populationBalanceModel: Added Prince and Blanch (1990) coalescence model

Patch contributed by Institute of Fluid Dynamics, Helmholtz-Zentrum
Dresden - Rossendorf (HZDR)

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/Make/files

@@ -21,6 +21,7 @@ populationBalanceModel/coalescenceModels/coalescenceModel/coalescenceModel.C
 populationBalanceModel/coalescenceModels/constantCoalescence/constantCoalescence.C
 populationBalanceModel/coalescenceModels/CoulaloglouTavlaridesCoalescence/CoulaloglouTavlaridesCoalescence.C
 populationBalanceModel/coalescenceModels/hydrodynamic/hydrodynamic.C
+populationBalanceModel/coalescenceModels/PrinceBlanch/PrinceBlanch.C
 
 populationBalanceModel/binaryBreakupModels/binaryBreakupModel/binaryBreakupModel.C
 populationBalanceModel/binaryBreakupModels/powerLawUniformBinary/powerLawUniformBinary.C

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/breakupModels/LaakkonenAlopaeusAittamaa/LaakkonenAlopaeusAittamaa.C

@@ -26,7 +26,6 @@ License
 #include "LaakkonenAlopaeusAittamaa.H"
 #include "addToRunTimeSelectionTable.H"
 #include "phaseCompressibleTurbulenceModel.H"
-#include "phaseSystem.H"
 
 // * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //
 
@@ -80,10 +79,6 @@ Foam::diameterModels::breakupModels::LaakkonenAlopaeusAittamaa::setBreakupRate
 {
     const phaseModel& continuousPhase = popBal_.continuousPhase();
     const sizeGroup& fi = *popBal_.sizeGroups()[i];
-    const volScalarField sigma
-    (
-        popBal_.fluid().sigma(phasePair(fi.phase(), continuousPhase))
-    );
 
     breakupRate =
         C1_*cbrt(continuousTurbulence().epsilon())
@@ -91,7 +86,7 @@ Foam::diameterModels::breakupModels::LaakkonenAlopaeusAittamaa::setBreakupRate
         (
             sqrt
             (
-                C2_*sigma
+                C2_*popBal_.sigmaWithContinuousPhase(fi.phase())
                /(
                     continuousPhase.rho()*pow(fi.d(), 5.0/3.0)
                    *pow(continuousTurbulence().epsilon(), 2.0/3.0)

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/breakupModels/LaakkonenAlopaeusAittamaa/LaakkonenAlopaeusAittamaa.H

@@ -27,18 +27,18 @@ Class
 Description
     Model of Laakkonen et al. (2006). The total breakup rate is calculated by
 
-        \f[
+    \f[
-            C_1 \epsilon_c^{1/3}
+        C_1 \epsilon_c^{1/3}
-            \mathrm{erfc}
+        \mathrm{erfc}
-            \left(
+        \left(
-                \sqrt
+            \sqrt
-                {
+            {
-                    C_2 \frac{\sigma}{\rho_c \epsilon_c^{2/3} (6v_i/\pi)^{5/9}}
+                C_2 \frac{\sigma}{\rho_c \epsilon_c^{2/3} (6v_i/\pi)^{5/9}}
-                  + C_3 \frac{\mu_c}{\sqrt{\rho_c \rho_d}
+              + C_3 \frac{\mu_c}{\sqrt{\rho_c \rho_d}
-                    \epsilon_c^{1/3} (6v_i/\pi)^{4/9}}
+                \epsilon_c^{1/3} (6v_i/\pi)^{4/9}}
-                }
+            }
-            \right)
+        \right)
-        \f]
+    \f]
 
     where
 

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/CoulaloglouTavlaridesCoalescence/CoulaloglouTavlaridesCoalescence.C

