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# Linear Momentum Balance in Aerodynamic Thrust

The force generated by a jet engine on a static thrust stand is determined using Newton's second law of motion. For a fixed, nondeforming control volume with uniform and steady flow, the sum of the forces is equal to the net rate of linear momentum flow through each control surface. Use sliders to change inlet and outlet radii, inlet velocity and inlet pressure. The anchoring force is the force to keep the jet engine stationary. The inlet and outlet pressures are gage pressures ().

### DETAILS

The linear momentum balance for a nondeforming, stationary control volume is:
.
Since the flow is steady, the momentum balance simplifies to:
,
where
is velocity (m/s),
is the density of the fluid (),
represents the product of the component of velocity normal to the control surface,
is the differential area of the control surface ().
Application of the momentum equation to the contents of the control volume yields:
,
where
is velocity (m/s),
is gage pressure (kPa),
is the thrust force (N).
For one-dimensional flow:
.
Due to conservation of mass :
,
,
using the ideal gas law:
,
where
is the mass flow rate (kg/s)
is the density of air at the inlet (),
is atmospheric pressure,
is the ideal gas constant,
is temperature (K).
The continuity equation is used to determine the velocity of air exiting the turbine:
,
,
where and is the radius (m).
Solving for the thrust force:
.
Reference
[1] B. R. Munson, T. H. Okiishi and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2009.

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