Contents 1 Propulsion Thermodynamics 1-1 1.1 Introduction.................................... 1-1 1.2 Thermodynamic cycles.............................. 1-8 1.2.1 The Carnot cycle............................. 1-8 1.2.2 The Brayton cycle............................ 1-11 1.3 The standard atmosphere............................ 1-14 1.4 Problems..................................... 1-15 2 Engine performance parameters 2-1 2.1 The definition of thrust............................. 2-1 2.2 Energy balance.................................. 2-6 2.3 Capture area................................... 2-8 2.4 Overall e ciency................................. 2-8 2.5 Breguet aircraft range equation......................... 2-10 2.6 Propulsive e ciency............................... 2-11 2.7 Thermal e ciency................................ 2-11 2.8 Specific impulse, specific fuel consumption................... 2-13 2.9 Dimensionless forms............................... 2-14 2.10 Engine notation.................................. 2-15 2.11 Problems..................................... 2-20 3 The ramjet cycle 3-1 3.1 Ramjet flow field................................. 3-1 3.2 The role of the nozzle.............................. 3-9 3.3 The ideal ramjet cycle.............................. 3-10 3.4 Optimization of the ideal ramjet cycle..................... 3-14 3.5 The non-ideal ramjet............................... 3-16 3.6 Ramjet control.................................. 3-16 3.7 Example - Ramjet with un-started inlet.................... 3-18 3.8 Very high speed flight - scramjets........................ 3-28 1
CONTENTS 2 3.8.1 Real chemistry e ects.......................... 3-32 3.8.2 Scramjet operating envelope....................... 3-32 3.9 Problems..................................... 3-35 4 The Turbojet cycle 4-1 4.1 Thermal e ciency of the ideal turbojet.................... 4-1 4.2 Thrust of an ideal turbojet engine....................... 4-4 4.3 Maximum thrust ideal turbojet......................... 4-9 4.4 Turbine-nozzle mass flow matching....................... 4-11 4.5 Free-stream-compressor inlet flow matching.................. 4-12 4.6 Compressor-turbine mass flow matching.................... 4-13 4.7 Summary - engine matching conditions..................... 4-14 4.7.1 Example - turbojet in supersonic flow with an inlet shock...... 4-14 4.8 How does a turbojet work?........................... 4-19 4.8.1 The compressor operating line..................... 4-20 4.8.2 The gas generator............................ 4-21 4.8.3 Corrected weight flow is related to f (M 2 )................ 4-22 4.8.4 A simple model of compressor blade aerodynamics.......... 4-24 4.8.5 Turbojet engine control......................... 4-29 4.8.6 Inlet operation.............................. 4-29 4.9 The non-ideal turbojet cycle........................... 4-33 4.9.1 The polytropic e ciency of compression................ 4-34 4.10 The polytropic e ciency of expansion..................... 4-37 4.11 The e ect of afterburning............................ 4-38 4.12 Nozzle operation................................. 4-39 4.13 Problems..................................... 4-40 5 The Turbofan cycle 5-1 5.1 Turbofan thrust.................................. 5-1 5.2 The ideal turbofan cycle............................. 5-3 5.2.1 The fan bypass stream.......................... 5-4 5.2.2 The core stream............................. 5-5 5.2.3 Turbine-compressor-fan matching.................... 5-6 5.2.4 The fuel/air ratio............................. 5-7 5.3 Maximum specific impulse ideal turbofan................... 5-7 5.4 Turbofan thermal e ciency........................... 5-9 5.4.1 Thermal e ciency of the ideal turbofan................ 5-12 5.5 The non-ideal turbofan.............................. 5-12 5.5.1 Non-ideal fan stream........................... 5-13 5.5.2 Non-ideal core stream.......................... 5-14 5.5.3 Maximum specific impulse non-ideal cycle............... 5-15
CONTENTS 3 5.6 Problems..................................... 5-17 6 The Turboprop cycle 6-1 6.1 Propellor e ciency................................ 6-1 6.2 Work output coe cient............................. 6-6 6.3 Power balance................................... 6-8 6.4 The ideal turboprop............................... 6-8 6.4.1 Optimization of the ideal turboprop cycle............... 6-10 6.4.2 Compression for maximum thrust of an ideal turboprop....... 6-11 6.5 Turbine sizing for the non-ideal turboprop................... 6-12 6.6 Problems..................................... 6-13 7 Rocket performance 7-1 7.1 Thrust....................................... 7-1 7.2 Momentum balance in center-of-mass coordinates............... 7-4 7.3 E ective exhaust velocity............................ 7-9 7.4 C e ciency.................................... 7-11 7.5 Specific impulse.................................. 7-11 7.6 Chamber pressure................................ 