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12. Consider a material that has a thermal conductivity varying with temperature as k = \beta T^2 and a constant thermal contraction coefficient, \alpha. Derive a relationship for the overall change in length of a rod of initial length L as a function of the temperature difference between the two ends of the rod. Show the result for the special case where the low temperature end is at 0 K. Compare the answer to that for k = constant. 13. Same as Problem 12 except let the thermal conductivity be a linear function of temperature, k = \beta T. 

          12. Consider a material that has a thermal conductivity varying with temperature as
k = \beta T^2 and a constant thermal contraction coefficient, \alpha. Derive a relationship
for the overall change in length of a rod of initial length L as a function of
the temperature difference between the two ends of the rod. Show the result
for the special case where the low temperature end is at 0 K. Compare the
answer to that for k = constant.
13. Same as Problem 12 except let the thermal conductivity be a linear function of
temperature, k = \beta T.
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12. Consider a material that has a thermal conductivity varying with temperature as k = βT^2 and a constant thermal contraction coefficient, α. Derive a relationship for the overall change in length of a rod of initial length L as a function of the temperature difference between the two ends of the rod. Show the result for the special case where the low temperature end is at 0 K. Compare the answer to that for k = constant. 13. Same as Problem 12 except let the thermal conductivity be a linear function of temperature, k = βT.

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University Physics with Modern Physics
University Physics with Modern Physics
Hugh D. Young 14th Edition
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Consider a material that has a linear thermal conductivity as a function of temperature, k=eta T and a constant thermal contraction coefficient, alpha . Derive a relationship for the overall change in length of a rod of initial length L as a function of the temperature difference between the two ends of the rod. Show the result for the special case where the low temperature end is at 0K. Compare the answer to that for k= constant. 12. Consider a material that has a thermal conductivity varying with temperature as k = T2 and a constant thermal contraction coefficient, . Derive a relationship for the overall change in length of a rod of initial length L as a function of the temperature difference between the two ends of the rod. Show the result for the special case where the low temperature end is at 0 K. Compare the answer to that for k =- constant. 13. Same as Problem 12 except let the thermal conductivity be a linear function of temperature, k = T.
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Transcript

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00:01 So in this problem, we're considering a uniform rod and material whose temperature varies only along its length in the x direction.
00:07 We're assuming that the only motion of energy is heat conduction within the rod, so no energy enters their leage from the side.
00:14 So we have a rod.
00:18 Here's our section.
00:20 This will be delta x, q1, q2, x minus delta x over 2, global midpoint x plus delta x over 2.
00:38 So considering the heat flowing from both directions into a small segment of length delta x, we want to derive this equation.
00:48 So the rate of heat flow into the segment from the left, according to newton's law of heat conduction, would be t1 over delta t, negative k -a -d -t over tx, where a is the cross -sectional area of the rod, and the heat flow into the section from the right.
01:23 So the total rate of heat flow into the segment.
01:49 We're simplifying that down gives us k a delta x d2t over dx squared.
01:59 So we want to rewrite the above equation in terms of heat capacity, so that would be q is equal to mc delta t and the density of the rod, which is m over a delta x.
02:14 So we'll give us mc delta t over delta t is mk over row.
02:28 Then we want to take the limit of delta t going to zero, and that will just give us k.
02:50 That's for part a.
02:53 And now for part b, we want to assume that k is independent of temperature and verify by substitution that the solution to the heat equation is valid.
03:02 So for this, we just want to evaluate the derivatives...
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