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MAE 6530 - Propulsion Systems II
Section 6.2: Optimal TurboFan Bypass Ratio
1
!macore
!mafan
!mafan
Vexitcore
Vexitfan
Vexitfan
!mafan
!mafan
!mcore
MAE 6530 - Propulsion Systems II
Review 1: Normalized Thrust and Isp
2
• Fully expanded nozzle & f >> 1 • Inlet, fan, compressor, turbine, and fan /core nozzles are isentropic• Combustor heat addition is as constant pressure and Low Mach
Toptimal( )turbofan
= γ ⋅M∞2 1
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitcore
V∞+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitfan
V∞−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
= γ ⋅M∞2 1
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
pexitfan= pexit
core= p∞ → πd = πb = πncore = πn
fan=1
πccore= τc
core
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
γγ−1,πc
fan= τc
fan
⎛
⎝⎜⎜⎜⎜⎞
⎠⎟⎟⎟⎟
γγ−1,πt = τt( )
γγ−1 .
Ioptimal( )turbofan
= Toptimal( )turbofan
⋅f
γ ⋅M∞
MAE 6530 - Propulsion Systems II
Review 2:
3
Fan/Core Summary
• Fan
• Core
Substitute
Vexitfan
V∞=Mexit
fan
M∞⋅Texitfan
T∞=
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅Texitfan
T∞→ isentropic fan→ pexit
fan
⎛
⎝⎜⎜⎜⎜⎞
⎠⎟⎟⎟⎟
γ−1γ
= p∞( )γ−1γ
→Texitfan=T∞
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc
fan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→ T( )
fan= γ ⋅M∞
2 β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= γ ⋅M∞2 β
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τr ⋅τc
fan−1
τr−1−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc
fan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
T( )core= γ ⋅M∞
2 11+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅
τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟−1
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
T( )turbofan
=Fthrustp∞A0
=!maV∞p∞A0
1+ f +βf 1+β( )
⋅Vexitcore
V∞+β1+β
⋅Vexitfan
V∞−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥+pexitcore
p∞⋅Aexitcore
A0+pexitfan
p∞
Aexitfan
A0−Aexitcore+ Aexit
fan
A0
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=
!maV∞p∞A0
=ρ∞ ⋅ A0 ⋅V∞ ⋅V∞
p∞A0=γV∞
2
γ ⋅Rg ⋅T∞= γ ⋅M∞
2
Aexitcore+ Aexit
fan= Aexit→
T( )turbofan
= γ ⋅M∞21+ 1
f1+β( )
1+β( )⋅Vexitcore
V∞+β1+β
⋅Vexitfan
V∞−1
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
+AexitA0
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟pexitp∞⋅Aexitcore
Aexit+pexitfan
p∞
Aexitfan
Aexit−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Review 3: TurboFan Matching Equations
4
τt =1−τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
τccore−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1+1+βf
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
• Turbine Work
• Fuel/Air Flow f =!ma!mfuel
If the bypass ratio goes to zero the matching condition reduces to the usual turbojet formula.
1f=
11+β
⋅τλ−τr ⋅τc
core
τ fuel−τλ
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
f = 1+β( )⋅τ fuel−τλτλ−τr ⋅τc
core
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan
5
• Recall the Definition of Thermal Efficiency
ηth =K .Eout−K .E.in!mfuel ⋅hfuel
=1− heat rejectedheat input
=1−!macore( )⋅ hexit
core−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+ !ma
fan⋅ hexit
fan−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+ !mfuel ⋅hexit
core
!macore + !mfuel( )⋅h04− !macore( )⋅h03
Core Flow Heat Rejection
Fan Flow Heat Rejection
Fuel Flow Heat Rejection
Head Added in Combustor• Rewrite efficiency by adding and subtracting
and dividing by !mfuel ⋅h∞ !macore
ηth =1−
1+!mfuel
!macore
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⋅ hexit
core−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+!mafan
!macore⋅ hexit
fan−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+!mfuel
!macore⋅h∞
1+!mfuel
!macore
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⋅h04 −h03
=1−
1+!mfuel
!macore
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⋅ hexit
core−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ hexit
fan−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+!mfuel
!macore⋅h∞
1+!mfuel
!macore
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⋅h04 −h03
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan (2)
6
• From the definition of bypass flow
• Substitution gives
!mafan
!macore= β
!mfuel
!macore=!mfuel
!matotal
!matotal!macore=!mfuel
!matotal
!macore + !mafan
!macore=1f⋅ 1+β( )
→ ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅ hexitcore
−h∞⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ hexit
fan−h∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+1f⋅ 1+β( )⋅h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04 −h03
