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Combustion Toolbox: An open-source thermochemical code for gas- and
condensed-phase problems involving chemical equilibrium
Alberto Cuadra∗, C´
esar Huete, Marcos Vera
Departamento de Ingenier´ıa T´ermica y de Fluidos, Escuela Polit´ecnica Superior, Universidad Carlos III de Madrid, 28911 Legan´es, Spain
Abstract
The Combustion Toolbox (CT) is a newly developed open-source thermochemical code designed to solve prob-
lems involving chemical equilibrium for both gas- and condensed-phase species. The kernel of the code is based
on the theoretical framework set forth by NASA’s computer program CEA (Chemical Equilibrium with Applica-
tions) while incorporating new algorithms that significantly improve both convergence rate and robustness. The
thermochemical properties are computed under the ideal gas approximation using an up-to-date version of NASA’s
9-coefficient polynomial fits. These fits use the Third Millennium database, which includes the available values from
Active Thermochemical Tables. Combustion Toolbox is programmed in MATLAB with an object-oriented archi-
tecture composed of three main modules: CT-EQUIL, CT-SD, and CT-ROCKET. The kernel module, CT-EQUIL,
minimizes the Gibbs/Helmholtz free energy of the system using the technique of Lagrange multipliers combined
with a multidimensional Newton-Raphson method, upon the condition that two state functions are used to define the
mixture properties (e.g., enthalpy and pressure). CT-SD solves processes involving strong changes in dynamic pres-
sure, such as steady shock and detonation waves under normal and oblique incidence angles. Finally, CT-ROCKET
estimates rocket engine performance under highly idealized conditions. The new tool is equipped with a versatile
Graphical User Interface and has been successfully used for teaching and research activities over the last four years.
Results are in excellent agreement with CEA, Cantera within Caltech’s Shock and Detonation Toolbox (SD-Toolbox),
and the Thermochemical Equilibrium Abundances (TEA) code. CT is available under an open-source GPLv3 li-
cense via GitHub https://github.com/CombustionToolbox/combustion toolbox, and its documentation can be found in
https://combustion-toolbox-website.readthedocs.io.
Keywords: Thermochemistry, shock wave, detonation, oblique front, rocket performance, GUI
1. Introduction
The computation of chemical equilibrium has been widely used during the last century to determine the compo-
sition of multi-component mixtures subject to complex thermochemical transformations. The resulting mathematical
problem is simple in systems involving only a few species, such as the complete combustion of rich hydrocarbon-air
mixtures, or the dissociation of diatomic gas mixtures, e.g., air at moderate temperatures. However, the incomplete
combustion of a typical hydrocarbon, for instance, methane, with air, involves hundreds of reactions and more than
fifty species [1], which makes finding the final equilibrium state of the products a challenging task.
Two equivalent methods can be employed to determine the composition of the products at equilibrium: using
equilibrium constants or minimizing the Gibbs/Helmholtz free energy of the system [2]. The first method requires
specifying a sufficiently large set of elementary reactions at equilibrium (see, e.g., Refs. [3–9]). This favors the second,
where each species is treated independently, and the focus is shifted to the chemical potentials of the different species
involved [10, 11]. The second method was first introduced by the pioneering work of White in 1958 [12] and has
become the cornerstone in the development of virtually all state-of-the-art thermochemical codes [13–30].
∗Corresponding author.
E-mail address: acuadra@ing.uc3m.es (A. Cuadra)
Preprint submitted to Elsevier September 24, 2024
arXiv:2409.15086v1 [physics.chem-ph] 23 Sep 2024
The solution to the resulting minimization problem requires the evaluation of the thermodynamic properties of all
species involved at a given temperature. For this purpose, extensive thermodynamic databases have been compiled,
such as the NIST-JANAF tables [31, 32], NASA’s polynomials [33] and, more recently, the Third Millennium (Burcat)
Database [34] with updates from Active Thermochemical Tables (ATcT) [35] to evaluate the enthalpies of formation.
The ATcT rely on the use of a complete thermochemical network (TN) instead of the more traditional datasets based
on individual reactions. The use of the full TN yields more accurate results that are, in addition, fully documented
(uncertainties included). But more important is the fact that databases are easily updated with new knowledge, and
readily provide new values for the thermochemical properties of all species [36].
A recent work by Scoggins et al. [37] reported an exhaustive review of over 1200 unique chemical species from
several databases that fully (Goldsmith [38] and Blanquart [39–42]), partially (Burcat), or did not (NASA) rely on
the ATcT. They identified significant differences between Burcat’s and NASA’s databases due to the inconsistency
of the species enthalpy of formation in the latter. Although this type of analysis is out of the scope of this work, all
chemical species from Burcat’s database are also available in Combustion Toolbox (CT) and can be identified by the
suffix “ M”. Thus, when both databases contain a given species, the final choice is left to the user.
Thermochemistry is firmly rooted in the study of combustion problems, high-speed flows, reactive and non-
reactive shocks, rocket engine performance, and high explosives [43, 44]. For instance, strong hypersonic shocks
involve changes in the molecular structure of the gas, including vibrational excitation leading to dissociation [45, 46],
and later electronic excitation leading to ionization [47], which eventually transform the gas into a plasma. Turbulent
combustion and gaseous detonations have also been the topic of intense research due to their high thermodynamic
efficiency in propulsion applications [48, 49]. But they often exhibit strong deviations from equilibrium due to the
wide range of length and time scales involved, making it necessary to rely on complex fluid dynamical analyses and
numerical simulations with a high computational cost [50, 51]. Despite the deep understanding provided by the latter
approach, there are still cases in which a proper physical explanation can not be found based only on numerical re-
sults. In these cases, separation of scales may allow to split the problem into simpler ones, where the assumption of
chemical equilibrium could be justified in some representative scenarios [45, 52, 53].
Chemical equilibrium can be formulated either for single-phase gas mixtures [7, 8, 12, 19, 21, 22, 25, 26], single-
phase gas mixtures with pure-condensed species—as in NASA’s CEA code [17], Refs. [3, 6, 14], and this work [30]—
or for multi-phase systems [4, 5, 9, 15, 16, 18, 20, 23, 24, 27–29, 54, 55]. The latter case requires special treatments
to ensure that the global minimum is reached, due to the non-convexity of the Gibbs free energy [20, 23, 56]. These
include global optimization methods like the tunneling method [57] or differential evolution [58, 59].
The above review has identified many thermochemical codes currently available to the community. Neverthe-
less, to the best of our knowledge, there is not yet an open-source code based on up-to-date databases, written in a
high-level programming language, fully documented, with high-performance computing capabilities, able to model a
wide variety of applications, and equipped with a user-friendly Graphic User Interface (GUI). Combustion Toolbox
was conceived with these long-term goals in mind, and is now presented and validated in this work. This MATLAB-
GUI thermochemical code represents the core of an ongoing research work and has been used to investigate a series
of problems during the last few years [45, 52, 53, 60–65]. Results are in excellent agreement with NASA’s CEA
code [17], Cantera [66] within Caltech’s Shock and Detonation Toolbox (SD-Toolbox) [67, 68], and the Thermo-
chemical Equilibrium Abundances (TEA) code [25].
The paper is structured as follows. Section 2 starts with an initial overview of CT. The equilibrium kernel (CT-
EQUIL) is presented in Section 3. The shock and detonation module (CT-SD) is discussed in Section 4, followed by
the rocket performance module (CT-ROCKET) in Section 5. The results of all modules are validated against other
codes in Sections 3 to 5. A detailed description of the GUI is given in Section 6. And, finally, the conclusions are
presented in Section 7.
2. Overview of Combustion Toolbox
Combustion Toolbox [30] is a GUI-based thermochemical code written in MATLAB with an equilibrium kernel
based on the mathematical formulation set forth by NASA’s CEA code [17]. The thermodynamic properties of the
gaseous species are modeled with the ideal gas equation of state (EoS), and an up-to-date version of NASA’s 9-
coefficient polynomial fits (NASA9) based on [33–35]. CT is a new thermochemical code written from scratch in a
object-oriented architectural format composed of three main modules: CT-EQUIL, CT-SD, and CT-ROCKET.
2
The kernel module, CT-EQUIL, computes the composition at the equilibrium of multi-component gas mixtures
that undergo canonical thermochemical transformations from an initial state (reactants), defined by its initial composi-
tion, temperature, and pressure, to a final state (products), defined by a set of chemical species (in gaseous—included
ions—or pure condensed phase) and two thermodynamic state functions, such as enthalpy and pressure, e.g., for
isobaric combustion processes. CT-SD solves steady-state shock and detonation waves in either normal or oblique
incidence. Finally, CT-ROCKET computes the theoretical performance of rocket engines under highly idealized con-
ditions. Even though all modules are enclosed in a user-friendly GUI, they can also be accessed from MATLAB’s
command line in plain code mode.
There is a fourth closed-source (i.e., proprietary) module, CT-EXPLO, that estimates the theoretical properties
of high explosive mixtures and multi-component propellants with non-ideal EoS. Although still under development,
CT-EXPLO is distributed in its current form as the thermochemical module of SimEx [60] subject to a proprietary
license. Further details on this module will be provided elsewhere.
