Utilities for the Fluid Properties Module

Newton method solver

The fluid properties module includes a 1D and a 2D Newton method solver.

1D solver

The 1D Newton method solver is used to solve the problem:

Find $var element = document.getElementById("moose-equation-cf2dd104-83e3-4652-a6c2-9084b6aacec3");katex.render("z", element, {displayMode:false,throwOnError:false});$ such that $var element = document.getElementById("moose-equation-f8d1f386-34c4-487a-b0b5-17532d10aed3");katex.render("f(x, z) = y", element, {displayMode:false,throwOnError:false});$ , with $var element = document.getElementById("moose-equation-db129b87-a03d-4487-b6da-4c5a113965e0");katex.render("x", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-efe0b678-d742-4b52-a65d-dae6128f28df");katex.render("y", element, {displayMode:false,throwOnError:false});$ known constants and $var element = document.getElementById("moose-equation-cdaae38b-d54a-4760-9275-2f59b76b004f");katex.render("f", element, {displayMode:false,throwOnError:false});$ a function.

$var element = document.getElementById("moose-equation-4910a63d-0613-480f-aaac-78872e9e72ec");katex.render("f", element, {displayMode:false,throwOnError:false});$ should compute both its return value and the derivatives of its value with regards to both its variables using references.

The routine will error if a NaN is encountered or if the maximum number of iterations is exceeded.

2D solver

The 2D Newton method solver is used to solve the problem:

$var element = document.getElementById("moose-equation-d2f3dd03-12fe-4c2c-8211-1854be723953");katex.render("\\begin{split} f1(x, y) &= a \\\\ f2(x, y) &= b \\end{split}", element, {displayMode:true,throwOnError:false});$

with $var element = document.getElementById("moose-equation-97b78c6c-376a-4de2-8ec0-7b22961c4536");katex.render("a", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a26492ec-aaf3-439e-aaf9-94474071bc13");katex.render("b", element, {displayMode:false,throwOnError:false});$ known constants, $var element = document.getElementById("moose-equation-38462be2-5e9b-492c-ab6a-520a5802daac");katex.render("f1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-cb91b193-6495-41de-b12f-b2e8b3c0feae");katex.render("f2", element, {displayMode:false,throwOnError:false});$ functions and $var element = document.getElementById("moose-equation-609e1a9d-4ef9-4705-89b4-1e0ef672e5e2");katex.render("x", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-2324734e-973b-4a5c-ab86-d2260047abc8");katex.render("y", element, {displayMode:false,throwOnError:false});$ the real-valued variables to solve for.

The functions should compute both their return value and the derivatives of their value with regards to both variables using references.

The routine will not error if a NaN is encountered or if the maximum number of iterations is exceeded. Instead, a converged boolean will be set to false, and the routine will return the last valid estimates of $var element = document.getElementById("moose-equation-d896113d-a464-406a-badb-ccc70e0cb489");katex.render("x", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-2cf642f6-5875-49b1-bddc-65d2446435e2");katex.render("y", element, {displayMode:false,throwOnError:false});$ . It is the responsibility of the user to handle this unconverged output.

Brent's method solver

Brent's method may be used to find roots of a general function. It was adapted from Numerical Recipes in C++.

More information may be found here.

NASA 7-coefficient polynomials

The NASA7 utilities, as used in NASA7IdealGasFluidProperties, evaluate the standard-state thermodynamic properties of ideal-gas species from NASA 7-coefficient polynomials (McBride et al., 1993). Each species has two sets of coefficients $var element = document.getElementById("moose-equation-4c671beb-3384-42f4-abbb-2c7c3a82e8f8");katex.render("a_0, \\ldots, a_6", element, {displayMode:false,throwOnError:false});$ , one for temperatures $var element = document.getElementById("moose-equation-02eb3bd0-db5e-4d66-a562-d64970816488");katex.render("T \\le T_\\text{mid}", element, {displayMode:false,throwOnError:false});$ and one for $var element = document.getElementById("moose-equation-3770c2d3-8f17-47da-8a32-80e3174f19f3");katex.render("T > T_\\text{mid}", element, {displayMode:false,throwOnError:false});$ :

var element = document.getElementById("moose-equation-1b364f54-cbb9-4a96-a6c2-a53d80833291");katex.render("\\frac{c_p^\\circ}{R} = a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4 \\,,", element, {displayMode:true,throwOnError:false});

