Section 6.2: Optimal TurboFan Bypass Ratio

MAE 6530 - Propulsion Systems II

Section 6.2: Optimal TurboFan Bypass Ratio

1

!macore

!mafan

!mafan

Vexitcore

Vexitfan

Vexitfan

!mafan

!mafan

!mcore

Stephen Whitmore


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⎛

⎝⎜⎜⎜

⎞


V∞−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟+

β1+β⎛

⎝⎜⎜⎜

⎞


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∞

Stephen Whitmore


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⎛

⎝⎜⎜⎜

⎞


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

⎛

⎝⎜⎜⎜⎜

⎞


⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟−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

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore

Stephen Whitmore

core

Stephen Whitmore

core

Stephen Whitmore

core


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

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore

Stephen Whitmore


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

Stephen Whitmore


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



7

• Collecting up h∞

ηth =1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


+β ⋅hexitfan− 1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−β−

1f⋅ 1+β( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥ h∞

1+ 1f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⋅h04−h03

=

1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


+β ⋅hexitfan− 1−β⎡⎣

⎤⎦ h∞

1+ 1f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⋅h04−h03

• Ideal fan is quasi isentropic …

hexitfan≈ h∞ → ηth =1−

1+ 1f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


−h∞

1+ 1f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⋅h04 −h03

Stephen Whitmore



8

• Factoring out

• From enthalpy cascade

h03h∞

ηth =1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


h∞−1

1+ 1f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⋅h04h∞−h03h∞

=1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


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

Stephen Whitmore



9

• Substituting

• As shown before during turbojet analysis

ηth =1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


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

Stephen Whitmore



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( )

⎛

⎝⎜⎜⎜

⎞


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

Stephen Whitmore

Stephen Whitmore

Stephen Whitmore


Example Turbofan Calculation

11

Vexitcore

V∞=


⎛

⎝⎜⎜⎜⎜

⎞


Vexitfan

V∞

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟=τr ⋅τc

fan−1

τr−1

Toptimal( )turbofan

= γ ⋅M∞2 1

1+β⎛

⎝⎜⎜⎜

⎞


V∞+

β1+β⎛

⎝⎜⎜⎜

⎞


V∞−1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥=

= γ ⋅M∞2 1

1+β⎛

⎝⎜⎜⎜

⎞


V∞−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟+

β1+β⎛

⎝⎜⎜⎜

⎞


V∞−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

γ ⋅M∞21+ 1

f1+β( )

1+β( )⋅Vexitcore

V∞+β1+β

⋅Vexitfan

V∞−1

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

~

Stephen Whitmore

beta=10.3333

Stephen Whitmore

Stephen Whitmore


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

Stephen Whitmore

beta=10.33333

Stephen Whitmore


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∞)

Sweeping Through Bypass Ratios

Stephen Whitmore

Stephen Whitmore

Stephen Whitmore

T


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

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore

Stephen Whitmore


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

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

Stephen Whitmore



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

Stephen Whitmore



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∞

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore



18

• Also … from earlier derivation

Vexitcore

V∞

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

2

=τr ⋅τc

core⋅τt−1

τr−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟→∂∂β


τr−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅τλτc ⋅τr

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

= 2 ⋅Vexitcore

V∞

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅ 1−

Vexitfan

V∞

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟∂∂β


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


⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⋅

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

τλτccore⋅τr

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅τr ⋅τc

core

τr−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅∂τt∂β

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟=

τλτr−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⋅∂τt∂β

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟



19

• Taking the derivtive

→∂∂β


⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⋅

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

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


⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=−

τλτr−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⋅τrτλ

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟τcfan−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟=

τrτr−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟⋅ τc

fan−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

substituting→

τrτr−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⋅ τc

fan−1

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

2 ⋅Vexitcore

V∞=Vexitfan

V∞−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore



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∞

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

Stephen Whitmore

Stephen Whitmore



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?

Stephen Whitmore



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−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−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⋅τλ

τccore⋅τr

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟=

14

τr ⋅τcfan−1

τr−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟+

12

τr ⋅τcfan−1

τr−1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟+1

Stephen Whitmore



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

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

Stephen Whitmore



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!

Stephen Whitmore


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.

Stephen Whitmore


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

Stephen Whitmore


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. …

Stephen Whitmore


Comparisons of Ideal and Non-Ideal TurboFan

28

τccore= 2.51

τcfan


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


Homework 6.2 (2)

30


fan3

Bypass


Fan

Bypass Exhaust

fan3

Fan


core4

core5

coreexit

fanexit

fanexit

∞

1

2

Stephen Whitmore


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


fan3

Bypass


Fan

Bypass Exhaust

fan3

Fan


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


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


fan3

Bypass


Fan

Bypass Exhaust

fan3

Fan


core4

core5

coreexit

fanexit

fanexit

∞

1

2

T= Fthrustp∞ ⋅ A∞

I=Isp ⋅g0c∞

TSFC= 1g0 ⋅ Isp

Stephen Whitmore

Stephen Whitmore

Verify Graphical peak is at optimal Bypass ratio


TurboFan Efficiencies

33

• Recall that

Thermal Efficiency =

Propulsive Efficiency =

Kinetic energy production rate

Kinetic energy production rate

Combustion Enthalpy of

Fuel


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( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=



35

Thermal Efficiency

• Also, recall from earlier …

ηth =1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


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

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥







35


ηpropulsive

=2⋅ T

fan+T

core( ) !macore

⋅V∞( )1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=2⋅ T

fan+T

core( ) ρ∞ ⋅A∞V∞2( )

1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

2⋅Rg ⋅T∞V∞

2Tfan+T

core

p∞A∞

⎛

⎝⎜

⎞

⎠⎟

1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

2γ⋅γ ⋅R

g⋅T∞

V∞2

Tfan+T

core

p∞A∞

⎛

⎝⎜

⎞

⎠⎟

1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

2γ ⋅M∞

2 ⋅Tfan+T

core

p∞A∞

⎛

⎝⎜

⎞

⎠⎟

1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

2γ ⋅M∞

2 ⋅ Tfan +Tcore( )1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥



36


T( )turbofan

= Tfan+T

core( )= γ ⋅M∞2 1

1+β⎛⎝⎜


V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+ β1+β

⎛⎝⎜

⎞⎠⎟⋅Vexitfan

V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥→

ηpropulsive

=

2γ ⋅M∞

2 ⋅ Tfan +Tcore( )1+ ff

⎛⎝⎜


V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

+β ⋅Vexit fan

V∞

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

− 1+β( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

2⋅ 11+β

⎛⎝⎜


V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+ β1+β

⎛⎝⎜


V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1+ ff

⎛⎝⎜


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∞

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

√



38


ηpropulsive

=

2⋅ 11+β

⎛⎝⎜


V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟+ β1+β

⎛⎝⎜


V∞

−1⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1+ ff

⎛⎝⎜


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



39

Thermal Efficiency

• Also, recall from earlier …

ηth =1−1+ 1

f⋅ 1+β( )

⎛

⎝⎜⎜⎜

⎞


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

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥





Stephen Whitmore

Stephen Whitmore

(7)


Questions??

32

Section 6.2: Optimal TurboFan Bypass Ratio

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Transcript of Section 6.2: Optimal TurboFan Bypass Ratio