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Proof of heat flow direction?

  1. Jan 19, 2008 #1
    "Proof" of heat flow direction?

    Hi everyone, I have a homework problem that basically says to prove that heat is conducted from a high temperature area to a low temperature area:

    1. The problem statement, all variables and given/known data

    Consider a one-dimensional conductor, stretching from x=0 to x=L. The two ends are maintained at T_0 and T_L. The four sides of the conductor are insulated. The temperature distribution along the conductor is steady.

    q_0 represents the heat that enters through the x=0 cross section. Assume q_0 is position, so heat is conducted in the positive x-direction.

    Invoke the 2nd law to prove that q_0 flows toward lower temperatures, for example, by showing that T_L cannot be greater than T_0

    2. Relevant equations

    2nd Law of thermodynamics

    3. The attempt at a solution

    My attempt is as follows:

    I have the 2nd Law in the following form:


    I have come across this equation in my text (Heat Transfer, Bejan, 1993) as well:


    I wish to substitute this equation into the "q" part of the 2nd Law. From here, in order for the "dS/dt" term to be greater than or equal to zero (Entropy always increasing), the T_0 - T_L term must be greater than zero, therefore, T_0 must be greater than T_L.

    Does this make sense? I'm not sure what to do about the summation term in the form of the 2nd law that I have. Do I only pick the "0" position and forget about the "L" position?

    Thanks in advance for your help.
    Last edited: Jan 19, 2008
  2. jcsd
  3. Jan 20, 2008 #2


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    Your question got me curious, so I pulled out my old statistical physics text (berkeley physics course-volume 5 by F. Reif). Heres a couple of quotes that may help?
    Last edited: Jan 20, 2008
  4. Jan 20, 2008 #3
    Thank you for the reply. I agree with the quotes you have listed, but I am unsure as to how they help. I agree that they discuss the means by which entropy increases, but I am unsure as to how the increase in entropy relates to the diffusion of heat from a high temperature area to a low temperature area.
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