Review of Fundamentals - Fluid Mechanics Introduction Properties of Compressible Fluid Flow Basics of One-Dimensional Gas Dynamics Nozzle Operating Characteristics Characteristics of Shock Wave A gas turbine cycle through the use of H-K Diagram
Compressible Flow Properties Approximate Mach zones: < M0.3 Subsonic, incompressible M0.3 M0.8 Subsonic, compressible M0.8 M1.2 Transonic, shock waves appear M1.2 M3 Supersonic M3 Hypersonic Normal to specify P, T, V to describe state In compressible flow, V is often replaced by Mach number, total pressure and total temperature. Total Enthalpy / Total Temperature Without gravity effects, the Steady Flow Energy Equation (SFEE) is For calorically perfect (constant c p, c v ) gas, The stagnation or total enthalpy h t is defined as The stagnation or total temperature T t is defined as
Total Enthalpy / Total Temperature For an aircraft in flight at velocity V a, the airstream velocity at the leading edge stagnation point is negligibly small Kinetic energy is brought to rest and produces a rise in temperature (aerodynamic heating) Adapted: Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly Watch video on aerodynamic heating: http://www.youtube.com/watch?v=rchlt5wdqbs Total Enthalpy / Total Temperature Inserting the total enthalpy into the SFEE: For calorically perfect gas, If there is no heat transfer and no work interactions, (i.e. q - w x = 0, or q = w x = 0), then and, for a calorically perfect gas,
Total Pressure The total pressure P t of a flowing gas is defined as the pressure obtained when the gas is brought to rest isentropically (s y s 1 = 0) (Note: recall that ) Source: Soon Kim Tat Stagnation Temperature and Pressure Stagnation temperature ratio Stagnation pressure ratio At M=1 (choked nozzle) for air (isentropic),
Stagnation Temperature and Pressure Source: Soon Kim Tat Total Pressure (Irreversible) Perfect gas is brought to stagnation (V 2 = 0) Under adiabatic (q = 0), no-shaft-work (w x = 0) Same final stagnation temperature will be attained whether it is irreversible or reversible process, i.e. T t2, irreversible = T t2,reversible However, the final total pressure will be lower, i.e. P t2, irreversible < P t2, reversible P y depends on the entropy increase, (s y s 1 ) - a measure of the degree of irreversibility Source: p 98, Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly
Total Pressure (Irreversible) Schlieren Imaging of Supersonic Inlet shock Waves Source: http://en.wikipedia.org/wiki/unstart Total pressure of air passing through an engine inlet and nozzle or a shock wave cannot increase and must decrease because of the irreversible effects of friction. One-Dimensional Gas Dynamics For a one-dimensional flow where q = w x = 0, For a calorically perfect gas, Rewrite in terms of dimensionless static enthalpy and dimensionless kinetic energy as We obtain or H + K = 1
H-K Diagram Dimensionless static enthalpy (H) and dimensionless kinetic energy (K) Useful for explaining the more complex internal flow behaviour of air breathing engines visualising the operation of propulsion devices Not a state diagram H-K Diagram M<1 M>1 1 1 Adapted: Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly
H-K Diagram Key: 0 = freestream reference state. Point c =choked condition at constant impulse. Points u and d denote end states of normal shock. Circled numbers denote isolines of constant property as follows: 1. Static enthalpy, static temperature 2. Kinetic energy, velocity, pressure (for frictionless heating or cooling only) 3. Isoline of constant Mach number Tt 4. Total enthalpy, total temperature (adiabatic), τ = 1 Tt0 5. Post-heat release adiabatic, τ > 1 6. Impulse function / stream thrust, area (for frictionless flow with heating or cooling only), case I = I 0 ; 7. Impulse function, case φ > φ 0 Scramjet H-K Diagram Adapted: Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly
Nozzle Design Source: The Jet Engine by Rolls Royce Nozzle Design From a large chamber, a gas flows through a nozzle with mass flow rate ṁ c chamber pressure P c = P t chamber temperature T c = T t Nozzle Gas Relationships: Graphics: Soon Kim Tat
Nozzle Design Approaches 4 variables: P, T, V, A Select one variable as independent and find the remaining Nozzle design To pass a given mass flow with minimum frictional losses between 2 regions of different pressure. (Independent variable: P) Nozzle operating characteristics Given a nozzle, determine the mass flow rates and pressure distribution for various nozzle pressure. (Independent variable: A) Nozzle Design - Example Design Objective: To expand exhaust gas to a target static pressure Source: Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly
Nozzle Flow and Shock Waves Consider a wind-tunnel nozzle (next slide) As air flows from storage chamber into evacuated receiver: raises the pressure in the nozzle exhaust region P a decreases the nozzle pressure ratio P n = P c / P a. 7 possible distinct nozzle pressure ratio operating conditions. Nozzle Flow and Shock Waves Nozzle Pressure Ratio P n =P c /P a 1. Underexpanded P n >P ṅ 2. Design expansion P n =P ṅ 3. Overexpanded P n <P ṅ 4. Normal shock at exit 5. Normal shock inside 6. Sonic at throat, subsonic elsewhere 7. Subsonic flow everywhere (mass flow below max) Adapted: Elements of Propulsion: Gas Turbines and Rockets by Jack D. Mattingly
Summary Total enthalpy, total temperature and total pressure, and their relationships Cycle of an air-breathing jet engine using H-K diagram Evaluating the issues related to nozzle design Reflection Question Evaluate what happens to the gas pressure, temperature and velocity as it passes through a convergent nozzle and a divergent duct, if the initial velocity is: a. Subsonic b. Supersonic