Section 5.4: Non-Ideal TurboJet Operation
Transcript of Section 5.4: Non-Ideal TurboJet Operation
MAE 6530 - Propulsion Systems II
Section 5.4: Non-Ideal TurboJet Operation
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MAE 6530 - Propulsion Systems II
Idealized Turbojet Model and The Brayton Cycle
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∞
T
S
Idealized Assumptions:1) Inlet and Diffuser are Isentropic2) Compressor, Turbine ~ Isentropic3) Burner @Low Mach Number, Constant
Pressure4) Turbine Work = Compressor Work5) Nozzle is Isentropic6) Ideally expanded nozzle where pexit = p∞
MAE 6530 - Propulsion Systems II
How is this Idealized Model Unrealistic?
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• Remember, everywhere there is an irreversibility, then we have entropy growth and a loss of stagnation pressure. Stagnation pressure losses limit the overall efficiency of the propulsion system,
• We have already studied two of most significant non-ideal mechanisms … inet shock waves and stagnation pressure losses across combustor
MAE 6530 - Propulsion Systems II
How is this Idealized Model Unrealistic? (2)
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• Across shock wave(s) entropy increases and we get a resulting stagnation pressure loss, and limits overall system efficiency
• Other sources of stagnation pressure losses include nozzle stagnation pressure losses are associated with viscous skin friction. • Stagnation pressure losses across the burner due to heat addition also cause pb to be always less than one. • Additional reduction of pb occurs due to wall friction, nonzero burner exit Mach number, and injector drag to to Reynolds stresses (right angle injection into flow).
MAE 6530 - Propulsion Systems II
Combustor Losses and Inefficiencies
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• Assuming Mean Values for Cp, g, Mw, Rg, Conservation of Mass and Momentum across the combustor gives (MAE 5420 Lecture 5.4)
• Conservation of Energy across combustor gives Gives
Cp ⋅ !mair + !mf( ) ⋅T04 = Cp ⋅ !mair( ) ⋅T03 + !mf ⋅hf ⋅ηburner
Cp ⋅ !mair + !mf( )Cp ⋅ !mair( )
T04T03
= 1+!mf ⋅hf ⋅ηburner
Cp ⋅ !mair( ) ⋅T03→ f = !mair
!mf
f +1f
⋅T04T03
= 1+ 1f!mf ⋅hf ⋅ηburner
Cp ⋅T03→ T04
T03= f
f +1⎛⎝⎜
⎞⎠⎟⋅ 1+ 1
fhf ⋅ηburner
Cp ⋅T03
⎛
⎝⎜⎞
⎠⎟
M 42 ⋅ 1+ γ −1
2M 4
2⎡⎣⎢
⎤⎦⎥
1+ γ M 42⎡⎣ ⎤⎦
2 = T04T03
⎡
⎣⎢
⎤
⎦⎥burner
⋅ f +1f
⎛⎝⎜
⎞⎠⎟
2
⋅M 3
2 ⋅ 1+ γ −12
M 32⎡
⎣⎢⎤⎦⎥
1+ γ M 32⎡⎣ ⎤⎦
2
MAE 6530 - Propulsion Systems II
Combustor Losses and Inefficiencies (2)
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• Finally, second law of thermodynamic gives
And …Solving for Stagnation Pressure Ratio
ΔsburnerCp
= ln T04T03
⎡
⎣⎢
⎤
⎦⎥ −
γ −1γ
⋅ ln P04P03
⎡
⎣⎢
⎤
⎦⎥
ΔsburnerCp
= ln M 42
M 321+ γ M 3
2
1+ γ M 42
⎛⎝⎜
⎞⎠⎟
γ +1γ⎛
⎝⎜⎜
⎞
⎠⎟⎟
→ P04P03
= T04T03
⎛⎝⎜
⎞⎠⎟⋅ M 3
2
M 421+ γ M 4
2
1+ γ M 32
⎛⎝⎜
⎞⎠⎟
γ +1γ⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
γγ −1
• In addition to loss of stagnation pressure, also necessary to account for incomplete combustion, and radiation/conduction heat losses to combustor walls. Combustor efficiency is defined directly from energy balance across burner …
ηc =
f +1f
⎛⎝⎜
⎞⎠⎟⋅h04 − h031fhf
=f +1( ) ⋅h04 − f ⋅h03
hf
MAE 6530 - Propulsion Systems II
Combustor Losses and Inefficiencies (3)
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• Thus the collected conservation equations are
P04P03
= ff +1
⎛⎝⎜
⎞⎠⎟⋅ 1+ 1
f!mf ⋅hf ⋅ηburner
Cp ⋅ !mair( ) ⋅T03⎛
⎝⎜⎞
⎠⎟⋅ M 3
2
M 421+ γ M 4
2
1+ γ M 32
⎛⎝⎜
⎞⎠⎟
γ +1γ⎛
⎝⎜⎜
⎞
⎠⎟⎟
T04T03
= ff +1
⎛⎝⎜
⎞⎠⎟⋅ 1+ 1
fhf ⋅ηburner
Cp ⋅T03
⎛
⎝⎜⎞
⎠⎟
M 42 ⋅ 1+ γ −1
2M 4
2⎡⎣⎢
⎤⎦⎥
1+ γ M 42⎡⎣ ⎤⎦
2 = T04T03
⎡
⎣⎢
⎤
⎦⎥burner
⋅ f +1f
⎛⎝⎜
⎞⎠⎟
2
⋅M 3
2 ⋅ 1+ γ −12
M 32⎡
⎣⎢⎤⎦⎥
1+ γ M 32⎡⎣ ⎤⎦
2
• Parametric equation set allows plot of stagnation pressure ratio as a function of compressor outlet Mach number (combustor inlet Mach number) M3
MAE 6530 - Propulsion Systems II
Combustor Losses and Inefficiencies (4)
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• Example
Combustor Inlet Mach Number Range from 0 to 0.5 …
MAE 6530 - Propulsion Systems II
Combustor Losses and Inefficiencies(4)
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• Stagnation pressure losses across the burner due to heat addition cause pb to be always less than one …rule of thumb is
MAE 6530 - Propulsion Systems II
Compressor and Turbine Losses and Inefficiencies
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• The shaft that connects the turbine and compressor is subject to frictional losses in the bearings that support the shaft and a shaft mechanical efficiency is defined using the work balance across the compressor and turbine. Typical shaft efficiencies are slightly less than unity.
