Additional drag models for the twoPhaseEulerFoam solver

1 Syamlal and O'Brien drag

1.1 Formulas

The Syamlal and O'Brien drag formula can be written as follows (according to the nomenclature used in the code):

$\beta' = \frac{3 \alpha \beta \rho_b}{4 V_r^2 d_a} C_{D_s} U_r$

The terminal velocity $V_r$ can be calculated using the formula:

$V_r = \frac{1}{2} \left ( A - 0.06 \texttt{Re} + \sqrt{\left( 0.06 \texttt{Re} \right)^2 + 0.12 \texttt{Re} \left( 2B - A \right) + A^2}\right)$

where $A = \beta^{4.14}$, $B = 0.8 \beta^{1.28}$ if $\beta \leq 0.85$ and $B = \beta^{2.65}$ if $\beta > 0.85$.

$C_{D_s}$ is found using the Della Valle formulation:

$C_{D_s} = \left ( 0.63 + 4.8 \sqrt{\frac{V_r}{\texttt{Re}}} \right )$

with

$\texttt{Re} = \frac{d_a U_r}{\nu_b}$

1.2 Implementation

The Syamlal and O'Brien drag model can be implemented in OpenFOAM by dividing the $\beta' [itex] by the product of the two volume fractions [itex] \alpha \beta$ to obtain $K$.

N.B. Every drag formulation has to be divided by the product $\alpha \beta$ before beeing implemented in twoPhaseEulerFoam because this product has been extracted for numerical reasons.

The code is the following.

• SyamlalOBrien.H

/*---------------------------------------------------------------------------*\
=========                 |
\\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
\\    /   O peration     |
\\  /    A nd           | Copyright (C) 1991-2004 OpenCFD Ltd.
\\/     M anipulation  |
-------------------------------------------------------------------------------
This file is part of OpenFOAM.

OpenFOAM is free software; you can redistribute it and/or modify it
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.

OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with OpenFOAM; if not, write to the Free Software Foundation,
Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

Class
SyamlalOBrien

Description
Syamlal, M., Rogers, W. and O'Brien, T. J. (1993) MFIX documentation,
Theory Guide. Technical Note DOE/METC-94/1004. Morgantown, West Virginia,
USA.

SourceFiles
SyamlalOBrien.C

\*---------------------------------------------------------------------------*/

#ifndef SyamlalOBrien_H
#define SyamlalOBrien_H

#include "dragModel.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

namespace Foam
{

/*---------------------------------------------------------------------------*\
Class SyamlalOBrien Declaration
\*---------------------------------------------------------------------------*/

class SyamlalOBrien
:
public dragModel
{

public:

//- Runtime type information
TypeName("SyamlalOBrien");

// Constructors

//- Construct from components
SyamlalOBrien
(
const dictionary& interfaceDict,
const volScalarField& alpha,
const phaseModel& phasea,
const phaseModel& phaseb
);

// Destructor

~SyamlalOBrien();

// Member Functions

tmp<volScalarField> K(const volScalarField& Ur) const;
};

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

} // End namespace Foam

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

#endif

// ************************************************************************* //
• SyamlalOBrien.C

/*---------------------------------------------------------------------------*\
=========                 |
\\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
\\    /   O peration     |
\\  /    A nd           | Copyright (C) 1991-2004 OpenCFD Ltd.
\\/     M anipulation  |
-------------------------------------------------------------------------------
This file is part of OpenFOAM.

OpenFOAM is free software; you can redistribute it and/or modify it
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.

OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with OpenFOAM; if not, write to the Free Software Foundation,
Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

\*---------------------------------------------------------------------------*/

#include "SyamlalOBrien.H"

// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //

namespace Foam
{
defineTypeNameAndDebug(SyamlalOBrien, 0);

(
dragModel,
SyamlalOBrien,
dictionary
);
}

// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //

Foam::SyamlalOBrien::SyamlalOBrien
(
const dictionary& interfaceDict,
const volScalarField& alpha,
const phaseModel& phasea,
const phaseModel& phaseb
)
:
dragModel(interfaceDict, alpha, phasea, phaseb)
{}

// * * * * * * * * * * * * * * * * Destructor  * * * * * * * * * * * * * * * //

Foam::SyamlalOBrien::~SyamlalOBrien()
{}

// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //

Foam::tmp<Foam::volScalarField> Foam::SyamlalOBrien::K
(
const volScalarField& Ur
) const
{
volScalarField beta = max(1.0 - alpha_, 1.0e-6);
volScalarField A = pow(beta, 4.14);
volScalarField B = 0.8*pow(beta, 1.28);

forAll (beta, celli)
{
if (beta[celli] > 0.85)
{
B[celli] = pow(beta[celli], 2.65);
}
}

volScalarField Re = max(Ur*phasea_.d()/phaseb_.nu(), 1.0e-3);
volScalarField Vr = 0.5*(A - 0.06*Re + sqrt(pow(0.06*Re,2.0) + 0.12*Re*(2.0*B-A) + pow(A,2.0)));
volScalarField Cds = pow(0.63 + 4.8*sqrt(Vr/Re),2.0);

return 0.75*Cds*phaseb_.rho()*Ur/(phasea_.d()*pow(Vr,2.0));
}

// ************************************************************************* //

In order to use this code:

• Add the files in a directory named SyamlalOBrien in /.../twoPhaseEulerFoam/interfacialModels/dragModels.
• Open the /../twoPhaseEulerFoam/interfacialModels/Make file and add to it the line

dragModels/SyamlalOBrien/SyamlalOBrien.C
• Use the wmake tool to rebuild twoPhaseEulerFoam

1.4 References

1. Dalla Valle, J.M., 1948, Micromeritics, Pitman, London.
1. Syamlal, M., The Particle-Particle Drag Term in a Multiparticle Model of Fluidization, Topical Report, DOE/MC/21353-2373, NTIS/DE87006500, National Technical Information Service, Springfield, VA, 1987.
1. Syamlal, M., Rogers, W. and O'Brien, T. J. (1993), MFIX documentation, Theory Guide. Technical Note DOE/METC-94/1004. Morgantown, West Virginia, USA, http://www.mfix.org.

