multiphaseEulerFoam: LaakkonenAlopaeusAittamaa: Updated to a more recent reference

The former implementation of the daughterSizeDistributionModel based on
Laakkonen et al. (2006) had the issue of not conserving the total
particle number and volume properly, except when the coefficient C4 is
set to a value of 2. The issue is solved by following the work of
Laakkonen et al. (2007).

The default coefficients in both the breakupModel and the
daughterSizeDistributionModel were updated as well according to the
suggestions of the authors.

Patch contributed by Institute of Fluid Dynamics,
Helmholtz-Zentrum Dresden - Rossendorf (HZDR)
and VTT Technical Research Centre of Finland Ltd.

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

@@ -64,7 +64,7 @@ LaakkonenAlopaeusAittamaa
             "C1",
             dict,
             dimensionSet(0, -2.0/3.0, 0, 0, 0),
-            6.0
+            2.25
         )
     ),
     C2_(dimensionedScalar::lookupOrDefault("C2", dict, dimless, 0.04)),

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

@@ -2,7 +2,7 @@
   =========                 |
   \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
    \\    /   O peration     | Website:  https://openfoam.org
-    \\  /    A nd           | Copyright (C) 2018-2020 OpenFOAM Foundation
+    \\  /    A nd           | Copyright (C) 2018-2021 OpenFOAM Foundation
      \\/     M anipulation  |
 -------------------------------------------------------------------------------
 License
@@ -25,7 +25,7 @@ Class
     Foam::diameterModels::breakupModels::LaakkonenAlopaeusAittamaa
 
 Description
-    Model of Laakkonen et al. (2006). The total breakup rate is calculated by
+    Model of Laakkonen et al. (2007). The total breakup rate is calculated by
 
     \f[
         C_1 \epsilon_c^{1/3}
@@ -44,7 +44,7 @@ Description
 
     \vartable
         \sigma      |  Surface tension [N/m]
-        v_i         |  Volume of mother bubble i [m]
+        v_i         |  Volume of mother bubble i [m3]
         \epsilon_c  |  Turbulent dissipation rate of continuous phase [m^2/s^3]
         \mu_c       |  Molecular dynamic viscosity of liquid phase [Pa s]
         \rho_c      |  Density of continuous phase [kg/m^3]
@@ -53,17 +53,15 @@ Description
 
     References:
     \verbatim
-        Laakkonen, M., Alopaeus, V., & Aittamaa, J. (2006).
+        Laakkonen, M., Moilanen, P., Alopaeus, V., & Aittamaa, J. (2007).
-        Validation of bubble breakage, coalescence and mass transfer models for
+        Modelling local bubble size distributions in agitated vessels.
-        gas-liquid dispersion in agitated vessel.
+        Chemical Engineering Science, 62, 721–740.
-        Chemical engineering science, 61(1), 218-228.
-        Eq. 2-3, p. 220.
     \endverbatim
 
 Usage
     \table
         Property     | Description             | Required    | Default value
-        C1           | coefficient C1          | no          | 6.0
+        C1           | coefficient C1          | no          | 2.25
         C2           | coefficient C2          | no          | 0.04
         C3           | coefficient C3          | no          | 0.01
         daughterSizeDistributionModel | inh. from breakupModel | inherited |
@@ -98,7 +96,7 @@ class LaakkonenAlopaeusAittamaa
 {
     // Private Data
 
-        //- Optional coefficient C1, defaults to 6.0
+        //- Optional coefficient C1, defaults to 2.25
         dimensionedScalar C1_;
 
         //- Optional coefficient C2, defaults to 0.04

applications/solvers/multiphase/multiphaseEulerFoam/phaseSystems/populationBalanceModel/daughterSizeDistributionModels/LaakkonenAlopaeusAittamaaDsd/LaakkonenAlopaeusAittamaaDsd.C

@@ -2,7 +2,7 @@
   =========                 |
   \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
    \\    /   O peration     | Website:  https://openfoam.org
-    \\  /    A nd           | Copyright (C) 2018-2020 OpenFOAM Foundation
+    \\  /    A nd           | Copyright (C) 2018-2021 OpenFOAM Foundation
      \\/     M anipulation  |
 -------------------------------------------------------------------------------
 License
@@ -57,7 +57,7 @@ LaakkonenAlopaeusAittamaaDsd::LaakkonenAlopaeusAittamaaDsd
 )
 :
     daughterSizeDistributionModel(breakup, dict),
-    C4_(dimensionedScalar::lookupOrDefault("C4", dict, dimless, 4.3))
+    C4_(dimensionedScalar::lookupOrDefault("C4", dict, dimless, 18.25))
 {}
 
