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Gas heat pump and refrigeration cycle
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*Gas heat pumps* and *gas regrigerators* are basically following the reverse **Brayton cycle** (or Joule cycle).

.. _Fig:chap5:BraytonReverse:

.. figure:: ./_static/fig/chap5/BraytonReverse.png
  :scale: 30%
  :align: center

  Reverse *Brayton* cycle for gas refrigeration.

As for any heat pump or refrigerator, (:eq:`COPr`, and :eq:`COPhp`) the *coefficients of performance* of *Reverse Brayton cycle* are:

.. math::

  COP_{R,Joule} = \frac{1}{\frac{|q_{23}|}{|q_{41}|}-1} \qquad \text{and} \qquad COP_{HP,Joule} = \frac{1}{1-\frac{|q_{41}|}{|q_{23}|}}

Considering the working fluid as an **ideal gas**, and because no machine is working in transformations 2-3 and 4-1, application of the balance energy equation reads:

.. math::

  |q_{23}| = \dot{m} c_p (T_2-T_3) \qquad \text{ and } \qquad |q_{41}| = \dot{m} c_p (T_1-T_4) 

such that the COPs of *Reverse Brayton cycle* become:

.. math::

  COP_{R,Joule} = \frac{1}{\frac{T_2-T_3}{T_1-T_4}-1} \qquad \text{and} \qquad COP_{HP,Joule} = \frac{1}{1-\frac{T_1-T_4}{T_2-T_3}}

As for gas turbine, it is possible to determine gas temperatures after the compressor and the turbine thanks to isentropic relations for ideal gases:

.. math::

  \frac{T_1}{T_2} = \frac{p_2}{p_1}^{\frac{1-\gamma}{\gamma}} = \frac{p_3}{p_4}^{\frac{1 - \gamma}{\gamma}} = \frac{T_4}{T_3}

we finally obtain:

.. math::
  :label: COPEffBraytonIGRev

  COP_{R,Joule} = \frac{1}{r_p^{\frac{\gamma-1}{\gamma}}-1} \qquad \text{and} \qquad COP_{HP,Joule} = \frac{1}{1-r_p^{\frac{1-\gamma}{\gamma}}}

where :math:`r_p=\frac{p_2}{p_1}` is the **pressure ratio**.

.. _Fig:chap5:COPJoule:

.. figure:: ./_static/fig/chap5/COPJoule.png
  :scale: 50%
  :align: center

  COP for *reverse Brayton cycle* (Joule cycle) for heat pump or refrigerators as a function of the pressure ratio for IG with :math:`\gamma=1.4`.

:numref:`Fig:chap5:COPJoule` reveals that the performance of such machine is good for low pressure ratio.

.. admonition:: Remarks

  * As for gas turbine, it is possible to easily account for the adiabatic compressor and adiabatic turbine irreversibilities by using the isentropic efficiency of both components (see :numref:`Sec:chap4:RealGasTurbineCycle`).
  * It is also possible to improve the cycle using **regeneration** (see :numref:`Sec:chap4:regeneration`)
  * Heat pumps and Refrigerators using the *reverse Brayton cycle* have low performances and are used only for specific applications.