{"id":17525,"date":"2018-04-04T17:08:50","date_gmt":"2018-04-04T17:08:50","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=17525"},"modified":"2022-11-12T08:55:06","modified_gmt":"2022-11-12T08:55:06","slug":"carnot-cycle-carnot-heat-engine","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-engineering\/thermodynamics\/thermodynamic-cycles\/carnot-cycle-carnot-heat-engine\/","title":{"rendered":"Carnot Cycle – Carnot Heat Engine"},"content":{"rendered":"
<\/div>\n
Carnot cycle is a theoretical cycle with the highest possible efficiency of all thermodynamic cycles. In a Carnot cycle<\/strong>, the system executing the cycle undergoes a series of four internally reversible processes<\/strong>: two isentropic processes<\/a><\/strong> (reversible adiabatic) alternated with two<\/strong> isothermal processes<\/a>.<\/strong>\n<\/div><\/div>\n
\"pV<\/a>
pV diagram of Carnot cycle. The area bounded by the complete cycle path represents the total work done during one cycle.<\/figcaption><\/figure>\n

The second law of thermodynamics<\/strong><\/a> places constraints upon the direction of heat transfer and sets an upper limit to the efficiency<\/a><\/strong> of conversion of heat to work in heat engines<\/strong><\/a>. So the second law is directly relevant for many important practical problems.<\/p>\n

In 1824, a French engineer and physicist, Nicolas L\u00e9onard Sadi Carnot,<\/strong> advanced the study of the second law by forming a principle (also called Carnot\u2019s rule<\/strong><\/a>) that specifies limits on the maximum efficiency any heat engine<\/strong> can obtain. In short, this principle states that the efficiency of a thermodynamic cycle<\/strong> depends solely on the difference between the hot and cold temperature reservoirs.<\/p>\n

Carnot\u2019s principle states:<\/strong><\/p>\n

    \n
  1. No engine can be more efficient than a reversible engine (Carnot heat engine<\/strong>) operating between the same high-temperature and low-temperature reservoirs.<\/li>\n
  2. The efficiencies of all reversible engines (Carnot heat engines<\/strong>) operating between the same constant temperature reservoirs are the same, regardless of the working substance employed or the operation details.<\/li>\n<\/ol>\n

    The cycle of this engine is called the Carnot cycle<\/strong>. A system undergoing a Carnot cycle<\/strong> is called a Carnot heat engine<\/strong>. It is not an actual thermodynamic cycle but is a theoretical construct and cannot be built in practice. All real thermodynamic processes are somehow irreversible<\/strong><\/a>. They are not done infinitely slowly,<\/strong> and infinitesimally small steps<\/strong> in temperature are also theoretical fiction. Therefore, heat engines must have lower efficiencies than limits on their efficiency due to the inherent irreversibility of the heat engine cycle they use.<\/p><\/div><\/div>

    <\/div>\n

    Carnot Cycle – Processes<\/h2>\n

    \"Carnot<\/a>In a Carnot cycle<\/strong>, the system executing the cycle undergoes a series of four internally reversible processes<\/strong>: two isentropic processes<\/a><\/strong> (reversible adiabatic) alternated with two<\/strong> isothermal processes<\/strong><\/a>:<\/p>\n

      \n
    1. Isentropic compression<\/strong> \u2013 The gas is compressed adiabatically from state 1 to state 2, where the temperature is TH<\/sub><\/strong>. The surroundings do work on the gas, increasing its internal energy and compressing it. On the other hand, the entropy<\/strong> remains unchanged<\/strong>.<\/li>\n
    2. Isothermal expansion<\/strong> \u2013 The system is placed in contact with the reservoir at TH<\/sub><\/strong>. The gas isothermally expands while receiving energy QH<\/sub> from the hot reservoir by heat transfer. The temperature of the gas does not change during the process. The gas does work on the surroundings. The total entropy change is given by: \u2206S = S<\/em><\/strong>1<\/sub><\/em><\/strong> \u2013 S<\/em><\/strong>4<\/sub><\/em><\/strong> = Q<\/em><\/strong>H<\/sub><\/em><\/strong>\/T<\/em><\/strong>H<\/sub><\/em><\/strong><\/li>\n
    3. Isentropic expansion<\/strong> \u2013 The gas expands adiabatically from state 3 to state 4, where the temperature is TC<\/sub><\/strong>. The gas works on the surroundings and loses an amount of internal energy equal to the work that leaves the system. Again the entropy remains unchanged.<\/li>\n
    4. Isothermal compression<\/strong> \u2013 The system is placed in contact with the reservoir at TC<\/sub><\/strong>. The gas compresses isothermally to its initial state while discharging energy QC<\/sub> to the cold reservoir by heat transfer. In this process, the surroundings do work on the gas. The total entropy change is given by: \u2206S = S<\/em><\/strong>3<\/sub><\/em><\/strong> \u2013 S<\/em><\/strong>2<\/sub><\/em><\/strong> = Q<\/em><\/strong>C<\/sub><\/em><\/strong>\/T<\/em><\/strong>C<\/sub><\/em><\/strong><\/li>\n<\/ol>\n<\/div><\/div>
      <\/div>\n

