Why is the rate of heat transfer constant throughout different materials?

A video I watched recently mentioned how if there was a system, let's say, a bar connecting a hot reservoir to a cold reservoir, 100 - copper - iron - 0, the rate of heat transfer between the hot reservoir and junction point between copper and iron is equal to the rate of heat transfer between the junction point and the cold reservoir. In short, Heat flow in = heat flows out. Can anyone explain why this is?

• It would violate the second law of thermodynamics. The asymmetric heat flow between two identical reservoirs would heat up one of the reservoirs to a higher temperature (without anything else happening). Commented Sep 28, 2022 at 8:34

The statement Heat flow in = heat flows out is made on the assumption that no heat escapes from the sides, ie the sides have an ideal thermal insulator around them.
It is a restatement of the law of conservation of energy which applies to this particular example.

First principle of thermodynamics, $$d E = \delta W + \delta Q$$ with no work $$\delta W = 0$$, and steady conditions $$d E = 0$$ (no variation of the energy of the system), give you $$\delta Q = 0$$.

If you write $$\delta Q$$ as the sum of all the contributions that would make the energy of the system increase (the positive contributions, with the convention used above, $$Q^{in} > 0$$) and the all the contributions that would make the energy of the system decrease (the negative one, with the convention used, $$Q^{out} < 0$$), you write

$$0 = \delta Q = \delta Q^{out} + \delta Q^{in}$$.

If you, as many engineers, don't like negative numbers, you can take the absolute values of the negative contribution, to write $$\delta Q^{out} = - | \delta Q^{out}|$$ and thus

$$\delta Q^{in} = | \delta Q^{out} |$$

Because of the 1st law of thermodynamics (which is also a law of conservation of energy).

• As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.
– Community Bot
Commented Sep 28, 2022 at 8:34