***********************
Compressions/Expansions
***********************

.. _Sec:chap3:technicalWork:

Expression of technical work
============================
For a **closed system** (cf. thermodynamics course), the work of reversible pressure forces for a transformation between state 1 and 2 expresses as:

.. math:: 
  :label: workClosedSystem

  w_{12} = -\int_1^2 pdv

For an **open system** (crossed by a fluid flow), the balance energy equation has been obtained in :numref:`Sec:chap1:BalanceEquations` and reads per unit of mass crossing the system:

.. math:: 
  :label: balanceEnergySpecific

  h_2 - h_1 = w_{t,12} + q_{12}

During the transformation and thanks to the *Gibbs equation*, we also have:

.. math::
  :label: gibbs

  dh = Tds + vdp

Such that between state 1 and 2:

.. math::
  :label: gibbsInt

  h_2 - h_1 = \int_1^2 Tds + \int_1^2 vdp

Relations :eq:`balanceEnergySpecific` and :eq:`gibbsInt` applied to a reversible transformation allows to express the technical work :math:`w_t`:

.. math:: 
  :label: techWorkInt

  w_{t,12} = \int_1^2 vdp

.. important::

  For a **reversible transformation**:

  * The specific pressure work in a *closed system* expresses as:

      .. math:: 

        w_{12} = - \int_1^2 pdv

  * The specific technical work in an *open system* expresses as:
    
      .. math:: 

        w_{t,12} = \int_1^2 vdp

According to the system under study, the work will be:

  * The opposite of the aera under the curve in (p,v) plane for *closed systems*
  * The aera on the left of the curve in the (p,v) plane for *open systems*

.. _Fig:chap3:workInClapeyron:

.. figure:: ./_static/fig/chap3/workInClapeyron.png
  :scale: 30%
  :align: center

  work for a closed system transformation between state 1 and 2 (pink aera) and for an open system between same states (green stripes aera).

Reversible compression/expansion
================================

Reversible compressions (or expansions) can be:

  * Isotherme: :math:`T=cte`
  * Polytropic: :math:`pv^k=cte`
  * Isentropic: :math:`s=cte`

A representation of these transformations in the *Clapeyron* diagram is presented in figure :numref:`Fig:chap3:compClapeyron`

.. _Fig:chap3:compClapeyron:

.. figure:: ./_static/fig/chap3/compClapeyron.png
  :scale: 30%
  :align: center

  Isotherme, Polytropic and Isentropic reversible compressions in (p,v).

For **ideal gas** equation of state, the expressions of specific works are recalled hereafter.

Isothermal transformation
-------------------------
.. math::
  :label: workIGIsothermal

  w_{12}^T = w_{t,12}^T = rT ln \left( \frac{p_2}{p_1} \right)

Isentropic transformation
-------------------------
.. math::
  :label: workIGIsent

  w_{12}^s = \frac{rT_1}{\gamma-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{\gamma-1}{\gamma}} -1  \right]

and 

.. math::
  :label: techWorkIGIsent

  w_{t,12}^s = \frac{\gamma rT_1}{\gamma-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{\gamma-1}{\gamma}} -1  \right]


Polytropic transformation
-------------------------
.. math::
  :label: workIGPolyp

  w_{12}^p = \frac{rT_1}{k-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{k-1}{k}} -1  \right]

and 

.. math::
  :label: techWorkIGPolyp

  w_{t,12}^p = \frac{krT_1}{k-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{k-1}{k}} -1  \right]

.. _Sec:chap3:efficiencies:

Real Compressions/Expansions
============================
If during a real compression (or expansion) the transformation remains *irreversible*, calculs are performed thanks to associated reversible transformations and the knowledge of corresponding *efficiencies* that are known as manufacturer's data.

For a *compression*, real work (positive) is always greater than the corresponding theoretical reversible transformation, such that we define the **compression efficiency** of a compressor by:

.. math::
  :label: compEfficiency

  \eta_{C} = \frac{w_{t,rev}}{w_{t,real}} \leq 1

For an *expansion*, the reversed situation appears as the technical work provided by the machine is always lower than the corresponding theoretical reversible transformation such that we define the **expansion efficiency** of a turbine by:

.. math::
  :label: turbEfficiency

  \eta_{T} = \frac{w_{t,real}}{w_{t,rev}} \leq 1

Thus depending on insulating of compressor/turbine, one will talk about isentropic, isothermic or polytropic efficiency.

