# In heat conduction, what does it actually mean to be in the steady state?

I have read about the method of heat conduction and I have some questions related to this topic:

If I consider a metal bar and supply heat in one end, the heat will flow through the bar and if I consider the bar to consist of many layers then each layer absorbs some amount of heat and the rest of the amount will flow to the next layer. So will all the layers attain the steady state at the same time?

And I have read that at steady state all the layers are no longer able to absorb any heat energy. Why can't the layers absorb any heat when they have reached steady state?

• The steady state will never be attained, but in practice one can get arbitrarily close. For a long bar that "arbitrarily close" will not be attained for all volume elements at the same time. Commented Jun 18, 2016 at 5:55
• I don't understand what"arbitrarily close" means Commented Jun 18, 2016 at 8:21
• The transient will decay and the temperature will be almost that of the theoretical steady state, but it won't ever be exactly the same. The difference will be unmeasurable small, though. If we set a limit for the difference, e.g. one part in thousand (that's already tough to measure), then the hot end of a long bar will get there first and the cold end will take a while longer. Another way of looking at it is that for a bar of infinite length the hot end will be quickly near steady state while the "far end" never will be. Commented Jun 18, 2016 at 8:42

Usually in these types of problems (if you want an analytical solution) you keep two ends of the bar at fixed temperatures and it is assumed that other methods of heat loss are not present (convection or radiation form the bar).

The temperature distribution depends on the fixed temperatures at the ends and the conductivity of the material.

The steady state is not attained instantaneously. The heat is transferred slowly and the transfer rate depends on the thermal conductivity.

If you take a section of the bar and the influx from one surface is equal to heat out-flux from other surface (in a time interval dt), which in turn equal to the total heat flow through the bar (in same time interval) the bar is said to be in steady state. In this condition temperature distribution along the bar will not change with time.

If bar is not in steady state the temperature along the bar will change with time and you have to use heat equation to calculate the temperature variation with time.

Usually this can be employed to 1D problems analytically (I am not sure if all the 1D problems can be solved analytically), to solve the real world 2D or 3D situations people usually use Finite element methods to get the transient and steady state solutions.

I hope this will help

regards,

• I asked why steady state is occurred not equilibrium state , please tell me why steady state is formed ,is it related with absorption of heat by the metal bar? Commented Jun 18, 2016 at 5:37
• I am sorry, I was also thinking that equilibrium and steady states are same but they are not as you have correctly pointed out. I have talked about steady state only but used the wrong word equilibrium. Equilibrium establish when the temperature difference between the two bodies is zero. I will edit my answer soon. Thanks for clearing my misconception Commented Jun 18, 2016 at 5:52
• But I asked that why at steady state no absorption takes place? Commented Jun 24, 2016 at 16:24
• That is because in the steady state the heat inflow in an infinitesimal volume is equal to heat out flow and that is by definition of steady state. A material will hit the steady state or not will be decided by the property of the medium. For example you take wood when you heat one end of the wood the steady state never achieved because the heat is not conducted and the temperature of very small portion increases to maximum, that portion keep on absorbing heat until it start to cool down by other means (convection). Commented Jun 24, 2016 at 17:14

When we say a steady state is reached, it means the temperature doesn't change. Along the bar, there is still temperature gradient, which drives the heat flow.

At steady state, each layer cannot absorb heat. If it can absorb heat, its temperature will increase. This contradicts to the steady state assumption.

• Exactly I agree with you.but I am very much eager to know how they can't absorb heat at that time? Commented Jun 19, 2016 at 2:12
• mathematically, the heat it gets from the upstream equals the heat it losses to the downstream because of temperature distribution. If it absorb heat, the temperature increases. The heat loss to the downstream is thus increases which will decrease the temperature. Thus it wouldn't absorb heat when it reaches steady state. Commented Jun 19, 2016 at 3:40

Any body can keep absorbing heat if it does not lose heat at all.

But if it constantly loses heat energy, then after some time, the amount of heat it absorbs is compensated by the heat it loses, and hence the difference in temperatures across its conductive surfaces is constant throughout time.

At this point, it achieves a steady state of heat transfer.