Gas Turbine Engines

Ideal Brayton Cycle

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

First Law of Thermodynamics:

Process	Specific Entropy
$2\to 3$	$h_{T,2}=h_{T,3}+w_{comp}$
$3\to 4$	$h_{T,4}=h_{T,3}+q_{in}$
$4\to 5$	$h_{T,4}=h_{T,5}+w_{turb}$

Second Law of Thermodynamics:

Process	Temperature	Specific Volume
$2\to 3$	$T_3=T_2{r}^{\frac{\gamma -1}{\gamma }}$	$v_3=v_2{r}^{\frac{-1}{\gamma }}$
$4\to 5$	$T_4=T_5{r}^{\frac{\gamma -1}{\gamma }}$	$v_4=v_5{r}^{\frac{-1}{\gamma }}$

Pressure Ratio:

$r=\frac{P_3}{P_2}$

Assumptions:

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

Works:

Stage	Specific Work
Compressor	$w_{comp}=C_p{T}_2[1-r^{\frac{\gamma -1}{\gamma }}]$
Turbine	$w_{turb}=C_p{T}_4[1-r^{\frac{\gamma -1}{\gamma }}]$
Net	$w_{net}=C_p{T}_2[(1-r^{\frac{\gamma -1}{\gamma }})+\frac{T_4}{T_2}(1-r^{\frac{\gamma -1}{\gamma }})]$

Heat:

• Combustion chamber $q_{in}=C_p{T}_2[\frac{T_4}{T_2}-r^{\frac{\gamma -1}{\gamma }}]$

Thermal Efficiency:

• Not a function of any temperature
• Independent of heat added
  

Real Brayton Cycle

Thermal Efficiency:

• 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
• $\eta =0.4$, $PR=12$

Civilian Aircraft:

• Low thrust, high efficiency
• $\eta =0.47$, $PR=40$

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)
  • $10K\Rightarrow$ life $\times$ 0.5
• Oxidation
• Thermal Fatigue