2.1. Software Architecture
As previously mentioned, the program was developed from scratch using MATLAB and an object-oriented archi-
tectural design. All the modules rely on CT-EQUIL to compute the thermodynamic properties of the species involved
and the chemical composition of the mixtures at equilibrium. The full package can be accessed from the GUI (see
Section 6) or directly from MATLAB’s command line. Additionally, the main computations performed using the GUI
are callbacks to the plain code. Consequently, any change made to the code is immediately reflected in the GUI,
leading to a more flexible and adaptable tool.
MATLAB was selected as programming language due to its excellent linear algebra, visualizing and debugging
capabilities, extensive documentation, active community, and dedicated app development framework (App Designer).
CT has significantly evolved since its version 1.0.5 [61, 69], which relied solely on procedural techniques for perfor-
mance advantages over MATLAB Object-Oriented Programming (OOP). However, the current version, v1.1.0 [30],
has been completely rewritten using the OOP paradigm due to the inherent limitations of procedural techniques, such
as reduced code modularity and scalability. After meticulous optimization, the new object-oriented-based code has
achieved substantial performance improvements, achieving between 1.3×and 1.9×better performance compared to
the previous version under identical test conditions. Even though MATLAB is an interpreted language, which intro-
duces a significant performance cost compared to other (compiled) languages [70], CT computational times are still
competitive compared to similar codes. For instance, as shown in Section 4, CT is about one order of magnitude faster
than the MATLAB version of Caltech’s SD-Toolbox used with Cantera (written in C++). However, there is room for
further improvement by isolating specific demanding tasks into C++ subroutines and accessing them from MATLAB
using MEX files. Moreover, the adoption of OOP in the current version facilitates the parallelization of the code.
These optimization techniques will be explored in future code releases.
The classes and methods included in CT’s modules (described in detail later) rely on several core and database
classes. The most relevant among these are:
•Species: stores the properties and thermodynamic functions of a chemical species.
•ChemicalSystem: encapsulates the chemical species and phases within a chemical system.
•Mixture: represents a multi-component mixture, including its thermodynamic properties and the solution of
the problem at hand.
•EquationState: defines the thermodynamic equation of state for a Mixture.
•Database: an abstract superclass with common methods and properties for database-related classes.
•NasaDatabase: constructs NASA’s thermochemical database [33] using a structure of Species objects. By
default, this class also includes data from Burcat’s database [34], obtained in its NASA9 format.
For efficient data access, the Species class employs griddedInterpolant objects (see MATLAB’s built-in func-
tion griddedInterpolant.m), which use piecewise cubic Hermite interpolating polynomials (PCHIP) [71]. This
approach speeds up the thermodynamic data access by a factor of 200% with respect to the evaluation of NASA’s
3
polynomials. Starting from v1.1.0, these objects are also temporally allocated in cache for even better performance.
Furthermore, for temperatures outside the bounds, we avoid the higher order terms of the polynomials by linear ex-
trapolation, similar to Ref. [7], extending the range of validity of the thermodynamic data available. It should be
emphasized that this extension is limited to a narrow temperature range and may not apply to temperatures signifi-
cantly outside the bounds.
The Combustion Toolbox is organized using namespaces. Hence, it is necessary to import the different subpack-
ages into MATLAB workspace as follows
1import combustiontoolbox.databases.NasaDatabase
2import combustiontoolbox.core.∗
The first statement imports a specific class or method, while the second imports an entire subpackage. To start working
with CT, we also need to load/generate a database and initialize the chemical system. This can be easily done with
one of the following sample statements:
1s y s t e m =ChemicalSystem (DB)
2s y s t e m =ChemicalSystem (DB, ' So o t f o r m a t i o n e x t e n d e d ' )
3s y s t e m =ChemicalSystem (DB, {' N2 ' ,' O2 ' ,'NO ' ,' N ' ,' O ' })
In these examples, DB represents an instance of the database class used, e.g., DB = NasaDatabase(). The first option
initializes the system with all the possible species that could arise based on the elements present in the reactants (see
class’s method findProducts). The second option initializes the system with an extended list of 94 species that
typically appear in CHON mixtures, including solid carbon C(gr). Finally, the third option restricts the system to a
specific list of species that may appear as products.
Figure 1: Combustion Toolbox simplified workflow.
Figure 1 summarizes the main steps required to solve a problem with CT. First, import the necessary packages.
Second, define the database and initialize the chemical system using one of the statements indicated above. Then,
specify the initial state of the mixture (temperature, pressure, and molar composition), select the solver, and define
the problem type. Depending on the problem’s constraints, additional parameters may be required, e.g., for planar
shocks, it is necessary to specify the pre-shock velocity in m/s or, equivalently, the pre-shock Mach number. Once
the setup is complete, invoke the solveArray method from the selected solver to compute the results. Finally, for
post-processing, CT solvers include a built-in report method that shows predefined plots. Results can be exported
using the Export class, which allows to save data into a spreadsheet or as a .mat file.
The CT source code is organized into several top-layer folders: +combustiontoolbox,databases,examples,gui,
installer,modules,utils, and validations. The +combustiontoolbox folder encapsulates the different modules (names-
paces) implemented in CT, such as CT-EQUIL (+equilibrium), CT-SD (+shockdetonation), and CT-ROCKET (+rocket).
The databases folder mainly consists of raw data and .mat files that contain the thermochemical properties of the in-
dividual chemical species [33–35]. The examples folder includes various examples that demonstrate the wide variety
of problems that can be solved with CT. The gui folder contains the routines that are specifically designed for the
GUI. The installer folder contains all the installation files of the GUI: the MATLAB toolbox and the royalty-free
stand-alone versions for different operative systems. This step is straightforward (see INSTALL.m), and further details
4
can be found on the CT website (https://combustion-toolbox-website.readthedocs.io/en/latest/install.html). Finally,
the validation folder includes the routines used to validate CT with the results obtained with other codes, the unit
testing files to ensure the correct functionality of the code, and all the graphs generated from these verifications.
2.2. Collaborative framework and version control system
The collaborative process of open-source code can greatly benefit from the contributions of other authors. How-
ever, in the absence of adequate tools, this process may become disorderly. To mitigate this risk, we employ Git as
version control system (VCS) and GitHub as online hosting service. These technologies enable comprehensive track-
ing of all changes made to the code while maintaining complete transparency throughout the development process
[72–75]. To further ensure the integrity of the package, all contributions are subject to rigorous testing before being
merged into the two primary branches (master and develop) using GitHub Actions, providing a powerful tool to
retrace errors in the code.
2.3. Documentation
A notable fraction of open-source codes lack sufficient documentation, which impedes the code’s usability and
accessibility. To amend this issue, we use Sphinx [76], a documentation generator written and used by the Python
community, along with its MATLAB-domain extension [77]. All function headers are written following Google’s
Python-style docstrings. The online documentation is hosted on Read the Docs and is regularly updated from the
GitHub repository. The new routines are automatically included in the online documentation using GitHub Actions.
The framework allows having specific documentation for each distributed version. Additionally, the package includes
several examples and all the validations carried out with other codes.
2.4. Benchmarks
The calculations presented in this work were performed on a laptop computer with the following specifications:
Intel(R) Core(TM) i7-11800H CPU @ 2.30GHz with 8 physical cores and 64GB of RAM, running on a 64-bit Win-
dows 11 Pro system and using MATLAB R2022b. The reported computation times represent the total elapsed time
from the initialization of CT to the completion of all calculations. As a reference, loading the databases (approxi-
mately 3600 species with their griddedInterpolant objects) takes an average of 0.7228 seconds, which was included
in the reported times. Unless otherwise specified, the tolerance for the molar composition was consistently set to 10−14
throughout the study.
3. Thermochemical equilibrium module
This section presents the thermochemical equilibrium module, CT-EQUIL, the kernel of the code, encapsulated
within the +equilibrium subpackage. This module relies on the EquilibriumSolver class, which is composed of
four main methods: equilibriumGibbs,equilibriumHelmholtz,equilibrateT, and equilibrate.
The first two, described in Sections 3.1 and 3.2, are used to compute the chemical equilibrium composition for
given temperature-pressure (TP), or temperature-volume (TV) states, respectively. The third is employed to compute
the thermodynamic properties of the mixture and is an upper layer of the previous routines. The last one, described in
Section 3.3, represents the top layer of CT-EQUIL and is implemented to compute the chemical equilibrium compo-
sition and thermodynamic properties for any of the following pairs of specified state functions: TP, HP, SP, TV, EV,
and SV, where T stands for temperature, P for pressure, H for enthalpy, S for entropy, E for internal energy, and V for
volume.
Combustion Toolbox enables the computation of chemical equilibrium under various caloric models regarding
the final gas mixture, including calorically perfect gas, calorically imperfect gas with frozen chemistry, or calorically
imperfect gas with equilibrium chemistry, including dissociation and ionization. An example of these calculations
is presented in Section 4. It is also possible to freeze the composition of a subset of the species at equilibrium by
defining them as inert species.
5
3.1. Equilibrium composition at specified temperature and pressure (TP)
The equilibriumGibbs routine computes the molar equilibrium composition n={n1,n2,...,nNS}of a mixture
S={S1,S2,...,SNS}of NS species at a given temperature Tand pressure pfor a closed system by minimizing the
Gibbs free energy G(T,p,n). This is an equality-constrained problem (ECP) subject to mass conservation, namely
min G(T,p,n)⇔dG(T,p,n)=X
j∈S
µj(T,p,n) dnj=0,(1a)
qi:=X
j∈S
ai jnj−b◦
i=0,∀i∈E(1b)
where njand µjare the number of moles and chemical potential of species j, respectively, ai j are the stoichiometric
coefficients, i.e., the number of atoms of element iper molecule of species j, and b◦
iis the number of mols of the i-th
element in the initial mixture. There are as many linear constraints qias NE elements E={E1,E2,...,ENE}involved.
