.. 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 `_ 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 `_ 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 `_ 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.