Regeneration Brayton cycle - irreversible An actual gas turbine differs from the ideal due to inefficiencies in the turbines and compressors and pressure losses in the flow passages (heat exchangers in closed cycle). The T - s diagram may be as shown: static data for plot T-s diagram temperature 1200 1000 800 600 400 1 1.2 1.4 1.6 entropy reversible cycle irreversible cycle regeneration inlet temperature reversible irreversible, heat exchanger maximum regeneration inlet temperature irreversible T 2s 2 3 4 1 6 7s 7 5s 5 s 7ss 8 1.8 2 2.2 state - reversible process 1 - start 2s - reversible compressor outlet 3 - outlet of heat addition T3 = Tmax 4 - outlet of turbine 5s - inlet to regenerator T5s = T4 irreversible 1 - start 2 - irreversible compressor outlet 6 - outlet of heat addition T6 = Tmax 4 - outlet of turbine 5 - inlet to regenerator T5 = T7 11/21/2005 1irreversible processes can be described by some efficiencies and heat transfer effectiveness: N.B. the efficiencies are defined wrt irreversible overall cycle h6 − h7 T6 − T7 turbine efficiency ηt = h6 − h7s = T6 − T7s ηt := 0.8 compressor efficiency ηc = h2s − h1 = T2s − T1 ηc := 0.78 h2 − h1 T2 − T1 T5 − T2 heat exchanger effectiveness ε = ε := 94% T7ss − T2 δpH  pressure loss in heater p6 = p3 − δpH = p3⋅1 − delta_p_over_p_H := 5%  p3  δpL  pressure loss (increase) p7 = p1 + δpL = p1⋅1 + delta_p_over_p_L := 3% in cooler, relative to p1  p1  we will combine these as follows as for efficiency only ∆p across turbine matters: δpH  p3⋅1 − p6 =  p3  = p2 ⋅(1 − δp%) delta_p_over_p := 1 −  1 − delta_p_over_p_H  p7 δpL  p1  1 + delta_p_over_p_L  p1⋅1 +  p1  this combines losses into effect on turbine delta_p_over_p = 7.767 % for these calculations taking advantage of constant cpo γ − 1γ := 1.4 power := T1 := 300 Tmax := 1200 maximum T3 := Tmax T6 := Tmax γ start with 1+ as η = 1Nc = 1 one compressor no 1.3 1.4 .. 5 mathematicallypr := , intercooling reversible relationships are developed in brayton_cycle_summary.mcd (may be 2005) reversible .... irreversible ..... () − () := pr ⋅ ( ) = 365.704 T2 pr := + ( ) = 384.236 T2s pr T1T2sprpowerT1 T2s2 () T1 ηc T22T4 pr :=  1 ⋅ ( ) = 984.402 p6_over_p7 pr) := pr⋅(1power ( − delta_p_over_p) () T3 T42 pr  p6_over_p7 2 = 1.845( ) reversible turbine calc in irreversible cycle ... T7s pr := T6⋅ 1 power ( ) = 1007()  T7s 2 p6_over_p7 pr()  T7()pr := − − () ⋅ ( ) = 1046T6 (T6 T7s pr)ηt T7211/21/2005 2at this point we can compute the thermal efficiency without regeneration reversible irreversible rev irrev + w −− − −− − ηth = wnet = wt c = T3 T4 (T2s T1) = T6 T7 (T2 T1) QH = T3 − T2s = (T6 − T2)qHqH T3 − T2s  T6 − T2  so thermal efficiency becomes − T4 pr − () − − T7 pr − ()T3 () (T2s pr T1) T6 () (T2 pr − T1)() := () :=ηth_basic_rev prT3 T2s prηth_basic_irr prT6 T2 pr− () − () 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 basic cycle - irreversible basic cycle - reversible efficiency Brayton cycles thermal efficiency ηt 0.8= ηc 0.78= ε 0.94= delta_p_over_p 7.8 % = pressure ratio with regeneration, all the states are the same with reversible - regen inlet temperature irreversible ... T5s := T4 T5 pr := T2 pr + ε⋅( () − ()) T5 2() () T7pr T2pr ( ) = 1006 with regeneration reversible irreversible rev irrev + − − − −−−wnet wt wc T3 T4 (T2s T1)T6 T7 (T2 T1) ηth_ic = qH = qH = T3 − T5s = T4 =  T6 − T5  QH = T3 − T5s = (T6 − T5) () − () := 1 − T6 () T2pr − T1 T2s pr T1ηth_reg_rev pr() := − T7 pr−( () ) T3 − T3⋅ 1 power ηth_reg_irr prT6 ()− T5 pr pr  11/21/2005 30.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 with regeneration - irreversible with regeneration - reversible thermal efficiency (ideal) efficiency regeneration irreversible 1 1.5 2 2.5 3 3.5 4 4.5 5 pressure ratio also look at magnitude of compressor work compared to turbine, say for pr = 2 (since these states are the same for w & w/o regeneration, the work is also the same workcomp  T2s − T1  workcomp  T2 − T1  ratiorev = =  ratioirr = = workturb  T3 − T4  workturb  T6 − T7  T2s pr − T1 T2pr − T1() () ratiorev()pr := () ratioirr pr := T3 − T4 pr − T7 pr() T6() ratiorev 2 = 30.5 % ratioirr 2 = 54.7 % ( ) ( ) Intercooled Irreversible (and reversible) 2s 2 7T := 0 T := 0 T := 0 reset to insure parameters from above ... 7s 4T := 0 T := 0 calculation γ = 1.4 power = 0.286 T1 = 300 T6 = 1.2 × 103 maximum for these calculations N := 1 one stage intercooling two compressors ηt = 0.8 ηc = 0.78 delta_p_over_p = 7.767 % efficiencies from above ... pr := 1.1 1.2 .. 5 rc(pr N) := pr + , range for assuming equal pressure ratios across multiple , N1pressure ratio compressors, the ratio for each is ... reversible ..... temperature out of all compressors (isentropic) intercooling occurs along p = constant to same T1. Subsequent compressions 2sT (pr N := rc(pr N)power T1 are at the same ratio so temperatures after each compression are the same., ) , ⋅ 11/21/2005 4 1T2s(pr N − T1, ) T2s(21, ) = 331.227 irreversible ... all compressors 2T (pr N, ) := T1 + T2(21, ) = 340.034 ηc p6_over_p7(pr) := pr⋅(1 − delta_p_over_p)  pr  T7s pr T6  p6_over_p7 pr()  T7s 2T4 pr := T3  1 power ( ) = () := ⋅ 1 power ( ) = 1.007 × 10() ⋅ T4 2 984.402  3 7T pr := − − () ⋅ ( ) = 1.046 × 10() T6 (T6 T7s pr)ηt T723 reversile .... irreversible wnet wt + wc T3 − T4 − (N + 1)⋅(T2s − T1) T6 − T7 − (N1)⋅(T2 − T1)+ = = = =ηth_ic qH qH T3 − T2s T6 − T2 (T6 − T7 pr)− (N + 1)⋅( ( , T1)() T2pr N) − ( , ) :=ηth_ic_irr pr NT6 − T2(pr N, ) (T3 − T4 pr)− (N + 1)⋅( ( , T1)() T2spr N) − ( , ) := ηth_ic_rev pr NT3 − T2s(pr N, ) efficiency Brayton cycles 1 1.5 2 2.5 3 3.5 4 4.5 5 pressure ratio workcomp (N1)⋅(T2s − T1) (N + …