaying factor, if I have done everything properly.

I have also seen there are solutions like
$\frac{A}{\sqrt{t}}e^{-x^2/4Kt}$

Also one can add a constant to any solution.

The problem I would like to solve is this: x = 0 is fixed at some temperature T_s. For x > 0 T(x, 0) is initially some other temperature T_i. If it made the solution simpler, the drop off needn't be so sharp. The point is that I would expect to see a solution so that every point with x>0 would become arbitrarily close to Ts if enough time was passed. Also, it should move along, so that it would take longer for a place with larger x to reach a given temperature than a place with smaller x.

The purpose of this thread is to solicit help in solving this problem, or any thoughts on my ideas about solutions to the heat equation.

 

[Chestermiller]

Is there another boundary, say at x = L, or is the system seminfinfinite? If there is another boundary, what is the boundary condition at x = L.

In most cases, unless you've studied PDEs, you are going to have trouble solving the transient heat conduction equation. However, how to solve these specific cases can be explained.

 

[MisterX]

I was considering a system that is seminfinite.

Actually in the system that inspired this there is a boundry but it has no fixed value (for example the end of a rod which is free to have any temperature). But I was interested in how heat would travel through the rod, so the seminfinite case would be fine I think.

 

[Chestermiller]

I was considering a system that is seminfinite.

Actually in the system that inspired this there is a boundry but it has no fixed value (for example the end of a rod which is free to have any temperature). But I was interested in how heat would travel through the rod, so the seminfinite case would be fine I think.

If the rod is of finite length, then that definitely matters. An appropriate boundary condition at the far end would probably be zero heat flux ( zero temperature gradient).

If you want to find out how to solve problems like these, and want to see some solutions to standard problems (such as yours), see Conduction of Heat in Solids by Carslaw and Jaeger. Another reference that shows similar results is Transport Phenomena by Bird, Stewart, and Lightfoot.

 

[sardar]

I am glad to see your interest in thermodynamics and your efforts in teaching yourself the subject. The heat equation is a fundamental equation in thermodynamics and it is used to describe the behavior of temperature in a given system. Your approach of assuming an exponential solution is a common method used in solving partial differential equations (PDEs).

I can confirm that your observation about the non-decaying traveling wave solutions is correct. This is because the complex exponential function has a decaying factor, which leads to a decaying solution. However, as you have mentioned, there are other types of solutions such as the Gaussian solution that does not decay in time. These solutions are important in understanding the behavior of temperature in a system.

I appreciate your efforts in trying to make the phase velocity and group velocity real. However, it is important to note that these velocities are not physical quantities but rather mathematical constructs. The physical quantity of interest in this equation is the temperature, which is a real and physical quantity.

Moving on to your proposed problem, it seems like you are interested in finding a solution that satisfies the boundary conditions of a fixed temperature at x=0 and an initially different temperature at x>0. This is a common type of problem in thermodynamics and there are well-established methods to solve it. One possible approach is to use the method of separation of variables, where the solution is expressed as a product of two functions, one depending only on x and the other only on t. This method has been extensively used in solving the heat equation and can be applied to your problem as well.

In conclusion, I am impressed by your interest and efforts in understanding the heat equation and finding solutions to it. However, as a scientist, I would suggest seeking guidance from experts in the field or taking a course in PDEs to gain a deeper understanding of the subject. I would be happy to provide further assistance or discuss any thoughts you may have on solutions to the heat equation. Keep up the good work!