Fourier's Law of Heat Conduction
Law
system에서 지점 $\mathbf{x}$ 의 local heat flux $\mathbf{q}$ 은 temperature $T$ 의 gradient $\nabla T$에 비례한다는 법칙을 Fourier’s law of heat conduction이라 한다. $$ \mathbf{q} \left( \mathbf{x} \right) = - k \nabla T \left( \mathbf{x} \right) $$ 여기서 $k$ 는 thermal conductivity이다.
Explanation
The negative sign on the right-hand side simply reflects that heat transfer always occurs from regions of higher temperature to regions of lower temperature, and it succinctly indicates that heat flows in the direction that lowers temperature.
Fick’s law: The diffusion flux is proportional to the change in density. $$ \mathbf{J} \left( \mathbf{x} \right) = - D \nabla \mathbf{u} \left( \mathbf{x} \right) $$
Fourier’s law of heat conduction corresponds to Fick’s law in mass transport; when expressed as equations they are formally identical, and their meaning and interpretation are likewise similar.
Newton’s law of cooling: At time $t$ the heat flux $q$ between an object and its surroundings is proportional to the temperature difference $\Delta T (t) = T(t) - T_{\infty}$. $$ q = h \Delta T (t) $$
Meanwhile, Newton’s law of cooling can be viewed as the special case of Fourier’s law in which heat transfer occurs only at a single point. While Fourier’s law describes how heat flows throughout space as a whole, Newton’s law of cooling describes how an object’s temperature changes in time.