Chalmers University of Technology Lecture 3 Some more thermodynamics : Brief discussion of cycle efficiencies - continued Ideal cycles II –Heat exchanger cycle Real cycles –Stagnation properties, efficiencies, pressure losses –The Solar Mercury 50 Real cycles –Mechanical efficiencies –Specific heats (temperature variation) –Fuel air ratio, combustion and cycle efficiencies –Bleeds Jet engine nozzles Radial compressor I

Chalmers University of Technology Generalization of the Carnot efficiency Is generalization of Carnot efficiency to Brayton cycle possible? Define average temp. to value that would give the same heat transfer, i.e.:

Chalmers University of Technology Generalization of Carnot efficiency But for the isobar we have, Thus, the average temperature is obtained from (dp=0): Derive an expression for the lower average temperature in the same way. Furthermore, we have Gibbs equation (Cengel and Boles) : as well as:

Chalmers University of Technology Generalization of Carnot efficiency

Chalmers University of Technology When T 4 > T 2 a heat exchanger can be introduced. This is true when: Heat exchange cycle

Chalmers University of Technology Here we obtain the efficiency:Not independent of T 3 !!! (simple cycle is independent of t 3 ) Power output is unaffected by heat exchangers since the turbine and compressor work are the same as in the simple cycle. Theory 3.1 – Ideal heat exchanger cycle

Chalmers University of Technology Heat exchange cycle Very high efficiencies can be theoretically be obtained! Heat exchanger metallurgical limits will be relevant. T4 = K =>

Chalmers University of Technology Heat exchange cycle Low pressure ratio => high efficiency What happens with the average temperature at which heat is added/rejected when the pressure ratio changes in heat exchange cycle? q in q out THTH THTH

Chalmers University of Technology Cycles with losses a.Change in kinetic energy between inlet and outlet may not be negligible : b.Fluid friction => - burners - combustion chambers - exhaust ducts

Chalmers University of Technology Cycles with losses c.Heat exchangers. Economic size => terminal temperature difference, i.e. T 5 < T 4. d.Friction losses in shaft, i.e. the transmission of turbine power to compressor. Auxiliary power requirement such as oil and fuel pumps. e.γ and c p vary with temperature and gas composition.

Chalmers University of Technology Cycles with losses f.Efficiency is defined by SFC (specific fuel consumption = fuel consumption per unit net work output). Cycle efficiency obtained using fuel heating value. g.Cooling of blade roots and turbine disks often require approximately the same mass flow of gas as fuel flow => air flow is approximated as constant for preliminary calculations. This is done in this course.

Chalmers University of Technology Stagnation properties For high-speed flows, the potential energy of the fluid can still be neglected but the kinetic energy can not! It is convenient to combine the static temperature and the kinetic energy into a single term called the stagnation (or total) enthalpy, h 0 =h+V 2 /2, i.e. the energy obtained when a gas is brought to rest without heat or work transfer

Chalmers University of Technology Stagnation properties For a perfect gas we get the stagnation temperature T 0, according to:

Chalmers University of Technology Stagnation pressure Defined in same manner as stagnation temperature (no heat or work transfer) with added restriction –retardation is thought to occur reversibly Thus we define the stagnation pressure p 0 by: Note that for an isentropic process between 02 and 01 we get

Chalmers University of Technology Compressor and turbine efficiencies Isentropic efficiency (compressors and turbines are approximately adiabatic => if expansion is reversible it is isentropic). The isentropic efficiency is for the compressor is: Where are the averaged specific heats of the temperature intervals ´ and respectively.

Chalmers University of Technology Compressor and turbine efficiencies Ideal and mean temperature differences are not very different. Thus it is a good approximation to assume: We therefore define: Similarly for the turbine:

Chalmers University of Technology Compressor and turbine efficiencies Using produces the frequently used expressions:

Chalmers University of Technology Turbine efficiency options If the turbine exhausts directly to atmosphere the kinetic energy is lost and a more proper definition of efficiency would be: In practice some of the kinetic energy is recovered in an exhaust diffuser => turbine pressure ratio increases. Here we put p 04 =p a for gas turbines exhausting into atmosphere and think of η t as taking both turbine and exhaust duct losses into account

Chalmers University of Technology Turbine diffusers Recovered energy

Chalmers University of Technology Heat-exchanger efficiency Conservation of energy (neglecting energy transfer to surrounding): In a real heat-exchanger T 05 will no longer equal T 04 (T 05 <T 04 ). We introduce heat exchanger effectiveness as: Modern heat exchangers are designed to for effectiveness values above 90%. Use of stainless steel requires T 04 around 900 K (or less). More advanced steal alloys can be used up to 1025 K.

