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In heat transfer across a rod, why actually steady state occur?

why temperature of sections of rod don't further increase as heat is even flowing through them and as whole rod is of same material then why the one section of rod absorbs more heat while other sections of same rod absorb less heat?

Kindly answer my question in terms of physical meaning not using equations , as equations also have some physical explanations and meanings.

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The steady state is reached when the rate at which heat is entering the rod is exactly equal to the rate at which heat is leaving the rod. The temperature of the rod is independent of the rate at which heat is flowing through the rod.

Think of it like a bucket with holes in it that is being filled from a tap. Once the rate at which water leaves the bucket equals the rate at which water enters the bucket, the height of the water in the bucket will remain the same - this is the steady state. If the bucket has many holes then height of the water in the steady state may be very low. If the bucket has only a few holes then the height of the water in the steady state will be higher - the bucket can hold more water if it is less leaky.

You also ask how different parts of the rod can have different steady state temperatures. Well, you can think of the rod as being made up of lots of little rods placed end to end. The steady state temperature of each little rod will depend on the temperatures of the rods on either side of it, which need not be the same. So each little piece of the rod may have a different steady state temperature, even though they all have the same amount of heat passing through them.

In the limit, as you make each little rod smaller and smaller, you end up with an equation that tells you how the temperature at each point in the rod depends on the temperatures on either side of that point - this is a partial differential equation called the heat equation. Solving this equation then tells you how the temperature at each point in the changes over time, and what the steady state temperature is at each point along the rod (if there is a steady state).

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  • $\begingroup$ i meant that why actually Heat incoming = Heat outgoing ( means why do different points of rod do not absorb any more heat after a constant temperature, due to which steady state is reasoned to be achieved)? $\endgroup$
    – Akarsh
    Jan 25, 2022 at 8:22
  • $\begingroup$ @Akarsh Because that's the definition of a steady state for this situation. Steady state means constant temperature; constant temperature means heat incoming= heat outgoing. If heat incoming > heat outgoing then temperature rises and we do not have a steady state. If heat incoming < heat outgoing then temperature falls and we also do not have a steady state. If heat incoming is fixed and heat outgoing increases with increasing temperature then the rod always approaches a steady state - but if heat incoming varies periodically then a steady state may not be possible. $\endgroup$
    – gandalf61
    Jan 25, 2022 at 8:58
  • $\begingroup$ How can it be actually, that different sections of rod absorb only certain amount of heat till a constant temperature accordingly such that rod attains a steady state ? how can it be explained logically that different points on rod arrange , set their temperatures for achieving the steady state ? And in this universe how can we just understand anything just by its definition their would be some basic reason behind that definition (phenomena) ? $\endgroup$
    – Akarsh
    Jan 27, 2022 at 4:20
  • $\begingroup$ @Akarsh I have expanded my answer above. Think of temperature as like the height of water in the leaky bucket - buckets with different heights of water can have the same amount of water flowing in and out of them. $\endgroup$
    – gandalf61
    Jan 27, 2022 at 8:07
  • $\begingroup$ It means then that as other part of rods also want heat to flow through them hence the preeceding sections of rod do absorb some of heat and then pass the left heat to next one and after some time their is created a condition in which all the heat passing through one section of rod passes without loss as this heat has to flow through other sections of rod and at last , heat transfers to surrounding (from other end ) without any loss . ? $\endgroup$
    – Akarsh
    Jan 28, 2022 at 8:23
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@gandalf61 has a good answer. I'll give you another example.

In your scenario, heat is flowing from one end of the rod to the other. Before steady state is reached, the rod is heating up. Therefore, the heat flowing in is greater than the heat flowing out. At steady state, the heat flowing in is equal to the heat flowing out.

At steady state, the rod cannot be at a uniform temperature. This is because heat only flows from high to low temperature. If the rod ends are at the same temperature, then no heat is flowing.

This is analogous to water pressure in a pipe. If the pressure at both ends are equal, then the water in the pipe is being pushed equally from both ends and no flow will occur.

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  • $\begingroup$ I asked that why actually steady state is achieved ? i mean why different points of rod do absorb different amount of heat , why those points do not absorb any more heat from heat flowing through them . Because when all points do not absorb any more heat after a constant temperature then the steady state is achieved and then , Heat incoming = Heat outgoing , but you have explained by taking steady state to be achieved , please answer why is steady state achieved while different sections of rod should absorb more heat like section at starting end (which attains max constant temp)? $\endgroup$
    – Akarsh
    Jan 25, 2022 at 8:19
  • $\begingroup$ @Akarsh: 1) The rod heats up when the heat starts entering one end. 2) The rod won't keep heating up forever. 3) Each part of the rod will eventually reach a steady temperature. 4) One end must be hotter than the other or else the heat can't flow. $\endgroup$
    – James
    Jan 26, 2022 at 12:29
  • $\begingroup$ @Akarsh: Think about it this way... There is something cold touching the other end of the rod (This must be so. If it was insulated then the rod would keep getting hotter until it melts as more heat is added.) So you have something hot touching one end, and something cold touching the other end. This is why the entire rod isn't at the same temperature. $\endgroup$
    – James
    Jan 26, 2022 at 12:48

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