OpenFOAM v9 User Guide

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7.1 Thermophysical models

Thermophysical models are concerned with energy, heat and physical properties. The thermophysicalProperties dictionary is read by any solver that uses the thermophysical model library. A thermophysical model is constructed in OpenFOAM as a pressure-temperature $\text{[math]}$ system from which other properties are computed. There is one compulsory dictionary entry called thermoType which speciﬁes the package of thermophysical modelling that is used in the simulation. OpenFOAM includes a large set of pre-compiled combinations of modelling, built within the code using C++ templates. This coding approach assembles thermophysical modelling packages beginning with the equation of state and then adding more layers of thermophysical modelling that derive properties from the previous layer(s). The keyword entries in thermoType reﬂects the multiple layers of modelling and the underlying framework in which they combined. Below is an example entry for thermoType:

thermoType
{
    type            hePsiThermo;
    mixture         pureMixture;
    transport       const;
    thermo          hConst;
    equationOfState perfectGas;
    specie          specie;
    energy          sensibleEnthalpy;
}

The keyword entries specify the choice of thermophysical models, e.g. constant transport (constant viscosity, thermal diﬀusion), Perfect Gas equationOfState , etc. In addition there is a keyword entry named energy that allows the user to specify the form of energy to be used in the solution and thermodynamics. The following sections explains the entries and options in the thermoType package.

7.1.1 Thermophysical and mixture models

Each solver that uses thermophysical modelling constructs an object of a speciﬁc thermophysical model class. The model classes are listed below.

ﬂuidThermo: Thermophysical model for a general ﬂuid with ﬁxed composition. The solvers using ﬂuidThermo are rhoSimpleFoam, rhoPorousSimpleFoam rhoPimpleFoam, buoyantSimpleFoam, buoyantPimpleFoam, rhoPorousSimpleFoam rhoParticleFoam and thermoFoam.
psiThermo: Thermophysical model for gases only, with ﬁxed composition, used by rhoCentralFoam and coldEngineFoam.
ﬂuidReactionThermo: Thermophysical model for ﬂuid of varying composition, including reactingFoam, chtMultiRegionFoam and chemFoam.
psiuReactionThermo: Thermophysical model for combustion solvers that model combustion based on laminar ﬂame speed and regress variable, e.g.XiFoam, XiEngineFoam, PDRFoam.
multiphaseMixtureThermo: Thermophysical models for multiple phases used by compressibleMultiphaseInterFoam.

The type keyword (in the thermoType sub-dictionary) speciﬁes the underlying thermophysical model used by t The user can select from the following.

hePsiThermo : available for solvers that construct ﬂuidThermo, ﬂuidReactionThermo and psiThermo.
heRhoThermo : available for solvers that construct ﬂuidThermo, ﬂuidReactionThermo and multiphaseMixtureThermo.
heheuPsiThermo : for solvers that construct psiuReactionThermo.

The mixture speciﬁes the mixture composition. The option typically used for thermophysical models without reactions is pureMixture , which represents a mixture with ﬁxed composition. When pureMixture is speciﬁed, the thermophysical models coeﬃcients are speciﬁed within a sub-dictionary called mixture .

For mixtures with variable composition, required by thermophysical models with reactions, the reactingMixture option is used. Species and reactions are listed in a chemistry ﬁle, speciﬁed by the foamChemistryFile keyword. The reactingMixture model then requires the thermophysical models coeﬃcients to be speciﬁed for each specie within sub-dictionaries named after each specie, e.g. O2 , N2 .

For combustion based on laminar ﬂame speed and regress variables, constituents are a set of mixtures, such as fuel , oxidant and burntProducts . The available mixture models for this combustion modelling are homogeneousMixture , inhomogeneousMixture and veryInhomogeneousMixture .

Other models for variable composition are egrMixture , multiComponentMixture and singleStepReactingMixture .

7.1.2 Transport model

The transport modelling concerns evaluating dynamic viscosity $\text{[math]}$ , thermal conductivity $\text{[math]}$ and thermal diﬀusivity $\text{[math]}$ (for internal energy and enthalpy equations). The current transport models are as follows:

const

assumes a constant $\text{[math]}$ and Prandtl number $\text{[math]}$ which is simply speciﬁed by a two keywords, mu and Pr , respectively.

sutherland

calculates $\text{[math]}$ as a function of temperature $\text{[math]}$ from a Sutherland coeﬃcient $\text{[math]}$ and Sutherland temperature $\text{[math]}$ , speciﬁed by keywords As and Ts ; $\text{[math]}$ is calculated according to:

√-- μ = -As--T--. 1+ Ts∕T \relax \special {t4ht=

(7.1)

polynomial

calculates $\text{[math]}$ and $\text{[math]}$ as a function of temperature $\text{[math]}$ from a polynomial of any order $\text{[math]}$ , e.g.:

N∑ −1 μ = aiTi. i=0 \relax \special {t4ht=

(7.2)

logPolynomial

calculates $\text{[math]}$ and $\text{[math]}$ as a function of $\text{[math]}$ from a polynomial of any order $\text{[math]}$ ; from which $\text{[math]}$ , $\text{[math]}$ are calculated by taking the exponential, e.g.:

N∑−1 i ln(μ) = ai[ln (T )]. i=0 \relax \special {t4ht=

(7.3)

tabulated

uses uniform tabulated data for viscosity and thermal conductivity as a function of pressure and temperature.

icoTabulated

uses non-uniform tabulated data for viscosity and thermal conductivity as a function of temperature.

