Skip to content

THEORY.md

Latest commit

 

History

History
117 lines (64 loc) · 3.65 KB

THEORY.md

File metadata and controls

117 lines (64 loc) · 3.65 KB

EnSim Theory and Mathematical Background

This document provides the theoretical foundation for the EnSim rocket engine simulation.

Table of Contents

  1. Thermodynamics
  2. Chemical Equilibrium
  3. Propulsion Performance
  4. References

Thermodynamics

NASA 7-Term Polynomials

EnSim uses NASA Glenn thermodynamic coefficients to calculate species properties. The polynomial form is:

$$\frac{C_p}{R} = a_1 + a_2T + a_3T^2 + a_4T^3 + a_5T^4$$

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

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

Where:

  • $R$ = Universal gas constant (8.3144621 J/mol·K)
  • $a_1...a_7$ = NASA polynomial coefficients
  • Two sets of coefficients: low-T (200-1000K) and high-T (1000-6000K)

Chemical Equilibrium

Gibbs Free Energy Minimization

The equilibrium composition is found by minimizing the total Gibbs free energy:

$$G = \sum_j n_j \left( \mu_j^{0} + RT \ln \frac{n_j P}{n_{total} P^{0}} \right)$$

Subject to element conservation:

$$\sum_j a_{ij} n_j = b_i \quad \forall \text{ elements } i$$

Gordon-McBride Method

EnSim uses the Newton-Raphson iteration method described by Gordon & McBride (NASA RP-1311):

  1. Initialize: Estimate species moles from stoichiometry
  2. Iterate: Solve the linearized system:

$$\begin{bmatrix} \nabla^{2} G & A^{T} \ A & 0 \end{bmatrix} \begin{bmatrix} \Delta n \ \lambda \end{bmatrix} = \begin{bmatrix} -\nabla G \ b - An \end{bmatrix}$$

  1. Convergence: Check $|\Delta n / n| < 10^{-7}$

Adiabatic Flame Temperature

Energy balance for adiabatic combustion:

$$\sum_j n_j H_j(T_{ad}) = \sum_i n_i^{0} H_i(T_0)$$

Solved iteratively with equilibrium composition at each temperature.

Propulsion Performance

Characteristic Velocity (C*)

$$C^{*} = \frac{\sqrt{\gamma R T_c}}{\Gamma}$$

where the vandenkerckhove function:

$$\Gamma = \sqrt{\gamma} \left( \frac{2}{\gamma + 1} \right)^{\frac{\gamma + 1}{2(\gamma - 1)}}$$

Exit Velocity

From isentropic expansion:

$$V_e = \sqrt{\frac{2\gamma}{\gamma - 1} R T_c \left[ 1 - \left( \frac{P_e}{P_c} \right)^{\frac{\gamma - 1}{\gamma}} \right]}$$

Thrust Coefficient

$$C_F = \Gamma \sqrt{\frac{2\gamma^2}{\gamma - 1} \left[ 1 - \left( \frac{P_e}{P_c} \right)^{\frac{\gamma - 1}{\gamma}} \right]} + \epsilon \frac{P_e - P_a}{P_c}$$

Specific Impulse

$$I_{sp} = \frac{C^{*} \cdot C_F}{g_0}$$

Area-Mach Relation

$$\frac{A}{A^{*}} = \frac{1}{M} \left[ \frac{2}{\gamma + 1} \left( 1 + \frac{\gamma - 1}{2} M^{2} \right) \right]^{\frac{\gamma + 1}{2(\gamma - 1)}}$$

Efficiency Corrections

Real-world performance includes losses:

$$C^{_}_{actual} = \eta_{C^{_}} \cdot C^{*}_{ideal}$$

$$C_{F,actual} = \eta_{C_F} \cdot C_{F,ideal}$$

Typical values:

  • $\eta_{C^{*}}$ = 0.94 - 0.99 (combustion efficiency)
  • $\eta_{C_F}$ = 0.96 - 0.99 (nozzle efficiency)

References

  1. Gordon, S. & McBride, B.J. (1994). Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications - Part I: Analysis. NASA Reference Publication 1311.

  2. McBride, B.J. & Gordon, S. (1996). Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications - Part II: Users Manual. NASA RP-1311.

  3. Sutton, G.P. & Biblarz, O. (2016). Rocket Propulsion Elements, 9th Edition. John Wiley & Sons.

  4. NASA Glenn Research Center. Thermodynamic Data. https://cearun.grc.nasa.gov/

  5. NIST Chemistry WebBook. Thermophysical Properties. https://webbook.nist.gov/chemistry/