      Isentropic Process<\/h2>\n

      An isentropic process<\/strong><\/a> is a thermodynamic process<\/strong><\/a>\u00a0in which the entropy<\/strong> <\/a>of the fluid or gas remains constant. It means the isentropic process<\/strong> is a special case of an adiabatic process<\/strong> in which there is no transfer of heat or matter. It is a reversible adiabatic process<\/strong>. The assumption of no heat transfer is very important since we can use the adiabatic approximation only in very rapid processes<\/strong>.<\/p>\n

      Isentropic Process and the First Law<\/strong><\/p>\n

      For a closed system, we can write the first law of thermodynamics in terms of enthalpy<\/a><\/strong>:<\/p>\n

      dH = dQ + Vdp<\/strong><\/p>\n

      or<\/strong><\/p>\n

      dH = TdS + Vdp<\/strong><\/p>\n

      Isentropic process (dQ = 0):<\/strong><\/p>\n

      dH = Vdp \u00a0\u00a0\u00a0\u00a0\u2192 \u00a0\u00a0\u00a0\u00a0W = H<\/strong>2<\/sub><\/strong> \u2013 H<\/strong>1<\/sub><\/strong> \u00a0\u00a0\u00a0\u00a0\u2192 \u00a0\u00a0\u00a0\u00a0H<\/strong>2<\/sub><\/strong> \u2013 H<\/strong>1<\/sub><\/strong> = C<\/em><\/strong>p<\/sub><\/em><\/strong> (T<\/em><\/strong>2<\/sub><\/em><\/strong> \u2013 T<\/em><\/strong>1<\/sub><\/em><\/strong>) \u00a0\u00a0\u00a0<\/em><\/strong>(for ideal gas<\/a>)<\/em><\/p>\n

      Isentropic Process of the Ideal Gas<\/strong><\/p>\n

      The isentropic process<\/strong> (a special case of the adiabatic process) can be expressed with the ideal gas law<\/strong><\/a> as:<\/p>\n

      pV\u03ba<\/sup> = constant<\/em><\/strong><\/p>\n

      or<\/p>\n

      p1<\/sub>V1<\/sub>\u03ba<\/sup> = p2<\/sub>V2<\/sub>\u03ba<\/sup><\/strong><\/em><\/p>\n

      in which \u03ba = cp<\/sub>\/cv<\/sub><\/strong> is the ratio of the specific heats<\/strong><\/a> (or heat capacities<\/strong>) for the gas. One for constant pressure (c<\/strong>p<\/sub><\/strong>)<\/strong> and one for constant volume (c<\/strong>v<\/sub><\/strong>)<\/strong>. Note that, this ratio \u03ba\u00a0<\/strong> = cp<\/sub>\/cv<\/sub><\/strong> is a factor in determining the speed of sound in a gas and other adiabatic processes.<\/p><\/div><\/div>

      <\/div>\n

      Isothermal Process<\/h2>\n

      An isothermal process<\/strong> is a thermodynamic process<\/strong><\/a>\u00a0in which the system\u2019s temperature<\/b>\u00a0remains constant<\/strong> (T = const). The heat transfer into or out of the system typically must happen at such a slow rate to continually adjust to the temperature of the reservoir through heat exchange. In each of these states, the thermal equilibrium<\/strong><\/a> is maintained.<\/p>\n

