Non-Ideal Cycle Analysis MAE 5350: Gas Turbines Non-Ideal Cycle Analysis Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

NON-IDEAL (“REAL”) CYCLES Principal deviations from ideal behavior: Imperfect diffusion of free-stream flow in engine inlet Non-isentropic compression and expansion in the turbomachinery (compressors and turbines) Stagnation pressure change in combustor Incomplete combustion in combustor Variation of gas properties (specific heat, g) through the engine Incomplete expansion, or over expansion, in the nozzle Extraction of compressor discharge air for turbine cooling and for airframe use (bleeds)

SOME COMMENTS ON THE WORKING FLUID We have assumed that the working fluid can be approximated as a perfect gas with constant specific heats In reality the specific heats, and specific heat ratio, , vary through the engine The effect of pressure is small (on the order of 0.1% for 20kPa to 45 Mpa [Cumpsty]) but the effect of temperature is appreciable The variation in cp and  is given in the next chart, which shows the dependence on temperature, for different values of the equivalence ratio, f, which is the ratio of the fuel air-ratio to the fuel air ratio for stoichiometric combustion For simplicity, the values of  will be taken to be different but constant in the different components. They will be denoted by subscripts The value for the compressor is denoted by c, the value for the turbine by t. Appropriate values are 1.4 and 1.3, respectively

VARIATION IN SPECIFIC HEAT AT CONSTANT PRESSURE, cp, AND SPECIFIC HEAT RATIO, , WITH TEMPERATURE FOR AIR AND FOR COMBUSTION PRODUCTS OF KEROSENE;  is the Equivalence Ratio [Cumpsty]

DEPARTURE FROM IDEAL BEHAVIOR: LOSSES IN ENGINE COMPONENTS Component efficiency has a large impact on cycle performance Characterizing losses in components (departures from ideal reversible processes) is a key aspect of real cycle analysis We will examine basic mechanisms and measures developed for assessing loss In this, it will be seen that entropy generated due to irreversibility is the most useful measure of loss (inefficiency) For an adiabatic flow the entropy increase translates to a stagnation pressure change pt can thus often be used as a loss indicator

LOSS SOURCES Viscous dissipation Boundary layers Shear layers (mixing) Heat transfer across a finite temperature difference Shocks

THERMODYNAMIC CYCLES [Walsh and Fletcher] Carnot Cycle Non-Ideal Brayton Cycle for turbojet, turboshaft, turboprop, ramjet

LOSSES AND STAGNATION PRESSURE CHANGES Consider a medium that undergoes an irreversible process 1--->2 Example: flow through a screen or throttle No work done, no heat transfer, therefore ht = constant Represent the states on a T-s diagram It is “conventional” to think about losses in terms of changes in stagnation pressure. Why?

FLOW THROUGH A SCREEN, THROTTLE, OR BLADE ROW

LOSSES IN A THROTTLING PROCESS (I) Losses are often expressed as a decrease in stagnation pressure Directly measurable quantity Related to the minimum work required to reverse the process to its original state 2 ---> 1 Lost work For the flow through a screen, examine the work required to reverse the process using an ideal process First Law of Thermodynamics: e = q - w, neglecting changes in all forms of energy except internal energy For perfect gas e = e(T) only, so for our example of the screen: e(Tt1) = e(Tt2)  q = w For a reversible process, the heat received per unit mass is: dqrev=Tds

LOSSES IN A THROTTLING PROCESS (II) Thus for the screen, qrev=Tt1s So wrev=Tt1s THE WORK REQUIRED TO RETURN THE SYSTEM TO INITIAL STATE (THE “LOST WORK”) IS DIRECTLY RELATED TO THE CHANGE IN ENTROPY Now we relate this entropy change to the change in total pressure If TT = constant and cp and cv are constant then

LOSSES IN A THROTTLING PROCESS (III) The minimum work required to restore the fluid to its initial state is thus directly connected to the change in stagnation pressure for a flow with constant stagnation temperature

