Brayton Cycle

Ideal Brayton Cycle

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Stages:

Compressor: $2 \to 3$

Increase pressure
Isentropic Compression

Combustion Chamber: $3 \to 4$

Fuel injection/heat addition
Constant Pressure

Turbine: $4 \to 5$

Extract work
Isentropic Expansion

Nozzle: $5 \to 1$

Reject entropy/heat (close cycle)
Constant Pressure?

Control Volume:

Not Control Mass/system.
Steady State operation

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First Law of Thermodynamics:

Process	Specific Entropy
$2 \to 3$	$h_{T, 2} = h_{T, 3} + w_{c o m p}$
$3 \to 4$	$h_{T, 4} = h_{T, 3} + q_{i n}$
$4 \to 5$	$h_{T, 4} = h_{T, 5} + w_{t u r b}$

Second Law of Thermodynamics:

Process	Temperature	Specific Volume
$2 \to 3$	$T_{3} = T_{2} r^{\frac{γ - 1}{γ}}$	$v_{3} = v_{2} r^{\frac{- 1}{γ}}$
$4 \to 5$	$T_{4} = T_{5} r^{\frac{γ - 1}{γ}}$	$v_{4} = v_{5} r^{\frac{- 1}{γ}}$

Pressure Ratio:

r = \frac{P_{3}}{P_{2}}

Assumptions:

Negligible changes in kinetic energy
(General case in ME 11b)

Works:

Stage	Specific Work
Compressor	$w_{c o m p} = C_{p} T_{2} [1 - r^{\frac{γ - 1}{γ}}]$
Turbine	$w_{t u r b} = C_{p} T_{4} [1 - r^{\frac{γ - 1}{γ}}]$
Net	$w_{n e t} = C_{p} T_{2} [(1 - r^{\frac{γ - 1}{γ}}) + \frac{T_{4}}{T_{2}} (1 - r^{\frac{γ - 1}{γ}})]$

Heat:

Combustion chamber $q_{i n} = C_{p} T_{2} [\frac{T_{4}}{T_{2}} - r^{\frac{γ - 1}{γ}}]$

Thermal Efficiency:

$η_{t h} = \frac{w_{n e t}}{q_{i n}} = 1 - r^{\frac{1 - γ}{γ}}$

Not a function of any temperature
Independent of heat added

Real Brayton Cycle

Thermal Efficiency:

$η_{t h} = \frac{w_{n e t}}{q_{i n}} = f (\frac{T_{4}}{T_{2}}, r)$

Temperature Ratio: $\frac{T_{4}}{T_{2}}$
Pressure Ratio: $r = \frac{P_{3}}{P_{2}}$

Best Conditions:

Large pressure ratio
- Better efficiency but low net power
Large $\frac{T_{4}}{T_{2}}$
- Better efficiency but more net power

Examples:

Military Aircraft:

High thrust, low efficiency
$η = 0.4$ , $P R = 12$

Civilian Aircraft:

Low thrust, high efficiency
$η = 0.47$ , $P R = 40$

Importance of $T_{4}$ :

$T_{4}$ is the Turbine Inlet Temperature (TIT)

Variations over flight:

Take off
- $T_{4}$ too high but temporary
Top of climb
- Melting point of turbine blade
Cruise
- Safe zone for sustainability

Blade lifetime limited by:

Creep
- Material Extension (high T)
- $10 K \Rightarrow$ life $\times$ 0.5
Oxidation
Thermal Fatigue

Ideal Brayton Cycle

Stages:

Compressor: 2→3

Combustion Chamber: 3→4

Turbine: 4→5

Nozzle: 5→1

Control Volume:

First Law of Thermodynamics:

Second Law of Thermodynamics:

Pressure Ratio:

Assumptions:

Works:

Heat:

Thermal Efficiency:

Real Brayton Cycle

Thermal Efficiency:

Best Conditions:

Examples:

Military Aircraft:

Civilian Aircraft:

Importance of T4:

Variations over flight:

Blade lifetime limited by:

Compressor: $2 \to 3$

Combustion Chamber: $3 \to 4$

Turbine: $4 \to 5$

Nozzle: $5 \to 1$

Importance of $T_{4}$ :