Chemical Rocket Performance

Stoichiometric Equations

Let a generic hydrocarbon-like fuel be ${\mathrm{C}}_{\mathrm{a}}{\mathrm{H}}_{\mathrm{b}}{\mathrm{O}}_{\mathrm{d}}{\mathrm{N}}_{\mathrm{c}}$ and oxidizer be ${\mathrm{O}}_2$ (or ${\mathrm{N}}_2{\mathrm{O}}_4$, $\mathrm{LOX}$, etc.). The stoichiometric reaction with ${\mathrm{O}}_2$ is

$${\mathrm{C}}_{\mathrm{a}}{\mathrm{H}}_{\mathrm{b}}{\mathrm{O}}_{\mathrm{d}}+{\nu }_O{\mathrm{O}}_2\to a\mathrm{C}{\mathrm{O}}_2+\frac{b}{2}{\mathrm{H}}_2\mathrm{O}(\text{plus species for }d\ne 0,c\ne 0).$$

Balance O atoms to get the stoichiometric oxidizer moles ${\nu }_O$, then compute:

• Stoichiometric O/F (by mass):

  $${\left(\frac{O}{F}\right)}_{\mathrm{sto}}=\frac{{\nu }_O{W}_{O_2}}{1\cdot W_F}.$$

• Equivalence ratio $\varphi$ (fuel-rich if $\varphi >1$):

  $$\varphi \equiv \frac{(F/O{)}_{\text{actual}}}{(F/O{)}_{\mathrm{sto}}}=\frac{(O/F{)}_{\mathrm{sto}}}{(O/F{)}_{\text{actual}}}.$$

• Mixture molecular weight (frozen composition assumption as a first cut):

  $$\bar{W}=\sum _i{Y}_i{W}_i,R=\frac{R_u}{\bar{W}}.$$

NOTE

• Performance linkage: $c^{\ast }\propto \sqrt{\frac{R{T}_0}{\gamma }}$ — thus $T_0(\varphi ,p_0)$, $R(\varphi )$, and $\gamma (\varphi ,T)$
  • from chemistry feed directly into $c^{\ast }$ (chamber)
  • and then: $I_{sp}=C_f{c}^{\ast }/g_0$ (system).

Chemical Equilibrium

In reality, hot products re-equilibrate in the chamber/nozzle. For a given $p_0$ and overall mixture ($\varphi$), the adiabatic flame temperature $T_0$ and composition $\{n_i\}$ satisfy:

• Element balances: $\mathbf{A}\vec{\mathbf{n}}=\vec{\mathbf{b}}$ (atoms conserved).
• Equilibrium at $(T,p)$: minimize total Gibbs free energy, $$\underset{\{n_i\ge 0\}}{\min }\mathrm{G}\left(T,p,\{n_i\}\right)=\sum _i{n}_i{\mu }_i(T,p),\text{s.t. element balances}.$$
• Energy closure (adiabatic): $$\sum _j{n}_{j,\text{react}}{h}_j(T_{\text{ref}})=\sum _i{n}_{i,\text{prod}}{h}_i(T_0).$$

Outputs: $\{n_i\}\Rightarrow \bar{W}(T_0,\varphi ,p_0),\gamma (T_0,\varphi ),R(T_0,\varphi ),$ then $c^{\ast }$ and (via nozzle isentropic relations) $C_f$ and $I_{sp}$.

Design Considerations

NOTE

• Fuel-rich mixtures (e.g., LOX/LH${}_2$ slightly rich) often yield higher $I_{sp}$ due to lower $\bar{W}$ and favorable $\gamma$, despite lower $T_0$ than exact stoichiometric.
• Chamber pressure $p_0$ strongly influences $c^{\ast }$ via $\dot{m}$ and level of dissociation; higher $p_0$ generally improves performance (and raises optimal area ratio).
• CEA-style analysis (NASA CEA): compute equilibrium/frozen properties at chamber, throat, and exit to get $c^{\ast },C_f,I_{sp}$ vs. $\varphi ,p_0,\epsilon ,p_a$.