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
1200
temperature
1000
800
600
400
1
1.2
1.4
1.6
1.8
2
2.2
entropy
reversible cycle
irreversible cycle
regeneration inlet temperature reversible
irreversible, heat exchanger maximum
regeneration inlet temperature irreversible
3
6
T
7ss
5
5s
2
2s
7s
7
4
4 - outlet of turbine
5s - inlet to regenerator T5s = T4
8
irreversible
1 - start
2 - irreversible compressor outlet
6 - outlet of heat addition T6 = Tmax
4 - outlet of turbine
5 - inlet to regenerator T5 = T7
1
s
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state - reversible process
1 - start
2s - reversible compressor outlet
3 - outlet of heat addition T3 = Tmax
1
irreversible processes can be described by some efficiencies and heat transfer effectiveness:
N.B. the efficiencies are defined wrt irreversible overall cycle
h6 − h7
ηt =
turbine efficiency
compressor efficiency
h 6 − h 7s
ηc =
h 2s − h 1
h2 − h1
T5 − T2
=
=
T6 − T7
η t := 0.8
T6 − T7s
T2s − T1
η c := 0.78
T2 − T1
heat exchanger effectiveness
ε =
pressure loss in heater
δp H 

p 6 = p 3 − δp H = p 3 ⋅  1 −
p3


delta_p_over_p_H := 5%
δp L 

p 7 = p 1 + δp L = p 1 ⋅  1 +
p1


delta_p_over_p_L := 3%
pressure loss (increase)
in cooler, relative to p 1
ε := 94%
T7ss − T2
we will combine these as follows as for efficiency only ∆p across turbine matters:

p6
p7
p 3 ⋅  1 −
δp H 
p3

 =
δp L 

p 1 ⋅  1 +
p1


=
p2
p1
⋅ ( 1 − δp%)
this combines losses into effect on turbine
for these calculations
γ := 1.4
power :=
γ−1
γ
one compressor no
intercooling
Nc = 1
 1 − delta_p_over_p_H 

 1 + delta_p_over_p_L 
delta_p_over_p := 1 −
delta_p_over_p = 7.767 %
taking advantage of constant c po
T1 := 300
Tmax := 1200
maximum
T3 := Tmax
T6 := Tmax
start with 1+ as η = 1
mathematically
pr := 1.3 , 1.4 .. 5
reversible relationships are developed in brayton_cycle_summary.mcd (may be 2005)
reversible ....
T2s ( pr) := pr
T4 ( pr) :=
irreversible .....
power
1

 pr 
⋅ T1
T2s ( 2 ) = 365.704
T2 ( pr) := T1 +
T2s ( pr) − T1
T2 ( 2 ) = 384.236
ηc
p6_over_p7 ( pr) := pr⋅(1 − delta_p_over_p)
power
⋅ T3
T4 ( 2 ) = 984.402
p6_over_p7 ( 2 ) = 1.845
1

reversible turbine calc in irreversible cycle ... T7s ( pr) := T6 ⋅ 
p6_over_p7 ( pr)

power

(
)
T7 ( pr) := T6 − T6 − T7s ( pr) ⋅ η t
11/21/2005
2
T7s ( 2 ) = 1007
T7 ( 2 ) = 1046
at this point we can compute the thermal efficiency without regeneration
reversible
η th =
wnet
qH
=
wt + wc
qH
irreversible
(
T3 − T4 − T2s − T1
=
T3 − T2s
rev
) =  T6 − T7 − ( T2 − T1 ) 


(
QH = T3 − T2s = T6 − T2


T6 − T2
irrev
)
so thermal efficiency becomes
η th_basic_rev ( pr) :=
(
T3 − T4 ( pr) − T2s ( pr) − T1
)
η th_basic_irr( pr) :=
T3 − T2s ( pr)
(
T6 − T7 ( pr) − T2 ( pr) − T1
)
T6 − T2 ( pr)
efficiency Brayton cycles
0.4
η t = 0.8
thermal efficiency
0.3
η c = 0.78
ε = 0.94
0.2
0.1
delta_p_over_p = 7.8 %
basic cycle - irreversible
basic cycle - reversible
0
2
2.5
3
3.5
4
4.5
5
pressure ratio
with regeneration, all the states are the same with
reversible - regen inlet temperature
irreversible ...
T5s := T4
(
T5 ( pr) := T2 ( pr) + ε ⋅ T7 ( pr) − T2 ( pr)
with regeneration
reversible
η th_ic =
wnet
qH
=
wt + wc
qH
η th_reg_rev ( pr) := 1 −
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=
(
T3 − T4 − T2s − T1
T3 − T5s = T4
irreversible
) =  T6 − T7 − ( T2 − T1 ) 


