Heat diffusion into an infinite rod

In summary, the conversation discussed the heat equation and heat diffusion in one dimension, as well as the concept of heat kernels and their mathematical function. The topic then shifted to the possibility of formulating an equation for temperature as a function of position and time for an infinitely sized rod with a constant temperature attached to one end. The solution for this scenario was mentioned as a complimentary error function. The conversation also touched on the backgrounds of the responders on Physics Forums, who are volunteers with a passion for helping others understand math and physics.
  • #1
kairama15
31
0
I have recently been curious about heat diffusion. If there is space in one dimension with any kind of temperature dispersed throughout, then the heat equation states that the derivative of the temperature with respect to time at any point equals some constant (k) multiplied by the second derivative of temperature with respect to space at that point... or:
dT/dt=k*d^2T/d^2x
This is expressed in the Wikipedia page regarding the heat equation. This is intuitive, and I understand this differential equation.
I was reading up on ‘heat kernels’, which seem to be a tiny point of heat inside an object (let’s say a long metal rod with a point somewhere in it that is very hot). Through a lot of math it can be shown that the function that describes the temperature as a function of position throughout the rod and time is:
(1/sqrt(4*pi*k*t)) * e^(-x^2/(4*k*t)) (found on wikipedia)
This function makes sense. At time t, the function expresses the temperature as being concentrated at a point in the rod and as time goes by the temperature moves out of the point along the length of the rod.
However, I am curious if anyone has ever attempted to formulate an equation for temperature as a function of position and time f(x,t) for an infinitely sized rod whose one end is attached to an object that remains at constant temperature? So, instead of a heat kernel being placed somewhere in the middle of the rod, an object of always constant temperature is placed at one end of the rod, slowly heating the rod from one end. I have a suspicion the function will likely be something like:
T=e^(-k*x/t)
but I don't know. Does anyone know if a solution was ever developed for this?
 
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  • #2
The solution is a complimentary error function involving ##x/\sqrt{t}##. See Transport Phenomena by Bird, Stewart, and Lightfoot or Conduction of Heat in Solids by Carslaw and Jaeger for the solution, or Heat Transmission by McAdams or any other heat transfer book.
 
  • #3
Thank you very much for your response. The complimentary error function of x/sqrt(t) does indeed satisfy the differential equation dT/dt=kd^2T/d^2x. How beautiful. This equation seems to suggest that heat diffusion throughout a rod or a substance is really slow, especially since the time component is 1/sqrt(t) inside of the complimentary error function which goes to 0 quite fast for increasing x.

Thank you so much for your response and helping me understand the beautiful mathematical description of this natural phenomenon. Are you 'responders' paid by physics forums, or do you respond as a hobby, or a little of both? I'm very happy there are people that can respond to my curious questions. I think math and physics are beautiful like art, so I like exploring them. And I want to know what your 'responders' backgrounds are.
 
  • #4
kairama15 said:
Thank you very much for your response. The complimentary error function of x/sqrt(t) does indeed satisfy the differential equation dT/dt=kd^2T/d^2x. How beautiful. This equation seems to suggest that heat diffusion throughout a rod or a substance is really slow, especially since the time component is 1/sqrt(t) inside of the complimentary error function which goes to 0 quite fast for increasing x.

Thank you so much for your response and helping me understand the beautiful mathematical description of this natural phenomenon. Are you 'responders' paid by physics forums, or do you respond as a hobby, or a little of both? I'm very happy there are people that can respond to my curious questions. I think math and physics are beautiful like art, so I like exploring them. And I want to know what your 'responders' backgrounds are.
Physics Forums staff are not paid. It is totally voluntary. Our only reward is being able to help members like yourself.

There are detailed bios of all the PF staff available in our forums.
 
  • #6
Thank you for your time helping people with math/physics. I think it's awesome. There should be more people like you folks.
 

FAQ: Heat diffusion into an infinite rod

What is heat diffusion into an infinite rod?

Heat diffusion into an infinite rod is a process in which heat is transferred from one end of a rod to the other end due to a temperature difference. This process occurs in a continuous manner until the temperature throughout the rod becomes uniform.

How does heat diffusion occur in an infinite rod?

Heat diffusion occurs in an infinite rod through a process called conduction, where heat is transferred from one molecule to another through direct contact. As the heated molecules vibrate, they transfer their energy to neighboring molecules, causing them to vibrate as well.

What factors affect heat diffusion in an infinite rod?

The rate of heat diffusion in an infinite rod is affected by several factors, including the thermal conductivity of the material, the temperature difference between the ends of the rod, and the length and cross-sectional area of the rod. The thermal conductivity of a material determines how easily heat can be transferred through it, while a larger temperature difference and a longer, thinner rod will result in a faster rate of heat diffusion.

What is the equation for heat diffusion in an infinite rod?

The equation for heat diffusion in an infinite rod is known as Fourier's law and is expressed as q = -kA(dT/dx), where q is the heat flow rate, k is the thermal conductivity, A is the cross-sectional area, and (dT/dx) is the temperature gradient along the rod.

How is heat diffusion into an infinite rod used in real-world applications?

Heat diffusion into an infinite rod has many practical applications, such as in cooking, where heat is transferred from a stove to a pot or pan, and in heating systems, where heat is distributed through pipes to warm buildings. It is also used in thermometers and other temperature measuring devices.

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