Application of Continuity Equation

Equation 3.30 gives

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

The first term in the equation cancels out because of the steady flow assumption (2 see Assumptions). Since all the flow takes place through (1) and (2) only the remaining term reduces to

$$- \rho~V_s~A~+~\rho~(V_s+dV_s)~(A+dA)~~=~~0$$ giving

$$\rho~(V_s+dV_s)~(A+dA)~~=~~ \rho~V_s~A~~=~~\dot{m}$$ (3.44)

where $\dot{m}$ is the mass flow rate through the control volume.

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