T6 − T5


T2s ( pr) − T1
1
T3 − T3 ⋅  
 pr 
power
η th_reg_irr( pr) :=
3
)
T5 ( 2 ) = 1006
rev
irrev
(
QH = T3 − T5s = T6 − T5
(
T6 − T7 ( pr) − T2 ( pr) − T1
T6 − T5 ( pr)
)
)
efficiency regeneration irreversible
0.8
thermal efficiency (ideal)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
with regeneration - irreversible
with regeneration - reversible
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
ratiorev =
workcomp
=
workturb
 T2s − T1 

 T3 − T4 
ratioirr =
T2s ( pr) − T1
ratiorev( pr) :=
T3 − T4 ( pr) T2s := 0
T2 := 0
 T2 − T1 

 T6 − T7 
T2 ( pr) −
T1
T6 − T7 ( pr)
T7 := 0
T7s := 0
parameters from above ...
power = 0.286
T1 = 300
3
T6 = 1.2 × 10
T4 := 0
maximum
reset to insure
calculation
for these calculations
one stage intercooling two compressors
η t = 0.8
pr := 1.1 ,1.2 .. 5
η c = 0.78
T2s (pr , N) := rc(pr , N)
power
efficiencies from above ...
delta_p_over_p = 7.767 %
range for
pressure ratio
reversible .....
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=
ratioirr( 2 ) = 54.7 %
Intercooled Irreversible (and reversible)
N := 1
workturb
ratioirr( pr) :=
ratiorev( 2 ) = 30.5 %
γ = 1.4
workcomp
assuming equal pressure ratios across multiple
compressors, the ratio for each is ...
1
rc(pr , N) := pr
N+
1
temperature out of all compressors (isentropic)
intercooling occurs along p = constant to same T 1 . Subsequent compressions
⋅ T1 are at the same ratio so temperatures after each compression are the same.
4
T2s (2 , 1 ) = 331.227
irreversible ... all compressors
T2 (pr , N) := T1 +
T2s (pr , N) − T1
ηc
T2 (2 , 1 ) = 340.034
p6_over_p7(pr) := pr⋅(1 − delta_p_over_p)
1
T4 ( pr) := T3 ⋅  
pr
power
1

T7s ( pr) := T6 ⋅ 
 p6_over_p7 ( pr) 
T4 ( 2 ) = 984.402
 
(
power
3
T7s ( 2 ) = 1.007 × 10
)
T7 ( pr) := T6 − T6 − T7s ( pr) ⋅ η t
reversile ....
η th_ic =
wnet
qH
=
wt + wc
qH
=
3
irreversible
(
T3 − T4 − (N + 1)⋅ T2s − T1
T3 − T2s
) = T6 − T7 − (N + 1)⋅ (T2 − T1)
T6 − T2
η th_ic_irr( pr , N) :=
η th_ic_rev( pr , N) :=
T7 ( 2 ) = 1.046 × 10
(T6 − T7(pr)) − (N + 1)⋅ (T2(pr , N) − T1)
T6 − T2 (pr , N)
(T3 − T4(pr)) − (N + 1)⋅ (T2s( pr , N) −
T1)
T3 − T2s (pr , N)
efficiency Brayton cycles
thermal efficiency
0.4
0.3
η t = 0.8
0.2
η c = 0.78
0.1
delta_p_over_p = 7.767 %
0
intercooled - irreversible
intercooled - reversible
0.1
1
1.5
2
2.5
3
3.5
4
4.5
5
pressure ratio
ratiorev =
workcomp
workturb
ratiorev( pr) :=
=
(
(N + 1)⋅ T2s − T1
)
ratioirr =
T3 − T4
(
(N + 1)⋅ T2s (pr , N) − T1
)
workcomp
workturb
ratioirr( pr) :=
T3 − T4 ( pr)
ratiorev( 2 ) = 29 %
(
(N + 1)⋅ T2 − T1
=
T6 − T7
(
(N + 1)⋅ T2 ( pr , N) − T1
)
T6 − T7 ( pr)
ratioirr( 2 ) = 52 %
calculations with reheat and multiple turbines are similar and will not be done here. see brayton_plot.mcd for
general calculations and plotting
11/21/2005
5
)