why when steady state conditions apply the rate of heat flow is the same?

In a steady state situation, the temperature of some arbitrary volume in the material is constant. This means that the net amount of heat that enters the volume must be zero. However, you may not have thermal equilibrium so there are temperature gradients and therefore also finite heat fluxes.

This means that while the total amount of heat that enters a volume is zero, heat does eneter through part of the boundary while it leaves again through other parts of the boundary.

If the heat is flowing perpendicular to a surface and is uniform along that surface, this means that it must be the same along the direction perpendicular to the surface. Otherwise the volume bounded by the surface and one parallel to it, would accumulate heat and thus its temperature would not stay constant.

thanks so much :)

When steady-state conditions apply, it means that a system has reached a state where all its properties and variables remain constant over time. In the context of heat transfer, it means that the system has reached a stable state where the rate of heat flow is constant.

The rate of heat flow, also known as heat transfer rate, is the amount of heat energy transferred per unit of time. In a steady-state condition, this rate remains constant because the system has achieved a thermal equilibrium, where the energy being received by the system is equal to the energy being dissipated.

To understand why the rate of heat flow is the same under steady-state conditions, let's consider an example. Imagine a metal rod with one end exposed to a heat source and the other end is cooled. Initially, there will be a temperature gradient along the rod, with higher temperatures near the heat source and lower temperatures near the cooling end.

As heat is transferred along the rod, some regions will be gaining heat energy from the heat source, while others will be losing heat energy to the cooling end. This heat transfer process will continue until the system reaches a steady state.

In steady state, the rate at which heat is being transferred from the heat source to the rod is equal to the rate at which heat is being transferred from the rod to the cooling end. This happens because temperature differences drive the heat transfer process.

If the rate of heat flow from the heat source was higher than the rate of heat flow to the cooling end, the temperature at the heat source would keep increasing, and the temperature at the cooling end would keep decreasing. This would lead to a non-steady state condition as the system tries to reach a new equilibrium.

Conversely, if the rate of heat flow from the heat source was lower than the rate of heat flow to the cooling end, the temperature at the heat source would keep decreasing, and the temperature at the cooling end would keep increasing. Again, this would not be a steady state as the system tries to establish a new equilibrium.

Therefore, in steady-state conditions, the rate of heat flow is balanced, and the system achieves a stable state where the rate of heat flow remains the same.