var element = document.getElementById("moose-equation-c1bc8fdd-f159-4cbc-836f-9f56d896ad67");katex.render("\\frac{h^\\circ}{R T} = a_0 + \\frac{a_1}{2} T + \\frac{a_2}{3} T^2 + \\frac{a_3}{4} T^3 + \\frac{a_4}{5} T^4 + \\frac{a_5}{T} \\,,", element, {displayMode:true,throwOnError:false});

var element = document.getElementById("moose-equation-d6b68948-de95-422c-ba93-bf13115004e2");katex.render("\\frac{s^\\circ}{R} = a_0 \\ln T + a_1 T + \\frac{a_2}{2} T^2 + \\frac{a_3}{3} T^3 + \\frac{a_4}{4} T^4 + a_6 \\,,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-75c90ce7-5016-498d-9706-af737344505e");katex.render("c_p^\\circ", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-365721aa-6ce2-43a5-be50-3fafe79d0e3b");katex.render("h^\\circ", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-90051675-cefa-4d82-bbd4-0dd22f664dcb");katex.render("s^\\circ", element, {displayMode:false,throwOnError:false});$ are the molar isobaric heat capacity, enthalpy, and entropy at the reference pressure of the data, and $var element = document.getElementById("moose-equation-41e8f62c-855a-41cd-abfe-7b25dbca04f9");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the universal gas constant. The enthalpy includes the enthalpy of formation, so the enthalpies of different species share a common reference and reaction enthalpies follow from their differences. The entropy is the absolute (third-law) entropy. The utilities return these dimensionless forms, so they do not depend on the value of $var element = document.getElementById("moose-equation-54e3812b-2898-488f-a83f-758224531b50");katex.render("R", element, {displayMode:false,throwOnError:false});$ , and they accept automatic differentiation types.

The built-in species are H $var element = document.getElementById("moose-equation-19f42e1e-a283-4cc0-b530-2b4107985607");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , O $var element = document.getElementById("moose-equation-ed043a37-a4c9-46b7-9f0b-c765adccaf8f");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , H $var element = document.getElementById("moose-equation-1fc211d6-0482-492b-bce0-047d8f2f1c14");katex.render("_2", element, {displayMode:false,throwOnError:false});$ O, CH $var element = document.getElementById("moose-equation-dc2dbfa5-513e-45fa-a107-5939a796e656");katex.render("_4", element, {displayMode:false,throwOnError:false});$ , CO, CO $var element = document.getElementById("moose-equation-aa65f7f4-0934-4a31-9f2b-bd9f49b511e5");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , and N $var element = document.getElementById("moose-equation-06bb5d4e-0da9-43e6-ba56-56b621c6a918");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , with the GRI-Mech 3.0 data (Smith et al., 1999) as distributed with Cantera 3.2.0 (Goodwin et al., 2025) in gri30.yaml. The reference pressure is 101325 Pa, and $var element = document.getElementById("moose-equation-c0f7a716-fddd-4afe-8ee7-fa21c7fee563");katex.render("T_\\text{mid} = 1000", element, {displayMode:false,throwOnError:false});$ K. The fits cover 200 K to 3500 K, except for N $var element = document.getElementById("moose-equation-f91d96a5-61f7-41b1-9181-8de33682cc10");katex.render("_2", element, {displayMode:false,throwOnError:false});$ , which covers 300 K to 5000 K. The utilities do not check these ranges. The molar masses are those Cantera computes from its atomic weights.

At $var element = document.getElementById("moose-equation-0ca63262-b1df-4760-a60e-4fd85eba9452");katex.render("T_\\text{mid}", element, {displayMode:false,throwOnError:false});$ the low-temperature polynomial applies, as in Cantera. The two polynomials of a species do not match exactly at $var element = document.getElementById("moose-equation-1ee35391-6dac-4445-a094-6fcc53e20f5a");katex.render("T_\\text{mid}", element, {displayMode:false,throwOnError:false});$ : for the built-in species, the mismatch is below $var element = document.getElementById("moose-equation-0ba6b4ba-5288-4ab0-9a9a-910db95c6493");katex.render("4 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$ relative in $var element = document.getElementById("moose-equation-f327c309-f0c7-4336-b43c-abeeb59a2879");katex.render("c_p^\\circ", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4b402ab3-fcde-465a-a69a-fe43dc32f7be");katex.render("6 \\times 10^{-3}", element, {displayMode:false,throwOnError:false});$ J/mol in $var element = document.getElementById("moose-equation-be71144d-687a-445e-b00f-821c16af1532");katex.render("h^\\circ", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-ad54205a-5002-4181-8757-c121a6a28f2b");katex.render("2 \\times 10^{-5}", element, {displayMode:false,throwOnError:false});$ J/(mol K) in $var element = document.getElementById("moose-equation-8cdb55d5-1012-4af7-91ea-22e676561061");katex.render("s^\\circ", element, {displayMode:false,throwOnError:false});$ .

