I have a wind tunnel (subsonic flow of air) which consists of a fan inside (and ahead of) a long cylinder of cross-sectional area of $A_e$, followed by a diffuser of $A_o$ c.s. area ($A_o>A_e$). The fan maintains a velocity of $V_e$ inside the tunnel. Assuming the flow inviscid and incompressible, I want to find the amount of force per unit area that the fan applies.
I chose the entire inside of the tunnel (fan included) as the fixed, non-deforming control volume.
Applying Conservation of Momentum for the c.v. (gage pressure used):
The volume integral was dropped since the flow is steady.
The last transition was done with the help of Conservation of Mass which yields:
The correct answer should be $(\rho/2)V_e^2A_e(A_e/A_o)^2$ and in the available solutions to me, Bernoulli was used on the stream line that begins after the fan and ends outside the tunnel at atm' pressure, and Momentum Conservation was applied to a control volume enclosing the fan alone.