• For the ideal case we have analyzed the compression and turbine cycles are isentropic, i.e.
• But with losses in these cycles, these relationships no longer strictly hold, and an adjustment is necessary to account for the losses
ηc/tMech
=h03 − h02
f +1f
⎛⎝⎜
⎞⎠⎟⋅ h04 − h05( )
π c = τ cγ / γ −1( ) . . . π t = τ t
γ / γ −1( )
MAE 6530 - Propulsion Systems II
Compressor and Turbine Losses and Inefficiencies (2)
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• Defining
• and
=P03 P02P03 P02
ηc =h03 |Δs=0 −h02h03 − h02
=P05 P04P05 P04
ηt =h04 − h05
h04 − h05 |Δs=0
MAE 6530 - Propulsion Systems II
Compressor and Turbine Losses and Inefficiencies (3)
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• The efficiencies { hpc , hpt } allow the relationship between compressor/turbine temperature and pressure ratios as a “polytropic process,” The polytropic process is a measure of the degree to which the compression process is isentropic,
and …
P03P02
= π c = τ cγγ −1( )⋅ηpc
. . . π t =P05P04
= τ tγγ −1( )⋅ηpt
ηc =h03 |Δs=0 −h02h03 − h02
=
P03P02
⎛⎝⎜
⎞⎠⎟
γ −1γ−1
P03P02
⎛⎝⎜
⎞⎠⎟
γ −1γ
⋅⋅ 1ηpc
−1
. . . ηt =h05 |Δs=0 −h04h5 − h05
=
P05P04
⎛⎝⎜
⎞⎠⎟
γ −1γ−1
P05P04
⎛⎝⎜
⎞⎠⎟
γ −1γ
⋅⋅ 1ηpt
−1
Modern compressors are designed to have values of hpc in the range 0.88 to 0.92.
Modern turbines are designed to values of hpt in the range 0.91 to 0.94.
ηc =
h03h02|Δs=0 −1
h03h02
−1==
P03P02
⎛⎝⎜
⎞⎠⎟
γ −1γ−1
P03P02
⎛⎝⎜
⎞⎠⎟
γ −1γ
⋅⋅ 1ηpc
−1
ηt =1−
h05h04
1−h05 |Δs=0h04
=
P05P04
⎛⎝⎜
⎞⎠⎟
γ −1γ
⋅⋅ 1ηpt
−1
P05P04
⎛⎝⎜
⎞⎠⎟
γ −1γ−1
MAE 6530 - Propulsion Systems II
Adjusted Brayton Cycle Plot for Non-Ideal TurboJet Operation
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h-s path of a turbojet with non-ideal compressor and turbine.
P03
P04
P05
P0e
MAE 6530 - Propulsion Systems II
Homework 5.4
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A2
Aexit
• Engine operates at a free stream Mach number, M∞ = 0.8. • Cruise Altitude is in the stratosphere, 11 km so T∞ = 216.65 K.• The design turbine inlet temperature, T04 = 1944 K • The design compressor ratio, pc = 20 . • Relevant area ratios are A2 /A*
4 = 10 and A2/A1throat = 1.2 . • Inlet throat area A1Throat = 20 cm2
• Assume the compressor, burner and turbine all operate ideally. • Nozzle is of a simple converging type with choked throat, A*
8=Aexit• Stagnation pressure losses due to wall friction in the inlet and nozzle are negligible. à CALCULATEà a) Correct Compressor massflow and M2 at compressor faceà b) Normalized exit pressure thrust, momentum thrust, and total thrustà c) Velocity ratio across Engine Vexit/V∞à d) Mach number at diffuser throat, M1throatà e) Inlet capture area
A*4
f +1f≈1
MAE 6530 - Propulsion Systems II
Homework 5.4 (2)
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A2
Aexit
• Now allow an expandable Nozzle where, Aexit > A*8
à CALCULATEà a) Optimal expansion ratio for nozzle Aexit /A*
8à b) c) Velocity ratio across Engine Vexit/V∞à c) thrust of optimal nozzleà d) Assuming the same combustor temperature and inlet throat area
à Plot the Compressor operating line à pc vs corrected massflow for 1 < pc < 15à Plot the capture area A∞ vs corrected massflow for 1 < pc < 15
A*4
MAE 6530 - Propulsion Systems II 16
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