=1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
+β ⋅hexitfan− 1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟−β−
1f⋅ 1+β( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥ h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04 −h03
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan (3)
7
• Collecting up h∞
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
+β ⋅hexitfan− 1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟−β−
1f⋅ 1+β( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥ h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04−h03
=
1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
+β ⋅hexitfan− 1−β⎡⎣
⎤⎦ h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04−h03
• Ideal fan is quasi isentropic …
hexitfan≈ h∞ → ηth =1−
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
−h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04 −h03
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan (4)
8
• Factoring out
• From enthalpy cascade
h03h∞
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04h∞−h03h∞
=1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
h03h∞
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅h04h∞
h∞h03−1
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
h03h∞=h03h02⋅h02h01⋅h01h0∞⋅h0∞h∞→
h03h02= τc
core
h02h01=1
h01h0∞= τr
h04h∞= τγ
→
h03h∞= τc
core⋅τr
h04h∞
h∞h03=
τγτccore⋅τr
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan (5)
9
• Substituting
• As shown before during turbojet analysis
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
hexitcore
h∞=h0exit
core
h∞⋅hexitcore
h0exitcore
= τγ ⋅1
1+ γ−12Mexit
core
2→1+ γ−1
2Mexit
core
2 = τccore⋅τr→
hexitcore
h∞=
τγτccore⋅τr
MAE 6530 - Propulsion Systems II
Thermal Efficiency of an Ideal TurboFan (6)
10
• Substituting
• This solution is identical to the turbo jet analysis with the core flow replacing the normal turbine flow path.
• This analysis shows for the ideal (isentropic) fan à hexitfan = h∞ --> the the heat rejected by the fan stream is zero.
• Therefore the thermal efficiency of the ideal turbofan is independent of the parameters of the fan stream.
• Only the mechanical efficiency (Thrust, Specific Impulse) are altered by bypass flow.
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1−
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1− 1τccore⋅τr
MAE 6530 - Propulsion Systems II
Example Turbofan Calculation
11
Vexitcore
V∞=
τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅τλτc ⋅τr
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=τr ⋅τc
fan−1
τr−1
Toptimal( )turbofan
= γ ⋅M∞2 1
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitcore
V∞+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitfan
V∞−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
= γ ⋅M∞2 1
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
γ ⋅M∞21+ 1
f1+β( )
1+β( )⋅Vexitcore
V∞+β1+β
⋅Vexitfan
V∞−1
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
~
MAE 6530 - Propulsion Systems II
Example Turbofan Calculation (2)
12
M∞ = 0.8
T0 4 =1750K
πccore =P03P02=T03T02
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
γγ−1
= τccore( )γγ−1
πc fan =P03P02
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟fan
=T03T02
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟fan
γγ−1
= τccore( )γγ−1
MAE 6530 - Propulsion Systems II
Example Calculation (3)
13
• Optimal Efficiency (Isp) occurs at a Bypass Ratio that results in most of the thrust produced by Fan and not Core Flow (assumed Nozzle exit pressure = P∞)
MAE 6530 - Propulsion Systems II
What Bypass Ratio Gives Optimized TurboFan Performance?
14
Normalized Specific Impulse→
I( )turbofan
=Isp ⋅g0
c∞= T( )
turbofan⋅
fγ ⋅M∞
fully expanded nozzle→
T( )turbofan
= γ ⋅M∞2 1
1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Air / Fuel Ratio→
f = 1+β( )⋅τ fuel−τλτλ−τr ⋅τc
⎛
⎝⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟f = 1+β( )⋅τ fuel−τλτλ−τr ⋅τc
core
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (2)
15
• what value of b maximizes the specific impulse?
Substitute Normalized Thrust and air / fuel ratio into Specific Impulse
I( )turbofan
=Isp ⋅g0
c∞= γ ⋅M∞
2 11+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+
β1+β⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⋅
1+β( )⋅τ fuel−τλτλ−τr ⋅τc
core
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
γ ⋅M∞
Simplify→
I( )turbofan
= M∞ ⋅τ fuel−τλτλ−τr ⋅τc
core
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+β ⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (3)
16
• What value of b maximizes the specific impulse?
