MAE 5350: Gas Turbines Ideal Cycle Analysis Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk INTRODUCTION TO CYCLE ANALYSIS Cycle Analysis What determines engine characteristics? Cycle analysis is study of thermodynamic behavior of air as it flows through engine without regard for mechanical means used to affect its motion Characterize components by effects they produce

Actual engine behavior is determined by geometry Cycle analysis is sometimes characterized as representing a rubber engine Main purpose is to determine which characteristics to choose for components of an engine to best satisfy a particular need Express T, , Isp, TSFC as function of design parameters Aircraft engines (and all gas turbine engines) operate on a Brayton Cycle 2 HEAT ENGINE: PROPULSION CHAIN Chemical Energy Heat (Thermal Energy) Combustion

Mechanical Power Thermal Mech. Power to GasFlow Mechanical Thrust Power Propulsive The overall efficiency for the propulsion chain is given by: U F What we get Thrust power 0 =

overall What we pay for Rate of chemical energy usage m h f comb Fuel flow rate m f h Heat of combustion (heat of reaction) = 4.3 x 107 J/kg for jet fuel comb u Flight speed o F = Thrust overall combustion thermalmech propulsive 3 CONCEPTS / TOOLS FOR ENGINE IDEAL CYCLE ANALYSIS Ideal gas equation of state, p = RT

One-dimensional gas dynamics Concepts of stagnation and static quantities (temperature, pressure, etc.) Relations between Mach number and thermodynamic properties Thermodynamics of propulsion cycle Make use of 1st and 2nd Laws of Thermodynamics Behavior of useful quantities: energy, entropy, enthalpy Relations between thermodynamic properties in a reversible (lossless) process Isentropic = reversible + adiabatic

Properties of cycles (it is cyclic) Air starts at atmospheric pressure and temperature and ends up at atmospheric pressure and temperature Definition of Open vs. Closed Cycles 4 STAGNATION QUANTITIES DEFINED Quantities used in describing engine performance are the stagnation pressure, enthalpy and temperature Stagnation enthalpy, ht , enthalpy state if stream is decelerated adiabatically to zero velocity u2 h h t 2 h c T Ideal gas p

u2 T T Stagnation temperature t 2c p 2 T t 1 u T (2c T ) p R c p 1 Speed of sound a 2 RT 2 T T t 1 1 u or t 1 1 M 2 T T

2 2 a2 Total to static temperature ratio in terms of Mach number 5 FOR REVERSIBLE + ADIABATIC = ISENTROPIC PROCESS P constant using p RT we find T p t p (p t p constant ( / 1)

T t 1 defines the stagnation pressure T is pressure if stream is decelerate d isentropically to zero velocity) 1 p 1 t 1 M2 p 2 For M 2 1, expand using the binomial theorem to get 1 p p u 2 t 2 " Bernouli Equation" for low speed flow 6

RECAP ON THERMODYNAMICS: 1st LAW First law (conservation of energy) for a system: chunk of matter of fixed identity E0 = Q - W Change in overall energy (E0 ) = Heat in - Work done E0 = Thermal energy + kinetic energy ... Neglecting changes in kinetic and potential energy E = Q - W ; (Change in thermal energy) On a per unit mass basis, the statement of the first law is thus: e = q - w 7 RECAP ON THERMODYNAMICS: 2nd LAW The second law defines entropy, s, by: dqreversible ds T Where dqreversible is the increment of heat received in a reversible

process between two states The second law also says that for any process the sum of the entropy changes for the system plus the surroundings is equal to, or greater than, zero ssystem ssurroundings0 Equality only exists in a reversible (ideal) process 8 REPRESENTING ENGINE PROCESS IN THERMODYNAMIC COORDINATES First Law: E = Q - W, where E is the total energy of the parcel of air. For a cyclic process E is zero (comes back to the same state) Therefore: Q (Net heat in) = W (Net work done) Want a diagram which represents the heat input or output. A way to do this is provided by the Second Law dq reversible

Tds where ds is the change in entropy of a unit mass of the parcel and dq is the heat input per unit mass Thus, one variable should be the entropy , s 9 STEADY FLOW ENERGY EQUATION (I) m ht 2 ht1 Q W shaft = Rate of heat in - Rate of shaft work done The quantity ht h u 2 / 2 is the stagnation enthalpy defined previously For any device in steady flow Heat input 1 Device 2 Per unit mass flow rate: Mass flow ht2 - ht1 =q-wshaft

Shaft work q is heat input/unit mass wshaft is the shaft work / unit mass 10 STEADY FLOW ENERGY EQUATION (II) The form of the steady flow energy equation shows that enthalpy, h: h = e + pv = e + p/ Natural variable to use in fluid flow-energy transfer processes. For an ideal gas with constant specific heat, dh = cpdT. Changes in enthalpy are equivalent to changes in temperature.

To summarize, the useful natural variables in representing gas turbine engine processes are h,s (or T, s). Represent thermodynamic cycle (Brayton) for gas turbine engine on a T,s diagram 11 THERMODYANMICS: BRAYTON CYCLE MODEL 12 GAS TURBINE ENGINE COMPONENTS Inlet: Slows, or diffuses, the flow to the compressor

Fan/Compressor: (generally two, or three, compressors in series) does work on the air and raises its stagnation pressure and temperature Combustor: Heat is added to the air at roughly constant pressure Turbine: (generally two or three turbines in series) extracts work from the air to drive the compressor or for power (turboshaft and industrial gas turbines) Afterburner: (on military engines) adds heat at constant pressure Nozzle: Raises the velocity of the exiting mass flow Exhaust gases reject heat to the atmosphere at constant pressure

13 THERMODYNAMIC CHARACTERISTICS OF COMPONENTS (IDEAL COMPONENTS) Inlet : sh t 0 Compressor: sh t w shaft w shaft 0, work input to compressor , ss 0 adiabatic, lossless Combustor and afterburner : sh t q (heat input) in Turbine : sh w t w shaft ss 0 adiabatic, lossless

shaft > 0, work output from turbine , Exhaust nozzle: No shaft work, no heat exchange, ss = 0 14 THERMODYNAMIC MODEL OF GAS TURBINE ENGINE CYCLE [Cravalho and Smith] 15 SCHEMATIC OF CONDITIONS THROUGH A GAS TURBINE [Rolls-Royce] 16 NOMINAL PRESSURES AND TEMPERATURES FOR A PW4000 TURBOFAN [Pratt&Whitney]

17 REVIEW OF STATION LOCATIONS 18