Turbojet cycle, thrust and propulsive efficiency

Cycle

Station numbers: 0 ambient, 1 inlet, 2 compressor inlet, 3 compressor exit, 4 turbine inlet, 5 turbine exit, 6 nozzle exit.

 flight->  [diffuser][C][ Comb ][T][nozzle]--> jet
            0-2      2-3  3-4   4-5   5-6

 T-s: 0-2 ram rise (isentropic, up), 2-3 compression (up),
      3-4 heat addition (right and up along the p3 line),
      4-5 turbine expansion (down; drop = 2-3 rise),
      5-6 nozzle expansion (down to p0), 6-0 exhaust cooling.

 T
 |              4
 |             /|
 |            / |
 |           /  5
 |          3   |
 |          |   |
 |          |   6
 |          2  /
 |          | /
 |          0
 +-------------------- s

Thrust

From the momentum equation on the engine, with fuel-air ratio $f$ and air mass flow $\dot m_a$:

$$F = \dot m_a\left[(1 + f)C_j - V\right] + (p_j - p_0)A_j$$

For a fully expanded nozzle ($p_j = p_0$) and small $f$, $F \approx \dot m_a (C_j - V)$. The specific thrust is $F/\dot m_a = C_j - V$.

Propulsive efficiency

Useful thrust power $= FV = \dot m_a V (C_j - V)$. Rate of increase of kinetic energy of the stream $= \tfrac12\dot m_a (C_j^2 - V^2)$.

$$\eta_p = \frac{\dot m_a V(C_j - V)}{\tfrac12\dot m_a (C_j^2 - V^2)} = \frac{2V}{C_j + V} = \frac{2}{1 + C_j/V}$$

It approaches 1 as $C_j \to V$, but then the thrust goes to zero. This is why the turbofan uses a large mass flow at a lower jet velocity.

Thermal and overall efficiency

$$\eta_{th} = \frac{\tfrac12\dot m_a (C_j^2 - V^2)}{\dot m_f \, LHV}, \qquad \eta_o = \eta_p \eta_{th} = \frac{F V}{\dot m_f \, LHV}$$

For the ideal Brayton-type engine, $\eta_{th} = 1 - T_0/T_{03}$, i.e. it depends on the overall pressure ratio (ram plus compressor). The thrust specific fuel consumption is $TSFC = \dot m_f/F = f/(C_j - V)$.

A higher turbine inlet temperature increases the specific thrust (higher $C_j$); a higher pressure ratio improves $\eta_{th}$ and so lowers the TSFC.