Adiabatic transformations/ isentropic efficiencies
--------------------------------------------------
When the compression/expansion machine is *thermically insulated*, we use the **isentropic efficiency** that provides the ratio between the real work and those associated to the corresponding reversbile transformation (isentropic). The real transformation induces an increase of the entropy.

.. _Fig:chap3:isentropicEfficiency:

.. figure:: ./_static/fig/chap3/isentropicEfficiency.png
  :scale: 30%
  :align: center

  Adiabatic compression and expansion. Comparaison between isentropic path and real path.

Because the transformations are adiabatics, the *isentropic efficiencies* for compressors or turbines become:

.. math::
  :label: IsentropicEfficiencies

  \eta_{C}^S = \frac{h_{t,1}^S-h_{t,0}}{h_{t,1}-h_{t,0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{h_{t,2}-h_{t,0}}{h_{t,2}^S-h_{t,0}}

.. admonition:: Simplifications

  - When Kinetic energy variations are negligible
  - When the working fluid is consiered as an *ideal gas*

  *Isentropic efficiencies* become:

  .. math::

    \eta_{C}^S = \frac{T_{1}^S-T_{0}}{T_{1}-T_{0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{T_{2}-T_{0}}{T_{2}^S-T_{0}}

  Using *Laplace* relations for isentropic transformation, we get:

  .. math::
    :label: IsentropicEfficienciesIG

    \eta_{C}^S = \frac{T_{0} r_p^{\frac{\gamma-1}{\gamma}}-T_{0}}{T_{1}-T_{0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{T_{2}-T_{0}}{T_{0} r_p^{\frac{\gamma-1}{\gamma}}-T_{0}}

  Then, it is possible to determine easily the final temperature of the working fluid by knowledge of the pressure ratio (:math:`r_p = p_{final}/p_{init}`) and isentropic efficiencies.

Non adibatic transformations/ polytropic efficiencies
-----------------------------------------------------
When the compression/expansion is not *thermically insulated*, isentropic efficiencies are no longer valid and we use the **polytropic efficiency** providing the ratio between the real work and those associated to the corresponding reversible polytropic transformation leading to the same thermodynamic state at the end of compression/expansion.

.. _Fig:chap3:polytropicEfficiency:

.. figure:: ./_static/fig/chap3/polytropicEfficiency.png
  :scale: 30%
  :align: center

  Non adiabatic compressions and expansions. Comparaison between polytropic path (lines) and real path (dash lines).

Possible transformation with heat exchanges during compression/expansions can lead to different possibilities summarized in :numref:`Fig:chap3:polytropicEfficiency`. Whatever the case is, we have for the polytropic (:math:`p`) and real transformation:

.. math::

  w_{t}^P = (h_{t,i}-h_{t,0}) - q_P \qquad \text{and} \qquad w_{t} = (h_{t,i}-h_{t,0}) - q

Such that the *polytropic efficiencies* express as:

.. math::
  :label: PolytropicEfficiencies

  \eta_{C}^P = \frac{(h_{t,1}-h_{t,0})-q_P}{(h_{t,1}-h_{t,0})-q} \qquad \text{and} \qquad \eta_{T}^P = \frac{(h_{t,2}-h_{t,0}) - q}{(h_{t,2}-h_{t,0}) -q_P}

.. admonition:: Simplifications

  - When Kinetic energy variations are negligible
  - When the working fluid is consiered as an *ideal gas*

  The technical work for a polytropic transformation is given by relation :eq:`techWorkIGPolyp`, and the real technical work expresses as:

  .. math::

    w_t = (h_{t,i}-h_{t,0}) - q = \frac{\gamma r}{\gamma -1} T_0 (r_p^{\frac{k-1}{k}}-1) - q

  *Polytropic efficiencies* thus become:

  .. math::
    :label: PolytropicEfficienciesIG

    \eta_{C}^P = \frac{ \frac{k}{k-1} (r_p^{\frac{k-1}{k}}-1)} { \frac{\gamma }{\gamma -1} (r_p^{\frac{k-1}{k}}-1) - \frac{q (\gamma-1)}{r T_0} } \qquad \text{and} \qquad \eta_{T}^P = \frac{  \frac{\gamma }{\gamma -1} (r_p^{\frac{k-1}{k}}-1) - \frac{q (\gamma-1)}{r T_0} } {\frac{k}{k-1} (r_p^{\frac{k-1}{k}}-1)}