@@ -84,7 +84,7 @@ addToCoalescenceRate
         (
           - C2_*continuousPhase.mu()*continuousPhase.rho()
            *continuousTurbulence().epsilon()
-           /sqr(sigma(fi.phase().name(), continuousPhase.name()))
+           /sqr(popBal_.sigmaWithContinuousPhase(fi.phase()))
            /pow3(1 + popBal_.alphas())
            *pow4(cbrt(fi.x())*cbrt(fj.x())/(cbrt(fi.x()) + cbrt(fj.x())))
         );

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/CoulaloglouTavlaridesCoalescence/CoulaloglouTavlaridesCoalescence.H

@@ -28,18 +28,18 @@ Description
     Model of Coulaloglou and Tavlarides (1977). The coalescence rate is
     calculated by
 
-        \f[
+    \f[
-            C_1 (v_i^{2/3} + v_j^{2/3}) (v_i^{2/9} + v_j^{2/9})^{1/2}
+        C_1 (v_i^{2/3} + v_j^{2/3}) (v_i^{2/9} + v_j^{2/9})^{1/2}
-            \frac{\epsilon_c^{1/3}}{1 + \alpha_d}
+        \frac{\epsilon_c^{1/3}}{1 + \alpha_d}
-            \text{exp}
+        \text{exp}
-            \left[
+        \left[
-              - C_2 \frac{\mu_c \rho_c}{\sigma^2}
+          - C_2 \frac{\mu_c \rho_c}{\sigma^2}
-                \frac{\epsilon_c}{(1 + \alpha_d)^{3}}
+            \frac{\epsilon_c}{(1 + \alpha_d)^{3}}
-                \left(
+            \left(
-                    \frac{v_i^{1/3} v_j^{1/3}}{v_i^{1/3} + v_j^{1/3}}
+                \frac{v_i^{1/3} v_j^{1/3}}{v_i^{1/3} + v_j^{1/3}}
-                \right)^{4}
+            \right)^{4}
-            \right]
+        \right]
-        \f]
+    \f]
 
     where
 

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/PrinceBlanch/PrinceBlanch.C

