05_part1c - 1.C Applications of the Second Law[VN-Chapter 6...

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1C-1 T H Q H W e Q L T L Carnot cycle 1.C Applications of the Second Law [VN-Chapter 6; VWB&S-8.1, 8.2, 8.5, 8.6, 8.7, 8.8, 9.6] 1.C.1 Limitations on the Work that Can be Supplied by a Heat Engine The second law enables us to make powerful and general statements concerning the maximum work that can be derived from any heat engine which operates in a cycle. To illustrate these ideas, we use a Carnot cycle which is shown schematically at the right. The engine operates between two heat reservoirs, exchanging heat Q H with the high temperature reservoir at T H and Q L with the reservoir at T L. . The entropy changes of the two reservoirs are: S Q T Q H H H H = < ; 0 S Q T Q L L L L = > ; 0 The same heat exchanges apply to the system, but with opposite signs; the heat received from the high temperature source is positive, and conversely. Denoting the heat transferred to the engines by subscript “e”, Q Q Q Q H e H L e L = − = − ; . The total entropy change during any operation of the engine is, S S S S total H servoir at T H L servoir at T L e Engine = + + Re Re { { { For a cyclic process, the third of these S e ( ) is zero, and thus (remembering that Q H < 0 ), S S S Q T Q T total H L H H L L = + = + (C.1.1) For the engine we can write the first law as U Q Q W e H e L e e = = + 0 (cyclic process) . Or, W Q Q e H e L e = + = Q Q H L . Hence, using (C.1.1) W Q T S Q T T e H L total H L H = − +
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1C-2 = − ( ) Q T T T S H L H L total 1 . The work of the engine can be expressed in terms of the heat received by the engine as W Q T T T S e H e L H L total = ( ) 1 . The upper limit of work that can be done occurs during a reversible cycle, for which the total entropy change ( S total ) is zero. In this situation: Maximum work for an engine working between T T H L and : W Q T T e H e L H = ( ) 1 Also, for a reversible cycle of the engine, Q T Q T H H L L + = 0 . These constraints apply to all reversible heat engines operating between fixed temperatures. The thermal efficiency of the engine is η = = Work done Heat received W Q H e = = 1 T T L H Carnot η . The Carnot efficiency is thus the maximum efficiency that can occur in an engine working between two given temperatures. We can approach this last point in another way. The engine work is given by W Q T S Q T T e H L total H L H = − + ( ) / or, T S Q Q T T W L total H H L H e = − + ( ) / The total entropy change can be written in terms of the Carnot cycle efficiency and the ratio of the work done to the heat absorbed by the engine. The latter is the efficiency of any cycle we can devise: S Q T T T W Q Q T total H e L L H e H e H e L Carnot Any other cycle = = 1 η η .
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