Theory and governing equations¶

This document describes the equations represented in EnSim and, equally important, the assumptions that close them.

Thermodynamic properties¶

For the seven-coefficient NASA form used by the packaged data,

\[ \frac{C_p^\circ}{R}=a_1+a_2T+a_3T^2+a_4T^3+a_5T^4, \]

\[ \frac{H^\circ}{RT}=a_1+\frac{a_2T}{2}+\frac{a_3T^2}{3} +\frac{a_4T^3}{4}+\frac{a_5T^4}{5}+\frac{a_6}{T}, \]

\[ \frac{S^\circ}{R}=a_1\ln T+a_2T+\frac{a_3T^2}{2} +\frac{a_4T^3}{3}+\frac{a_5T^4}{4}+a_7. \]

The parser selects the coefficient interval declared for each species and rejects unavailable species. The model is ideal gas and does not calculate fugacity or high-pressure real-fluid injection states.

Chemical equilibrium¶

At fixed temperature and pressure, the gas composition minimizes

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

subject to elemental conservation

\[ \sum_j a_{ij}n_j=b_i,\qquad n_j\ge 0. \]

The chamber problem is adiabatic. EnSim therefore iterates temperature until reactant and product enthalpy agree while resolving equilibrium at every trial temperature. Species selection bounds the physical solution: an omitted species cannot appear, even if thermodynamically favorable.

Frozen ideal-nozzle performance¶

The chamber mixture is treated as calorically perfect during nozzle expansion; gamma and molar mass are frozen at the chamber solution. The characteristic velocity is

\[ c^*=\frac{\sqrt{R_sT_c}} {\sqrt{\gamma}\left(2/(\gamma+1)\right)^{(\gamma+1)/(2(\gamma-1))}}. \]

The supersonic exit Mach number is obtained from

\[ \frac{A_e}{A_t}=\frac{1}{M_e} \left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M_e^2\right) \right]^{(\gamma+1)/(2(\gamma-1))}. \]

Then

\[ \frac{P_e}{P_c}=\left(1+\frac{\gamma-1}{2}M_e^2\right)^{-\gamma/(\gamma-1)}, \]

\[ C_F=\sqrt{\frac{2\gamma^2}{\gamma-1} \left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)} \left[1-\left(\frac{P_e}{P_c}\right)^{(\gamma-1)/\gamma}\right]} +\frac{(P_e-P_a)A_e}{P_cA_t}, \]

and Isp = C_F c* / g0. For a conical nozzle, the momentum component of C_F is multiplied by (1 + cos(alpha))/2; the user-supplied nozzle-efficiency factor is then applied to the complete coefficient. Boundary-layer loss, finite-rate chemistry, two-phase flow and separated-flow thrust are outside this calculation.

Planar minimum-length nozzle¶

The advanced contour tool solves the two-dimensional compatibility relations for an irrotational, calorically perfect supersonic flow. Along the two characteristic families,

\[K_- = \theta + \nu = \mathrm{constant},\qquad K_+ = \theta - \nu = \mathrm{constant},\]

where \(\nu(M)\) is the Prandtl-Meyer function and \(\mu=\sin^{-1}(1/M)\) is the Mach angle. A centered expansion begins at a sharp throat corner with \(\theta_{w,\max}=\nu(M_e)/2\); reflected characteristics then turn the wall back to axial flow. The plotted transverse coordinate is a planar half-height. Consequently, \(A_e/A_t=y_e/y_t\), not the squared radius ratio used for an axisymmetric nozzle. Characteristic count controls discretization only; the model does not include axisymmetric source terms, throat rounding, viscosity or variable thermochemistry.

Regenerative cooling¶

Gas-side convection uses the standard Bartz correlation with throat diameter, local area ratio, chamber pressure, characteristic velocity, gas properties and the Bartz property-variation factor. Coolant convection uses Gnielinski in its turbulent validity range; smooth-channel Darcy friction uses a corresponding correlation. Wall conduction and the two convective resistances form a local thermal-resistance network. See Cooling model.

Chamber acoustics¶

Longitudinal, tangential and radial acoustic frequencies use ideal cylindrical cavity eigenvalues. Frequency proximity alone is not a growth model. EnSim only classifies a mode when the user supplies compatible driving and damping rates. See Combustion instability.

Flight dynamics¶

The rigid-body translational and rotational equations are

\[ \dot{\mathbf r}=\mathbf v,\qquad \dot{\mathbf v}=\frac{\mathbf F}{m}+\mathbf g, \]

\[ \mathbf I\dot{\boldsymbol\omega} =\mathbf M-\boldsymbol\omega\times(\mathbf I\boldsymbol\omega). \]

A unit quaternion maps body coordinates to the propagation frame. The WGS-84 path evaluates gravity in ECEF, including axisymmetric J2, and accounts for Earth rotation when converting atmospheric velocity. The standard aerodynamic model uses dynamic pressure, air-relative velocity and preliminary Barrowman normal-force and center-of-pressure derivatives. Axial drag uses the explicit coefficient supplied by the user, referenced to body frontal area; EnSim does not synthesize a Mach-dependent drag polar.

Uncertainty propagation¶

Positive quantities such as pressure and area ratio use lognormal factors; bounded quantities use explicitly truncated samples. Sample standard deviations use N-1. A two-dimensional normal confidence ellipse is scaled with the chi-square quantile for two degrees of freedom. These assumptions describe the chosen input model, not epistemic certainty about a real engine or vehicle.

Primary sources¶

Gordon and McBride, NASA RP-1311 Part I (1994) and Part II (1996).
McBride, Zehe and Gordon, NASA/TP-2002-211556 (2002).
Bartz, Jet Propulsion, Vol. 27, No. 1 (1957), pp. 49-51.
Gnielinski, International Chemical Engineering, Vol. 16 (1976), pp. 359-368.
Jackson, Murri and Shelton, NASA/TM-2015-218675 (2015).
Barrowman and Barrowman, The Theoretical Prediction of the Center of Pressure, NARAM-8, 1966.
Shames and Seashore, Design Data for Graphical Construction of Two-Dimensional Sharp-Edge-Throat Supersonic Nozzles, NACA RM E8J12 (1948).
Goldman and Vanco, Computer Program for Design of Two Dimensional Supersonic Nozzle with Sharp Edged Throat, NASA TM X-1502 (1968).