Public Member Functions | List of all members
LuoSvendsen Class Reference

Model of Luo and Svendsen (1996). The breakup rate is calculated by. More...

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Public Member Functions

 TypeName ("LuoSvendsen")
 Runtime type information. More...
 
 LuoSvendsen (const populationBalanceModel &popBal, const dictionary &dict)
 
virtual ~LuoSvendsen ()=default
 Destructor. More...
 
virtual void correct ()
 Correct diameter independent expressions. More...
 
virtual void addToBinaryBreakupRate (volScalarField &binaryBreakupRate, const label i, const label j)
 Add to binary breakupRate. More...
 
- Public Member Functions inherited from binaryBreakupModel
 TypeName ("binaryBreakupModel")
 Runtime type information. More...
 
 declareRunTimeSelectionTable (autoPtr, binaryBreakupModel, dictionary,(const populationBalanceModel &popBal, const dictionary &dict),(popBal, dict))
 
 binaryBreakupModel (const populationBalanceModel &popBal, const dictionary &dict)
 
autoPtr< binaryBreakupModel > clone () const
 
virtual ~binaryBreakupModel ()=default
 Destructor. More...
 

Additional Inherited Members

- Static Public Member Functions inherited from binaryBreakupModel
static autoPtr< binaryBreakupModel > New (const word &type, const populationBalanceModel &popBal, const dictionary &dict)
 
- Protected Attributes inherited from binaryBreakupModel
const populationBalanceModel & popBal_
 Reference to the populationBalanceModel. More...
 

Detailed Description

Model of Luo and Svendsen (1996). The breakup rate is calculated by.

\[ C_4 \alpha_c \left(\frac{\epsilon_c}{d_j^2}\right)^{1/3} \int\limits_{\xi_{min}}^{1} \frac{\left(1 + \xi\right)^{2}}{\xi^{11/3}} \mathrm{exp} \left( - \frac{12c_f\sigma}{\beta\rho_c\epsilon_c^{2/3}d_j^{5/3}\xi^{11/3}} \right) \mathrm{d} \xi \]

where

\[ c_f = \left(\frac{v_i}{v_j}\right)^{2/3} + \left(1 - \frac{v_i}{v_j}\right)^{2/3} - 1 \]

\[ \xi_{min} = \frac{\lambda_{min}}{d_j}\,, \]

and

\[ \lambda_{min} = C_5 \eta\,. \]

The integral in the first expression is solved by means of incomplete Gamma functions as given by Bannari et al. (2008):

\[ \frac{3}{11 b^{8/11}} \left( \left[\Gamma(8/11, b) - \Gamma(8/11, t_{m})\right] + 2b^{3/11} \left[\Gamma(5/11, b) - \Gamma(5/11, t_{m})\right] + b^{6/11} \left[\Gamma(2/11, b) - \Gamma(2/11, t_{m})\right] \right) \]

where

\[ b = \frac{12c_f\sigma}{\beta\rho_c\epsilon_c^{2/3}d_j^{5/3}} \]

and

\[ t_{min} = b \xi_{min}^{-11/3}\,. \]

Note that in the code, the upper incomplete gamma function is expressed as

\[ \Gamma(a,z) = Q(a,z) \Gamma(a) \]

\( \alpha_c \) = Void fraction of continuous phase [-]
\( \epsilon_c \) = Turbulent dissipation rate of continuous phase [m2/s3]
\( d_j \) = Diameter of mother bubble j [m3]
\( v_i \) = Volume of daughter bubble i [m3]
\( v_j \) = Volume of mother bubble j [m3]
\( \xi \) = Integration variable [-]
\( \xi_{min} \) = Lower bound of integral [-]
\( c_f \) = Increase coefficient of surface area [-]
\( \sigma \) = Surface tension [N/m]
\( \rho_c \) = Density of continuous phase [kg/m3]
\( \eta \) = Kolmogorov length scale [m]
\( \Gamma(a,z) \) = Upper incomplete gamma function
\( Q(a,z) \) = Regularized upper incomplete gamma function
\( \Gamma(a) \) = Gamma function

References: 

        Luo, H., & Svendsen, H. F. (1996).
        Theoretical model for drop and bubble breakup in turbulent dispersions.
        AIChE Journal, 42(5), 1225-1233.
        Eq. 27, p. 1229.
        Bannari, R., Kerdouss, F., Selma, B., Bannari, A., & Proulx, P. (2008).
        Three-dimensional mathematical modeling of dispersed two-phase flow
        using class method of population balance in bubble columns.
        Computers & chemical engineering, 32(12), 3224-3237.
        Eq. 49, p. 3230.
Usage
Property Description Required Default value
C4 Coefficient C4 no 0.923
beta Coefficient beta no 2.05
C5 Minimum eddy ratio no 11.4
Source files

Definition at line 230 of file LuoSvendsen.H.

Constructor & Destructor Documentation

◆ LuoSvendsen()

LuoSvendsen ( const populationBalanceModel &  popBal,
const dictionary &  dict 
)

Definition at line 58 of file LuoSvendsen.C.

References List< T >::append(), Foam::constant::electromagnetic::e, and Foam::Math::incGammaRatio_Q().

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◆ ~LuoSvendsen()

virtual ~LuoSvendsen ( )
virtualdefault

Destructor.

Member Function Documentation

◆ TypeName()

TypeName ( "LuoSvendsen"  )

Runtime type information.

◆ correct()

void correct ( )
virtual

Correct diameter independent expressions.

Reimplemented from binaryBreakupModel.

Definition at line 157 of file LuoSvendsen.C.

References populationBalanceModel::continuousPhase(), populationBalanceModel::continuousTurbulence(), phaseModel::nu(), binaryBreakupModel::popBal_, Foam::pow025(), and Foam::pow3().

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◆ addToBinaryBreakupRate()

void addToBinaryBreakupRate ( volScalarField &  binaryBreakupRate,
const label  i,
const label  j 
)
virtual

Add to binary breakupRate.

Implements binaryBreakupModel.

Definition at line 173 of file LuoSvendsen.C.

References Foam::constant::physicoChemical::b, Foam::cbrt(), sizeGroup::d(), forAll, sizeGroup::phase(), Foam::pow(), phaseModel::rho(), Foam::sqr(), and sizeGroup::x().

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The documentation for this class was generated from the following files: