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equilibrium-c: A Lightweight Modern Equilibrium Chemistry Calculator for Hypersonic Flow Applications

Nicholas N. Gibbons

TL;DR

eqc addresses the need for a modern, open-source chemical equilibrium solver suitable for hypersonic and multi-physics simulations by employing an element-potential framework that minimises Gibbs energy under elemental constraints. The code combines a rigorously derived thermodynamic foundation with a robust, analytically-informed Newton solver, including under-relaxation and log-space handling to ensure convergence even with many species and high-temperature ionisation. Validation spans analytic benchmarks (e.g., CO₂ dissociation), cross-code comparisons with NASA CEA, and experimental data from shock-tunnel tests, demonstrating accuracy and robustness across representative regimes. The result is a lightweight, minimal-dependency tool with a Python interface and unit-tested C core, intended for easy integration into complex fluid-dynamics, combustion, and hypersonic design workflows, with broad applicability beyond its primary focus.

Abstract

equilibrium-c (eqc) is a program for computing the composition of gas mixtures in chemical equilibrium. In typical usage, the program is given a known thermodynamic state, such as fixed temperature and pressure, as well as an initial composition of gaseous species, and computes the final composition in the limit of a large amount of time relative to the reaction speeds. eqc includes a database of thermodynamic properties taken from the literature, a set of core routines written the C programming language to solve the equilibrium problems, and a Python wrapper layer to organise the solution process and interface with user code. Dependencies are extremely minimal, and the API is designed to be easily embedded in multi-physics codes that solve problems in fluid dynamics, combustion, and chemical processing. In this paper, I first introduce the equations of chemical equilibrium, then spend some time discussing their numerical solution, and finally present a series of example problems, with an emphasis on verification and validation of the solver.

equilibrium-c: A Lightweight Modern Equilibrium Chemistry Calculator for Hypersonic Flow Applications

TL;DR

eqc addresses the need for a modern, open-source chemical equilibrium solver suitable for hypersonic and multi-physics simulations by employing an element-potential framework that minimises Gibbs energy under elemental constraints. The code combines a rigorously derived thermodynamic foundation with a robust, analytically-informed Newton solver, including under-relaxation and log-space handling to ensure convergence even with many species and high-temperature ionisation. Validation spans analytic benchmarks (e.g., CO₂ dissociation), cross-code comparisons with NASA CEA, and experimental data from shock-tunnel tests, demonstrating accuracy and robustness across representative regimes. The result is a lightweight, minimal-dependency tool with a Python interface and unit-tested C core, intended for easy integration into complex fluid-dynamics, combustion, and hypersonic design workflows, with broad applicability beyond its primary focus.

Abstract

equilibrium-c (eqc) is a program for computing the composition of gas mixtures in chemical equilibrium. In typical usage, the program is given a known thermodynamic state, such as fixed temperature and pressure, as well as an initial composition of gaseous species, and computes the final composition in the limit of a large amount of time relative to the reaction speeds. eqc includes a database of thermodynamic properties taken from the literature, a set of core routines written the C programming language to solve the equilibrium problems, and a Python wrapper layer to organise the solution process and interface with user code. Dependencies are extremely minimal, and the API is designed to be easily embedded in multi-physics codes that solve problems in fluid dynamics, combustion, and chemical processing. In this paper, I first introduce the equations of chemical equilibrium, then spend some time discussing their numerical solution, and finally present a series of example problems, with an emphasis on verification and validation of the solver.

Paper Structure

This paper contains 20 sections, 211 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Thermodynamics of Chemical Equilibrium
  3. Governing Equations
  4. Numerical Method
  5. Numerical Convergence and Robustness
  6. Example Problems
  7. Carbon Dioxide Textbook Problem
  8. Equilibrium Air Chemistry
  9. Equilibrium Normal Shock with Ionisation
  10. Reflected Shock Tunnel Analysis
  11. Conclusions
  12. Formula for Dice Roll Microstates
  13. Partial Derivative of Entropy w.r.t Temperature
  14. Partial Derivative of Entropy w.r.t Volume
  15. Partial Derivatives of the Gibbs Energy wrt. Composition
  16. ...and 5 more sections

Figures (10)

  • Figure 1: Chemically reacting flow passing through a shockwave.
  • Figure 2: Illustration of particles and energy levels in an ideal gas.
  • Figure 3: Discrete PDFs of multiple dice rolls, normalised across domain and range.
  • Figure 4: Comparison of eqc numerical solutions with CO2 dissociation problem from turns, Chapter 2.
  • Figure 5: Post-normal shock electron number density, computed vs. Altitude and Flight velocity
  • ...and 5 more figures