Details

The force on the engine is found by integrating over the control surface:

.

This gives

,

which simplifies to:

.

Conservation of mass is used to find , , , :

,

,

.

By combining the given equations, the force in the engine can be calculated using any one of the following equations:

,

,

.

is the force on the engine

is the density of air

is the velocity of air entering engine

is the area of the entrance of engine

are the velocities of air exiting engine

are the areas at the exits, where the air exits the engine

is the velocity of air exiting the engine through the front of the engine

is the area of the exit at the front

is the direction from the horizontal that air exits the engine

is the mass flow rate of air entering the engine

are the mass flow rates of air exiting the engine

is the mass flow rate of air exiting the front of the engine

Reference

[1] B. R. Munson, D. F. Young, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2009.