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Lecture 21
• heat engines and refrigerators using ideal gas as working substance
• Brayton cycle
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Ideal gas Heat Engines
Ideal gas summary I
• closed cycle trajectory: clockwise for Wout > 0Wout = Wexpand ! |Wcompress| = area inside closed curve
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Ideal gas summary II• depends only on T
• identify each process, draw pV diagram
• use ideal gas law to know n, p, V, T at one point
• use ideal gas law and equations for specific processes for p, V, T at beginning/end of each process
• calculate for each process
• by adding `s: confirm by area within curve
• add positive values of Q to find
• check:
Strategy for heat engine problems
(!Eth)net = 0, ! < 1, signs of Ws and Q...
QH
WsWout
Q, Ws and !Eth
Eth
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Brayton cycle (heat engine)• adiabatic compression (1 2): raises
T; isobaric expansion (2 3): raises T further, heat by fuel; adiabatic expansion (3 4): spins turbine, T still high; isobaric expansion (4 1): heat transferred to cooling fluid
• Thermal efficiency:Process 2! 3 (isobaric):QH = nCP (T3 " T2)Process 4! 1 (isobaric):QC = |Q41| = nCP (T4 " T1)# !B = 1" T4!T1
T3!T2Use pV = nRT and pV ! = constant (adiabatic)to give p(1!!)/!T = constant
# T1 = T2
!p2p1
"(1!!)/!= T2r
(1!!)/!p ,
where rp $ pmax
pminand T4 = T3r
(1!!)/!p
# !B = 1" 1
r(!!1)/!p
(increases with rp)
TH ! T3; TC " T1
! = 1! QC
QH
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Brayton cycle (refrigerator)
• heat engine backward, ccw in pV: low-T heat exchanger is “refrigerator”
• sign of W reversed, area inside curve is : used to extract from cold reservoir and exhaust to hot...
• gas T lower than (1 4), higher than (3 2) gas must reach by adiabatic expansion, by adiabatic compression
QH
Win QC
TC TH
TCTH
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Comparison of Brayton cycle heat engine and refrigerator
• Brayton cycle refrigerator is not simply heat engine run backward, must change hot and cold reservoir: heat transferred into cold reservoir for heat engine , from cold reservoir in refrigerator ; heat transferred from hot reservoir for heat engine , into hot reservoir for refrigerator
• heat engine: heat transfer from hot to cold is spontaneous, extract useful work in this process via system...
• refrigerator: heat transfer from cold to hot not spontaneous, make it happen by doing work via system...
(TC ! T1)(TC ! T4)
(TH ! T3)(TH ! T2)