--- title: "Turbojet cycle, thrust and propulsive efficiency" subject: "Turbo Machines" chapter: 6 key: "semester_7/turbo_machines:turbojet-cycle-thrust-propulsive-efficiency" source: https://www.bctengineer.com/q/semester_7/turbo_machines/turbojet-cycle-thrust-propulsive-efficiency site: BCT Engineering --- # Turbojet cycle, thrust and propulsive efficiency Turbo Machines · Chapter 6: Gas Turbine Cycles of Aircraft Propulsion - Asked in: year not given, 6 marks - Video: [Turbojets: Thermodynamics for Mechanical Engineers](https://www.youtube.com/watch?v=Ir4zlj45vRY) — Mike Schertzer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/turbo_machines/turbojet-cycle-thrust-propulsive-efficiency - Whole chapter: https://www.bctengineer.com/md/semester_7/turbo_machines/chapter-6 ## Question Draw the schematic and T-s diagram of the ideal turbojet engine cycle. Derive the expressions for the thrust, the propulsive efficiency and the overall efficiency of a turbojet in terms of flight and jet velocities. ## Short answer Thrust is F = mdot_a[(1 + f)C_j - V] plus the pressure term, approximately mdot_a(C_j - V). Propulsive efficiency is eta_p = 2V/(C_j + V) = 2/(1 + C_j/V). Overall efficiency is eta_o = eta_p eta_th = FV/(mdot_f LHV), with ideal thermal efficiency depending on overall pressure ratio. ## Answer #### Cycle Station numbers: 0 ambient, 1 inlet, 2 compressor inlet, 3 compressor exit, 4 turbine inlet, 5 turbine exit, 6 nozzle exit. ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a fully expanded nozzle and small f, specific thrust is approximately A. C_j + V B. C_j V C. C_j - V D. C_j / V Answer: C. F = m_a[(1+f)C_j - V] reduces to m_a(C_j - V), so F/m_a = C_j - V. (Key point: Thrust for fully expanded nozzle) 2. Propulsive efficiency of the turbojet in terms of velocities is A. 2V/(C_j + V) B. (C_j - V)/V C. V/C_j D. (C_j + V)/2V Answer: A. Thrust power m_a V (C_j - V) divided by KE rate 1/2 m_a (C_j^2 - V^2) gives 2V/(C_j + V). (Key point: Propulsive efficiency) 3. Why not make C_j equal to V to get eta_p = 1? A. Thermal efficiency becomes infinite B. The nozzle would choke C. Fuel flow becomes zero D. Thrust becomes zero Answer: D. F = m_a(C_j - V) goes to zero, hence turbofans use large mass flow with lower jet velocity. (Key point: Trade-off) 4. Overall efficiency of a turbojet is A. eta_p / eta_th B. eta_p x eta_th = F V/(m_f LHV) C. eta_p + eta_th D. F/(m_f LHV) Answer: B. Overall efficiency is thrust power over fuel energy rate, which is the product of propulsive and thermal efficiencies. (Key point: Overall efficiency) 5. Higher pressure ratio and higher turbine inlet temperature respectively A. lower eta_th; lower specific thrust B. raise eta_th (lower TSFC); raise specific thrust through higher C_j C. raise specific thrust; lower eta_th D. have no effect on either Answer: B. Ideal eta_th = 1 - T0/T03 grows with pressure ratio, while higher T04 gives more expansion energy and jet velocity. 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