@@ -0,0 +1,146 @@
+/*---------------------------------------------------------------------------*\
+  =========                 |
+  \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
+   \\    /   O peration     |
+    \\  /    A nd           | Copyright (C) 2018 OpenFOAM Foundation
+     \\/     M anipulation  |
+-------------------------------------------------------------------------------
+License
+    This file is part of OpenFOAM.
+
+    OpenFOAM is free software: you can redistribute it and/or modify it
+    under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 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, see <http://www.gnu.org/licenses/>.
+
+\*---------------------------------------------------------------------------*/
+
+#include "PrinceBlanch.H"
+#include "addToRunTimeSelectionTable.H"
+#include "mathematicalConstants.H"
+
+// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //
+
+namespace Foam
+{
+namespace diameterModels
+{
+namespace coalescenceModels
+{
+    defineTypeNameAndDebug(PrinceBlanch, 0);
+    addToRunTimeSelectionTable
+    (
+        coalescenceModel,
+        PrinceBlanch,
+        dictionary
+    );
+}
+}
+}
+
+
+// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //
+
+Foam::diameterModels::coalescenceModels::PrinceBlanch::
+PrinceBlanch
+(
+    const populationBalanceModel& popBal,
+    const dictionary& dict
+)
+:
+    coalescenceModel(popBal, dict),
+    C1_("C1", dimless, dict.lookupOrDefault<scalar>("C1", 0.356)),
+    h0_("h0", dimLength, dict.lookupOrDefault<scalar>("h0", 1e-4)),
+    hf_("hf", dimLength, dict.lookupOrDefault<scalar>("h0", 1e-8)),
+    turbulentCollisions_(dict.lookup("turbulentCollisions")),
+    buoyantCollisions_(dict.lookup("buoyantCollisions")),
+    laminarShearCollisions_(dict.lookup("laminarShearCollisions"))
+{}
+
+
+// * * * * * * * * * * * * * * * Member Functions  * * * * * * * * * * * * * //
+
+void
+Foam::diameterModels::coalescenceModels::PrinceBlanch::
+addToCoalescenceRate
+(
+    volScalarField& coalescenceRate,
+    const label i,
+    const label j
+)
+{
+    const phaseModel& continuousPhase = popBal_.continuousPhase();
+    const sizeGroup& fi = *popBal_.sizeGroups()[i];
+    const sizeGroup& fj = *popBal_.sizeGroups()[j];
+    scalar pi = constant::mathematical::pi;
+    const uniformDimensionedVectorField& g =
+        popBal_.mesh().lookupObject<uniformDimensionedVectorField>("g");
+
+    dimensionedScalar rij = 1.0/(1.0/fi.d() + 1.0/fj.d());
+
+    volScalarField collisionEfficiency =
+        exp
+        (
+          - sqrt
+            (
+                pow3(rij)*continuousPhase.rho()
+               /(16.0*popBal_.sigmaWithContinuousPhase(fi.phase()))
+            )
+           *log(h0_/hf_)
+           *cbrt(continuousTurbulence().epsilon())/pow(rij, 2.0/3.0)
+        );
+
+    if (turbulentCollisions_)
+    {
+        coalescenceRate +=
+            (
+                C1_*pi*sqr(fi.d() + fj.d())
+               *cbrt(continuousTurbulence().epsilon())
+               *sqrt(pow(fi.d(), 2.0/3.0) + pow(fj.d(), 2.0/3.0))
+            )
+           *collisionEfficiency;
+    }
+
+    if (buoyantCollisions_)
+    {
+        dimensionedScalar Sij = pi/4.0*sqr(fi.d() + fj.d());
+
+        coalescenceRate +=
+            (
+                Sij
+               *mag
+                (
+                    sqrt
+                    (
+                        2.14*popBal_.sigmaWithContinuousPhase(fi.phase())
+                       /(continuousPhase.rho()*fi.d()) + 0.505*mag(g)*fi.d()
+                    )
+                  - sqrt
+                    (
+                        2.14*popBal_.sigmaWithContinuousPhase(fi.phase())
+                       /(continuousPhase.rho()*fj.d()) + 0.505*mag(g)*fj.d()
+                    )
+                )
+            )
+           *collisionEfficiency;
+    }
+
+    if (laminarShearCollisions_)
+    {
+        FatalErrorInFunction
+            << "Laminar shear collision contribution not implemented for "
+            << this->type() << " coalescence model."
+            << exit(FatalError);
+    }
+}
+
+
+// ************************************************************************* //

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/PrinceBlanch/PrinceBlanch.H

