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

Answer 1

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.

Answer 2

thanks so much :)