      Isothermal\u00a0Process and the First Law<\/strong><\/p>\n

      The classical form of the first law of thermodynamics <\/a>is the following equation:<\/p>\n

      dU = dQ \u2013 dW<\/strong><\/p>\n

      In this equation, dW is equal to dW = pdV<\/strong> and is known as the boundary work<\/a>.<\/p>\n

      In the isothermal process<\/strong> and the ideal gas<\/strong><\/a>, all heat added to the system will be used to do work:<\/p>\n

      Isothermal process (dU = 0):<\/strong><\/p>\n

      dU = 0 = Q \u2013 W \u00a0\u00a0\u00a0\u2192 \u00a0\u00a0\u00a0\u00a0W = Q \u00a0\u00a0\u00a0\u00a0\u00a0<\/em><\/strong>(for ideal gas)<\/em><\/p>\n

      Isothermal Process of the Ideal Gas<\/strong><\/p>\n

      The isothermal\u00a0process\u00a0<\/strong>can be expressed with the ideal gas law<\/strong> as:<\/p>\n

      pV\u00a0= constant<\/em><\/strong><\/p>\n

      or<\/p>\n

      p1<\/sub>V1<\/sub>\u00a0= p2<\/sub>V2<\/sub><\/strong><\/em><\/p>\n

      On a p-V diagram, the process occurs along a line (called an isotherm) that has the equation p = constant \/ V<\/strong>.<\/p>\n

      See also: Boyle-Mariotte Law<\/a>.<\/p>\n<\/div><\/div>

      \n
      \"Isentropic<\/a>
      Isentropic process –\u00a0main characteristics<\/figcaption><\/figure>\n<\/div><\/div>
      \n
      \"Isothermal<\/a>
      Isothermal process – main characteristics<\/figcaption><\/figure>\n<\/div><\/div>
      <\/div>\n

      Carnot Cycle – pV, Ts diagram<\/h2>\n
      \"Ts<\/a>
      Ts diagram of Carnot cycle. The area under the Ts curve of a process is the heat transferred to the system during that process.<\/figcaption><\/figure>\n

      The Carnot cycle is often plotted on a pressure-volume diagram<\/strong> (pV diagram<\/strong>) and a temperature-entropy diagram<\/strong> (Ts diagram<\/strong>).<\/p>\n

      When plotted on a pressure-volume diagram<\/strong>, the isothermal processes follow the gas\u2019s isotherm lines, adiabatic processes move between isotherms, and the area bounded by the complete cycle path represents the total work that can be done during one cycle.<\/p>\n

      The temperature-entropy diagram<\/strong> (Ts diagram), in which a point specifies the thermodynamic state on a graph with specific entropy<\/a> (s) as the horizontal axis and absolute temperature (T) as the vertical axis, is the best diagram to describe the behavior of a Carnot cycle<\/strong>.<\/p>\n

      It is a useful and common tool, particularly because it helps to visualize the heat transfer during a process. For reversible (ideal) processes, the area under the T-s curve of a process is the heat transferred to the system during that process.<\/p><\/div><\/div>

      <\/div>\n

      Carnot Cycle Efficiency<\/h2>\n

      In general, the thermal efficiency<\/strong><\/a>, \u03b7<\/em><\/strong>th<\/sub><\/em><\/strong>, of any heat engine is defined as the ratio of the network<\/a>\u00a0it does, W<\/strong>, to the heat<\/a> input at the high temperature, QH<\/sub>.<\/p>\n

      \"thermal<\/a><\/p>\n

      Since energy is conserved according to the first law of thermodynamics<\/strong> <\/a>and energy cannot be converted to work completely, the heat input, QH<\/sub>, must equal the work done, W, plus the heat that must be dissipated as waste heat Q<\/strong>C<\/sub><\/strong> into the environment. Therefore we can rewrite the formula for thermal efficiency as:<\/p>\n

      \"thermal<\/a><\/p>\n

      Since Q<\/em><\/strong>C<\/sub><\/em><\/strong> = \u2206S.T<\/em><\/strong>C<\/sub><\/em><\/strong> and Q<\/strong><\/em>H<\/sub><\/em><\/strong> = \u2206S.T<\/em><\/strong>H<\/sub><\/em><\/strong>, the formula for this maximum efficiency is:<\/p>\n

      \"Carnot<\/a><\/p>\n

      where:<\/p>\n