Derivation of the Continuity Equation
(Section 9-2, Çengel and Cimbala)
We summarize the second derivation in the text – the one that uses a differential control
volume. First, we approximate the mass flow rate into or out of each of the six surfaces of the
control volume, using Taylor series expansions around the center point, where the velocity
components and density are u, v, w, and
. For example, at the right face,
The mass flow rate through each face is equal to
times the normal component of velocity
through the face times the area of the face. We show the mass flow rate through all six faces
in the diagram below (Figure 9-5 in the text):
Next, we add up all the mass flow rates through all six faces of the control volume in order to
generate the general (unsteady, incompressible) continuity equation:
Ignore terms higher than order dx.
Infinitesimal control volume
of dimensions dx, dy, dz.
Mass flow rate through
the right face of the
control volume.
Area of right
face = dydz.