The NS species can be either gaseous or condensed, assuming pure components. Thus, there are NG gaseous species
SG={S1,S2,...,SNG}and NC =NS −NG condensed species SC={SNG+1,SNG+2,...,SNS }, where SGand SCare
the respective subsets of S. In addition, the system must comply with the Gibbs phase rule [78], ensuring that the
number of elements does not exceed the number of chemical species present. This condition can be expressed as
NCs+1≤NE ≤NS, where NCsdenotes the number of condensed species that remain stable while coexisting with
the gas phase at chemical equilibrium. Equation (1a) must be supplemented with an EoS to define the thermodynamic
functions. Our code implements the ideal gas EoS for the chemical potential
µj(T,p,n)=µ◦
j(T)+κjRT
ln nj−ln X
j∈SG
nj+ln p
p◦
,∀j∈S(2)
where µ◦
j(T) is the chemical potential of species jat the reference pressure ( p◦=1 bar) and the specified temperature,
κjis either one or zero depending on whether the species is in gaseous or condensed phase, and Ris the universal
gas constant. As stated above, the CT-EQUIL module computes all thermochemical properties within the ideal gas
approximation using an up-to-date version of NASA’s 9-coefficient polynomial fits [33] that incorporates the Third
Millennium database [34], including the available values from Active Thermochemical Tables. For more details on
the calculation of the thermochemical properties from the values contained in these databases, the reader is referred
to the original sources [33, 34].
The Gibbs phase rule constrains the number of condensed species that can appear at equilibrium, but it does not
specify which of these NC species will be stable [9]. To determine the stability of the condensed phases, we directly
impose the non-negative condition nj>0,∀j∈SCfollowing the approach of Kulik et al. [79] and Leal et al. [56, 80].
These inequalities are reformulated as a set of equality constraints by introducing slack variables ϕ, resulting in
rj:=njϕj−τ=0,∀j∈SC(3)
where we have included a constant τ=10−25 min(b◦) for numerical stability purposes [80]. In Eq. (3), a condensed
species is considered stable when ϕj≈0 and nj>0, whereas unstable condensed species is characterized by ϕj>0
and nj≈0.
The ECP formulated in (1)-(3) is solved using the method of Lagrange multipliers (an extended description thereof
can be found in [81]), which introduces the Lagrangian function
L(T,p,n,λ,φ)=G(T,p,n)+λq(n)−φr(n),(4)
where λand φrepresent the multiplier vectors, of length NE and NC, respectively. If there is an infinitesimal change,
dnand dλ, the differential of Lcan be obtained from Eq. (4) with use made of (1) to yield dL=0, namely
X
j∈S
µj(T,p,n)+X
i∈E
ai jλi
+X
j∈SC
ϕj
dnj+X
i∈E
X
j∈S
ai jnj−b◦
i
dλi=0,(5)
6
which has to satisfy the complementary condition given in (3). Additionally, the sum of the molar compositions must
equal the total number of moles of gaseous species in the system, thus
X
j∈SG
nj=n.(6)
Considering that dnj, dλi,ϕj, and nare independent, Eqs. (3), (5), and (6) constitute a system of NS non-linear
equations subject to a set of NE +NC +1 linear constraints. Furthermore, to ensure that the molar composition njis
strictly positive, it is convenient to work with the functions ln njand ln nin the gaseous species. In our implementation,
we found that excluding this approach for the condensed species resulted in a more efficient algorithm.
Using these definitions, Eqs. (3), (5), and (6) can be rearranged in the form f(x)=0, where xis the vector of
unknowns composed of ln nG
j,nC
jand ϕC
j, ln n, and πi=−λi/RT . For the sake of conciseness, variables specific to
gaseous species are denoted with the superscript G, indicating ∀i∈SG, while those related to condensed species are
denoted with the superscript C, indicating ∀j∈SC. To solve this system of equations, Combustion Toolbox uses a
multidimensional Newton-Raphson (NR) method
J·δx=−f(x),(7)
where Jis the Jacobian matrix with components Ji j ≡∂fi/∂ xj,δxis the correction vector composed of ∆ln nG
j,∆nC
j,
∆ϕC
j,∆ln nand ∆πi, and fis the vector function. Equation (7) can be expanded as the following system of NS +NC
+NE +1 linear equations
∆ln nj−∆ln n−X
i∈E
ai j∆πi=−µj
RT ,∀j∈SG(8a)
X
i∈E
ai j∆πi+ ∆ϕj=µj
RT −ϕj,∀j∈SC(8b)
ϕj∆nj+nj∆ϕj=τ−njϕj,∀j∈SC(8c)
X
j∈SG
ai jnj∆ln nj+X
j∈SC
ai j∆nj=b◦
i−X
j∈S
ai jnj,∀i∈E(8d)
X
j∈SG
nj∆ln nj−n∆ln n=n−X
j∈SG
nj,(8e)
where the dimensionless Lagrange multiplier πihas been taken equal to zero in the right hand side of Eqs. (8a) and
(8b). This is a suitable simplification as long as λiappears linearly in the first bracket of Eq. (5), as discussed in
Ref. [13]. Algebraic manipulation of (8) allows to reduce the system’s dimensions due to the spareness of the upper
left corner of the Jacobian matrix J. Thus, substituting ∆ln njfrom (8a) in (8d) and (8e), and ∆ϕjfrom (8c) in (8b),
NS equations are drop out from (8), providing a reduced symmetric system of NE +NC +1 equations, namely
X
i∈EX
j∈S
al jai j nj∆πi+X
j∈SC
al j∆nj+
X
j∈SG
al jnj
∆ln n=b◦
l−X
j∈S
al jnj+X
j∈SG
al jnjµj
RT ,∀l∈E(9a)
X
i∈E
ai j∆πi−ϕj
nj
∆nj=µj
RT −τ
nj
,∀j∈SC(9b)
X
i∈EX
j∈SG
ai jnj∆πi+
X
j∈SG
nj−n
∆ln n=n−X
j∈SG
nj+X
j∈SG
njµj
RT ,(9c)
which is solved using MATLAB’s LU factorization with partial pivoting routine built on LAPACK [82]. The updated
solution vector xat the (k+1)-th iteration is given by xk+1=xk+τkδxk, where τkis the step size, or relaxation,
parameter for the k-th iteration, defined as in Refs. [17, 80] for the gaseous and condensed phase, respectively. As
previously indicated, the solution vector xkhas πi=0; consequently, it is not necessary to relax the new value obtained,
7
i.e., πi,k+1= ∆πi,k. Note that in Eq. (9), xis composed of nC
j, ln n, and πi=0. Hence, to update the terms ln nG
jand
ϕC
j, the corrections ∆ln nG
jand ∆ϕC
jmust be obtained from (8a) and (8c) after each solution of (9), which requires
defining a set of initial estimates x0for the molar number njof all the possible products and for the slack variables
ϕC
j. We proceed using the max-min method [21, 22] implemented in Mutation++ [24, 55]. The initial composition is
obtained from n0=(1 −α)nmajor
0+αnminor
0with a merging factor α=0.01. Here, nmajor
0represents the composition
of the major chemical species and is obtained by solving Eq. (1) and n≥0 while considering µ≈µ◦. The initial
composition of the minor species nminor
0is defined as the composition that maximizes the smallest number of moles
of a single species while still satisfying (1). Both problems are solved using simplex algorithms (see simplex.m and
simplexDual.m routines, respectively). However, when performing parametric sweeps, CT starts with the moles
of the gaseous species njobtained from the previous calculation, provided that their magnitudes exceed a predefined
threshold value of 10−6. This approach facilitates the convergence of the iterative procedure and accelerates the overall
computational efficiency.
The algorithm first solves the ECP problem for the gas phase, removing species with molar compositions below a
certain tolerance to prevent an ill-conditioned Jmatrix. Once convergence is achieved (by default, the tolerance for
the molar composition is set to 10−14 and for the NR is 10−5), if there are condensed species in the set of products,
a second iteration process is conducted. The procedure is similar, but now we include in the set of unknowns all the
condensed species that satisfy the vapor pressure test, namely
1
RT
∂L
∂nj
=
µ◦
j
RT −X
i∈E
∆πiai j <0,∀j∈SC(10)
whose addition to the system will reduce the Gibbs free energy of the system even further, corresponding with the first
bracket of Eq. (5) in dimensionless form. The term ϕC
jis omitted because it is effectively zero when the condensed
species is stable. Note that if (10) yields negative values, the Lagrange function may not be at equilibrium (dL=0),
which means that the added species can appear at the final state of equilibrium. If, in the new equilibrium state, the
slack variable yields |log[exp (−ϕC
j/RT )]|>10−2, the molar composition nC
jof the added species drops below the
tolerance threshold, or the Jacobian matrix Jbecomes singular, these species are considered unstable at equilibrium
and are consequently removed from the set SC. This process is repeated until all the condensed species in SCthat
satisfy T∈[Tmin,Tmax ], i.e., whose temperature range is compatible with the system’s temperature, have been tested.
For any added condensed species whose initial compositions were not predefined from the max-min method, we
assign a starting value of nC
0=10−6. Similarly, for the slack variables, we initially assume that all species are stable,
setting their values to ϕC
0=10−14.
In the presence of ionized gases, the algorithm neglects the Coulombic interactions associated with ideal plasmas.
In this case, there is only an additional restriction that is given by the electroneutrality of the mixture [83]
X
j∈SG
ae jnj=0,(11)
where the stoichiometric coefficient ae j represents the number of electrons in ion jrelative to the neutral species.
Thus, for {e−,N+
2}we have aej ={1,−1}. This means that the electron Ee, with index e, is treated as an element and
has an associated Lagrange multiplier πe(λe) in dimensionless form (dimensional form). This assumption is valid
only when the ion density is sufficiently small, i.e., for weakly ionized gases. The code directly detects if there are
ions in the set of possible products Sand calls another subroutine that ensures that condition (11) is met.
3.2. Equilibrium composition at specified temperature and volume (TV)
For calculating the molar equilibrium composition of the mixture at a given temperature Tand volume v, we have
to minimize the Helmholtz free energy of the system, defined as F=G−pv. Upon use of this relation into the
minimization condition dF(T,v,n)=0, we get
X
j∈S
µj(T,v,n)nj−pv =0 (12)
8
to be used in substitution of Eq. (1a). For convenience, the chemical potential of species jis now expressed as a
function of the mixture’s volume v. For an ideal gas EoS we have
µj(T,v,n)=µ◦
j(T)+κjRT
ln nj
P
j∈SG
nj
+ln njRT
p◦v
,∀j∈S(13)
which ultimately gives a different reduced system of equations, to be solved instead of (9), namely
X
i∈EX
j∈S
al jai j nj∆πi+X
j∈SC
al j∆nj=b◦
l−X
j∈S
al jnj+X
j∈SG
al jnjµj
RT ,∀l∈E(14a)
X
i∈E
ai j∆πi−ϕj
nj
∆nj=µj
RT −τ
nj
,∀j∈SC.(14b)
Unlike in the TP calculations, the linear system no longer includes the total number of moles, n, and its correction
factor, ∆ln n, as they are drop out of the system of NE +NC dimensions written above. Here, µjis given by Eq. (13)
and the correction values ∆ln nG
jdo not depend of ∆ln n, yielding
∆ln nj=X
i∈E
ai j∆πi−µj
RT ,∀j∈SG.(15)
On the other hand, the values of ∆ϕC
jare provided by the same expression (8c). The computation of the chemical
composition and thermodynamic properties of a given mixture at specified temperature and volume is performed by
the routine equilibriumHelmholtz.
3.3. Equilibrium composition for other pairs of state functions
In many practical applications, the equilibrium temperature of the system is not known a-priori, which requires
the provision of supplementary information to close the problem. This additional information may be obtained from
an enthalpy, internal energy, or entropy conservation equation, subject to the requirement that the corresponding state
function fremains unchanged, namely
∆f(T)≡fF(T)−fI(TI)=0,(16)
where the subscripts F and I refer here to the final and initial states of the mixture, respectively. Unlike in NASA’s
CEA code, we have increased the flexibility of the CT-EQUIL module by decoupling this additional equation and
retrieved the new condition by using a second-order NR method
Tk+1=Tk−f(Tk)
f′(Tk).(17)
The derivatives of the state functions f′(T)involved in the different transformations can be expressed analytically
in the form: (∂h/∂T)p=cp,(∂e/∂T)v=cv,(∂s/∂T)p=cp/T, and (∂s/∂T)v=cv/T, for HP, EV, SP, and
SV transformations, respectively. Following common practice, h,e,s,cp, and cvdenote here the enthalpy, internal
energy, entropy, and the specific heats at constant pressure and constant volume, respectively. It is worth noting that
that although they are written in lower case letters, these variables refer here to extensive magnitudes.
The initial estimate T0is computed using a regula falsi method. Nevertheless, when carrying out parametric
studies, the program uses the temperature obtained in the previous calculation as an initial estimate to accelerate
convergence toward the new solution. However, if Tkis significantly distant from the actual solution, this approach
can lead to unsatisfactory convergence. To overcome this issue, we can select a more robust root-finding method,
as Eq. (16) has been purposely decoupled. In particular, the code has implemented an implicit third-order Newton-
Steffensen root-finding algorithm [84], defined as follows:
Tk+1=Tk−f2(Tk)
f′(Tk)hf(Tk)−fT∗
k+1i,(18)
where the temperature at the (k+1)-th iteration, Tk+1, is re-estimated by using the provisional value T∗
k+1provided by
the application of the classical method defined in (17). The convergence criterion max{|(Tk+1−Tk)/Tk+1|,|∆f/fF|} < 0
is set by default to 10−3in both methods and is generally reached in two to five iterations.
9
3.4. Validations
To illustrate the wide variety of applications of Combustion Toolbox and asses the capabilities of the CT-EQUIL
module, several validation tests have been conducted. In this work, we provide three of them in which only a reduced
set of species are presented for clarity. Further validation tests can be easily accessed through the CT website or by
just utilizing the user-interface validation add-on, uivalidation, which is implemented in the GUI. Alternatively, the
user can also run the scripts included in the validations folder.
First test: In planetary science, thermochemical equilibrium codes like TEA [25], Fastchem [7–9], and GGCHEM [6],
are used to model the atmospheric composition of giant planets, brown dwarfs, and other celestial bodies. Further
examples can be found in Refs. [85–88]. Such models can help to reveal the physicochemical processes (both chem-
ical and radiative) that drive the evolution of these atmospheres, including the formation of clouds and the escape
of atmospheric gases into space [89]. This motivates the first validation case: the composition of the hot-Jupiter
exoplanet WASP-43b’s atmosphere. To this end, it is necessary to provide temperature and pressure profiles, as
well as the planet’s metallicity, which refers to the abundance of elements heavier than hydrogen and helium present
in its composition. For example, Fig. 2 shows the variation of the species molar fractions Xj=nj/Pj∈Snjwith
pressure (right panel) corresponding to the temperature-pressure profile given in [90] (left panel), and assuming an
atmospheric 50×solar metallicity. This value is needed to determine the mixture’s initial number of moles nj, upon
knowledge of the elemental solar abundances that are here estimated from [91], which considers H the reference
element. The readAbundances method, encapsulated within the SolarAbundances class, facilitates the retrieval
of solar mass abundances. These quantities are then processed into molar abundances through the class method
abundances2moles, where a default unity metallicity is assumed.
Figure 2: Variation of molar fraction with pressure (right panel) for the temperature-pressure profile of exoplanet WASP-43b (left panel) with an
atmospheric 50×solar metallicity; solid line: numerical results obtained with CT; symbols: numerical results obtained with TEA [25]. The species
denoted with subscript M is obtained from Burcat’s database [34].
It is important to note that different exoplanets may necessitate the utilization of distinct solar abundances. Nev-
ertheless, the program is compatible with additional datasets that follow the same format as the original dataset
abundances.txt (located in the databases folder). The results (solid lines) show an excellent agreement with those
obtained with the Thermochemical Equilibrium Abundances (TEA) code [25] (symbols) even down to the µbar level.
The minor differences in HCN, C2H2(acetylene), and HSM(the subscript M denotes that is obtained from Burcat’s
database) come from the discrepancies of the free energies compared to the NIST-JANAF database [31, 32] that is
implemented in TEA. In this test, the computation time with a tolerance of 10−32 for the molar composition was 0.9
seconds (TEA: 6.42 seconds) for a set of 26 species considered and a total of 90 case studies, which represents a 7×
speed-up factor for our code (36×loading a specific database for this case).
10
Figure 3: Variation of the molar fractions, Xj(top), and of different thermodynamic mixture properties: (a) temperature, T, (b) density, ρ, (c)
enthalpy, h, (d) internal energy, e, (e) Gibbs energy, g, (f) entropy, s, (g) specific heat capacity at constant pressure, cp, and (h) adiabatic index,
γs, for an HP transformation in lean-to-rich acetylene (C2H2)-air mixtures at standard conditions (T1=300 K, p1=1 bar); solid line: numerical
results obtained with CT; symbols: numerical results obtained with NASA’s CEA code [17].
Figure 4: Variation of the molar fractions Xjfor a Silica-Phenolic mixture at atmospheric pressure ( p=1 atm) with T∈[200,5000]; solid line:
numerical results obtained with CT; symbols: numerical results obtained with NASA’s CEA code [17].
11
Second test: As previously indicated, thermochemical codes are essential for understanding and predicting the
intricate chemical transformations that take place in combustion processes. Therefore, validating CT with a canonical
combustion test is critically important. To this end, the second validation test involves the adiabatic isobaric com-
bustion of acetylene and air, a promising mixture for advanced internal combustion engines due to acetylene’s high
energy density and low carbon content. Specifically, the investigation focuses on the isobaric reactive mixture at an
initial pressure of p1=1 atm and temperature of T1=300 K, and considers a wide range of equivalence ratios
φ∈[0.5,4]. Figure 3 shows the variation of the molar composition of the products with φ(top panel) along with other
mixture properties (Figs. 3a-h). It should be noted that, unlike the previous test, in this case the results obtained with
CT (represented by solid lines) are compared to those of NASA’s CEA code [17] (represented by symbols). Once
again, the results are in excellent agreement with the benchmark code, including the generation of solid carbon C(gr)
when the equivalence ratio reaches or exceeds φ≈2.6. The computation time elapsed 3.34 seconds for a set of 94
species and 351 case studies, which is 1.4×faster compared to the previous release [61, 69].
