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Cycles
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Thermal machines cycles and efficiency (def)
============================================
We consider in this section thermal machines working between two thermal reservoirs. In these machines, the *working fluid* is the thermodynamic system that exchange thermal energies between a hot reservoir (at temperature :math:`T_H`) and a cold one (at temperature :math:`T_C`). 
Following results of :numref:`Sec:chap2:EnergyConversion`, it can supply mechanical energy (it is thus a *heat engine*) or absorb mechanical energy (it can be a *heat pump* or *refrigerator*).

Evolutions during a cycle
-------------------------
In a thermal machine, the *working fluid* is evolving on a thermodynamic cycle. Let us consider a transformation :math:`a` from a state 1 to a state 2 as represented on :numref:`Fig:chap2:cycle`

.. _Fig:chap2:cycle:

.. figure:: ./_static/fig/chap2/cycle.png
  :scale: 30%
  :align: center

  Entropic diagram representing thermal energy exchanged during a cycle.

During reversible transformation from 1 to 2, the heat exchanged is positive and corresponds to the hatched area. During this transformation, the fluid is absorbing thermal energy :math:`Q_{1\rightarrow 2}>0`.

For the return from state 2 to 1, the fluid will supply thermal energy :math:`Q_{2\rightarrow 1}<0`. Thus two possibilities occurs:

  * The fluid follows transformation :math:`m` (in blue): :math:`|Q_{2\rightarrow 1}|<|Q_{1\rightarrow 2}|` such that the thermal energy exchanged during cycle is *positive*. The application of first principle immplies that the mechanical work will be negative: this is a **heat engine**.
  * The fluid follows transformation :math:`h` (in red): :math:`|Q_{2\rightarrow 1}|>|Q_{1\rightarrow 2}|` such that the thermal energy exchanged during cycle is *negative*. The application of first principle immplies that the mechanical work will be positive: this can be a **heat pump** or a **refirgerators**.

From this observation, one can observe several things:

.. admonition:: Remarks

  * When a cycle is followed clockwise in the entropic diagram (T,S), the cycle is describing a *heat engine*
  * When a cycle is followed reversed clockwise in the entropic diagram (T,S), the cycle is describing a *heat pump* or a *refrigerator*
  * Any cycle can be used as a basis to develop a thermal machine.

Energy conversion efficiency
----------------------------
The economice notion of efficiency is generally defined by the ratio bewteen the *usefull energy provided* by the machine and its *energy absorption*, or in other words the ratio between benefits of a thermodynamic system and costs:

.. math::

  \eta \equiv \frac{\text{benefits}}{\text{costs}}

Heat engine and thermal efficiency
**********************************
For a heat engine, a mechanical energy :math:`-W` is produced (benefit) from a thermal energy absorption from *hot source* :math:`Q_H` (cost). This defines the **thermal efficiency** of the *heat engine*:

.. math::

  \eta_{th} = \frac{-W}{Q_H}

Thanks to the first principle, it can also be written as:

.. math::
  :label: thEfficiency

  \eta_{th} = \frac{Q_H + Q_C}{Q_H} = 1 + \frac{Q_C}{Q_H} = 1 - \frac{|Q_C|}{|Q_H|}

.. admonition:: Remark

  * For internal combustion engines, the thermal efficiency is about 25% for gasoline engines and 40% for diesel engines. For gas turbine, the thermal efficiency can reach 60%.
  * A great part of chemical energy (from fuel combustion) is then dissipated in heat and released in environnement (to the *cold sink*).

Heat generators and coefficient of performance (COP)
****************************************************
For a *heat pump* or *refrigerators*, a mechanical energy :math:`W` is consumed (cost) and the benefits depend on the usefull thermal energy exchanged. Because these ratio are generally greater than 1, we talk about **coefficient of performance**:

For a *heat pump* : 

.. math::
  :label: COPhp

  COP_{HP} = \frac{-Q_H}{W} = \frac{Q_H}{Q_H+Q_C} = \frac{1}{1 + \frac{Q_C}{Q_H}} = \frac{1}{1 - \frac{|Q_C|}{|Q_H|}}

For a *refrigerator* : 

.. math::
  :label: COPr

  COP_{R} = \frac{Q_C}{W} = \frac{Q_C}{-Q_H-Q_C} = \frac{1}{-\frac{Q_H}{Q_C}-1} = \frac{1}{\frac{|Q_H|}{|Q_C|}-1}

Carnot cycle
============
If a thermal machine is working between two TER in a reversible maneer, then it must exchange with both TER following isothermal transformation. These two isothermal transformation can thus be linked using:

  - 2 isentropes: this is the **Carnot cycle**
  - 2 isochores: this is the **Stirling cycle**
  - 2 isobares: this is the **Ericsson cycle**

These three cycles are ideal cycles for a thermal machine working between two TER and serve as reference cycle providing the maximum thermal efficiencies/COP.