• Since fan is ~ Isentropic only bypass nozzle and not bypass massflow affectsBypass exit velocity ratio
Necessary Condition
→∂ I( )
turbofan
∂β= 0→ ∂
∂β
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+β ⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 0
Thus→ ∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=1−
Vexitfan
V∞
I( )turbofan
= M∞ ⋅τ fuel−τλτλ−τr ⋅τc
core
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+β ⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Thus→ ∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=1−
Vexitfan
V∞
Vexitfan
V∞= 0
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (4)
17
• Find …
• Start with
Thus→ ∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=1−
Vexitfan
V∞
∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 2 ⋅Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→∂∂β
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 2 ⋅Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ 1−
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
from earlier derivation
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc
core⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟→∂∂β
τr ⋅τccore⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 2 ⋅Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ 1−
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (5)
18
• Also … from earlier derivation
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc
core⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟→∂∂β
τr ⋅τccore⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 2 ⋅Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ 1−
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟∂∂β
τr ⋅τccore⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
= 2 ⋅Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ 1−
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
τ t is a function of β
since turbine powers both fan and compressor
→∂∂β
τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
τλτccore⋅τr
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τr ⋅τc
core
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=
τλτr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (6)
19
• Taking the derivtive
→∂∂β
τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
τλτccore⋅τr
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τr ⋅τc
core
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=
τλτr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=∂∂β
1−τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
τccore−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1+1+βf
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=−τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
τcfan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1+1+βf
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
+τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
τcfan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟τccore−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
1+1+βf
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
2 ⋅1f
1f≈ 0→
∂τt∂β
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟τcfan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟→∂∂β
τr ⋅τc ⋅τt−1τr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=−
τλτr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟τcfan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟=
τrτr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
substituting→
τrτr−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
2 ⋅Vexitcore
V∞=Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (7)
20
from earlier→Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
=τr ⋅τc
fan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=τr ⋅τc
fan−1
τr−1−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+1=
τr ⋅τcfan−1− τr−1( )τr−1
+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟=τr ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
τr−1+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Simplifying
→Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
−1=τr ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
τr−1→ Substituting→
VexitfanV∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
2
−1
2 ⋅Vexitcore
V∞=Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Re arranging and expanding diffference of squares
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
−1= 2 ⋅Vexit
core
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→
Vexitfan
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
2
−1=Vexit
fan
V∞+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟= 2 ⋅
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
Cancelling Terms
Vexitfan
V∞+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟= 2 ⋅
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→
Vexitfan
V∞+1−2
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟= 2 ⋅
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟−2
→Vexit
fan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟= 2 ⋅
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→ Vexit
fan−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= 2 ⋅ Vexit
core−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (8)
21
Optimization Criterion
Vexitfan−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= 2 ⋅ Vexit
core−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
Optimal turbofan design delivers twice the velocity increment across the fan compared to velocity increment across the turbine (core flow)
How does this criterion relate to the fan bypass ratio?
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (9)
22
Optimization Criterion
2 ⋅ Vexitcore−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟= Vexit
fan−V∞
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟→ 2 ⋅
Vexitcore
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→ 2 ⋅
Vexitcore
V∞−2=
Vexitfan
V∞−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟→Vexitcore
V∞=
12
Vexitfan
V∞+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
From Earlier Derivations
Vexitcore
V∞
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=
τr ⋅τccore⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλ
τccore⋅τr
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Vexitfan
V∞=
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
→ Substitute→τr ⋅τc
core⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλ
τccore⋅τr
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=
12
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
Square Both Sides→
τr ⋅τccore⋅τt−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅τλ
τccore⋅τr
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟=
14
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+
12
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+1
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (10)
23
• Solve for tt ….
• From Earlier Derivation
• Substitute into above
τt = 1+ 14
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+12
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⋅τccore⋅τr
τλ
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ τr−1( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⋅1
τr ⋅τccore
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
τt =1−τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅ τc
core−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
τrτλ
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟⋅ τc
core−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+β ⋅ τc
fan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥=1− 1+ 1
4
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+12
τr ⋅τcfan−1
τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⋅τccore⋅τr
τλ
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ τr−1( )
⎧
⎨
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⋅1
τr ⋅τccore
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟
MAE 6530 - Propulsion Systems II
Optimized TurboFan Performance (10)
24
• Finally …. Solving for the optimal bypass ratio gives
Whew! .. Check Numerical Result agans earlier simulationβoptimal =
βoptimal =1
τcfan−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟
⋅τλ
τccore⋅τr−1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⋅ τc
core−1
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟+
τλτr2 ⋅τc
core
τr−1( )− 14τr−1τr
⎛
⎝⎜⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⎟
τr ⋅τcfan−1
τr−1+1
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎧
⎨
⎪⎪⎪⎪
⎩⎪⎪⎪⎪
⎫
⎬
⎪⎪⎪⎪
⎭⎪⎪⎪⎪
=10.16
Check!
