.. _sphx_glr_api_gallery_thermodynamics_cycles:

Cycles
------

Heat engines: Carnot's bound and the idealized cycles of real engines.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Sadi Carnot (1824) argued that a heat engine&#x27;s work comes from heat falling from a hot body to a cold one, as a water wheel&#x27;s comes from falling water, and that a reversible engine between two temperatures is the best possible. If some engine did better, it could drive the reversible engine backwards as a heat pump and, with no work left over, move heat from cold to hot. In modern form the bound is \eta_C = 1 - T_c/T_h (~engineeringkit.thermodynamics.carnot_efficiency), with absolute temperatures. The example runs Carnot&#x27;s argument with numbers and shows how the bound rises with the boiler temperature.">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_01_carnot_efficiency_thumb.png
    :alt:

  :doc:`/api/gallery/thermodynamics/cycles/plot_01_carnot_efficiency`

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      <div class="sphx-glr-thumbnail-title">Carnot's Réflexions: no engine beats the reversible one (1824)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="William Rankine&#x27;s Manual of the Steam Engine (1859) analysed the steam engine as a cycle of four steady-flow steps: a pump raises condensed water to boiler pressure, the boiler turns it into superheated steam, the steam expands through the engine (now a turbine), and a condenser turns the exhaust back into water. Because the water is pumped as a liquid, the pump needs about 1% of the turbine work, which is the cycle&#x27;s great advantage over gas cycles. The efficiency (~engineeringkit.thermodynamics.rankine_ideal_efficiency) is computed here with IAPWS-IF97 steam properties (~engineeringkit.thermodynamics.steam_properties) for a plant at 10 MPa and 500 °C condensing at 10 kPa.">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_02_rankine_cycle_thumb.png
    :alt:

  :doc:`/api/gallery/thermodynamics/cycles/plot_02_rankine_cycle`

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      <div class="sphx-glr-thumbnail-title">Rankine's steam cycle (1859)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="George Brayton&#x27;s &quot;Ready Motor&quot; (patented 1872) compressed air in one cylinder, burned fuel in it at constant pressure, and expanded the gases in another. The engine itself lost out to Otto&#x27;s, but its cycle (also called Joule&#x27;s) is the cycle of every gas turbine and jet engine: isentropic compression, heat addition at constant pressure, isentropic expansion and heat rejection at constant pressure. With a calorically perfect gas the efficiency depends only on the pressure ratio, \eta = 1 - r_p^{(1-\gamma)/\gamma} (~engineeringkit.thermodynamics.brayton_efficiency). The example rebuilds that result from the four state temperatures.">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_03_brayton_cycle_thumb.png
    :alt:

  :doc:`/api/gallery/thermodynamics/cycles/plot_03_brayton_cycle`

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      <div class="sphx-glr-thumbnail-title">Brayton's constant-pressure cycle (1872)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Nicolaus Otto&#x27;s 1876 engine compressed its charge before igniting it, a principle Alphonse Beau de Rochas had described in 1862. Its four strokes (intake, compression, power, exhaust) are idealized as the air-standard Otto cycle: isentropic compression through a ratio r, heat added at constant volume, isentropic expansion and heat rejected at constant volume. The efficiency 1 - r^{1-\gamma} (~engineeringkit.thermodynamics.otto_efficiency, OttoCycle) depends only on the compression ratio. Otto&#x27;s first engine had r \approx 2.5; knock limits petrol engines to about 10-12 today.">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_04_otto_cycle_thumb.png
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  :doc:`/api/gallery/thermodynamics/cycles/plot_04_otto_cycle`

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      <div class="sphx-glr-thumbnail-title">Otto's four-stroke cycle (1876)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Rudolf Diesel&#x27;s Theorie und Konstruktion eines rationellen Wärmemotors (1893) set out to approach Carnot&#x27;s efficiency by compressing air alone so hard that injected fuel ignites by itself. Because there is no fuel in the cylinder during compression, a diesel cannot knock and can use compression ratios of 15-22. The fuel burns while the piston moves, which the air-standard Diesel cycle idealizes as heat added at constant pressure up to a cutoff ratio r_c = v_3/v_2 (~engineeringkit.thermodynamics.diesel_efficiency).">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_05_diesel_cycle_thumb.png
    :alt:

  :doc:`/api/gallery/thermodynamics/cycles/plot_05_diesel_cycle`

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      <div class="sphx-glr-thumbnail-title">Diesel's compression-ignition cycle (1893)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A Carnot engine reaches its efficiency only by running infinitely slowly, so it delivers no power. Frank Curzon and Boye Ahlborn (1975), after Novikov and Chambadal (1957), let a reversible engine exchange heat with its reservoirs through finite conductances, \dot Q = K\,\Delta T. Run slowly, it is efficient but weak; run fast, the temperature drops eat the whole temperature difference. At maximum power the efficiency is 1 - \sqrt{T_c/T_h} (~engineeringkit.thermodynamics.curzon_ahlborn_efficiency), whatever the conductances, and it describes real power plants better than Carnot&#x27;s bound does.">

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  .. image:: /api/gallery/thermodynamics/cycles/images/thumb/sphx_glr_plot_06_curzon_ahlborn_thumb.png
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  :doc:`/api/gallery/thermodynamics/cycles/plot_06_curzon_ahlborn`

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      <div class="sphx-glr-thumbnail-title">Curzon and Ahlborn: efficiency at maximum power (1975)</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/thermodynamics/cycles/plot_01_carnot_efficiency
   /api/gallery/thermodynamics/cycles/plot_02_rankine_cycle
   /api/gallery/thermodynamics/cycles/plot_03_brayton_cycle
   /api/gallery/thermodynamics/cycles/plot_04_otto_cycle
   /api/gallery/thermodynamics/cycles/plot_05_diesel_cycle
   /api/gallery/thermodynamics/cycles/plot_06_curzon_ahlborn