.. _Sec:chap2:Carnot:

Carnot heat engine
------------------

.. _Fig:chap2:CarnotCycle:

.. figure:: ./_static/fig/chap2/CarnotCycle.png
  :scale: 30%
  :align: center

  Carnot cycle (for a heat engine).

It is easy to remark that in the reversible isothermes, we have :math:`Q_C = T_C (S_1-S_2)` and :math:`Q_H = T_H (S_2-S_1)`, such that thanks to equation :eq:`thEfficiency`, it is easy to show that:

.. math::
  :label: CarnotThEff

  \eta_{th,C} = 1 - \frac{T_C}{T_H}

As :math:`T_H>T_C`, the carnot thermal efficiency is always lower than 1.

.. admonition:: Remark

  * The Carnot thermal efficiency gives the maximum efficiency that can be reached by a heat engine between two TER.
  * For example, the Carnot thermal efficiency of a thermal power plant beteen :math:`T_H = 1000 K` (combustion temperature) and :math:`T_C = 300 K` (ambiant) is about 70%. Real thermal efficiency of such power plant reaches 40% because of non reversibility.

.. _Sec:chap2:CarnotHeatGen:

Carnot heat pump/refrigerators
------------------------------

.. _Fig:chap2:CarnotCycleGen:

.. figure:: ./_static/fig/chap2/CarnotCycleGen.png
  :scale: 30%
  :align: center

  Carnot cycle (for a heat generator: heat pump or refrigerator).

This times, in the reversible isothermes, we have :math:`Q_C = T_C (S_2-S_1)` and :math:`Q_H = T_H (S_1-S_2)`.

If this is a heat pump, the coefficient of performance :eq:`COPhp` becomes:

.. math::
  :label: CarnotCOPhp

  COP_{HP,C} = \frac{1}{1 - \frac{T_C}{T_H}} > 1

If this is a refrigerator, the coefficient of performance :eq:`COPr` becomes:

.. math::
  :label: CarnotCOPr

  COP_{R,C} = \frac{1}{\frac{T_H}{T_C}-1}

If the :math:`COP_{HP,C}` of a heat pump is greater than 1, we cannot conclude for a refrigerator.

Other cycles
============
It is shown that the thermal efficiency of a Carnot's cycle is the better thermal efficiency a heat engine can reach between two TER. It is let to the reader to apply the same reasonement to a heat pump or refrigerator.

Heat engine Reversible cycles
-----------------------------
If we consider any **reversible cycle**, there are two possibilities:

  1. If the cycle is working between the two TER :math:`T_C` and :math:`T_H`, then it will have the same thermal efficiency as Carnot cycle.
  2. Otherwise, it can be decomposed in an infinity of reversible cycles :math:`i` working between two TER whith temperatures :math:`T_{Ci} > T_C` and :math:`T_{Hi} < T_H` who have a thermal efficiency lower than Carnot efficiency between :math:`T_C` and :math:`T_H`. As a result, the global efficiency will be lower than Carnot's efficiency.

Heat engine irreversible cycles
-------------------------------
Thanks to second principle of thermodynamics, a *heat engine* that exchange between two TER during a cycle will guarantee that (equation :eq:`secondPpeCycle`):

.. math::

  \frac{Q_H}{T_H} + \frac{Q_C}{T_C} \le 0 \text{ (0 if reversible cycle)}

That can be rewritten as:

.. math::

  \frac{|Q_H|}{T_H} - \frac{|Q_C|}{T_C} \le 0

Or:

.. math::

  -\frac{T_C}{T_H} \ge -\frac{|Q_C|}{|Q_H|}

And finally, thanks to relation :eq:`thEfficiency`, the thermal efficiency of a heat pump reads:


.. math::

    \eta_{th} = 1 - \frac{|Q_C|}{|Q_H|} \le 1 - \frac{T_C}{T_H}

In other words, the thermal efficiency of any irreversible cycle between two TER will be lower than the Carnot's thermal efficiency.

Importance of Carnot cycle
==========================

.. caution::

	**Carnot cycle**, as any reversible cycle working between two TER (Ericsson, Stirling, etc.) may serve as reference for real cycles describing thermal machines :

* In a *heat engine*, thermal exchanged cannot be represented by isothermal transformations. It is more often isobaric/isochore transformations (or a combination of both).
* In a *heat pump* (or *refrigerator*), it is possible to be close to isothermal transformations during heat exchanges using phase change properties of a liquid-vapor couple (:math:`p^{sat}(T)`).
* Whatever the system is, transformations will be non reversibles.