Third test: Recent advancements in chemical equilibrium solvers have opened up new opportunities for studying
the complex phenomena that take place on the surface of ablative materials during atmospheric reentry. For instance,
Helber et al. [92] used a chemical equilibrium code to investigate the ablation of carbon-based materials under con-
ditions experienced during Earth’s atmospheric reentry. Other studies focused on carbon-fiber-reinforced-polymers
(CFRP) [93] or silicon-based ablative materials [94]. Regardless of the composition of the ablator, predicting the
equilibrium species generated during the ablation process is a challenging task, as it involves the evaluation of a large
number of condensed species at very high enthalpies. To assess the capabilities of CT under more demanding sce-
narios than those of the previous tests, we recall in this third test the example presented in Ref. [24]. The problem
consists of a parametric study of a Si-C6H5OH mixture at atmospheric pressure (p=1 atm) for a wide range of
temperatures T∈[200,5000] K. Figure 4 illustrates the variation with temperature of the equilibrium molar fractions
Xjfor the Silica-Phenolic mixture, along with the results computed using NASA’s CEA code [17]. It can be seen that
there is total agreement even for the multiple condensed species. Previous work [24] reported that NASA’s CEA code
was not able to converge for T<400 K. However, this is only true when computing the parametric study, not the
individual cases, whereas CT converges in both situations. In this test, the computation time was 3.05 seconds for a
set of 177 species (40 in condensed phase) and 481 case studies, making this version 1.67×faster than the previous
release [61, 69].
4. Shock and detonation module
This section presents the routines of the shock and detonation module, CT-SD, which are encapsulated in the
ShockSolver and DetonationSolver classes, respectively. These solvers depend on the CT-EQUIL module de-
scribed above, and therefore, each object is associated with an instance of the EquilibriumSolver class. The
CT-SD module determines the post-shock equilibrium state of steady non-reactive and reactive shocks with arbitrary
incidence angles, β(see inset on the right panel of Fig. 5b). The methods to solve normal shocks and detonation
waves, β=π/2, are based on the algorithm outlined in NASA’s Reference Publication 1311 [17, Chapters 7-9].
4.1. Oblique shocks
First, consider the problem of an undisturbed, planar, normal shock wave. The pre-shock density, pressure, en-
thalpy, and velocity (in the reference frame attached to the shock) are denoted, respectively, as ρ1,p1,h1, and u1. The
corresponding flow variables in the post-shock gases are denoted as ρ2,p2,h2, and u2. The well-known Rankine-
Hugoniot (RH) relations for the variation of pressure and enthalpy are, respectively
p2=p1+ρ1u2
1 1−ρ1
ρ2!and h2=h1+u2
1
2
1− ρ1
ρ2!2
.(19)
These equations must be supplemented by the equation of state, in our case, the ideal EoS p=ρRT/W, where
W=Pj∈SG(nj/Pj∈SGnj)Wjstands for the average molecular mass of the gaseous mixture computed in terms of
the gas-phase molar fractions, nj/Pj∈SGnj, and the molecular masses of the gaseous species, Wj. Also required is
the caloric EoS, h=Pj∈SnjH◦
j(T), that gives enthalpy in terms of the temperature and gas mixture composition.
12
Figure 5: Pressure-deflection (a) and wave angle-deflection (b) shock polar diagrams for air (78% N2, 21% O2, and 1% Ar) at pre-shock temperature
T1=300 K and pressure p1=1 atm, and a range of pre-shock Mach numbers M1between 2 and 14; solid line: calorically imperfect gas with
ionization/dissociation; dashed: calorically imperfect gas with frozen chemistry; circles: results obtained with Cantera [66] within Caltech’s SD-
Toolbox [68]; diamonds: maximum deflection angle θmax.
As previously discussed, the molecular masses, Wj, and the molar specific enthalpies, H◦
j(T), are evaluated from a
combination of NASA’s [33] and Burcat’s (Third Millennium) [34] thermochemical databases.
The solver employs a NR method (see Ref. [17, Chapters 7-9] for more information) to determine the roots
of the system of equations governing both reactive and non-reactive shocks (routines detCJ and shockIncident,
respectively). Consequently, the jump relationships, such as p2/p1and ρ2/ρ1, can be calculated with ease. In this
subsection, we focus on the oblique shock configuration that implicitly includes the solution of the normal shock
wave. When the angle formed by the shock plane and the upstream flow is not π/2, in the RH equations (19) the
pre- and post-shock velocities, u1and u2, must be replaced by their respective components normal to the shock,
u1nand u2n. These equations can be readily reformulated in terms of the magnitudes of the pre- and post-shock
velocities upon direct substitution of the trigonometric relationships u1n=u1sin βand u2n=u2sin (β−θ), where
βis the shock incidence angle and θis the flow deflection angle, both measured with respect to the upstream flow
direction. For oblique shocks, the continuity of the tangential velocity across the shock, u1t=u2t, or, equivalently,
u1cos β=u2cos (β−θ), is also required.
In the case of a normal shock wave, the gas properties downstream of the shock are determined by two factors: the
upstream thermodynamic state and a parameter that characterizes the intensity of the shock, usually either the shock
speed u1relative to the upstream gas or the shock Mach number, M1=u1/a1, where a1represents the speed of sound
upstream the shock. The speed of sound, a, is formally defined as a2=p/ρ (∂ln p/∂ ln ρ)s. Other problems, such as
those concerning blast waves, impose the post-shock pressure p2as the initial input parameter describing the shock
intensity. Oblique shocks, on the other hand, require an additional geometrical restriction given by the value of the
incidence angle βor the flow deflection angle θ. It is well known that when the value of βis specified, there exists
a unique solution for the post-shock fluid state (see routine shockObliqueBeta). However, in the case where θis
given, there are two possible solutions for β: a weak shock solution linked to a smaller value of β, representing the
weakest possible shock, and a strong solution corresponding to a larger β(see routine shockObliqueTheta) [95].
13
The two solutions converge for θ=θmax , which corresponds to the maximum deflection angle of the shock for a given
M1(see Fig. 5), above which the problem does not admit any solution. Remarkably close to the maximum deflection
angle, in the weak-shock branch, we find the sonic condition M2=1, below and above which the post-shock flow
is supersonic (weak shocks) and subsonic (strong shocks, and weak shocks between the sonic line and the maximum
deflection angle), respectively. Then, the supersonic branch always corresponds to the weak shock solution [96].
In either scenario, when the value of θis specified, βbecomes an implicit variable that must be determined
numerically. To this end, CT employs an iterative procedure based on the continuity of the tangential velocity across
the shock, which can be manipulated using the above trigonometric identities to give
f(β)≡θ+tan−1 u2n
u1cos β!−β=0.(20)
This equation must be solved for the shock incidence angle βwith use made of the RH relations (19), the ideal gas EoS,
and provided that f′(β)and f′′ (β)can be written as explicit functions. This enables the use of Halley’s third-order
iterative method [97]
βk+1=βk−2f(βk)f′(βk)
2f′(βk)−f(βk)f′′ (βk)(21)
to find the root of Eq. (20). This approach exhibits rapid convergence and meets the default convergence criterion
of 10−3within two or three iterations. However, it is worth noting that like other root-finding methods, Halley’s
method only provides one of the possible roots of the nonlinear system, which generally corresponds to the root
closest to the initial guess used during the iterative process. Therefore, we must supply sufficiently accurate guesses
to cover the whole set of solutions that include both the weak- and strong-shock branches. One straightforward
approach to acquiring such guesses is to anticipate the solution domain bounded by the acoustic weak-shock limit
βmin =sin−1(1/M1)and the normal shock configuration βmax =π/2. In particular, we choose β0=0.5(βmin +βmax)
for the weak-shock branch and β0=0.97βmax for the strong-shock branch.
The left and right plots in Fig. 5 depict the pressure ratio-deflection angle and the incidence angle-deflection
angle shock polar diagrams for dry air (consisting of 78% N2, 21% O2, and 1% Ar) initially at room conditions
(T1=300 K, p1=1 atm). These results were obtained using the shockPolar method. It is worth noting that
CT provides users with a variety of caloric models to choose from: i) calorically perfect gas with frozen chemistry
(constant specific heat at constant pressure cp, adiabatic index γ, and nj), ii) calorically imperfect gas with frozen
chemistry (constant nj), and iii) calorically imperfect gas with variable composition, including dissociation, ionization,
and recombination reactions at equilibrium. These models are incorporated by specifying a sufficiently large set of
species for the calculations, and using NASA’s [33] and Burcat’s (Third Millennium) [34] databases for evaluating the
thermodynamic properties.
Figure 5 displays the results obtained with models ii) and iii) spanning a set of pre-shock Mach numbers M1
ranging from 2 to 14. The results are compared with Caltech’s Shock and Detonation Toolbox [67], which uses
Cantera [66] as kernel for the computations of chemical equilibrium. It is found that the lobes in the pressure ratio-
deflection angle diagram expand due to dissociation/ionization effects, particularly in the hypersonic flow regime,
M1>5. As a result, weak oblique shocks exhibit smaller pressure ratios while strong ones exhibit larger pressure
ratios for the same deflection angle. Moreover, the endothermic (cooling) effect caused by dissociation/ionization in
hypersonic oblique shocks leads to an increase in the post-shock density that also increases the wave deflection angle
at all incidence angles. The results obtained from both codes are in complete agreement for all conditions tested.
