Section 5.4: Non-Ideal TurboJet Operation

MAE 6530 - Propulsion Systems II

Section 5.4: Non-Ideal TurboJet Operation

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Idealized Turbojet Model and The Brayton Cycle

2

∞

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∞


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


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).


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


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



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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



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• Example

Combustor Inlet Mach Number Range from 0 to 0.5 …


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


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( )


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


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


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


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


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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Section 5.4: Non-Ideal TurboJet Operation