{"id":1834,"date":"2021-07-27T20:21:56","date_gmt":"2021-07-28T00:21:56","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/6-4-carnot-cycles\/"},"modified":"2022-08-11T20:05:15","modified_gmt":"2022-08-12T00:05:15","slug":"6-4-carnot-cycles","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/6-4-carnot-cycles\/","title":{"raw":"6.4 Carnot cycles","rendered":"6.4 Carnot cycles"},"content":{"raw":"
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6.4.1 Irreversibilities<\/h2>\r\n

The Kelvin-Planck and Clausius statements of the second law indicate that for heat engines, [latex]\\eta_{th}< 1[\/latex], and for refrigerators and heat pumps, [latex]COP< \\infty[\/latex]. But can [latex]\\eta_{th} \\to 1[\/latex] <\/span>and [latex]COP \\to \\infty[\/latex] in an ideal condition? How is an ideal process or cycle defined? What is the theoretical limit of [latex]\\eta_{th} [\/latex] or [latex]COP [\/latex] <\/span>in an ideal cycle? To answer these questions, we will first need to understand the concepts of reversible processes and irreversibilities.<\/p>\r\n \r\n

A [pb_glossary id=\"2493\"]reversible process[\/pb_glossary] is a process that can be reversed without leaving any change in either the system or <\/em>its<\/em> surroundings<\/em>, <\/em>which means both the system and its surroundings always return to their original states during a reversible process. A fictitious, frictionless pendulum\u00a0can be treated as a reversible process, see Figure 6.4.1<\/a>; it will never stop and always returns to its original state. However, in reality, friction always exists. A pendulum will gradually slow down and <\/a>eventually stop.<\/p>\r\n \r\n

\r\n\r\n[caption id=\"attachment_1201\" align=\"aligncenter\" width=\"226\"]\"Frictionless Figure 6.4.1 <\/strong>Frictionless pendulum as an example of a reversible process
<\/em>[\/caption]\r\n

Factors that render a process irreversible are called [pb_glossary id=\"2499\"]irreversibilities[\/pb_glossary]. Friction, unrestrained expansion, mixing of fluids, heat transfer through a finite temperature difference, electric resistance, inelastic deformation of solids, and chemical reactions are some examples of irreversibilities. All processes occurring in nature are irreversible due to the existence of irreversibilities. Some processes are more irreversible than others. A reversible process is commonly used as an idealized model to which an actual process can be compared. The efficiency of a reversible process is the theoretical limit that an actual, irreversible process can possibly achieve in a nearly ideal condition.<\/p>\r\n\r\n

6.4.2 Carnot heat engine<\/h2>\r\n

A Carnot heat engine is an idealized cycle, which consists of four reversible processes. Figures 6.4.2<\/a>-6.4.4<\/a> show the schematic of a Carnot heat engine and its pressure-volume, [latex]P-\\mathbb{V}[\/latex], and temperature-entropy, [latex]T-S[\/latex], diagrams. The cycle consists of the following four reversible processes:<\/p>\r\n\r\n<\/div>\r\n

\r\n
    \r\n \t
  1. Process A[latex]\\to[\/latex]B: a reversible isothermal heat addition process of [latex]Q_H[\/latex] from a heat source at constant temperature [latex]T_H[\/latex].\r\n<\/em><\/li>\r\n \t
  2. Process B[latex]\\to[\/latex]C: a reversible adiabatic expansion process, during which the temperature drops from [latex]T_H[\/latex] to\u00a0<\/span> [latex]T_L[\/latex].<\/li>\r\n \t
  3. Process C[latex]\\to[\/latex]D: a reversible isothermal heat removal process of [latex]Q_L[\/latex] from a heat sink <\/span>at constant temperature [latex]T_L[\/latex].<\/li>\r\n \t
  4. Process D[latex]\\to[\/latex]A: a reversible adiabatic compression process, in which the temperature increases from [latex]T_L[\/latex] <\/a>to <\/span>[latex]T_H[\/latex].<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_2681\" align=\"aligncenter\" width=\"500\"]\"A<\/a> <\/a>Figure 6.4.2<\/strong>\u00a0Carnot heat engine<\/em>[\/caption]\r\n\r\n[caption id=\"attachment_2535\" align=\"aligncenter\" width=\"300\"]\"P-V<\/a> <\/a>Figure 6.4.3<\/strong>\u00a0Carnot heat engine: [latex]P-\\mathbb{V}[\/latex] diagram<\/em>[\/caption]\r\n\r\n[caption id=\"attachment_2536\" align=\"aligncenter\" width=\"300\"]\"T-s<\/a> Figure 6.4.4<\/strong><\/em>\u00a0Carnot heat engine: T-S diagram<\/em>[\/caption]\r\n\r\nBecause the Carnot heat engine cycle is an ideal cycle consisting of only reversible processes, it produces the maximum power output and has the maximum thermal efficiency among all heat engines operating between the same heat source at [latex]T_H[\/latex] and the same heat sink at [latex]T_L[\/latex]. The thermal efficiency of a Carnot heat engine can be expressed as<\/span>\r\n\r\n \r\n

    [latex]\\eta_{th,\\ rev}= 1-\\left(\\displaystyle\\frac{Q_L}{Q_H}\\right)_{rev} = 1-\\displaystyle\\frac{T_L}{T_H}[\/latex]<\/p>\r\nwhere\r\n

    [latex]T_H[\/latex]: absolute temperature of the heat source, in Kelvin<\/p>\r\n

    [latex]T_L[\/latex]: absolute temperature of the heat sink, in Kelvin<\/p>\r\n

    [latex]Q_H[\/latex]: heat supplied by the heat source to the Carnot heat engine, in kJ<\/p>\r\n

    [latex]Q_L[\/latex]: heat rejected from the Carnot heat engine to the heat sink, in kJ<\/p>\r\n

    [latex]\\eta_{th,\\ rev}[\/latex]: thermal efficiency of the Carnot heat engine, dimensionless<\/p>\r\n \r\n\r\nThe thermodynamic temperature scale, or absolute temperature scale is defined so that [latex]\\left(\\displaystyle\\frac{Q_L}{Q_H}\\right)_{rev} = \\displaystyle\\frac{T_L}{T_H}[\/latex].\r\n\r\n \r\n\r\nIn summary, the thermal efficiency of an actual heat engine is always less than that of the Carnot heat engine. It is impossible for any heat engine to achieve a better performance than the Carnot heat engine operating between the same heat source at [latex]T_H[\/latex] and the same heat sink at [latex]T_L[\/latex].\r\n