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I am having a bit of difficulty understanding the statement of Fourier's law of heat conduction.

As mentioned in Georg Joos , Theoretical Physics

The quantity of heat passing per second through unit area of a surface normal to the direction of flow, is proportional to the negative of the temperature gradient.

My conceptual difficulty is with the phrase 'direction of flow'. Now heat is just energy, a scalar and this phrase somehow seems to imply that this energy has some kind of velocity, which seems weird.

Can someone explain how this should be interpreted.

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Consider two regions $R_1$ and $R_2$ separated by an interface consisting of a planar surface. Let $\mathbf n_{21}$ denote the unit normal vector along the interface pointing from volume $1$ to volume $2$. If energy (in the form of heat conduction) is being transferred between these two systems, then this transfer has a direction in the sense that heat can either be flowing from region $1$ to region $2$, or vice versa. We can associate a vector $\mathbf q$ to the heat flow as follows. If the amount of energy flowing between the two regions per unit time per unit area on the interface in the form of heat is $q>0$, then $$ \mathbf q = q\mathbf n_{21} $$ if the energy is flowing from region $1$ to region $2$, and $$ \mathbf q = -q\mathbf n_{21} $$ if the energy is flowing from region $2$ to region $1$. The vector $\mathbf q$ is sometimes called the heat flux density.

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If you have a glass of cold water and drop hot water into it, you can say there is a flow of heat from the hot water into the cold water.

You are forgetting that every vector has a scalar part. In this case it is (roughly) the amount of heat in a given chunk of moving material. Precisely, the scalar is proportional to the quantity of heat per unit volume and the speed the material is moving. The vector part is the direction it flows in.

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"Direction of flow" is used in the sense of 'gradient'. The temperature gradient is responsible for heat flow from region of higher temperature to region of lower temperature. It is like saying my top speed is 15 m per sec when I run(flow) downwards from the top of a slope. Hope this helps

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