Chalmers University of Technology Pressure losses – burners & heat-exchangers Burner pressure losses –Flame stabilizing & mixing –Fundamental loss (Chapter 7 + Rayleigh- line appendix A.4) Heat exchanger pressure loss –Air passage pressure loss ΔP ha –Gas passage pressure loss ΔP hg –Losses depend on heat exchanger effectiveness. A 4% pressure loss is a reasonable starting point for design.

Chalmers University of Technology The Solar Mercury MW output η = 40.5 % System was designed from scratch to allow high performance integration of heat-exchanger

Chalmers University of Technology Mechanical losses Turbine power is transmitted directly from the turbine without intermediate gearing => (only bearing and windage losses). We define the transmission efficiency η m : Usually power to drive fuel and oil pumps are transmitted from the shaft. We will assume η m =0.99 for calculations.

Chalmers University of Technology Temperature variation of specific heat We have already established: c p =f 1 (T) c v =f 2 (T) Since γ =c p /c v we have γ=f 3 (T) The combustion product thermodynamic properties will depend on T and f (fuel air ratio)

Chalmers University of Technology Pressure dependency? At 1500 K dissociation begins to have an impact on c p and γ. Detailed gas tables for afterburners may include pressure effects. We exclude them in this course.

Chalmers University of Technology Temperature variation of specific heat In this course we use: Since gamma and c p vary in opposing senses some of the error introduced by this approximation is cancelled.

Chalmers University of Technology Calculate f that gives T 03 for given T 02 ? Use first law for control volumes (q=w=0) and that enthalpy is a point function (any path will produce the same result) Determining the fuel air ratio f is small (typically around 0.02) and c pf is also small => last term is negligible. The equation determines f.

Chalmers University of Technology Combustion temperature rise Hypothetic fuel: 86.08% carbon 13.92% hydrogen ΔH 25 = kj/kg Curves ok for kerosene burned in dry air. Not ok in afterburner (f in ≠0).

Chalmers University of Technology Shaft cycle performance parameters

Chalmers University of Technology Bleeds Combustor and turbine regions require most of the cooling air. Anti-ice Rule of thumb: take air as early as possible (less work put in) Accessory unit cooling (oil system, aircraft power supply (generator), fuel pumps) Air entering before rotor contributes to work!

Chalmers University of Technology Aircraft propulsion – thrust generation

Chalmers University of Technology Jet engine – principles of thrust generation

Chalmers University of Technology Jet engine – principles of thrust generation No heat or work transfer in the jet engine nozzle Stagnation temperature is constant

Chalmers University of Technology Mach number relations for stagnation properties We have already introduced the stagnation temperature as : and shown that (revision task) : The Mach number is defined as : The specific heat ratio γ is defined:

Chalmers University of Technology Mach number relations for stagnation properties Thus: but we defined: which directly gives:

Chalmers University of Technology Nozzle efficiencies Nozzle may operate choked or unchoked:

Chalmers University of Technology Nozzle efficiencies Critical pressure for irreversible nozzle is obtained from: which gives:

Chalmers University of Technology Impeller - work is transferred to accelerate flow and increase pressure Diffuser - recover high speed generated in impeller as pressure Basic operation of radial compressor

Chalmers University of Technology Radial compressor operation Typical design takes 50 % of increase in static pressure in diffuser Conservation of angular momentum governs performance:

Chalmers University of Technology Slip factor Due to inertia of flow C w2 < U: Stanitz formula for estimating σ

Chalmers University of Technology Power input factor -  Power is put into overcoming additional friction not related to the flow in the impeller channels Converts energy to heat => additional loss =>

Chalmers University of Technology Overall pressure rise: P 03 is here used to denote the pressure at compressor exit. P 02 is reserved for the stagnation pressure between the impeller and the diffuser vanes

Chalmers University of Technology Example 4.1a ψ =1.04, σ = 0.90 N = rev/s, D = 0.5 m D eye,tip = 0.3, D eye,root = 0.15 m = 9.0 kg/s T 01 = 295 K P 01 = 1.1 bar η c = 0.78 Compute pressure ratio and power required

Chalmers University of Technology Learning goals Understand why the Carnot cycle can be used for qualitative arguments also for the Joule/Brayton cycle Be able to state reasonable loss levels for gas turbine components (turbine and compressor performance are given in Lecture 4) and include them in cycle analysis Know how to compute cycle efficiencies for the heat exchanger cycle