 
@@ -81,12 +81,18 @@ LaakkonenAlopaeusAittamaaDsd::antiderivative
 ) const
 {
     return
-        (
+        (4.0/3.0 + C4_/3)
-            pow(xk, -C4_ - 3)*pow(xk - v, C4_)*(v - xk)*((C4_ + 1)*(C4_ + 2)
+       *(
-           *(C4_ + 3)*pow3(v) - (C4_ + 1)*(C4_ + 2)*(bndr*(C4_ + 4) - 3.0*xk)
+            pow(xk, -C4_ - 3)*pow(xk - v, C4_)*(v - xk)
-           *sqr(v) - 2.0*xk*(C4_ + 1)*(bndr*(C4_ + 4) - 3.0*xk)*v - 2.0*bndr
+           *(
-           *sqr(xk)*C4_ + 6.0*pow3(xk) - 8.0*bndr*sqr(xk))
+                (C4_ + 1)*(C4_ + 2)*(C4_ + 3)*pow3(v)
-        )/(2.0*(range)*(C4_ + 4));
+              - (C4_ + 1)*(C4_ + 2)*(bndr*(C4_ + 4) - 3*xk)*sqr(v)
+              - 2*v*xk*(C4_ + 1)*(bndr*(C4_ + 4) - 3*xk)
+              - 2*bndr*C4_*sqr(xk)
+              + 6*pow3(xk)
+              - 8*bndr*sqr(xk)
+            )
+        )/(2*range*(C4_ + 4));
 }
 
 

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

@@ -26,12 +26,12 @@ Class
     LaakkonenAlopaeusAittamaaDsd
 
 Description
-    Daughter size distribution model used by Laakkonen et al. (2006). Note that
+    Daughter size distribution model of Laakkonen et al. (2007). Note that the
-    the diameters in the original expression was substituted by bubble volumes
+    diameters in the original expression were substituted by bubble volumes
     giving
 
     \f[
-        \left(9 + \frac{33}{2}C_4 + 9C_4^2 + \frac{3}{2}C_4^3\right)
+        (1 + C_4)(2 + C_4)(3 + C_4)(4 + C_4)
         \times \left(\frac{1}{3}\right) \left(\frac{1}{v_j}\right)
         \left(\frac{v_i}{v_j}\right)^{2} \left(1 - \frac{v_i}{v_j}\right)^{C_4}
     \f]
@@ -39,23 +39,28 @@ Description
     where
 
     \vartable
-        v_i         |  Volume of daughter bubble i [m]
+        v_i         |  Volume of daughter bubble i [m3]
-        v_j         |  Volume of mother bubble j [m]
+        v_j         |  Volume of mother bubble j [m3]
     \endvartable
 
+    The total number of daughter bubbles generated depends on C4 and evaluates
+    to
+
+    \f[
+        \frac{4}{3} + \frac{C_4}{3}
+    \f]
+
     References:
     \verbatim
-        Laakkonen, M., Alopaeus, V., & Aittamaa, J. (2006).
+        Laakkonen, M., Moilanen, P., Alopaeus, V., & Aittamaa, J. (2007).
-        Validation of bubble breakage, coalescence and mass transfer models for
+        Modelling local bubble size distributions in agitated vessels.
-        gas-liquid dispersion in agitated vessel.
+        Chemical Engineering Science, 62, 721–740.
-        Chemical engineering science, 61(1), 218-228.
-        Eq. 3, p. 220.
     \endverbatim
 
 Usage
     \table
         Property     | Description             | Required    | Default value
-        C4           | Coefficient C4          | no          | 4.3
+        C4           | Coefficient C4          | no          | 18.25
     \endtable
 
 SourceFiles
@@ -87,7 +92,7 @@ class LaakkonenAlopaeusAittamaaDsd
 {
     // Private Data
 
-        //- Optional coefficient C4, defaults to 4.3
+        //- Optional coefficient C4, defaults to 18.25
         dimensionedScalar C4_;