iMechanica - Comments for "Heat Conduction"
https://www.imechanica.org/node/4942
Comments for "Heat Conduction"enRe Re: 2 questions on heat conduction lectures
https://www.imechanica.org/comment/16196#comment-16196
<a id="comment-16196"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/10008#comment-10008">2 questions on heat conduction lectures</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Tony: Following your suggestions, I have streamlined the notes. I am teaching the <a href="http://imechanica.org/node/725">course on advanced elasticity</a> again, and have been updating other sections of the notes as well.</p>
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</ul>Wed, 02 Feb 2011 13:13:25 +0000Zhigang Suocomment 16196 at https://www.imechanica.orgRe: 2 questions on heat conduction lectures
https://www.imechanica.org/comment/10029#comment-10029
<a id="comment-10029"></a>
<p><em>In reply to <a href="https://www.imechanica.org/comment/10008#comment-10008">2 questions on heat conduction lectures</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><ol><li>Complete set of equations. You are right: the full list of equations are the same as before. The calculation of entropy shows that, if we adopt Fourier's law, the second law of thermodynamics is satisfied. In this derivation, we indeed have used energy conservation. The situation might be understood in a familiar context. Let's say we want to solve a set of 2 by 2 algebraic equations. We can use one equation to simplify another, but we will still have two equations.</li>
<li>Independent internal variables. A more systematic way to select independent variables is to list all the variables and then list all the constraints. Then decide which variables can vary independently. In practice, we can think over this matter in several ways, some physical and some mathematical. </li>
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</ul>Sun, 08 Mar 2009 01:48:57 +0000Zhigang Suocomment 10029 at https://www.imechanica.org2 questions on heat conduction lectures
https://www.imechanica.org/comment/10008#comment-10008
<a id="comment-10008"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/4942">Heat Conduction</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
I have two questions stemming from the lectures on heat conduction, which may be trivial in nature, but forgive me: blame it on the fact I am a biologist!
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First, I am very impressed that by our success in class in applying the fundamental postulate to a variational statement of total entropy and recovering boundary conditions and Fourier's law. The fact that we stopped there suggests that at that point everything is known.
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But, to be completely explicit and proceed to solve for the details of the temperature field, it would seem to me that the next step would be to (as usual) combine fourier’s law and an energy conservation statement to find a pde for the temperature field. But, did we not already use a conservation of energy statement for a material particle in our variational expression of the change in entropy of the composite? That is, it seems like we already used conservation of energy in finding our fourier law, but now the only thing I can see to do is to use it again, and thereby derive the usual pde – but this makes me nervous as I have a vague notion one ought not use the same relation twice in deriving a governing equation.
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My second confusion: How do we ensure a good choice of independent variables? In the notes and lecture, you state that a good choice of independent variables is I,q, Q. The defense of your choice of seems to rest on physical intuition –
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Once we know the energy added to a material particle from the reservoir and from neighboring material particles, the conservation of energy determines the variation of the energy of the material particle, and the thermodynamic model of the material determines the entropy of and the temperature of the material particle.” (p.6).
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This makes sense, but is not the requirement of independence a mathematical one? And if that is so, it ought to be evident from the mathematical structure what an appropriate choice would be. The physical argument makes a lot of sense, but I worry as the physical picture gets more complex it won’t be as easy to see.
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</ul>Wed, 04 Mar 2009 18:31:00 +0000Tony Rockwellcomment 10008 at https://www.imechanica.org