Lecture 15 heat engines and refrigerators using ideal gas as working substance Brayton cycle Refrigerators closed cycle uses external work to remove heat from cold reservoir and exhaust heat to hot reservoir 2nd law does not allow spontaneous e g air conditioner or kitchen make air that is cooler than environment even colder Eth 0 cyclical QH QC Win exhaust more heat than removed from inside cool room by leaving refrigerator door open coefficient of performance perfect refrigerator Win 0 K forbidden by 2nd law informal statement 3 real refrigerator uses work K Example 1 0 L of 20 degree Celsius water is placed in a refrigerator The refrigerator s motor must supply an extra 8 0 W power to chill the water to 5 degree Celsius in 1 0 hr What is the refrigerator s coefficient of performance No perfect Heat Engine 1 connect perfect engine to refrigerator no net work for 2 combined but heat transferred from cold to hot not allowed by 2nd law informal statement 4 no perfect heat engine must waste heat Wout QH Wout QH 1 1 from energy conservation Example of Proof by Contradiction want to prove statement A e g there exists a perfect engine is not true assume A is true find a violation of basic law e g 2nd law of thermodynamics assumption is incorrect A is not true Summary Using only energy conservation and heat not transferred from cold to hot deduce heat engines and refrigerators exist must use closed cycle processes no perfect Unanswered questions upper limit on K Ideal gas Heat Engines closed cycle trajectory clockwise for Wout 0 Wout Wexpand Wcompress area inside closed curve Ideal gas summary I Ideal gas summary II Eth depends only on T Strategy for heat engine problems 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 Q Ws and Eth for each process Wout by adding Ws s confirm by area within curve add positive values of Q to find QH check E 0 1 signs of W and Q th net s 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 compression 4 1 heat transferred to cooling fluid TH T3 TC T1 Thermal efficiency 1 QC QH Process 2 3 isobaric QH nCP T3 T2 Process 4 1 isobaric QC Q41 nCP T4 T1 1 B 1 TT43 T T2 Use pV nRT and pV constant adiabatic to give p 1 T constant 1 1 T1 T2 pp21 T 2 rp 1 and T T r where rp ppmax p 4 3 min 1 B 1 1 increases with rp rp Brayton cycle refrigerator heat engine backward ccw in pV low T heat exchanger is refrigerator sign of W reversed area inside curve is Win used to extract QC from cold reservoir and exhaust QH to hot gas T lower than TC 1 4 higher than TH 3 2 gas must reach T1 TC by adiabatic expansion T3 TH by adiabatic compression 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 TC T1 from cold reservoir in refrigerator TC T4 heat transferred from hot reservoir for heat engine TH T3 into hot reservoir for refrigerator TH T2 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
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