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.. 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.
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