How can the thermal conductivity of a uniform rod be constant while also decreasing in the direction of heat transfer? A teacher told me this, but why isn't temperature at all points in the rod being equal at steady state?
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3$\begingroup$ maybe he's muddling up k and dT/dt? $\endgroup$– Mohammad AtharCommented Jun 13, 2018 at 15:13
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$\begingroup$ I think it has something to do with varying temperature in the rod at different points in steady state..maybe but I don't know the reason $\endgroup$– Hydrous CaperillaCommented Jun 13, 2018 at 15:14
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1$\begingroup$ The question may've been misworded. I mean, if something's constant, it can't also be decreasing. So what was actually said? $\endgroup$– NatCommented Jun 13, 2018 at 22:49
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1 Answer
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I guess that I don't totally understand your question. I think that you may be confusing several concepts.
First, the thermal conductivity is generally temperature dependent. So, the hot end of the rod can have a different thermal conductivity than the cold end of the rod. However, unless there is a really large temperature difference across the rod, the thermal conductivity shouldn't change much.
As for your last sentence, steady state does not mean equilibrium. Equilibrium means no energy is moving -- the heat current is zero. Steady state means that the heat current isn't changing in time, but the heat current doesn't have to be zero. For example, you could stick one end of your rod in a furnace at 500 degrees and put the other end in a cooler at 0 degrees. Heat will flow thru the rod, and if you wait a little while, the heat current will stabilize and become constant as long as the hot and cold ends are maintained at a constant temperature.
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2$\begingroup$ To add to this, I would say that steady state means that the temperature is not changing with time at each location along the rod, even though it can be varying with location along the rod. $\endgroup$ Commented Jun 13, 2018 at 22:38
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$\begingroup$ So the heat current is constant at each point on the rod but they may have different temperatures? $\endgroup$ Commented Jun 14, 2018 at 0:43
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$\begingroup$ Basically yes. If you want to get pedantic, I'd say that the heat current thru any cross-section of the rod is constant, not that the heat current is constant at each point, but that's not a big difference. $\endgroup$– lnmaurerCommented Jun 14, 2018 at 3:14