However, CT-SD exhibits superior performance compared to Caltech’s SD-Toolbox with Cantera. Our code performs
26×faster on average (CT-SD: 3.78 seconds vs. Caltech’s SD-Toolbox & Cantera: 99.72 seconds; for a large subset
that contains 1200 points of all the cases represented in Fig. 5b, with both codes running on the same platform and
with the same subset of 13 chemical species), which demonstrates the excellent performance of the CT-SD module.
4.2. Regular reflections
Understanding the reflection of shock waves offflat surfaces is a problem of great relevance to high-speed flows.
The angle subtended by the incident shock and the flat surface determines the type of shock reflection, with β=0
representing normal reflections and 0 < β < π/2 oblique reflections. In a reference frame with origin at the point of
contact of the shock with the wall, the latter exhibit an incoming free stream parallel to the wall, with M1>1. For
14
sufficiently small incidence angles, the incident shock deflects the free stream uniformly towards the wall an angle θ.
The reflected shock then deflects back the perturbed stream to its original flow direction parallel to the wall. This type
of reflection is called regular, and is depicted in Fig. 6b. Regular reflections leave uniform flow patterns behind both
the incident and reflected shocks, which can be thoroughly computed using Combustion Toolbox.
By contrast, for incidence angles above a certain critical value, β > βmax(M1), the reflected wave, characterized
by M2<M1and consequently a lower maximum deflection angle, is not able to deflect the flow back to its upstream
direction parallel to the wall. This leads to the so-called irregular, or Mach, reflections, where the reflected and
incident shocks merge into a single wave called the Mach stem, which connects the wall to the triple point where the
three shocks meet, as illustrated in Fig. 6c. These reflections produce non-uniform flows that include a high-speed
shear layer, or slipstream, emanating from the triple point [98]. The properties of these flows cannot be determined
solely by the polar-plot charts or the zero-dimensional RH equations and are therefore out of the scope of this work.
Figure 6: Pressure-deflection shock polar diagrams for a regular shock reflection in atmospheric air (78% N2, 21% O2, and 1% Ar) at 30 km above
sea level (pre-shock temperature T1=226.51 K and pressure p1=1.181 ·10−2atm), pre-shock Mach number M1=20, and deflection angle
θ=35◦; solid line: calorically imperfect gas with dissociation/ionization; dashed line: calorically imperfect gas with frozen chemistry; dotted line:
axes of symmetry; circles: results of Zhang et al. [99]; diamonds: states 1, 2, and 3.
To calculate regular reflections, CT-SD uses the routine shockObliqueReflectedTheta to compute the incident
wave by specifying the wave angle β(or the flow deflection θ) and the pre-shock velocity u1. This results in the
calculation of the post-shock state (2), which serves as pre-shock state for the reflected wave (see sketch in Fig. 6b).
If the incident shock is sufficiently strong, the transition to state (2) can result in significant thermochemical effects
that may cause changes in the aerothermal properties compared to those of a calorically perfect fixed-composition
gas. In this case, the values of M2and θmay change accordingly. The reflected shock increases the gas pressure
and temperature even further, and the properties in state (3) can be determined using the code routine employed for
single oblique shocks imposing the counter deflection angle θcalculated for the incident shock. This guarantees that
the streamlines in state (3) are parallel to the reflecting surface. If there is no solution for the reflected shock (which
occurs for sufficiently large values of θ), irregular Mach reflections occur. As discussed above, these reflections
involve non-uniform flow properties and non-steady solutions, and thus their computation is beyond the capabilities
of our code.
As an illustrative example, Fig. 6 represents the pressure-deflection shock polar diagrams for a regular shock
reflection in atmospheric air at 30 km above sea level with an incident Mach number of M1=20 and a deflection
angle of θ=35◦under the same gas models ii) and iii) used in the oblique shock charts presented previously. The
polar plot starting from state (2), obtained by increasing the incident wave angle from 0 to π/2, determines the solution
of state (3) for the deflection angle θ=35◦. This, in turn, determines the reflected shock at the intersection of the
second polar with the vertical axis (θ=0). As can be seen, the dissociation and ionization effects are more pronounced
in the reflected shock, as the accumulated temperature jump in both shocks amplifies the endothermicity associated
15
with the dissociation reactions, resulting in substantially higher overall pressure ratios. Finally, the outcomes are
compared with those acquired by Zhang et al. [99] under the same flow conditions, revealing excellent agreement
in all instances. In our calculations, the computation time was 1.75 seconds for a group of 28 species and 200 case
studies. These values depend on the tolerance, which was set to 10−5for the root-finding method.
As previously discussed, for each M1, there exists a maximum value of β(and, consequently, of θ) beyond which
regular reflection is impossible. This maximum value is determined using the shockPolarLimitRR method, which
identifies the point where the polar diagram of the reflected shock becomes tangent to the θ=0 axis. Our code is
able to compute βmax and θmax in cases involving high-temperature thermochemical effects. To determine this limit,
we impose the condition θ3,max −θ=0 and employ an iterative algorithm based on Broyden’s method [100], which
makes use of the set of routines described above.
4.3. Planar gaseous detonations
The thermochemical framework used to describe shock waves involving endothermic molecular transformations
can be easily extended to account for exothermic reactions, as occurs in planar detonations. As with shock waves, the
computation of detonations requires knowledge of the pre-shock state and the degree of overdrive that measures the
contribution of the external supporting mechanism. However, unlike shock waves, detonations can be self-sustained,
i.e., propagate without any external contribution exerting additional pressure from behind. This propagation mode,
named after Chapman-Jouguet (CJ), involves the maximum possible expansion of the hot products. Therefore, the
burnt-gas state is obtained by imposing the sonic condition in the post-wave flow M2=1. If the pressure behind the
detonation wave is larger than what is anticipated by the CJ condition, which can only be achieved using an external
forcing mechanism, the detonation is considered over-driven and results in subsonic downstream conditions, with
M2<1. Conversely, under-driven detonations occur when the burnt gas is in a supersonic state with M2>1. How-
ever, these types of waves are typically disregarded because the post-shock Mach waves within the non-equilibrium
region diverge from the oblique shock wave, leading to acoustic decoupling from the wall and making wedge at-
tachment impossible [95]. Further studies also suggest that under-driven detonations may violate the second law of
thermodynamics [101]. Nevertheless, since post-detonation equilibrium conditions are independent of the internal
structure, under-driven detonations are not excluded by CT when calculating the complete spectrum of possibilities.
The over-driven/under-driven solutions can be determined numerically for a defined upstream mixture and a given
degree of overdrive η=u1/ucj (see functions detOverdriven and detUnderdriven) by using the routines designed
for CJ detonations (see detCJ) and normal shocks (see shockIncident). The first routine is needed to determine the
minimum velocity u1=ucj (or η=1) required for a planar detonation to propagate, while the second is used to obtain
the post-detonation state for a given degree of overdrive η.
Careful selection of the initial guesses is also required to obtain the solutions for over-driven and under-driven
detonations. For instance, T2,guess and p2,guess denote the estimated temperature and pressure after the detonation front,
and should be anticipated considering the significant variations arising from the degree of overdrive and the type
of propagation mode. To obtain the initial guesses for over-driven detonations, p2,guess is computed using Eq. (19),
assuming a constant γ=γ1=γ2,guess. This gives p2,guess =p1(2γM2
1−γ+1)/(γ+1). The temperature guess is
based on the fact that, for sufficiently strong shocks, the kinetic energy downstream is much lower than upstream of
the shock, u2
2/u2
1∼(ρ1/ρ2)2≪1. This simplifies Eq. (19) to h2,guess =h1+u2
1/2, thereby enabling the computation
of T2,guess by solving the thermochemical equilibrium problem at specified enthalpy and pressure, h2,guess and p2,guess.
This approximation becomes more accurate with increasing degrees of overdrive, as the differences between u1and
u2become more significant. This estimation method is the same as the one utilized in the incident normal shocks
routine. For under-driven detonations, a reasonable initial guess can be obtained by considering the range of acceptable
values for the mean post-shock density. By defining the dimensionless parameter ζ∈(0,1) to measure how close
ρ2,guess is to the CJ state compared to the initial state, we can construct an initial guess for the density as follows:
ρ2,guess =[ζ/ρ2,cj +(1 −ζ)/ρ1]−1. The pressure and temperature values can then be determined using Eq. (19) and the
ideal gas EoS, respectively. It has been found that a value of ζ=0.1 is suitable for the set of Mach numbers tested.
4.4. Oblique gaseous detonations
Detonation waves can also occur in oblique configurations, although they are much less common than oblique
shocks. However, oblique detonations are critical in Oblique Detonation Wave Engines (ODWE) [102, 103], where the
combustion process occurs across an oblique detonation wave that revolves around a cylindrical combustion chamber.
16
Figure 7: Pressure-deflection (a) and wave angle-deflection (b) detonation polar diagrams for a stoichiometric hydrogen (H2) air (79% N2, 21%
O2) mixture at pre-shock temperature T1=300 K and pressure p1=1 atm, and a range of pre-shock Mach numbers M1between 5 and 10; solid
line: results from CT considering a calorically imperfect gas with dissociation; circles: results from Zhang et al. [99].