MAE 6530 - Propulsion Systems II
Discussion of Optimal TurboFan Performance
25
core
fan
• Variation of Optimal Bypass Ratio as a Function of Freestream Mach Number• At Higher Mach Numbers optimum bypass ratio decreases until the fan disappears altogether and we basically convert the engine to a turbojet.
MAE 6530 - Propulsion Systems II
Discussion of Optimal TurboFan Performance (2)
26
core
Increasing Mach Number
Ideal turbofan bypass ratio for maximum specific impulse as a function of fan pressure ratio. Plot shown for several Mach numbers
MAE 6530 - Propulsion Systems II
Discussion of Optimal TurboFan Performance (3)
27
• From this plot is is observed that increasing fan pressure ratio leads to an optimum at a lower bypass ratio. • Curves all allow for optimum systems at very low fan pressure ratios and high bypass ratios. • This result is an artifact of the assumptions underlying the ideal turbofan. • Once non-ideal effects are included, low fan pressure ratio solutions reduce to much lower bypass ratios. …
MAE 6530 - Propulsion Systems II
Comparisons of Ideal and Non-Ideal TurboFan
28
τccore= 2.51
τcfan
MAE 6530 - Propulsion Systems II
Homework 6.2
29
Consider the TurboFan Engine whose Block Diagram is Shown Below
Diffuser Compressorcore3
fan3
Bypass
TurbineCombustorCore Exhaust
Fan
Bypass Exhaust
fan3
Fan
Bypass ExhaustBypass
core4
core5
coreexit
fanexit
fanexit
∞
1
2
With Following Conditions Aircraft Mach Number: 0.70Aircraft Altitude: 6 kmBypass Ratio: 2Fan Pressure Ratio: 2Compressor Ratio: 6Burner Outlet Temperature: 1700 K Fuel: JP4 à hf = 42.68 MJ/kg
MAE 6530 - Propulsion Systems II
Homework 6.2 (2)
30
Diffuser Compressorcore3
fan3
Bypass
TurbineCombustorCore Exhaust
Fan
Bypass Exhaust
fan3
Fan
Bypass ExhaustBypass
core4
core5
coreexit
fanexit
fanexit
∞
1
2
MAE 6530 - Propulsion Systems II
Homework 6.2 (3)
31
Assume the Following Component Propertiesi) Diffuser, Compressor, Fan, Turbine, Nozzle ~ Isentropicii) Nozzle exit flow is NOT mixediii) Combustor is 35% efficient àiv) Fuel massflow is NOT negligiblev) Mean specific heats, gamma are constant across engine constant àvi) Fan, Core Nozzle Exits Optimized for Altitude
Diffuser Compressorcore3
fan3
Bypass
TurbineCombustorCore Exhaust
Fan
Bypass Exhaust
fan3
Fan
Bypass ExhaustBypass
core4
core5
coreexit
fanexit
fanexit
∞
1
2
ηb =!Q3−4!mfuel ⋅hf
= τ f ⋅Cp ⋅T∞hf γ≈1.4
Cp ≈1004.96J /kg−K
MAE 6530 - Propulsion Systems II
Homework 6.2 (4)
32
Calculatei) Normalized Thrustii) % of Thrust delivered by Core Flowiii) % of Thrust delivered by Bypass Flowii) Ratio of Bypass Thrust to Core Thrustiii) Normalized Specific Impulseiv) TSFC lbm/lbf-hrv) Bypass Ratio for Optimal Ispvi) Optimal TSFCvii) Thermal, Propulsive, and Total Efficiency
Diffuser Compressorcore3
fan3
Bypass
TurbineCombustorCore Exhaust
Fan
Bypass Exhaust
fan3
Fan
Bypass ExhaustBypass
core4
core5
coreexit
fanexit
fanexit
∞
1
2
T= Fthrustp∞ ⋅ A∞
I=Isp ⋅g0c∞
TSFC= 1g0 ⋅ Isp
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies
33
• Recall that
Thermal Efficiency =
Propulsive Efficiency =
Kinetic energy production rate
Kinetic energy production rate
Combustion Enthalpy of
Fuel
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (2)
34
Propulsive Efficiency
ηpropulsive
ηpropulsive
=
V∞ ⋅ Tfan +Tcore( )12 !macore
⋅ 1+ ff
⎛⎝⎜
⎞⎠⎟⋅V
exitcore
2 −V∞2⎛
⎝⎜⎞
⎠⎟+ 12 !mafan
⋅ Vexit fan
2 −V∞2( )=
V∞ ⋅ Tfan +Tcore( )12 !macore
⋅ 1+ ff
⎛⎝⎜
⎞⎠⎟⋅V
exitcore
2 −V∞2 +β ⋅ V
exit fan
2 −V∞2( )⎡
⎣⎢
⎤
⎦⎥
=
2⋅V∞ ⋅ Tfan +Tcore( )!macore
⋅ 1+ ff
⎛⎝⎜
⎞⎠⎟⋅V
exitcore
2 +β ⋅Vexit fan
2 − 1+β( )⋅V∞2⎡
⎣⎢
⎤
⎦⎥
=2⋅ T
fan+T
core( ) V∞
!macore
⋅ 1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (2)
35
Thermal Efficiency
• Also, recall from earlier …
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1−
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1− 1τccore⋅τr
• This solution is identical to the turbo jet analysis with the core flow replacing the normal turbine flow path.
