.. finite_area_comb

.. _`finite_area_comb`:

Finite Area Combustor
=====================

In a rocket engine, the ratio of chamber cross-sectional area to throat area is called
the **contraction ratio**,  **CR**.  

Unless otherwise directed, CEA runs rocket 
calculations assuming an infinite CR.

For an infinite contraction ratio, the pressure at the injector face, **Pcinj_face**, is the same as the 
pressure in the chamber combustion end plenum, **Pcomb_end**.::

    Pcinj_face = Pcomb_end       (where: CR = infinite)


In a real chamber, however, as the chamber cross-sectional area gets smaller,
(as CR gets closer to 1.0), the pressure drop from Pcinj_face to Pcomb_end increases.::

    Pcinj_face > Pcomb_end       (where: CR < infinite)


This pressure drop is called the Rayleigh line loss.
It is the stagnation pressure loss associated
with the heat transfer effects in a duct of constant area and is the locus
of points on an enthalpy-entropy diagram defined by the momentum equation,
continuity equation, and the equation of state. 

A discussion of this phenomenon is included in the classic design manual 
`Design of Liquid Propellant Rocket Engines by Huzel and Huang <https://ntrs.nasa.gov/search.jsp?N=0&Ntk=All&Ntx=mode+matchallany&Ntt=19710019929>`_ 
on page 6, which simplifies the ratio of the injector face pressure to the plenum pressure with the equation::

    Pcinj_face / Pcomb_end = 1 + gamma * MachNumber**2
    
The above equation, equation *(1-15)* in 
`Huzel and Huang  <https://ntrs.nasa.gov/search.jsp?N=0&Ntk=All&Ntx=mode+matchallany&Ntt=19710019929>`_ 
is shown in the right-hand image below.

.. image:: ./_static/compare_rayleigh.png
    :width: 60%

.. image:: ./_static/Pinj_over_Pc_Huzel_and_Huang.jpg
    :width: 39%

The above graph shows the CEA/RocketCEA calculation of the Rayleigh line loss as well as
a simple approximation equation for estimating that loss::

    Pcinj_face/Pcomb_end = 1.0 + 0.54 / CR**2.2

Since different propellant combinations have nearly identical Rayleigh line loss (see above graph),
it is often sufficient, for engineering purposes, to approximate the Rayleigh line loss
with a simple correlating equation such as the one shown here.

This is especially convenient when designing to a known plenum pressure, **Pcomb_end**,
and deriving the injector face pressure, **Pcinj_face**, that CEA and RocketCEA require 
as an input.

One could also get the chamber gamma and mach number from **RocketCEA** and plug those values
into the equation from 
`Huzel and Huang  <https://ntrs.nasa.gov/search.jsp?N=0&Ntk=All&Ntx=mode+matchallany&Ntt=19710019929>`_ 


CEA fac Option
--------------

The CEA program offers the option to calculate the Rayleigh line loss for you
by using the **fac** option. (The above chart was generated with RocketCEA using the fac option).

.. image:: ./_static/fac_manual_option.jpg
    :width: 65%

A traditional CEA run that sets **fac** has an extra column of data called
**COMB END** that indicates what the chamber plenum pressure, **Pcomb_end**, would be 
if the injector face pressure, **INJECTOR** pressure, were specified.
An example of that extra column is shown below.::

                     INJECTOR  COMB END  THROAT     EXIT
     Pinj/P            1.0000   1.0692   1.7921   473.77
     P, ATM            68.046   63.643   37.970  0.14363
     T, K             3483.35  3467.55  3288.16  1441.62
     RHO, G/CC       3.2038-3 3.0113-3 1.9141-3 1.7133-5
     H, CAL/G         -235.74  -253.44  -509.60 -2372.05
     U, CAL/G         -750.09  -765.27  -990.00 -2575.07
     G, CAL/G        -15090.3 -15057.2 -14547.5 -8526.66
     S, CAL/(G)(K)     4.2644   4.2692   4.2692   4.2692

     M, (1/n)          13.458   13.463   13.602   14.111
     (dLV/dLP)t      -1.02525 -1.02508 -1.01972 -1.00000
     (dLV/dLT)p        1.4496   1.4485   1.3717   1.0001
     Cp, CAL/(G)(K)    2.0951   2.0962   1.9277   0.7309
     GAMMAs            1.1401   1.1398   1.1401   1.2387
     SON VEL,M/SEC     1566.3   1562.3   1513.8   1025.8
     MACH NUMBER        0.000    0.246    1.000    4.122


In **RocketCEA** the fac option is implemented by specifying the fac contraction ratio, **fac_CR** 
when creating a CEA_Obj. For example:
    
All calls to the **ispObj** will assume the input contraction ratio, **fac_CR**, and
use the input **Pc** as the **Pcinj_face**.

