Application to an one-dimensional control volume

Consider an one-dimensional stream tube flow as shown in Fig.3.20. Let us mark a control volume bound by surface 1and surface 2. We know that there is any inflow/outflow of mass only through these two surfaces. The remaining surface S being made up of streamlines does not allow any mass flow through it.

Figure 3.20 : Control Control Volume for an one-dimensional steady flow

We assume that a uniform flow prevails at surfaces 1 and 2, V₁ and V₂ being the velocities. If the areas of cross section are A₁ and A₂, an application of the continuity equation 3.31 gives

$$\int_{CS}~ \rho \overrightarrow{V}.d\overrightarrow{A}~=~ \int_{... ...d\overrightarrow{A}~+\int_{2}~ \rho \overrightarrow{V}.d\overrightarrow{A}~=~0$$ (3.37)

simplifying to

$$\rho_1~V_1~A_1 ~=~ \rho_2~V_2~A_2$$ (3.38)

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