• This analysis shows for the ideal (isentropic) fan à hexitfan = h∞ --> the the heat rejected by the fan stream is zero.
• Therefore the thermal efficiency of the ideal turbofan is independent of the parameters of the fan stream.
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (3)
35
Propulsive Efficiency
ηpropulsive
=2⋅ T
fan+T
core( ) !macore
⋅V∞( )1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=2⋅ T
fan+T
core( ) ρ∞ ⋅A∞V∞2( )
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2⋅Rg ⋅T∞V∞
2Tfan+T
core
p∞A∞
⎛
⎝⎜
⎞
⎠⎟
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2γ⋅γ ⋅R
g⋅T∞
V∞2
Tfan+T
core
p∞A∞
⎛
⎝⎜
⎞
⎠⎟
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2γ ⋅M∞
2 ⋅Tfan+T
core
p∞A∞
⎛
⎝⎜
⎞
⎠⎟
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2γ ⋅M∞
2 ⋅ Tfan +Tcore( )1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (4)
36
Propulsive Efficiency
T( )turbofan
= Tfan+T
core( )= γ ⋅M∞2 1
1+β⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟+ β1+β
⎛⎝⎜
⎞⎠⎟⋅Vexitfan
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥→
ηpropulsive
=
2γ ⋅M∞
2 ⋅ Tfan +Tcore( )1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2⋅ 11+β
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟+ β1+β
⎛⎝⎜
⎞⎠⎟⋅Vexitfan
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
Checkà forTurbojetβ =0
1+ ff
⎛⎝⎜
⎞⎠⎟=1 →η
propulsive=
2⋅Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
−1⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
=
2⋅Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟⋅Vexitcore
V∞
+1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
= 2
1+Vexitcore
V∞
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
√
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (6)
38
Propulsive Efficiency
ηpropulsive
=
2⋅ 11+β
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟+ β1+β
⎛⎝⎜
⎞⎠⎟⋅Vexitfan
V∞
−1⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
1+ ff
⎛⎝⎜
⎞⎠⎟⋅Vexitcore
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+β ⋅Vexit fan
V∞
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
− 1+β( )⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
→Vexitcore
V∞
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
τr⋅τ
c⋅τ
t−1
τr−1
⎛
⎝⎜⎞
⎠⎟⋅
τ λ
τc⋅τ
r
→Vexitfan
V∞
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟=
τr⋅τ
cfan
−1
τr−1
MAE 6530 - Propulsion Systems II
TurboFan Efficiencies (2)
39
Thermal Efficiency
• Also, recall from earlier …
ηth =1−1+ 1
f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅hexitcore
h∞−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1−
1+ 1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
τccore⋅τr 1+
1f⋅ 1+β( )
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟⎟⋅τγ
τccore⋅τr−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=1− 1τccore⋅τr
• This solution is identical to the turbo jet analysis with the core flow replacing the normal turbine flow path.
• This analysis shows for the ideal (isentropic) fan à hexitfan = h∞ --> the the heat rejected by the fan stream is zero.
• Therefore the thermal efficiency of the ideal turbofan is independent of the parameters of the fan stream.
MAE 6530 - Propulsion Systems II
Questions??
32