@@ -0,0 +1,205 @@
+/*---------------------------------------------------------------------------*\
+  =========                 |
+  \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
+   \\    /   O peration     |
+    \\  /    A nd           | Copyright (C) 2018 OpenFOAM Foundation
+     \\/     M anipulation  |
+-------------------------------------------------------------------------------
+License
+    This file is part of OpenFOAM.
+
+    OpenFOAM is free software: you can redistribute it and/or modify it
+    under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 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, see <http://www.gnu.org/licenses/>.
+
+Class
+    Foam::diameterModels::coalescenceModels::PrinceBlanch
+
+Description
+    Model of Prince and Blanch (1990). The coalescence rate is calculated by
+
+    \f[
+        \left( \theta_{ij}^{T} + \theta_{ij}^{B} + \theta_{ij}^{LS} \right)
+        \lambda_{ij}
+    \f]
+
+    with the coalescence efficiency
+
+    \f[
+        \lambda_{ij} =
+            \mathrm{exp}
+            \left(
+              - \sqrt{\frac{r_{ij}^3 \rho_c}{16 \sigma}}
+                \mathrm{ln} \left(\frac{h_0}{h_f}\right)
+                \epsilon_c^{1/3}/r_{ij}^{2/3}
+            \right)\;,
+    \f]
+
+    the turbulent collision rate
+
+    \f[
+        \theta_{ij}^{T} =
+            C_1 \pi (d_i + d_j)^{2} \epsilon_c^{1/3}
+            \sqrt{d_{i}^{2/3} + d_{j}^{2/3}}\;,
+    \f]
+
+    and the buoyancy-driven collision rate
+
+    \f[
+        \theta_{ij}^{B} = S_{ij} \left| u_{ri} - u_{rj} \right|\;,
+    \f]
+
+    where the rise velocity of bubble i is calculated by
+
+    \f[
+        u_{ri} = \sqrt{2.14 \sigma / \left(\rho_c d_i \right) + 0.505 g d_i}\;,
+    \f]
+
+    the equivalent radius by
+
+    \f[
+        r_{ij} = \left( \frac{1}{d_i} + \frac{1}{d_j} \right)^{-1}
+    \f]
+
+    and the collision cross sectional area by
+
+    \f[
+        S_{ij} = \frac{\pi}{4} \left(d_i + d_j\right)^{2}\;.
+    \f]
+
+    Note that in equation 2, the bubble radius has been substituted by the
+    bubble diameter. Also the expression for the equivalent radius r_ij
+    (equation 19 in the paper of Prince and Blanch (1990)) was corrected.
+    The collision rate contribution due to laminar shear in the continuous phase
+    is currently neglected.
+
+    \vartable
+        \theta_{ij}^{T}   |  Turbulent collision rate [m3/s]
+        \theta_{ij}^{B}   |  Buoyancy-driven collision rate [m3/s]
+        \theta_{ij}^{LS}  |  Laminar shear collision rate [m3/s]
+        \lambda_{ij}      |  Coalescence efficiency [-]
+        r_{ij}            |  Equivalent radius [m]
+        \rho_c            |  Density of continous phase [kg/m3]
+        \sigma            |  Surface tension [N/m]
+        h_0               |  Initial film thickness [m]
+        h_f               |  Critical film thickness [m]
+        \epsilon_c        |  Continuous phase turbulent dissipation rate [m2/s3]
+        d_i               |  Diameter of bubble i [m]
+        d_j               |  Diameter of bubble j [m]
+        u_{ri}            |  Rise velocity of bubble i [m/s]
+        S_{ij}            |  Collision cross sectional area [m2]
+        g                 |  Gravitational constant [m/s2]
+    \endvartable
+
+    References:
+    \verbatim
+        "Bubble Coalescence and Break-Up in Air-Sparged Bubble-Columns"
+        Prince, M.J., Blanch, H.W.
+        AIChE Journal, Vol. 36, Issue 10, 1990, pp. 1485-1499
+    \endverbatim
+
+Usage
+    \table
+        Property     | Description             | Required    | Default value
+        C1           | Coefficient C1          | no          | 0.089
+    \endtable
+
+SourceFiles
+    PrinceBlanch.C
+
+\*---------------------------------------------------------------------------*/
+
+#ifndef PrinceBlanch_H
+#define PrinceBlanch_H
+
+#include "coalescenceModel.H"
+
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+namespace Foam
+{
+namespace diameterModels
+{
+namespace coalescenceModels
+{
+
+/*---------------------------------------------------------------------------*\
+                        Class PrinceBlanch Declaration
+\*---------------------------------------------------------------------------*/
+
+class PrinceBlanch
+:
+    public coalescenceModel
+{
+    // Private data
+
+        //- Optional coefficient C1, defaults to 0.089
+        dimensionedScalar C1_;
+
+        //- Initial film thickness, defaults to 1e-4m
+        dimensionedScalar h0_;
+
+        //- Critical film thickness, defaults to 1e-8m
+        dimensionedScalar hf_;
+
+        //- Switch for considering turbulent collisions
+        Switch turbulentCollisions_;
+
+        //- Switch for considering buoyancy-induced collisions
+        Switch buoyantCollisions_;
+
+        //- Switch for considering buoyancy-induced collisions
+        Switch laminarShearCollisions_;
+
+
+public:
+
+    //- Runtime type information
+    TypeName("PrinceBlanch");
+
+    // Constructor
+
+        PrinceBlanch
+        (
+            const populationBalanceModel& popBal,
+            const dictionary& dict
+        );
+
+
+    //- Destructor
+    virtual ~PrinceBlanch()
+    {}
+
+
+    // Member Functions
+
+        //- Add to coalescenceRate
+        virtual void addToCoalescenceRate
+        (
+            volScalarField& coalescenceRate,
+            const label i,
+            const label j
+        );
+};
+
+
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+} // End namespace coalescenceModels
+} // End namespace diameterModels
+} // End namespace Foam
+
+// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
+
+#endif
+
+// ************************************************************************* //

