************
Applications
************

Balance equations presented in :numref:`Sec:chap1:BalanceEquations` are applied here to the constituting elements of engineering machines: nozzles, heat exchangers, compressors and turbines, etc. These elements are made for continuous processes (steady flow) and generally present one fluid entry and one fluid exit.

.. _Fig:chap1:twoFluidSec:

.. figure:: ./_static/fig/chap1/twoFluidSec.png
  :scale: 20%
  :align: center

  A fluid system with one entry and one exit

In that specific case, balance equations :eq:`massEquation` and :eq:`energyEquation` simplify in:

.. math:: 
	:label: massTwoFluidSec

	\dot{m_2} = - \dot{m_1} = \dot{m}

and 

.. math:: 
  :label: energyTwoFluidSec

  \dot{m} (h_{t,2} - h_{t,1}) = \dot{Q} + \dot{W}_{t}

Acceleration in a nozzle
========================
**Nozzles** can be found in gas turbine or on aircrafts/rocket engines. They are basic components used to accelerate/decelerate a flow.

.. _Fig:chap1:nozzle:

.. figure:: ./_static/fig/chap1/nozzle.png
  :scale: 30%
  :align: center

  Left: subsonic convergent nozzle, Middle: subsonic divergent nozzle, Right: Ariane'5 Vulcain engine nozzle.

In nozzle systems:

  * It is commonly accepted that no thermal energy is exchanged (:math:`\dot{Q}=0`) due to important fluid velocities. 
  * Moreover, no working machine is present (:math:`\dot{W}_{t}`)
  * Potential energy is negligible.

such that relation :eq:`energyTwoFluidSec` becomes:

.. math:: 
  :label: nozzleEq

  h_{2} - h_{1} = - \frac{1}{2}(u_2^2-u_1^2)

Heat exchanger
==============
**Heat exchangers** allow to exchange a thermal energy between two fluids without mixing. The simpler heat exchanger is the *double-tube* presented in :numref:`Fig:chap1:heatExchanger`.

.. _Fig:chap1:heatExchanger:

.. figure:: ./_static/fig/chap1/heatExchanger.png
  :scale: 30%
  :align: center

  Double-tube heat exchanger. The cold fluid is absorbing thermal energy provided by the hot fluid.

In heat exchanger systems:

  * Kinetic energy variation is commonly negligible.
  * Potential energy is negligible.
  * No working machine is present (:math:`\dot{W}_{t}`)

Such that for example if considering the cold fluid system, the balance energy equation :eq:`energyTwoFluidSec` becomes:

.. math:: 
  :label: heatExchangeEq

  \dot{m}_C (h_{2}^C - h_{1}^C) = \dot{Q}

If the heat exchanger is insulated, the hot fluid system balance energy will read:

.. math:: 

  \dot{m}_H (h_{2}^H - h_{1}^H) = -\dot{Q}

Compressor/Turbine
==================
These elements contains a rotary mechanical device to convert flow energy into mechanical work (turbine) and reversely (compressor). The mechanical work is transmitted thanks to a shaft.

.. _Fig:chap1:compTurb:

.. figure:: ./_static/fig/chap1/compTurb.png
  :scale: 30%
  :align: center

  Left: schematic representation of a compressor and a turbine. Right: multi-stage compressor.

In these elements, this is commonly accepted that:

  * Kinetic energy variation is negligible.
  * Potential energy negligible.
  * No heat exchanges unless they are cooled (or heated) :math:`\dot{Q} =0`.

Balance energy equation becomes:

.. math:: 
  :label: turbCompEq

  \dot{m} (h_{2} - h_{1}) = \dot{W}_{t}

In a **turbine**, a work is produced on the shaft (:math:`W_t < 0` because lost by the turbine), and the flow enthalpy is decreasing because of fluid expansion resulting in a lower pressure at the turbine exit than at the entry.

In a **compressor**, as for a pump or a ventilator, the fluid's enthalpy is increasing because of fluid compression resulting in an increase of flow pressure as a work is provided on the shaft (:math:`W_t > 0` because earned by the compressor).

.. _Sec:chap1:laminating:

Throttling Valves
=================
**Throttling valves** produce a pressure drop in a flow. It can be obtained thanks to adjustable valve or thanks to a porous.

.. _Fig:chap1:laminating:

.. figure:: ./_static/fig/chap1/laminating.png
  :scale: 20%
  :align: center

  A high pressure gas is expanded through a hole. This kind of expansion is isenthalpic.

Common hypothesis are:

  * No heat echanges (insulated walls),
  * No working machine,
  * Kinetic energy variation is negligible.

Such that the first principle reduces to:

.. math::
  :label: isenthalpicExpansion

  h_1 = h_2

If the fluid can be considered as ideal gas, the isenthalpic expansion is also isothermal:

.. math::

  T_1 = T_2