7-12 7.7 Combustion chamber stagnation pressure drop................ 7-14 7.8 The Tsiolkovsky rocket equation........................ 7-15 7.9 Reaching orbit.................................. 7-17 7.10 The thrust coe cient............................... 7-18 7.11 Problems..................................... 7-20 8 Multistage Rockets 8-1 8.1 Notation...................................... 8-1 8.2 The variational problem............................. 8-3 8.3 Example - exhaust velocity and structural coe cient the same for all stages 8-6 8.4 Problems..................................... 8-7 9 Thermodynamics of reacting mixtures 9-1 9.1 Introduction.................................... 9-1 9.2 Ideal mixtures................................... 9-2 9.3 Criterion for equilibrium............................. 9-5 9.4 The entropy of mixing.............................. 9-5 9.5 Entropy of an ideal mixture of condensed species............... 9-10 9.6 Thermodynamics of incompressible liquids and solids............. 9-12 9.7 Enthalpy..................................... 9-14 9.7.1 Enthalpy of formation and the reference reaction........... 9-15 9.8 Condensed phase equilibrium.......................... 9-17
CONTENTS 4 9.9 Chemical equilibrium, the method of element potentials........... 9-22 9.9.1 Rescaled equations............................ 9-28 9.10 Example - combustion of carbon monoxide................... 9-31 9.10.1 CO Combustion at 2975.34K using Gibbs free energy of formation. 9-36 9.10.2 Adiabatic flame temperature...................... 9-38 9.10.3 Isentropic expansion........................... 9-40 9.10.4 Nozzle expansion............................. 9-41 9.10.5 Fuel-rich combustion, multiple phases................. 9-42 9.11 Rocket performance using CEA......................... 9-45 9.12 Problems..................................... 9-45 10 Solid Rockets 10-1 10.1 Introduction.................................... 10-1 10.2 Combustion chamber pressure.......................... 10-2 10.3 Dynamic analysis................................. 10-4 10.3.1 Exact solution.............................. 10-6 10.3.2 Chamber pressure history........................ 10-8 10.4 Problems..................................... 10-9 11 Hybrid Rockets 11-1 11.1 Conventional bi-propellant systems....................... 11-1 11.2 The hybrid rocket idea.............................. 11-3 11.2.1 The fuel regression rate law....................... 11-4 11.2.2 Specific impulse.............................. 11-7 11.2.3 The problem of low regression rate................... 11-7 11.3 Historical perspective.............................. 11-9 11.4 High regression rate fuels............................ 11-12 11.5 The O/F shift.................................. 11-15 11.6 Scale-up tests................................... 11-16 11.7 Regression rate analysis............................. 11-17 11.7.1 Regression rate with the e ect of fuel mass flow neglected....... 11-17 11.7.2 Exact solution of the coupled space-time problem for n = 1/2.... 11-18 11.7.3 Similarity solution of the coupled space-time problem for general n and m.................................... 11-19 11.7.4 Numerical solution for the coupled space-time problem, for general n and m and variable oxidizer flow rate................. 11-20 11.7.5 Example - Numerical solution of the coupled problem for a long burning, midsize motor as presented in reference [1]............. 11-22 11.7.6 Sensitivity of the coupled space-time problem to small changes in a, n, and m.................................. 11-24 11.8 Problems..................................... 11-26
CONTENTS 5 A Thermochemistry A-1 A.1 Thermochemical tables.............................. A-1 A.2 Standard pressure................................ A-2 A.2.1 What about pressures other than standard?.............. A-4 A.2.2 Equilibrium between phases....................... A-5 A.2.3 Reference temperature.......................... A-7 A.3 Reference reaction and reference state for elements.............. A-7 A.4 The heat of formation.............................. A-8 A.4.1 Example - heat of formation of monatomic hydrogen at 298.15 K and at 1000 K.................................. A-9 A.4.2 Example - heat of formation of gaseous and liquid water....... A-10 A.4.3 Example - combustion of hydrogen and oxygen diluted by nitrogen. A-13 A.4.4 Example - combustion of methane................... A-14 A.4.5 Example - the heating value of JP-4.................. A-16 A.5 Heat capacity................................... A-17 A.6 Chemical bonds and the heat of formation................... A-20 A.6.1 Potential energy of two hydrogen atoms................ A-20 A.6.2 Atomic hydrogen............................. A-22 A.6.3 Diatomic hydrogen............................ A-23 A.7 Heats of formation computed from bond energies............... A-26 A.8 References..................................... A-28 B Selected JANAF data B-1