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/coalescenceModel/coalescenceModel.C

@@ -24,7 +24,6 @@ License
 \*---------------------------------------------------------------------------*/
 
 #include "coalescenceModel.H"
-#include "phaseSystem.H"
 
 // * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //
 
@@ -38,25 +37,6 @@ namespace diameterModels
 }
 
 
-// * * * * * * * * * * * * Protected Member Functions  * * * * * * * * * * * //
-
-const Foam::tmp<Foam::volScalarField>
-Foam::diameterModels::coalescenceModel::sigma
-(
-    const word& phase1,
-    const word& phase2
-) const
-{
-    const phasePairKey key
-    (
-        phase1,
-        phase2
-    );
-
-    return popBal_.fluid().sigma(key);
-}
-
-
 // * * * * * * * * * * * * * * * * Selector  * * * * * * * * * * * * * * * * //
 
 Foam::autoPtr<Foam::diameterModels::coalescenceModel>

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/coalescenceModels/coalescenceModel/coalescenceModel.H

@@ -59,16 +59,6 @@ protected:
         const populationBalanceModel& popBal_;
 
 
-    // Protected member functions
-
-        //- Return the surface tension coefficient
-        const tmp<volScalarField> sigma
-        (
-            const word& phase1,
-            const word& phase2
-        ) const;
-
-
 public:
 
     //- Runtime type information

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/daughterSizeDistributionModels/LaakkonenAlopaeusAittamaaDsd/LaakkonenAlopaeusAittamaaDsd.H

@@ -32,10 +32,10 @@ Description
     mass conservation following Kumar and Ramkrishna (1996). Also the internal
     coordinate was changed to particle volume, which gives
 
-        \f[
+    \f[
-            \frac{60}{v_j} \left(\frac{v_i}{v_j}\right)^{2}
+        \frac{60}{v_j} \left(\frac{v_i}{v_j}\right)^{2}
-            \left(1 - \frac{v_i}{v_j}\right)^{2}
+        \left(1 - \frac{v_i}{v_j}\right)^{2}
-        \f]
+    \f]
 
     where
 

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/populationBalanceModel/populationBalanceModel.C

@@ -1145,6 +1145,22 @@ Foam::diameterModels::populationBalanceModel::gamma
 }
 
 
+const Foam::tmp<Foam::volScalarField>
+Foam::diameterModels::populationBalanceModel::sigmaWithContinuousPhase
+(
+    const phaseModel& dispersedPhase
+) const
+{
+    const phasePairKey key
+    (
+        dispersedPhase.name(),
+        continuousPhase_.name()
+    );
+
+    return fluid_.sigma(key);
+}
+
+
 void Foam::diameterModels::populationBalanceModel::solve()
 {
     const dictionary& solutionControls = mesh_.solverDict(name_);

applications/solvers/multiphase/reactingEulerFoam/phaseSystems/populationBalanceModel/populationBalanceModel/populationBalanceModel.H

@@ -412,6 +412,13 @@ public:
             const dimensionedScalar& v
         ) const;
 
+        //- Return the surface tension coefficient between a given dispersed
+        //  and the continuous phase
+        const tmp<volScalarField> sigmaWithContinuousPhase
+        (
+            const phaseModel& dispersedPhase
+        ) const;
+
         //- Solve the population balance equation
         void solve();
 };