To compute oblique detonations, knowledge of the pre-shock state and the degree of overdrive caused by the sup-
porting mechanism, typically a wedge deflecting a reactive supersonic stream and generating the oblique detonation, is
required, just like in oblique shocks. Thus, given the temperature, pressure, composition, and pre-shock velocity, one
can calculate the detonation polar diagrams with the particularity that now the exothermicity of the reaction increases
the number of possible solutions, as occurs for planar detonations [104, 105]. Then, for a given detonation angle β(or
deflection angle θ), two solutions for the burnt-gas state can be found, associated with under-driven (M2n>1) and
over-driven (M2n<1) conditions (see routines detonationObliqueBeta and detonationObliqueTheta).
At the Chapman-Jouguet regime the two solutions merge into a single solution. This state is characterized by a
sonic normal component of the downstream velocity vector (M2n=1). The corresponding values for the upstream
Mach number and shock angle are M1=M1,cj and β=βcj. Both in the under-driven and over-driven cases, the
RH-equations only produce real solutions if the values for the upstream Mach number and shock angle are larger than
the corresponding values for the CJ condition, M1≥M1,cj and β≥βcj. For oblique detonations with angles in the
range βcj < β < π/2, a given shock angle corresponds to two different deflection angles, namely θover and θunder.
In the over-driven branch, the solution resembles that of an oblique shock. The condition M2=1 is reached just
below the maximum deflection angle separating the strong and weak solutions. Since the solution is multi-valued,
the computation of the different branches requires accurate initial guesses regardless of the input parameter, be it the
shock or the flow deflection angle (see Section 4.3).
CT provides embedded functionalities for obtaining polar diagrams that characterize incident oblique detonations
(see function detPolar). These functionalities perform a direct computation of a set of cases (100 by default)
by sweeping the range of possible solutions for both the under-driven and over-driven branches. As an illustrative
example, Fig. 7 shows the pressure-deflection (a) and wave angle-deflection (b) polar diagrams for detonations in
stoichiometric hydrogen (H2)-air (79% N2, 21% O2) mixtures at pre-shock temperature T1=300 K and pressure
p1=1 atm, for a range of pre-shock Mach numbers M1between 5 and 10. The results are compared with those of
Zhang et al. [99], which are found to be in remarkable agreement. The computation time was 5.57 seconds for a set
of 26 species and 1500 case studies and a tolerance of 10−5for the root-finding method. For comparative purposes,
17
the computation time required by Caltech’s SD Toolbox [68] with Cantera [66], which only provides the over-driven
branch, was 101.77 seconds. This represents a 18×speed-up factor for our code.
The blue-shaded area shown in Fig. 7 corresponds to weak over-driven conditions, which are the most likely to
occur in oblique detonations. This region, bounded by θcj < θ < θmax, is of significant interest due to its applicability
to the study of ODWE systems [106, 107]. For instance, recent research by Guo et al. [107] investigated the impact
of pre-shock conditions on the stationary window for CH4-air oblique detonations. The authors discovered that the
limits θcj and θmax depend strongly on the pre-shock velocity and heat release associated with the mixture, while
variations with the upstream temperature and pressure are not as prominent. Combustion Toolbox allows to perform
these calculations easily, highlighting its relevance in carrying out preliminary studies before tackling more complex
flow configurations.
5. Rocket module
The calculation of the theoretical performance of rocket engines has drawn renewed attention in recent times due
to the emergence of new private space companies such as Virgin Galactic, SpaceX, Blue Origin, Rocket Lab, or PLD
Space, focused on the development of low-cost and reusable launch vehicles [108]. Despite the inherent complexities
of these systems associated with the variety of physicochemical phenomena involved, a reasonably accurate estimation
of engine performance can be achieved with fairly simple thermochemical calculations. In particular, since rocket
engines typically operate at moderate pressures, the ideal gas assumption can be applied without the need for more
complex equations of state. Additionally, the long residence times of the reacting gases in the combustion chamber
compared to the chemical reaction times allow for further simplification of the calculations. This simplification allows
for the utilization of thermochemical equilibrium tools such as CT.
The initial release of CT-ROCKET incorporates the mathematical description proposed in [17], implemented
within the RocketSolver class. This method relies on several simplifying assumptions, including one-dimensional
flow, uniform cross-sectional area, negligible flow velocity in the combustion chamber inlet, adiabatic combustion,
isentropic expansion in the nozzle, homogeneous flow, the ideal gas law, and continuity of temperature and velocity
between gaseous and condensed species. Further details on the numerical implementation can be found in Ref. [17,
Chapter 6].
Figure 8: Sketch of the cross section of a finite area chamber (FAC) rocket engine. The dashed line represents the difference with an infinite area
chamber (IAC), which is only included for the top region for clarity. Chemical transformations: (p-inj) and (p-inf) instant adiabatic combustion
at constant pressure (HP); (inj-c) entropic process; (inf-t), (c-t), and (t-e) isentropic process at defined pressure (SP). Right: variation of the
temperature ( ) and Mach number ( ) from the combustor end (c) to the exit (e) for a LOX/RP1 mixture with equivalence ratio φ=1.5 in a
high-pressure combustion chamber p1=100 atm considering calorically imperfect gas with dissociation (line) and calorically imperfect gas with
frozen chemistry (dashed).
CT-ROCKET leverages the ability of the CT-EQUIL module to determine the gas composition within the rocket
engine at various points of interest, such as the injector (inj), the combustion chamber outlet (c), the nozzle throat (t),
and different points between (t) and (c/inf) where the hot gases are compressed (subsonic region) or between (t) and
the nozzle outlet (e) where the hot gases undergo the final expansion (supersonic region), as illustrated in Fig. 8 (left).
18
The right panel shows the temperature and the Mach number of the fluid particles from the combustion chamber outlet
(c) to the exit (e), passing through the throat (t) with At=Ac/3, for a LOX/RP1 mixture with equivalence ratio φ=1.5
(representing a 2.27 oxidizer/fuel weight ratio) in a high-pressure combustion chamber at p1=100 atm. The area
ratio is taken as the control variable, upon condition that thermochemical equilibrium is achieved at each position.
Additionally, the module calculates the thrust generated by the rocket engine. CT-ROCKET allows for calcula-
tions using either frozen chemistry or chemical equilibrium, accounting for combustion chambers with both finite
(entropic process) and infinite (isentropic process) dimensions. The frozen chemistry and chemical equilibrium ap-
proaches provide an estimate of the performance limits of rocket engine nozzles, as demonstrated by Grossi et al. [109]
through two-dimensional numerical simulations based on finite-rate kinetics. This feature enables the performance
of parametric analyses to determine the optimal theoretical configuration for a given launch condition or to evaluate
the environmental impact at various stages of the rocket vehicle. For instance, with the increasing number of space
launches [110, 111], there has been a shift away from highly toxic fuels such as unsymmetrical dimethylhydrazine
(UDMH) during the early phases of launch, towards so-called “green” propellants like kerosene (RP1) [112]. It is
expected that the use of toxic fuels will be further restricted or even prohibited in the near future, thus driving the need
to gain more experience with alternative “green” propellants [113]. The CT-ROCKET module can therefore prove to
be a valuable tool in this endeavor.
As previously discussed, this module also includes various routines to compute the state of the mixture at different
points of the rocket engine. These states can be modeled using either an infinite area chamber (IAC) or a finite area
chamber (FAC) implemented through the rocketIAC and rocketFAC methods, respectively. For example, in the FAC
model, an iterative procedure is employed to determine the mixture states at the chamber outlet (c) and at the throat (t).
This is achieved by utilizing the IAC model upon defining the initial fresh mixture’s composition, temperature, and
pressure, along with the area ratios Ac/Atand Ae/At.
Figure 9: Thermodynamic properties at the nozzle exit of a rocket engine with aspect ratios Ac/At=2 and Ae/At=3 for different liquid bi-
propellant mixtures in a high-pressure combustion chamber, p1=100 atm, with equivalence ratios φ∈[0.5,4]: temperature, T(a), pressure, p(b),
enthalpy, h(c), specific heat capacity at constant pressure, cp(d), adiabatic index, γs(e), gas velocity, u(f), specific impulse at sea level, Isp (g),
specific impulse in a vacuum, Ivac (h); solid line: results obtained with CT; symbols: results obtained with the NASA’s CEA [17]: LOX/LH2(♢),
LOX/RP1 (), LOX/LCH4(△), N2O4/MMH (□).
Numerous validations were conducted using NASA’s CEA code to ensure the reliability and robustness of CT-
ROCKET. As an example, Fig. 9 displays a range of thermodynamic properties computed at the nozzle exit (e). The
geometrical aspect ratios defining the combustion chamber and the nozzle are Ac/At=2 and As/At=3. Several
reacting mixtures were utilized in the computations, including LOX/LH2, LOX/RP1, LOX/LCH4, and N2O4/MMH,
the latter consisting of nitrogen tetroxide and monomethyl-hydrazine. The reactants were introduced in a combus-
tion chamber where they reacted isobarically under high-pressure conditions, p1=100 atm. The inlet temperature
of the propellants was set to their respective boiling points, except for N2O4, which was evaluated at 300 K. The
computations were performed over a wide range of equivalence ratios, φ∈[0.5,4], to investigate the impact of the
fuel-to-oxidizer ratio on the combustion process and the resulting reaction products.
