For our balance, we will choose an **arbitrary** fixed (in
space) control volume. Thus, it is an *open* system (i.e.,
one through which mass may flow). Because of this, the total
material within our control volume can change with time as stuff
flows in and out.

A "word equation" for this process may be written as:

rate of change of mass in CV | = | net rate of mass flow
into CV |

We note that the amount of mass contained in a small section of volume, dV, can be written as $\displaystyle{\rho}$ dV. In order to calculate all of the material in our control volume we simply add all of the contributions from each section dV (i.e., we integrate), so that the rate of change of mass in the CV is:

$\displaystyle{\frac{d}{dt}\int\int\int \rho dV}$

The easiest way to capture the amount of material flowing
into/out of the CV is to watch the stuff that crosses the
boundaries. That is, if we look at the velocity of the mass and see
what portion of that velocity is *perpendicular* to the
surface, we can then add up all the contributions of stuff crossing
the surfaces, so that our final equation can be written as:

$\displaystyle{\frac{d}{dt}\int\int\int \rho dV = - \int\int\rho\left (\mathbf{v}\cdot \mathbf{n} \right ) dA}$

Functionally, what this equation means is that, in order to perform a macroscopic mass balance, we simply look at the inlets and outlets of a process and multiply the (area) average velocity times the density and the cross-sectional area of that orifice.

Derive and use a macroscopic mass balance (continuity equation)