WLF

(Williams-Landel-Ferry) calculates $\text{[math]}$ as a function of temperature from coeﬃcients $\text{[math]}$ and $\text{[math]}$ and reference temperature $\text{[math]}$ speciﬁed by keywords C1 , C2 and Tr ; $\text{[math]}$ is calculated according to:

( ) μ = μ0exp −-C1(T-−-Tr) C2 + T − Tr \relax \special {t4ht=

(7.4)

7.1.3 Thermodynamic models

The thermodynamic models are concerned with evaluating the speciﬁc heat $\text{[math]}$ from which other properties are derived. The current thermo models are as follows:

eConst

assumes a constant $\text{[math]}$ and a heat of fusion $\text{[math]}$ which is simply speciﬁed by a two values $\text{[math]}$ , given by keywords Cv and Hf .

eIcoTabulated

calculates $\text{[math]}$ by interpolating non-uniform tabulated data of $\text{[math]}$ value pairs, e.g.:
( (200 1005) (400 1020) );

ePolynomial

calculates $\text{[math]}$ as a function of temperature by a polynomial of any order $\text{[math]}$ :

N −1 ∑ i cv = aiT . i=0 \relax \special {t4ht=

(7.5)

ePower

calculates $\text{[math]}$ as a power of temperature according to:

( )n0 cv = c0 -T-- . Tref \relax \special {t4ht=

(7.6)

eTabulated

calculates $\text{[math]}$ by interpolating uniform tabulated data of $\text{[math]}$ value pairs, e.g.:
( (200 1005) (400 1020) );

hConst

assumes a constant $\text{[math]}$ and a heat of fusion $\text{[math]}$ which is simply speciﬁed by a two values $\text{[math]}$ , given by keywords Cp and Hf .

hIcoTabulated

calculates $\text{[math]}$ by interpolating non-uniform tabulated data of $\text{[math]}$ value pairs, e.g.:
( (200 1005) (400 1020) );

hPolynomial

calculates $\text{[math]}$ as a function of temperature by a polynomial of any order $\text{[math]}$ :

N∑−1 cp = aiT i. i=0 \relax \special {t4ht=

(7.7)

hPower

calculates $\text{[math]}$ as a power of temperature according to:

( ) -T-- n0 cp = c0 Tref . \relax \special {t4ht=

(7.8)

hTabulated

calculates $\text{[math]}$ by interpolating uniform tabulated data of $\text{[math]}$ value pairs, e.g.:
( (200 1005) (400 1020) );

janaf

calculates $\text{[math]}$ as a function of temperature $\text{[math]}$ from a set of coeﬃcients taken from JANAF tables of thermodynamics. The ordered list of coeﬃcients is given in Table 7.1. The function is valid between a lower and upper limit in temperature $\text{[math]}$ and $\text{[math]}$ respectively. Two sets of coeﬃcients are speciﬁed, the ﬁrst set for temperatures above a common temperature $\text{[math]}$ (and below $\text{[math]}$ ), the second for temperatures below $\text{[math]}$ (and above $\text{[math]}$ ). The function relating $\text{[math]}$ to temperature is:

cp = R ((((a4T + a3)T + a2)T + a1)T + a0). \relax \special {t4ht=

(7.9)

In addition, there are constants of integration, $\text{[math]}$ and $\text{[math]}$ , both at high and low temperature, used to evaluating $\text{[math]}$ and $\text{[math]}$ respectively.

Description	Entry	Keyword

Lower temperature limit	$\text{[math]}$	Tlow
Upper temperature limit	$\text{[math]}$	Thigh
Common temperature	$\text{[math]}$	Tcommon
High temperature coeﬃcients	$\text{[math]}$	highCpCoeffs (a0 a1 a2 a3 a4...
High temperature enthalpy oﬀset	$\text{[math]}$	a5...
High temperature entropy oﬀset	$\text{[math]}$	a6)
Low temperature coeﬃcients	$\text{[math]}$	lowCpCoeffs (a0 a1 a2 a3 a4...
Low temperature enthalpy oﬀset	$\text{[math]}$	a5...
Low temperature entropy oﬀset	$\text{[math]}$	a6)

Table 7.1: JANAF thermodynamics coeﬃcients.

7.1.4 Composition of each constituent

There is currently only one option for the specie model which speciﬁes the composition of each constituent. That model is itself named specie, which is speciﬁed by the following entries.

nMoles: number of moles of component. This entry is only used for combustion modelling based on regress variable with a homogeneous mixture of reactants; otherwise it is set to 1.
molWeight in grams per mole of specie.

7.1.5 Equation of state

The following equations of state are available in the thermophysical modelling library.

adiabaticPerfectFluid

Adiabatic perfect ﬂuid:

( ) p-+-B-- 1∕γ ρ = ρ0 p0 + B , \relax \special {t4ht=

(7.10)

where $\text{[math]}$ are reference density and pressure respectively, and $\text{[math]}$ is a model constant.

Boussinesq

Boussinesq approximation

ρ = ρ [1− β(T − T )] 0 0 \relax \special {t4ht=

(7.11)

where $\text{[math]}$ is the coeﬃent of volumetric expansion and $\text{[math]}$ is the reference density at reference temperature $\text{[math]}$ .

icoPolynomial

Incompressible, polynomial equation of state:

N∑ −1 ρ = aiTi, i=0 \relax \special {t4ht=

(7.12)

where $\text{[math]}$ are polynomial coeﬃcients of any order $\text{[math]}$ .

icoTabulated

Tabulated data for an incompressible ﬂuid using $\text{[math]}$ value pairs, e.g.
rho ( (200 1010) (400 980) );

incompressiblePerfectGas

Perfect gas for an incompressible ﬂuid:

ρ = -1-pref, RT \relax \special {t4ht=

(7.13)

where $\text{[math]}$ is a reference pressure.

linear

Linear equation of state:

ρ = ψp + ρ , 0 \relax \special {t4ht=

(7.14)

where $\text{[math]}$ is compressibility (not necessarily $\text{[math]}$ ).

PengRobinsonGas

Peng Robinson equation of state:

1 ρ = zRT--p, \relax \special {t4ht=

(7.15)

where the complex function $\text{[math]}$ can be referenced in the source code in PengRobinsonGasI.H, in the $FOAM_SRC/thermophysicalModels/specie/equationOfState/ directory.

perfectFluid

Perfect ﬂuid:

1 ρ = ---p+ ρ0, RT \relax \special {t4ht=

(7.16)

where $\text{[math]}$ is the density at $\text{[math]}$ .

perfectGas

Perfect gas:

ρ = -1-p. RT \relax \special {t4ht=

(7.17)

rhoConst

Constant density:

ρ = constant. \relax \special {t4ht=

(7.18)

rhoTabulated

Uniform tabulated data for a compressible ﬂuid, calculating $\text{[math]}$ as a function of $\text{[math]}$ and $\text{[math]}$ .

rPolynomial

Reciprocal polynomial equation of state for liquids and solids:

1= C + C T + C T 2 − C p− C pT ρ 0 1 2 3 4 \relax \special {t4ht=

(7.19)

where $\text{[math]}$ are coeﬃcients.

7.1.6 Selection of energy variable

The user must specify the form of energy to be used in the solution, either internal energy $\text{[math]}$ and enthalpy $\text{[math]}$ , and in forms that include the heat of formation $\text{[math]}$ or not. This choice is speciﬁed through the energy keyword.

We refer to absolute energy where heat of formation is included, and sensible energy where it is not. For example absolute enthalpy $\text{[math]}$ is related to sensible enthalpy $\text{[math]}$ by

∑ i h = hs + ciΔh f i \relax \special {t4ht=

(7.20)

where $\text{[math]}$ and $\text{[math]}$ are the molar fraction and heat of formation, respectively, of specie $\text{[math]}$ . In most cases, we use the sensible form of energy, for which it is easier to account for energy change due to reactions. Keyword entries for energy therefore include e.g. sensibleEnthalpy, sensibleInternalEnergy and absoluteEnthalpy.

7.1.7 Thermophysical property data

The basic thermophysical properties are speciﬁed for each species from input data. Data entries must contain the name of the specie as the keyword, e.g. O2, H2O, mixture, followed by sub-dictionaries of coeﬃcients, including:

specie: containing i.e. number of moles, nMoles , of the specie, and molecular weight, molWeight in units of g/mol;
thermodynamics: containing coeﬃcients for the chosen thermodynamic model (see below);
transport: containing coeﬃcients for the chosen tranpsort model (see below).

The following is an example entry for a specie named fuel modelled using sutherland transport and janaf thermodynamics:

fuel
{
    specie
    {
        nMoles       1;
        molWeight    16.0428;
    }
    thermodynamics
    {
        Tlow         200;
        Thigh        6000;
        Tcommon      1000;
        highCpCoeffs (1.63543 0.0100844 -3.36924e-06 5.34973e-10
                      -3.15528e-14 -10005.6 9.9937);
        lowCpCoeffs  (5.14988 -0.013671 4.91801e-05 -4.84744e-08
                      1.66694e-11 -10246.6 -4.64132);
    }
    transport
    {
        As           1.67212e-06;
        Ts           170.672;
    }
}

The following is an example entry for a specie named air modelled using const transport and hConst thermodynamics:

air
{
    specie
    {
        nMoles          1;
        molWeight       28.96;
    }
    thermodynamics
    {
        Cp              1004.5;
        Hf              2.544e+06;
    }
    transport
    {
        mu              1.8e-05;
        Pr              0.7;
    }
}

OpenFOAM v9 User Guide - 7.1 Thermophysical models