How is thermal conduction represented for a given temperature difference for any material by the rate of heat conduction?

(N/A) Consider a metallic bar of length $L$ and uniform cross-sectional area $A$ with its two ends maintained at different temperatures. This can be done by placing the ends in thermal contact with large reservoirs at temperatures $T_{C}$ and $T_{D}$ respectively,where $T_{C} > T_{D}$.
Assuming the ideal condition that the sides of the bar are fully insulated,no heat is exchanged with the surroundings. Initially,the temperatures of different parts of the bar increase with time. After some time,a 'steady state' is reached where the temperature of the bar decreases uniformly with distance from $T_{C}$ to $T_{D}$.
In this steady state,the reservoir at $C$ supplies heat at a constant rate,which is transferred through the bar and given out at the same rate to the reservoir at $D$. Both the rate of heat flow $\frac{dQ}{dt}$ and the temperature gradient $\frac{dT}{dx}$ remain constant with time.
Experimentally,it is found that the rate of heat flow (or heat current) $H$ is given by:
$H = \frac{dQ}{dt} = KA \frac{T_{C} - T_{D}}{L}$
where $K$ is the thermal conductivity of the material.