As illustrated in the different subplots of Fig. 9, CT-ROCKET accurately predicts the properties of primary interest
at the nozzle exit, including temperature (T) in (a), pressure (p) in (b), enthalpy (h) in (c), specific heat capacity at
19
constant pressure (cp) in (d), adiabatic index (γs) in (e), gas velocity (u) in (f), specific impulse at sea level (Isp )
in (g), and specific impulse in a vacuum (Ivac) in (h). The results demonstrate excellent agreement with the CEA
code, with uniform convergence. However, the NASA code exhibited numerical instabilities for certain cases, such as
LOX/RP1 at φ=3 and LOX/LCH4 at φ=4. The computation time for LOX/H2 was 11.85 seconds for a set of 11
species and a total of 351 cases. The other mixtures were computed using 94 species, with an average computation
time of 30.88 seconds. It is noteworthy that the computation time per species is almost three times less for the latter
cases. The performance of the CT-ROCKET module has improved by a factor of 1.9×compared with the previous
release [61, 69].
6. Graphic User Interface
This section presents a detailed overview of the Graphic User Interface (GUI) developed for CT. The GUI is
intended to provide a user-friendly and intuitive interface for the visualization and analysis of data. Figures 10 and 11
depict the fundamental elements of the GUI, described in detail below:
•The menu bar comprises predefined actions such as clear, save, snapshot, and check for updates, along with
options to access online documentation, tutorials, examples, and license. It also provides access to additional
tools, or add-ons, that can expand the GUI’s capabilities. These add-ons include controls for numerical errors
and visualization settings, species selection, as well as the ability to perform code validations and provide
feedback to the development team (see CT documentation or its website).
•The interface is divided into two main tabs (setup and results) and additional sub-tabs designed to organize the
content and prevent the user from feeling overwhelmed. The setup tab contains a control panel to configure
the problem to be addressed. The results tab contains a data visualization area for post-processing all collected
data.
•The control panel is a crucial part of the GUI that enables users to configure the problem conditions. It provides
a range of controls and options to adjust parameters such as the chemical species (reactants and products), the
initial state (composition, temperature, and pressure), the type of problem to be solved, and other parameters
for the setup of single-case or parametric studies.
•The command window provides a command-line interface that allows users to interact with the GUI through a
series of text commands. This feature is particularly useful for advanced users who prefer to work with code or
scripts. Through this tool, users can input commands and execute scripts, while the dialog box prompts the user
for further input or confirmation before executing a command. The dialog box displays practical information
like warnings, errors, and execution time, providing valuable feedback to the user.
•The lamp component serves as a visual indicator of the analysis status. When the analysis is complete, the
lamp emits a green light, while a yellow light indicates that the computations are still in progress. A red light
indicates an error in the analysis.
•The data visualization area displays the computed results in a visual format, such as plots or tables. It allows
users to interact with and explore the data in a way that’s intuitive and easy to understand.
•The tree component collects all the data and exhibits the hierarchical organization of the obtained outcomes,
which enables users to explore and access distinct aspects of each case.
•Additional features have been incorporated to improve the GUI’s usability, including context menus and key-
board shortcuts. These functionalities enable users to perform intricate tasks with ease and speed.
For illustrative purposes, Figs. 10 and 11 show several Combustion Toolbox GUI screenshots captured during
the parametric study of adiabatic isobaric combustion of acetylene-air mixtures reported as second validation test in
Section 3.4. This case corresponds to the data displayed in Fig. 3. As observed in Fig. 10, the first step involves
setting up the problem conditions, which includes i) the initial mixture (composition, temperature, and pressure), that
20
Figure 10: Example of GUI configuration (a) and result post-processing (b) to reproduce the case of Fig. 3. In particular, the thermodynamic
properties correspond to the case selected in the tree object (φ=0.5).
can be chosen from the predefined mixtures via the reactants drop-down menu, or by manually adding the appropriate
species name one-by-one to the same object; ii) the problem configuration (adiabatic at constant pressure, etc.); iii)
the control parameter for a parametric or individual study, such as the equivalence ratio; and iv) additional input
parameters that may be required based on the type of problem.
If a parametric study is selected with the equivalence ratio as the control parameter, the post-processing step can
be initiated as depicted in Fig. 10b. If the problem is well-posed, the lamp object will turn green, meaning that the
computations were carried out successfully, and a message will appears in the dialog box of the GUI. Subsequently,
the obtained dataset is re-structured in the background to fit into the tree object for result post-processing. By selecting
each solution of the tree object, the GUI automatically updates the thermodynamic properties of the mixtures in the
data visualization area (see Fig. 10). Additionally, in the results →custom figures tab, all mixture properties can be
analyzed by plotting the results, as illustrated in Fig. 11. The datasets collected using the GUI can also be exported to
a structured spreadsheet or .mat file.
In brief, the GUI offers a broad range of tools for examining problems related to chemical equilibrium. The user-
friendly design and intuitive features make it accessible to a variety of users, including those with a limited technical
background. The GUI is an essential tool for researchers and practitioners who need to perform and analyse extensive
parametric studies of the wide range of problems that can be addressed by the code. However, it is important to note
that the GUI is intended to supplement traditional coding approaches instead of replacing them. Despite its ability to
streamline tasks for non-experts, plain code actually exhibits greater versatility. The GUI itself is constructed upon
an existing codebase, whose fundamental functions and calculations are still available for access and manipulation
via the command line interface. In fact, proficient users interested in intricate analytical requirements may find that
direct coding and execution is a more efficient and effective method than relying exclusively on the GUI. Thus, the
GUI ought to be regarded as a tool that assists users with specific tasks rather than as a replacement for the potent and
flexible nature of traditional coding.
21
Figure 11: Post-processing the results of Fig. 3 through the GUI: (a) chemical composition and (b) custom plots. The tab results →mixture
composition shows the mixture composition for the case selected in the tree object (φ=0.5).
7. Conclusions
In this work, we present the Combustion Toolbox (CT), an innovative open-source thermochemical code devel-
oped for the calculation of equilibrium states in gaseous reacting systems. While CT primarily focuses on combustion
problems that may involve the formation of condensed-phase species, its abilities extend to other areas of interest, in-
cluding the calculation of the atmospheric compositions of gaseous exoplanets, ablation processes, hypersonic shocks,
and detonations. CT has been implemented in MATLAB and designed with an object-oriented modular architecture,
making it both user- and developer-friendly. Additionally, CT is equipped with an advanced Graphic User Inter-
face that encapsulates the three modules and multiple built-in functions, providing users with a convenient operating
experience.
At present, the three modules included in CT are CT-EQUIL, CT-SD, and CT-ROCKET. The first module, CT-
EQUIL, is the kernel of the Combustion Toolbox and is responsible for solving the chemical composition of the system
at equilibrium. This is achieved by minimizing the Gibbs/Helmholtz free using the Lagrange multiplier approach
coupled with a multidimensional Newton-Raphson method. The second module, CT-SD, solves post-shock/detonation
states for normal and oblique incident flows, including the computation of reflected waves. The third module, CT-
ROCKET, is designed to determine the mixture composition at various points of interest within rocket engines, along
with the calculation of the theoretical rocket performance.
The modules have been validated against existing state-of-the-art codes, including NASA’s Chemical Equilibrium
with Applications (CEA) [17], Cantera [66] within Caltech’s Shock and Detonation Toolbox (SD-Toolbox) [67, 68],
and the newly developed Thermochemical Equilibrium Abundances (TEA) code [25]. All tests showed excellent
agreement. As a matter of fact, CT exhibits superior computational performance in terms of cost and time, outper-
forming Caltech’s SD-Toolbox and TEA by a significant margin. Additional validations can be accessed through the
web at https://combustion-toolbox-website.readthedocs.io, which also provides further documentation and examples.
The tool is actively maintained and can be accessed at https://github.com/CombustionToolbox/combustion toolbox.
22
While the Combustion Toolbox has shown promising results, it is still an ongoing research project that requires
additional development to enhance its capabilities. We aim to introduce additional functionalities in future versions of
the code, such as the incorporation of non-ideal equations of state (currently under implementation), the analysis of
multi-phase systems, a more accurate model for rocket propellant performance, and the extension of the database to
include transport properties. We are also considering expanding the code to other well-known open-source program-
ming languages, such as C++ and Python. The former is preferred due to its exceptional performance compared to
MATLAB [70], while the latter is desired due to its simplicity [114]. An intermediate step will involve using MEX
functions in the kernel of the code to combine C++ and MATLAB for calculating chemical equilibrium at defined
temperature and pressure/volume, which is anticipated to substantially improve the speed of the code.
Acknowledgements
The authors acknowledge funding from Comunidad de Madrid under the Multiannual Agreement H2SFE-CM-
UC3M. C.H. also received support from projects TED2021-129446B-C41 and PID2022-139082NB-C5 (MICIN-
N/FEDER, UE). M.V. would like to express his lifetime gratitude for the invaluable and inspiring collaborations
with Professors Amable Li˜
n´
an and Antonio L. S´
anchez over the past twenty-five years. Their seminal contributions,
insight, motivation, and long term support played a pivotal role in initiating and promoting this research effort.
Declaration of Competing Interest
The authors declare that they have no competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
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