My textbook preliminarily describes the governing equation for heat transfer as follows:
$$\text{Energy In - Energy Out + Energy Generated = Energy Stored}$$
It then goes on to derive the heat transfer equation by only considering heat flow in the $x$-direction:
$$\text{Energy in during time $\Delta t$} = (q''_x \Delta y \Delta z + [u \Delta y \Delta z \rho c_p(T - T_R)]_x) \Delta t$$
$$\text{Energy out during time $\Delta t$} = (q''_{x + \Delta x} \Delta y \Delta z + [u \Delta y \Delta z \rho c_p(T - T_R)]_{x + \Delta x}) \Delta t$$
$$\text{Energy generated during time $\Delta t$} = Q \Delta x \Delta y \Delta z \Delta t$$
$$\text{Energy stored during time $\Delta t$} = \Delta x \Delta y \Delta z \rho c_p \Delta T$$
If we are only considering heat flow in the $x$-direction, then why are all $\Delta y$ and $\Delta z$ included in these equations? What purpose do they serve?
I would greatly appreciate it if people could please take the time to clarify this.
The heat equation is for a volume of material. Say we use a small parallelipiped, with sides $\Delta x$, $\Delta y$, and $\Delta z$. The energy "in" is the going through the surface perpendicular to the $x$-axis at $x$, and it is proportional to the size of the face, $\Delta y\Delta z$. There are quantities that depend on the volume, the energy generated and the energy stored. The volume is $\Delta x\Delta y\Delta z$. All these quantities are described in such a way to make sense macroscopically (for the entire object), especially when we talk about the dimensionality. But, if you look carefully, $\Delta y\Delta z$ appears in every term, so it can be ignored (you can divide the entire equation by it).