Linear Momentum Balance in Aerodynamic Thrust

​
inlet radius (m)
1
outlet radius (m)
0.75
inlet velocity (m/s)
150
inlet pressure (kPa)
-25
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
F
is the force to keep the jet engine stationary. The inlet and outlet pressures are gage pressures (
P
gage
=
P
abs
-
P
atm
).

Details

The linear momentum balance for a nondeforming, stationary control volume is:
∂
∂t
∫
CV
VρdV+
∫
CS
VρV·

n
dA=∑F
.
Since the flow is steady, the momentum balance simplifies to:
∫
CS
VρV·

n
dA=∑F
,
where
V
is velocity (m/s),
ρ
is the density of the fluid (
kg
3
m
),
V·

n
represents the product of the component of velocity normal to the control surface,
dA
is the differential area of the control surface (
2
m
).
Application of the momentum equation to the contents of the control volume yields:
∫
CS
uρV·

n
dA=
P
1
A
1
-
P
2
A
2
+
F
t
,
where
u
is velocity (m/s),
P=
P
abs
-
P
atm
is gage pressure (kPa),
F
t
is the thrust force (N).
For one-dimensional flow:
(+
u
1
)(-
m
1
)+(+
u
2
)(+
m
2
)=
P
1
A
1
-
P
2
A
2
+
F
t
.
Due to conservation of mass
m=
m
1
=
m
2
:
m=
m
1
=
ρ
1
A
1
u
1
,
m(
u
2
-
u
1
)=
P
1
A
1
-
P
2
A
2
+
F
t
,
using the ideal gas law:
ρ
1
=
P
1
+
P
atm
R
T
1
,
where
m
is the mass flow rate (kg/s)
ρ
1
is the density of air at the inlet (
kg
3
m
),
P
atm
=101.3kPa
is atmospheric pressure,
R
is the ideal gas constant,
T
1
is temperature (K).
The continuity equation is used to determine the velocity of air exiting the turbine:
A
1
u
1
=
A
2
u
2
,
u
2
=
u
1
A
1
A
2
,
where
A=π
2
r
and
r
is the radius (m).
Solving for the thrust force:
F
t
=-
P
1
A
1
+
P
2
A
2
+m(
u
2
-
u
1
)
.

References

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

Permanent Citation

Michael Wrobel, Ryan Hollenbaugh, Rachael L. Baumann, Garret D. Nicodemus, Nick Bongiardina
​
​"Linear Momentum Balance in Aerodynamic Thrust"​
​http://demonstrations.wolfram.com/LinearMomentumBalanceInAerodynamicThrust/​
​Wolfram Demonstrations Project​
​Published: October 7, 2013