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Chamber thermochemistry#

The chamber is treated the way NASA CEA treats it: an adiabatic, constant-pressure reactor that reaches chemical equilibrium before the gas enters the nozzle. Two numbers come out of it that everything downstream depends on, the flame temperature and the product composition.

Propellants in storage state: LH2 20 K, LOX 90 K, RP-1 298 K Gibbs minimum at (T, pc) element balance, nj ≥ 0, NASA Glenn polynomials adjust T until H(products) = H(reactants) Chamber state Tc, composition, molar mass, frozen gamma Isentrope s(T, p) = sc p(T) in closed form, cp(T) of every species, frozen Throat and exit throat where ρu peaks, exit where (ρu)t / (ρu) = ε Performance c* × ηc*, CF × ηCF momentum × (1 + cos α) / 2
From propellants to performance. The chamber problem (left) is solved once; the nozzle (right) reuses its composition.

Species data#

Each species carries NASA Glenn seven-coefficient polynomials in two temperature ranges:

\[ \frac{c_p^\circ}{R}=a_1+a_2T+a_3T^2+a_4T^3+a_5T^4 \]
\[ \frac{h^\circ}{RT}=a_1+\frac{a_2T}{2}+\frac{a_3T^2}{3}+\frac{a_4T^3}{4}+\frac{a_5T^4}{5}+\frac{a_6}{T} \]
\[ \frac{s^\circ}{R}=a_1\ln T+a_2T+\frac{a_3T^2}{2}+\frac{a_4T^3}{3}+\frac{a_5T^4}{4}+a_7 \]

The standard state is 1 bar, as in the Glenn database. The parser reads the range limits from each record and refuses a malformed one rather than guessing.

Reactants#

Propellants enter at their storage state. Liquid hydrogen at 20.27 K carries −4.47 MJ/kg relative to the gas at 298 K; that energy is spent warming and evaporating the propellant and never reaches the flame. The packaged table uses the enthalpies CEA assigns to its liquid reactants:

Key Formula Storage temperature Enthalpy, kJ/kg
LH2 H2 20.27 K −4470.5
LCH4 CH4 111.64 K −5562.3
RP-1 CH1.9423 298.15 K −1768.6
N2H4 N2H4 298.15 K +1572.2
MMH CH6N2 298.15 K +1176.4
UDMH C2H8N2 298.15 K +813.7
ETHANOL C2H6O 298.15 K −6023.9
LOX O2 90.17 K −405.6
NTO N2O4 298.15 K −190.7
HTP H2O2 298.15 K −5520.6
N2O N2O (vapour) 298.15 K +1854.0

The difference matters. LH2/LOX at O/F 6, 70 bar and area ratio 60 burns at 3485 K with a vacuum Isp of 436.9 s. Treated as gases at room temperature, the same mixture gives 3591 K and 445.9 s, two percent too optimistic. EnSim 3.x did exactly that.

Equilibrium#

At temperature \(T\) and pressure \(p\) the product mixture minimises the Gibbs energy

\[ G=\sum_j n_j\left[\frac{g_j^\circ(T)}{RT}+\ln\frac{n_j}{n}+\ln\frac{p}{p^\circ}\right] \]

subject to the element balance \(\sum_j a_{ij}n_j=b_i\) and \(n_j\ge 0\). The candidate products are every gas in the database made only of the elements the reactants bring in.

The minimum is found in two passes. SLSQP works on the mole numbers with the element balance as a linear constraint; a least-squares polish then solves the optimality conditions in element potentials, which drives trace species to the precision CEA reports. A linear program supplies a feasible starting point.

Flame temperature#

The chamber is adiabatic, so the products must carry the reactants' enthalpy:

\[ H_\text{products}(T_c)=H_\text{reactants}. \]

The residual is evaluated with a full equilibrium at each trial temperature. The search starts from an estimate, steps outward by 400 K until the residual changes sign and then closes the bracket with Brent's method. Every step solves a new composition, so dissociation is accounted for at the temperature it happens.

What the nozzle receives#

The chamber result carries the temperature, the mole fractions, the molar mass and the frozen ratio of specific heats,

\[ \gamma_f=\frac{c_p}{c_p-R/M},\qquad c_p=\sum_j x_j c_{p,j}^\circ(T_c)/M. \]

\(\gamma_f\) is reported for reference and used by the simpler studies (sweeps of nozzle shape, uncertainty, the MOC contour). The engine result itself does not use a constant \(\gamma\): see nozzle expansion.

Agreement with CEA#

For the five liquid propellant pairs in the validation set (hydrolox, methalox, kerolox and two hypergolic pairs), chamber temperature agrees with NASA CEA to within 0.01 % and vacuum Isp to within 0.05 %. The comparison and how to repeat it are on the validation page.

Sources#

  • S. Gordon and B. J. McBride, Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications, NASA RP-1311, Part I (1994) and Part II (1996).
  • B. J. McBride, M. J. Zehe and S. Gordon, NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species, NASA/TP-2002-211556 (2002).