# Why does the steady state exist In thermal conduction?

In thermal conduction,why doesn't transient state go on continuously,why does steady state exist?why after some considerable time A metal bar sttached to a sink and source comes to steady state where the the temperature of different points on bar is different but it doesn't change with time?I want to know what actually happens at molecular level?

• What research have you done? Jan 15, 2022 at 11:14
• @Bob D,I Am still In learning phase,this Topic Is introduced To us now,But They didn't gave any justification about it,I tried thinking About many different answers On PSE,but many of them said it is just definition of steady state,While others said,It is just because Heat In=heat out But My question Is why doesn't transient state Is permanent Forever,Why is steady state achieved?why doesn't the different sections of rod Want to absorb anymore heat ? Mar 13, 2022 at 7:56
• I think I understand your question better now. Before I contemplate posting an answer, do you understand that steady state conditions are achieved when temperatures along the rod are not changing in time? And, if so, is your actual question why temperatures stop changing in time, which is equivalent to asking why does heat stop being "absorbed"? Mar 13, 2022 at 19:32
• Did you not understand my last comments? Mar 16, 2022 at 22:39

Heat flow is governed by the heat equation $$\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2}$$ where $$u(x,t)$$ is the temperature at a point and $$D$$ is a constant. The term $$\frac{\partial^2 u}{\partial x^2}$$ describes the curvature of $$u$$ at a point. The reason that the heat equation goes to a steady state is that moving in the direction of curvature makes the curvature smaller. It will act something like this In the left bump the curvature is negative and in the right bump the curvature is positive. We can show this slightly more rigorously. We know the function $$\sin kx$$ is an eigenfunction of $$\frac{\partial^2 u}{\partial x^2}$$ so let's guess a solution of the form $$u(x,t)=A(t)\sin kx.$$ Plugging this in the heat equation gives $$A'(t)=-Dk^2A(t)$$ which has solution $$A(t)=e^{-Dk^2t}.$$ Using a Fourier transform we can write a general function $$u$$ as a sum of infinitely many sines and cosines. Since we now know how each of these functions behaves we can predict how the function as a whole will behave. The amplitude of each sine wave goes down over time and the sines with the smallest wave length (small wave length means large $$k$$) disappear the fastest.

I didn't take into account boundary conditions here so if you want to predict what happens when you keep two points at a fixed temperature you would have to do some more work. But the result is the same: curvature decreases over time.

A final note on what happens on a molecular level. The important thing that happens on a molecular level is that heat tends to spread out. This makes the heat diffuse: the heat equation is mathematically the same as the diffusion equation.

• ,Thanks For your Effort,But I think I am lacking Knowledge Of what You have mentioned equations about,But I just need To know what actually Is happening to rod that it tends to achieve Steady state Mar 13, 2022 at 8:01

In thermal conduction, why doesn't transient state go on continuously, why does steady state exist?

As you already know, steady state conditions exist when the temperatures in the bar are no longer changing (increasing) in time. For the transient state to go on continuously, the temperatures in the bar would need to continually increase in time. But the temperature in the bar cannot exceed the temperature of the heat source, $$T_H$$. To do so would require heat to spontaneously flow from a lower temperature region to a higher temperature region in violation of the second law of thermodynamics.

Bottom line: Temperatures cannot continue to rise because the maximum possible temperature is limited to the temperature of the heat source.

Hope this helps.

• can we have a chatroom?I have some Doubts Which might make this comment section Too lengthy? Mar 18, 2022 at 9:11
• Sure. You establish the room and I’ll join Mar 18, 2022 at 10:16
• I have created the room,Please join,chat.stackexchange.com/rooms/134927/… Mar 21, 2022 at 4:59
• @DheerajGujrathi I’m in. Describe your doubts Mar 22, 2022 at 12:37
• as the chatroom is now closed,I will ask my doubt anyways if that's okay,What if heat source is maintained at constant temperature and sink is also maintained at constant temperature,hence is there some kind of mechainsm that initially rate of increase in temperature by source dominates rate of decrease in temperature by sink,later at some stage,both rates become equal,and hence the temperature becomes constant?I remembered this analogy due to recent chemical equillibrium classes,where initially forward rate dominates backward rate,then both become equal at some point,if it is true,why so? Sep 12 at 8:12