Cyclic Processes A cyclic process is one that periodically returns to its initial state Change of a state function in one cycle is zero U U f U 0 0 in particular U Q W 0 1. law: Q W In a cyclic process heat transferred to a system equals the work done by it Cyclic equilibrium processes are represented by closed lines in state space P Work done during one cycle from a->b and back a b V W P dV = area of Heat Engines Heat engine: any cyclic process that absorbs heat |Qin|, rejects heat |Qout|, and does a positive amount of work W in each cycle Steam engine Total heat transferred to the system In one cycle Work out Q Q in Q out W 0 Q W
Heat in |Qin| Heat out |Qout| What you get Energy efficiency: work out W heat in Q in (in the textbook the efficiency is called e, I prefer ) What you have to invest With W Q in Q out W Q in Q in Q out Q in 1 Q out Qin Carnot engine in an ideal gas 1 1 Heat absorbed at the hotter isotherm Th T1 T2
adiabats Th isotherms 2 4 Tc 3 2 Heat rejected at the colder isotherm Tc T3 T4 3 No heat transferred during adiabatic processes from state 1 2 Heat absorbed at the hotter isotherm Th T1 T2 V Q in Q1 2 nRTh ln 2 V1 from state 2 3 No heat exchange because of adiabatic change Q 2 3 0 from state 3 4
Qout Q3 4 nRTc ln from state 4 1 Tc T3 T4 Heat rejected at the colder isotherm V4 V3 No heat exchange because of adiabatic change Q 4 1 0 1 Efficiency of the Carnot engine: Q out Qin V3 nRTc ln V4 1 V nRTh ln 2 V1 For the adiabatic processes we obtain: 2 3 4 1
V3 V2 V4 V1 Carnot Tc 1 Th (Although derived for the special case when the working substance is an ideal gas, this result is true for all Carnot engines) Nicolas Lonard Sadi Carnot (17961832) Stirling engines Flywheel Four processes of a Stirling engine 11 Heating at constant volume Piston 22 expansion 33 Cooling at constant volume Displacer 44 compression displacer displacer displacer displacer Efficiency of the ideal Stirling engine: Constant volume processes need not to be considered in the
net heat transfer when isochoric heat transfer processes take place inside a regenerator such that Q2 3 Q4 1 0 1 Q out Qin isotherms 1 V1 V4 4 2 V2 V3 3 V nRTc ln 3 V4 T 1 1 c C V Th nRTh ln 2 V3 V2 V1 V4 V1 Ideal Stirling engine has the same energy efficiency as the Carnot cycle
Gas turbine used in aircraft propel ships electrical power generator etc. Work provided by the turbine is used to drive the compressor and to do useful work like: Turbo-prop engine - rotate a propeller or - increase the kinetic energy of the outgoing gas Tu gine n e et rboj Fuel in Heat in Steam engine Work out closed system Air in gas turbine open system Work out Exhaust gas Heat out No particle exchange with surrounding
Matter exchange with surrounding Idealization of the gas turbine as a closed system Heat in Fuel in Heat exchanger burner gas turbine gas Exhaust gas gas gas turbine Heat exchanger Heat out PV diagram of the gas turbine (Brayton or Joule cycle) 1 2 Process in the compressor 6 adiabatic compression V1 V2 2 3 2
3 Heating the gas (by burning the fuel) adiabates 4 P V2 V3 3 2 0 0 4 Adiabatic expansion V3 V4 in the turbines 4 1 2 4 V 4 1 Real engine: exhaust gases out fresh air in Here: cooling exhaust gas V4 V1 return it to the engine at atmospheric pressure 1
Efficiency of the gas turbine: Q out (in general) Qin 2 3 Heating the gas (by burning the fuel) Adiabatic changes Q=0 Q exchange only at 4 1 cooling 6 3 Q in Q 2 3 nc P (T3 T2 ) 2 P2 P3 Ph 3 4 P 2 4
1 Q out Q 4 1 nc P (T1 T4 ) T T 1 4 1 T3 T2 1 Pl 1 Ph T2 Ph ( 1) / T3 T1 Pl and Ph ( 1) / ( 1) / T4 Pl ( 1) / 2
P1 P4 Pl 1 0 0 2 4 V Adiabatic changes: PV const. V Efficiency of gas turbine determined by pressure ratio 4 nRT P T P 1 const T P ( 1) / const
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