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

Inheritance diagram for LuoSvendsen:

Collaboration diagram for LuoSvendsen:

Public Member Functions
	TypeName ("LuoSvendsen")
	Runtime type information. More...

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

virtual	~LuoSvendsen ()
	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 ()
	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 [m^2/s^3]
	=	Diameter of mother bubble j [m^3]
	=	Volume of daughter bubble i [m^3]
	=	Volume of mother bubble j [m^3]
$\xi$	=	Integration variable [-]
$\xi_{min}$	=	Lower bound of integral [-]
	=	Increase coefficient of surface area [-]
$\sigma$	=	Surface tension [N/m]
$\rho_c$	=	Density of continuous phase [kg/m^3]
$\eta$	=	Kolmogorov length scale [m]
$\Gamma(a,z)$	=	Upper incomplete gamma function
	=	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 228 of file LuoSvendsen.H.

Constructor & Destructor Documentation

◆ LuoSvendsen()

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

◆ ~LuoSvendsen()

virtual ~LuoSvendsen ( )

inlinevirtual

Destructor.

Definition at line 274 of file LuoSvendsen.H.

References LuoSvendsen::addToBinaryBreakupRate(), and LuoSvendsen::correct().

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Member Function Documentation

◆ TypeName()

TypeName ( "LuoSvendsen" )

Runtime type information.

◆ correct()

virtual void correct ( )

virtual

Correct diameter independent expressions.

Reimplemented from binaryBreakupModel.

Referenced by LuoSvendsen::~LuoSvendsen().

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◆ addToBinaryBreakupRate()

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

virtual

Add to binary breakupRate.

Implements binaryBreakupModel.