For example, the above CEA output was generated with the code.

.. code-block:: python

    from rocketcea.cea_obj import CEA_Obj
    ispObj = CEA_Obj( oxName='LOX', fuelName='LH2', fac_CR=2.5)
    s = ispObj.get_full_cea_output( Pc=1000.0, MR=6.0, eps=40.0)
    print( s )

The chamber plenum pressure, **Pcomb_end**, will be determined by applying the 
Rayleigh line loss to **Pcinj_face**.

It is also possible to calculate **Pcinj_face / Pcomb_end** for any contraction ratio using the following:

.. code-block:: python

    PinjOverPcomb = ispObj.get_Pinj_over_Pcomb( Pc=Pc, MR=MR, fac_CR=CR )

The graph in the 1st section above was created using this approach.

.. literalinclude:: ./_static/example_scripts/compare_rayleigh.py

Specify Plenum Pressure
-----------------------

Since it is more common to specify a plenum pressure, **Pcomb_end**, 
and calculate an injector face pressure, **Pcinj_face**,
The following script will use **RocketCEA** to calculate the required **Pcinj_face** 
that gives **Pcomb_end**.

.. code-block:: python

    """
        figure out Pcinj_face to get desired Pcomb_end (100 atm in example)
    """
    from rocketcea.cea_obj import CEA_Obj

    cr = 2.5 # contraction ratio
    ispObj = CEA_Obj( oxName='LOX', fuelName='LH2', fac_CR=cr)

    # Use 100 atm to make output easy to read
    Pc = 100.0 * 14.6959

    # use correlation to make 1st estimate of Pcinj_face / Pcomb_end
    PinjOverPcomb = 1.0 + 0.54 / cr**2.2 

    # use RocketCEA to refine initial estimate
    PinjOverPcomb = ispObj.get_Pinj_over_Pcomb( Pc=Pc * PinjOverPcomb, MR=6.0 )

    # print results (noting that "COMB END" == 100.00 atm)
    s = ispObj.get_full_cea_output( Pc=Pc * PinjOverPcomb, MR=6.0, eps=40.0)
    print( s )

Output from the above script::

                     INJECTOR  COMB END  THROAT     EXIT
     Pinj/P            1.0000   1.0693   1.7944   479.16
     P, ATM            106.93   100.00   59.593  0.22317
     T, K             3532.34  3516.04  3327.30  1432.71
     RHO, G/CC       4.9892-3 4.6890-3 2.9813-3 2.6786-5
     H, CAL/G         -235.74  -253.64  -512.51 -2378.56
     U, CAL/G         -754.77  -770.11  -996.59 -2580.33
     G, CAL/G        -15064.1 -15030.3 -14496.0 -8399.74
     S, CAL/(G)(K)     4.1979   4.2026   4.2026   4.2026

     M, (1/n)          13.524   13.529   13.659   14.111
     (dLV/dLP)t      -1.02259 -1.02243 -1.01737 -1.00000
     (dLV/dLT)p        1.3977   1.3966   1.3245   1.0001
     Cp, CAL/(G)(K)    1.9426   1.9433   1.7862   0.7293
     GAMMAs            1.1431   1.1429   1.1435   1.2393
     SON VEL,M/SEC     1575.6   1571.5   1521.9   1022.9
     MACH NUMBER        0.000    0.246    1.000    4.140


System Performance
------------------

Choosing a contraction ratio is part of an overall system performance trade.

A smaller CR gives a smaller, lighter engine, but leads to a heavier pressurization system
and perhaps heavier tankage.

Focusing on engine thrust to weight ratio completely ignores system implications.

That said, the most common contraction ratio is **2.5**.
Very large booster engines tend to have smaller CR, small engines tend to have larger CR.