2 Gidaspow's drag (Ergun - Wen&Yu)

2.1 Formulas

The Gidaspow formulation of the drag factor uses the Ergun correlations if $\beta < 0.8$:

$\beta' = 150 \frac{\alpha^2 \nu_b}{\beta^2 \rho_b d_a^2} + 1.75 \frac{\rho_b \alpha}{\beta d_s} U_r$

If $\beta \geq 0.8$, the Wen and Yu correlation is used:

$\beta' = \frac{3 C_{D_s} \alpha \beta \rho_b U_r}{4 d_a} \beta^{-2.65}$

where $C_{D_s} = 0.44$ if $\texttt{Re} \geq 1000$, and

$C_{D_s} = \frac{24}{\texttt{Re}} \left( 1 + 0.15 \texttt{Re}^{0.687} \right)$

elsewhere ($\texttt{Re} < 1000$).

2.2 Implementation

• GidaspowErgunWenYu.H

/*---------------------------------------------------------------------------*\
=========                 |
\\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
\\    /   O peration     |
\\  /    A nd           | Copyright (C) 1991-2004 OpenCFD Ltd.
\\/     M anipulation  |
-------------------------------------------------------------------------------
This file is part of OpenFOAM.

OpenFOAM is free software; you can redistribute it and/or modify it
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.

OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with OpenFOAM; if not, write to the Free Software Foundation,
Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

Class
GidaspowErgunWenYu

Description
D. Gidaspow, Multiphase flow and fluidization,

SourceFiles
GidaspowErgunWenYu.C

\*---------------------------------------------------------------------------*/

#ifndef GidaspowErgunWenYu_H
#define GidaspowErgunWenYu_H

#include "dragModel.H"

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

namespace Foam
{

/*---------------------------------------------------------------------------*\
Class GidaspowErgunWenYu Declaration
\*---------------------------------------------------------------------------*/

class GidaspowErgunWenYu
:
public dragModel
{

public:

//- Runtime type information
TypeName("GidaspowErgunWenYu");

// Constructors

//- Construct from components
GidaspowErgunWenYu
(
const dictionary& interfaceDict,
const volScalarField& alpha,
const phaseModel& phasea,
const phaseModel& phaseb
);

// Destructor

~GidaspowErgunWenYu();

// Member Functions

tmp<volScalarField> K(const volScalarField& Ur) const;
};

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

} // End namespace Foam

// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

#endif

// ************************************************************************* //
• GidaspowErgunWenYu.C

/*---------------------------------------------------------------------------*\
=========                 |
\\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
\\    /   O peration     |
\\  /    A nd           | Copyright (C) 1991-2004 OpenCFD Ltd.
\\/     M anipulation  |
-------------------------------------------------------------------------------
This file is part of OpenFOAM.

OpenFOAM is free software; you can redistribute it and/or modify it
Free Software Foundation; either version 2 of the License, or (at your
option) any later version.

OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with OpenFOAM; if not, write to the Free Software Foundation,
Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

\*---------------------------------------------------------------------------*/

#include "GidaspowErgunWenYu.H"

// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //

namespace Foam
{
defineTypeNameAndDebug(GidaspowErgunWenYu, 0);

(
dragModel,
GidaspowErgunWenYu,
dictionary
);
}

// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //

// Construct from components
Foam::GidaspowErgunWenYu::GidaspowErgunWenYu
(
const dictionary& interfaceDict,
const volScalarField& alpha,
const phaseModel& phasea,
const phaseModel& phaseb
)
:
dragModel(interfaceDict, alpha, phasea, phaseb)
{}

// * * * * * * * * * * * * * * * * Destructor  * * * * * * * * * * * * * * * //

Foam::GidaspowErgunWenYu::~GidaspowErgunWenYu()
{}

// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //

Foam::tmp<Foam::volScalarField> Foam::GidaspowErgunWenYu::K
(
const volScalarField& Ur
) const
{
volScalarField beta = max(1.0 - alpha_, 1.0e-6);

volScalarField bp = pow(beta, -2.65);
volScalarField Re = max(Ur*phasea_.d()/phaseb_.nu(), 1.0e-3);

volScalarField Cds = 24.0*(1.0 + 0.15*pow(Re, 0.687))/Re;

forAll(Re, celli)
{
if(Re[celli] > 1000.0)
{
Cds[celli] = 0.44;
}
}

// Wen and Yu (1966)
volScalarField KWenYu = 0.75*Cds*phaseb_.rho()*Ur*bp/phasea_.d();

// Ergun
volScalarField KErgun = 150.0*alpha_*phaseb_.nu()*phaseb_.rho()
/(pow(beta*phasea_.d(), 2.0))
+ 1.75*phaseb_.rho()*Ur/(beta*phasea_.d());

forAll (beta, cellj)
{
if (beta[cellj] <= 0.8)
{
KWenYu[cellj] = KErgun[cellj];
}
}

return 1.0*KWenYu;
}

// ************************************************************************* //

2.3 References

1. D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, New York, 1994.
2. C. Y. Wen, Y. H. Yu, Mechanics of fluidization, Chemical Engineering Progress Symposium Series, 62:100-111, 1966.

--Alberto 03:21, 29 Aug 2005 (CEST)