CONNECTION BETWEEN ENTROPY CHANGES AND TURBOMACHINERY COMPONENT EFFICIENCY Compressor Turbine Efficiency for compressor: For given pt2 / pt1 , how much shaft work is done Shaft work / unit mass flow rate = ht2 - ht1 (assuming adiabatic)

ht2 = ht2 + ht2 - ht2 = ht2 + Dht Along pt2 = const. curve ht2 = ht2 + ht2 - ht2 = ht2 + Dht Isentropic compression At const pt But or Thus at const pt

CONNECTION BETWEEN ENTROPY CHANGES AND COMPONENT EFFICIENCY ht2 = ht2 + Tt2 Ds or 2 ; Entropy rise directly tied to efficiency, similarly for turbine ; Turbine

SUMMARY: LOSSES AND STAGNATION PRESSURE Entropy is the basic measure of loss - Entropy is not measured directly For adiabatic processes, we can relate entropy changes and changes in stagnation pressure - Stagnation pressure is measured Stagnation pressure is often used as the figure of merit for component loss (or component efficiency) The next several slides show the application for an inlet/diffuser combination d = pt out / pt in

INLET AND DIFFUSER LOSS Subsonic diffusers Need to supply air to the engine at the Mach number the compressor demands Need to be efficient over range of free-stream Mach numbers from take-off to cruise Modern computational tools enable efficient inlets with stagnation pressure recoveries greater than 0.95 Supersonic diffusers Shock waves exist and introduce a loss mechanism Very large variations in capture stream tube area Inlet compression is a larger fraction of the overall compression process and overall cycle efficiency is thus more sensitive to inlet design References provide detailed information about inlet design

SCHEMATIC DIAGRAMS OF SUBSONIC AND SUPERSONIC INLETS AND DIFFUSERS [Kerrebrock]

REPRESENTATIVE VALUES OF INLET/DIFFUSER STAGNATION PRESSURE RECOVERY AS A FUNCTION OF FLIGHT MACH NUMBER [Kerrebrock]

EFFICIENCIES IN TURBOMACHINERY COMPONENTS: COMPRESSOR AND TURBINE Consider the compression process through a compressor stage The goal is to achieve a given stagnation pressure ratio, and to do this at minimum work We need a relation involving dh and dp to capture this dh = Tds + dp/r Apply this to the stagnation conditions: dht= Ttds + dpt/t The flow in the compressor is essentially adiabatic The second law says that for a fluid particle ds > 0 for all real processes For given change in pt, as ds increases, so does ht , the stagnation enthalpy Thus, for a given change in stagnation pressure, the change in stagnation temperature, which is a direct measure of the work we must do, reflects how “good” we are at compression

THE ADIABATIC (OR ISENTROPIC) EFFICIENCY For a compressor, the comparison of ideal to actual work for a given stagnation pressure rise or ratio furnishes the metric known as adiabatic (or isentropic) efficiency Pt2 Ideal work for pt change: Ideal Dht = ht2s - ht1 2s 2 Tt2 Actual Actual work for pt change: Actual Dht = ht2 - ht1 T or h Ideal ht Pt1 Tt1 h = Ideal work/actual work 1 s Ideal work for pt change: Ideal ht = ht25 - ht1 Actual work for pt change: Actual ht = ht2 - ht1

COMPRESSOR ADIABATIC EFFICIENCY The adiabatic (sometimes called isentropic) efficiency is the ratio of the ideal work for a given pt to the actual work needed Definition:  = Ideal work / Actual work There is a difference between this quantity and the cycle efficiency: - The cycle efficiency can describe an ideal situation - Cycle efficiency is set by the second law --a fundamental limitation on the conversion of heat to work - The adiabatic efficiency is a measure of “how well we did the design” and reflects our capabilities

