Momentum Equation

Considering first the forces the only force that acts upon the control volume is the net force on the disc or the Thrust, F. Pressures being equal at (1) and (4) does not contribute to the surface force. Since the flow takes place in a horizontal direction there is no body force to be considered. Accordingly,

$$F~=V_4~\rho~V_4~A_4~-~V_1~\rho~V_1~A_1~=~\dot{m}~(V_4~-~V_1)$$ (3.102)

Noting that V₂ = V₃, this force F is equal to A(p₃ - p₂), where A is the area of cross section of the disc. As a consequence we have,

$$F~=A~(p_3~-~p_2)=~\dot{m}~(V_4~-~V_1)$$

$$(p_3~-~p_2)~=~{{\dot{m}} \over A}(V_4~-~V_1)$$

$$(p_3~-~p_2)~=~\rho~A_2~(V_4~-~V_1)$$ (3.103)

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