 Heat flux density

Heat flux density is a vector $\mathbf{q}$ describing the rate of heat transfer per unit area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Through a planar face of area $A$ with uniform normal component $q$, the signed rate is $\dot{Q}=Aq$ in $\mathrm{W}$; positive $q$ means heat travels in the chosen positive normal direction. Fourier's law determines the conductive contribution. In a bar with a constant cross-sectional area, steady thermal conduction without internal sources or side losses makes both $q$ and $\dot{Q}$ constant along the bar. Constant $\dot{Q}$ need not mean constant $q$ when the cross-sectional area varies.