BEHAVIOR OF STAGNATION PRESSURE AND TEMPERATURE IN A COMPRESSOR STAGE Stagnation pressure and temperature rise in rotor Shaft work is done on fluid Stagnation temperature is constant in the stator Forces on fluid, and angular momentum changes, exist, but no shaft work is done Stagnation pressure falls in stator Constant stagnation temperature, but entropy rises; Tds = dh - dp/r again Ideal work/unit mass is cp(Tt3’ - Tt1) [Using notation below] Actual work/unit mass is cp(Tt2 - Tt1) > cp(Tt3’ - Tt1) hcompressor = Ideal work/Actual work for same pressure ratio = Adiabatic/isentropic efficiency

THERMODYNAMIC STATES IN A COMPRESSOR STAGE [Cohen, Rodgers, Savaranamutoo]

 is stagnation temperature ratio: p is stagnation pressure ratio COMPRESSOR ADIABATIC EFFICIENCY IN TERMS OF PRESSURE AND TEMPERATURE RATIOS The adiabatic efficiency (and the corresponding quantity for turbines) is a metric of how effectively we are able to raise the stagnation pressure It is useful to put it in terms of the stagnation pressure and temperature ratios, which are the quantities actually measured  is stagnation temperature ratio: p is stagnation pressure ratio

ACTUAL AND IDEAL WORK FOR A TURBINE pt1 1 ht actual pt2 h ht ideal 2 2s s

ADIABATIC EFFICIENCY FOR A TURBINE For a turbine the “reverse” situation occurs For a given pressure ratio (expansion ratio), the work extracted in the real process is less for the actual process than for the reversible, adiabatic (isentropic process) Adiabatic (isentropic) efficiency for the turbine is defined as the ratio of Actual work / Ideal work, i.e., the ratio of the amount we actually received, compared to the amount we could have received in an isentropic process In terms of temperature and pressure ratios:

PARAMETERS AFFECTING CYCLE POWER AND EFFICIENCY The ratio Tt4/Tt2, the ratio of turbine entry temperature to compressor inlet temperature is an important parameter For a given , i.e., given cycle pressure ratio, increasing Tt4/Tt2 brings a rapid rise in net power (this is effectively how the engine is controlled) Compressor and turbine efficiencies have a marked effect on overall cycle efficiency and work. This is because for a Brayton cycle, much of the turbine work goes to drive the compressor The next two pages show plots of net power per unit of enthalpy flow and cycle efficiency for different values of the temperature ratio Tt4/Tt2 as well as the effects of component efficiency on cycle efficiency

ENGINE CYCLE (THERMAL) EFFICIENCY VARIATIONS [Philpot]

POWER AND CYCLE EFFICIENCY TRENDS WITH TURBINE TEMPERATURE AND COMPONENT EFFICIENCY [Cumpsty] Tt4 / Tt2 Tt4 / Tt2 Tt4 / Tt2

BRAYTON CYCLE FOR SIMPLE GAS TURBINE Pressure ratio 40, inlet temperature =288K, turbine temperature 1700K, turbine and compressor adiabatic efficiencies both 0.9 [Cumpsty]

BRAYTON CYCLE FOR GAS TURBINE WITH SEPARATE POWER TURBINE Pressure ratio 40, inlet temperature =288K, turbine temperature 1700K, turbine and compressor adiabatic efficiencies 0.9 [Cumpsty]

ENGINE PERFORMANCE SET BY Basic cycle selection Pressure ratio Turbine inlet temperature Bypass ratio Technology levels available Achievable component efficiencies Achievable work per stage Mechanical design and materials selection

SUMMARY A gas turbine engines can be regarded as “a core with different loads fitted to it” [Cumpsty] It can be analyzed in an approximate and useful manner by replacing the combustion by an equivalent heat transfer The thermodynamic cycle efficiency, and the cycle power, are strongly dependent on: Adiabatic component efficiencies Ratio of turbine entry stagnation temperature (max temperature in the cycle) to compressor inlet stagnation temperature