References

David G. Goodwin, Harry K. Moffat, Ingmar Schoegl, Raymond L. Speth, and Bryan W. Weber. Cantera: an object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes. 2025. Version 3.2.0. URL: https://www.cantera.org.

BibTeX

@misc{cantera320,
    author = "Goodwin, David G. and Moffat, Harry K. and Schoegl, Ingmar and Speth, Raymond L. and Weber, Bryan W.",
    title = "Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes",
    note = "Version 3.2.0",
    url = "https://www.cantera.org",
    year = "2025"
}

RIS

TY  - GEN
AU  - Goodwin, David G.
AU  - Moffat, Harry K.
AU  - Schoegl, Ingmar
AU  - Speth, Raymond L.
AU  - Weber, Bryan W.
TI  - Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes
PY  - 2025
UR  - https://www.cantera.org
ER  -

Plain Text

David G. Goodwin, Harry K. Moffat, Ingmar Schoegl, Raymond L. Speth, and Bryan W. Weber. Cantera: an object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes. 2025. Version 3.2.0. URL: https://www.cantera.org.

Bonnie J. McBride, Sanford Gordon, and Martin A. Reno. Coefficients for calculating thermodynamic and transport properties of individual species. Technical Report NASA TM-4513, NASA Lewis Research Center, 1993.

BibTeX

@techreport{mcbride1993,
    author = "McBride, Bonnie J. and Gordon, Sanford and Reno, Martin A.",
    title = "Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species",
    institution = "NASA Lewis Research Center",
    number = "NASA TM-4513",
    year = "1993"
}

RIS

TY  - RPRT
AU  - McBride, Bonnie J.
AU  - Gordon, Sanford
AU  - Reno, Martin A.
TI  - Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species
PY  - 1993
IS  - NASA TM-4513
ER  -

Plain Text

Bonnie J. McBride, Sanford Gordon, and Martin A. Reno. Coefficients for calculating thermodynamic and transport properties of individual species. Technical Report NASA TM-4513, NASA Lewis Research Center, 1993.

Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. Moriarty, Boris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson, Soonho Song, William C. Gardiner, Jr., Vitali V. Lissianski, and Zhiwei Qin. GRI-Mech 3.0. 1999. URL: http://combustion.berkeley.edu/gri-mech/.

BibTeX

@misc{smith_grimech30,
    author = "Smith, Gregory P. and Golden, David M. and Frenklach, Michael and Moriarty, Nigel W. and Eiteneer, Boris and Goldenberg, Mikhail and Bowman, C. Thomas and Hanson, Ronald K. and Song, Soonho and Gardiner, Jr., William C. and Lissianski, Vitali V. and Qin, Zhiwei",
    title = "{GRI-Mech} 3.0",
    url = "http://combustion.berkeley.edu/gri-mech/",
    year = "1999"
}

RIS

TY  - GEN
AU  - Smith, Gregory P.
AU  - Golden, David M.
AU  - Frenklach, Michael
AU  - Moriarty, Nigel W.
AU  - Eiteneer, Boris
AU  - Goldenberg, Mikhail
AU  - Bowman, C. Thomas
AU  - Hanson, Ronald K.
AU  - Song, Soonho
AU  - Gardiner, William C., Jr.
AU  - Lissianski, Vitali V.
AU  - Qin, Zhiwei
TI  - GRI-Mech 3.0
PY  - 1999
UR  - http://combustion.berkeley.edu/gri-mech/
ER  -

Plain Text

Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. Moriarty, Boris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson, Soonho Song, William C. Gardiner, Jr., Vitali V. Lissianski, and Zhiwei Qin. GRI-Mech 3.0. 1999. URL: http://combustion.berkeley.edu/gri-mech/.

Newton method solver
Brent's method solver
NASA 7-coefficient polynomials
References

Theory

Usage

Development

Utilities for the Fluid Properties Module

Newton method solver

1D solver

2D solver

Brent's method solver

NASA 7-coefficient polynomials

References