.. _Sec:chap3:optimisation:

*************************************
Optimisation of Compression/Expansion
*************************************

As presented in :numref:`Sec:chap3:technicalWork`, the technical work can be represented by the aera on the left of the curve corresponding to the reversible transformation in the *Clapeyron* diagramm.

.. _Fig:chap3:optimisationWork:

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

  For a compression, the required technical work is minimum for the isothermal transformation. For an expansion, the provided technical work is maximum for the isothermal transformation.

The lower mechanical work needed for a compressor is the better and the bigger is the mechanical work supplied by a turbine is the better.

.. important::

  As shown on :numref:`Fig:chap3:optimisationWork`, it is clear that:

    * For a compressor, optimisation will be obtain by **cooling**
    * For a turbine, optimisation will be obtain by **heating**

Compression optimisation
========================
The minimum *technical work* for a compressor corresponds to isothermal transformation. There are two techniques to approach isothermal transformation in a compressor:

  * Continuous cooling of the compressor 
  * Multi-stage compressions with intercooling

Continuous cooling
------------------
**Continuous cooling** can be performed with surrounding the compressor with a cooling pipe. It is possible to quantify the cooling interest of a compression by comparison between the technical work needed for an isentropic and those obtained for an isothermal transformation. We define the **theoretical cooling efficiency**:

.. math::
  :label: coolingEfficiency

  e_{th,cool} = \frac{w_{t}^S}{w_t^T}

Under *ideal gas* assumption, using relations :eq:`techWorkIGIsent` and :eq:`workIGIsothermal`, it becomes:

.. math::
  :label: coolingEfficiencyIG

  e_{th,cool} = \frac{\gamma}{\gamma-1} \frac{r_p^{\frac{\gamma-1}{\gamma}}-1}{ln (r_p)}

Where :math:`r_p=p_2/p_1` represent the compression *pressure ratio*.

.. _Fig:chap3:coolingEfficiency:

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

  For a pressure ratio of 14, the mechanical energy earned for an isothermal compression is 50% compared to an isentropic one.

Efficiency of cooling is theroretical since it is difficult to ensure an isothermal compression using continuous cooling and heat transfer through the compressor carter.

Multi-stage cooling
-------------------
**Multi-stage cooling** is often used for multi-stage compression. It consists in cooling the fluid using a heat exchanger between two compression stages.

.. _Fig:chap3:twoStagesCompression:

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

  Two-stage compression. Transformations in compressors are polytropic (adiabatic or not) and a constant pressure heat exchange occurs between the two stages of compression. The earned work is clearly visible on the *Clapeyron* diagramm.

How to choose the compression ratio of each stage ?

The technical work for the global two-stage compression is:

.. math::

  w_{t} = w_{t}^{C_1} + w_{t}^{C_2}

If the two compression stages perform a polytropic compression, thanks to :eq:`techWorkIGPolyp` we get:

.. math::

  w_{t} = \frac{krT_0}{k-1} \left[ \left( \frac{p_X}{p_0} \right)^{\frac{k-1}{k}} -1  \right] + \frac{krT_0}{k-1} \left[ \left( \frac{p_1}{p_X} \right)^{\frac{k-1}{k}} -1  \right]

Derivation of this expression using :math:`p_X` variable leads to:

.. math::

  \frac{d w_{t}}{d p_X} = \frac{krT_0}{k-1} p_X^{\frac{1}{k-1}} \left[ p_0^{\frac{-k}{k-1}} - p_1^{\frac{k}{k-1}} p_X^{\frac{-2k}{k-1}}  \right]

such that the technical work is minimum for:

.. math::

  p_X = (p_0 p_1)^{1/2} \qquad \text{i.e.} \qquad  \frac{p_X}{p_0} = \frac{p_1}{p_X}

This result can be extand to multi-stages compressions.

.. important::

  The pressure ratio in each stage of a **multi-stage compression with intercooling** should be equal in order to minimize the technical work.

.. _Fig:chap3:multiStagesCompression:

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

  Multi-stage compression with intercooling.

Turbine optimisation
====================
The maximum *technical work* for a turbine correspond to isothermal transformation. This is generally obtain by multi-stage expansions with **reheat**.

Conclusions are similar than those obtain with multi-stage compression with intercooling and are not developped here.