Conservation of Mass

First we apply the Reynolds Transport theorem, Eq. 3.27 to derive an equation for conservation of mass. We note that in the equation, N is the extensive property of interest which now is mass m. The corresponding intensive property is

$$\eta =~{{N} \over {m}}~=~ {{m} \over {m}}~=~1$$ (3.28)

Accordingly we substitute for m and $\eta$ in Eq.3.27. We have

$$\left. {{dm} \over {dt}} \right)_s~=~~{\partial \over \partial t}... ...} \rho d ~ \forall~~+~~ \int_{CS}~ \rho \overrightarrow{V}.d\overrightarrow{A}$$ (3.29)

By definition that a system is an entity of fixed mass, the left hand side of the above equation is zero, thus giving the equation for conservation of mass as

$$~~{\partial \over \partial t} \int_{CV} \rho d ~ \forall~~+~~ \int_{CS}~ \rho \overrightarrow{V}.d\overrightarrow{A}~=~0$$ (3.30)

which expresses that the rate of accumulation of mass within a control volume is equal to the net rate of flow of mass into the control volume. This equation is also called the Continuity Equation.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney