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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 p - T 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 μ, thermal conductivity κ and thermal diffusivity α (for internal energy and enthalpy equations). The current transport models are as follows:
const
assumes a constant μ and Prandtl number P r = c μ∕κ p which is simply specified by a two keywords, mu and Pr, respectively.
sutherland
calculates μ as a function of temperature T from a Sutherland coefficient As and Sutherland temperature Ts, specified by keywords As and Ts; μ is calculated according to:
        √ -- As   T μ = –––––––––. 1 + Ts∕T
(7.1)
polynomial
calculates μ and κ as a function of temperature T from a polynomial of any order N, e.g. :
    N -1 ∑      i μ =     aiT . i=0
(7.2)
logPolynomial
calculates ln(μ) and ln (κ) as a function of ln(T ) from a polynomial of any order N; from which μ, κ are calculated by taking the exponential, e.g. :
        N∑- 1 ln(μ) =     ai[ln(T )]i. i=0
(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 μ as a function of temperature from coefficients C1 and C2 and reference temperature Tr specified by keywords C1, C2 and Tr; μ is calculated according to:
           (              ) - C1-(T–––Tr) μ = μ0 exp   C  + T -  T 2        r
(7.4)

7.1.3 Thermodynamic models

The thermodynamic models are concerned with evaluating the specific heat cp from which other properties are derived. The current thermo models are as follows:
eConst
assumes a constant c v and a heat of fusion H f which is simply specified by a two values cv Hf, given by keywords Cv and Hf.
eIcoTabulated
calculates cv by interpolating non-uniform tabulated data of (T,cp) value pairs, e.g. :
( (200 1005) (400 1020) );
ePolynomial
calculates cv as a function of temperature by a polynomial of any order N:
     N-1 ∑      i cv =     aiT  . i=0
(7.5)
ePower
calculates cv as a power of temperature according to:
       (     )n0 c  = c   -T--    . v    0  Tref
(7.6)
eTabulated
calculates cv by interpolating uniform tabulated data of (T, cp) value pairs, e.g. :
( (200 1005) (400 1020) );
hConst
assumes a constant cp and a heat of fusion Hf which is simply specified by a two values cp Hf, given by keywords Cp and Hf.
hIcoTabulated
calculates cp by interpolating non-uniform tabulated data of (T,cp) value pairs, e.g. :
( (200 1005) (400 1020) );
hPolynomial
calculates cp as a function of temperature by a polynomial of any order N:
     N∑-1 cp =     aiT i. i=0
(7.7)
hPower
calculates cp as a power of temperature according to:
       (     )n0 -T-- cp = c0  Tref    .
(7.8)
hTabulated
calculates cp by interpolating uniform tabulated data of (T, cp) value pairs, e.g. :
( (200 1005) (400 1020) );
janaf
calculates cp as a function of temperature T 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 T l and T h respectively. Two sets of coefficients are specified, the first set for temperatures above a common temperature Tc (and below Th), the second for temperatures below Tc (and above Tl). The function relating cp to temperature is:
cp = R ((((a4T + a3)T + a2 )T + a1)T + a0).
(7.9)
In addition, there are constants of integration, a5 and a6, both at high and low temperature, used to evaluating h and s respectively.

Description Entry Keyword



Lower temperature limit T  (K ) l Tlow
Upper temperature limit T  (K ) h Thigh
Common temperature T  (K) c Tcommon
High temperature coefficients a ...a 0    4 highCpCoeffs (a0 a1 a2 a3 a4...
High temperature enthalpy offset a5 a5...
High temperature entropy offset a6 a6)
Low temperature coefficients a0...a4 lowCpCoeffs (a0 a1 a2 a3 a4...
Low temperature enthalpy offset a5 a5...
Low temperature entropy offset a6 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:
       (        )1∕γ p-+-B-- ρ = ρ0   p0 + B     ,
(7.10)
where ρ0,p0 are reference density and pressure respectively, and B is a model constant.
Boussinesq
Boussinesq approximation
ρ = ρ0[1 - β (T  - T0)]
(7.11)
where β is the coeffient of volumetric expansion and ρ0 is the reference density at reference temperature T0.
icoPolynomial
Incompressible, polynomial equation of state:
    N -1 ∑       i ρ =     aiT , i=0
(7.12)
where ai are polynomial coefficients of any order N.
icoTabulated
Tabulated data for an incompressible fluid using (T, ρ) value pairs, e.g.  
rho ( (200 1010) (400 980) );
incompressiblePerfectGas
Perfect gas for an incompressible fluid:
     1 ρ = –––-pref, RT
(7.13)
where p ref is a reference pressure.
linear
Linear equation of state:
ρ =  ψp + ρ0,
(7.14)
where ψ is compressibility (not necessarily (RT )-1).
PengRobinsonGas
Peng Robinson equation of state:
      1 ρ =  –––-p, zRT
(7.15)
where the complex function z =  z(p,T) can be referenced in the source code in PengRobinsonGasI.H, in the $FOAM_SRC/thermophysicalModels/specie/equationOfState/ directory.
perfectFluid
Perfect fluid:
     -1-- ρ =  RT p + ρ0,
(7.16)
where ρ0 is the density at T =  0.
perfectGas
Perfect gas:
     1 ρ = RT--p.
(7.17)
rhoConst
Constant density:
ρ = constant.
(7.18)
rhoTabulated
Uniform tabulated data for a compressible fluid, calculating ρ as a function of T and p.
rPolynomial
Reciprocal polynomial equation of state for liquids and solids:
1-= C  + C  T + C  T2 - C  p - C pT ρ     0    1      2       3     4
(7.19)
where Ci 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 e and enthalpy h, and in forms that include the heat of formation Δhf 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 h is related to sensible enthalpy hs by

         ∑       i h = hs +     ciΔh f i
(7.20)
where ci and hif are the molar fraction and heat of formation, respectively, of specie i. 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;
    }
}

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OpenFOAM v9 User Guide: 7.1 Thermophysical models