PRODUCTION OF THRUST
Newton’s 2nd Law (e.g. S F = d/dt (mv))
for a control volume of fixed mass with steady flow in and out
and no acceleration of the frame of reference relative to inertial
coordinates:
ÂF =
Sum of external forces
on control volume, e.g.
Pressure forces
Shear forces
Body forces
Reaction forces
Û
Ù u (r u ) ⋅ nds
ıs
Net flux of momentum through
surface of control volume
For x-component of vectors:
Û
F
=
 x Ùı ux ru ⋅ nds
s
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1
PRODUCTION OF THRUST
Û
T - D + Â Pr essure Forces = Ù ru x (u ⋅ n )dA
ıs
È
˘ Û
Û
Ù
(PR - Po )dA R ˙ = Ù ru x
(u ⋅ n )dA
T - D + Í -PeAe + Po Ae ıA R
Î
˚ ıs
Û
Ù ru x u ⋅ ndA = r eue Aeue - ro uo Aouo + Û
Ù
rux u ⋅ ndA
ıs
ıC s - A o -A e
Û
˙ ou o + Ù
˙ eue - m
ru u ⋅ ndA
=m
ıCs -A o -A e x
Û
Û
˙ eu e - m
˙ o uo + Ù
T - D - (Pe - Po )Ae - Ù (PR - Po )dAR = m
ru u ⋅ ndA
ıC s - Ae - Ao x
ıAR
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2
PRODUCTION OF THRUST
Everything that relates to flow through the engine is conventionally
called thrust. Everything that relates to the flow on the outside of the
engine is conventionally call drag. Therefore, gathering only those
terms that relate to the fluid that passes through the engine, we have:
˙ eue - m
˙ ouo + (Pe - Po )Ae
T=m
The thrust is largely composed of the net change in momentum of the
air entering and leaving the engine, with a typically small adjustment
for the differences in pressure between the inlet and the exit.
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3
EFFICIENCY
We have related the thrust of a propulsion system to the net changes
in momentum, pressure forces, etc. Now we will look at how
efficiently the propulsion system converts one form of energy to
another on its way to producing thrust:
propulsive power
what you get
=
overall efficiency =
fuel power
what you pay for
propulsive power = thrust⋅ flight velocity = Tu o
˙ fh
fuel power = fuel mass flow rate ⋅ fuel energy per unit mass = m
Thus,
Tuo
hoverall =
˙ fh
m
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4
EFFICIENCY
It is often convenient to break the overall efficiency into:
thermal efficiency and propulsive efficiency where
2 ˆ
Êm
˙
˙ e u2
u
m
e- o o
Á
˜
2
2
rate of production of propellant k.e. Ë
¯
hthermal =
=
˙ fh
fuel power
m
hprop =
Tuo
propulsive power
=
2
ˆ
rate of production of propellant k.e.
Ê
ṁe u2
˙
u
m
e- o o
Á
˜
2
2
Ë
¯
Such that,
h
overall =
h
thermal h
propulsive
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5
OVERALL PROPULSION SYSTEM EFFICIENCY
Trends in thermal efficiency are driven by increasing compression
ratios and corresponding increases in turbine inlet temperature.
Whereas trends in propulsive efficiency are due to generally higher
bypass ratio engines
(After Koff, 1991)
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6
EFFICIENCY
The thermal efficiency is the same as that used in
thermodynamics. For an ideal Brayton cycle it is a function of
the temperature ratio across the compressor
Tatm
Wnet
= 1= 1hth-idealBraytoncycle =
Tcomp. exit
Qin
1
g -1
(PR) g
Higher temperature ratio = higher pressure ratio = higher
thermal efficiency
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PRESSURE RATIO TRENDS
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8
TURBINE INLET TEMPERATURE
• Desire for higher turbine inlet temperature is driven by
desire for high specific work
• Specific work is work per unit of airflow
• High work per unit of airflow = smaller engine, lower weight,
etc.
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9
Work WORK
BRAYTONBrayton
CYCLECycle
SPECIFIC
3
2.5
TR=TT4/T0
2
TR = 4
TR = 5
1.5
TR = 6
TR = 7
1
0.5
For a given turbine inlet temperature,
there is a compressor pressure ratio
that maximizes the work per unit of airflow
0
0
10
20
Compressor
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30
Pressure
40
50
Ratio
10
TURBINE INLET TEMPERATURE TRENDS
Waitz, 2002
(After Koff, 1991)
11
PROPULSIVE EFFICIENCY
We can use our expression for thrust to rewrite the equation
for propulsive efficiency in a more convenient form
˙ (ue - uo )
T ªm
(sin ce m˙ e ª m˙ o )
Then,
ṁ uo (u e - uo )
2uo
2
hp =
=
=
ṁ 2
uo + ue 1+ ue
2
ue - uo
2
uo
(
Waitz, 2002
)
12
TRENDS IN BYPASS RATIO
Waitz, 2002
(After Koff, 1991)
13
OTHER EXPRESSIONS FOR OVERALL EFFICIENCY
Specific Impulse (I or Isp):
thrust
I=
fuel weight flow rate
(units: seconds)
Thrust Specific Fuel Consumption (SFC or TSFC):
mass flow rate of fuel
SFC =
thrust
(units: lbm/hr/lbf or kg/s/N)
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IMPLICATIONS FOR ENGINE DESIGN
Considering jointly the expressions for thrust and propulsive
efficiency,
hprop
˙ (ue - uo )
F @m
2
=
ue
1+
uo
As
ue
↑
uo
F
↑
˙
m
hprop Ø
As
ue
Æ1
uo
F
Ø
˙
m
hprop ↑
Also, as
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F
↑
ṁuo
Ainlet Ø
Drag Ø
15
PROPULSIVE EFFICIENCY AND SPECIFIC THRUST
1.0
1.5
�p
2
3
0.5
6
ue / u0
11
0
0
1
5
10
F / m u0
Image adapted from: Kerrebrock, J. L. Aircraft Engines and Gas Turbines. 1991.
Waitz, 2002
16
PROPULSIVE EFFICIENCY VS. FLIGHT SPEED
Reproduced with the kind permission of Rolls-Royce plc.Copyright Rolls-Royce plc 1986. Used with permission.
Waitz, 2002
17
PROPULSIVE EFFICIENCY AND SPECIFIC THRUST
For fighter aircraft that need high thrust/weight and fly at high
speed, it is typical to employ engines with smaller inlet areas
and higher thrust per unit mass flow
However, transport aircraft that require higher efficiency and
fly at lower speeds usually employ engines with relatively
larger inlet areas and lower thrust per unit mass flow
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PROPULSIVE EFFICIENCY
Image removed due to copyright considerations.
At low flight velocities, the highest propulsive
efficiency is typically obtained with a propeller or an
unducted fan
Image removed due to copyright considerations.
Waitz, 2002
. with permission.
Reproduced with the kind permission of Rolls-Royce plc.Copyright Rolls-Royce plc 1986. Used
20
AIRCRAFT AND ENGINE DESIGN ISSUES
•
Thrust ≈ (mass flow) x (change in velocity across engine)
• Range ~ fuel efficiency (commercial and military)
– Thermal efficiency
High pressures and
temperatures
– Propulsive efficiency
Large mass flow with small
velocity change
• Maneuverability (military)
– High thrust-per-weight,
small compact engine
• Supersonic flight (military)
– Low drag, small compact
engine
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High energy conversion per
unit volume (high
temperatures and
pressures)
Small mass flow with large
velocity change
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AN EXAMPLE OF CYCLE PERFORMANCE IMPROVEMENT
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