A THERMODYNAMIC CYCLE 25 A HEAT ENGINE operates on a three stage cycle Process 1 2 is an isothermal compression from state p1 V1 to state p2 V2 Process 2 3 an isochoric process from pressure p2 to pressure p3 Process 3 1 an adiabatic expansion from state p3 V3 to state p1 V1 The working substance is n moles of an IDEAL DIATOMIC GAS Cv 5 2 R Denote the initial STATE 1 by the thermodynamic variables p1 V1 and T1 the compression ratio r V1 V2 where r 1 Draw the CYCLE on the accompanying p V Diagram labeling each STATE and each STAGE process clearly 3 only in terms of n C Perform a complete STATE ANALYSIS Express all answers v p1 a b V1 T1 r and 6 STATE ANALYSIS STATE 1 2 3 P p1 rp1 r p1 V V1 V1 r V1 r T T1 T1 ETH nCv T1 nCv T1 r 1 T1 r 1 nCv T1 c Perform a FIRST LAW ANALYSIS for the cycle Express all answers in terms of n only Cv p1 V1 T1 r and FIRST LAW ANALYSIS 10 PROCESS Qi f 1 2 2 3 3 1 CYCLE 1 nCV T1 ln r 1 nCV T1 ln r r 1 1 nCv T1 0 Wi f 0 1 r 1 nCv T1 ETH 0 r 1 1 nCv T1 1 r 1 nCv T1 0 W1 2 WORK INPUT W3 1 WORK OUTPUT so NET WORK OUTPUT W1 2 W3 1 Efficiency NET WORK OUTPUT HEAT INPUT for this cycle W1 2 W3 1 Q2 3 yielding r 1 1 nCv T1 1 nCV T1 ln r r 1 1 nCv T1 simplified 1 ln r 1 r 1 1 and CARNOT 1 Tmin Tmax 1 1 r 1 d only the efficiency of the cycle for converting INPUT Derive in terms of r and HEAT ENERGY into NET MECHANICAL ENERGY WORK OUTPUT Simplify the expression for the efficiency 3 only Derive in terms of r and between the extreme temperatures of this cycle 2 Finally compare the efficiency of this cycle to the corresponding CARNOT efficiency for a compression ratio r 5 1 the efficiency of a CARNOT HEAT ENGINE operating r 5 Efficiency Corresponding CARNOT efficiency 0 285 28 6 0 475 47 5 NOTE The SECOND LAW efficiency of this cycle 0 285 0 475 0 60 the cycle operates at 60 of the efficency of a CARNOT cycle operating between the temperature extremes Tmin and Tmax Possibly useful HINTS nRT R Cv nCvT and R Cv 1 k ln x ln xk NET WORK OUTPUT Wout Win
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