Temperature at different points in a metal rod during heat conduction

Question

$$ k = \frac{\frac{Q}{t}}{A(\frac{T_1 - T_2}{L})} $$

where k is thermal conductivity of the solid, Q is total amount of heat transferred, t is time taken for the heat transfer, A is area of the cross section, L is the length of the solid and T1 and T2 are the temperatures at the hotter end and the colder end respectively.

According to this formula, when a metal rod is getting heated through conduction, the temperature gradient $$\frac{T_1 - T_2}{L}$$ decides what the temperature will be at different points down the length. However, physics also states that when heat transfer takes place, it goes on until the temperature of the hotter object and the cooler object becomes equal. So, how can heat conduction stop before temperature becomes the same throughout the length of the metal rod without contradicting the basic theory of heat transfer?

Answer 1

"However, physics also states that when heat transfer takes place, it goes on until the temperature of the hotter object and the cooler object becomes equal."

That's only true in equilibrium.

In this case, we know the system is not in equilibrium because we have heat being added ($Q$). If $Q = 0$ after some amount of time, $\frac {T_1 - T_2}{L}$ will go to $0$.

I believe this even fits with the wording you chose:

"According to this formula, when a metal rod is getting heated through conduction, the temperature gradient decides what the temperature will be at different points down the length."

(emphasis mine) Basically, the gradient can only exist when there is heat being transferred through the rod. When there is no net heat transfer through the rod, the gradient becomes a flat line due to the bar having a uniform temperature.

Another thing to point out is that they aren't talking about the temperature gradient when convection stops. They are talking about the temperature gradient at some time during the heat transfer. This could either be a steady-state conduction or a snapshot of a transient process at one point in time.

Answer 2

Hmm. Yes what you said is right that heat transfer goes on until there is temperature difference but net heat exchange becomes 0. Let me explain. Suppose a rod and it's ends are kept at different temperatures. At a particular point of the rod , in steady state, the heat going forward becomes equal to the heat coming from backward so temperature remains same.Although heat exchange continually happens from the that point to a cooler part but a hotter part is also giving that point heat. I hope you understand.