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Nozzle expansion#

The gas leaves the chamber with the composition it reached there and keeps it to the exit plane. This is CEA's frozen rocket problem. It is one percent or two pessimistic for hot, dissociated mixtures, where real recombination in the nozzle returns some energy, and close to exact for cooler ones.

Dc Dt De chamber: volume = L* × throat area 80 % Rao bell c t e injector face 30 deg cone, throat arc 1.5 Rt exit plane, Ae / At = ε
Stations used throughout: chamber (c), throat (t) and exit (e).

The isentrope#

With the mole fractions \(x_j\) fixed, entropy per unit mass is

\[ \frac{s(T,p)}{R/M}=\sum_j x_j\frac{s_j^\circ(T)}{R}-\sum_j x_j\ln x_j-\ln\frac{p}{p^\circ}. \]

Holding it at the chamber value \(s_c\) gives the pressure on the isentrope in closed form:

\[ \ln\frac{p}{p^\circ}=\sum_j x_j\frac{s_j^\circ(T)}{R}-\sum_j x_j\ln x_j-\frac{s_c}{R/M}. \]

So every state along the expansion is a function of temperature alone. The velocity follows from the energy equation, \(u=\sqrt{2\,[h(T_c)-h(T)]}\), and the density from the ideal-gas law. Both \(h\) and \(c_p\) keep their full temperature dependence; nothing is frozen except the composition.

Throat and exit#

The throat is where the mass flux \(\rho u\) peaks along the isentrope, found by a bounded one-dimensional search. Continuity then fixes the exit:

\[ \frac{A_e}{A_t}=\frac{(\rho u)_t}{(\rho u)_e}, \]

solved for the supersonic root. From the throat state,

\[ c^*=\frac{p_c}{(\rho u)_t},\qquad C_F=\frac{u_e}{c^*}+\frac{(p_e-p_a)}{p_c}\frac{A_e}{A_t},\qquad I_{sp}=\frac{C_F\,c^*}{g_0}. \]

The difference from the textbook constant-\(\gamma\) relations is not small. For LCH4/LOX at O/F 3.4, 100 bar and area ratio 35:

Isp vacuum Exit pressure
Constant \(\gamma_f = 1.1955\) 346.5 s 25.2 kPa
EnSim, frozen with \(c_p(T)\) 343.25 s 22.6 kPa
NASA CEA 3.3.2, frozen 343.25 s

The gas cools from 3598 K to 1210 K in that nozzle and its \(c_p\) falls with it; a constant \(\gamma\) misses that.

Losses#

Three corrections are applied, in this order:

  • Divergence. A conical nozzle of half-angle \(\alpha\) wastes the radial part of the momentum: the momentum term of \(C_F\) is multiplied by \(\lambda=(1+\cos\alpha)/2\). Set \(\alpha=0\) for a contoured nozzle whose losses you prefer to put in \(\eta_{C_F}\).
  • Combustion efficiency \(\eta_{c^*}\) scales \(c^*\) and therefore the mass flow at a given throat area and chamber pressure.
  • Nozzle efficiency \(\eta_{C_F}\) scales the complete thrust coefficient, pressure term included.

Thrust is \(F=C_F\,p_c\,A_t\) and mass flow \(\dot m=p_cA_t/c^*\), both after the efficiencies.

Back pressure#

Performance at an ambient pressure \(p_a\) changes only the pressure term, so performance_at re-uses the solved nozzle. EnSim does not model flow separation: a strongly over-expanded nozzle is reported with the attached-flow thrust, and the Engine page warns when \(p_e/p_a\) falls below the Summerfield estimate of 0.4. If \(p_a\) exceeds the throat pressure the nozzle cannot run choked and no result is returned.

Wall contour#

The drawn and exported wall is a cylinder, a 30° cone, an upstream throat arc of \(1.5R_t\), a downstream arc of \(0.382R_t\) and a parabolic Rao bell at 80 % of the length of a 15° cone, with entry and exit angles of 35° and 8°. The cylinder length makes the volume up to the throat equal to \(L^* A_t\). The bell is a geometric approximation: its angles are fixed rather than optimised for the area ratio, and the performance calculation does not depend on it.

Planar MOC contour#

The MOC contour tool designs a minimum-length nozzle by the method of characteristics for a calorically perfect gas. Along the two families of characteristics

\[ \theta+\nu=\text{const},\qquad \theta-\nu=\text{const}, \]

where \(\nu(M)\) is the Prandtl–Meyer function. A centred expansion at a sharp throat corner turns the wall to \(\theta_\text{max}=\nu(M_e)/2\), and the reflected characteristics bring it back to axial flow. The geometry is planar: the reported coordinate is a half-height and \(A_e/A_t=y_e/y_t\).

Sources#

  • Gordon and McBride, NASA RP-1311 Part I (1994), section 6.
  • G. V. R. Rao, "Approximation of Optimum Thrust Nozzle Contour", ARS Journal 30(6), 1960.
  • D. K. Huzel and D. H. Huang, Modern Engineering for Design of Liquid-Propellant Rocket Engines, AIAA, 1992.
  • Shames and Seashore, NACA RM E8J12 (1948); Goldman and Vanco, NASA TM X-1502 (1968).